Structural Equation Modeling

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Structural Equation Modeling: A Multidisciplinary Journal

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Residual Structures in Latent Growth Curve Modeling Kevin J. Grimma; Keith F. Widamana a University of California, Davis

Online publication date: 08 July 2010

To cite this Article Grimm, Kevin J. and Widaman, Keith F.(2010) 'Residual Structures in Latent Growth Curve Modeling',

Structural Equation Modeling: A Multidisciplinary Journal, 17: 3, 424 — 442 To link to this Article: DOI: 10.1080/10705511.2010.489006 URL: http://dx.doi.org/10.1080/10705511.2010.489006

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Structural Equation Modeling, 17:424–442, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 1070-5511 print/1532-8007 online DOI: 10.1080/10705511.2010.489006

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Residual Structures in Latent Growth Curve Modeling Kevin J. Grimm and Keith F. Widaman University of California, Davis

Several alternatives are available for specifying the residual structure in latent growth curve modeling. Two specifications involve uncorrelated residuals and represent the most commonly used residual structures. The first, building on repeated measures analysis of variance and common specifications in multilevel models, forces residual variances to be invariant across measurement occasions. The second, generally arising from structural equation modeling perspectives, allows residual variances to be freely estimated across occasions. Usually an afterthought, alternate specifications of the residual structure can have sizable effects on variance and covariance estimates for the intercept and slope factors (Ferron, Dailey, & Yi, 2002; Kwok, West, & Green, 2007; Sivo, Fan, & Witta, 2005), the fit of the model, and model convergence. We propose additional residual structures in latent growth modeling arising from ideas regarding growth curve reliability. These structures allow residual variances to change across time, but in constrained ways. We provide three illustrations and highlight the importance of focusing on residual structures in latent growth curve modeling.

Residual structures in latent growth curve modeling are often either overlooked or invoked with little deliberation, even though several options for residual structures have been proposed (see Wolfinger, 1993, 1996). Options for residual structures come in two general forms: independent and autocorrelated. Independent residual structures force residual covariances to zero, whereas autocorrelated residual structures allow for nonzero covariation among residuals. Independent residual structures are either invariant (i.e., homogeneous, or constrained equal) or unstructured (i.e., freely estimated); and autocorrelation structures can have several different structures that include compound symmetry, first-order autoregressive, moving average, and autoregressive moving average, to name a few. Simulation studies using Monte Carlo methods have shown that different specifications of residual structures can have a large impact on variance and covariance estimates (see Ferron, Dailey, & Yi, 2002; Kwok, West, & Green, 2007; Sivo, Fan, & Witta, 2005), especially when an autocorrelated structure is used to generate data and an independent structure is imposed in the growth model. Residual structures can also Correspondence should be addressed to Kevin J. Grimm, University of California Davis, Department of Psychology, One Shields Avenue, Davis, CA 95616, USA. E-mail: [email protected]

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have sizable effects on model fit and convergence. In applied research, the residual structure underlying empirical data is unknown, and the researcher must select a residual structure that is appropriate for the data. Most commonly, an independent residual structure is specified, and residual variances are often forced to be invariant across occasions, rather than unstructured. In this article, we review growth curve modeling in a structural modeling framework, describe two common independent residual structures and their foundations, propose new residual structures, illustrate their use with longitudinal achievement data from the National Longitudinal Survey of Youth and the Early Childhood Longitudinal Study, and highlight the effects that alternative residual structures have on parameter estimates.

LATENT GROWTH CURVE MODELING Latent growth curve modeling (e.g., McArdle & Epstein, 1987; Meredith & Tisak, 1990; Rogosa & Willett, 1985) is a contemporary approach to model systematic within-person change and between-person differences in within-person change for a series of repeated measurements. The latent growth curve model can be specified in the structural equation modeling (SEM) framework using a restricted common factor model (see Meredith & Tisak, 1984, 1990) and in the multilevel modeling framework (see Bryk & Raudenbush, 1987, 1992) as repeated measures are nested within individuals. Following the common factor approach (see Meredith & Tisak, 1990), the latent growth curve model can be written as yi D ƒ˜i C ©i ;

(1)

where yi is a t 1 vector of scores for individual i on variable y measured at t occasions, ƒ is a t r matrix of factor loadings with columns of “basis functions” (Meredith & Tisak, 1990) that represent specific change aspects, ˜i is an r 1 vector of factor scores (e.g., intercept and slope scores) for individual i , and ©i is a t 1 vector of residual scores for individual i . The elements of ƒ describe the structure of changes, can be fixed to test specific hypotheses (e.g., linear growth), can be estimated to allow the data to best determine the change function (e.g., Meredith & Tisak, 1990), or can include multiple change components (e.g., Ram & Grimm, 2007). The factor scores can be written as deviations from the group mean, such as ˜i D ’ C ¨i ;

(2)

where ’ is an r 1 vector of latent factor means and ¨i is an r 1 vector of mean deviations for individual i . The expectations of the population mean vector, , and covariance matrix, †, based on the latent growth curve model are D ƒ’ † D ƒˆƒ0 C ‚

(3)

where ˆ is an r r latent variable covariance matrix and ‚ is a t t matrix of residual variances and covariances.

