Chapter 1

GEOMETRY OF INDIVIDUAL VARIATION IN PERSONALITY AND SLEEP-WAKE ADAPTABILITY

GEOMETRY OF INDIVIDUAL VARIATION IN PERSONALITY AND SLEEP-WAKE ADAPTABILITY

ARCADY A. PUTILOV

Nova Science Publishers, Inc. New York

Copyright © 2011 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA ISBN: 978-1-61668-840-0

Published by Nova Science Publishers, Inc. New York

CONTENTS Preface

vii

Introduction

Geometry and Applications of the Spherical Cube Model

Part 1

Spherical Cube Representation of the Structure of Sleep-Wake Adaptability

Chapter 1

Taxonomy of Chronotypes, Trototypes and Somnotypes: History and State of the Art

1

Three-Dimensionality of the Structure of Sleep-Wake Adaptability

17

Chapter 2

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Chapter 3

Validation of the Sleep-Wake Adaptability Scales Predicted by the Spherical Cube Model 73

Part 2

Spherical Cube Representation of the Structure of Personality Lexicon

Chapter 4

Taxonomy of Personality Traits and Emotional States: History and State of the Art

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Three-Dimensionality of the Cross-Culturally Universal Structure of Personality Lexicon

125

Evolutionary Psychology Perspectives on the Spherical Cube Model of Personality Lexicon

221

A General Approach to Uncovering Spherical Cube Structures

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Chapter 5 Chapter 6 Conclusion

vi Appendix 1 Appendix 2

Contents Russian Version of the SWPAQ (Sleep-Wake Pattern Assessment Questionnaire)

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Russian Version of the SCIoPS (Spherical Cube Inventory of Personality Structure)

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References

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Index

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PREFACE Scientific investigation is often aimed on generation and description of a low dimensional simple form that is, however, an accurate representation of the structure of numerous empirically obtained variables. In despite of this aim, some scientific descriptions of real world structures are difficult to visualize due to their dimensional complexity. The book considers two such structures, the structure of personality lexicon and the structure of adaptive ability of the sleep-wake cycle. When factor analysis, most widely used method of data reduction, was applied to the empirical data sets, it revealed six factorial dimensions of personality lexicon (i.e., Extraversion, Agreeableness, Conscientiousness, Emotional Stability, Intelligence and Self-Assurance), and six factorial dimensions of sleep-wake adaptability (Morning and Evening Lateness, Anytime and Daytime Wakeability, and Anytime and Nighttime Sleepability). A principal question arises as to whether these six factorial dimensions might be visualized in a three-dimensional space. The author proposed the spherical cube model to explain why the answer to this question must be yes. The model provides a way of replacement of a sixfactor representation by a more realistic representation with only three spatial (underlying) dimensions. In geometric terms, the model has a shape of spherical cube. The six pairs of this cube’s edges on the surface of the sphere represent the six largest factorial dimensions. This cube serves as a system of coordinates for mapping any of a large number of narrow individual traits on the surface of this sphere. The author proposed the original circumplex criteria for testing in quantitative terms whether the structure revealed by empirical study confirms well to the structure predicted by the model. The model was first introduced for the structuring adaptive ability of the sleep-wake cycle. It helped to identify the exact number of broad adaptive traits of this cycle and to determine their relationship with the narrower and broader traits. Further research demonstrated that the same model can contribute to the

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continuing debate about the structure of personality. The spherical cube model was able to account for the correlations between 5-7 factorial dimensions that personality psychologists manage to recover by factor analysis of the long lists of personality descriptive terms or questionnaire items. The numerous personalityrelevant words were mapped on the surface of spherical cube with the cube’s edges representing the six broad personality traits. The author provided an explanation of why the formally identical model demonstrated the applicability to two rather distinct real structures. Furthermore, the model opens a new perspective of uncovering three-dimensional structures of different kinds of reality described by using the multi-scale inventories. The book consists of six chapters. Two chapters include reviews of modern literature on individual differences. One concerns the studies of variation in chronotype (morning-evening preference), trototype (wakeability), and somnotype (sleepability), and another concerns the debates around the scientific taxonomies of personality traits and emotional states. Four other chapters of the book present the results of original researches of individual variation. They were aimed on the structuring sleep-wake adaptability, the quantification of relationship between the individual traits of sleep- and wake-related behavior and the responsiveness of the sleep-wake regulation to sleep loss, the structuring Russian personality lexicon, and the explaining origin and function of personality differences. The book appeals to lay audience and scientists who are interested in learning new ideas and provocative observations about individual differences in human personality and sleep-wake traits. The audience of this book also includes the researches from those numerous branches of science that recognize the structuring individual differences as an actual issue. Furthermore, original modeling and empirical results presented in the book can be used for teaching the university courses on statistical analysis, personality, emotion, neurophysiology, and psychophysiology. In short, this book addresses an issue of uncovering the true shape (topology) of a multi-dimensional structure of the differences between individuals. It is aimed at demonstrating that the shape of the real structure might be more parsimonious than a shape yielded by factor analysis, a most widely used method of data reduction. In addition, the book introduces an original approach to analyzing empirical data sets that uncover realistic representations of the structure of individual variation. Finally, it shows a perspective of the implication of these representations for the search for evolutionary origins and functions of individual differences. The book is structured as follows.

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In a brief introduction section I describe the spherical cube model in general terms and review the history of its application in the fields of differential chronobiology and personality psychology. Thereafter, I use the data on sleep-wake adaptability (Part 1) and personality lexicon (Part 2) to test the predictions of the spherical cube model. The analysis confirms that: (1) the six largest factorial dimensions can actually have a threedimensional structure; (2) this structure can be directly revealed by performing three-dimensional scaling; and (3) evidence for similarity between empirically and theoretically predicted structures can be provided by applying circumplex criteria. Parts 1 and 2 are independent of one another. This means that if a reader is mostly interested in understanding the structural features of personality, he/she can read the second rather than the first part. The book’s conclusion includes a brief discussion of the perspective of applying a general approach to the analysis of empirical data sets in the framework of the spherical cube model.

To Nastya, of course

INTRODUCTION: GEOMETRY AND APPLICATIONS OF THE SPHERICAL CUBE MODEL This book is centered on uncovering a real structure underlying a large set of individual trait variables. The spherical cube model can explain a link between the actual (three-dimensional) and factorial (six-factor) structures of individual variation in sleep-wake adaptability (Part 1) and personality lexicon (Part 2).

General Methodology of Structuring Individual Variation in Trait Variables A scientific taxonomy offers a standard vocabulary that facilitates accumulation and communication of empirical findings. Such a taxonomy must provide an understanding of a large number of specific instances in a simplified way that makes unnecessary separate examination of each specific instance. Any taxonomy proposes a structure that has a shape (topology). The most scientifically valuable shapes are those that are parsimonious in terms of geometry and dimensionality. Consequently, a scientific investigation focuses on discovering a low dimensional simple form representation (approximation) of the empirically revealed structure (see Maraun, 1997 for detail). The most preferable shapes are just two- or three-dimensional ones because they can be visualized. Furthermore, these shapes must remain simple forms, for example two-dimensional shapes such as squares or circles or threedimensional shapes such as cubes or spheres. More generally, the aim of scientific investigation is to generate and describe a low dimensional simple form that is an adequately accurate

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representation of the structure of numerous empirically obtained variables. In other words, the scientific representation must, despite its parsimony, retain the most important features of the empirical structure. The shape of the empirical structure can be revealed by the analysis of the measures of relatedness (or proximity or association) between the variables constituting the structure under investigation (Maraun, 1997). The measure of relatedness that has dominated scientific research for a century is the intercorrelation among variable (e.g. between individual trait variables). The structure of variables is usually revealed by the analysis of the pattern of correlations among them. In a good deal of empirical studies such analysis is limited to factor analysis of a matrix of pairwise coefficients of correlation. For at least the last six decades the factor-analytic approach has become the most influential method for generating low dimensional representations of individual trait variables. By applying factor analysis researchers hope to reduce the dimensionality of representation of the empirical structure without great loss of information. This is a variable-reduction procedure in which many variables are organized by a few factors that summarize the interrelations among them. This analysis provides coordinates called “factor loadings” that locate the variables as points in the n-dimensional common factor space. Usually, n is much smaller than the number of individual trait variables (e.g. lists of questionnaire items or of personality-relevant words). However, the practice of factoring individual trait variables indicates that the original set often cannot be replaced by only two or three factors. In this sense the structure under investigation is more than three-dimensional and thus cannot be visualized. The model presented in this book can be visualized in three-dimensional space as a spherical cube. This is a cube that divides the surface of a sphere into six equal parts (six spherical squares). The major prediction of the model – a prediction that is fully testable – is that any individual trait variable can be mapped on the surface of this spherical cube. Another important prediction is that the six largest factors yielded by the factor analysis can be also accommodated by this model. To emphasize this feature of the model such factors will be specified below as factorial dimensions. They are visualized as the six pairs of edges opposing one another on the surface of the spherical cube. These predictions will be tested here by analyzing two empirical structures, the structure of sleep-wake adaptability and the structure of personality lexicon.

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Main Ideas of the Book When one performs factor analysis of a long set of trait variables, the reality might be confused with the appearance of 5-7 orthogonal factors. One can interpret this factor-analytic result as revealing a 5-7-dimensional reality. However, the real dimensionality might be much simpler. The spherical cube model postulates the possibility of replacing trait taxonomies with 6±1 factors by the three-dimensional taxonomy in the shape of a spherical cube. Namely, the space structure of trait variables might be represented in a shape composed of six spherical squares with corners of 120 each. Such representation assumes the rather unusual angular location of six broad factorial dimensions revealed by factor analysis of the individual trait variables. They are visualized as the six pairs of edges of the spherical cube. Such locations of factorial dimensions might be proven by performing a multidimensional scaling analysis or by applying a set of circumplex criteria proposed here. Furthermore, the results of either factor or multidimensional scaling analyses might be used to map a huge number of separate individual trait variables on the surface of the spherical cube. The empirical evidence supporting the spherical cube model is provided by detailed analysis of the original data. They were collected in studies aimed at uncovering trait structure in two domains of individual differences. One structure was developed to understand the diversity of the adaptive abilities of the sleep-wake cycle (Part 1). Another structure offers an explanation of the diversity of personality traits (Part 2). The findings indicate that the spherical cube model can help detect, describe and understand the natural structure of individual traits. In particular, the empirical results suggest that, although the traditional factor analysis tends to overestimate the underlying dimensionality of the structure of individual trait variation, its results can be easily explained in the frame of the spherical cube model and confirmed by the results of multidimensional scaling. In general, this book illustrates the heuristic potential of the spherical cube model for research aimed at discovering the precise nature of linkages between individual traits. The present modeling and empirical findings allow us to conclude that a spherical cube representation of individual traits variation is a more elegant, parsimonious, universal and insightful model compared to traditional multi-factor representations, and that these multi-factor representations can be incorporated in the framework of the spherical cube model.

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Geometrical Features of the Spherical Cube Model To obtain a spherical cube, a cube is inscribed in a sphere so that the central points of the sphere and cube coincide and all corners of the cube lie on the spherical surface. In other words, if the edges of the cube are projected onto the sphere by tracing radii that pass through the cube’s edges, then great circle arcs are formed on the sphere that divide it into six equal parts, each part being a spherical square (Figures 1.1, 1.2, 1.17, 1.21, 1.22, 2.1, 2.2, 2.17, and 2.21-2.24). The geometry of a cube (Figures 1.1, 1.2, 1.6-1.11, 1.13, 1.15, 1.17, 2.1, 2.2, 2.6-2.11, 2.13, 2.15, 2.17, 2.21 and 2.24) assumes that it has: 1) three axes or vectors running through the center to connect above with below, left with right, and front with back (i.e., axis A, axis B, and axis C); 2) three pairs of faces or squares, top opposes bottom, left opposes right, and front opposes back (i.e., A opposes a, B opposes b, and C opposes c); 3) four pairs of vertices or corners opposing one another (i.e., AbC opposes aBc, Abc opposes aBC, ABc opposes abC, and ABC opposes abc); and 4) six pairs of edges or ribs (lines connecting the corners) opposing one another (i.e., AB opposes ab, aB opposes Ab, AC opposes ac, aC opposes Ac, BC opposes bc, and bC opposes Bc).

Spherical Cube Representation of the Structures of Individual Trait Variables The spherical cube model is based on the assumption that the real structure of individual traits can be represented as a sphere or three-axis circumplex (see, for instance, Figure 2.21, top). Any element of this structure – i.e. any specific individual trait – can be located on the surface of the sphere or three-dimensional circular shape. The cube is inscribed in the sphere to accommodate six factorial dimensions that are not proposed to be fully orthogonal, and to allow the possibility of relating any specific individual trait to one or two or three or four of these six factorial dimensions. In geometric terms, the spherical cube is defined by only three orthogonal axes (see, for instance, Figure 2.17). They are named spatial or underlying

Introduction

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dimensions A, B, and C, or A/a, B/b, and C/c. There is one very important deduction which follows from the geometry of the spherical cube model. Factor analysis performed to reveal from 5 to 7 rotated factors fails to provide the direct measurements of coordinates of the hundreds of specific traits on three spatial (underlying) dimensions, A, B, and C. The model’s view of factorial dimensions rests on an assumption that the six largest rotated factors are visually represented by six pairs of the edges (ribs) of the spherical cube, AB/ab, aB/Ab, AC/ac, aC/Ac, BC/bc, and bC/Bc. In other words, since the spherical cube has only three axes, there are no psychometric axes in this model which can be directly provided by 5-7-factor solutions. Instead, each of the six largest factorial dimensions (for instance, AB/ab) is jointly determined by combining the poles (high or low) of two spatial (underlying) dimensions (for instance A/a and B/b). Due to the broad variation of the 3rd spatial dimension (for instance, C/c), each of six factorial dimensions can be interpreted as actually “Big” (broad) individual trait. It can be characterized by a broad bandwidth of content mapping along two antipodal edges (ribs) of the spherical cube (for instance, AB and ab). Since the six factorial dimensions are conceptualized as the edges (ribs) of the cube, one can transform the information on factor loadings of any specific trait variable in information on position of this variable relative to the edges of the cube. Thus, these edges can be used as a system of coordinates for mapping any specific trait exemplified by one or several closely related trait variables (see, for instance, Figure 2.21, top). Namely, 6 specific dimensions might be related to only one factorial dimension (e.g., AB/ab), 24 specific dimensions might be related to two factorial dimensions (e.g., ABAC/abac), 4 specific dimensions might be related to three factorial dimensions (e.g., ABC/abc), and 3 specific dimensions might be related to four factorial dimensions (e.g., A/a). Moreover, at is shown in Parts 1 and 2, there exists a direct way to obtain the coordinates of specific trait variables on the surface of the spherical cube. The coordinates can be calculated by performing three-dimensional scaling. The results of multidimensional scaling demonstrate the possibility of representing the whole domain of individual trait variables in terms of three polarized spatial dimensions, A/a, B/b, and C/c. Comparison of the results of factor and multidimensional scaling analyses indicates that the differences between them can be conceptualized in terms of the spherical cube model. In particular, the locations of the six largest dimensions yielded by factor analysis roughly coincide with the locations of the edges (ribs) of the cube, while three

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dimensions revealed by three-dimensional scaling correspond to three axes of the sphere and cube (see, for instance, Figure 2.6).

Development and Applications of the Spherical Cube Model I introduced the spherical cube model initially as a structural representation of adaptive ability of the sleep-wake cycle. The earlier studies on this representation are briefly reviewed in the first chapter of the book among other investigations of individual variation in sleep-wake behavior. The model was applied for prediction of a new scale and new subscales of the sleep-wake pattern questionnaires. The predicted constructs were developed and validated in both questionnaire and experimental studies. In the questionnaire study, the tetra-circumplex criterion was introduced to provide a quantitative approach to testing whether the structure of a questionnaire corresponds to the structure predicted by the model. In the experimental study, subjects were deprived of sleep for one night in order to compare the subjectively assessed measures (i.e., the traits of sleep-wake behavior, levels of sleepiness and performance, sleep history, etc.) with the principal components of waking EEG (i.e., those components that provide the objective markers of the parameters of sleep-wake regulation such as sleep debt and sleep pressure). The results of these studies showed that the model offers a common theoretical and methodological framework for the development of the unified taxonomy of the individual chronobiological variation associated with a person’s chronotype (morning-evening preference), trototype (wakeability) and somnotype (sleepability). The same model was also applied to structuring personality traits. Many personality researches share the belief that if factor analysis of the long list of personality attributes reliably yields a set of 6±1 orthogonal factors then the real dimensionality of the structure of personality lexicon cannot be smaller than 5. At the same time they often report the significant inter-correlations among these factors, as well as among the 5 or more scales developed on the basis of these factors. These inter-correlations indicate that the dimensionality of the model of personality structure can be reduced. The original results presented in the book suggest the possibility of structuring the long lists of personality-relevant terms with only three orthogonal dimensions. The particular way of such reduction of dimensionality was exemplified by the structure of Russian personality terms. Again, the correspondence between

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empirically derived structure and the structure predicted by the model was confirmed by applying the tetra-circumplex criterion. The spherical cube model was also supported by the comparison between dimensions revealed by factor and multidimensional scaling analyses. It was shown that, geometrically, the edges and axes of the spherical cube can be related, respectively, to the six factorial dimensions yielded by the former analysis and to the three spatial dimensions yielded by the latter analysis. Moreover, three-dimensional scaling provided the possibility to introduce new circumplex criteria (tri-circumplex criterion and hexa-circumplex criterion) for quantitative comparison of empirically and theoretically derived structures. In addition, the attempts to develop and implicate the spherical cube model in the field of personality psychology led to the following findings. The structural representations of personality traits and emotional states were unified within the same three-dimensional structure proposed by the model. It was shown that the model offers the external criteria for detecting convergence between the competing 5-7 factor models of personality structure. Moreover, it was shown that three axes of the spherical cube structure might represent the rotational variants of the well-known three-dimensional classifications, such as the three Osgood dimensions of meaning (Evaluation, Activity, and Potency) and the three broadest personality factors (i.e., broad Extraversion, Agreeableness, and Conscientiousness, or Dynamism, Affiliation, and Structure). Finally, the spherical cube model was applied to explain the natural origin and function of personality structure and to explain the evolutionary mechanisms responsible for the maintenance of individual variation in personality traits.

