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January 2011
New Column!
Instructor’s Corner
The focus of this new column is to inform and educate Division 5 members on topics of interest. Each issue will contain a column written by a member with expertise in a particular area. The intent is to educate Division 5 members on topics and issues spanning the core missions of the division. If you have a topic you would like to write about for an upcoming issue, please contact Julie Lackaff at
[email protected].
Conveying Complex Statistical Concepts as Metaphors1 Todd D. Little University of Kansas “But the greatest thing by far is to have a command of metaphor. This alone cannot be imparted by another; it is the mark of genius, for to make good metaphors implies an eye for resemblances.” —Aristotle, Poetics In 2006, I was asked to write an essay on my use of metaphor to communicate complex statistical concepts. We have all used simple metaphors for simple concepts such as the balance point of a teeter totter to convey the concept of the mean of a distribution, or the concept of degrees of freedom being like money (you pay one for each estimate or you save one for each estimate you don’t make). For advanced statistical concepts, however, I also find that a good metaphor can open up vast learning opportunities for students. Metaphors can be quite complex, ranging from computer metaphors for understanding human cognition to simple clichés such as “a bird in the hand is twice as good as two in the bush.” They can come in mixed forms as well, such as “a stone in the hand will trump two birds formerly of the bush.” And regardless of whether the attempt at metaphor, analogy, or simile is technically a metaphor, analogy, or simile is less important than the basic discovery that I have made in my 20 years of teaching advanced, mathematically based concepts; namely, a good metaphor is worth a thousand numbers, equations, and formulae. I use terms like metaphor and analogy loosely. I am generally trying to capture the idea of making a comparison, such as when a wellunderstood word, phrase, concept, or image that ordinarily designates one thing is used in reference to another. Such comparisons are drawn in order to show a similarity in some respect, usually under the assumption that if things agree in some respects, they probably agree in others. I regularly teach a course on Structural Equation Modeling. As most of you know SEM involves a lot of mathematically based concepts and ideas.2 A handful of students will instantly resonate with the material presented in footnote 2. Most students, on the other hand, would find this material bewildering at best and more likely beyond comprehension. My struggle as a teacher of such material has always been to figure out ways to make the concepts and issues that surround them accessible to the math phobic.
As a graduate student, I took a course on the general use of metaphor in science. It struck me that drawing such parallels among things that exist in nature, or “kinds,” the precise terms created to capture the essence of the “kinds,” and the imprecise (but rich-by-analogy) metaphors used to represent the relations between them, might be beneficial in my teaching. One clearly runs the risk of imprecision and nearly all good metaphors eventually fall apart. However, a good metaphor or analogy taps existing knowledge structures about how a set of things relate, thereby aiding in learning the precise “term” for the “kind” and the relational nature among the “terms” for the “kinds.” In recent years, I have routinely queried my classes on my use of metaphor to gauge if the metaphors that I use are helpful. The student responses that I have received convey well the reasons for why metaphors can be particularly useful in teaching advanced concepts. This first response captures the scaffolding idea when introducing imprecise terms as proxies of the precise (which in turn reflect the mathematical underpinnings): One of the first things I noticed about your lectures from the onset was your use of concrete examples and analogies. This was especially useful in the initial lectures in which you described what SEM was all about. Also I noticed in your earlier lectures you used very little SEM terminology and more analogies which I think helped to lay a foundation for the uses and purposes of SEM. Later, I noticed you started pairing the analogies along with the terminology used in SEM. When used in tandem, it provides students with multiple avenues for understanding. For example, if one student does not understand the technical terms/ description, at least she can still grasp the basics of the lecture. Not all students come to the class with the same level of understanding and background so analogies help to put all students on the same playing field, so to speak.
There are very few times when PowerPoint and its animation capabilities actually enhance a lecture. However, when carefully considered and applied, visual metaphors (animated or not) can be tremendously helpful. Student B captures this idea. This student also captures well my reasons for using metaphor and analogy in conveying math intensive concepts to the “less-mathematically inclined”: I’ve never had a stats teacher use metaphors to explain key concepts. Instead, they have always thrown a formula at me. I am a little mathphobic and when I see a really complicated formula, I usually drift off. I much prefer to think of things at the conceptual level and metaphors can be very helpful in understanding concepts, even if I don’t completely
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understand the math behind it all. Given the conceptual nature of SEM, your metaphors have helped me to grasp its basic principles. Perhaps the most helpful metaphor has been the example of the three-legged stool. If you had given me just the math of why three indicators of a construct are more stable than four indicators, I doubt I would have gotten it. I probably still don’t completely get the math, but I understand the principle at a conceptual level. The use of visual metaphors has also been extremely helpful. The metaphor of the construct as a circle with a centroid has helped me visualize the concepts of measurement error and triangulating around the construct centroid. These demonstrations have also been helpful in literally seeing the relationship between reliability and validity in a new way. Seeing the concepts allowed me to ease my way into the class and tackle the math later with more of a conceptual background. Honestly, if the first day was all formulas and no metaphors, I am not sure I would have stuck with the class.
