Introduction Types of mediation Causal steps approach SPSS example Miscellaneous topics Proportion mediated Sobel test. Mediation analysis

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1 Mediation analysis Seminar General Statistics Matthijs J Warrens GION

2 Multiple regression analysis 1 dependent variable (INT) Several independent variables 1, 2, 3 (INT) (predictors) Research question Can be predicted from 1, 2 and/or 3? All predictors have the same status

3 Path analysis General technique Variables arranged into a path structure Variables can have different roles: dependent/independent/both Structural Equation Modeling (SEM) without latent variables A path structure allows us to study other types of questions Mediation analysis Simple mediation is a special case of path analysis Series of regression analyses

4 Simple mediation Three variables = dependent variable = independent variable M = mediator M Research question Is the relationship between and mediated by M? M = mediator if it accounts for the relationship between and Mediation analysis important since this type of research question occurs quite often

5 Illustrative example Research question: Is the relationship between age and blood pressure mediated by weight? Variables = (systolic) blood pressure; = age; M = weight Weight (M) Age () Blood pressure () People tend to have higher blood pressure as they get older (direct effect) People tend to gain weight as they get older Heavy people tend to have high blood pressure (indirect effect)

6 Full mediation We distinguish three cases Full mediation Partial mediation No mediation Full mediation Indirect effect: influence of on goes via mediator M No direct effect of on M

7 Partial or no mediation Partial mediation Indirect effect: part of influence of on goes via mediator M Direct effect of on M No mediation No indirect effect Only a direct effect of on M M

8 Causal steps approach Baron RM, Kenny DA (1986) The moderator-mediator variable distinction in social psychological research. Journal of Personality and Social Psychology 51: Mediation requires causal steps 2 necessary steps Can be tested by a series of regression analyses Unmediated model: Mediated model: c a M c b Two models 4 regression coefficients or betas: a, b, c and c Can be estimated with 3 regression analyses

9 Step 1 Unmediated model: Mediated model: c a M c b Is related to? Logic: without a relationship between and there is nothing to mediate (Not always a necessary step: direct and indirect effect may have opposite signs and cancel each other out) Regression analysis Predict from Provides an estimate of coefficient c Tests if c 0

10 Step 2 Unmediated model: Mediated model: c a M c b Is related to M? Logic: if does not cause M, there is no mediation Regression analysis Predict M from Provides an estimate of coefficient a Tests if a 0

11 Step 3 Unmediated model: Mediated model: c a M c b Is M related to, if we control for? Logic: if M does not cause, there is no mediation Regression analysis Predict from M, with as other predictor Provides an estimate of coefficient b Tests if b 0 must be in regression (to rule out M and both caused by ) Steps 1 to 3 OK mediation

12 Step 4 Unmediated model: Mediated model: c a M c b Is related to, if we control for M? Logic: if is related, there is partial mediation If is no longer related, there is full mediation Regression analysis Predict from, with M as other predictor Provides an estimate of coefficient c Tests if c 0

13 Illustrative example Research question: Is the relationship between age and blood pressure mediated by weight? Age () a Weight (M) c BP () Age () c b BP () Three regression analyses in SPSS To estimate a, b, c and c We can express effects in either regression or beta coefficients We will use beta coefficients first

14 Regression analysis 1 Age () a Weight (M).500 BP () Age () c b BP () Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Dependent Variable: Blood pressure Is related to? es. t = 6.275, p <.001 Coefficient c =.500 is significant: step 1 OK

15 Regression analysis 2 Age () Weight (M).536 b.500 BP () Age () c BP () Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Dependent Variable: Weight Is related to M? es. t = 6.893, p <.001 Coefficient a =.536 is significant: step 2 OK

16 Regression analysis 3 Age () Weight (M) BP () Age () c BP () Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Weight Dependent Variable: Blood pressure Is M related to, if we control for? es. t = 4.812, p <.001 Coefficient b =.416 is significant: step 3 OK Steps 2 and 3 OK there is mediation

17 Regression analysis 3 Age ().500 BP () Weight (M) Age () BP () Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Weight Dependent Variable: Blood pressure Steps 1 to 3 OK Is related to, if we control for M? es. t = 3.196, p =.002 Coefficient c =.277 is significant Beta decreases from.500 to.277 partial mediation If c not significant full mediation If beta would have increased suppression

18 3 MRAs We require 3 MRAs to check the causal steps Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Dependent Variable: Blood pressure Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Dependent Variable: Weight Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Weight Dependent Variable: Blood pressure