Acknowledgements Preparation of this book was facilitated by grant 01-06-85009а/У from the Russian Foundation for the Humanities. Empirical research was partly supported by grant 06-06-00375а from the Russian Foundation for the Humanities, and by grants 07-06-00263a and 10-06-00114а from the Russian Foundation for Basic Research. I am indebted to Olga Donskaya for her participation in data collection and to Dr. Evgeniy Verevkin for his help in data analysis. I am also very grateful to Dmitriy Putilov who helped me with so many things, including collecting data, data analysis, and drawing some figures for this book. Finally, I have benefited greatly from the support, encouragement and editorial help of Dr. Frank Salter.

PART 1. SPHERICAL CUBE REPRESENTATION OF THE STRUCTURE OF SLEEP-WAKE ADAPTABILITY This Part 1 is centered on uncovering the topology of a multi-dimensional structure of between-individual differences in behavior associated with the alternations of sleep and wakefulness. The spherical cube model was employed to discover the precise nature of the linkages between the individual adaptive traits of the sleep-wake cycle. The first chapter serves as an introduction to the second chapter containing the results of questionnaire research concerning the structuring of adaptability of sleep-wake habits. The empirical research presented in the third chapter provides evidence for the validity of the scales of multi-dimensional questionnaire predicted by the spherical cube model.

Chapter 1

TAXONOMY OF CHRONOTYPES, TROTOTYPES AND SOMNOTYPES: HISTORY AND STATE OF THE ART ABSTRACT Individual variation in human sleep-wake behavior is mostly studied by researchers in the field of sleep physiology and chronobiology – the scientific study of biological rhythms. This chapter contains a brief review of the literature in these fields on questionnaire studies of individual variation in sleep-wake behavior. This review also includes the first publications of the author on the development and application of the spherical cube model.

QUESTIONNAIRE ASSESSMENT OF MORNING-EVENING PREFERENCE Everyone recognizes the differences between individuals in their everyday habits related to the sleep-wake cycle. Compared to the rather intense interest shown by lay people in their individual sleep-wake pattern, the individual variation in this pattern is rarely considered to be an important topic of scientific research. Rather, the investigators in such fields as chronobiology and sleep physiology tend to treat the study of individual traits as an area of applied rather than fundamental research. For example, the assessment of individual differences can be aimed at prediction of tolerance to shift and night

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work. In one of our previous publications (Putilov and Putilov, 2005), we noted that the scientific literature lacks publications aimed at explaining the structure of variation in human sleep-wake behavior in terms of a chronobiological model. Moreover, nobody yet had tried to use the theoretical knowledge and empirical facts about individual variation to develop a generalized model of physiological mechanisms regulating sleep and wake states. Any research aimed at ranking and typing people makes no sense without specifying the dimensions of traits on which the individuals differ. The first scientifically recognized dimension of individual differences in human daily rhythms has become the preference for timing of sleep, wake and work. The two extreme chronotypes on this dimension are often nicknamed “larks” and “owls”. One of the tools for assessment of morning-evening preference, the 19item Questionnaire for Self-Assessment of Morningness-Eveningness, was produced by Horne and Östberg in 1976. It has been translated into a dozen languages and applied in thousands studies (i.e., Kerkhof et al., 1981; Ishihara et al., 1984; Adan and Almirall, 1991; Taillard et al., 2004). The research indicates that this and other similar scales for distinguishing between morning types and evening types (i.e., Torsvall and Åkerstedt, 1980; Smith et al., 1989; Brown, 1993; Bohle et al., 2001) can be associated with individual circadian phase position and with individual tolerance to shift and night work (Breithaupt et al., 1978; Åkerstedt and Torsvall, 1981; Kerkhof, 1985; Härmä et al., 1988; Bohle and Tilley, 1989; Costa et al., 1989; Moog and Hildebrandt, 1989; Bailey and Heitkemper, 2001; Duffy et al., 2001). However, the morning-evening questionnaires were criticized for poor or lack of statistical analyses that could provide the basis for their psychometric evaluation (i.e., Smith et al., 1989; Brown, 1993). It is noteworthy that for many years the researchers of chronotypological differences did not asked such questions as: (1) is the preference for sleep-wake (rest-work) timing a single trait; (2) what are other individual traits of sleep-wake patterns; and (3) can researchers measure them with multi-dimensional questionnaires? Factor analysis is the instrument most frequently used for the delineation of the dimensions on which the individuals differ. This mathematic technique can distill large numbers of separate questionnaire items associated with specific traits into a smaller number of higher order traits (called factors) that account for most of the differences between individuals. For a long period of time factor analysis was not applied to evaluate the instruments developed for assessing morning-evening preference, probably because of the belief that

Taxonomy of Chronotypes, Trototypes and Somnotypes

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morningness-eveningness is a unitary construct which simply reflects the position of the individual phase of the circadian rhythm. However, when factor analysis and other conventional psychometric methods were employed to evaluate earlier published questionnaires, the results pointed to the need to improve them (Larsen, 1985; Smith et al., 1989; Brown, 1993). For instance, instruments such as the 19-item Questionnaire for Self-Assessment of Morningness-Eveningness (Horne and Östberg, 1976) – still widely used – have been found to be somewhat imperfect as they contain questions with low item-total correlations and several separate dimensions (Moog et al, 1982; Larsen, 1985; Brown, 1993). The multi-dimensionality of the morningness-eveningness construct was confirmed by many other studies. Even factor analysis of a shortened (7-item) version of the morning-evening questionnaire, Diurnal Type Scale (Torsvall and Åkerstedt, 1980), sorted out two groups of items. One group was associated with morning questions, and another group mostly included evening questions. Furthermore, multidimensionality was reported for the modified variants of earlier proposed questionnaires. For instance at least three broad factors were revealed by factor analyzing the 13-item modified version of morningness-eveningness scale called the Composite Scale of Morningness (Smith et al., 1989; Caci et al., 2005; Randler, 2009). Furthermore, two broad factors were revealed in the reduced (7-item) version of this scale (Randler, 2009).

MULTI-DIMENSIONAL APPROACH TO ASSESSMENT OF ADAPTIVE ABILITY OF THE SLEEP-WAKE CYCLE The attempts to introduce a multi-dimensional approach to questionnaire studies of individual variation in adaptive features of circadian rhythms were pioneered by Folkard, Monk and Lobban (1979). Their research indicated that morning-evening preference is not the only individual trait determining the success or failure of biological adaptation to night and shift work. Three high order factors were yielded by factor analysis of responses to the 20 items of the Questionnaire for Prediction of Adjustment to Shift Work (Folkard et al., 1979). In addition to the traditional morning-evening factor, Folkard et al (1979) defined two new factors, rigidity-flexibility (of sleeping habits) and languidness-vigorousness (or inability-ability to overcome drowsiness). Later, the initial questionnaire was developed in Circadian Type Inventory (Di Milia et al., 2005).

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I applied a psychometrically based approach to two Russian language chronobiological questionnaires (Putilov, 1987, 1997a). One, the 13-item Scale for Assessment of Circadian Lateness, was designed as a uni-dimensional morning-evening scale. It was developed through a series of four surveys involving 66, 155, 343 and 442 respondents. The number of questions and the number of answers were both consequently reduced (20161513 and 10765, respectively) after evaluations of the distribution of responses to each question and inter-relations among responses to different questions (see Putilov and Putilov, 2005, for more detail). By contrast the second instrument, the Sleep-Wake Pattern Assessment Questionnaire (SWPAQ), was originally constructed as a multi-dimensional inventory (Putilov, 1990, 1993a). It is designed to self-assess the adaptive ability of sleep-wake patterns (“sleep-wake adaptability”). An initial list of 200 statements with 5 response choices was reduced to 40 statements with 2 choices (yes or no) through a series of four surveys involving 117, 221, 306 and 356 participants (see Putilov, 1993a, 2000; Putilov and Onishenko, 2005; Putilov and Putilov, 2005, for details of the construction process). These 40 statements were selected to represent a 5-factor solution yielded by applying varimax rotation aimed at obtaining uncorrelated factors of the 5 largest principal components (see also Chapters 4 and 5 of this book for a more detailed description of the factor-analytic technique). Evidence that a variety of adaptive abilities are engaged in the sleep-wake cycle was provided by applying cluster- and factor-analytic methods for elaborating the relationship between the items of several chronotypological questionnaires (Putilov, 1990, 1993a; Putilov D. et al., 2007). For example, the 40-item version of the SWPAQ (Putilov, 1990) includes 10 groups of 4 closely related items (subscales or tetrads) reflecting the abilities to wake late in the evening, to wake at night, to sleep in the morning, to get up at fixed times, to wake early in the morning, to wake at anytime, to fall asleep at anytime, to fall asleep in the evening, to sleep deeply after midnight and to sleep until morning (Putilov, 1990, 1993ab, 1997a, 2003b, 2005). The factor analysis of the SWPAQ sorted 40 items into the first five factors. Consequently the tetrads were combined to form five scales for assessing more general abilities named Evening Lateness, Morning Lateness, Anytime Wakeability, Anytime Sleepability and Nighttime Sleepability (scales E, M, W, F and S, respectively). The psychometrical evaluation of the 40-item SWPAQ revealed that some of its 5 scales have substandard levels of reliability (most probably due to too small numbers of items). Therefore, 12 new items were added to


5

improve the reliability of three of the five SWPAQ scales (Putilov, 1993b, 2000; Putilov and Onishenko, 2005). The resulting (52-item) version of the SWPAQ consisted of the same 5 scales. Each of these scales, E, M, W, F and S, included two or three subscales (tetrads) each of which consists of two positively and two negatively keyed statements. Each subscale is meant to be associated with a specific adaptive ability of the sleep-wake cycle. Namely, three abilities – to wake in the evening rather than in the morning, to wake late in the evening, and to wake at night – were recognized as falling within the broad sleep-wake category of Evening Lateness (scale E). Four other broader traits were interpreted as follows. Morning Lateness (scale M) includes the inability to get up at fixed times, to wake early in the morning, and to delay sleep on weekends. Anytime Wakeability (scale W) includes the abilities to shift sleep-wake timing, and to wake at anytime. Anytime Sleepability (scale F) includes the abilities to fall asleep at anytime and to nap regularly. And Nighttime Sleepability (scale S) includes the abilities to sleep deeply either in the evening or in the middle of the night or in the morning. American and Russian students completed the English and Russian versions of a battery of four chronobiological questionnaires (the 19-item Questionnaire for Self-Assessment of Morningness-Eveningness, the 13-item Scale for Assessment of Circadian Lateness, the 20-item Questionnaire for Prediction of Adjustment to Shift Work, and the 40-item SWPAQ). The scores on 10 scales were subjected to factor analysis. The results suggested the possibility of grouping these scales in at least three groups (Putilov and Putilov, 2005). More specifically, the three-factor varimax solution yielded three very broad factors: the Lateness factor (E and M scales of the SWPAQ and the single scales of the 19-item Questionnaire for Self-Assessment of Morningness-Eveningness and the 13-item Scale for Assessment of Circadian Lateness), the Wakeability factor (W scale of the SWPAQ and the scale for ability to overcome drowsiness of the Questionnaire for Prediction of Adjustment to Shift Work), and the Sleepability factor (F and S scales of the SWPAQ and the scale for flexibility of sleeping habits of the Questionnaire for Prediction of Adjustment to Shift Work). Such grouping was confirmed in further questionnaire studies (i.e., Putilov and Onishenko, 2005; Putilov A. et al., 2007).

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MODELING SELF-ASSESSED DIFFERENCES IN SLEEPWAKE PATTERNS In order to explain the factorial structure of sleep-wake adaptability a three-dimensional model was propounded (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007). It postulates that the structure of individual variation in sleep-wake adaptive behavior originates from the interaction between only three underlying parameters. In terms of this model any subjectively assessed trait at any level of generality might be located in a three-dimensional space determined by these three parameters. Geometrically, a trait occupies a certain area on the surface of a sphere (a three-dimensional circumplex) formed by three orthogonal spatial dimensions representing these three underlying parameters (Putilov, 2006, 2007). The difference in generality of sleep-wake traits might simply reflect the difference in the size of the areas occupied by these traits on the surface of three-dimensional circumplex. If all three dimensions are fixed, this is a narrow (specific) trait. It may correspond to a subscale of a scale (i.e. to a single tetrad of the SWPAQ). If two of three dimensions are fixed, while the 3rd dimension varies, this is a more general (broad) trait. It may be represented by a questionnaire scale (i.e. a scale of the SWPAQ). If only one of three dimensions is fixed, this is one of the three most general traits. It may be associated with one of three “superfactors” revealed by a three-factor solution. For instance, the Lateness “superfactor” includes the items of the scales E, Evening Lateness, and M, Morning Lateness of the SWPAQ. Figures 1.1, 1.2, 1.6, 1.7, 1.17, 1.21, 1.22 and Table 1.6 illustrate the main idea behind the model: the possibility of reduction of all subjectively assessed traits to three hypothetical underlying parameters which are the orthogonal axes of sphere. The model might be visualized in the form of a spherical cube, as shown in Figures 1.1, 1.2, 1.19, 1.21, and 1.22, or in the form of a cube inscribed in a sphere (“cube-in-globe”), as shown in Figures 1.6, 1.7, 1.13 and 1.15. The inscription of the cube in the sphere is necessary to show the locations of the largest factors revealed by factor analysis and for locating the narrow traits of sleep-wake adaptability (Putilov, 2007). The six largest factors (factorial dimensions) are represented by the six pairs of edges of the cube. They are drawn by fixing two spatial (underlying) parameters and allowing the 3rd underlying parameter to vary (Table 1.6). In other words, the model predicts that any of a number of subjectively assessed traits of different levels of generality might be conceptualized as a


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certain combination of only three underlying dimensions. The traits can form a hierarchy of sleep-wake adaptability traits by imposing certain limitations on the range of variation along these dimensions. For instance, three “superfactors” (broadest traits) can be distinguished by limiting variation along one dimension and allowing variation along the two other dimensions. Six factors (broad traits) can be distinguished by limiting variation along two of the three dimensions and allowing variation along the third dimension. Finally, a much larger number of “subfactors” (narrow traits) can be delineated by limiting variation along all three dimensions (Figures 1.1, 1.2, 1.6, 1.7, 1.13, 1.15, 1.17-1.22 and Table 1.6). Possibly each of the three orthogonal dimensions proposed by the model might be associated with a separate parameter of chronophysiological regulation of the sleep-wake cycle. For example, the first two parameters can be related to two processes proposed as two major components (circadian and homeostatic) in the two-process models of regulation of the sleep-wake cycle (Borbély, 1982; Daan et al., 1984; Putilov, 1995a). As for the third parameter, it can represent the circadian clock arousal process suggested by Edgar et al. in 1993 or it can be viewed as a more general arousal dimension that was recognized by a number of models explaining the phychophysiological basis of personality and emotion (see, i.e., Panksepp, 1982; Eysenck and Eysenck, 1985; Ellis, 1987, and Chapter 4). Because factor analysis says nothing about the fundamental processes underlying the proposed three-dimensional structure, the validation of the chronophysiological background requires experimental research. Such research can be aimed at testing correlations between subjectively assessed traits and objectively measured chronophysiological signatures of the sleepwake regulatory mechanisms.

TESTING PREDICTIONS CONCERNING THE NUMBER AND SIZE OF FACTORIAL DIMENSIONS In our first publications (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007), we recognized that some predictive features of the three-dimensional model might be tested without elaborating its chronophysiological background. These fully testable and falsifiable predictions include suggestions about the exact numbers of adaptabilities of different levels of generality that can be examined by factoring chronobiological questionnaires

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(Putilov and Putilov, 2005). In 2005, Putilov and Putilov compared the numbers of abilities predicted by the model and assessed by the questionnaires. It was noted that, although the results of factor analysis of the chronotypological questionnaires provided an empirical basis for the model, there exists a certain disagreement between the factorial structure proposed by the model and the factorial structure of the 52-item SWPAQ. The geometry of the model predicts six factorial dimensions of similar size while the empirically developed 52-item SWPAQ includes five rather than six scales, and these scales were dissimilar on the number of 4-item subscales (two for the scales W and F and three for the scales E, M, and S). Consequently, a questionnaire study was undertaken to test these particular predictions of the model concerning the number and features of factorial dimensions of the SWPAQ (Putilov and Putilov, 2006; Putilov, 2006, 2007). The relationships between the five scales helped to specify the features of the sleep-wake cycle’s adaptive abilities that were predicted but not yet assessed by means of the SWPAQ. One new scale and two new subscales were constructed through a series of three questionnaire surveys (Putilov and Putilov, 2006). The SWPAQ was enlarged to 72-items by adding a second wakeability scale (Daytime Wakeability, V) and third subscales for the W and F scales (Putilov, 2007). The items of the SWPAQ are listed in Appendix 1 and their English translations are given in Figure 1.20. In addition to earlier studies aimed at validating the first five scales of the SWPAQ (Cherepanova and Putilov, 1993; Melnikov et al., 1999; Putilov, 2000; Putilov et al., 2002; Danilenko et al., 2004), a special experimental study was undertaken. Its goal was to demonstrate that the new wakeability scale (V) shows the expected pattern of association with the objective electrophysiological indices of wakefulness level under the condition of sleep restriction (Putilov D. et al., 2007; Verevkin et al., 2008; Putilov A. et al., 2009ab, 2010). In general, the questionnaire and experimental studies (Putilov and Putilov, 2005, 2006; Putilov, 2007; Putilov D. et al., 2007; Putilov A. et al., 2009ab; Verevkin et al., 2008) demonstrated that the modeling approach to the study of individual variation in sleep-wake habits might be employed to develop chronobiological instruments for conducting fundamental and applied research. The findings of these studies provided support for the assumption that subjectively assessed features of the sleep-wake cycle reflect underlying inter-individual differences in parameters of chronophysiological regulation mechanisms and can be used for evaluating biological risks of night and shift work (Putilov et al., 2009ab, 2010).


