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A. Nayena Blankson, Ph.D.Spelman College

University of Southern CaliforniaGC3 Lecture SeriesSeptember 6, 2013

Plot data, descriptives, etc.

Check for outliers

T i i d Treat missing data◦ Listwise◦ Pairwise◦ Imputation

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When the effect of one variable (X) on another variable (Y) is through the effect of a third variable (M).

Understanding processes the “how” Understanding processes—the how

Indirect effect◦ Sometimes used interchangeably with mediation.

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X YcTotal Effect Model

Total Effect= c= c’ + ab

X M Yba

c’

Mediation Model

Indirect Effect= a x b

Direct Effect= c’

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Shyness ExecutiveFunctioning

Vocabulary

Mediation question: Does executive functioning mediate the relation between shyness and vocabulary?

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Four conditions1. There is a relation between X and Y. (path c)2. There is a relation between X and the

mediator. (path a)3. There is a relation between the mediator

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 c’). (compare c to c’) Complete mediation vs. partial mediation

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Three equations1. Y= i1 + cX + r1 Condition 1

2. M= i2 +aX + r2d Condition 2

3. Y= i3 + c’X + bM + r3 Condition 3 Condition 4: Compare c to c’.

i1, i2, and i3 are intercepts; r1, r2, and r3 are residuals

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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

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REGRESSION/DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE/STATISTICS COEFF OUTS BCOV R ANOVA CHANGE ZPPC G/CRITERIA=PIN(.05) POUT(.10)/NOORIGIN /DEPENDENT VocabS/METHOD=ENTER Covar1 Covar2 ShyS/METHOD=ENTER EFS .

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REGRESSION/DESCRIPTIVES MEAN STDDEV CORR SIG N /MISSING LISTWISE/STATISTICS COEFF OUTS BCOV R ANOVA CHANGE ZPPCHANGE ZPP/CRITERIA=PIN(.05) POUT(.10)/NOORIGIN /DEPENDENT EFS /METHOD=ENTER Covar1 Covar2 ShyS.

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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 Y3. Test the relation between mediator and Y with X in the model◦ path b = 5.44, p<.05◦ path c’ = -.13, ns

4. Compare path c and c’

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X Y-3.57, p< .05

X M Y

-.13, ns

-.63, p< .05 5.44, p< .05

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Step 1 is not required.

There is not a joint test of the conditions.

D id di i f h i Does not provide a direct estimate of the size of the indirect effect, nor the significance of the indirect effect.

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A statistic for a x b is derived by dividing the effect by its standard error.

Significance of the test indicates significant mediationmediation

Low power

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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.

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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 A sampling distribution of the indirect effect is obtained, and confidence intervals can be derived based on this distribution.

Produces more robust estimates

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300

-2.00 -1.00 0.00

Bootstrap Sampling Distribution of Indirect Effect

0

100

200

Co

un

t

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SPSS macros/scripts (PROCESS) can be downloaded from Hayes’ website

Other statistical programs can also be used◦ E g Mplus SAS◦ E.g., Mplus, SAS

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process vars = ShyS EFS VocabS Covar1 Covar2 /y=VocabS/x=ShyS

/m=EFS/total=1/model=4/boot=5000/effsize=1/.

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Model = 4Y = VocabSX = ShySM = EFS

Statistical Controls:CONTROL= Covar1 Covar2

Sample size200

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Outcome: EFSModel SummaryR R-sq F df1 df2 p.47 .22 18.56 3 196 .00

Modelffcoeff se t p

constant 133.36 5.29 25.19 .00ShyS -.63 .09 -6.68 .00 (Path a)Covar1 .32 .17 1.86 .06Covar2 -.48 .17 -2.79 .01

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Outcome: VocabSModel SummaryR R-sq F df1 df2 p.81 .66 93.35 4 195 .00Model

ff tcoeff se t pconstant -54.92 48.34 -1.14 .26EFS 5.44 .32 17.16 .00 (Path b)ShyS -.13 .47 -.28 .78 (Path c’)Covar1 .13 .76 .17 .87Covar2 1.60 .78 2.05 .04