19 Correlation table + 1 MRA If we are only interested in the beta coefficients The correlation table and 1 MRA also suffice Since in simple regression the correlation is the beta coefficient Correlations BP Age Weight BP Pearson Correlation Sig. (2-tailed) N Age Pearson Correlation Sig. (2-tailed) N Weight Pearson Correlation Sig. (2-tailed) N Coefficients Unstandardized Standardized Coefficients Coefficients Model B Std Error Beta t Sig. 1 (Constant) Age Weight Dependent Variable: Blood pressure

20 Correlation and causality M Mediated model is a directed graph (arrows) Arrows suggest a causal model Correlation does not imply causation Correlational data cannot prove a causal model We can present data that support the model But there are almost always alternative models Causal steps approach can be used to reject model If we have data that reject the model, then the model is highly unlikely

21 2 or more mediators M 1 M 2 Mediated model can be extended to 2 or more mediators Mediated model with 2 mediators 1 direct effect 2 indirect effects 4 regression analyses

22 When is nominal M If is nominal (instead of interval) : ANOVA M: ANOVA, M : ANCOVA

23 Effect sizes Causal steps approach for mediation Based on 3 regression analyses Standard effect size in MRA is explained variance (VAF) R squared, squared semi-partial correlations VAF is quite a complicated concept in mediation analysis See for details De Heus P. (2012) R squared effect size measures and overlap between direct and indirect effect in mediation analysis. Behavior Research Methods 44: To compare the strength of the direct and indirect effect use, e.g., the proportion mediated

24 Proportion mediated c a M c b Total effect (= c) can be split into direct effect = c indirect effect = ab such that total effect = direct effect + indirect effect Thus, c = c + ab or c c = ab Proportion mediated P med = indirect total = ab c = c c c and P dir = direct total = c c

25 Proportion mediated with betas Age ().500 BP () Weight (M) Age () BP () Total effect = c =.500 Direct effect = c =.277 Indirect effect = ab = (.536)(.416) =.223 Indirect effect = c c = =.223 P med = indirect total = =.446 and P dir = 1 P med =.554

26 Proportion mediated with regression coefficients Age ().822 (.131) BP () Age () Weight (M).891 (.129).412 (.086).455 (.142) BP () Total effect = c =.822 Direct effect = c =.455 Indirect effect = ab = (.891)(.412) =.367 Indirect effect = c c = =.367 P med = indirect total = =.446 and P dir = 1 P med =.554

27 Properties proportion mediated Proportion mediated Does not refer to variance explained Not an effect size Its size depends on size of the total effect If total effect small, proportion mediated is part of something small (which is small itself) Can have values below 0 and above 1 e.g., in the case of suppression

28 2 or more mediators c a a 2 M 1 c M 2 b 1 1 b 2 Total effect (= c) can be split into direct effect = c first indirect effect = a 1 b 1 second indirect effect = a 2 b 2 such that total effect = direct effect + indirect effect 1 + indirect effect 2 or, c = c + a 1 b 1 + a 2 b 2

29 Sobel test a M c b Indirect effect ab Consists of two parts a and b Coefficients are tested separately in regression analyses Effect has not been tested directly Sobel test Test of indirect effect ab Several versions exist Not in SPSS; calculate by hand

30 Sobel test (Aroian version) a M c b Each Sobel test has a form z = ab SE ab Versions differ in how SE ab is estimated Best way to obtain SE ab is to use bootstrapping Aroian ab z = b 2 SE 2 a + a 2 SE 2 b + SE 2 ase 2 b where a and b are regression coefficients and SE a and SE b are the associated standard errors

31 Sobel test (Aroian version) Example Age ().822 (.131) BP () Age () Weight (M).891 (.129).412 (.086).455 (.142) BP () ab z = b 2 SE 2 a + a 2 SE 2 b + SE 2 ase 2 b = (.891)(.412) (.412)2 (.129) 2 + (.891) 2 (.086) 2 + (.129) 2 (.086) 2 = 3.91 We have p <.05 if z > 1.96 indirect effect is significant

32 Conclusion Simple mediation Baron and Kenny: series of regression analyses Sobel test Proportion mediated More complicated mediation models Directed acyclic graphs (DAGs) Structural equation modeling (SEM) Iacobucci D (2008) Mediation Analysis. Quantitative Applications in the Social Sciences 156, Sage.