Emil Coman 1, Eugen Iordache 2, Maria Coman 3 1. SESSION: Extensions to Mediational Analyses

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1 Testing mediation the way it was meant to be: Changes leading to changes then to other changes. Dynamic mediation implemented with latent change scores SESSION: Extensions to Mediational Analyses Emil Coman 1, Eugen Iordache 2, Maria Coman 3 1 U. of Connecticut Health Center, 2 Transilvania U., Romania. 3 Eastern Conn State U. Modern Modeling Methods Conference, Storrs, CT, May 21-22, 213

2 Acknowledgment David Kenny s training ?- My work Good job DK: we all male mistakes Modern Modeling Methods Conference, Storrs, CT, May 21-22, 213

3 Plan The case for mediation as a dynamic concept. From linking successive levels/states of variables to actual changes: methods to test change mediation, i.e. true mediation. From change scores to Latent Change Scores (LCS) Application of LCS to test true dynamic mediation Self-mediation The meaning of changes-leading-to-changes-leading-to-changes Conclusions

4 Change and regressions Mediation is by definition a dynamic concept. The initial definitions of mediation were centered on analyzing the change processes [1] leading to specific effects: change is central to the concept of mediation [2]. Coleman distinctly spelled out long ago [3] how longitudinal regressions derive from modeling the changes in outcomes, as time differentials of an outcome specified as a function of simultaneous variables: dx 1 /dt = a + b x 1, which leads by integration to x 1t = a/b (e bδt -1)+ x 1t-1 e bδt, which for equally spaced observation Δt = [t-(t-1)] are all the same: (3:435) x 1t = a + b x 1t-1 a a & b b 1. Judd, C.M. and D.A. Kenny, Process Analysis: Estimating Mediation in Treatment Evaluations. Evaluation Review, (5): p Maxwell, S.E. and D.A. Cole, Bias in Cross-Sectional Analyses of Longitudinal Mediation. Psychological Methods March, (1): p Coleman, J.S., The mathematical study of change, in Methodology in social research, J. H. M. Blalock, Editor. 1968, McGraw-Hill: New York. p

5 Change and regressions 2 Arminger also showed the example of attitudinal change over time [4], by specifying a differential equation model whereby time-varying variables predict the derivative of the outcome with respect to time: dy/dt. Integrating this equation leads to regression equations of prior time variables predicting later outcomes (p. 22). This point has been reiterated in the work of McArdle over several decades [5, 6]. 1. Arminger, G., Linear stochastic differential equation models for panel data with unobserved variables. Sociological methodology, : p McArdle, J.J., Dynamic but structural equation modeling of repeated measures data, in The handbook of multivariate experimental psychology, J.R. Nesselroade and R.B. Cattell, Editors. 1988, Plenum Press: New York. p McArdle, J.J., Cautiously adding dynamics to longitudinal models, in Modern Modeling Methods (M3) Conference212: Storrs, CT.

6 Brief study description Theory of Planned Behavior (TPB) based health intervention aimed at improving physical activity (PA) using new technologies, in two randomized conditions: and SMS [5]. Five waves of data, here focusing on variables: 1. Attitudes about PA; 2. Intention to engage in PA; and Perceived Behavioral Control. Participants got 12 s, and in the second condition an additional 24 personalized reinforcement SMS (text messages). Suggs, L. S., Blake, H., Bardus, M., & Lloyd, S. (213). Effects of text-messaging in addition to s on physical activity among university and college employees in the UK. Journal of Health Services Research. doi: /

7 Pair Link Diagram An alternative to a scatterplot Average change is +.6 Changes Means Pre Post The pre and post scores show individual differences, the lines differences in changes. The red double interrupted line links the average changes pre->post. This is the ingredient in linear growth models (LGM) and Latent Change/Difference Score (LCS) models. Credit: Jeremy Miles PPT Post (X 1Ave =5.2) vs. Pre (X 2Ave =5.8) Means

8 Pair Link Diagram Average change is still +.6; σ X1X2 =1 Perfect stability, but with individual changes Changes Means Pre Post Post (X 1Ave =5.2) vs. Pre (X 2Ave =5.8) Means The red double interrupted line links the average changes pre->post. This is the ingredient in linear growth models (LGM) and Latent Change/Difference Score (LCS) models

