A Longitudinal Look at Longitudinal Mediation Models David P. MacKinnon, Arizona State University Causal Mediation Analysis Ghent, Belgium University of Ghent January 8-9, 03 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 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, 990) The generative mechanism through which the focal independent variable is able to influence the dependent variable (Baron & Kenny, 986) 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, 988)
Single Mediator Model MEDIATOR a M b INDEPENDENT VARIABLE X c DEPENDENT VARIABLE Y 3
Directed Acyclic Graph M X Y 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: () Mediation for Explanation () 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 Action theory MEDIATORS M, M, M 3, Conceptual Theory INTEREVENTION PROGRAM X OUTCOMES Y If the mediators selected are causally related to Y, then changing the mediators will change Y. Test of each theory is important when 8 total effect is nonsignificant.
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, 955; Judd & Kenny, 98; Baron & Kenny, 986) 9
Equation Social Science MEDIATOR M INDEPENDENT VARIABLE X c DEPENDENT VARIABLE Y. The independent variable is related to the dependent variable: 0
Equation Epidemiology MEDIATOR M INDEPENDENT VARIABLE A ф DEPENDENT VARIABLE Y. The independent variable is related to the dependent variable:
Equation Social Science MEDIATOR a M INDEPENDENT VARIABLE X DEPENDENT VARIABLE Y. The independent variable is related to the potential mediator:
Equation Epidemiology MEDIATOR β M INDEPENDENT VARIABLE A DEPENDENT VARIABLE Y. The independent variable is related to the potential mediator: 3
Equation 3 Social Science MEDIATOR a M b INDEPENDENT VARIABLE X c DEPENDENT VARIABLE Y 3. The mediator is related to the dependent variable controlling for exposure to the independent variable: 4
Equation 3 Epidemiology MEDIATOR M θ INDEPENDENT VARIABLE A θ DEPENDENT VARIABLE Y 3. The mediator is related to the dependent variable controlling for exposure to the independent variable: 5
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 θ = ф - θ Direct effect= θ Total effect= β θ + θ = ф 6
Social Science Equations with Covariate C. E[Y X=x, C=c] = i + c X + c C E[Y X=x, M=m, C=c] = i + c X + b M + c 3 C E[M X=x, C=c] = i 3 + a X + a C With XM interaction E[Y X=x, M=m, C=c] = i 4 + c X + b M + h XM + c 4 C 7
Epidemiology Equations with Covariate C. E[Y A=a, C=c] = ф 0 + ф A + ф C E[Y A=a, M=m, C=c] = θ 0 + θ A + θ M + θ 4 C E[M A=a, C=c] = β 0 + β A + β C With AM interaction E[Y A=a, M=m, C=c] = θ 0 + θ A + θ M + θ 3 AM + θ 4 C VanderWeele (00) 8
Identification Assumptions. No unmeasured X to Y confounders given covariates.. 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 (009) 9
Omitted Variables/Confounders (Judd & Kenny, 98 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, 984 p. 37-38): 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, 980; James et al., 0 98).
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.
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, 99; Pearl, 00).
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. 3
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 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. 4
Judd & Kenny (98) (Judd & Kenny, p. 63): 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. 5
Judd & Kenny (98) (Judd & Kenny, 98 p. 6): 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 variables are mediated through the Time variables 6
Mediation is a Longitudinal Model A mediator is a variable in a chain whereby an independent variable causes the mediating variable which in turn causes the outcome variable these are longitudinal relations. X, M, and Y in single mediator model imply longitudinal relations even if measured at the same time. For a single mediator model, temporal order for X is clear when it represents random assignment, but the temporal order of M and Y must be based on prior research or theory. 7
Timing of Relations When does X affect M or M affect Y? Triggering, cascading, and other timing processes (Tang & DeRubeis, 999; Howe et al., 00) Tang & DeRubeis (999) found evidence that change in therapy occurs within the first few sessions. How are decisions made about timing? Not often considered in research projects except with respect to when a manipulation is made and the easiest time for data collection. Timing is crucial for deciding when to collect 8 longitudinal measures (Collins & Graham, 003).