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In its recent (72-item) form, the SWPAQ alone or as part of a battery of chronobiological questionnaires can be applied to quantify individual differences in such most broad sleep-wake adaptability traits as Lateness (i.e. morning-evening preference), Wakeability (i.e. tolerance to sleep pressure), and Sleepability (i.e. propensity to fall asleep and to sleep deeply). Following the terminological distinction suggested by Lavie and Zwuluni (1992) and further developed by Van Dongen et al. (2005), any individual can be classified as representing, at least, three different types of variation in sleepwake behavior. Chronotype can be determined by Lateness score (i.e. the sum of E and M scores of the SWPAQ) which reflects later or earlier timing of wakefulness and sleep. Trototype can be self-assessed with wakeability score (the sum of W and V scores) which provides information about greater or smaller vulnerability to sleep loss. Somnotype can be self-assessed with sleepability score (sum of F and S scores) reflecting higher or lower sleep propensity (Putilov et al., 2009ab, 2010).

CIRCUMPLEX CRITERION BASED ON THE SPHERICAL CUBE MODEL Paradoxically, the results of factor analysis of chronobiological questionnaires served as an empirical basis for developing the model, but the model disagrees with the factor-analytic assumption that six varimax-rotated factors might represent six fully orthogonal dimensions. Instead, threedimensionality rather than six-dimensionality of the structure of sleep-wake adaptability is one of the most important predictions of the model. To evaluate the theoretical model empirically its constructs must be translated into geometrically based measurement procedures. The question of whether the structure of sleep-wake adaptability fits the three-dimensional model was addressed with the tetra-circumplex criterion (Putilov, 2007). This criterion allows us to empirically test whether actual structure (i.e. a structure of chronobiological questionnaires) exhibits such predicted geometrical features as circumplexity and three-dimensionality. Geometrically, the proposed theoretical structure, a sphere, is a threedimensional circumplex. An infinite number of two-dimensional circumplexlike shapes might be obtained by slicing this sphere into two approximately equal halves. However, when the cube is inscribed in this sphere, it can be used as a system of coordinates for cutting the sphere such that it produces a

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small number two-dimensional circumplex-like shapes that optimally represent its circumplexical features (Putilov, 2007). The sets of twodimensional representations of the three-dimensional circumplex permit avoidance of difficulties in displaying and evaluating an empirical structure in three-dimensional space. Indeed, as can be seen in Figure 1.7, there exist only four two-dimensional circumplex-like shapes each of which connects in a circle the centers of six cube’s edges. The vast majority of the narrow traits distributed on the surface of the sphere can be arranged in these four circumplexes (Figures 1.8-1.12) in order to provide the possibility of empirically examining the relations between the traits included in each of four circumplexes. It is assumed (Putilov, 2007) that these relations reveal the principle of circumplex structure. This circumplex principle was proposed by Guttman (1954) and introduced for the first time in the area of interpersonal research by Leary (1957). It contends that trait variables are arranged around a circle in two-dimensional space. As a consequence, it is expected that the correlations between trait variables gradually change in accordance with gradual changes in the traits’ meaning. For instance, the adjacent trait variables must demonstrate the highest positive correlation and the highest extent of similarity of their meaning, while the traits opposing one another in the circle must demonstrate the lowest (negative) correlation and antipodal meanings (for more details about circumplexes see Chapters 2, 4 and 5). Thus, the tetra-circumplex criterion (Putilov, 2007) assumes that a set of four almost two-dimensional representations can be used to examine the extent of similarity between the actual and theorized structures of sleep-wake adaptability. The criterion was developed and applied (Putilov, 2007) for determination of the extent of deviation of the SWPAQ structure from the structure of sleep-wake adaptability predicted by the model (Putilov and Putilov, 2005, 2006; Putilov, 2006). The results of the analysis of inter-correlations among 18 subscales (tetrads) of the 72-item SWPAQ indicated that these subscales might be configured into four two-dimensional circumplex-like shapes. It was concluded (Putilov, 2007) that, at least at the level of scales, the questionnaire exhibits a clear three-dimensional circumplex structure in accordance with theoretical expectations, and that this finding provides empirical evidence of similarity between the actual and theorized structures of sleep-wake adaptability (Putilov, 2007). However, although the empirical testing mostly yielded promising findings, it also showed some measurable difference between the SWPAQ structure and the theoretically predicted structure of


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sleep-wake adaptability. It was suggested that applying the tetra-circumplex criterion may accelerate the process of developing more accurate instruments for assessing the broad and narrow traits of sleep-wake adaptability (Putilov, 2007). In sum, the proposed taxonomic model of individual differences in adaptive abilities of the sleep-wake cycle (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007) postulates that the structure of sleep-wake adaptability can be visually represented by a sphere (three-dimensional circumplex) formed by three orthogonal axes. The model predicts that: (1) any ability can be located on the surface of this three-dimensional sphere; and that (2) the six pairs of edges of the cube inscribed in this sphere represent six broad abilities corresponding to the six largest factors (factorial dimensions) yielded by factor analysis of a set of chronobiological questionnaires. The tetra-circumplex criterion was introduced to examine the correspondence between the structure predicted by the model and an empirically derived structure of a multidimensional chronobiological questionnaire.

LIMITATIONS OF FACTOR ANALYSIS AS A TOOL FOR UNCOVERING ORTHOGONAL DIMENSIONS Different statistical approaches can be employed to group the studied phenomena, such as the numerous individual traits of the sleep-wake cycle. Factor analysis is most widely used statistical method for this purpose. This method divides up the total amount of variation into a small number of dimensions called factors. Such division might help to evaluate the relative importance of each particular individual trait represented by a separate factor. For instance, the analysis determines which factors account for more variance than others. Such differences are often interpreted as indicating that the factors differ in the extent of their generality (i.e. in the number of different questionnaire items or questionnaire subscales in which the factor accounts for some variance). The application of factor analysis in questionnaire studies began with attempts to determine the structure of personality traits descriptors. In the recent literature, the empirically derived 5-7 factor models, such as the Big Five model, have become the most popular representations of personality structure (see Chapter 4 for detail).

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However, the theoretical basis for such factorial structures has not been provided. One of the main methodological problems is rooted in the long term tradition of using factor analysis as a tool for inductive “discovery” of the dimensions of personality structure. An empirically supported criticism points to inherent limitations of this method. For instance, when a linear factor analysis is used with dichotomous (nonlinear) data, there is a tendency to overestimate dimensionality (e.g. van Schuur and Kiers, 1994). The overestimation of dimensionality of personality structure is evident from the observation that, although the factors are forced into orthogonality by means of a varimax rotation, significant inter-correlations are always found among questionnaire scales constructed for measurement of the individual differences on these factors (see Block, 1995). Notably, Maraun (1997) empirically demonstrated that the Five Factor model might simply be a methodological artifact of imposed constraints of factor analysis. By changing the method of analysis to better suit personality data (i.e., when multidimensional scaling is applied), the five factors can be reduced to just two dimensions (Maraun, 1997). It seems that multidimensional scaling may produce analyses that are clearer and more parsimonious than those of factor analytic solutions (MacCallum, 1974; Davison, 1985; Fitzgerald and Hubert, 1987). Particularly, it is superior compared to factor-analytic method in the analysis of correlation matrices that have circumplex structure (Kluger and Tikochinsky, 2001). In fact, the above mentioned weaknesses of factor analysis as a tool for uncovering orthogonal dimensions of circumplex structure were addressed by the invention of the spherical cube model. The model limits the concept of sleep-wake adaptability to three dimensions in despite of factor-analytic results yielding six factorial dimensions. It holds that as few as three orthogonal dimensions are required to explain the six factorial dimensions of sleep-wake adaptability. Therefore, the question arises whether the application of other methods of data reduction, such as multidimensional scaling, can lead to the isolation of only three fully orthogonal dimensions.

SUMMARY OF PREVIOUS RESULTS OF THE STRUCTURAL MODEL OF SLEEP-WAKE ADAPTABILITY Summing up, the proposed formal geometric model (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007) describes a three dimensional structure in the shape of a cube inscribed in a sphere, with three underlying parameters as


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vertical, width, and depth dimensions (supposedly they are associated with the three independent parameters of chronophysiological regulation of the sleepwake cycle). The surface of the sphere formed by these dimensions represents all observed variability of subjectively assessed adaptive traits of the sleepwake cycle. The cube is inscribed in the globe to visualize the way by which these subjectively assessed traits of sleep-wake adaptability are related to the underlying parameters. Namely, it is postulated that the six pairs of edges of the cube correspond to the six largest factors revealed by factor analysis of multi-dimensional chronobilogical questionnaires, such as the 72-item SWPAQ. The predictions of the model concern both the structure of questionnaires for assessment of subjectively recognized traits of sleep-wake behavior and the nature of objective chronophysiological variables behind these traits. The tetra-circumplex criterion was introduced (Putilov, 2007) to provide empirical evidence based on the results of comparing the questionnaire structure with the theoretically predicted structure. Empirical support for the model was provided by both questionnaire (Putilov, Putilov, 2006; Putilov, 2007) and experimental studies (Putilov D. et al., 2007, Putilov A. et al., 2009ab, 2010; Verevkin et al., 2008). The preliminary results (Putilov, 2007) indicate that the formal geometrical model reflects the natural structure of sleep-wake adaptability and that the questionnaire structure meets such predictions of the model as threedimensionality and circumplexity of sleep-wake adaptability. The results of the application of the tetra-circumplex criterion can be useful for further improving the instruments used for assessment of individual habitual traits of the sleep-wake cycle. It remains, however, to be clarified: (1) why the six factorial dimensions suggested by this model are not fully orthogonal; and (2) why, unlike these six factorial dimensions, the three underlying dimensions of the structure of sleepwake adaptability (its three axes) are not revealed by factor analysis of questionnaire data.

SOME GENERAL CONCLUSIONS 

Individual variation in adaptive features of the human sleep-wake cycle has been studied in the field of chronobiology and sleep physiology for several decades.

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The psychometrical evaluation of chronobiological questionnaires revealed the multi-dimensional nature of individual adaptive ability of the sleep-wake cycle. Such a finding suggests the necessity of clarifying both the exact structure of this ability and the chronophysiological background of subjectively assessed variability in the adaptive features of the sleepwake cycle. The results of factor analysis of a battery of chronobiological questionnaires served as an empirical basis for developing the threedimensional (spherical cube) model that explain the multidimensional nature of variation in human sleep-wake behavior. The model postulates the possibility of reducing all varieties of individual habitual traits of the sleep-wake cycle to three underlying sources of variance. The model also provides a valuable taxonomy of adaptive abilities of the sleep-wake cycle. It clarifies the relationship between actual (spatial) and artificial (factorial) dimensions of the structure of inter-individual variation in sleep-wake adaptability. The six factorial dimensions appear to originate from pairwise combinations of polarized spatial dimensions. The model offers the testable assumptions concerning the structure of trait variables designed to represent individual variation in the sleepwake pattern. For instance, it predicts the three-dimensional circumplexity of this structure and the possibility of distinguishing six factorial dimensions of sleep-wake adaptability. When the model was developed, its predictions were used to provide empirical evidence that the model yields the natural structure of sleepwake adaptability and to further enlarge multi-dimensional instruments for assessing sleep-wake patterns (i.e. such as the SWPAQ). By using the SWPAQ, individuals can be differentiated on the three most broad individual traits of sleep-wake adaptability, Lateness (i.e. morning-evening preference or chronotype), Wakeability (i.e. tolerance to sleep pressure or trototype), and Sleepability (i.e. sleep propensity or somnotype).

Taxonomy of Chronotypes, Trototypes and Somnotypes 

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The geometry of the spherical cube model suggests the possibility of applying the tetra-circumplex criterion to examine the correspondence between theoretically and empirically derived structures, such as the factorial structure of the SWPAQ, and the possibility of applying a theoretically based approach to further improve the structures of chonobiological questionnaires. The preliminary results from application of the tetra-circumplex criterion for testing the SWPAQ structure indicate that, in corroboration of the spherical cube model, the SWPAQ comprises four two-dimensional circumplex-like shapes. It was concluded that future revisions of the SWPAQ may primarily focus on improving its structural characteristics by examining and correcting the precise location of some of items, subscales (tetrads) and scales with respect to their structural fit to the four twodimensional circumplexes. The model postulates the three-dimensionality of the structure of sleep-wake adaptability and, hence, disagrees with an assumption that the six largest varimax-rotated factors can represent six fully orthogonal dimensions of sleep-wake adaptability. The question arises whether the application of other than factor analysis methods of data reduction, such as multidimensional scaling, can lead to the isolation of only three fully orthogonal dimensions of the SWPAQ structure.

Chapter 2

THREE-DIMENSIONALITY OF THE STRUCTURE OF SLEEP-WAKE ADAPTABILITY ABSTRACT This chapter includes some new results obtained by applying the spherical cube model to further explore the structure of adaptive ability of the sleep-wake cycle. The model postulates that the six broad traits of sleep-wake adaptability revealed as the six largest factors can be visualized as the six pairs of edges of a cube inscribed in a sphere formed by three orthogonal dimensions (a three-axis circumplex), and that any adaptive trait of the sleep-wake cycle can be mapped at the surface of this sphere. This research seeks: (1) to identify the three dimensions underlying the structure of sleep-wake adaptability; and (2) to suggest ways of further improving tools for assessing this ability. The responses to 72 items of the SWPAQ provided by 1068 adults and adolescents were subjected to both factor and multidimensional scaling analyses. In general, the results provided empirical support for the applicability of the model to structuring sleep-wake adaptations. Specifically, multidimensional scaling helped: (1) to identify three orthogonal axes of the spherical cube representation of the structure of sleep-wake adaptability; and (2) to locate narrow adaptive traits of this ability on the surface of the spherical cube. The three-dimensional coordinates of items, subscales and scales of the SWPAQ were mapped on the surface of the spherical cube, and directions for further developing the questionnaire were determined.

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METHOD Theoretical Framework The three-dimensional (spherical cube) model was introduced in a search for the natural structure of sleep-wake adaptability (Putilov and Putilov, 2005). It depicts all subjectively assessed behavior traits in spherical form and visualizes the six broad traits of sleep-wake adaptability as the six pairs of edges of the cube inscribed in this sphere (“cube-in-globe”). The originality of the model is rooted in distinguishing two kinds of dimensionality, spatial and factorial (Putilov and Putilov, 2006; Putilov, 2006, 2007). The model implies that variation exists along only three underlying parameters (spatial dimensions) which determine the three-dimensional space for all variability of adaptive traits of the sleep-wake cycle. By combining these three orthogonal dimensions, a larger number of the subjectively recognized dimensions of sleep-wake adaptability might be generated. In particular, to generate any of the six broad traits revealed by factor analysis of chronobiological questionnaires as one of the six largest factors (factorial dimensions), two underlying parameters must be set to minimal or maximal value, while the third parameter might be left to vary between minimal and maximal values (Table 1.6). Geometrically, this varying parameter “draws” a pair of edges of a cube on a surface of a sphere to represent one of the six broad abilities of the sleep-wake cycle (Figures 1.1 and 1.17). Further, by fixing this parameter, any broad trait might be divided into narrower traits (Figures 1.1, 1.6, 1.21 and 1.22). One of them might be named a core (or pure) trait, since its two poles are located around the centers of the edges. Other might be called borderline (or mixed or blended) traits because their poles’ locations are shifted relative to a center of edge. For instance, the traits shifted along the edges toward the vertices of the cube might be considered to be admixtures of three adjacent broad abilities (Figures 1.1 and 1.6).

Preliminary Results of Comparing Theoretical and Empirical Structures of the SWPAQ In the earlier studies (Putilov and Putilov, 2005), the results of factor analysis of the battery of chronotypological questionnaires were used as the


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empirical facts supporting the model of individual variability of the adaptive behaviors related to the alternations of the sleep and wake states. It was found that assessing the maximal number of broad and narrow traits predicted by the model can be done with one of the questionnaires included in this battery, the SWPAQ (Putilov, 1990, 2000; Putilov and Onischenko, 2005). The model was then successfully applied for prediction of the properties of the omitted broad and narrow traits. In order to assess these traits one new scale and two new subscales (tetrads) were constructed (Putilov and Putilov, 2006). Twenty new questionnaire items relevant to these predicted abilities were proposed and corrected through a series of three questionnaire surveys. The new version of the SWPAQ comprises 72 true-false items. They are grouped into 6 scales for measuring 6 broad traits of the sleep-wake cycle. Each of these scales is composed of 3 subscales (tetrads) representing narrow traits (Putilov, 2007). Factor analysis of the 72-item SWPAQ revealed the six-factor structure of sleep-wake adaptability. The six-dimensional space is impossible to visualize and represent pictorially. However, the model predicts that, despite the application of varimax rotation, the six largest factorial dimensions are not fully orthogonal dimensions. Rather, these six factorial dimensions might represent a three-dimensional reality (Putilov and Putilov, 2006; Putilov, 2006, 2007). The model hypothesizes a three-dimensional structure with only three (vertical, width, and depth) axes. Geometrically, this structure can be visualized as a spherical cube (Figures 1.2, 1.17, 1.21 and 1.22) or as a combination of two simple forms, a cube included in a sphere (metaphorically named “cube-in-globe” in earlier publications; Figure 1.6). The six pairs of edges on the surface of this spherical cube (Figure 1.2, 1.17, 1.21 and 1.22) are the locations of the six broad adaptive traits of the sleep-wake cycle. Factor analysis of the responses to the items of the chronobiological questionnaires can permit the isolation of these six traits as the six largest rotated factors. Thereafter, these edges of the cube can serve as a system of coordinates for mapping the more numerous narrow adaptive traits (i.e., core subtraits of broad traits and the mixtures of 2-3 broad traits). It was suggested that the geometric assumptions underlying the model permit the possibility of applying a test, the tetra-circumplex criterion, for justifying the correspondence between the empirical and theoretical structures of sleep-wake adaptability. The criterion (Putilov, 2007) was used to compare the model predictions with the results of organization of the SWPAQ tetrads in four circumplex-like shapes. These four empirically constructed circumplexlike structures showed the desirable circumplex properties (Putilov, 2007).

