Plot data, descriptives, etc. Check for outliers A. Nayena Blankson, Ph.D. Spelman College University of Southern California GC3 Lecture Series September 6, 2013 Treat missing i data Listwise Pairwise Imputation 2 When the effect of one variable (X) on another variable (Y) is through the effect of a third variable (M). Understanding processes the how Indirect effect Sometimes used interchangeably with mediation. 4 X c Y Total Effect Model Total Effect= c= + ab Shyness Executive Functioning Vocabulary X a M b Y Mediation Model Indirect Effect= a x b Direct Effect= Mediation question: Does executive functioning mediate the relation between shyness and vocabulary? 5 6 1
Four conditions 1. There is a relation between X and Y. (path c) 2. There is a relation between X and the mediator. (path a) 3. There eeis sa relation eato between ee the mediator edato and Y when both X and the mediator are predictors. (path b) 4. The X Y relation (path c) is reduced when the mediator is included (path ). (compare c to ) Complete mediation vs. partial mediation 1. Y= i 1 + cx + r 1 Condition 1 2. M= i 2 +ax + r 2 Condition 2 Three equations 3. Y= i 3 + X + bm + r 3 Condition 3 Condition 4: Compare c to. i 1, i 2, and i 3 are intercepts; r 1, r 2, and r 3 are residuals 7 8 Shyness ShyS M= 50, SD = 10 Vocabulary VocabS M= 500, SD = 100 Executive Functioning EFS M= 100, SD = 15 Home environment HomeS M= 50, SD = 10 2 covariates Covar1, Covar2 N= 200 REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS BCOV R ANOVA CHANGE ZPP /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT VocabS /METHOD=ENTER Covar1 Covar2 ShyS /METHOD=ENTER EFS. 9 10 REGRESSION /DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE /STATISTICS COEFF OUTS BCOV R ANOVA CHANGE ZPP /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT EFS /METHOD=ENTER Covar1 Covar2 ShyS. 11 12 2
1. Test the relation between X and Y. path c = -3.57, p<.05 2. Test the relation between X and mediator. path a = -.63, p<.05 3. Test the relation between mediator and Y with X in the model path b = 5.44, p<.05 path = -.13, ns 4. Compare path c and 13 14 X -3.57, p<.05 Y Step 1 is not required. There is not a joint test of the conditions. X M Y -.63, p<.05 5.44, p<.05 Does not provide a direct estimate of the size of the indirect effect, nor the significance of the indirect effect. -.13, ns 15 16 A statistic for a x b is derived by dividing the effect by its standard error. Significance of the test indicates significant mediation Low power A resampling strategy where k samples (usually 1000 or greater) of N units are drawn, with replacement from the original sample of N units. Generates a reference distribution for significance testing and confidence interval estimation. 17 18 3
Count The indirect effect is calculated for each of the k bootstrap samples. The average of the k indirect effects is taken as the estimated indirect effect. A sampling distribution of the indirect effect is obtained, and confidence intervals can be derived based on this distribution. Produces more robust estimates 300 200 100 0-2.00-1.00 0.00 Bootstrap Sampling Distribution of Indirect Effect 19 20 SPSS macros/scripts (PROCESS) can be downloaded from Hayes website Other statistical programs can also be used Eg E.g., Mplus, SAS process vars = ShyS EFS VocabS Covar1 Covar2 /y=vocabs/x=shys /m=efs/total=1/model=4/boot=5000/effsize=1/. 21 22 Model = 4 Y = VocabS X = ShyS M = EFS Statistical Controls: CONTROL= Covar1 Covar2 Sample size 200 Outcome: EFS Model Summary R R-sq F df1 df2 p.47.22 18.56 3 196.00 Model coeff se t p constant 133.36 5.29 25.19.00 ShyS -.63.09-6.68.00 (Path a) Covar1.32.17 1.86.06 Covar2 -.48.17-2.79.01 23 24 4