Cross-sectional models Cross-section is a snapshot of relations. Models assume that a system has reached an equilibrium so observed relations are not just due to the particular point of observation. But systems may be dynamic and change over time in complicated ways. Meaning of cross-sectional relations (relation of rank order of level) is different from longitudinal relations (relation of rank order of change). Cross-sectional mediation may differ in many ways from longitudinal mediation. May take time for effects to occur. Size of effect depends on time lageffect day apart is likely different from an effect year apart. (Cole & Maxwell, 003; Gollob & Reichardt, 99; MacKinnon, 008; Maxwell & Cole 007; Maxwell et al., 0 and Commentaries in Multivariate Behavioral Research) 9
Benefits of Longitudinal Data Time-ordering of X to M to Y is investigated. Can shed light on whether changes in M precede changes in Y. Both cross-sectional and longitudinal relations can be examined. Removes some alternative explanations of effects, e.g., effects of static variables can be removed. 30
What if repeated measures of X, M, and Y are available? Measures of X, M, and Y at two time points allow for several options; difference score, ANCOVA, residualized change score, relative change Measures of X, M, and Y at three or more time points allow for many alternative longitudinal models. For many examples, X is measured once and represents random assignment of participants to one of two groups. Other variables often do not represent random assignment. 3
Stability, Stationarity, and Equilibrium Stability-the extent to which the mean of a measure is the same across time. Stationarity-the extent to which relations among variables are the same across time. Equilibrium-the extent to which a system has stabilized so that the relations examined are the same over time. Cole & Maxwell, 003; Dwyer, 983; Kenny, 979; MacKinnon, 008; Wohlwill, 973 3
Models for Two Waves Difference Scores for X, M, and Y in the mediation regression equations. Analysis of Covariance where the baseline value of X, M, and Y is included as a predictor of the follow-up value of X, M, and Y. Residual Change. Predict time with time and use the difference between the time score and predicted time score as the dependent variable. Relative Change. The change divided by the baseline measure or the natural logarithm of time divided by time (Tornqvist et al., 985). Controversy over difference score versus ANCOVA models. 33
Regression to the Mean Galton s regression to mediocrity. Tall parents tend to have shorter children. Short parents tend to have taller children. Occurs when two variables are imperfectly related. Examples are the sophomore jinx and spontaneous remission. Galton squeeze diagrams to investigate regression to the mean. Lord s (967) paradox 34
Two-wave Longitudinal Model BASELINE OUTCOME b POST-TEST OUTCOME BASELINE MEDIATOR b 5 b POST-TEST MEDIATOR c 3 PROGRAM a 4 Mediated effect=a 4 b 5 Direct effect = c 3 35
DAG with Confounder U U Y Y M M X
Summary of Two-Wave Models Difference score versus ANCOVA models. For randomized X, ANCOVA has more statistical power. If there is a difference in the results between the two models, check for baseline differences. Difference score and residualized change measures are useful because they transform two measures to one measure, i.e., the difference score combines the time mediator and time mediator so all the models can be applied. Meaning of mediation with the different models differ: Correlated change scores, correlated adjusted time scores. Note issue of Lord s paradox for the M to Y relation because M is not randomized. More options with more waves of data. More complexity too though. 37
Models for Three or More Waves Autoregressive Models Latent Growth Curve Models (LGM) Latent Change Score Models (LCS) Autoregressive and Latent Growth Curve Models (ALT) Differential Equation Models (DEM) Others: Area Under the Curve, Multilevel Structural Equation Models, Survival Analysis, fractional polynomial (Royston & Altman, 994), spline (Borghi et al., 006), functional data analysis (Ramsay, 005) 38
39 Χ 3 Χ Χ b b 3 3 3 3,,,, b b X b X X b X x Autoregressive (Jöreskog, 974)
Autoregressive Model with Time-Ordered Mediation, Cole & Maxwell, (003); MacKinnon (994, 008) X a c c s s 3 b b s3 s3 3 Note: Residuals at the same time are correlated 40
Autoregressive Models Many mediated effects. Standard error of the sum of (or any function) the indirect effects can be derived with the multivariate delta method or resampling methods. Model does not allow for random effects for individual change and does not include modeling of means. Change in growth of means is an important aspect of longitudinal data. 4