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In particular, it was demonstrated (Putilov, 2007) that, in accordance with the circumplex expectations, the subscales (tetrads) forming four circles showed rather gradual changes of their meaning. Moreover, the results mostly confirmed the numerical supposition of gradual reduction of correlation coefficients with increase of the inter-subscale distance.

Figure 1.1. Spherical cube visualization of the structure of sleep-wake adaptability. The three-dimensional spherical cube model was applied for explanation of the variability of sleep-wake traits. If one assumes that the individual variation in only three underlying parameters is responsible for the appearance of any subjectively assessed trait of sleep-wake adaptability, as few as three spatial (underlying) dimensions might be sufficient for graphing structural representation of this adaptability. These three spatial dimensions (three axes) of the adaptability structure were determined by means of three-dimensional scaling (see Table 1.2). They were alphabetically labeled A, B, and C. The vertical dimension can be interpreted as Arousing or Wakeability (i.e., easy/hard, A/a), the width dimension can be interpreted as Bedtiming or Sleepability (i.e., easy/hard, B/b), and the depth dimension can be interpreted as Clock delaying or Lateness (i.e., easy/hard, C/c). The spherical cube model postulates that the three- and six-factor solutions yielded by factor analysis (Table 1.1) can be visualized on the surface of the three-dimensional spherical cube. The six pairs of cube’s edges correspond to the approximate locations of six factorial dimensions (Table 1.1). Such geometrical interpretation assumes that these dimensions might be regarded as the pairwise combinations of the extremes of the three spatial (underlying) dimensions (Table 1.6). These six factorial dimensions constitute the six broad traits of the sleep-wake adaptability assessed with six scales of the Sleep-Wake Pattern Assessment Questionnaire (SWPAQ), E, M, W, V, F, and S, or Evening Lateness, BC/bc, Morning Lateness, aC/Ac, Anytime Wakeability, AB/ab, Daytime Wakeability, AC/ac, Anytime Sleepability, aB/Ab, and Nighttime Sleepability, Bc/bC (Table 1.4). Left: the three “superscales” and six scales of the SWPAQ labeled near along the vertices and edges of the cube, correspondingly. Right: 13 of 18 subscales (tetrads) that are linked to six pairs of edges of the cube, three pairs of faces of the cube, and four pairs of vertices of the cube.


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The tetra-circumplex criterion permits estimation of the extent of inconsistency between theoretical and empirical structures of sleep-wake adaptability. Although the results on four circumplexes indicated that, in general, the 72-item SWPAQ structure meets the assumption of circumplexity and three-dimensionality, they also revealed that the location of some of the SWPAQ subscales is not optimal. Therefore, the conclusion was drawn (Putilov, 2007) that some details of the SWPAQ structure might desire further correction, if one wants to increase the level of fit of this structure to the ideal three-dimensional circular structure.

Figure 1.2. Spherical cube structure of sleep-wake adaptability: opaque and transparent versions. The two-dimensional representation of the SWPAQ (bottom) can be regarded as a transparent version of the three-dimensional spherical cube structure (top). Left and right top shapes are, correspondingly, front and rear views of the threedimensional spherical cube representation of the six-scale SWPAQ structure, left bottom shape presents a transparent version of the top representation, and right bottom shape presents a transparent version of the three-dimensional representation of thirteen subscales of the SWPAQ shown in Figure 1.1 (bottom). Two spatial dimensions (Arousing or Wakeability, A, and Bedtiming or Sleepability, B) are identical in twoand three-dimensional versions, while the third spatial dimension C (Clock delaying or Lateness) is absent in the two-dimensional version. This interpretation of twodimensional representation of the adaptability structure was confirmed by empirical results illustrated in Table 1.2 and Figures 1.3-1.5.

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Despite the promising results of previous research on structuring sleepwake adaptability in the framework of the model, a detailed analysis of the SWPAQ structure is still required with a new data set. It is expected that such analysis will, again, generally confirm the correspondence between the theoretical and empirical structures. However, it is also expected to be revealed that the SWPAQ is not an optimal representation of the structure predicted by the spherical cube model, because some area of this structure might be overrepresented, while other might be underrepresented by the SWPAQ’s subscales.

Empirical Data The SWPAQ was designed to assess the individual traits of the sleepwake cycle (Putilov, 2000, 2007). Each of 72 true-false items is a short statement describing sleep-wake habitual behavior. Appendix 1 provides the Russian version of the two-sided single-page questionnaire. The English translations of the 72 statements are shown in Figure 1.20. The SWPAQ comprises six scales (E, M, W, V, F and S) that were empirically derived by means of factor analysis. The six scales of the SWPAQ represent six broad traits of sleep-wake adaptability. Each is assessed by the equal number of positively and negatively keyed items. The scales are labeled Evening Lateness (E), Morning Lateness (M), Anytime Wakeability (W), Daytime Wakeability, (V), Anytime Sleepability (F), and Nighttime Sleepability (S). Any one scale includes three subscales or tetrads (four closely related items) that are meant to represent 18 narrow traits. Earlier studies (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007) demonstrated that six varimax rotated factors derived by factor analysis of chronobiological questionnaires roughly correspond to the six scales (E, M, W, V, F and S), and that three varimax rotated factors (“superfactors”) roughly correspond to three couples of scales (“superscales” EM, WV, and FS). These three most general traits were named Lateness (Morning-Evening Preference), Wakeability (Languidness-Vigorousness), and Sleepability (FlexibilityRigidity of Sleep Habits). Following the terminological distinction suggested by Lavie and Zwuluni (1992) and further developed by Van Dongen et al. (2004), the EM score can be interpreted to indicate subject’s chronotype (later or earlier timing of wakefulness and sleep), WV score can represents the subject’s trototype (smaller or greater vulnerability to sleep loss), and FS score can represent the subject’s somnotype (higher or lower sleep propensity).


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Figure 1.3. Two-dimensional scaling plots of the 72 items of the SWPAQ. Twodimensional scaling of Z-scored responses to 72 statements of the SWPAQ (see also Table 1.2). Top: combined sample (all data sources, n=1068); bottom: only new sample (n=718). Spatial dimension A corresponds to “West-East” direction in top plot, and to the “South-North” direction in bottom plot. Spatial dimension B corresponds to “North-South” direction in top plot, and to the “East-West” direction in bottom plot. Abbreviated labels of items: first letter signifies scale (i.e., e belongs to the scale E, Evening Lateness), second letter signifies positive (p) or negative scoring (n) of an item.

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Figure 1.4. Two-dimensional scaling plots of the 18 subscales of the SWPAQ. Twodimensional scaling of Z-scored sums of responses to two items of the same pole of a subscale (tetrad). Top: combined sample (all data sources, n=1068); bottom: only new sample (n=718). Spatial dimension A corresponds to the “West-East” direction, and spatial dimension B corresponds to the “South-North” direction. Abbreviated labels of the halves of subscales (sums of responses to two items): first letter signifies scale (i.e., e belongs to scale E, Evening Lateness), number signifies subscale of a given scale (i.e., e3 is evening-morning preference subscale of the scale E), and second letter signifies positive (p) or negative scoring (n) of two items of the half of subscale.


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Figure 1.5. Two-dimensional scaling plots of the 6 scales of the SWPAQ. Twodimensional scaling of Z-scored sums of responses to six items (a half of a scale characterizing one of its two poles). Top: combined sample (all data sources, n=1068); bottom: only new sample (n=718). Spatial dimension A corresponds to the “WestEast” direction, and spatial dimension B corresponds to the “South-North” direction. Abbreviated labels for the halves of scales (sums of responses to six items): first letter signifies positive (p) or negative scoring (n) of items, second letter signifies scale (i.e., E is the Evening Lateness scale).

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Two questionnaire surveys were employed in the present study. In both surveys, the subjects completed the 72-item SWPAQ by indicating their agreement or disagreement with each statement (either “Yes” or “No”). The respondents of the first survey completed the final version of SWPAQ (Putilov, 2007). An earlier survey was conducted (Putilov, 2007) with the version which only slightly differs from the final version (20 of 72 items were subjected to minor corrections, and the order of items was changed). The new sample consisted of 358 female and 360 male respondents with mean age of 24.6 (SD=10.8) and 22.7 (SD=7.0), respectively. An older sample included 212 female respondents with mean age of 24.9 (SD=14.5), and 138 male respondents with mean age of 23.7 (SD=13.0). In the majority of Tables (1.11.5, 1.7-1.10 and 1.14), the results are presented for the combined sample and separately for the new sample.

Performing Factor Analysis Factor analysis was employed to the data set as a tool for identifying the factorial dimensions of sleep-wake adaptability. Because the model makes direct predictions about the numbers of spatial and factorial dimensions (three and six, respectively), a set of responses to 72 items was subjected to principal component analysis with three and six factors respectively rotated to the varimax criterion. Rotated factor solutions were examined and interpreted by inspecting the high-loading questionnaire items. In particular, such inspection was aimed at examining: (1) the correspondence of the six rotated factors to the six scales of the SWPAQ; and (2) the correspondence between the three rotated factors to the three “superscales” of the SWPAQ. Table 1.1 summarizes the results of factor analysis of the new data set and the combined data set consisting of both new and older samples (n=718 and n=1068, respectively).

Performing Multidimensional Scaling The aim of applying the multidimensional scaling analysis was to test the dimensionality and circumplexity of the SWPAQ structure. Such scaling provides coordinates of each item or each pole of a subscale or each pole of scale in n-dimensional space. Two- and three-dimensional spaces were of


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major interest because they provide the structures that can be visualized. The solutions with two, three and four dimensions were obtained and compared on stress and fit measures. The results are reported in Table 1.2. Each of three coordinates of each item provided by three-dimensional scaling was compared with the standard deviation (SD) calculated for all items to determine whether it could be considered as characterizing the pole of a dimension predicted by the spherical cube model (i.e., these might be the items with, at least, one coordinate either higher than +1SD or lower than -1SD).

Figure 1.6. “Cube-in-globe” representation of scales and subscales of the SWPAQ. Cube is inscribed in sphere (“globe”) to depict the six broad traits of sleep-wake adaptability. The six pairs of cube’s edges correspond to the six broad traits conceptualized as pairwise combinations of the extremes of the three spatial (underlying) dimensions A, B, and C (Table 1.6). They are labeled along the edges similarly to the corresponding SWPAQ scales (E, M, W, V, F, and S, or Evening Lateness, Morning Lateness, Anytime Wakeability, Daytime Wakeability, Anytime Sleepability, and Nighttime Sleepability). Each scale includes three subscales (tetrads e1-3, m1-3, w1-3, v1-3, f1-3, and s1-3). Some of them correspond to core (pure) narrow traits located near the centers of the edges (e2, m2, w2, v2, f2, and s2). Some of other subscales correspond to mixed (blended or intermittent) narrow traits located near the vertexes (corners), the meeting points of three edges (i.e., e1, m1, v1, and f1). The remaining subscales (shown in Figure 1.1, bottom) may correspond to narrow traits located near the centers of faces (i.e., e3, v3, and s3). Top “globe” also illustrates the locations of hypothetical narrow traits that are predicted but not assessed with the 18 SWPAQ’s subscales (i.e., e?, m?, w?, v?, f?, and s?). Bottom “globe” shows only those 10 SWPAQ’s subscales that were included in four two-dimensional circumplexes shown in Figures 1.8-1.11.

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Figure 1.7. Tetra-circumplexical slicing of “cube-in-globe” representation of the SWPAQ. “Cube in globe” shape (Figure 1.6, top) was reproduced four times to illustrate the way by which four almost two-dimensional slices (circumplexes) were obtained by connecting the centers of three factorial dimensions in the twodimensional circles (see also Figure 1.12). Each of these four quasi-circumferences runs through the areas of three SWPAQ’ scales. Because each scale of the SWPAQ subsumes three subscales (tetrads), the sequence of, at least, 6 of these 9 subscales can be ordered without long gaps. In general, such ordering suggests possibility of inclusion of, at least, 10 SWPAQ’s tetrads (shown in Figure 1.6, bottom) into four circumplexes (shown in Figures 1.8-1.11). It is predicted that the subscales included in each circumplex must gradually change their meaning (Figures 1.8-1.11), and that the association between them must decline gradually with the increase of inter-subscale interval, up to turning into the negative association when the inter-subscale distance exceeds ¼ of circumference. It is expected that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales (tetrads), and that, hence, a simple chain of correlations can represent the pattern of decline of association among the subscales included in each of four circumplexes (Table 1.7).


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Comparison of Inter-Correlations Among and Between Spatial and Factorial Dimensions To show that, despite applying the procedure of varimax rotation, the factorial dimensions are still inter-correlated, the matrix of inter-correlations among factor loadings was calculated (Tables 1.3 and 1.4). The matrix of inter-correlations among three factors was compared with the matrix of intercorrelations among the three coordinates obtained by multidimensional scaling which should have been fully orthogonal (Table 1.3). The matrix of intercorrelations among factorial loadings on six factors is presented along with the matrix of inter-correlations among the scores on six scales in Tables 1.4. Additionally, Table 1.5 illustrates the pattern of inter-correlations among pure subscales of the six scales.

Mapping Narrow, Broad, and Broadest Traits in ThreeDimensional Space The model does not provide any keys for determining which pair of cube’s edges represents a particular SWPAQ scale. Therefore it is necessary to construct such a cube-like shape that fits well with the empirical relationships between questionnaire scales. First, the inter-correlations among the factors, scales, and subscales (tetrads) might indicate which of them are the most probable neighbors. Second, item loadings and the loadings averaged over scales or subscales might be used as indicators of their proximity to one or more edges representing the six broad factorial dimensions of sleep-wake adaptability. Third, the three coordinates of any item, subscale or scale might be directly calculated by performing three-dimensional scaling. Consequently in order to determine the positions of each of six factorial dimensions relative to the positions of the edges of the cube, the patterns of inter-correlations among scales and subscales and the patterns of intercorrelations among factors’ loadings were examined (Tables 1.3-1.5). The results of the assignment of the scales to edges on a basis of intercorrelations among factor’s loadings, scales and subscales were then confirmed by checking the coordinates of the scales and subscales of the SWPAQ provided by the three-dimensional scaling (Tables 1.11-1.13). Table

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1.6 explains the associations of the poles of six factorial dimensions with the pairwise combinations of the poles of three polarized spatial dimensions.

Figure 1.8. Tetra-circumplex representation of the SWPAQ’ subscales: first circumplex. Figure illustrates the gradient of meaning of items included in the first circumplex of the tetra-circumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.7.

Figures 1.1 and 1.2 illustrate the empirically suggested relationship between the SWPAQ constructs in terms of the spherical cube model. They show how the three-dimensional solutions yielded by multidimensional scaling and the three- and six-factorial solutions yielded by factor analysis can be visualized on the surface of the three-dimensional spherical cube.


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Mapping Narrow, Broad, and Broadest Traits in TwoDimensional Space Figure 1.2 illustrates how the three-dimensional representation (dimensions A, B, and C) might be transformed into the two-dimensional version of the spherical cube model (when the third orthogonal dimension, C, is ignored). This Figure suggests the possibility of comparison with the empirical results shown in the three next Figures 1.3-1.5 providing the empirically determined positions of the items, subscales (tetrads) and scales, respectively, in two-dimensional space.

Figure 1.9. Tetra-circumplex representation of the SWPAQ’ subscales: second circumplex. Figure illustrates the gradient of meaning of items included in the second circumplex of the tetra-circumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.7.

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Application of the Circumplex Criteria Not surprisingly, the predicted locations of the SWPAQ constructs (Figures 1.1, 1.2, and 1.6) tend to deviate more or less from their empirically obtained coordinates (i.e., Tables 1.11-1.13). The question arises whether these deviations are random or assume that the empirical structure does not fit well in the three-dimensional circumplexical structure predicted by the model.

Figure 1.10. Tetra-circumplex representation of the SWPAQ’ subscales: third circumplex. Figure illustrates the gradient of meaning of items included in the third circumplex of the tetra-circumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.7.

The correctness of mapping of six scales (and corresponding factors) was tested with the tetra-circumplex criterion (Putilov, 2007). Its application is


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permitted by the geometric assumptions underlying the spherical cube model. It postulates that the three-dimensional structure (three-axis circumplex) can be adequately presented by a set of four almost two-dimensional circumplexes (see the previous chapter for details).

Figure 1.11. Tetra-circumplex representation of the SWPAQ’ subscales: fourth circumplex. Figure illustrates the gradient of meaning of items included in the fourth circumplex of the tetra-circumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.7.

Figures 1.6 and 1.7 illustrate the way in which a three-dimensional circumplex can be sliced into two halves to obtain each of four twodimensional circumplexes with identical geometrical features.

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Figure 1.12. Four two-dimensional slices of the SWPAQ’s scales. The spherical cube model predicts that the content of six largest factorial dimensions can be organized in four two-dimensional circumplexes each of which includes three factorial dimensions (each of six factorial dimensions is included in two circumaplexes, and, in each circumplex, a factorial dimension has one left and one right neighbor). For instance, the factorial dimension E is a neighbor of factorial dimensions V (on the left) and F (on the right) in the bottom left circumplex and of factorial dimensions M (on the left) and W (on the right) in the bottom right circumplex (see also Figure 1.7). The plane of each of four circumplexes might be partitioned into 6 segments of 60 degree each (this is a theoretically predicted distance between adjacent poles of factorial dimensions). Such two-dimensional tetra-circumplex representation of the three-dimensional shape provides a possibility to evaluate whether the scales of the SWPAQ confirm well to the theoretically expected circular structure. It is predicted that the scales included in each circumplex must gradually change their meaning, and that the association between them must decline with the increase of inter-scale interval, up to turning into the negative association when the inter-subscale distance exceeds ¼ of circumference. It is expected that random errors in selection of items for the scales are likely to average out when mean correlations are calculated for equidistant subscales (tetrads), and that the pattern of decline of association among the subscales can be represented for each of four circumplexes by a simple chain of correlations (Table 1.8).


