A Longitudinal Look at Longitudinal Mediation Models David P. MacKinnon, Arizona State University Causal Mediation Analysis Ghent, Belgium University of Ghent January 28-29, 2013 Introduction Assumptions Unique Issues with Longitudinal Relations Two-wave Mediation Models Three or more wave Mediation Models Application to a Health Promotion Study *Thanks to National Institute on Drug Abuse and Yasemin Kisbu-Sakarya and 1 Matt Valente.
Mediator Definitions A mediator is a variable in a chain whereby an independent variable causes the mediator which in turn causes the outcome variable (Sobel, 1990) The generative mechanism through which the focal independent variable is able to influence the dependent variable (Baron & Kenny, 1986) A variable that occurs in a causal pathway from an independent variable to a dependent variable. It causes variation in the dependent variable and itself is caused to vary by the independent variable (Last, 1988) 2
Single Mediator Model MEDIATOR M a b INDEPENDENT DEPENDENT VARIABLE VARIABLE c’ X Y 3
Directed Acyclic Graph M Y X 4
Mediation is important because … Central questions in many fields are about mediating processes Important for basic research on mechanisms of effects Critical for applied research, especially prevention and treatment Many interesting statistical and mathematical issues 5
Applications Two overlapping applications of mediation analysis: (1) Mediation for Explanation (2) Mediation by Design 6
Mediation by Design • Select mediating variables that are causally related to an outcome variable. • Intervention is designed to change these mediators. • If mediators are causally related to the outcome, then an intervention that changes the mediator will change the outcome. • Common in applied research like prevention and treatment. 7
Intervention Mediation Model MEDIATORS Conceptual Action M 1 , M 2 , M 3 , Theory theory … INTEREVENTION OUTCOMES PROGRAM X Y If the mediators selected are causally related to Y, then changing the mediators will change Y. Test of each theory is important when total effect is nonsignificant. 8
Mediation Regression Equations Tests of mediation for a single mediator use information from some or all of three equations. The coefficients in the equations may be obtained using methods such as ordinary least squares regression, covariance structure analysis, or logistic regression. The following equations are in terms of linear regression and expectations. (Hyman, 1955; Judd & Kenny, 1981; Baron & Kenny, 1986) 9
Equation 1 Social Science MEDIATOR M INDEPENDENT DEPENDENT VARIABLE VARIABLE c X Y 1. The independent variable is related to the dependent variable: 10
Equation 1 Epidemiology MEDIATOR M INDEPENDENT DEPENDENT VARIABLE VARIABLE ф 1 A Y 1. The independent variable is related to the dependent variable: 11
Equation 2 Social Science MEDIATOR M a INDEPENDENT DEPENDENT VARIABLE VARIABLE X Y 2. The independent variable is related to the potential mediator: 12
Equation 2 Epidemiology MEDIATOR M β 1 INDEPENDENT DEPENDENT VARIABLE VARIABLE A Y 2. The independent variable is related to the potential mediator: 13
Equation 3 Social Science MEDIATOR M a b INDEPENDENT DEPENDENT VARIABLE VARIABLE c’ X Y 3. The mediator is related to the dependent variable controlling for exposure to the independent variable: 14
Equation 3 Epidemiology MEDIATOR M θ 2 INDEPENDENT DEPENDENT VARIABLE VARIABLE θ 1 A Y 3. The mediator is related to the dependent variable controlling for exposure to the independent variable: 15
Effect Measures Natural Indirect Effect = ab = c- c’ ab = c- c’ for ordinary least squares regression not nonlinear models like logistic regression. Direct effect = c’ Total effect = ab + c’ = c Natural Indirect Effect = β 1 θ 2 = ф 1 - θ 1 Direct effect = θ 1 Total effect = β 1 θ 2 + θ 1 = ф 1 16
Social Science Equations with Covariate C. E[Y|X=x, C=c] = i 1 + c X + c 2 C E[Y|X=x, M=m, C=c] = i 2 + c’ X + b M + c 3 C E[M|X=x, C=c] = i 3 + a X + a 2 C With XM interaction E[Y|X=x, M=m, C=c] = i 4 + c’ X + b M + h XM + c 4 C 17
Epidemiology Equations with Covariate C. E[Y|A=a, C=c] = ф 0 + ф 1 A + ф 2 C E[Y|A=a, M=m, C=c] = θ 0 + θ 1 A + θ 2 M + θ 4 C E[M|A=a, C=c] = β 0 + β 1 A + β 2 C With AM interaction E[Y|A=a, M=m, C=c] = θ 0 + θ 1 A + θ 2 M + θ 3 AM + θ 4 C VanderWeele (2010) 18