Student C and D capture perhaps one of the key reasons why I think metaphors work so well in my courses; namely, they give the student something richer to hang the ideas on: Your teaching style makes things very visual and puts complicated concepts into the common English language. It’s as if the metaphors and analogies point to a familiar cognitive framework that can be used to make sense of far less familiar, complicated statistical concepts.
And more than one way to understand the concepts: I like analogies in general, and in multivariate statistics classes specifically… I think presenting things in more than one way helps students who learn differently better understand the concepts.
Some metaphors work better than others and some work better for some people in different ways, but I was also very pleased to see that some of these metaphors might actually proliferate and help others learn some of the key concepts of the area: The “water in the ice cube tray” metaphor worked very well for me today. I also actually used your ‘pinning down a cloud’ analogy to a friend of mine who is also taking SEM (not here). I did not do it justice, but it seemed to have sunk in with her.
In SEM, we attempt to represent latent constructs from numerous measured indicators. The latent construct is like a floating cloud. To measure it and quantify it, one simply defines a point somewhere in the cloud to pin it down. Once pinned down, one can measure from that point to the edges of the cloud and quantify it. The point where the measurements are taken is arbitrary. You simply need to pick a point and pin it down. This analog roughly communicates the often difficult concept of setting the scale of measurement for a latent construct. Turning to the final student comment, this student emphasizes the point that one cannot survive on metaphor alone (and the point that even the math able like them): I don’t think analogies can substitute completely for math. I am more of a math-oriented learner than verbal. So while the analogies help, the
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math helps me understand concepts just as much. I wouldn’t want to see math replaced completely with analogies and metaphors.
As for how I find metaphors, I don’t know if it’s just a gift (which I doubt) or if it’s just a straight forward process of when you want one, it will come to you. I suspect that many readers of this essay will find that metaphors are rather easy to come by if they decide to try using them (visual or verbal) as part of their curriculum content. I also suspect that you will find your students to be favorably inclined. Returning to the equations in footnote 2, these formulae convey the fundamental features for understanding the concept of factorial invariance. When I teach SEM or Longitudinal SEM, I typically use a metaphor to convey the concept of factorial invariance (Little, in press), for example: Two botanists see two plants. One plant is on the south side of a hill; it is tall has broad leaves, and a slender stem. The other is on the north side of a hill; it is short, thin leaved, and has a thick stem. Although these outward characteristics and differences are easily observed, they wonder whether the two plants are the same species or not. If the plants are the same species, they argue, then the different growth patterns have been shaped by the context of being on the north or south side of a hill; otherwise the differences are because they are different species of plant. To test their conjectures, they carefully dig down to the roots of the plants. If each plant is the same species, it should have a signature root system. In this case, they see that the first plant has three roots, which follow a particular pattern of length and thickness: the first is medium long and medium thick, the second is longest and thickest, and the third is shortest and thinnest. They carefully dig up the other plant and see that it too has three roots that follow the same pattern of relative length and thickness, except that, for this plant, they see that, like the plant itself, all the roots are longer and thinner by about the same proportions. Because both plants have the same number of roots that follow the same pattern of length (loadings) and thickness (intercepts), they conclude that the plants (constructs) are fundamentally the same species (factorially invariant) and the observed differences (cross-time differences or group differences) are due to the context. They also notice that the amounts of dirt and spindly bits still on the roots appear to be about the same, but they ignore this information because it’s just dirt and spindly bits (residuals). Indicators are like the roots of a construct. Any observed differences at the level of the construct can be attributed to the construct only if the relative root structure of the construct can be shown to be the same across constructs. That is, the lengths (loadings) and thicknesses (intercepts) of each corresponding indicator (root structure) must be proportionally about the same in order to conclude that the construct is fundamentally the same and that the apparent differences are true construct-level differences. This plant metaphor works well for thinking about between group differences. A longitudinal metaphor that we have used to convey the same idea is to imagine a small constellation of birds. Are they a flock of birds traveling as a connected continued on p. 8
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January 2011
Instructor’s Corner continued from p. 7
unit, or are we witnessing a temporary confluence? If the birds (indicators) are a flock (a construct) then they travel across the vista maintaining their relative relationship to one another (longitudinal factorial invariance) even as the flock as a whole rises and falls (the construct can change over time). If, on the other hand, one or more of the birds veer away from the others, the flock (construct) is no longer a flock (it lacks factorial invariance). In longitudinal SEM we can see clearly when the flock is no longer a flock because our tests for longitudinal factorial invariance show us when an indicator no longer has the same relationships with the other indicators. In other analytic techniques (e.g., repeated measures ANOVA), we can only assume the flock remained a flock, which can be a dubious assumption. If you have a useful metaphor that you would like to convey, please post it to our web page by sending it to Alan Reifman or contact Julie Lackaff for a possible contribution to our instructor’s corner. References Little, T. D. (in press). Longitudinal structural equation modeling: Individual difference panel models. New York: Guilford Press. Little, T. D., & Slegers, D. W. (2005). Factor analysis: Multiple groups. In B. Everitt & D. Howell (Eds.) & D. Rindskopf (Section Ed.), Encyclopedia of statistics in behavioral science (Vol. 2, pp. 617–623). Chichester, UK: Wiley.
