methods for integrating moderation and mediation: a...

Methods for IntegratingModeration and Mediation:

A General Analytical FrameworkUsing Moderated Path Analysis

Jeffrey R. EdwardsUniversity of North Carolina

Lisa S. LambertGeorgia State University

Research that ExaminesModeration and Mediation

• Many streams of research combine moderation and mediation. Some classic examples:§ Job characteristics affect critical psychological states,

which in turn affect motivation, and both of these links are moderated by growth needs strength.§ The interactive effect of expectancy, instrumentality,

and valence on performance is mediated by motivation.

• More generally, moderation and mediation are combined whenever causal paths are contingent on some variable or an interaction effect is transmitted through an intervening variable.

Approaches for CombiningModeration and Mediation

• Researchers have used various approaches for combining moderation and mediation. Among these, the most common are the following:– Piecemeal analyses of moderation and mediation

followed by joint interpretation.– Subgroup analyses in which the sample is split on the

moderator variable and mediation is tested separately in the resulting subgroups.

– Moderated causal steps analyses in which the steps are applied to the product of the independent variable and moderator variable.

Piecemeal Approach• Moderation is analyzed with ANOVA or moderated

regression, testing whether b3 ≠ 0:Y = b0 + b1X + b2Z + b3XZ + eY

• Mediation is tested with the causal steps approach:Y = b0 + b1X + eY

M = a0 + a1X + eM

Y = b0 + b1M + b2X + eY• Mediation is evaluated based on four conditions:

(1) b1 ≠ 0 in the first equation; (2) a1 ≠ 0 in the second equation; (3) b1 ≠ 0 in the third equation; and (4) b2 = 0 in the third equation.

Problems with the Piecemeal Approach• The piecemeal approach does not reveal which

paths connecting X, M, and Y vary as a function of Z.

• Most studies that use the piecemeal approach apply the causal steps procedure for testing mediation, thereby inheriting its limitations.§ Requiring a significant relationship between X and Y in

the first step can obscure mediated and direct effects with opposite signs.§ Although the causal steps procedure tests the paths

from X to M and from M to Y, it does not directly test the mediated (i.e., indirect) effect represented by the product of these two paths.

Subgroup Approach

• The sample is split into subgroups representing different levels of the moderator variable, based on a priori considerations or results from tests of moderation.

• Mediation is analyzed separately in each subgroup, typically with the causal steps procedure.

• Moderated mediation is inferred if support for mediation differs across subgroups.

Problems with the Subgroup Approach• Analyses conducted within each subgroup have

lower statistical power than would be available from the full sample.

• Subgroups are often formed by dichotomizing a continuous moderator variable.

• Differences in mediation across levels of the moderator variable are usually not tested.

• Within each subgroup, mediation is typically evaluated using the causal steps procedure, which introduces its attendant problems.

Moderated Causal Steps Approach

• The causal steps approach to testing mediation is adapted to test mediation of a product term. The relevant equations are as follows:Y = b0 + b1X + b2Z + b3XZ + eY

M = a0 + a1X + a2Z + a3XZ + eM

Y = b0 + b1M + b2X + b3Z +b4XZ + eY• Mediated moderation is evaluated based on four

conditions: (1) b3 ≠ 0 in the first equation; (2) a3 ≠0 in the second equation; (3) b1 ≠ 0 in the third equation; and (4) b4 = 0 in the third equation.

Problems with the Moderated Causal Steps Approach

• Problems with the basic causal steps approach apply to the moderated causal steps approach.§ A nonsignificant interaction between X and Z in the first

step does not rule out moderating effects of opposite sign on the direct and indirect effects relating X to Y.§ The influence of Z on the indirect effect of X on Y through

M is not directly tested.• Moderation is usually examined for only a subset of the

paths relating X, M, and Y.• Testing the coefficient on XZ controlling for M does not

reveal how M alters the relationship between X and Y.• Simple paths relating X, M, and Y at levels of Z are

rarely reported.

Path Analytic Framework for IntegratingMediation and Moderation

• The framework developed here combines moderated regression and path analysis.– The paths of the model are expressed as a series of

regression equations.– The regression equations are supplemented with

moderator variables and their products with the independent variables.

– The equation for the mediating variable is substituted into the equation for the dependent variable to obtain a reduced form equation.

– The reduced form equation is used to identify direct, indirect, and total effects and show how these effects vary across levels of the moderator variable.