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These four circumplexes are unique in that each of them organizes in a circle three core subscales (tetrads) located around the centers of cube’s edges. By summarizing the measurements of the degree to which each of four two-dimensional slices of a three-dimensional circumplex fits the circular model, the criterion provides an answer to the question whether a set of individual trait variables comprises a three-axis circumplex. It is expected that the circular nature of an empirical circumplex will be indicated by gradual change in traits meaning, and that this change will be confirmed by quantitative estimations of change in the strength of correlation between traits included in each circumplex. Specifically, it is predicted that traits adjacent to each other are more highly correlated than traits two steps apart on the circle, which are, in turn, more highly correlated than traits three steps apart, which in turn exhibit higher correlations than traits four steps apart, which in turn exhibit higher correlations compared to the antipodal traits five steps apart that show the lowest negative correlations. The majority of 18 tetrads of the SWPAQ (Figure 1.6) were represented as vectors in four almost two-dimensional circular spaces formed by the coordinates of core tetrads (Figure 1.7). First, a set of four such circular structures was drawn in Figure 1.7. Second, the meanings of all adjacent tetrads in each of four circumplexes were compared in Figures 1.8-1.11 (these tetrads can represent either the same or different scales). Third, additional comparison was made for the meanings of the scales included in four circumplexes in Figure 1.12. Fourth, the inter-correlations among six tetrads forming each of four circumplexes were calculated. Fifth, the inter-correlations representing the equal distances among subscales were averaged within each of four circumplexes, and these intercorrelations were then averaged over four circumplexes with the expectation that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales. These mean inter-correlations are presented in Table 1.7. Sixth, the inter-correlations among scales were computed. Seventh, the inter-correlations representing the equal distances among scales were averaged within each of four circumplexes, and these intercorrelations were then averaged over four circumplexes. These mean intercorrelations are presented in Table 1.8.

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Figure 1.13. Tri-circumplexical slicing of “cube-in-globe” representation of the SWPAQ. The spherical cube model predicts that only three couples of six factorial dimensions are centered at right angles to each other on the circumplexes formed by the axes A, B, and C. Namely, there are right angle relationships between factorial dimensions M and V on the axes A and B, between the factorial dimensions F and W on the axes A and B, and between the factorial dimensions S and E on axes B and C. Consequently, these pairs of factorial dimensions can be organized in three twodimensional circumplexes. Such three two-dimensional circular spaces were produced by slicing the “cube-in-globe” shape through the centers of two pairs of cube’s edges. It is suggested that each of these three circumferences runs over the areas of four SWPAQ’ subscales included in, at least, two scales. Such two-dimensional tricircumplex representation of the three-dimensional shape provides a possibility to evaluate whether the subscales of the SWPAQ confirm well to the theoretically expected circular structure. It is predicted that the resemblance and inter-correlations among the subscales must decline with the increase of interval between them (Figure 1.14 and Table 1.9, respectively).


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Figure 1.14. Tri-circumplex representation of the SWPAQ’ subscales. Figure illustrates the gradient of meaning of items included in three circumplexes of the tricircumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.13.

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Figure 1.15. Hexa-circumplexical slicing of “cube-in-globe” representation of the SWPAQ. “Cube-in-globe” shape was sliced up along each pair of cube’s edges to obtain six two-dimensional circular spaces. It is suggested that each of the six circumferences runs over the areas of four SWPAQ’ subscales belonging to, at least, three scales. Such two-dimensional hexa-circumplex representation of the threedimensional shape provides a possibility to evaluate whether the subscales of the SWPAQ confirm well to the theoretically expected circular structure. It is predicted that the subscales included in each circumplex must gradually change their meaning (Figures 1.16), and that the association between them must decline gradually with the increase of inter-subscale interval, up to turning into the negative association when the inter-subscale distance exceeds ¼ of circumference. It is expected that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales (tetrads), and that the pattern of decline of association among the subscales can be represented for each of six circumplexes by a simple chain of correlations (Table 1.10).


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Figure 1.16. Hexa-circumplex representation of the SWPAQ’ subscales. Figure illustrates the gradient of meaning of items included in six circumplexes of the hexacircumplexical representation of the SWPAQ’s subscales. See also notes to Figure 1.15.

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Thus, a set of Figures (1.7-1.12) and Tables (1.7 and 1.8) permitted comparison of the empirical pattern of meanings and correlations with the pattern predicted by the circumplex structure. The tetra-circumplex criterion was introduced for testing the structures yielded by factor analysis (Putilov, 2007). The four circumplexes do not cross the centers of 8 squares constituting the spherical cube shape (A and a, B and b, and C and c). It is problematic to determine factor loadings of items located in these areas (i.e., they might be equally distant from four cube’s edges which are the poles of four factors). However, because three-dimensional scaling permits the evaluation of coordinates of the items in these locations, two more circumplexical criteria might be proposed for testing three-dimensionality and circumplexity of empirical structure. The model postulates that the threedimensional structure (three-axis circumplex) can be adequately presented by two sets of two-dimensional circumplexes (Figures 1.13-1.16). Three circumplexes of one set connect the centers of two pairs of edges as shown in Figure 1.13, and six circumplexes of another set connect the opposing edges as shown in Figure 1.15. These circumplexes are similar to the circumplexes of the tetracircumplexical representation in that the predicted theoretical order of the scales’ relations requires that closeness of meanings and correlations between subscales adjacent on the circle are greater than closeness of meanings and correlations between subscales one step apart, closeness of meanings and correlations between subscales one step apart are greater than closeness of meanings and correlations between subscales two steps apart, and closeness of meanings and correlations between subscales two steps apart are greater than closeness of meanings and correlations between subscales directly across from one another (Figures 1.14 and 1.16 and Tables 1.9 and 1.10). In general, the circumplex criteria are aimed at justifying the correspondence between the empirical and theoretical structures of sleep-wake adaptability by avoiding the difficulty of displaying circular features of the structure of sleep-wake adaptability in three-dimensional space. The question arises whether the deviations from circular structure are random or whether they are systematically due to the empirical structure not fitting well in the three-dimensional circumplexical structure predicted by the model. It is required that closeness of meanings and correlations between scales adjacent on the circle are greater compared to closeness of meanings and correlations between more distant scales; that closeness and correlation gradually decrease with increase of the inter-scale distance, and that they are the least similar for the last of the compared pairs of scales.


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Figure 1.17. Six squares constituting spherical cube structure of sleep-wake adaptability. The spherical cube shape is shown from the top (face or square A), bottom (face or square a), left (face or square b), right (face or square B), front (face or square C), and back (face or square c). The spherical cube model postulates that the spherical cube structure of sleep-wake adaptability is generated by vertical, width, and depth dimensions. They were interpreted as Arousing or Wakeability (i.e., easy/hard, A/a), Bedtiming or Sleepability (i.e., easy/hard, B/b), and Clock delaying or Lateness (i.e., easy/hard, C/c). The cube divides the surface of the sphere into six equal parts (six spherical squares) to illustrate the assumption of the model that the six-factor solution yielded by factor analysis (Table 1.1) can be visualized on the surface of the three-dimensional spherical cube as the six pairs of cube’s edges. Such geometrical interpretation assumes that the six factorial dimensions are pairwise combinations of the extremes of the three spatial (underlying) dimensions (Table 1.6). These six factorial dimensions constitute six broad traits of the sleep-wake adaptability. These traits can be assessed by 6 SWPAQ’ scales, E, M, W, V, F, and S, or Evening Lateness, BC/bc, Morning Lateness, aC/Ac, Anytime Wakeability, AB/ab, Daytime Wakeability, AC/ac, Anytime Sleepability, aB/Ab, and Nighttime Sleepability, Bc/bC (see also Table 1.4).

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Figure 1.18. SWPAQ subscales organized around six squares of spherical cube structure. Subscales (tetrads) of the SWPAQ were organized in the six circles around three pairs of faces (six squares) of the cube, v3=A/a, s3=B/b, and e3=C/c (see notes to Figure 1.17).


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Figure 1.19. SWPAQ subscales organized around eight vertexes of spherical cube structure. Subscales (tetrads) of the SWPAQ were organized around four pairs of vertices (corners) of the cube, w1=ABC, e1=AbC, m1=abC, and f1=aBC (see notes to Figure 1.17).

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Figure 1.20. SWPAQ subscales organized around twelve edges of spherical cube structure. Subscales (tetrads) of the SWPAQ were organized around six pairs of edges (ribs) of the cube, e2=BC/bc, m2=aC/Ac, w2=AB/ab, v2=AC/ac, f2=aB/Ab, and s2=Bc/bC (see notes to Figure 1.17).


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Detailed Classification of Trait Variables in Geometrical Terms of the Spherical Cube Model The space coordinates of each of 72 items in three dimensions provided by the three-dimensional scaling of the new data set were analyzed in more detail. Some of these results are presented in Tables 1.11-1.13.

Figure 1.21. Map of 37 subscales predicted by the spherical cube model. Spherical cube model postulates that all traits of sleep-wake adaptability might be located in three-dimensional space. Specifically, the surface of sphere includes all individual variation generated by three spatial (underlying) parameters. At least, 37 subscales might be required to achieve even distribution of the SWPAQ constructs over the whole surface of the spherical cube representation of sleep-wake adaptability. Each of these subscales must be designed to assess a separate narrow trait of this ability.

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The coordinates of the subscales and scales were determined by averaging the coordinates of separate items (Table 1.11) and by three-dimensional scaling of subscales’ and scales’ scores (Tables 1.12 and 1.13, respectively). These coordinates provided information on the possible locations of subscales and scales on the surface of the spherical cube shown in Figures 1.1 and 1.2.

Figure 1.22. Map of SWPAQ’s items exemplifying 18 of 37 predicted subscales. The spherical cube model predicts that more than a half of hypothetical narrow traits of sleep-wake adaptability (19 of 37) cannot be yet assessed with the SWPAQ subscales (tetrads). These subscales are abbreviated as e?, m?, w?, v?, f?, and s?. Eighteen items of the SWPAQ illustrate the locations of 18 already established subscales of the SWPAQ (tetrads e1-3, m1-3, w1-3, v1-3, f1-3, and s1-3).

These results were also used to select subsets of subscales (tetrads) for circumplexes shown in Figures 1.6-1.13 and 1.15. Additionally, the coordinates of items characterizing 18 subscales (tetrads), 6 scales, 6 factors, and 3 underlying dimensions were compared with the loadings of these items


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on three and six factors yielded by the three- and six-factor solutions (data are not shown). The internal consistency of the scales and “superscales” was estimated using Cronbach’s alpha coefficient and the correlation between two halves of a scale (Table 1.18). Figures 1.17-1.22 summarize the results on checking the correspondents between the structure of the SWPAQ and structure predicted by the spherical cube model. Figure 1.17 presents a view of the structure at the level of the scales from top, bottom, right, left, front and back. Figure 1.18 presents the items associated with these views, displaying the three pairs of faces of the spherical cube (six squares). Figure 1.19 shows the items associated with four pairs of the cube’s vertices. Figure 1.20 contains all 72 items of the SWPAQ divided into six scales each of which was associated with a pair of edges of the cube. Finally, Figures 2.21 and 2.22 provide the general view of locations of the SWPAQ subscales in comparison with the locations of a bigger number of subscales predicted by the spherical cube model. Given that there is no error-free questionnaire and that the largest part of the SWPAQ was constructed on the basis of factor-analytic results obtained prior to the introduction of the spherical cube model, it should come as no surprise that the spherical cube structure as it is depicted in Figures 1.1, 1.2, 1.6, 1.7, 1.17 and 1.21 differs somewhat from the actual SWPAQ structure illustrated in Tables 1.3-1.5 and 1.7-1.11. The comparison of ideal and real structures allows the quantitative detection of the deviations of the SWPAQ constructs from the constructs predicted by the model. Therefore, one of the important questions of the present analysis was the following. If the structure of the 72-iten SWPAQ fits the spherical cube structure in general, can some details of the questionnaire structure be further improved to produce a closer fit between the empirical structure and the spherical cube model?

RESULTS Results of Factor Analysis The spherical cube representation is a rather restrictive model. When applied to the multi-factor structure of the SWPAQ, it postulates threedimensionality and circumplexity of this questionnaire (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007). Furthermore, the limitations are imposed on

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the number of the broadest and broad traits of sleep-wake adaptability (Figure 1.1). This theoretical prediction aims the factor analysis at the rotation of the sets of three and six largest factors. Table 1.1. Variance accounted for by each factor of the rotated 6- and 3factor solutions Data source Solution Six-factor Component 1 2 Inter-pretation -E +M Rotation sums of squared loadings Total 4.73 4.67 % of variance 6.57 6.49 Cumulative % 6.57 13.05 Data source Solution Component 1 2 Interpretation +F +M Rotation sums of squared loadings Total 4.73 4.68 % of variance 6.57 6.50 Cumulative % 6.57 13.07

Combined sample Three-factor 1 2 -WV +FS

3 -EM

4.30 3.88 2.28 5.97 5.39 3.16 25.49 30.87 34.04 New sample Six-factor 3 4 5 6 -E +V -S +W

7.58 10.53 10.53

4.84 6.73 24.74

4.59 6.37 19.44

7.67 10.66 10.66

3 +F

4 +W

5 -S

6 -V

4.65 6.46 19.52

4.57 6.34 25.78

3.87 5.38 31.16

2.12 3.02 34.18

1 -WV

5.39 7.49 18.02

Three-factor 2 3 +FS -EM 5.51 6.66 18.31

4.58 6.35 24.67

Notes. Factor analysis of responses to 72 items was performed separately for New sample (n=718) and for the whole (new and older) sample (Combined sample, n=1068). Six factors of the rotated six-factor solution (it accounts for about a third of the total variance) roughly correspond to the six scales of the Sleep-Wake Pattern Assessment Questionnaire (SWPAQ), E, M, W, V, F, and S (Evening Lateness, Morning Lateness, Anytime Wakeability, Daytime Wakeability, Anytime Sleepability, and Nighttime Sleepability). Three “superfactors” of the three-factor solution (it accounts for about a quarter of the total variance) correspond to the pairs of scales (“superscales”), EM, WV, and FS (Lateness, Wakeability, and Sleepability). The hypothetical locations of six factors (scales) and three “superfactors” (“superscales”) in three- and two-dimensional spaces are shown in Figures 1.1, 1.2 and 1.17.

A classification of the sleep-wake cycle’s adaptive traits proposed by the model is based on distinguishing between three spatial (underlying) dimensions and six factorial dimensions. In particular, the model predicts the coupling of six factorially distinguishable dimensions of sleep-wake adaptability in the three factor solution. This prediction was tested by inspecting the three- and six-factor solutions based on two data sets, the new and combined. Both solutions were found to be easily interpretable. A set of six factors resembling the corresponding six scales of the SWPAQ (E, M, W,


49

V, F, and S) emerged in the rotated six-factor solution, and the rotated threefactor solution pointed to the factors corresponding to the three “superscales” (EM, WV, and FS). Table 1.1 shows that the correspondence between factorial dimensions extracted from new and combined data sets appears fairly similar. Table 1.2. Stress and fit measures for 2-, 3-, and 4-dimensional scaling of the SWPAQ Data source Number of dimensions Interpretation of dimensions Normalized Raw Stress Dispersion Accounted For Tucker's Coefficient of Congruence

Combined sample 2 3 A, B A, B, C 0.047 0.013 0.952 0.987

4 A, B, C,+? 0.009 0.991

New sample 2 3 A, B A, B, C 0.048 0.016 0.952 0.983

4 A, B, C,+? 0.009 0.991

0.976

0.996

0.976

0.996

0.993

0.992

Notes. Multidimensional scaling of Z-scored responses to 72 statements was performed for all data sources (Combined sample, n=1068) and separately for New sample (n=718). The coordinates of statements in 2-dimensional space are shown in Figure 1.3. The axes A, B, and C revealed by three-dimensional solution were interpreted as Arousing (i.e., easy/hard, A/a), Bedtiming (i.e., easy/hard, B/b), and Clock delaying (i.e., easy/hard, C/c). The hypothetical locations of these dimensions in three-dimensional space of the spherical cube are shown in Figures 1.2, 1.3 and 1.17.

The particular contents of factors extracted from the new data set were interpreted as highly congruent with the contents of factors extracted from the combined data set. The following results illustrate that each factor extracted from the new and combined data sets was substantially covered by a corresponding scale of the SWPAQ. Among 12 items with 6 highest positive and 6 lowest negative loadings on factor E there were 10 items from E-scale in results on the new sample set and 8 items from this scale in the results on the combined sample. Among 12 items with the highest and lowest loadings on factors M, W, V, F and S there were, correspondingly, 11 and 11 items of Mscale, 6 and 7 items of W scale, 6 and 5 items of V scale, 11 and 11 items of F scale, and 12 and 12 items of S scale. Moreover, among 24 items with the highest and lowest loadings on the EM “superfactor” in the new and combined samples, there were, respectively, 22 and 20 items of E and M scales. The 24 items with the highest and lowest loadings on the WV “superfactor” included 18 and 18 items from the scales W and V. The scales F and S were represented by 22 and 20 items in the lists of

50

Arcady A. Putilov

24 items having the highest and lowest loadings on the FS “superfactor”. The most remarkable deviation from the expected results was the underrepresentation of the items of V and W scales among the items with the highest and lowest loadings on the factors V and W, respectively. However, the vast majority of other items with the highest and lowest loadings on these factors were from the same “superscale” WV. Overall, the geometrical features of the spherical cube model suggest certain limitations of the number of traits of sleep-wake adaptability of different levels of generality. The empirical data provide strong support for six factorial dimensions, and for the possibility of their coupling in three factorial dimensions in the three-factor solution. The factorial structure of the SWPAQ was replicated with the new data set. All six expected broad factorial dimensions were among the six largest factors (E, M, W, V, F, and S), and all three of the expected broadest factorial dimensions were among the three largest factors (EM, WV, and FS).