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RESIDUAL STRUCTURES IN LATENT GROWTH CURVES Applications of growth modeling tend to focus on ƒ, ’, and ˆ as these matrices describe the pattern of changes, the average change function, and between-person differences in the factors representing change and their relations to one another, respectively. Of importance here is the structure of ‚, which has been shown to have substantial effects on latent variable covariances (i.e., ˆ; Ferron et al., 2002; Kwok et al., 2007; Sivo et al., 2005). The first question regarding the structure of ‚ is whether it is specified as diagonal, reflecting the assumption that residuals are mutually uncorrelated, or whether ‚ is allowed to be nondiagonal. If ‚ is diagonal, the most common specification in empirical research, diagonal elements are typically either constrained to be equal (i.e., invariant) or freely estimated (i.e., unstructured). In either case, specifying ‚ to be diagonal forces longitudinal covariances among manifest variables to arise from the growth factors (˜i )—their structure (ƒ), and their variances and covariances (ˆ). Ferron et al. (2002), Sivo et al. (2005), and Kwok et al. (2007) examined effects of specifying ‚ to be diagonal (and invariant) when an autocorrelation structure among residuals had been used to generate data. In all cases, variances and covariances of the growth factors (ˆ) and the residual variances (diagonal elements of ‚) were found to be biased, at times severely biased; however, the means of the growth factors (’; fixed effects) were unbiased. Finally, standard errors of the fixed effects were generally shown to be overestimated. Here, we focus on alternative specifications of ‚, all of which employ diagonal ‚ matrices. As mentioned, diagonal elements of ‚ are usually forced to be invariant across time or unstructured. However, in modeling reaction time data on a complex learning task, Browne and du Toit (1991) constrained elements of ‚ to be diagonal and to follow a two-parameter monotonically decreasing exponential trend because few fluctuations in scores were observed after the task had been mastered. The opposing specifications of diagonal and invariant versus diagonal and unstructured arise from different statistical foundations (repeated-measures analysis of variance [ANOVA], multilevel models, and structural equation models) and different theoretical expectations regarding how growth processes unfold. In repeated-measures ANOVA, homogeneity of variance is often assumed; because the latent growth curve can be conceived as an expansion of repeated-measures ANOVA, forcing residual variances to be equal across time points is reasonable, especially because the same variable is measured repeatedly. Additionally, when growth models are fit in a multilevel modeling framework, the default of a single residual (Level 1) variance is equivalent to forcing diagonal elements of ‚ to be equal (see Ferrer, Hamagami, & McArdle, 2004). Willett and Sayer (1994) considered independent and homoscedastic, or invariant, residual variances across time to “obey stringent classical assumptions” (p. 368). On the other hand, latent growth models are often fit within a structural modeling framework, providing additional flexibility in fitting models to data. Allowing the diagonal elements of ‚ to be freely estimated often results in a better fitting model; because most growth models remain identified when relaxing this homogeneity constraint, requiring homogeneity of variance seems overly restrictive, especially when data lead us to believe such a specification is inappropriate. However, with many time points, scarce data, or complex change models (e.g., multiphase/spline models), allowing diagonal values in ‚ to be freely estimated might lead to model underidentification or convergence problems. Wolfinger (1996) suggested that


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researchers should choose the most parsimonious covariance structure that reasonably fits a given set of data. Theoretical arguments for an equality constraint on diagonal elements of ‚ have also been made. These arguments include the fact that the same variable is measured over time, and “the same variable” should have the same residual variance across time. Additionally, the equality constraint on diagonal values in ‚ forces all variance changes in manifest variables to arise from the growth factors—a very parsimonious choice in modeling. In contrast, arguments against the equality constraint are often based on improved statistical fit. Additionally, if observed variance of the manifest variable increases over time (and this often occurs in empirical data), an equality constraint on the residual variance indicates the observed variable is more reliable over time. On this point, Willett and Sayer (1994) said that researchers often have “no reason to believe a priori that the precision with which an attribute can be measured is identical at all ages” (p. 368), a remark in favor of an unstructured, or, at least, a noninvariant, residual structure.

