Mediator analysis within field trials Laura Stapleton UMBC

Session outline Basic mediation model Comment on causality Tests of the hypothesized mediation effect Examples of mediation models for cluster randomized trials Brief preview of advanced issues and software

Basic mediation model Outcome Y Mediator M Treatment T ab c’ Outcome Y Treatment T c total effect = indirect effect + direct effect c = ab + c’

Causality concerns Because the mediator is not manipulated, causal interpretations are limited Possible misspecification Outcome Y Mediator M Treatment T ab Ok!  In future research, manipulate mediator For now, assume or hypothesize that M causes Y Z

Tests of the hypothesized mediation effect The estimate of the indirect effect, ab, is based on the sample To infer that a non-zero ab exists in the population, a test of the significance of ab is needed Several approaches have been suggested and differ in their ability to “see” a true effect (power) Outcome Y Mediator M Treatment T ab c’

Tests of the hypothesized mediation effect z test of ab (with normal theory confidence interval) Asymmetric confidence interval (Empirical M or distribution of the product) Other tests not considered today:  Causal steps approach (Baron & Kenny)  Test of joint significance  Bootstrap resampling

z test of ab product Calculate z = Compare z test value to critical values on the normal distribution Can also calculate confidence interval around ab CI = One of the least powerful approaches Problem is that the ab product is not normally distributed, so critical values are inappropriate se ab =

I simulated 1,000 estimates of a and 1,000 estimates of b where mean = 0 and SD=1 Distribution of path aDistribution of path b Distribution of product of axb

Empirical M-test (asymmetric CI) Determines empirical distribution of z of the ab product (not assuming normality) Distribution is leptokurtic and symmetric when αβ =0, but is skewed if αβ > 0 or αβ < 0 Given a, b, and their SEs, PRODCLIN determines the distribution of ab and critical values Confidence interval limits: If CI does not include zero, then “significant”

Mediation models for cluster randomized trials Extend basic model to situations when treatment is administered at group level Model depends on whether mediator is measured at group or individual level  Upper-level mediation (2→2→1 Design)  Cross-level mediation (2→1→1 Design)  Cross-level and upper-level mediation (2→1 / 2→1 Design)

Measured variable partitioning First, consider that any variable may be partitioned into individual level components and cluster level components

Mediation model possibilities

Data Example Context Cluster randomized trial (hierarchical design) 14 pre-schools: ½ treatment, ½ control Socio-emotional curriculum Outcome is child behavior Possible mediators: teacher attitude, child socio- emotional knowledge Sample data are on posted handout (n=84) Analyses with SPSS (HLM and SAS available)

Total effect of treatment Before we examine mediation, let’s examine the total effect of treatment on the outcome… Outcome CLUSTER Outcome INDIVIDUAL Treatment CLUSTER Treatment γ’ 01

Total effect of treatment: Results Given that SD of Y is 4.381, effect size of treatment is large:.97. MIXED Y WITH T /FIXED= T | SSTYPE(3) /RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC) /METHOD = REML /CRITERIA = CIN(95) MXITER(100) MXSTEP(5) SCORING(1) SINGULAR( ) HCONVERGE (0,ABSOLUTE) LCONVERGE(0,ABSOLUTE) PCONVERGE( ,ABSOLUTE) /PRINT = CPS G SOLUTION TESTCOV.

Upper-level mediation model (2→2→1) Outcome CLUSTER Outcome INDIVIDUAL Mediator CLUSTER Mediator Treatment CLUSTER Treatment γ 01 γ’ 02 γ’ 01

Upper-level mediation model: Results To estimate the a path, I ran an OLS regression is SPSS using a file from the 14 schools The estimate is.714 with a standard error of.628

Upper-level mediation model: Results To estimate the b path, I ran a mixed model The estimate is.795 with a SE of.656 MIXED Y WITH T M1 /FIXED= T M1 | SSTYPE(3) /RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC) >

Upper-level mediation model: Results Outcome CLUSTER Outcome INDIVIDUAL Mediator CLUSTER Mediator Treatment CLUSTER Treatment  Direct effect =  Indirect effect = (.714)(.795) =.568  Total effect = DE + IE = = 4.239

Upper-level mediation model: Results Significance test of the indirect effect PRODCLIN

Cross-level mediation model (2→1→1) Outcome CLUSTER Outcome INDIVIDUAL Mediator INDIVIDUAL Treatment CLUSTER Treatment γ’ 01 γ’ 10 Mediator CLUSTER Mediator INDIVIDUAL Treatment CLUSTER Treatment γ 01 Model AModel B

Cross-level mediation model: Results To estimate the a path: The estimate is with SE of MIXED M2_GrandC WITH T /FIXED= T | SSTYPE(3) /RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC) >

Cross-level mediation model: Results To estimate the b path: The estimate is.592 with a SE of.143 MIXED Y WITH M2_GrandC T /FIXED= M2_GrandC T | SSTYPE(3) /RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC) >

Cross-level mediation model: Results Outcome CLUSTER Outcome INDIVIDUAL Mediator INDIVIDUAL Treatment CLUSTER Treatment Mediator CLUSTER Mediator INDIVIDUAL Treatment CLUSTER Treatment Model AModel B  Direct effect =  Indirect effect = (2.643)(.592) =  Total effect = = 4.239

Cross-level mediation model: Results Test of the indirect effect

Cross-level and upper-level mediation model (2→1 / 2→1) Mediator CLUSTER Mediator INDIVIDUAL Treatment CLUSTER Treatment γ 01 Model A Model B Outcome INDIVIDUAL M Mediator INDIVIDUAL Treatment γ’ 10 Outcome CLUSTER Mediator CLUSTER Treatment CLUSTER γ’ 01 γ’ 02 Ave. M

Cross-level and upper-level mediation model: Results Path a is the same as in the prior model. For the b paths: MIXED Y WITH M2_AVE M2_GrandC T /FIXED= M2_AVE M2_GrandC T | SSTYPE(3) /RANDOM = INTERCEPT | SUBJECT(J) COVTYPE(VC) >

Cross-level and upper-level mediation model (2→1 / 2→1) Outcome INDIVIDUAL M Mediator INDIVIDUAL GRAND_C Treatment.600 Outcome CLUSTER Mediator CLUSTER Treatment CLUSTER Ave. M Mediator CLUSTER Mediator INDIVIDUAL Treatment CLUSTER Treatment Note that there are now TWO mediation paths:  ab individual = (2.643)(.600) =  ab cluster = (2.643)(-.041) = -.109

Test of the indirect effect at the individual level: Cross-level and upper-level mediation model (2→1 / 2→1)

Test of the indirect effect at the cluster level: Cross-level and upper-level mediation model (2→1 / 2→1)

Outcome INDIVIDUAL M Mediator INDIVIDUAL GROUP_C Treatment.600 Outcome CLUSTER Mediator CLUSTER Treatment CLUSTER Ave. M Mediator CLUSTER Mediator INDIVIDUAL Treatment CLUSTER Treatment  ab cluster = (2.643)(.559) = with GROUP_C  ab cluster = (2.643)(-.041) = with GRAND_C The level-2 effect of the mediator differs with group- versus grand-mean centering:

Brief preview of advanced issues Multisite / randomized blocks (1→1 →1) Testing mediation in 3-level models Including multiple mediators Examining moderated mediation Dichotomous or polytomous outcomes Measurement error in mediation models Bayesian estimation of indirect effects

Notes on software SPSS HLM SAS (PROC MIXED) MLwiN Mplus