9 What are the changes and how do they relate to the level variables and other changes?.6 Difference scores for consecutive time lags for the attitudes, intentions, and perceived behavioral control variables Note: Last PBC change was zero Attitudes(2-1) Attitudes(3-2) Attitudes(4-3) Attitudes(5-4) Intentions(2-1) Intentions(3-2) Intentions(4-3) Intentions(5-4) PercBehContr(2-1) PercBehContr(3-2) PercBehContr(4-3) PercBehContr(5-4)

10 Changes-with-changes difference scores cross-correlations Wave 1 Wave2 Wave3 Wave4 Int Int Δ Att NS Δ Int NS +.17 NS Δ PA Att Att 3 PA PA NS NS Correlations between relevant raw and one-leg difference variables chosen for comparative modeling; p values in subscripts, unless NS or. The changes-leading-to-changes is captured by the co-variability between consecutive ΔInt and ΔAtt; the covariabilities between Int & ΔInt, and Att &ΔInt, indicate within-variable self-feedback, while those between Int &ΔAtt, and Att &ΔInt cross-variable couplings.

11 Wave1 Changes-to-changes model shown with pair link diagram Wave2 Wave2 Wave3 Y 2 Y 3 μ M1C = μ M1T μ M2T μ M2C μ Y2T μ Y3T μ Y2C μ Y3C M 1 M 2 We depict the case where 2 groups start off at the same average level of M. The M variable changes from wave 1 to wave 2, and the Y variable changes from wave 2 to wave 3. The red and green lines link the ΔM 21 changes (slopes, angles in the figure) to the ΔY 32 changes. The double lines show the links between the average changes/angles in two groups: C(control) and T (Treatment).

12 Lagged autoregressive model e ρ X 1 X 2 Hexagons show means and intercepts; is freely estimated, where it was set to zero, all regression coefficients set to unity unless labeled otherwise; ρ is the stability coefficient; the model as such cannot capture change.

13 Define LCS 1 st MPLUS code to set up LCS scores! Set up latent variable dlx21 by X2@1;!defined by the LATTER variable X1 (X1Var); dlx21 (DLXVar);!Set up model X2 on X1@1; X2 on dlx21@1 ; dlx21 on X1(DLXonX1); 2 nd MPLUS code dlx21 BY X1@;! defined by the former variable X2 ON X1@1 the difference X2@; [X2@ dlx21];

14 Define LCS 2 nd MPLUS code measurement errors LX1 by ;!define LVs LX2 by ;!define LVs dlx21 by LX2@1 ;!define LCS scores LX2 on ;! Auto-regressions AR [LX1-LX2@ dlx21@ ] ; LX1 (ExogLXVar); LX2@ dlx21@ ; X1 X2 (EqualErrors) ; 3 rd MPLUS code!define LCS scores dlx21 by; dl21@; dl21 on X1@-1 X2@1; Linda Muthen Mplus discussion forum

15 Lagged autoregressive and Latent Change Score example e X2 4.5 Att 1 Att Att e X2 Att ΔL Att e ΔLX21 Attitude about Physical Activity (PA) at baseline (1) and post (2): Stability is: 1 plus proportional growth (self-feedback) coefficient β: ρ = (β + 1).29 =

16 Change and stability Average change is +.6, but varies Post (X 1Ave =5.2) vs. Pre (X 2Ave =5.8) Means e ΔLX21 X 1 β ΔL X21 e X2 X ρ = β + 1 When β, we have instability, because ρ = (β + 1) >< 1

17 Change and stability Average change is +.6, but change is constant Post (X 1Ave =5.2) vs. Pre (X 2Ave =5.8) Means X 1 ΔL X21 e X2 X 2 4 e ΔLX When β =, we have perfect stability, because ρ = (β + 1) = 1, so change is constant.