ALTERNATIVE RESIDUAL STRUCTURES In this article, we offer two alternative specifications of independent residual structures in latent growth curve modeling, each of which is based on considerations related to growth curve reliability or reliability of the manifest variable given the growth curve. Growth curve reliability (¡lgc ; Meredith & Tisak, 1990; Tisak & Tisak, 1996; Willett & Sayer, 1994) is usually based on the first measurement occasion and refers to the ratio of variance attributable to the intercept (i.e., true variance) to total variance (i.e., variance attributable to the intercept plus residual variance). We use the term growth curve reliability, but recognize that it represents a lower bound on reliability of a measure given a specific growth curve and that this value can change depending on the chosen growth model. Growth curve reliability after the initial time of measurement reflects true score variance arising from both the intercept and slope components and this is why growth curve reliability is often based on only the first measurement occasion. As a first alternative residual structure, we propose that ‚ be specified sohthat the reliability i ƒˆƒ0 to be of the manifest variable is constant across time. This would force diag ƒˆƒ 0 C‚ invariant across time. Under this approach, if the variance of the manifest variable increases across time, residual variance should increase proportionally to total variance to maintain constant reliability. This specification would be especially appropriate in situations in which computerized adaptive testing (CAT) is used because stopping criteria in CAT often involve meeting a particular standard error of measurement around a person’s estimated latent trait score. If latent trait score variance increases (or decreases) over time or age and the standard error of measurement is calibrated in a metric consonant with latent trait variance, then residual variance should rise (or fall) proportionally to the rise (or fall) in latent trait variance, yielding constant reliability across occasions. As a second alternative, constraining the reliability of the manifest variable to increase over time might be a reasonable assumption, especially in developmental research contexts. Many constructs are difficult to measure reliably in young children (or older adults), so

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increased reliability when testing older children (or younger adults) would not be unexpected. A psychological test is often assumed, naively, to have a single reliability. However, advances in item response theory (IRT) have shown how the standard error of measurement can be dependent on level of ability or aptitude, an individual’s response pattern, and the match between item difficulty and ability (see Embretson & Reise, 2000). As a result, systematic increases in reliability across the years of childhood and adolescence are a likely outcome in empirical research applications in which the ability level of the sample increases or the consistency of behavior in a domain increases.

CURRENT PROJECT In this project, we compare model fit, parameter estimates, and model convergence with different independent residual structures when fitting simple growth curves to longitudinal achievement data focused on early reading and vocabulary skills. For each outcome, we fit growth models with invariant residual variances, freely estimated residual variances, invariant growth curve reliability, and linearly changing growth curve reliability. We highlight differences in estimated parameters, model fit, and model convergence. We conclude with additional considerations in fitting latent growth curves.

METHOD Illustrative Data Data for this project were derived from the National Longitudinal Survey of Youth–Children and Young Adults (NLSY–CYA) and the Early Childhood Longitudinal Study–Kindergarten Cohort (ECLS–K; National Center for Education Statistics, 2001). The NLSY was initiated in 1979 with a multistage stratified random sample of 12,686 participants between the ages of 14 and 21 (Center for Human Resource Research, 2004). In 1986, children of female participants were recruited for the NLSY–CYA, and data collection began. This sample of children was assessed every 2 years through 2006. Measures of verbal ability include the Reading Comprehension test from the Peabody Individual Achievement Test (PIAT; Dunn & Markwardt, 1970) and the Peabody Picture Vocabulary Test (PPVT; Dunn & Dunn, 1981). Figures 1a and 1b are longitudinal plots for n D 200 participants on the PIAT Reading Comprehension test and the PPVT, respectively, from age 5 through 14. The longitudinal trends displayed in Figure 1a fan out over time such that the observed variance was small at age 5 and gradually increased with age. The longitudinal trends in Figure 1b, on the other hand, appear to exhibit similar amounts of variance in the early years compared with the later years. The ECLS–K is a nationally representative sample of 21,260 children attending kindergarten in 1998–1999. The ECLS–K follows these children from the fall of kindergarten to the end of eighth grade. Currently, data from the beginning of kindergarten through the end of fifth grade are publicly available. A measure of reading achievement was collected in the fall and spring of kindergarten and first grade and the springs of third and fifth grade. The reading test scores were


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(a)

(b) FIGURE 1 Longitudinal plots for a random sample of n D 200 participants on the (a) Peabody Individual Achievement Test Reading Comprehension test, (b) Peabody Picture Vocabulary Test, and (c) Early Childhood Longitudinal Study Reading test. (continued )


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(c) FIGURE 1

(Continued ).

vertically linked IRT scores, which are appropriate for examining change. Longitudinal trends are plotted for a random sample of n D 200 participants in Figure 1c. As seen in Figure 1a, relatively little variance in scores was present at the beginning of kindergarten compared with the remaining measurement occasions. Additionally, the change pattern appears to be more complex than those seen in Figure 1a and 1b, with little change during kindergarten, followed by accelerated change from kindergarten into third grade, before a final decelerated change from third through fifth grades. Longitudinal Models Four series of growth models were fit that differed in the constraints imposed on the diagonal structure of ‚. In the first series, residual variances were forced to be equal (i.e., homogeneous) across time; in the second series, residual variances were freely estimated; in the third series, the growth curve reliability was forced to be invariant across time; and in the fourth, the growth curve reliability was allowed to change linearly with age and grade. Within each series, we also examined different change patterns (ƒ). We fit the (a) level only, (b) level plus linear slope, and (c) level plus latent slope, where minimum constraints are imposed on the second column of ƒ (Meredith & Tisak, 1990). Mplus v5.0 (Muthén & Muthén, 1998–2008) was used to estimate all models. The Model Constraints command of Mplus was necessary to fit the latent growth curves with constant and linearly changing growth curve reliability.

