Analysing change between two variables measured at 3 points

Question

I have two variables measured concurrently at 3 time points, let's call them wealth (W1, W2, W3) and anxiety (A1, A2, A3). Suppose that wealth and anxiety are uncorrelated. Now, I'm interested whether change in anxiety is related to change in wealth. What would be the recommended way of testing this? Naively I would just create 4 new variables W2-W1, W3-W2, A2-A1 and A3-A2 and look at correlation coefficients between W2-W1 vs. A2-A1 and between W3-W2 vs. A3-A2. Somehow this approach seems silly, since I want a single model that would look at the overall change. Should I be looking at mixed/multilevel models? Specifically which model? Thanks!

Answer 1

I would look at latent growth curves, or specifically, multivariate latent growth curves. In those models, you have "slope" factors (linear, quadratic, or even "latent basis", where the shape of the change is determine by the data). The correlation between these slope factors can be freely estimated if you decide so.

Software that can run those are virtually any structural equation modeling software, such as Mplus, Amos, the packages lavaan (recommended) or sem in R. I think PROC CALIS in SAS can do it as well, though I'm not certain.

References on latent growth curves:

• Kline, R. B. (2010). Principles and practice of structural equation modeling (3rd ed.). New York: Guilford. (see chapter 11) (on Amazon)
• Preacher, K. J., Wichman, A. L., MacCallum, R. C., & Briggs, N. E. (2008). Latent growth curve modeling. Thousand Oaks, California: SAGE. (on Amazon)

Answer 2

One big issue will be level of measurement, since continuous-data (e.g., count-data) require different models than discrete-data (e.g., a single likert scale rating of anxiety, or a choice of a single income bracket where wealth has been polytomized). poisson distributions are often useful for counts, whereas irt models (e.g., graded response, partial credit, rating scale) may better suit polytomous data (see Templin, 2007; Linacre, 2000; Wright, 1998; Ostini & Nering, 2006; Nering & Ostini, 2011).

As for the matter of modeling change, I highly recommend reading McArdle's (2009) great review of latent change models, including those @PatrickCoulombe has wisely suggested. I agree that multivariate latent growth-models seem suitable to modeling overall change, but it's not silly to consider whether changes between consecutive time points are more closely related than overall changes. You can even do both! Take a look at autoregressive latent trajectory (ALT) models (Bollen & Zimmer, 2010), particularly Figure 4 in Bollen and Curran (2004). If you've got plenty of continuous data, or can model continuous latent factors from plenty of discrete data (cf. Regression testing after dimension reduction and Factor analysis of questionnaires composed of Likert items), you might be able to fit some pretty mind-boggling models like these...but that's a big "if"—I've seen rules of thumb for model fitting suggest anywhere from 5 to 20 observations per parameter that needs estimating.

Even without sufficient data tonnage, you might consider a different approach. Comparing nested models with various constraints can produce interesting results for general model comparison hypotheses (e.g., Grant, Langan–Fox, & @JeromyAnglim, 2009). In your case, you might try constraining correlations between first-order latent changes to be equal, i.e. (roughly): $$\text{covariance}(W_2-W_1\ \text{with}\ A_2-A_1)=\text{covariance}(W_3-W_2\ \text{with}\ A_3-A_2)$$ in one model, and then fitting the unconstrained model in which the above is nested. You can compare these nested models with a $\chi^2$ test to see whether the unconstrained model fits significantly better, compare other fit indices to see whether the constrained model might make more efficient (not necessarily in the proper statistical sense of "efficiency", which I honestly don't understand myself) use of parameter estimates. See Wikipedia's section on assessing model fit and Lei and Wu (2007) for some discussion of fit indices (Lei and Wu give a great introduction to sem in general), many of which penalize model fit by the number of parameters estimated. Freely estimating two identical model parameters will thus reduce such fit statistics relative to a model that constrains them to be equal; such a constrained model only freely estimates one parameter for both paths. You might also try looking at modification indices for the constrained model to see whether the constraint is a major source of model misfit to your data's variance-covariance structure.

References

Bollen, K. A., & Zimmer, C. (2010). An overview of the autoregressive latent trajectory (ALT) model. In Longitudinal research with latent variables, pp. 153–176. Springer: Berlin, Heidelberg.

Bollen, K. A., & Curran, P. J. (2004). Autoregressive latent trajectory (ALT) models: A synthesis of two traditions. Sociological Methods & Research, 32(3), 336–383. Retrieved from: http://www.odum.unc.edu/content/pdf/Bollen%20Curran%20%282004%20SMR%29.pdf.

Grant, S., Langan–Fox, J., & Anglim, J. (2009). The Big Five traits as predictors of subjective and psychological well-being. Psychological Reports, 105(1), 205–231.

Lei, P. W., & Wu, Q. (2007). Introduction to structural equation modeling: Issues and practical considerations. Educational Measurement: Issues and Practice, 26(3), 33-43.

Linacre, J. M. (2000). Comparing "partial credit models" (PCM) and "rating scale models" (RSM). Rasch Measurement Transactions, 14(3), 768. Retrieved from: http://www.rasch.org/rmt/rmt143k.htm.

McArdle, J. J. (2009). Latent variable modeling of differences and changes with longitudinal data. Annual Review of Psychology, 60, 577–605.

Nering, M. L., & Ostini, R. (Eds.). (2011). Handbook of polytomous item response theory models. Taylor & Francis.

Ostini, R., & Nering, M. L. (2006). Chapter 3 – Polytomous Rasch models. In R. Ostini & M. L. Nering, Polytomous item response theory models, pp. 26–64. Quantitative Applications in the Social Sciences, 144. Sage. Retrieved from: http://www.sagepub.com/upm-data/5197_Ostini_Chapter_3.pdf.

Templin, J. (2007). IRT models for polytomous response data. Item Response Theory Stats Camp '07, University of Kansas. Retrieved from: http://jonathantemplin.com/files/irt/irt07ku/irt07ku_lecture03.pdf.

Wright, B. D. (1998). Model selection: Rating scale model (RSM) or partial credit model (PCM)? Rasch Measurement Transactions, 12(3), 641–642. Retrieved from: http://www.rasch.org/rmt/rmt1231.htm.