Results of Multidimensional Scaling Multidimensional scaling analysis was used to determine how many dimensions are required to explain the empirical SWPAQ structure, and to spatially represent its dimensions. As it is shown in Table 1.2, the scaling procedures yielded the normalized raw stress values of 0.048 (the new sample) and 0.047 (the combined sample) for the two-dimensional model, 0.016 and 0.013 for three-dimensional model, and 0.009 and 0.009 for the fourdimensional model. The values for dispersion accounting for these three models were, respectively, 0.952 and 0.952, 0.983 and 0.987, and 0.991 and 0.991. Tucker's coefficients of congruence values were 0.976 and 0.976, 0.992 and 0.993, and 0.996 and 0.996, respectively. It appears that a threedimensional solution is quite appropriate to account for the questionnaire data. For instance, the value of the normalized raw stress statistic decreases substantially (more than three times) and drops below 0.02 when the number of dimensions is increased from 2 to 3. Further increasing dimensionality does not decrease stress substantially. A high goodness of fit in three dimensions is also indicated by the fact that the model accounts for more than 98% of the data dispersion. The addition of the fourth dimension improved the fit by less than 1%. In a perfect three-dimensional circumplex, variables should be equidistant from the center of the sphere. In other words, any two-dimensional circles


51

obtained by slicing a three-dimensional shape should have a constant radius. If all SWPAQ items can be mapped on the surface of the sphere, this is a case of three-dimensional circumplex structure. If only some of them can be mapped on the surface, while other can be located inside the sphere, this is a case of three-dimensional radix structure. Inspection of three coordinates of each of 72 items of the SWPAQ showed that any item had at least one coordinate either above 1SD or below -1SD in magnitude. This result suggests that all 72 items are located at a substantial distance from the center of the three-dimensional sphere (i.e., near the surface of the circumplex and away from its center). Table 1.3. Correlations between and among three spatial and three factorial dimensions of the SWPAQ Solution

Rotated three-factor solution

Three-dimensional solution

Dimension A

EM 0.232*

A

B

C

WV

A

1st

0.002

0.010

0.977*** -0.203

B

0.008

2nd

-0.012

0.065

0.879***

0.617*** B

C

-0.017

-0.031

3rd

-0.139

-0.363**

0.716*** C

WV

0.966***

0.203~

-0.131

1st

-0.085

0.147

WV

-0.118

2nd

0.244*

FS

FS

-0.330

**

0.875

*** ***

EM

0.073

0.495

New sample

A

B

-0.289 0.822

*

***

C

Three-dimensional solution

FS

Combined sample

rd

0.072

0.188

3

WV

FS

EM

Rotated three-factor solution

EM Dimension Solution

Notes. Three spatial dimensions (1st, 2nd, 3rd) were revealed by multidimensional scaling (A, B, and C of Three-dimensional solution), and three factorial dimensions (1st, 2nd, 3rd) were revealed by factor analysis (WV, FS, and EM of varimax Rotated three-factor solution). Three spatial dimensions are represented by averaged coordinates of the responses to 72 SWPAQ’s items, and three factorial dimensions are represented by averaged factor loadings for the responses to 72 SWPAQ’s items. Above diagonal: Combined sample (n=1068). Below diagonal: New sample (n=718). Levels of significance of two-tailed Spearman rank coefficient of correlation: * p < 0.05;** p < 0.01, ***p < 0.001. All Pearson coefficients of correlation among spatial dimensions were equal to zero.

Thus, the results on the three-dimensional solution provide evidence for the existence of three measurable underlying dimensions predicted by the spherical cube model. They also demonstrate that the empirically revealed locations of 72 items fit a three-dimensional circumplex configuration better than a three-dimensional radix configuration.

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Arcady A. Putilov

Results of Comparing Inter-Correlations among and between Spatial and Factorial Dimensions Tables 1.3-1.5 suggest that the results on the inter-correlations among the dimensions of the SWPAQ were almost identical for the new and combined sets. Table 1.3 illustrates a very important fact that the three-dimensional solution obtained by performing multidimensional scaling is not identical to the three-factor solution. Three “superfactors” provided by the three-factor solution can be obtained by reorienting the axes of the spherical cube (Figure 1.1). Table 1.4. Inter-correlations among scores on six scales and among loadings on six factors of the SWPAQ E-scale E-factor -0.055 (-0.030) 0.525 (0.197) 0.161 (0.423) 0.092 (0.614) 0.398 (0.270)

0.147 (0.152) M-scale M-factor -0.666 (-0.211) -0.110 (-0.711) 0.232 (0.086) -0.125 (-0.038)

0.355 (0.317) -0.366 (-0.378) W-scale W-factor 0.476 (-0.093) -0.292 (0.428) 0.261 (0.228)

0.177 (0.182) -0.413 (-0.453) 0.446 (0.445) V-scale V-factor -0.295 (0.034) -0.085 (0.103)

0.023 (0.011) 0.139 (0.176) 0.032 (0.058) -0.318 (-0.323) F-scale F-factor 0.404 (0.368)

0.122 (0.079) 0.087 (0.105) 0.199 (0.178) 0.068 (0.065) 0.258 0.281 S-scale S-factor

Note. Value without parentheses shows correlation for the combined sample (n=1068), and value in parentheses shows correlation for the new sample (n=718). Those inter-correlations that are based on factor loadings are shown below diagonal, and those based on scales’ scores are shown above diagonal. Spearman coefficients of correlation higher than 0.20 in magnitude are printed in boldface, and those lower -0.20 in magnitude are printed in italics.

However, it is necessary to stress that, unlike the factor loadings on three factors, the three coordinates provided by multidimensional scaling are not inter-correlated. Table 1.3 illustrates the relationship between three dimensions obtained by factor and scaling analyses. Table 1.4 indicates that the suggestion of inter-correlation among six factorial dimensions better accounts for the results of factor analysis than does the suggestion of orthogonal relationship between them. Indeed, both matrices - of inter-correlations among loadings and of inter-correlations among scales - suggest that the six factorial


53

dimensions cannot be considered as fully orthogonal. This Table also illustrates the close match between the matrix of inter-correlations among factorial loadings on six factors and the matrix of inter-correlations among the scores on six scales. Table 1.5. Inter-correlations among scores on core subscales of six scales of the SWPAQ 0.098

0.364

0.060

0.075

0.054

0.095

m2

-0.025

-0.226

0.050

0.093

0.321

-0.025

w2

0.117

0.131

0.102

0.041

-0.259

0.112

v2

-0.138

0.030

0.056

0.070

0.144

-0.161

f2

0.147

-0.016

0.123

0.072

0.007

0.191

s2

e2

Note. The correlations among subscales (tetrads) for the new sample (n=718) are shown below diagonal, and those for the combined sample (n=1068) are shown above diagonal. Spearman coefficients of correlation higher than 0.20 in magnitude are printed in boldface, and those lower than -0.20 in magnitude are printed in italics.

Importantly, the inter-correlations might be lowered if the matrix is based on only those six subscales (tetrads) of the six scales that represent core (pure) traits of broad traits (Table 1.5). This result supports the asumption about the geometry of the factorial dimensions predicting the absence of highly or even moderately inter-correlated core subscales, because they are located near the centers of the cube edges (Figures 1.1, 1.2, 1.6, 1.21 and 1.22). Although the majority of scales or core subscales correlated with one another in a predictable manner, some of the coefficients of correlation were higher or lower than the expected coefficients. Summing up, the inter-correlations obtained among the SWPAQ constructs support the model’s prediction that the application of factor analysis yields biased estimates of the dimensionality of the structure of sleep-wake adaptability. They suggest that three-dimensional scaling rather than factor analysis reveals truly orthogonal dimensions of sleep-wake adaptability.

Table 1.6. Presumed three-axis representation of six factorial dimensions of adaptability structure High and low poles of the six factorial dimensions (Label of the pole) High poles of six factorial dimensions V+ (Daytime Wakeability) AC M+ (Morning Lateness) aC E+ (Evening Lateness) BC S+ (Nighttime Sleepability) Bc F+ (Anytime Sleepability) aB W+ (Anytime Wakeability) AB Low poles of six factorial dimensions V- (Daytime Wakeinability) ac M- (Morning Earliness) Ac E- (Evening Earliness) bc S- (Nighttime Sleepinability) bC F- (Anytime Sleepinability) Ab W- (Anytime Wakeinability) ab

Pole of spatial dimension Arousing (A) Bedtiming Clock de(B) laying (C)

Interpretation of the poles of six factorial dimensions in terms of three spatial dimensions

High (A) Low (a) Various Various Low (a) High (A)

Various Various High (B) High (B) High (B) High (B)

High (C) High (C) High (C) Low (c) Various Various

Easy Arousing & Easy Delaying Hard Arousing & Easy Delaying Easy Bedtiming & Easy Delaying Easy Bedtiming &Hard Delaying Hard Arousing & Easy Bedtiming Easy Arousing & Easy Bedtiming

Low (a) High (A) Various Various High (A) Low (a)

Various Various Low (b) Low (b) Low (b) Low (b)

Low (c) Low (c) Low (c) High (C) Various Various

Hard Arousing & Hard Delaying Easy Arousing & Hard Delaying Hard Bedtiming & Hard Delaying Hard Bedtiming & Easy Delaying Easy Arousing & Hard Bedtiming Hard Arousing & Hard Bedtiming

Notes. The spherical cube model postulates that three vectors (spatial dimensions) determine the coordinates of six personality factors (factorial dimensions). Each pole of such factorial dimension is associated with the pairwise combination of the poles of two spatial dimensions (either High or Low), while the 3rd spatial dimension varies between high and low poles (Various). Each factorial dimension can be visualized as a pair of cube’s edges generated by spatial dimensions A, B, and C (Figures 1.1, 1.2, top, 1.6, 1.7, 1.13, 1.15 and 1.17). It is suggested that these three dimensions roughly correspond to the dimensions revealed by threedimensional scaling. They were interpreted as Arousing (i.e., easy/hard, A/a), Bedtiming (i.e., easy/hard, B/b), and Clock delaying (i.e., easy/hard, C/c).

Table 1.7. Inter-correlations among the selected SWPAQ subscales organized in 4 circumplexes Distance, degree Data source Four Separate Circumplexes Circumplex 1 Circumplex 2 Circumplex 3 Circumplex 4 General Circumplex Data source Four Separate Circumplexes Circumplex 1 Circumplex 2 Circumplex 3 Circumplex 4 General Circumplex

30 60 Combined sample

90

120

150

180

0.157 0.124 0.186 0.197 0.166 New sample

0.042 0.062 0.074 0.111 0.072

0.020 -0.015 0.009 0.020 0.008

-0.017 -0.007 -0.047 -0.083 -0.038

-0.120 -0.101 -0.145 -0.169 -0.134

-0.244 -0.236 -0.237 -0.303 -0.255

0.166 0.108 0.188 0.187 0.162

0.064 0.061 0.086 0.107 0.080

0.018 -0.027 0.009 0.022 0.006

-0.028 -0.003 -0.054 -0.071 -0.039

-0.124 -0.091 -0.156 -0.162 -0.133

-0.252 -0.211 -0.240 -0.293 -0.249

Note. Table contains mean inter-correlations among items exemplifying equidistant poles of subscales. The plane of each of Four Separate Circumplexes (see also Figures 1.7 and 1.12) is partitioned into 12 segments of 30 degree each (this is a theoretically predicted distance between adjacent subscales’ poles). Each segment is represented by a response to a single item (all included items are listed in Figures 1.8-1.11). General Circumplex was obtained by averaging over Four Separate Circumplexes. It is expected that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales (tetrads). Printed in boldface: correlations of 0.15 and above in magnitude, boldface italics: correlations between 0.05 and 0.15 in magnitude, italics: correlations between -0.10 and -0.20 in magnitude, and underlined italics: correlations below -0.20 in magnitude.

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Arcady A. Putilov

Results of Mapping Narrow, Broad, and Broadest Traits in Three-Dimensional Space The applied methods of statistical analyses provided several keys for assigning the six factorial dimensions to the six pairs of cube’s edges. They include the patterns of inter-correlations, loadings, and coordinates of scales, subscales and items. The obtained results were used to draw the spherical cube shown in Figures 1.1, 1.2, 1.6, 1.21 and 1.21. This assignment differs somewhat from that based on the earlier obtained preliminary results (Putilov and Putilov, 2005, 2006; Putilov, 2007). Namely, the locations of two of six scales were changed to better agree with the results of the inter-correlations, loadings, and coordinates. Table 1.6 contains a classification of the six factors assuming that they are the combinations of the three polarized spatial (underlying) dimensions (Figures 1.1 and 1.6). Moreover, Table 1.6 corroborates the results on the coordinates of the 72 items in that the coordinates of the six scales of SWPAQ might be mapped on or near the surface of the three-axis circumplex rather than inside it. Generally, none of the questionnaire items, subscales and scales can be located near the origin of the sphere. The vertical dimension of the three-dimensional structure can be interpreted as Wakeability or Arousing (i.e., easy/hard, A/a), the width dimension can be interpreted as Sleepability or Bedtiming (i.e., easy/hard, B/b), and the depth dimension can be interpreted as Lateness or Clock delaying (i.e., easy/hard, C/c). To summarize, the results confirm the prediction of the model concerning the possibility of visualizing the six broad traits of sleep-wake adaptability as the six pairs of edges of a spherical cube.

Results of Mapping Narrow, Broad, and Broadest Traits in TwoDimensional Space The results of multidimensional scaling indicate that the two-dimensional solution seems to be also acceptable. At least, it accounts for more than 95%


57

of the data dispersion. Figure 1.2 provides an explanation of the twodimensional representation of the SWPAQ suggesting that it is simply a transparent version of the three-dimensional spherical cube structure. An attractive feature of two-dimensional representation of sleep-wake adaptability is the clear visibility of the location of each item in twodimensional space. Figures 1.3, 1.4, and 1.5 show the empirical results on the two-dimensional structure of the SWPAQ’s items, subscales and scales, respectively. In all these Figures, the two-dimensional representation differentiates Wakeability and Sleepability dimensions (A and B, respectively), whereas the third dimension, Lateness (C), is missed. Table 1.8. Inter-correlations among the SWPAQ scales organized in 4 circumplexes Distance, degree 60 120 Data source Combined sample Four Separate Circumplexes Circumplex 1 -0.038 0.109 Circumplex 2 -0.119 0.192 Circumplex 3 -0.076 0.159 Circumplex 4 -0.199 0.269 -0.108 General Circumplex 0.182

180

60 120 New sample

-0.578 -0.571 -0.542 -0.624 -0.579

0.114 0.182 0.160 0.262 0.180

-0.044 -0.122 -0.075 -0.194 -0.109

180

-0.578 -0.565 -0.538 -0.615 -0.574

Note. Table contains mean inter-correlations among six-item groups exemplifying equidistant poles of scales. The plane of each of Four Separate Circumplexes (see also Figure 1.12) is partitioned into 6 segments of 60 degree each (this is a theoretically predicted distance between adjacent scales’ poles). Each segment is represented by a sum of responses to 6 items characterizing one of two poles of a 12-item scale. General Circumplex was obtained by averaging over Four Separate Circumplexes. Printed in boldface: correlations of 0.15 and above in magnitude, boldface italics: correlations between 0.05 and 0.15 in magnitude, italics: correlations between -0.10 and -0.20 in magnitude, and underlined italics: correlations below -0.20 in magnitude.

In fact, these results indicate that only two dimensions are required to adequately represent the circular relations among the scales of the SWPAQ (Figure 1.5). However, the results of scaling of subscales and items suggest that the third dimension might be necessary to produce a fully circular representation. At least, Figures 1.3 and 1.4 reveal some items and subscales’ poles that are shifted to some extent toward the center of the two-dimensional circumplex.

Table 1.9. Inter-correlations among the selected SWPAQ subscales organized in 3 circumplexes Distance, degree Data source Three Separate Circumplexes Circumplex A&B Circumplex A&C Circumplex B&C General Circumplex

45 90 135 Combined sample 0.172 0.167 0.257 0.199

0.025 0.014 0.019 0.019

-0.126 -0.143 -0.148 -0.139

180

45 90 New sample

-0.408 -0.337 -0.400 -0.382

0.176 0.185 0.172 0.177

0.030 0.008 0.014 0.017

135

180

-0.118 -0.142 -0.137 -0.132

-0.405 -0.333 -0.383 -0.373

Note. Table contains mean inter-correlations among two-item groups exemplifying equidistant poles of core (pure) traits of 3 spatial dimensions and 6 factorial dimensions. The plane of each of Three Separate Circumplexes (see also Figure 1.13) is partitioned into 8 segments of 45 degree each (this is a theoretically predicted distance between adjacent subscales’ poles). Each segment is represented by a sum of responses to two items (listed in Figure 1.14). General Circumplex was obtained by averaging over Three Separate Circumplexes. It is expected that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales (tetrads). Printed in boldface: correlations of 0.15 and above in magnitude, italics: correlations between -0.10 and -0.20 in magnitude, and underlined italics: correlations below -0.20 in magnitude.

Table 1.10. Inter-correlations among the selected SWPAQ subscales organized in 6 circumplexes Distance, degree Data source Six Separate Circumplexes Circumplex A&S Circumplex A&E Circumplex B&M Circumplex B&V Circumplex C&W Circumplex C&F General Circumplex

45 90 135 Combined sample 0.168 0.208 0.175 0.142 0.259 0.181 0.189

0.024 0.024 0.027 0.009 0.024 0.005 0.019

-0.128 -0.163 -0.137 -0.116 -0.172 -0.137 -0.142

180

45 90 New sample

-0.400 -0.437 -0.411 -0.402 -0.379 -0.442 -0.412

0.151 0.208 0.159 0.126 0.251 0.182 0.179

0.028 0.025 0.020 -0.002 0.029 0.000 0.017

135

180

-0.123 -0.162 -0.136 -0.110 -0.167 -0.144 -0.140

-0.393 -0.434 -0.405 -0.404 -0.369 -0.445 -0.408

Note. Table contains mean inter-correlations among two-item groups exemplifying equidistant poles of pure traits of three spatial dimensions and of core (pure) and mixed (blended) traits of six factorial dimensions. The plane of each of Six Separate Circumplexes (see also Figure 1.15) is partitioned into 8 segments of 45 degree each (this is a theoretically predicted distance between adjacent subscales’ poles). Each segment is represented by a sum of responses to two items (listed in Figure 1.16). General Circumplex was obtained by averaging over Six Separate Circumplexes. It is expected that random errors in selection of items for the questionnaire are likely to average out when mean correlations are calculated for equidistant subscales (tetrads). Printed in boldface: correlations of 0.15 and above in magnitude, boldface italics: correlations between 0.05 and 0.15 in magnitude, italics: correlations between -0.10 and -0.20 in magnitude, and underlined italics: correlations below -0.20 in magnitude.

