The paired t-test as a simple latent change score model

Abstract

The paired t-test as a simple latent change score model The t-test is a common statistical test of differences in means. Despite the fact that its extension, the paired t-test (t-testP), appears in most introductory statistics textbooks, it is less known that for repeated variables the t-testP is in fact a model of change that can be replicated within the Structural Equation Modeling (SEM) framework. We show how to perform the t-testP with Latent Change Scores (LCS) models, which allow for direct testing of significance of mean changes, and moreover can explain inter-individual (and group) differences in changes over time. Modeling backgroundStructural models with latent variables can nowadays replicate virtually any statistical test (Skrondal & Rabe-Hesketh, 2004). It has been shown in particular that a latent growth curve model (LGM) with df = 0 fully replicates the t-testP statistical tests for two waves of data (Voelkle, 2007) and yields asymptotically identical results. LCS models themselves fully replicate the repeated measures Anova and LGMs models and results, and additionally are much broader and capable to relax their assumptions (McArdle, 2009). We show how a more flexible Latent Change Score (LCS) model with changes correlated with initial levels fully replicates the t-testP, and moreover can be expanded to test more complex hypotheses of change. We demonstrate our proposal with a dataset on changes in Hemoglobin A1c (HgA1c) levels among diabetics who participated in a quality improvement study using peer supporters (Phase II – Diabetes eCo-System, Villagra, 2013). HgA1c values range between 5% and 12% (HgA1c >6.5% defines a diabetic patient, NGSP, 2013). Many studies reporting changes in HgA1c values use the t-testP to test for significance of changes (e.g. Satoh-Asahara et al., 2013). Hence, we tested the changes in HgA1c in a group of diabetics using the t-testP, and replicated these results in the LCS framework. The key feature of the t-testP is the computation of the sample average of differences (ΔY21i = Y2i-Y1i) which is compared to zero, using an estimate of its standard error that accounts for the covariance between initial levels and changes. Because the t-testP is testing the significance of Y1->Y2 changes, a natural alternative is to directly specify the change score (Y2-Y1 or ΔY21) in a model of change. Because of known issues around the reliability of the change score (King et al., 2006) and estimation problems due to adding a variable that is a linear combination of two other variables in the same model, the Latent Change Score (LCS) directly inserts the change score in a structural model, however as a latent variable. Rewriting the ΔY21 = Y2-Y1 equation as LCS21 = Y2-Y1, one can define the true changes (McArdle & Nesselroade, 1994) with the multiple regression: Y2 = 0 + 1 * Y1 + 1 * LCS21 + 0 (1)where we specifically indicate that the two regression coefficients are fixed to 1 (unity), and the intercept of Y2 is set to zero, as are the mean and variance of the Y2 residual error, i.e. the entire variance of Y2 is explained by the true change LCS21 (see Figure 1). - Insert Figure 1 about here -While t-testp statistical outputs from various software report the correlation between the two repeated measures, the role of this indicator of measure stability is not directly obvious in the t-testP model. There are two ways in which one can incorporate stability information into the LCS model of change: by specifying a correlation between Y1 and LCS21 (like McArdle, 2001, Fig. 16.3), or with a directed path from Y1 to LCS21. There is a clear difference in terms of the test of significance of changes in outcome Y between the two setups: the correlation specification tests the significance of the mean of the LCS21 score and yields identical results with the t-testP, while the γ path specification is testing the significance of the LCS21 intercept; hence, differences in significance level will be observed. The simple model in Fig. 1 is asymptotically equivalent to the t-testP, when Y1 is correlated with LCS21, or when the γ path is set to zero (in the directed path setup). The γ path has been called proportional growth (McArdle, 2009), immanent change (Coleman, 1968), self-feedback (McArdle, 2009), or self-mediation (Coman et al., 2013); we label it here as Self-Feedback/Mediation (SF/M). It can be easily shown that the γ coefficient’s absolute value is the complement (1- ρ) of the common auto-regressive (AR1) path, or stability (Kenny, 1979, p. 287), from a simple lagged model Y1->Y2 (Coman et al., 2013): when γ = 0, there is perfect stability (ρ = 1). When an outcome changes in level but under perfect stability (equivalent to the LCS model with zero SF/M), the knowledge of one’s initial outcome value Y1 means perfect prediction of his/her Y2 follow-up value; the Y2 intercept in this case tells us by how much Y increases (or decreases) uniformly across the entire sample, i.e. the change is t