18 Cross-Lag Path model (CLPM) of intervention effects α X1 β AR,X α X2 X 1 X 2 /1= Int β effectx γ XY β effecty γ YX α Y1 Y 1 β AR,Y Y 2 α Y2 X 2 = α X2 + β AR,X X 1 + γ XY Y 1 + β effectx Int + 1e X2 Y 2 = α Y2 + β AR,Y Y 1 + γ YX X 1 + β effecty Int + 1e Y2 Note: All unlabeled regression coefficients are set to 1 (unity).

19 Univariate intervention effect LCS model e X 1 β X1 α ΔLY21 X 2 β effectx ΔL X21 e ΔLX21 /1= Int X 2 = 1ΔL X21 + 1X 1 + & ΔL X21 = α ΔLX21 + β effect Int + β X1 X 1 + e ΔLX21 The true ΔLX is in fact a rate of change (ΔX/Δt), but time t is excluded from equations and models by ensuring that all Δt =1. Note: All unlabeled regression coefficients are set to 1 (unity). McArdle, J. J., & Nesselroade, J. R. (23). Growth curve analysis in contemporary psychological research. In J. Schinka & W. Velicer (Eds.), Handbook of psychology (Vol. 2, pp ). New York:: Pergamon.

20 /1 Bivariate concurrent (LCS) model with proportional growth effect (β) and coupling (γ) Effect on X Effect on Y X 1 γ YX γ XY Y 1 β X ΔL X21 ΔL Y21 β Y X 2 Y 2 Y 2 = 1ΔL Y21 + 1Y 1 + & err ΔLX21 err ΔLY21 ΔL Y21 = α ΔLY21 + β Y Y 2 +γ YX X 1 + 1err ΔLX Notes: All unlabeled regression coefficients are set to 1 (unity); the LCS setup with proportional growth term implies apparently counterintuitively that the post outcome values are the result of a mediation mechanism: an auto-regressive effect (from prior values), a change mechanism, and an indirect effect of prior values through the change mechanism itself. McArdle, J. J., & Prindle, J. J. (28). A latent change score analysis of a randomized clinical trial in reasoning training. Psychology and aging, 23(4), 72.

21 Bivariate sequential LCS model X 1 e ΔLX21 β X ΔL X21 e X The changes-leading-to-changes mechanism is captured by the new ξ coefficient between ΔL X21 and ΔL Y32 ; the coupling effect can be specified as γ XY between X 1 & ΔL Y32 (γ YX between Y 2 & ΔL X21 can not be specified on causality logical grounds). X 2 e ΔLY21 ξ YX Y 2 β Y ΔL Y32 e Y Y 3

22 Bivariate LCS example of a Theory of Planned Behavior (TPB)- based Physical Activity (PA) intervention group Attitude 2 e Att Attitude R 2 = e ΔLAtt ΔL Att e ΔLY Intent 3.7 NS ΔL Int43 R 2 =.38 Intent 4 e Int Larger changes in attitudes lead to larger changes in intent (+.19), which adds to the effect of the prior level in attitudes (+.41); unstandardized estimates; input covariance matrix.

23 Bivariate LCS example of a Theory of Planned Behavior (TPB)-based Physical Activity (PA) intervention group Attitude 2 e Att Attitude R 2 =.36 e ΔLAtt ΔL Att NS ΔL Int43 R 2 = e ΔLY Intent 3 e Int Intent 4 Larger changes in attitudes lead to larger changes in intent (+.19), which adds to the effect of the prior level in attitudes (+.41); unstandardized estimates; input covariance matrix.

24 Barron-Kenny vs Dynamic Mediation model of intervention effects (no measurement errors) Wave1 Wave2 Wave3 M 1 a M 2 b Intervention Y 2 c Y 3 Barron-Kenny mediation Intervention group Individual differences in M 2 Individual differences in Y 3 Intervention group Differences in M 1 ->M 2 changes Individual differences Y 2 -> Y 3 changes M 1 β X M 2 Dynamic mediation ΔL M21 a ch b ch Intervention c ch β Y ΔL Y32 Y 2 Y 3

25 Comparison between static and dynamic mediation models Indirect effect: +.13 Attitude Intent PBControl Indirect effect: -.13 ΔL Attitude ΔL Intent ΔL PBControl 54 Coefficients are standardized. -.22