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Input and output scripts are available from http://psychology.ucdavis.edu/labs/Grimm/personal/ downloads.html.

RESULTS


PIAT Reading Comprehension Regardless of the imposed residual structure, changes in reading comprehension were best characterized with a latent basis growth model—an increasing function with a decelerating rate of change. Parameter estimates and fit statistics for the latent basis growth model with different residual structures are contained in Table 1. For these models, the latent variable means and basis coefficients were similar regardless of residual structure, supporting previous work that residual structures have little effect on estimates of these parameters. However, convergence and the latent variable covariances depended heavily on the chosen residual structure. When an equality constraint was imposed on the residual variance (Model M1; ¦2 D 1,691, parameters D 14; comparative fit index [CFI] D .856, Tucker–Lewis Index [TLI] D .874, root mean squared error of approximation [RMSEA] D .060), convergence issues were encountered as a negative estimate was obtained for the intercept variance. Centering the intercept at age 10 or age 14 did not solve the convergence issue; although the estimate for the intercept variance was now positive, the intercept–slope correlation was greater than 1. Convergence issues were not encountered when residual variances were freely estimated (Model M2), when the growth curve reliability was held constant (Model M3), or when the growth curve reliability changed linearly (Model M4). Model M2 (¦2 D 427, parameters D 23; CFI D .966, TLI D .964, RMSEA D .032), with freely estimated residual variances, fit the data much better than Model M1 (¦2 .9/ D 1,264), providing support for the notion that freeing the residual variance often improves model fit. Residual variances were 6.84 and 10.40 at ages 5 and 6, respectively, whereas residual variances ranged from 35.82 to 51.19 at ages 7 through 14, indicating at least a two-tier structure for residual variance. The intercept and slope variances were 16.79 and 90.33, respectively, and the intercept–slope correlation was .15, positive and statistically significant, but small, under Model M2. Model M3 (¦2 D 1,480, parameters D 14; CFI D .875, TLI D .890, RMSEA D .056), with constant growth curve reliability, fit well—better than M1, but not as well as M2 with a freely estimated residual structure. The estimate of intercept variance (18.98) was similar to that obtained from M2, but the slope variance (52.32) was considerably less than the estimate from M2 (and similar to the estimate from M1). The intercept–slope correlation was .47— larger than the estimate from M2. Growth curve reliability was .597 indicating that about 60% of the variance at each occasion was due to the growth factors, leaving 40% of the variance in the residual. The estimate of residual variance was 12.80 at age 6, increased gradually, and was 68.22 at age 14. Model M4 (¦2 D 862, parameters D 15; CFI D .929, TLI D .936, RMSEA D .043), with linearly changing reliability, fit better than Models M1 and M3, but not as well as Model M2. In Model M4, the intercept and slope variances were 10.06 and 71.64, respectively. The intercept–slope correlation was .65, the largest positive intercept–slope correlation of the models fit. The estimate of growth curve reliability was .398 at age 5, indicating a low level

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TABLE 1 Parameter Values and Fit Statistics for Latent Basis Growth Curves with Different Residual Structures Fit to the Peabody Individual Achievement Test Reading Comprehension Scores

M1: Equality Constraint

M2: Freely Estimated

M4: Linearly Changing Reliability

M3: Constant Reliability

M

SE

M

SE

M

SE

M

SE

12.55 40.34

.138 .229

12.44 40.43

.104 .222

12.24 40.49

.114 .230

12.43 40.61

.107 .210


Latent variable means ’1 ’2 Basis coefficients œ1;2 œ2;2 œ3;2 œ4;2 œ5;2 œ6;2 œ7;2 œ8;2 œ9;2 œ10;2

0 .097 .276 .468 .611 .719 .814 .889 .951 1

Fixed .004 .004 .004 .004 .004 .005 .004 .006 Fixed

0 .102 .280 .466 .612 .720 .817 .891 .951 1

Fixed .003 .003 .004 .004 .004 .005 .004 .006 Fixed

0 .103 .289 .476 .619 .723 .819 .889 .957 1

Fixed .003 .003 .003 .004 .004 .005 .005 .006 Fixed

0 .099 .284 .469 .608 .712 .809 .883 .952 1

Fixed .003 .003 .003 .004 .004 .005 .004 .006 Fixed

Latent variable covariances ¥1;1 ¥2;1 ¥2;2

.46 30.92 56.69

.797 1.120 2.833

16.79 5.82 90.33

.991 1.423 3.117

18.98 14.94 52.32

.538 .746 2.204

10.06 17.48 71.64

.453 .728 2.616

39.39 39.39 39.39 39.39 39.39 39.39 39.39 39.39 39.39 39.39

.440 .440 .440 .440 .440 .440 .440 .440 .440 .440

6.84 10.40 35.82 46.31 41.45 39.42 36.73 36.62 51.19 47.22

.906 .812 1.094 1.334 1.318 1.309 1.448 1.492 2.064 1.994

12.80 15.25 21.55 30.38 38.81 45.78 52.93 58.62 64.35 68.22

.287 .285 .330 .416 .509 .597 .697 .787 .883 .953

15.20 18.30 28.25 39.60 46.26 48.64 49.29 47.06 43.77 38.59

.370 .358 .552 .719 .713 .647 .635 .710 .897 1.105

Residual variances ™1;1 ™2;2 ™3;3 ™4;4 ™5;5 ™6;6 ™7;7 ™8;8 ™9;9 ™10;10 Fit statistics ¦2 No. of parameters RMSEA CFI TLI