Todd D. Little is the winner of the 2009 University wide Kemper Award for Teaching Excellence given by the University of Kansas. He is a Professor in the Department of Psychology, where he directs the Quantitative Training program, and he is Director of the Center for Research Methods and Data Analysis, the KU summer institutes in statistics (Stats Camps), and the Social and Behavioral Sciences Methodology minor for undergraduates, all at the University of Kansas.
Parts of this essay originally appeared as Little, T. D. (2006). Thought Candy: Metaphor in Education. Reflections from the Classroom, 8, 13–15, which is a publication of the Center for Teaching Excellence at the University of Kansas, Daniel K Bernstein, director, Judy Eddy, editor. I’m grateful to Dan and Judy for allowing me to modify and reprint it here. 1
For example, multiple group mean and covariance structures models would begin with the matrix algebra notations for the general factor model, which, for multiple samples of g = 1, 2 ... g, is represented by:
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xg = τg + Λg ξg + δg(1) E(xg) = μg = τg + Λg κg (2) Σg = Λg Φg Λ’g + Θg (3) where x is a vector of observed or manifest indicators, ξ is a vector of latent constructs, τ is a vector of intercepts of the manifest indicators, Λ is the factor pattern or loading matrix of the indicators, κ represents the means of the latent constructs, Φ is the variance-covariance matrix of the latent constructs, and Θ is a symmetric matrix with the variances of the error terms along the diagonal and possible covariances among the residuals in the off diagonal. All of the parameter matrices are subscripted with a g to indicate that the parameters may take different values in each population. For the common factor model, we assume that the indicators (i.e., items, parcels, scales, responses, etc.) are continuous variables that are multivariate normal in the population and the elements of Θ have a mean of zero and are independent of the estimated elements in the other parameter matrices (see Little & Slegers, 2005). Factorial invariance holds when model fit is good and the parameter matrices Λ and τ are mathematically equal across groups and only ξ and κ are allowed to vary across groups (and thereby convey any group differences as construct differences across groups).
Coming Soon! We hope to introduce another new column where Division 5 members have the opportunity to express their opinions about an interesting and/or controversial topic related to topics and issues spanning the core missions of the division. Look for more information about this on the listserv and upcoming issues of the Score.
APA Division 51 Society for the Psychological Study of Men and Masculinity (SPSMM) A one year, free membership for 2011 is being offered by Society for the Psychological Study of Men and Masculinity (SPSMM), Division 51 of APA. SPSMM advances knowledge in the psychology of men through research, education, training, public policy, and improved clinical services for men. Benefits of membershipi nclude: (1) Free subscription to Psychology of Men and Masculinity (the official empirical journal of Division 51); (2) Participation in SPSMM Listserv where members exchange information and ideas, discuss research and practice, and network with colleagues; (3) Opportunities to serve in leadership roles in Division 51’s Committees and Task Forces; (4) Involvement with Divisional Web page on your interests and expertise in psychology of men; (5) Opportunities to meet, network, and socialize with over 500 psychologists committed to advancing the psychology men and gender. For further information about the free membership application process, go to Division 51’s website: http://www.apa. org/divisions/div51/. For electronic application, go to www.apa.org/divapp. Contact Keith Cooke at
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