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********TOTAL EFFECT MODEL ****************Outcome: VocabSModel SummaryR R-sq F df1 df2 p.37 .14 10.52 3 196 .00M d lModel

coeff se t pconstant 670.32 37.12 18.06 .00ShyS -3.57 .66 -5.38 .00 (Path c)Covar1 1.84 1.19 1.55 .12Covar2 -1.02 1.21 -.84 .40

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*********** 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 c’)Effect SE t p-.1310 .4654 -.2815 .7786

Indirect effect of X on YEffect Boot SE BootLLCI BootULCI

EFS -3.4407 .5386 -4.4892 -2.3717

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Shyness Executive Functioning

Vocabulary5.44-.63

Indirect effect= -.63 x 5.44 = -3.44 95% bias corrected CI = -4.49, -2.37

-.13

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Multiple time points and experimental designs◦ Allows more stringent test of mediation

Categorical mediation Categorical mediation◦ Mediator or dependent variable is categorical◦ Macro by Nathaniel Herr (UCLA)

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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

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When the strength or direction of an association between two variables is dependent on another variable.

Moderation tells us in what contexts and Moderation tells us in what contexts and under what conditions a relationship holds.

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W

X Y

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Executive Functioning Vocabulary

Home

Does home environmental stimulation moderate the association between executive functioning and vocabulary?

Functioning Vocabulary

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Not necessary but helpful for interpretation◦ E.g., age

No centering needed for categorical moderatorsmoderators

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Center or not Create product terms Regression analysis◦ Step 1(Block 1): Covariates and IV◦ Step 2: Moderator◦ Step 2: Moderator◦ Step 3: Interaction term – is this term significant?

Test simple slopes

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process vars = EFS VocabS HomeS Covar1 Covar2/y=VocabS/x=EFS/m=HomeS

/model=1/center=1/plot=1/quantile=1/.

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Model = 1Y = VocabSX = EFSM = HomeS

Statistical Controls:CONTROL= Covar1 Covar2

Sample size200

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Outcome: VocabS

Model SummaryR R-sq F df1 df2 p998 996 8531 67 5 194 00.998 .996 8531.67 5 194 .00

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Moderation Output• 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        .0040

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 .008 10 6 95 04 164 87 008.10 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

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Data for visualizing conditional effect of X of YEFS HomeS yhat

-19.6928 -14.4525 380.0990-12.0184 -14.4525 399.2749

-1.4327 -14.4525 425.72531.4327 14.4525 425.7253... ... ...... ... ...... ... ...... ... ...... ... ...

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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.

O t tOutputConditional effect of X on Y at values of the moderator(s)

HomeS Effect se t p-10.00 3.38 .05 72.56 .00

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Calculating and plotting simple slopes◦ Excel◦ ModGraph

Multiple moderators◦ If 3-way interaction is not significant remove fromIf 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

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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.

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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 blvariables

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!

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An interaction between X and moderator in predicting M-- Path a moderated

W

X M Ya b

c’49

age

gender depression diabetesmanagement

a b

c’

Korbel et al. (2007)

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An interaction between M and moderator in predicting Y - Path b moderated

W

X M Ya b

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Home

Shyness Executive Functioning

Vocabulary

a b

c’

Blankson et al. (2011)

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Paths a and b moderated by different variables

ZW

X M Ya b

c’

ZW

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Sensation seeking

Exposure to media

Slater et al. (2007)

Negativeexperiences around alcohol

Attention to media crime and accidents

Alcohol-relatedriskjudgmentsa b

c’

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Paths a and b moderated by the same variable

W W

X M Ya b

c’

W

d

W

e

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Race Race

Attachment Social Anxiety

Friendship Satisfaction

Parade, Leerkes, & Blankson (2010)

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Mediator variable model ◦ The prediction of the mediator variable from the

independent variable.

Dependent variable modelp◦ The prediction of the outcome variable from the

predictor and mediator variable.

Relevant moderators and interaction terms are included in each regression.

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The coefficients obtained from the regressions are used to calculate estimates of the indirect effect.

Significant interactions are probed similar to Significant interactions are probed similar to Aiken and West (1991).

Bootstrapping is used to calculate confidence intervals.

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Shyness ExecutiveFunctioning

Vocabulary

Home

Moderated mediation question: Does the relation between shyness and vocabulary through executive functioning differ for different home environments?