1,691 14 .060 .856 .874

427 23 .032 .966 .964

1,480 14 .056 .875 .890

862 15 .043 .929 .936

Note. RMSEA D root mean squared error of approximation; CFI D comparative fit index; TLI D Tucker–Lewis Index.

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of reliability at this age, and growth curve reliability increased .039 each year, leading to a reliability of .75 at age 14.


Peabody Picture Vocabulary Test Changes in the PPVT were also best characterized using a latent basis model that had an increasing function with a decelerating rate of change, regardless of the imposed residual structure. Parameter estimates and fit statistics for the latent basis growth models with different residual structures fit to the PPVT are contained in Table 2. As with Reading Comprehension, latent variable means and basis coefficients were similar regardless of residual structure, and latent variable covariances depended on the residual structure, but no convergence issues were encountered with any model fit to PPVT data. When an equality constraint was imposed on the residual variance (Model M1), the model fit well (¦2 D 186, parameters D 14; CFI D .973, TLI D .977, RMSEA D .018), and fit improved when no equality constraints were imposed on residual variances (Model M2: ¦2 D 106, parameters D 23; CFI D .997, TLI D .986, RMSEA D .014). Fit was slightly worse when a constant growth curve reliability was imposed (Model M3: ¦2 D 214, parameters D 14; CFI D .968, TLI D .972, RMSEA D .020). When growth curve reliability was allowed to change linearly with age (Model M4: ¦2 D 162, parameters D 15; CFI D .978, TLI D .980, RMSEA D .017), the fit was better than Models M1 and M3, but worse than Model M2. However, all models showed close fit to the data with RMSEAs less than .05. Differences in the latent variable covariance were sizable, although similarities were evident for Models M1 and M3. Overall, intercept and slope variances were estimated to be larger when restricted residual structures were imposed (Models M1, M3, and M4) compared to when residual variances were unstructured (Model M2). Standard deviations of the intercept from Models M1, M3, and M4 were 7% to 15% larger than the standard deviation of the intercept from Model M2. For the slope, standard deviations from M1 and M3 were 25% to 37% larger than M2. Additionally, in Models M1, M3, and M4, the intercept–slope correlations were negative and statistically significant ( .30, .33, and .17, respectively), whereas in Model M2 the intercept–slope correlation was positive and nonsignificant (.05). In terms of residual variance, the estimates in M2 were dependent on age, ranging from 136.42 at age 6 to 55.06 at age 9 and did not appear to change systematically. For M3, growth curve reliability was .783 and, coupled with the negative intercept–slope correlation, led to a decrease in residual variance from age 5 to 7 before a gradual increase from age 7 to 14. For M4, growth curve reliability was .691 at age 5 and increased .019 each year, resulting in a growth curve reliability of .86 at age 14, which led to a decrease in residual variance from age 5 through 14.

ECLS–K Reading Achievement The reading achievement data from the ECLS–K were best characterized by the latent basis model with periods of slow and rapid change, regardless of the residual structure. Parameter estimates and fit statistics for the latent basis growth models with different residual structures are contained in Table 3. Similar to results from Reading Comprehension and PPVT, the latent variable means and basis coefficients were similar regardless of residual structure. However,

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TABLE 2 Parameter Values and Fit Statistics for Latent Basis Growth Curves with Different Residual Structures Fit to the Peabody Picture Vocabulary Test Scores