Outcome: VocabS Model Summary R R-sq F df1 df2 p.81.66 93.35 4 195.00 Model coeff se t p constant -54.92 48.34-1.14.26 EFS 5.44.32 17.16.00 (Path b) ShyS -.13.47 -.28.78 (Path ) Covar1.13.76.17.87 Covar2 1.60.78 2.05.04 ********TOTAL EFFECT MODEL **************** Outcome: VocabS Model Summary R R-sq F df1 df2 p.37.14 10.52 3 196.00 Model coeff se t p constant 670.32 37.12 18.06.00 ShyS -3.57.66-5.38.00 (Path c) Covar1 1.84 1.19 1.55.12 Covar2-1.02 1.21 -.84.40 25 26 *********** TOTAL, DIRECT, AND INDIRECT EFFECTS ************ Total effect of X on Y (path c) Effect SE t p -3.5717.6639-5.3802.0000 Direct effect of X on Y (path ) Effect SE t p -.1310.4654 -.2815.7786 Indirect effect of X on Y Effect Boot SE BootLLCI BootULCI EFS -3.4407.5386-4.4892-2.3717 Shyness -.63 Executive Functioning -.13 5.44 Indirect effect= -.63 x 5.44 = -3.44 95% bias corrected CI = -4.49, -2.37 Vocabulary 27 28 Multiple time points and experimental designs Allows more stringent test of mediation Categorical mediation Mediator or dependent variable is categorical Macro by Nathaniel Herr (UCLA) Multiple mediators Can be accommodated by the PROCESS macro In other programs, you would calculate the indirect effect using the product rule and determine the significance Multilevel mediation Dr. Page-Gould 29 30 5
When the strength or direction of an association between two variables is dependent on another variable. Moderation tells us in what contexts and under what conditions a relationship holds. 32 Home X W Y Executive Functioning Vocabulary Does home environmental stimulation moderate the association between executive functioning and vocabulary? 33 34 Not necessary but helpful for interpretation E.g., age No centering needed for categorical moderators Center or not Create product terms Regression analysis Step 1(Block 1): Covariates and IV Step 2: Moderator Step 3: Interaction term is this term significant? Test simple slopes 35 36 6
process vars = EFS VocabS HomeS Covar1 Covar2/y=VocabS/x=EFS/m=HomeS /model=1/center=1/plot=1/quantile=1/. Model = 1 Y = VocabS X = EFS M = HomeS Statistical Controls: CONTROL= Covar1 Covar2 Sample size 200 37 38 Moderation Output Outcome: VocabS Model Summary R R-sq F df1 df2 p.998.996 8531.67 5 194.00 39 Model coeff se t p constant 474.01 1.34 354.48.00 HomeS 4.83.05 99.43.00 EFS 5.35.03 162.42.00 int_1.20.00 60.38.00 Covar1 1.31.09 14.92.00 Covar2 1.16.09 12.82.00 Interactions: int_1 EFS X HomeS R square increase due to interaction(s): R2 chng F df1 df2 p int_1.09 3646.10 1 194.00 40 Conditional effect of X on Y at values of the moderator(s) == Simple slopes HomeS Effect se t p -14.45 2.50.06 43.26.00-7.86 3.80.04 90.74.00.20 5.39.03 163.61.00 8.10 6.95.04 164.87.00 13.18 7.95.05 147.01.00 Values for quantitative moderators are 10th, 25th, 50th, 75th, and 90th percentiles Data for visualizing conditional effect of X of Y EFS HomeS yhat -19.6928-14.4525 380.0990-12.0184-14.4525 399.2749-1.4327-14.4525 425.7253............................................. 41 42 7
To get confidence interval at specific levels of the moderator: process vars = EFS VocabS HomeS Covar1 Covar2/y=VocabS/x=EFS/m=HomeS /model=1/center=1/plot=1/quantile=1 /mmodval = -10. Output t Conditional effect of X on Y at values of the moderator(s) HomeS Effect se t p -10.00 3.38.05 72.56.00 43 44 Calculating and plotting simple slopes Excel ModGraph Multiple moderators If 3-way interaction is not significant, remove from model Longitudinal data Moderation of change in a variable over time growth curve analyses including interaction terms 45 Moderated mediation When a mediation effect differs across levels of another variable. The process underlying the relation between two variables is different in different contexts/conditions. Conditional indirect effect Specifies what the mediation effect is for different levels of the moderator. Path a moderated -- an interaction between X and moderator in predicting M Path b moderated -- an interaction between M and moderator in predicting Y Both paths a and b moderated by different variables Paths a and b moderated by the same variable Paths a, b, and c moderated by the same variable Hayes (2013) presents over 70 different models! 47 48 8