60

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To summarize, the spherical cube model facilitates understanding of the relationship between the two- and three-dimensional representations of sleepwake adaptability. The results of multidimensional scaling indicate that the two-dimensional solution is somewhat worse than the three-dimensional solution but, nevertheless, it is of special interest due to its practical utility. It allows a clear visualization of the degree to which the SWPAQ scales are arranged in a circular fashion around the two of three largest dimensions, Wakeability and Sleepability.

Results of Applying the Circumplex Criteria The results of applying the circumplex criteria indicate that, irrespective of the direction of the SWPAQ’s slicing (producing the sets of either 4 or 3 or 6 circumplexes), each slice represents a two-dimensional circumplex structure rather than a radix structure. In accordance with the circumplex expectations, the SWPAQ’s subscales (tetrads) forming each circle (Figures 1.7, 1.13 and 1.15) showed a gradual change of meaning (Figures 1.8-1.11, 1.14 and 1.16) and gradual reduction of correlation coefficients with increase of the inter-subscale distance (Tables 1.7, 1.9 and 1.10). The results of organizing the SWPAQ’s scales in four circumplexes suggest that their locations, in general, confirm the prediction of the model. For instance, the pattern of correlations among the items included in each of four circumplexes was as expected (Table 1.8). The inspection of the locations of subscales and scales within the two-dimensional circumplexical representations of the SWPAQ (Figures 1.8-1.11, 1.14 and 1.16) also showed that the vast majority of them might be mapped near the circumference of the circle, and none can be mapped near the center of the circle. This result indicates that the empirically revealed locations of the scales and subscales fit a three-dimensional circumplex rather than a three-dimensional radix configuration. In sum, the tri-, tetra-, and hexa-circumplex criteria were introduced to evaluate the circumplexity and three-dimensionality of the SWPAQ structure. These criteria assume the possibility of quantitative confirmation of the conceptually derived relations between six broad traits of sleep-wake adaptability.Testing the predictions of circumplexity and three-dimensionality with these criteria demonstrated that, at the levels of subscales (tetrads) and scales, the SWPAQ has the desirable circumplex properties.

Results of Classification of Trait Variables in Geometrical Terms of the Spherical Cube Model Figure 1.1 illustrates the geometry of the SWPAQ. Each of the three factorial dimensions of the three-factor solution includes a pair of factorial dimensions of the six-factor solution. They are assessed by summing scores on Lateness scales, EM, Wakebility scales, WV, and Sleepability scales, FS (Table 1.3). The mapping of “superscales” (3), scales (6) and subscales (18) is mostly based on the results of three-dimensional scaling (Tables 1.2, 1.111.13).Top of the Figure 1.1 shows the six scales of the SWPAQ labeled along the edges of the cube. Bottom of the Figure shows 13 of 18 subscales (tetrads) that are linked to six pairs of edges of the cube, e2=BC/bc, m2=aC/Ac, w2=AB/ab, v2=AC/ac, f2=aB/Ab, and s2=Bc/bC, three pairs of faces of the cube, v3=A/a, s3=B/b, and e3=C/c, and four pairs of vertices of the cube, w1=ABC, e1=AbC, m1=abC, and f1=aBC. Figures 1.17-1.20 clarify the prediction that, at least, thirteen subscales (tetrads) of the SWPAQ can be organized in 26 configurations around three pairs of faces of the cube, v3=A/a, s3=B/b, and e3=C/c (Figure 1.18), four pairs of the cube’s vertices, w1=ABC, e1=AbC, m1=abC, and f1=aBC (Figure 1.19), and six pairs of the cube’s edges, e2=BC/bc, m2=aC/Ac, w2=AB/ab, v2=AC/ac, f2=aB/Ab, and s2=Bc/bC (Figure 1.20). Within each such configuration, the gradual change of subscale meaning must be observed. However, an empirically developed list of the 72 SWPAQ items is not necessary to provide the optimal measurement of the structure of sleep-wake adaptability predicted by the spherical cube model (Figures 1.1 and 1.2). Both combined and new samples appear to fit a three-dimensional spherical structure, although none of the analyzed samples gives a perfect fit. Table 1.11 contains the estimations of mean coordinates of items grouped in 18 subscales (tetrads) and 6 scales. Tables 1.12 and 1.13 also include the coordinates of the subscales and scales, respectively, obtained by three-dimensional scaling of the scores calculated for the positive and negative halves of subscales and scales. The results point to those elements of the SWPAQ structure that require further correction in order to increase the SWPAQ structure’s degree of fit with the ideal three-dimensional circular model. Comparing the actual structure (i.e., Figures 1.3-1.5 and Tables 1.11-1.13) with the idealized two- and three-dimensional structures leads to two suggestions for improving the SWPAQ. On one hand, some of the 18 subscales (tetrads) of the SWPAQ can be replaced by new ones since not all of them might be easily separated from one another in two- or three-dimensional

space (i.e., Tables 1.11-1.13). However, on the other hand many new subscales might be required to achieve an even distribution of the SWPAQ subscales over the surface of the spherical cube (i.e., Figures 1.21 and 1.22). As shown in Figures 1.6 and 1.7, the spherical cube model predicts the possibility of constructing at least 7 additional subscales that can replace 4 old subscales that do not differ very much spatially from other subscales of the same scale. However, as shown in Figures 1.21 and 1.22, there is also the possibility of constructing 19 new subscales to achieve an even distribution of subscales over the surface of the spherical cube. In particular a better fit can be obtained by slightly reorganizing the SWPAQ structure. As indicated by the coordinates reported in Tables 1.11 and 1.12, the subscales of the scales M (m3), W (w3), F (f3), and S (s1) are located rather close to another subscale shown in Figures 1.1 and 1.6 (m2, w2, f1, and s2, respectively). By contrast, there are several empty areas of the three-dimensional structure. These are seven areas marked as “?” in Figures 1.6 and 1.7. They might be filled with seven new subscales of the scales E (1), M (1), W (1), V (1), F (1), and S (2). Because all these 7 new subscales are the borderline (mixed) subscales, their possible content can be predicted by inspecting the content of the surrounding old items from other scales. Their inclusion might strengthen the links between the scales of the SWPAQ which will lead to increased circumplexity of the empirical structure. For instance, the following suggestions can be made about the possible relationships of the seven new subscales (Figures 1.1, 1.6, and 1.7). A new wsubscale (+w?) can be related to the scales E(+e1) and V(+v?). A new vsubscale (+v?) can be a neighbor of the scales E(+e1) and W(+w?). A new esubscale (+e?) can be located near the border with M(+m1) and F(+f?) scales. A new f-subscale (+f?) can be adjacent to the scales E(+e?) and M(+m1). The negative pole of a new m-subscale (-m?) can be adjacent to the opposite poles of the scales W(+w1) and S(+s?). One of two new S-subscales (+s?) can be located on the border with two scales, W(+w1) and M(-m?), while another Ssubscale (+s?) can be positioned on the border with the scales V(-v1) and F(+f1). The differences between ideal and real structures become evident when the predicted and observed coordinates of the scales or factors are compared with the help of nomenclature offered by the spherical cube model (Figures 1.21 and 1.22). This comparison indicates that a way to radically revise the SWPAQ is to considerably enlarge it. Such revision will lead to departure

from the factor-analytic structure toward a fully circumplexical three-axis structure. The reliability estimates reported in Table 1.14 indicate that all scales and “superscales” of the SWPAQ demonstrate an acceptable level of internal consistency. Hence, this level might not alter dramatically until the enlargement of the questionnaire, taking into account the exclusion of 4 old tetrads (m3, w3, f3, and s1) and the adding of 7 new subscales leading to 84 items (72+7*4-4*4). Furthermore, a marked increase must be expected after enlargement of the number of subscales from 18 to 37. In general, the results of the questionnaire study indicate that the SWPAQ fits a theoretical circular model, but not ideally. Some discrepancies were revealed between observed and ideal locations of the subscales (tetrads) and scales. The spherical cube structure formed by the subscales (tetrads) of the SWPAQ still seems to be gappy. Some areas of the empirically derived structure are worse filled with items compared to other areas. Further revision of this tool could focus on creating new subscales that fill the largest gaps in the empirical structure in order to improve its circumplexical features. Specifically, as it was noted on the basis of comparison of theoretical and empirical structures, one can predict that 7 new subscales can be added to the recent SWPAQ version after excluding 4 subscales whose coordinates appear to be too close to the coordinates of other subscales. However, considerable further enlargement is required to depart completely from the factor-analytic past of the SWPAQ, and to achieve fully circumplexical structure.

General Conclusion Concerning Results To conclude, the spherical cube model postulates that the six broad traits of sleep-wake adaptability derived by factor analysis of questionnaire data cannot be associated with the three spatial (underlying) dimensions of the structure of this adaptability. The multidimensional scaling analysis was applied to uncover three questionnaire-based dimensions with one-to-one relationships to the three dimensions predicted by the model. The results supported the assumption that the three dimensions yielded by the three-dimensional scaling are truly orthogonal dimensions (unlike the three or six dimensions revealed by factor analysis) and that they might be aligned with the three axes of the spherical cube representation of sleep-wake adaptability.

Table 1.11. Coordinates of items grouped in eighteen subscales of the SWPAQ High pole Pole of Spatial Dimension Subscale A B 6 pairs of edges (ribs) of the spherical cube (factorial dimensions) (e2 Evening Lateness) BC/bc 0.23 1.18 (m2 Morning Lateness) aC/Ac -1.26 0.33 (m3 Morning Lateness) aC/Ac -1.30 0.19 (w2 Anytime Wakeability) AB/ab 0.93 1.28 (w3 Anytime Wakeability) AB/ab 1.51 0.69 (v2 Daytime Wakeability) AC/ac 1.39 -0.31 (f2 Anytime Sleepability) aB/Ab -0.86 1.17 (s2 Nighttime Sleepability) Bc/bC -0.01 0.90 (s1 Nighttime Sleepability) Bc/bC -0.28 1.66 3 pairs of spherical squares (faces) of spherical cube (v1 Daytime Wakeability) A/a 1.44 -0.22 (v3 Daytime Wakeability) A/a 1.56 -0.06 (s3 Nighttime Sleepability) B/b 0.05 1.83 (e3 Evening Lateness) C/c -0.23 0.53 4 pairs of vertices (corners) of spherical cube (f1 Anytime Sleepability) aBc/AbC -0.49 1.40 (f3 Anytime Sleepability) aBc/AbC -0.21 1.41 (m1 Morning Lateness) aBC/Abc -1.03 0.63 (w1 Anytime Wakeability) ABc/abC 1.44 0.66 (e1 Evening Lateness) ABC/abc 0.43 0.58

Low pole b c

A-a

Polar Distance B-b C-c

C

a

1.52 1.01 0.95 -0.04 0.06 0.89 -0.66 -1.72 -0.56

-0.63 1.14 1.37 -0.88 -1.30 -1.08 0.98 -0.45 -0.02

-0.81 -0.41 -0.04 -1.34 -0.64 0.39 -1.23 -1.37 -1.33

-1.61 -1.34 -1.25 -0.48 -0.20 -0.12 0.10 1.03 0.86

0.86 -2.40 -2.66 1.81 2.81 2.47 -1.84 0.43 -0.25

1.99 0.74 0.23 2.62 1.33 -0.70 2.40 2.27 3.00

3.12 2.35 2.20 0.44 0.26 1.01 -0.76 -2.74 -1.42

0.27 -0.12 0.10 2.08

-1.25 -1.35 0.44 -0.07

-0.24 0.08 -1.60 -0.76

-0.28 -0.14 -0.19 -1.64

2.68 2.91 -0.38 -0.16

0.02 -0.15 3.44 1.29

0.55 0.02 0.30 3.72

-0.87 -0.57 0.80 0.22 2.00

-0.09 1.17 1.00 -1.22 -1.04

-1.32 -0.82 -0.93 -0.84 -0.66

0.89 0.96 -0.99 0.13 -1.08

-0.41 -1.38 -2.03 2.66 1.47

2.73 2.23 1.56 1.50 1.24

-1.76 -1.53 1.79 0.09 3.08

Notes. Three-dimensional coordinates of each item were standardized (SD=1 for each of three sets of 72 coordinates) and averaged over items of each subscale’s pole (i.e., the items ## 13 and 49 exemplify high pole of core trait +e2 of broad trait E or BC/bc). Polar Distance: difference between opposite poles is calculated by subtracting mean values of coordinates of two items characterizing low pole from those characterizing high pole.

Table 1.12. Coordinates of scores on eighteen subscales of the SWPAQ High pole Low pole Pole of Spatial Dimension Subscale A B C a b Coordinates of six subscales representing six pairs of cube’s edges (factorial dimensions) e2 (Evening Lateness) BC/bc 0.07 0.51 0.50 -0.27 -0.35 m2 (Morning Lateness) aC/Ac -0.59 0.08 0.34 0.58 -0.12 w2 (Anytime Wakeability) AB/ab 0.38 0.56 -0.05 -0.37 -0.60 v2 (Daytime Wakeability) AC/ac 0.64 -0.25 0.25 -0.57 0.20 f2 (Anytime Sleepability) aB/Ab -0.47 0.42 -0.23 0.51 -0.50 s2 (Nighttime Sleepability) Bc/bC -0.03 0.36 -0.58 -0.17 -0.55 Coordinates of other twelve subscales e1 (Evening Lateness) ABC/abc 0.21 0.28 0.66 -0.47 -0.33 e3 (Evening Lateness) C/c -0.11 0.24 0.69 -0.01 -0.33 m1 (Morning Lateness) aBC/Abc -0.53 0.23 0.27 0.53 -0.32 m3 (Morning Lateness) aC/Ac -0.62 0.01 0.34 0.65 0.02 w1 (Anytime Wakeability) ABc/abC 0.65 0.33 0.05 -0.55 -0.39 w3 (Anytime Wakeability) AB/ab 0.67 0.34 0.00 -0.58 -0.32 v1 (Daytime Wakeability) A/a 0.74 0.16 0.09 -0.61 -0.18 v3 (Daytime Wakeability) A/a 0.74 0.02 -0.06 -0.62 -0.04 f1 (Anytime Sleepability) aBc/AbC -0.33 0.54 -0.31 0.60 -0.27 f3 (Anytime Sleepability) aBc/AbC -0.17 0.54 -0.20 0.02 -0.61 s1 (Nighttime Sleepability) Bc/bC -0.21 0.69 -0.17 0.06 -0.50 s3 (Nighttime Sleepability) B/b -0.05 0.76 0.03 0.27 -0.63

Polar Distance B-b C-c

c

A-a

-0.52 -0.44 -0.10 -0.16 0.08 0.34

0.34 -1.17 0.75 1.21 -0.97 0.14

0.86 0.20 1.15 -0.44 0.93 0.91

1.01 0.78 0.04 0.41 -0.32 -0.92

-0.32 -0.54 -0.31 -0.42 0.07 -0.05 -0.02 -0.06 0.33 0.31 0.25 -0.07

0.68 -0.10 -1.06 -1.26 1.20 1.25 1.35 1.35 -0.93 -0.19 -0.27 -0.32

0.61 0.57 0.55 -0.02 0.71 0.66 0.34 0.06 0.80 1.15 1.19 1.39

0.98 1.22 0.58 0.75 -0.02 0.06 0.11 0.00 -0.63 -0.52 -0.41 0.10

Notes. Scores were calculated separately for the halves of the subscales characterizing their high and low poles. Multidimensional scaling of the scores on 18 subscales (tetrads) yielded three dimensions A, B and C and gave the coordinates of each pole’s score in three-dimensional space. Polar Distance: Difference between coordinates of scores characterizing high and low pole of subscale.

Table 1.13. Coordinates of six scales of the SWPAQ Pole of Spatial Dimension Scale E (Evening Lateness) BC/bc M (Morning Lateness) aC/Ac W (Anytime Wakeability) AB/ab V (Daytime Wakeability) AC/ac F (Anytime Sleepability) aB/Ab S (Nighttime Sleepability) Bc/bC

High pole A 0.03 -0.58 0.54 0.72 -0.41 -0.14

B 0.26 0.04 0.45 0.05 0.49 0.65

C 0.64 0.33 0.03 0.03 -0.11 -0.25

Low pole a -0.27 0.60 -0.48 -0.59 0.49 0.09

b -0.31 -0.06 -0.44 -0.06 -0.43 -0.63

c -0.49 -0.40 -0.03 -0.07 0.18 0.14

Polar Distance A-a B-b 0.30 0.58 -1.19 0.10 1.03 0.89 1.31 0.11 -0.90 0.92 -0.23 1.29

C-c 1.13 0.72 0.06 0.10 -0.30 -0.40

Notes. Scores were calculated separately for the halves of the scales characterizing their high and low poles. Multidimensional scaling of the scores on six scales yielded three dimensions A, B and C and gave the coordinates of each pole’s score in three-dimensional space. Polar Distance: Difference between coordinates of scores characterizing high and low pole of scale.

Table 1.14. Reliability of the scales and pairs of scales (“superscales”) of the SWPAQ Scale and location Number of items Data source Combined sample New sample Data source Combined sample New sample

E BC/bc M aC/Ac W AB/ab 12 12 12 Alpha 0.73 0.80 0.74 0.78 0.80 0.81 Correlation between halves A and B 0.65 0.67 0.70 0.64 0.66 0.70

V AC/ac 12

F aB/Ab 12

S Bc/bC 12

EM C/c 24

WV A/a 24

FS B/b 24

0.73 0.73

0.80 0.79

0.77 0.76

0.80 0.80

0.84 0.83

0.81 0.81

0.59 0.61

0.66 0.66

0.66 0.63

0.71 0.69

0.74 0.73

0.72 0.71

Note. Calculations were performed for the whole (new and older) sample (Combined sample, n=1068) and separately for New sample (n=718). Spherical cube’s edges might be represented by six SWPAQ scales, and cube’s faces might be represented by three pairs of scales (“superscales”, a sum of two scales’ scores).