M1: Equality Constraint M


SE

M


M3: Constant Reliability SE

M

SE

M

SE


Latent variable means ’1 ’2

44.64 77.90

.310 .684

44.67 77.89

.310 .632

44.63 77.69

.315 .701

44.74 77.99

.315 .648

Basis coefficients œ1;2 œ2;2 œ3;2 œ4;2 œ5;2 œ6;2 œ7;2 œ8;2 œ9;2 œ10;2

0 .194 .370 .493 .609 .716 .798 .873 .951 1

Fixed .006 .006 .006 .007 .006 .007 .008 .011 Fixed

0 .195 .369 .490 .607 .716 .798 .872 .953 1

Fixed .006 .006 .006 .006 .005 .006 .007 .010 Fixed

0 .194 .370 .495 .611 .718 .801 .875 .950 1

Fixed .006 .006 .006 .007 .006 .007 .008 .012 Fixed

0

1

Fixed .006 .006 .006 .006 .006 .007 .007 .011 Fixed

.192 .369 .490 .605 .713 .796 .871 .951


262.49 69.83 213.86

7.662 8.493 14.741

202.65 7.49 112.84

14.554 18.030 25.492

274.53 77.64 201.32

6.757 7.222 13.942

232.68 34.13 176.82

8.46 8.99 14.361

73.86 73.86 73.86 73.86 73.86 73.86 73.86 73.86 73.86 73.86

1.960 1.960 1.960 1.960 1.960 1.960 1.960 1.960 1.960 1.960

122.76 136.42 95.95 89.17 55.06 59.04 82.81 83.73 55.49 54.21

13.946 10.700 7.535 6.662 5.055 4.434 5.661 6.059 9.525 8.757

76.19 69.92 67.89 68.55 70.71 74.06 77.52 81.28 85.69 88.96

2.277 2.106 2.041 2.021 2.016 2.034 2.067 2.117 2.227 2.314

104.25 92.44 86.08 81.34 77.64 74.31 70.01 65.22 60.51 54.00

4.757 4.115 3.644 3.007 2.415 1.999 2.023 2.601 3.524 4.578

Residual variances ™1;1 ™2;2 ™3;3 ™4;4 ™5;5 ™6;6 ™7;7 ™8;8 ™9;9 ™10;10 Fit statistics ¦2 No. of parameters RMSEA CFI TLI

186 14 .018 .973 .977

106 23 .014 .997 .986

214 14 .020 .968 .972

162 15 .017 .978 .980


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TABLE 3 Parameter Values and Fit Statistics for Latent Basis Growth Curves with Different Residual Structures Fit to the Early Childhood Longitudinal Study Reading Scores

M1: Equality Constraint



M3: Constant Reliability

M

SE

M

SE

M

SE

M

SE

28.85 108.66

.116 .202

29.00 108.80

.087 .192

28.80 107.80

.095 .231

28.84 108.10

.095 .226

Fixed .001 .001 .001 .001 Fixed

0

1

Fixed .001 .001 .001 .001 Fixed

Latent variable means


’1 ’2 Basis coefficients œ1;2 œ2;2 œ3;2 œ4;2 œ5;2 œ6;2

0 .106 .168 .393 .815 1

Fixed .001 .001 .001 .001 Fixed

0 .106 .169 .392 .813 1

.108 .173 .402 .816

0 .107 .172 .401 .814 1

Fixed .001 .001 .001 .001 Fixed


139.25 74.16 417.91

1.986 2.480 6.764

110.37 59.19 425.47

1.417 2.122 6.503

116.96 119.21 476.31

1.510 2.260 7.863

108.67 117.54 537.44

1.460 2.250 9.156

76.30 76.30 76.30 76.30 76.30 76.30

.514 .514 .514 .514 .514 .514

11.46 39.21 75.56 164.40 118.75 61.26

.550 .681 1.835 2.150 2.259 2.678

31.04 39.32 45.77 76.97 116.88 220.73

.323 .338 .372 .570 1.47 2.083

36.09 44.03 49.10 81.25 143.56 142.89

.450 .446 .439 .656 1.550 3.528

Residual variances ™1;1 ™2;2 ™3;3 ™4;4 ™5;5 ™6;6 Fit statistics ¦2 No. of parameters RMSEA CFI TLI

22,269 10 .274 .675 .713

10,579 15 .225 .845 .807

16,510 10 .236 .759 .787

16,101 11 .240 .765 .780


unlike the previous results, the latent variable covariances did not show sizable changes when different residual structures were imposed. Another difference between the results for the reading achievement data from the ECLS–K and the verbal ability data from the NLSY– CYA was that none of the latent basis models adequately fit the data. Therefore, the structure of changes and between-person differences therein were more complex than allowed by the latent basis model. As a result, a more complex set of change components are needed to