An interaction between X and moderator in predicting M-- Path a moderated age W X M Y a b gender depression diabetes management a b Korbel et al. (2007) 49 50 An interaction between M and moderator in predicting Y - Path b moderated Home W X M Y a b Shyness a Executive Functioning b Vocabulary Blankson et al. (2011) 51 52 Paths a and b moderated by different variables Sensation seeking Exposure to media W Z X M Y a b Negative experiences around alcohol a Attention to media crime and accidents b Alcohol-related risk judgments Slater et al. (2007) 53 54 9
Paths a and b moderated by the same variable Race Race W W d e Attachment Social Anxiety Friendship Satisfaction X M Y a b Parade, Leerkes, & Blankson (2010) 55 56 Mediator variable model The prediction of the mediator variable from the independent variable. Dependent variable model The prediction of the outcome variable from the predictor and mediator variable. Relevant moderators and interaction terms are included in each regression. The coefficients obtained from the regressions are used to calculate estimates of the indirect effect. Significant interactions are probed similar to Aiken and West (1991). Bootstrapping is used to calculate confidence intervals. 57 58 Shyness Executive Functioning Home Vocabulary process vars = ShyS EFS VocabS HomeS Covar1 Covar2 /y=vocabs/x=shys /m=efs/v=homes/model=14/center=1 /quantile=1/boot=5000. Moderated mediation question: Does the relation between shyness and vocabulary through executive functioning differ for different home environments? model with path b moderated 59 60 10
**PROCESS Procedure for SPSS Beta Release 130612** Written by Andrew F. Hayes, Ph.D. http://www.afhayes.com *************************************************** Model = 14 Y = VocabS X = ShyS M = EFS V = HomeS Statistical Controls: CONTROL= Covar1 Covar2 Sample size 200 61 Moderated Mediation Output Outcome: EFS (Mediator variable model) Model Summary R R sq F df1 df2 p.47.22 18.56 3 196.00 Model coeff se t p constant 33.3559 5.2939 6.3008.0000 ShyS.6327.0947 6.6817.0000 Covar1.3152.1695 1.8595.0645 Covar2.4812.1728 2.7850.0059 62 Moderated Mediation Output Outcome: VocabS (Dependent variable model) Model Summary R R sq F df1 df2 p.9979.9958 7634.2097 6 193.0000 Model coeff se t p constant 483.9972 2.8641 168.9900.0000 EFS 5.2931.0353 150.1316.0000 ShyS.2022.0518 3.9041.0001 HomeS 4.8361.0469 103.0832.0000 int_1.1971.0032 62.4317.0000 Covar1 1.3398.0850 15.7556.0000 Covar2 1.1397.0874 13.0451.0000 63 Moderated Mediation Output ******************** DIRECT AND INDIRECT EFFECTS ************************* Direct effect of X on Y Effect SE t p.2022.0518 3.9041.0001 Conditional indirect effect(s) () of X on Y at values of the moderator(s) () Mediator HomeS Effect Boot SE BootLLCI BootULCI EFS 14.4525 1.5463.2322 1.9931 1.0790 EFS 7.8558 2.3690.3540 3.0425 1.6533 EFS.2005 3.3738.5045 4.3328 2.3510 EFS 8.0978 4.3587.6528 5.5985 3.0294 EFS 13.1792 4.9924.7483 6.4083 3.4689 Values for quantitative moderators are 10th, 25th, 50th, 75th, and 90th percentiles 64 Mplus code for Model 14 TITLE: Mod Med with covariates (Using centered data) CONTINUED DATA: FILE IS 'USC ModMed data.dat ; VARIABLE: NAMESShyness Covar1 Covar2 EF Home!Covariates; EFHome Vocab VocabS EFS HomeS ShyS Mod2 EFSHomeS VocabS EFSC ON Covar1 Covar2 ; HomeSC EFSC ShySC HomeEFSC; MODEL CONSTRAINT: USEVARIABLES ShySC EFSC HomeSC VocabS Covar1! Create indirect effects at levels of the moderator. Covar2 HomeEFSC;!Insert values for moderator below-- here we have 50 th!25th & 75th percentile; IndEff= a(b + e*w) ANALYSIS: TYPE = MEANSTRUCTURE; NEW(M3i_M);!Indirect effect