4 Latent Growth Model (LGM) I S X X X 3 ε ε ε 3 Meredith & Tisak (990) Means S I X S I X S I X IS S I,,,,,, 0 3 3 3
Latent Growth Curve (Model Cheong et al., 003) Χ c ' b a i y s y Υ Υ Υ 3 i m s m Μ Μ Μ 3 43
Latent Growth Models (LGM) LGM models change over time by estimating an intercept and slope for change in variables. These models can be used to investigate mediation by estimating change over time for the mediator and change over time for the outcome. The relation between the change in the mediator and change in the outcome represents the b path (Cheong et al. 003). The causal direction of correlated change is ambiguous. Another LGM estimates change in the mediator at earlier time points and relates to change in the outcome at later time points providing more evidence for temporal precedence of the mediator. 44
D.P. MacKinnon Latent Change Score (LCS) McArdle (00) ε ε 3 β D β D X X X 3 Means X B D X B D D X X D X X x,,,,, 3 3 3
Latent Change Score Mediation Model X M M a c c a b b M M M M 3 M M 3 Y Y b 3 Y3 Y 3 Y Y Y Y 3 D.P. MacKinnon
Latent Change Score Models (LCS) LCS parameterizes models so that change between adjacent waves is analyzed. Really a special case of latent growth curve modeling but with growth between adjacent waves. More complicated change over time can be made by picking different coefficients and second order factors. Second order factors can be change in change or second derivatives. Promising model not often used for mediation analysis. Promising in that often theory predicts effects at different change points. 47
Longitudinal models for a steroid prevention project (ATLAS) Adolescents Teaching and Learning to Avoid Steroids (ATLAS). Randomized high school football teams in Oregon and Washington to receive the steroid prevention program or an information only group. Just individual data here. Measured the same persons over repeated occasions. Here we will look at preliminary models for four repeated measures. The dependent variable is intentions to use steroids. Linn Goldberg (OHSU) principal investigator. For more on the program see Goldberg et al. (996) and for mediation see MacKinnon et al., (00). LGM Cheong et al., (003). Program delivered after baseline measurement. In general, timing of the mediators should be relatively quick for knowledge and beliefs measures. It may take longer for norms measures. Four waves of measurement for the models studied. 48
Analysis decisions LGM model, slope coded as 0 * where * indicates a free parameter. Note that there was a booster after the 3rd measurement. If the model was not identified, then loadings were 0.5 * 4.5 to represent the months from baseline. All LGM models had RMSEA lower than.04 (lowest.09). Autoregressive model. Tested for stationarity in the a and b paths. Stationarity observed more often for b paths and less often for a paths, as expected. All RMSEAs lower than.088 (lowest was.068). 49
LGM and Autoregressive mediation effects Mediator LGM ab(se) z Autoregressive ab(se) z Knowledge -.8(.) -4.88 -.08(.0) -4.90 Coach Tol -.(.05) -.7 -.0(.0) -3.4 Team as info -.(.06) -3.4 -.04(.0) -3.30 Peer as info -.(.05) -.43 -.04(.0) -.30 Reasons not use -.(.04) -.98 -.0(.0) -3.0 Normative bel -.(.07) -.64 -.00(.00) -0.4 50
Modern Causal Inference for Longitudinal Data Time varying effects lead to complexities when interpreting causal effects. Changes at earlier waves could cause subsequent variables that complicate model interpretation. For example, the relation of M to Y at each wave can lead to complications. Should earlier measures of M or Y be included in the prediction of later waves of data? Problem of collider bias. 5
Modern Causal Inference for Longitudinal Data Specify longitudinal models in a potential outcome and causal framework. G-computation ~ standardization where predictions are made for factual and counterfactual data. G-estimation to obtain a parameter value that removes effect of interest. Marginal Structural Model with inverse probability weighting to weight observations by amount of confounding. (Robins 986, 989, 999 and colleagues) 5
Marginal Structural Model IPW Example Obtain predictors of M that will render M unaffected by confounders L. Note that this assumes that all confounders are in the statistical model-the no unmeasured confounders assumption. The method uses inverse probability weighting to reweight participants according to exposure to treatment and values of confounders. (Coffman, 0; Robins, Hernan, & Brumbeck, 000). 53