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model these data adequately. For example, change during kindergarten might be fundamentally different from changes after kindergarten, so a spline or multiphase model (Cudeck & Klebe, 2002; Ram & Grimm, 2007) that allows for different phases of development might be more appropriate. Or, a structured latent curve model (Browne, 1993; Browne & du Toit, 1991) might be required, with one component controlling the amount of change and a second component controlling when these changes are quickest. These more complex models were not examined because the spline model with a kindergarten change component and an after-kindergarten change component was not identified with unstructured residual variances and the estimation of structured latent curves (Browne, 1993; Browne & du Toit, 1991) is outside the scope of this article. Across alternate residual structures, large differences in parameter estimates were found for residual variances, but small differences were found in the latent variable covariance matrix. When an invariant residual structure was imposed in Model M1, residual variances were estimated to be 76.30. The intercept variance was 139.25, the latent slope variance was 417.91, and the intercept–slope correlation was .31. When residual variances were unstructured in Model M2, their estimates ranged from 11.46 in the fall of kindergarten to 164.40 in the spring of first grade. Residual variance gradually increased until the spring of first grade and then decreased through the spring of fifth grade. Intercept variance was 110.37, latent slope variance was 425.47, and the intercept–slope correlation was .27. Next, constant growth curve reliability was imposed in Model M3, leading to growth curve reliability of .790. Residual variances increased over time, beginning at 31.04 in the fall of kindergarten and ending at 220.73 in the spring of fifth grade. Intercept variance was 116.96, slope variance was 476.31, and the intercept–slope correlation was .51. Finally, growth curve reliability increased linearly across time in Model M4, and growth curve reliability increased from .751 in the fall of kindergarten to .860 in the spring of fifth grade. Residual variance was 36.09 in the fall of kindergarten and increased to 143.56 in the spring of third grade before a small decrease to 142.89 in the spring of fifth grade. Intercept variance was 108.67, slope variance was 537.44, and the intercept–slope correlation was .49. Interestingly, Model M1 (invariance) and M2 (unstructured) had the most similar latent variable covariance matrix, but exhibited the greatest difference in model fit. Model-Based Reliability Estimates In this section, we highlight how growth curve reliability differed when different residual structures were imposed. Growth curve reliabilities of the variables at each age or grade for each model are displayed in Table 4. Examining reliabilities over time based on the residual structure gives us another aspect of the model to consider, which could help inform model choice. For example, the reliabilities for PIAT Reading Comprehension change drastically when different residual structures were imposed. When an invariant residual structure was imposed, reliability was undefined at age 5 (i.e., .96) and lower intercept–slope correlation. However, this choice would lead to selection of a model having latent variable variances and covariances and manifest variable residual variances estimated with much less precision (i.e., larger SEs) than in Model M4, supporting our choice of Model M4 as the optimal model. For the PPVT, we recommend Model M1 with an invariant residual structure because this residual structure is the simplest a priori structure to specify, all models fit well, and only small differences in fit arose when different residual structures were imposed. Finally, for ECLS Reading Achievement, we recommend Model M2 with unstructured residual variances because of its superior fit (i.e., the only model with CFI and TLI greater than .80). However, we are cautious in this recommendation because of the rather poor overall fit, and we believe additional change components or an autocorrelated residual structure might be necessary to fit these data adequately.

DISCUSSION The structure of residual variances and covariances in growth curve modeling is often seen as relatively unimportant, even though several researchers have shown, through Monte Carlo simulations, that considerable bias in the random effects of the growth factors (i.e., latent variable covariance matrix) can arise from incorrect specification of autocorrelated residual structures. That is, if the residual structure used to generate data was an autocorrelated structure, but an independent residual structure was employed in modeling, bias in estimates of random effects was a common outcome. In this article, we extended this work by considering additional residual structures and showing how different model constraints on independent residual structures can also affect


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the latent variable covariance matrix, sometimes rather dramatically. In our work, we did not simulate data, so the population model underlying the data was unknown, and we were therefore unable to fit the known population model with assurance. However, the relatively simple differences in residual structure specification we considered were found to have, at times, sizable effects on intercept and slope variances and their covariance. The effects of residual structures on the latent variable covariance matrix become magnified when predictors of intercepts and slopes are entered into the model. We often found changes (sometimes larger, sometimes smaller) in the variances of intercepts and slopes when the residual structure was invariant or followed a constant or linearly changing growth curve reliability, compared to when residual variances were freely estimated. Larger variances of intercept or slope latent variables could lead to smaller standardized effect sizes (i.e., Cohen’s d , “, R2 ) for predictors of intercepts and slopes, leading to underestimation of covariate effects when imposing certain residual structures. Additionally, because alternate residual structures affected the correlation between intercepts and slopes, associations between intercepts and slopes with covariates are likely to vary in magnitude and possibly direction depending on which residual structure is imposed. For example, when the latent basis growth model was fit to the PPVT with different residual structures, the intercept–slope correlation varied from .33 to .05, which could change the valence of associations of the intercept or slope with other variables. These results highlight the importance of thinking carefully about the selection of an appropriate residual structure.

Correlated Residuals in Growth Curve Modeling Correlated residuals could be theoretically meaningful as they can represent carryover effects from the previous occasion(s) that were not accounted for by the covariance structure of the latent variables (e.g., intercept and slope), especially if measurement occasions are closely spaced in time. Or, correlated residuals might represent unmodeled associations that would not be present with a more complex growth curve, with a different (or more appropriate) time structure, or by modeling practice effects (see McArdle & Woodcock, 1997). In applied research utilizing latent growth curve models, linear models are often fit because of their ease in estimation and their intuitive interpretation. However, many researchers have examined more complex models, including nonlinear change models, multiphase (spline) models, or latent basis models (see Browne, 1993; Browne & du Toit, 1991; Cudeck & Klebe, 2002; Grimm & Ram, 2009; Meredith & Tisak, 1990; Ram & Grimm, 2007), as these models have interpretable parameters, can capture complex developmental patterns, and can be fit with a variety of available software (see Blozis, 2004, 2007; Blozis, Harring, & Mels, 2008; Grimm & Ram, 2009). The way in which time is structured is another important consideration in latent growth modeling that deserves greater attention. Many researchers use measurement occasion as the structure for time in latent growth curves even though age or grade might be more appropriate in structuring change across time. Additional time structures, such as pubertal or biological age and time to death (see McArdle & Bell, 2000), might be needed to get a more complete understanding of developmental processes and, in turn, might lead to a simpler residual structure than would otherwise be required.