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process vars = ShyS EFS VocabS HomeS Covar1 Covar2 /y=VocabS/x=ShyS

/m=EFS/v=HomeS/model=14/center=1/quantile=1/boot=5000.

model with path b moderated

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**PROCESS Procedure for SPSS Beta Release 130612**Written by Andrew F. Hayes, Ph.D. http://www.afhayes.com

***************************************************Model = 14

Y = VocabSX ShySX = ShySM = EFSV = HomeS

Statistical Controls:CONTROL= Covar1 Covar2Sample size

200

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Moderated Mediation OutputOutcome: 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       .005962

Moderated Mediation OutputOutcome: VocabS  (Dependent variable model)

Model Summary

R        R‐sq          F         df1        df2          p

.9979      .9958   7634.2097          6         193      .0000

Model

ff tcoeff          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)DATA: FILE IS 'USC ModMed data.dat’;VARIABLE: NAMESShyness Covar1 Covar2 EF Home EFHome Vocab VocabS EFS HomeS ShyS Mod2 EFSHomeS HomeSC EFSC ShySC HomeEFSC;

USEVARIABLES ShySC EFSC HomeSC VocabS Covar1 Covar2 HomeEFSC;

ANALYSIS: TYPE = MEANSTRUCTURE;

BOOTSTRAP = 5000;

MODEL:

CONTINUED

!Covariates;VocabS EFSC ON Covar1 Covar2 ;

MODEL CONSTRAINT:! Create indirect effects at levels of the moderator.!Insert values for moderator below-- here we have 50th

!25th & 75th percentile; IndEff= a(b + e*W)NEW(M3i_M); !Indirect effect at 50th percentile.M3i_M = a*(b + (e*.2005));NEW(M3i_1SB); !Indirect effect at 25th percentile;M3i_1SB = a*(b + (-7.8558*e));NEW(M3i1SA); !Indirect effect at 75th percentile;

!Mediator variable model.EFSC ON ShySC* (a); ! Path a.EFSC; !Dependent variable model.VocabS ON EFSC* (b); ! Path b.VocabS ON ShySC*; VocabS ON HomeSC;VocabS ON HomeEFSC* (e); !Interaction .

VocabS; [ShySC]; ! Intercept of XShySC; [HomeSC]; HomeSC;

NEW(M3i1SA); !Indirect effect at 75th percentile;M3i1SA= a*(b + (8.0978*e));

OUTPUT: CINTERVAL(bcbootstrap); ! bias corrected CI

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Mplus outputTwo-Tailed

Estimate S.E. Est./S.E. P-Value

EFSC ONSHYSC -0.633 0.094 -6.749 0.000COVAR1 0.315 0.160 1.969 0.049COVAR2 -0.481 0.185 -2.597 0.009

VOCABS ONEFSC 5.293 0.035 149.134 0.000SHYSC -0.202 0.050 -4.029 0.000HOMESC 4.836 0.048 100.507 0.000O SC 836 0 0 8 00 50 0 000HOMEEFSC 0.197 0.003 60.191 0.000COVAR1 1.340 0.086 15.627 0.000COVAR2 1.140 0.091 12.479 0.000

New/Additional ParametersM3I_M -3.374 0.500 -6.742 0.000M3I_1SB -2.369 0.351 -6.748 0.000M3I1SA -4.359 0.647 -6.732 0.000

CONFIDENCE INTERVALS OF MODEL RESULTSLower 2.5% Upper 2.5%

New/Additional ParametersM3I_M -4.342 -2.380 M3I_1SB -3.050 -1.667 M3I1SA -5.606 -3.072

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“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 b t t d l R lt t d i T bl 2 Abootstrapped 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”.

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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 mediationto 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

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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 suppressionEquivalence 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.

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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 frommodeling [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.

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Hayes- PROCESS macro◦ http://www.afhayes.com/spss-sas-and-mplus-

macros-and-code.html Graphing moderation◦ http://www.jeremydawson.co.uk/slopes.htm

htt // i t i / / l j◦ http://www.victoria.ac.nz/psyc/paul-jose-files/modgraph/modgraph.php

Graphing mediation◦ http://www.victoria.ac.nz/psyc/paul-jose-

files/medgraph/download.php Categorical mediation◦ http://www.nrhpsych.com/mediation/logmed.html

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