ATLAS IPW Analysis Confounders may explain the relation of M to Y in these data. It would be useful to apply a method that adjusts for this discrepancy. A large number of measures were used in the propensity model. The effects of the intervention were not large so it is possible that these effects would be attenuated after adjustment. Team social norm mediator and nutrition behavior outcome. 54
Predictors of M Ethnicity, grade in school, school has a gym Baseline grade point average, peers as an information source, body image, intent to use steroids, communication skills, perceived susceptibility to steroid use, knowledge of anabolic steroid effects, positive aspects of steroids, self esteem, depression, win-at-all costs attitude, perceived severity of steroid use, coach tolerance for steroids, attitudes about steroid users. 55
ATLAS Data No Adjustment Standard 95% Confidence Parameter Estimate Error Limits Z Pr > Z Intercept 0.008 0.0389-0.068 0.0844 0. 0.835 X 0.46 0.0668 0.954 0.557 6.38 <.000 M 0.75 0.035 0.098 0.33 5.45 <.000 56
ATLAS IPW Results Standard 95% Confidence Parameter Estimate Error Limits Z Pr > Z Intercept -0.0035 0.0395-0.080 0.0739-0.09 0.990 X 0.437 0.068 0.903 0.557 6. <.000 M 0.87 0.033 0.0655 0.90 3.99 <.000 Weights ranged from. to about 7 IPW Confidence Limits UCL=.3 LCL=.034 Usual Confidence Limiits UCL=.35 LCL=.059 57
More Complexities of Longitudinal Modeling Does the measure have the same meaning at each wave? X, M, and Y at an earlier developmental stage may differ from X, M, and Y at a later stage. Relations between change in X on M and change in M on Y may differ. Timing of measurement should match theoretical change. Transitions are important, e.g., home to elementary school, elementary to high school, high school to workforce/college. Sleeper Effects. Measure at appropriate times to capture effects. 58
Types of change over time Cumulative: There may be cumulative effects such that more M yields more Y. Threshold: Once a mediator gets to a certain level, then it will change Y. Once a level of a mediator is reached, the individual changes to a new level, e.g., learning a concept in algebra. Saltation and stasis: Change occurs rapidly after no previous change, e.g., human growth (Lampl et al., 99) The types of changes may differ over time. And change for X to M to may differ for M to Y nonlinear relations. 59
Person-oriented Methods Focus on patterns of responses by individuals. Classifies individuals based on their responses, such as whether their responses are consistent with mediation or not. Configural Frequency Analysis, Latent Class Analysis, Markov Models, are examples Complement for variable-oriented methods, may provide different information. Combination of both approaches for mediation in mixture models is an active area of research. 60
Summary Mediation is a Longitudinal model. Many alternative models that provide different information about mediation effects. Often requires an iterative process to model longitudinal data. Lots of methods work needs to be done to understand these models: causal inference, model equivalence, validity of assumptions. Need examples of applying the models to real data. 6
Thank You References available by sending me an e-mail at David.MacKinnon@asu.edu Most topics are covered in MacKinnon (008). Introduction to Statistical Mediation Analysis, Erlbaum; Mahwah, NJ. e.g., Causal Inference circa 008 Chapter 3, Longitudinal Mediation models in Chapter 8, and background for mediation in Chapters and. See website for Research In Prevention Laboratory http://www.public.asu.edu/~davidpm/ 6
Hypothesized Effects of Ghent Mediation Analysis Presentation Mediation Awareness Ghent Presentation Knowledge of Longitudinal Models Models for different number of waves # Accurate Mediation Analyses Importance of Causal Inference Methods 63
General Autoregressive Model X X X 3 3 3 Note: All residuals are correlated at the same time are correlated 64
Autoregressive Model with Time-Ordered Mediation, Cole & Maxwell (003) X s X s X 3 a a c c s s 3 b b s3 s3 3 Note: All residuals at the same time are correlated 65
Latent Growth Curve Model Χ Χ Χ 3 c b i y s y i x s x a Υ Υ Υ 3 i m s m Μ Μ Μ 3 66
PerIn PerIn PerIn3 PerIn4 PerIn5 PerIn6 0 Program a b c 3 4 0 Nutrit Nutrit Nutrit3 Nutrit4 Nutrit5 Nutrit6
PerIn PerIn PerIn3 0 Program a b c 3 4 0 Nutrit3 Nutrit4 Nutrit5 Nutrit6 68
Latent Change Score Mediation Model X X 3 X X X X 3 x x Y3 Y 3 //03 Y Y 3 Y a a c c 3 c Y Y Y b b a 3 b 3 M M M M M 3 D.P. MacKinnon M M M 3