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Proposed Residual Structures The residual structures proposed here, based on the idea of growth curve reliability, are theoretically based and provide additional levels of flexibility compared with the common notion that residuals should be uncorrelated and have equal variance across times of measurement. Constant growth curve reliability is based on the idea that the growth curve accounts for the same proportion of true score variance at each occasion. As with an invariant residual structure, this structure is simple, has only a single parameter—growth curve reliability—that governs the estimation of residual variance. Therefore, the constant-reliability constraint might be particularly useful when models are complex and data are scarce. The constant-reliability constraint is also flexible because it allows residual variance to change across time, but in a structured way. The second alternate constraint on residual variance—linearly changing growth curve reliability—is based on the idea that the construct might have differing levels of precision over time. If so, reliability of the manifest variable can increase or decrease over time, which might be reasonable, depending on the developmental outcome under consideration. For example, whereas height might well be measured with similar reliability between the ages of 3 and 12 years, vocabulary skills are likely to be measured with differential reliability across the same age span. A linear change in growth curve reliability is a testable assumption that might or might not hold for a set of data. Additionally, researchers can examine these residual structures using many available SEM programs. We used Mplus, but constraints we imposed can be implemented in any program that allows nonlinear equality constraints.

Concluding Remarks Growth curve analysis has been pursued primarily using SEM or multilevel modeling approaches (see Willett & Sayer, 1994). SEM initially yielded flexibility in modeling residual structures that was not then available under a multilevel modeling approach. When Meredith and Tisak (1984, 1990) introduced growth curves, they presumed mutual independence of the residuals, but mentioned that relaxation of this assumption was possible and discussed tridiagonal, block diagonal, and autoregressive structures. At this point in time, growth modeling in the multilevel modeling framework did not have this flexibility, and residuals were required to be independent and invariant over time. Now, multilevel modeling programs can fit latent growth models with different residual structures (see Kwok et al., 2007), allowing researchers to evaluate different residual structures when fitting growth curves. On this issue of residual structures, Browne and du Toit (1991) noted that it is desirable for residual variances in repeated measures data to “be required to follow smooth trends over time and not fluctuate wildly” (p. 50). Further, Browne and du Toit argued that repeated measures models should not have too many parameters, because “meaningless wastebasket parameters employed only to make the model appear to fit well should be avoided” (1991, p. 50). These suggestions are important to consider when modeling longitudinal data, and researchers must strike a balance between “optimal” residual structures and statistical model fit. Researchers who favor the theoretical underpinnings and simplicity of invariant residual variances believe that this part of the growth model is not up for debate and that a theoretical (i.e., not statistical) reason for freeing up residual variances must exist for this to change. In this sense, freeing


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up residual variances or allowing for an autocorrelational structure can be seen as an “easy” way out to improve model fit. Rather than taking this easy way out, perhaps the investigator should consider alternative growth structures (e.g., additional change components) as the current growth model for the covariances might be incorrect. Correctly specifying the growth model might lead to a very simple, even invariant set of residual variances. Simplicity or parsimony is a desirable property of a model, as it requires fewer ad hoc explanations for parameter estimates in the model. Improperly imposed constraints, although embodying simplicity in one part of a model, might require unplanned, ad hoc complexity in another part of the structural model. For example, constraining residual variances to equality across times of measurement led to a latent basis model for PIAT Reading Comprehension that was unacceptable in several ways: The intercept latent variable had negative variance, and the CFI and TLI indexes of practical fit were unacceptable. Some researchers might have responded to this outcome by adding complexity to the growth model, positing additional trends (e.g., quadratic trends or spline components) or multiple latent groups. However, by relaxing the equality constraints on residual variances across time and thereby introducing complexity in this part of the model, the chi-square index of fit was reduced by almost 75%, practical fit indexes fell into a clearly acceptable range, and intercept and slope latent variables had reasonable and robust estimates of variance and covariance. In effect, we could retain simplicity of the growth model in terms of the latent growth factors—which would be the focus of attention in explaining growth and development in reading comprehension—by introducing complexity in our residual structure, which was still reasonably simple in being diagonal, although with changing variance across time. Our results lead to an important recommendation for applications of latent growth models: Researchers should pay at least as much attention to the structure of residuals specified in their growth models as they do to the nature of the growth processes specified to represent psychological growth and change optimally.

ACKNOWLEDGMENTS This research was supported by a National Science Foundation REECE Program Grant (DRL0815787) and the National Center for Research on Early Childhood Education, Institute of Education Sciences, U.S. Department of Education (R305A06021). The opinions and views expressed in this article are those of the authors and do not necessarily represent the views and opinions of the U.S. Department of Education.

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