Identification Assumptions 1. No unmeasured X to Y confounders given covariates. 2. No unmeasured M to Y confounders given covariates. 3. No unmeasured X to M confounders given covariates. 4. There is no effect of X that confounds the M to Y relation. VanderWeele & VanSteelandt (2009) 19
Omitted Variables/Confounders (Judd & Kenny, 1981 p. 607): “… a mediational analysis may also yield biased estimates because of omitted variables that cause both the outcome and one or more of the mediating variables. If variables that affect the outcome and ….mediating variables are not controlled in the analysis, biased estimates of the mediation process will result, even .. a randomized experimental research design ...” (James & Brett, 1984 p. 317- 318): “… misspecification due to a "serious" unmeasured variables problem. By a serious unmeasured variables problem is meant that a stable variable exists that (a) has a unique, nonminor, direct influence on an effect (either m or y, or both); (b) is related at least moderately to a measured cause of the effect (e.g., is related to x in the functional equation for m); and (c) is unmeasured — that is, is not included explicitly in the causal model and the confirmatory analysis (James, 1980; James et al., 20 1982).
Assumptions Reliable and valid measures. Data are a random sample from the population of interest. Coefficients, a , b , c’ reflect true causal relations and the correct functional form. Mediation chain is correct. Temporal ordering is correct: X before M before Y. No moderator effects. The relation from X to M and from M to Y are homogeneous across subgroups or other participant characteristics. 21
Significance Testing and Confidence Limits Product of coefficients estimation, ab , of the mediated effect and standard error is the most general approach with best statistical properties for the linear model given assumptions. Best tests are the Joint Significance, Distribution of the Product, and Bootstrap for confidence limit estimation and significance testing again under model assumptions. For nonlinear models and/or violation of model assumptions, the usual estimators are not necessarily accurate. New developments based on potential outcome approaches provide more accurate estimators (Robins & Greenland, 1992; Pearl, 2001). 22
Testing Mediation When the Total Effect is Not Statistically Significant Test of ab can be more powerful than test of c , i.e., mediation more precisely explains how X affects Y. Lack of statistically significant c is very important for mediation analysis because failure of treatment, relapse, or both theories is critical for future studies. Inconsistent mediation relations are possible because adding a mediator may reveal a mediation relation. Note the test of c is important in its own right but is a different test than the test for mediation. It is also a causal estimator. 23
More on Temporal Order Assumption Assume temporal ordering is correct: X before M before Y. Assume that relations among X, M, and Y are at equilibrium so the observed relations are not solely due to when they are measured, i.e., if measured 1 hour later a different model would apply. Assume correct timing and spacing of measures to detect effects. But manipulations target specific times with many patterns of change over time. 24
Judd & Kenny (1981) • (Judd & Kenny, p. 613): While the estimation of longitudinal multiple indicator process models is complex, it is also likely to be quite rewarding, since only through such an analysis can we glimpse the process whereby treatment effects are produced. Without knowledge of this process, generalizing treatment effects may be difficult. 25
Judd & Kenny (1981) • (Judd & Kenny, 1981 p. 611): Specifically we might include the mediational and outcome constructs assessed at a point in time prior to the delivery of the treatment. … Here again we are assuming a randomized experimental research design, so that treatment is not related to any of the pretreatment measures. … we are reducing bias in the estimation of the mediational process by controlling for pretreatment differences on all mediating and outcome variables. … The success of this strategy depends on meeting two assumptions besides the usual assumptions of ANCOVA … constructs must be assessed without error in order to adequately control for them. Second, assuming that the effects of all omitted variables that cause … Time 2 variables are mediated through the Time 1 variables 26
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