26 Reminder: Bivariate dual Latent Change Score (LCS) model with constant and proportional changes L IX X X 1 X 2 β X β X β X X 3 γ YX ΔL X1 γ YX ΔL X21 γ YX ΔL X32 L SX L SY ξ YX ξ XY ξ YX ξ XY ΔL Y1 ΔL Y21 γ XY γ XY γ XY ΔL Y32 L IY β Y β Y β Y Y Y 1 Y 2 Y 3 Note: Parameters represent: β - the proportional growth (dotted arrows); γ coupling (interrupted double lines); ξ - changes-to-changes (double line arrows); arrows from slope factors Ls to ΔL s are constant change parameters α (here set to unity). Grimm, K. J., An, Y., McArdle, J. J., Zonderman, A. B., & Resnick, S. M. (212). Recent Changes Leading to Subsequent Changes: Extensions of Multivariate Latent Difference Score Models. Structural Equation Modeling: A Multidisciplinary Journal, 19(2), doi: 1.18/ McArdle, J. J. (29). Latent Variable Modeling of Differences and Changes with Longitudinal Data. Annual Review of Psychology, 6,

27 Bivariate LGM modeling of sequential (dynamic) mediation X 1 X 2 LI X /1 effect LS X21 β YX err SX α SY LI Y LS Y32 err SY Y 2 Y 3 Y 2 = 1I Y + 1S Y + 1e Y2 S Y = α SY + β YX S X +1err SX Note: All unlabeled regression coefficients are set to 1 (unity); residual error variances set to zero, per Voelkle, p. 381.

28 Mediation model of intervention effects specified as a piecewise Latent Growth Model (LGM) LI M M 1 M 2 M 3 1 A LS M11 a c b c Intervention c c LS Y1 Y 1 Y 2 Y 3 LI Y Wave1 Wave2 Wave3 Note: Paths set to zero are not shown, and all unlabeled paths are set to unity; A this path can be freed to allow the mediator trajectory to change freely after wave 2.

29 Original latent difference score mediation model MacKinnon, D. P. (28). Introduction to statistical mediation analysis: Lawrence Erlbaum Associates. p. 216

30 Latent change score mediation model a c' ΔLY 43 b Y 43 Y 4 If we had a 4 th measurement point for Y, we could in fact have tested the model: changes in X subsequent changes in M subsequent changes in Y.

31 Latent difference score mediation model 2 Selig, J. P., & Preacher, K. J. (29). Mediation Models for Longitudinal Data in Developmental Research. Research in Human Development, 6(2-3), doi: 1.18/ p. 16

32 Mediation changes-to-changes model under the Latent Change Score (LCS) specification (no measurement errors) Wave 1 Wave2 Wave3 Wave4 M 2 M 3 ΔL M32 a c b c ΔL X21 c c ΔL Y43 X 1 X 2 Y 3 Y 4 Many indirect effects to be tested: e.g. X 1 ΔL X21 M 2 ΔL M32 ΔL Y43 Y 4 Note: The mediation effects leading to changes in the Y outcome consists of: a s (e.g. Δ X Δ M) then b s (e.g. ΔM ΔY changes-to-changes); causal paths follow left right direction, except the links from LCS to their observed values which are nearly vertical to indicate non-causal nature of the paths.

33 Suggested extensions 1. Measurement error should be included in any model; with 2 waves, the setup is equal measurement error variance across waves. 2. Multi-group LCS models can test a variety of across-group hypotheses; they are difficult to specify and fit well. 3. LCS mediation models can test whether interventions lead to changes in one outcome, then to subsequent changes in another; few other models have done this. -

34 Conclusions 1. The original meaning of structural modeling contained the change with time component, which needs to be reinstated. 2. Latent change score modeling allows for separating out the change process itself and investigating it directly; in particular it brings to surface the concept of self-mediation, of a variable unto its subsequent levels through its own change mechanism. 3. LCS mediation models can test whether interventions lead to changes in one outcome, then to subsequent changes in another; few other models have done this. -

35 Thanks! Questions? -