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Empirical evidence favored the results of three- and two-dimensional scaling but not four-dimensional scaling. The circumplexity and threedimensionality of the empirical structure of sleep-wake adaptability was confirmed by applying tetra-, tri-, and hexa-circumplex criteria. It was also demonstrated that the six questionnaire scales or the six corresponding factorial dimensions can be visualized as the edges of a spherical cube. The computations based on several methods of statistical analysis (i.e., intercorrelations, factor loadings, factors’ and scales’ coordinates) revealed that the locations of some of the SWPAQ constructs are not optimal. Such results can direct further revision of the questionnaire content in order to achieve a better fit to the theoretically predicted structure.

DISCUSSION The research presented in this chapter advances earlier studies aimed at exploring the structure of adaptive ability of the sleep-wake cycle. The spherical cube model allows the visualization of sleep-wake adaptability as a three-dimensional structure. It explains how the six broad adaptive traits corresponding to the six largest factors (or to the six scales of a multidimensional questionnaire) can emerge from only three spatial (underlying) dimensions. The spherical cube representation also explains why factor analysis can yield the traits of different levels of generality (i.e., the broadest, broad and narrow). In particular, it postulates that the cube inscribed in the sphere shows the locations of the six broad traits, and that this cube can provide a taxonomic coordinate system for mapping narrow traits. Any narrow trait can be located on the surface of the sphere (three-axis circumplex) formed by the three spatial dimensions. In other words, any such trait is simply a blending of vertical, depth and width dimensions of the spherical cube. The empirical examinations of the SWPAQ structure using correlation analysis, factor analysis, multidimensional scaling and circumplex criteria provided results resembling the results of our previous studies (Putilov and Putilov, 2006; Putilov, 2007). In general, the collected empirical findings confirm the similarity of the natural and theoretically predicted (spherical cube) structures of sleep-wake adaptability. Since the model postulates that there are only three axes of the studied structure, the recent analysis of questionnaire data was centered on revealing these three spatial dimensions of sleep-wake patterns by statistical methods. Moreover, it was necessary to prove that these dimensions are sufficient to

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model the structure of all subjectively assessed traits despite factor analytic evidence for the existence of at least six rather broad traits corresponding to the six scales of the SWPAQ. It was demonstrated that the structures of sleep-wake adaptability yielded by factor analysis and multidimensional scaling of the responses to the 72 items of the SWPAQ differ in some important respects. However, this difference between the dimensions of the SWPAQ structure revealed by two methods can be explained in terms of the spherical cube model. For instance, the model explains the correlations between factorial dimensions. Since the six largest factorial dimensions can be associated with the six pairs of edges of the cube inscribed in the sphere, the inter-correlations among the loadings on six factors and among the scores on the six scales of the SWPAQ must be expected (Putilov, Putilov, 2006; Putilov, 2006, 2007). As for the three factorial dimensions obtained by varimax rotation of the three largest principal components, they also were found to inter-correlate with one another. These explanations of the results of correlation and factor analyses were confirmed by multidimensional scaling analysis of the SWPAQ items, subscales, and scales. Each of the three factors of the three-factor solution can be interpreted as a rotational variant of one of the three spatial dimensions yielded by three-dimensional scaling. However, the dimensions yielded by three-dimensional scaling appear to be fully orthogonal and they rather than the dimensions yielded by three-factor solution might represent three axes of the spherical cube representation of sleep-wake adaptability. The spherical cube model predicts that all individual trait variables are organized in a three-dimensional circumplex structure. Indeed, it was demonstrated that the results of factor and multidimensional scaling analyses argue very strongly for three-dimensionality and circumplexity of the SWPAQ structure. The results of multidimensional scaling provided a considerably better approximation of the SWPAQ structure using the three-dimensional solution compared to the two-dimensional solution. However, these results do not exclude the possibility of development of a more parsimonious twodimensional representation of the SWPAQ structure. At least, the twodimensional solution accounts for more than 95% of the data dispersion. Theoretically, such representation is easily interpreted as a transparent version of the three-dimensional spherical cube structure. The contents of three and six factors yielded by factor analysis were found to be consistently replicable. For instance, analysis of the new data set produced a six-factor structure that closely resembles the structure obtained in the analysis of previous data sets (Putilov, 2007). In accordance with earlier


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studies (Putilov and Putilov, 2006; Putilov, 2006, 2007; Putilov A. et al., 2007), the six broad traits, Evening Lateness (E), Morning Lateness (M), Anytime Wakeability (W), Daytime Vakeability (V), Anytime Sleepability (F) and Nighttime Sleepability (S), were unmistakably identified among the six largest factors. The content of the factors showed correspondence to the same name scales, E, M, W, V, F, and S. However, although an almost complete match was demonstrated for four scales, there were two exceptions, scales W and V (Anytime and Daytime Wakeabilities). Sampling biases could be responsible for some minor differences noted in the content of these factors in the new data set. It has to be noted, however, that these contents still agrees with the expectations, because they point on the corresponding broad trait of sleep-wake adaptability. For instance, the majority of the items of these two scales (V and W) belong to the same pair of Wakebility scales of the “superscale” WV. The results supported the conclusion made previously (Putilov, 2007) that the four circles each defined by three scales must show circumplex characteristics. The empirical evidence was obtained by applying the tetracircumplex criterion to the results of correlation analysis of the relationships between responses provided by the new sample of adult and adolescents. Also it was demonstrated that, in corroboration of the spherical cube model, the SWPAQ not only comprises four circumplex structures proposed by the tetracircumplex criterion, but also comprises three and six circumplexes proposed by two new, tri- and hexa-circumplex, criteria. In the present study, the factor analysis was enriched for the first time by multidimensional scaling. It helped not only to prove the three-dimensionality and circumplexity of the SWPAQ, but also to map directly each of its 72 items, 18 subscales (tetrads), and 6 scales on the surface of a spherical cube. In accordance with the assumptions of the model, the items’ coordinates calculated on the basis of the three-dimensional solution did not provide evidence for a three-dimensional radix structure of sleep-wake adaptability. The results of three-dimensional scaling might be interpreted as indicating that all of the 72 SWPAQ items reliably represent either a high or low pole of an individual trait of sleep-wake adaptability, and, hence, the three-dimensional circumplex is the most likely candidate for the structure of sleep-wake adaptability. The present results allowed suggestions about how to correct the earlier proposed empirical structure of the SWPAQ. They highlighted the areas of the spherical cube which are too densely populated by the items and subscales, and the gaps that can be filled with new items and subscales. This is not a

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surprising result considering that the empirical SWPAQ structure was developed by means of traditional factor analysis. The results are interpreted as indicating that a better correspondence between the theoretical and empirical structures can be achieved by the SWPAQ’s revision at the level of subscales (tetrads). For four scales the content of one of three subscales was found to resemble the content of another subscale of this scale. Consequently, further possible revision of the SWPAQ scales could be aimed on replacing these subscales by 7 additional subscales of the same scales with somewhat distant meaning. The possible meaning of each of the predicted subscales can be prompted by analysis of the meaning of the adjacent subscales of two other scales. It is expected that further development of the SWPAQ by revision of a smaller portion of its subscales cannot challenge the factor-analytical background of this instrument. However, neither the existing 72-item version nor the revised 84-item version can cover the whole surface of the structure predicted by the spherical cube model. Therefore, a more radical enlargement is recommended to achieve the closest resemblance between the empirical structure of the SWPAQ and the theoretically predicted spherical cube representation of sleep-wake adaptability. The number of subscales might be markedly increased in order to obtain a fully circumplexical structure with 37 subscales (tetrads) that are evenly distributed over the surface of the spherical cube representation of sleep-wake adaptability. Originally, the model also addressed the issues of causality and the chronophyiological mechanisms underlying differences between people on sleep-wake behavior. The assumption was made (Putilov and Putilov, 2005, 2006; Putilov, 2006, 2007) that the factors revealed by factor analysis refer to overt (observable) rather than to underlying (causal) characteristics of sleepwake adaptability. The results of the present study lend support to the suggestion that even questionnaire data alone, being appropriately analyzed, might yield the three dimensions underlying the structure of this adaptability. In particular, application of multidimensional scaling provided the demonstration that individuals can vary along only three dimensions of sleepwake adaptability, and that this method can reveal pure questionnaire measures of these three causal dimensions. Further experimental research is necessary to test the assumption that the model can highlight the nature of the relationships between observed individual traits of sleep-wake behavior and their chronophysiological background.


71

SOME GENERAL CONCLUSIONS    















When six squares clad a sphere with 3 squares around each vertex and 120 degree internal angles, this is called a spherical cube. Such a shape was proposed as a structural model of sleep-wake adaptability. Empirical proof of the spherical cube representation of sleep-wake adaptability was obtained by analyzing the structure of the SWPAQ. The analysis of inter-correlations, factors analysis, multidimensional scaling, and circumplex criteria were applied in the present analysis of a new questionnaire data set. In particular the supposition that the subscales (tetrads) of the SWPAQ exhibit a circular ordering was supported by demonstrating that the SWPAQ structure meets the tetra-, tri-, and hexa-circumplex criteria. A proposed configuration of the scales on the edges of a spherical cube provided a good fit between the observed and hypothetical patterns. The most insightful finding is the demonstration that the truly orthogonal dimensions were yielded by three-dimensional scaling, and that the traits constituting sleep-wake adaptability can be directly mapped onto the surface of a spherical cube. Such empirical demonstration provides strong support for the claim that the spherical cube model yields the natural structure of sleepwake adaptability. The research also detected some measurable difference between the SWPAQ and the theoretically predicted structure of sleep-wake adaptability. The theory-based methodological approach can be employed for improving circumplexical features of the SWPAQ by means of revising this instrument at the subscale level. In general, the spherical cube representation of sleep-wake adaptability paves the way for moving from a descriptive to an explanatory model of multi-dimensional variation in sleep-wake behavior.

Chapter 3

VALIDATION OF THE SLEEP-WAKE ADAPTABILITY SCALES PREDICTED BY THE SPHERICAL CUBE MODEL ABSTRACT In the analysis presented in this chapter the external validity of the SWPAQ’s scales and “superscales” was established by using selfreports on sleep history and day- and nighttime alertness in 130 sleepdeprived individuals. Moreover, individual differences along these scales and “superscales” were interpreted in terms of a general model of sleep-wake regulation.

INTRODUCTION People experience the deterioration of alertness and performance throughout one night of sleep deprivation. They considerably vary on their capability to adapt to the extension of waking in night hours. Van Dongen and Dinges (2001) proposed that, at least, three independent parameters can be estimated for the model-based prediction of reduced alertness and performance in the conditions of sleep loss. These are: (1) the timing and/or rate of circadian adjustment (i.e., circadian phase); (2) the amount of sleep needed per day (i.e., sleep need); and (3) the rate of impairment per hour of sleep loss (i.e., vulnerability to sleep deprivation). There is solid evidence for substantial and clear distinguishable inter-individual differences in each of these three

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parameters. This inter-individual variation is characterized by withinindividual stability (i.e., replicability), and robustness (i.e., insensitivity to experimental manipulations). Such features suggest that the inter-individual differences represent systematic trait-like variability (see Van Dongen and Dinges, 2001, Van Dongen et al., 2004, for details). Understanding the basis of this variability may yield new insight into sleep-wake regulation and sleepwake pathology (Van Dongen et al., 2005). The general model for explaining daytime and nighttime courses of alertness/drowsiness was proposed by Borbély (1982). It was named the twoprocess model because it postulates the homeostatic and circadian processes as two major determinates of the duration and timing of states of sleep and wakefulness (Borbély , 1982; Borbély and Achermann, 2005). The original version of this model suggests superposition (Borbély, 1982; Daan et al., 1984) and the modified version predicts the interaction (Putilov, 1995ab) between these two processes. The first – homeostatic – process is represented by the drive for sleep. It accumulates during wakefulness and dissipates during sleep episode (Borbély, 1982; Daan et al., 1984). The modeling results and some empirical evidence suggest that the second – circadian – process might modulate the parameters of the homeostatic process with an approximately 24hour period (Putilov, 1995ab). The two-process model of sleep-wake regulation predicts that two processes simultaneously influence the level of alertness/drowsiness in order to promote relatively stable wakefulness during an approximately 16-hour time interval from morning to evening. Thereafter they mutually promote a rapid increase in the drive for sleep lasting until the end of the night and causing a progressive decline in the level of alertness. The next morning, irrespective of whether the subject slept, the circadian process stimulates a spontaneous increase in the level of alertness (i.e., this effect remains notable after sleep deprivation despite further increase of the homeostatic drive for sleep). The present experimental research sought to identify objective markers of alertness level during sleep deprivation. The results on associations between the objective and subjective measures have been published previously (Putilov D. et al., 2007; Verevkin et al., 2008; Putilov A. et al., 2009ab, 2010). They provided experimental evidence for psychophysiological underpinning of individual state and trait differences in sleep-wake adaptability. The recent analysis focuses on providing further evidence for the validity of the SWPAQ’s scales and “superscales” by using the subjective state self-ratings

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and self-reported sleep patterns, and by applying model-based simulations for explaining individual differences along these scales and “superscales”.

METHOD Theoretical Framework The present validation of the SWPAQ is based on relating its scales and “superscales” to the self-reported sleep history and time course of alertness and energy throughout more than 24 hours of sustained wakefulness. The analysis of self-ratings on the Karolinska Sleepiness Scale (KSS) and selfreported bed and rising times are widely accepted as the external criteria for assessing the validity of the morningness-eveningness scales (i.e., Kerkhof, 1985; Diaz-Morales and Sanchez-Lopez, 2005; Randler, 2009). In addition to such pure empirical approaches, the general model of sleepwake regulation (Daan et al., 1984; Putilov, 1995ab) was applied in the present analysis to obtain simulations of the time course of alertness. The goal was to explain individual differences in the scores on the SWPAQ scales and “superscales” in terms of the parameters of the processes underlying the sleepwake cycle.

Subjects and Protocol Data of this experimental study were collected over a three-year period (years 2006, 2007, and 2008). One hundred and thirty healthy subjects were studied as paid volunteers. The mean and median ages of 54 male participants were 27.41 years (SD=10.10) and 23 years, respectively. For 76 female subjects, the mean and median ages were 30.84 years (SD=13.37) and 24 years, respectively. None of the study participants reported any physical or mental health problems or a history of psychiatric or sleep disorders. The experiments were performed in accordance with the ethical standards laid down in the Declaration of Helsinki. The experimental protocol was approved by the Ethics Committee of the Siberian Branch of the Russian Academy of Medical Sciences. Informed written consent was obtained from all participants. In the experimental morning (between 08:00 and 8:30 o’clock), the subjects were admitted to a research unit of the institute and remained there

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until approximately 11:00 the next morning. The experimental procedure starts with brief instructions. Over the next 24 hours the study participants completed 9 EEG recording sessions divided by 3-hour intervals and 5 performance trials. The EEG measures were reported elsewhere (Putilov D. et al., 2007; Verevkin et al., 2008; Putilov A. et al., 2009ab, 2010). Subjects also completed several questionnaires (one is reported in Chapter 5). This task was scheduled at the time intervals between the performance and EEG measurements. When subjects were not participating in the research procedures they were engaging in such activities as reading, writing, playing board and computer games, surfing the Internet, watching TV, listening to music, consuming light snacks and drinks (but not alcohol or caffeinated beverages), etc. The participants were also asked to avoid any medications, vigorous physical activity and exposure to light brighter than 500 lux. The study personnel ensured that participants always remained awake.

Sleep History and State Self-Ratings During the week preceding the experimental morning, the participants were asked to record their individual bed and wake times. Sleep logs of each participant consisted of the morning recalls of sleep history covering a period of up to 6 nights prior to the experimental morning. The reports include ratings on a 5-step Sleep Satisfaction Score (ranged from 1=”not satisfied at all” to 5=”excellent”), on clock times for going to bed and for complete awakening (wake up), on nap episodes and their duration, and on sleep latency (the selfperceived time interval between going to bed and falling asleep). Additionally, total time spent in bed (time in bed) was calculated as the difference between the clock times for going to bed and complete awakening, and mid sleep time (mid sleep) was calculated as the clock times midway between going to bed and complete awakening. Self-reported sleep history was analyzed separately for the day of experiment (Day 0) and the preceding days (Days 1-5), because in the experimental morning the clear difference was produced between late and early chronotypes in the extent of night sleep restriction (see Tables 1.15. 1.16 and 1.19, and Putilov et al., 2009ab, 2010, for more details). After each EEG recording the participants were asked to determine their self-perceived sleepiness/alertness on the 9-point Karolinska Sleepiness Scale (KSS; Åkerstedt and Gillberg, 1990; Gillberg et al., 1994). It is verbally anchored at the following steps: 1=”very alert”, 3=”alert”, 5=”neither alert nor

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sleepy”, 7=”sleepy, but no difficulty staying awake”, and 9=”very sleepy, with great difficulty keeping awake”. The intermediate steps are not anchored verbally. Additional self-ratings included the assessment of mood and energy expressed on the 100-mm Visual Analog Scales (VAS). Mood varies from “very bad” to “very good” and energy varies from “very low” to “very high”. To compare the changes in subjective state scores with the trait scores the mean scores were calculated for the late nighttime period (the time points from 7th to 9th in Table 1.17).

Trait Self-Ratings The 72-item Sleep-Wake-Pattern Assessment Questionnaire (SWPAQ; Putilov, 2007) was administered twice: initially in the week prior to the experiment and soon after subjects arrived in the research unit. The scores on the six scales and three “superscales” of the SWPAQ were averaged within subjects (see two previous chapters for more detail about SWPAQ selfassessments). The sample was divided into three groups representing either different chronotypes or different trototypes (Tables 1.18-1.20). The scores of 0 and 6 were chosen as the borderlines between the three groups (Table 1.18). The subjects were identified as representing extreme types, when they had the negative averaged scores on both scales of the same “superscale” (E≤0 and M≤0 or W≤0 and V≤0), or high positive scores on both scales of the same “superscale” (E≥6 and M≥6 or W≥6 and V≥6). Other combinations of scores were regarded as intermediate (0