at 50 th percentile. M3i_M = a*(b + (e*.2005)); BOOTSTRAP = 5000; NEW(M3i_1SB);!Indirect effect at 25th percentile; M3i_1SB = a*(b + (-7.8558*e)); MODEL: NEW(M3i1SA);!Indirect effect at 75th percentile;!mediator variable model. M3i1SA= a*(b + (8.0978*e)); EFSC ON ShySC* (a);! Path a. EFSC; OUTPUT:CINTERVAL(bcbootstrap);! bias corrected CI!Dependent variable model. VocabS ON EFSC* (b);! Path b. VocabS ON ShySC*; VocabS ON HomeSC; VocabS ON HomeEFSC* (e);!interaction. Mplus output Two-Tailed Estimate S.E. Est./S.E. P-Value EFSC ON SHYSC -0.633 0.094-6.749 0.000 COVAR1 0.315 0.160 1.969 0.049 COVAR2-0.481 0.185-2.597 0.009 VOCABS ON EFSC 5.293 0.035 149.134 0.000 SHYSC -0.202 0.050-4.029 0.000 HOMESC 4.836 0.048 0 100.507 0.000 000 HOMEEFSC 0.197 0.003 60.191 0.000 COVAR1 1.340 0.086 15.627 0.000 COVAR2 1.140 0.091 12.479 0.000 New/Additional Parameters M3I_M -3.374 0.500-6.742 0.000 M3I_1SB -2.369 0.351-6.748 0.000 M3I1SA -4.359 0.647-6.732 0.000 VocabS; [ShySC]; ShySC; [HomeSC]; HomeSC;! Intercept of X CONFIDENCE INTERVALS OF MODEL RESULTS Lower 2.5% Upper 2.5% New/Additional Parameters M3I_M -4.342-2.380 M3I_1SB -3.050-1.667 M3I1SA -5.606-3.072 65 66 11
The interaction between executive functioning and home environment was significant in the prediction of vocabulary (see Table 1). To further probe the moderated mediation effect, estimates of the indirect effect along with bias corrected bootstrap confidence intervals were calculated at the mean as well as at ±1 SD from the mean, based on 5000 bootstrapped t samples. Results are presented in Table 2. As can be seen, the indirect effect was more strongly negative in home environments that are more stimulating. Theory or prior research should be used to determine which model(s) to test. A significant simple mediation is not required to test for moderated mediation. More complex models: Multiple mediators, IVs, and DVs. Up to 10 mediators in parallel or 4 in sequence and multiple moderators using PROCESS 67 68 Aiken, L. S., & West, S. G. (1991). Multiple regression: Testing and interpreting interactions. Thousand Oaks: Sage. Baron, R. M., & Kenny, D. A. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182. MacKinnon, D. P., Krull, J., & Lockwood, C. M. (2002). Equivalence of the mediation, confounding, and suppression effect. Prevention Science, 1, 173-181. Preacher, K. J., & Hayes, A. F. (2004). SPSS and SAS procedures for estimating indirect effects in simple mediation models. Behavior Research Methods, Instruments, & Computers, 36, 717-731. Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40, 879-891. Edwards, J. R., & Lambert, L. S., (2007). Methods for integrating moderation and mediation: A general analytical framework using moderated path analysis. Psychological methods, 12, 1-22. Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis. New York: Guilford Press. Hayes, A. F. (2012). PROCESS: A versatile computational tool for observed variable mediation, moderation, and conditional process modeling [White paper]. Retrieved from http://www.afhayes.com/public/process2012.pdf Preacher, K. J., Rucker, D. D., & Hayes, A. F. (2007). Addressing moderated mediation hypotheses: Theory, methods, and prescriptions. Multivariate Behavioral Research, 42, 185-227. Rose, B. M., Holmbeck, G. N., Coakley, R. M., Franks, E. A. (2004). Mediator and moderator effects in developmental and behavioral pediatric research. Developmental and Behavioral Pediatrics, 25, 58-67. 69 70 Hayes- PROCESS macro http://www.afhayes.com/spss-sas-and-mplusmacros-and-code.html Graphing moderation http://www.jeremydawson.co.uk/slopes.htm http://www.victoria.ac.nz/psyc/paul-jose- t i / / files/modgraph/modgraph.php Graphing mediation http://www.victoria.ac.nz/psyc/paul-josefiles/medgraph/download.php Categorical mediation http://www.nrhpsych.com/mediation/logmed.html 71 12