Basic Mediated Model

• Equations for the model:M = a0 + a1X + eM

Y = b0 + b1M + b2X + eY

• Reduced form equation:Y = b0 + b1(a0 + a1X + eM) + b2X + eY

= (b0 + a0b1) + (b2 + a1b1)X + eY + b1eM

• b2 = direct effect, a1b1 = indirect effect

X

M

Y

a1 b1

b2

First Stage Moderation Model

• Equations for the model:M = a0 + a1X + a2Z + a3XZ + eM

Y = b0 + b1M + b2X + eY

X

M

Y

a1

Z

a3

b1

b2

Reduced Form Equation for theFirst Stage Moderation Model

• Substitute for M:Y = b0 + b1(a0 + a1X + a2Z + a3XZ + eM) + b2X + eY

• Distribute and collect like terms:Y = (b0 + a0b1) + (b2 + a1b1)X + a2b1Z + a3b1XZ

+ b1eM + eY

• Write in terms of simple slopes:Y = [b0 + (a0 + a2Z)b1] +

[b2 + (a1 + a3Z)b1]X + b1eM + eY

• [b0 + (a0 + a2Z)b1] = simple intercept• [b2 + (a1 + a3Z)b1] = simple slope

• b1eM + eY = simple residual

Simple Effects for theFirst Stage Moderation Model

simpledirecteffect

simplefirst

stage

simplesecondstage

simple indirect effect


Y = b0 + b1M + b2X + b3Z + b4MZ + eY

Second Stage Moderation Model

X

M

Y

a1

Z

b4

b1

b2

Reduced Form Equation for theSecond Stage Moderation Model

• Substitute for M:Y = b0 + b1(a0 + a1X + eM) + b2X + b3Z

+ b4(a0 + a1X + eM)Z + eY

• Distribute and collect like terms:Y = (b0 + a0b1) + (b2 + a1b1)X + (b3 + a0b4)Z

+ a1b4XZ + eY + b1eM + b4ZeM

• Write in terms of simple slopes:Y = [(b0 + b3Z) + a0(b1 + b4Z)]

+ [b2 + a1(b1 + b4Z)]X + eY + (b1 + b4Z)eM

• [(b0 + b3Z) + a0(b1 + b4Z)] = simple intercept• [b2 + a1(b1 + b4Z)] = simple slope

• eY + (b1 + b4Z)eM = simple residual

Simple Effects for theSecond Stage Moderation Model

simpledirecteffect

simplefirst

stage

simplesecondstage


First and Second Stage Moderation Model


Y = b0 + b1M + b2X + b3Z + b4MZ + eY

X

M

Y

a1 b1

b2

Z

b4a3

Reduced Form Equation for theFirst and Second Stage Moderation Model

• Substitute for M:Y = b0 + b1(a0 + a1X + a2Z + a3XZ + eM) + b2X

+ b3Z + b4(a0 + a1X + a2Z + a3XZ + eM)Z + eY• Distribute and collect like terms:

Y = b0 + a0b1 + (b2 + a1b1)X + (b3 + a2b1 + a0b4)Z+ a2b4Z2 + (a3b1 + a1b4)XZ + a3b4XZ2

+ eY + b1eM + b4ZeM• Write in terms of simple slopes:

Y = [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)]+ [b2 + (a1 + a3Z)(b1 + b4Z)]X+ eY + (b1 + b4Z)eM

• [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)] = simple intercept• [b2 + (a1 + a3Z)(b1 + b4Z)] = simple slope


Simple Effects for theFirst and Second Stage Moderation Model

simpledirecteffect

simplefirst

stage

simplesecondstage


Total Effect Moderation Model


Y = b0 + b1M + b2X + b3Z + b4MZ + b5XZ + eY

X

M

Y

a1 b1

b2

Z

b4a3

b5

Reduced Form Equation for theTotal Effect Moderation Model

• Substitute for M:Y = b0 + b1(a0 + a1X + a2Z + a3XZ + eM) + b2X + b3Z

+ b4(a0 + a1X + a2Z + a3XZ + eM)Z + b5XZ + eY• Distribute and collect like terms:

Y = b0 + a0b1 + (b2 + a1b1)X + (b3 + a2b1 + a0b4)Z+ a2b4Z2 + (b5 + a3b1 + a1b4)XZ + a3b4XZ2

+ eY + b1eM + b4ZeM• Write in terms of simple slopes:

Y = [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)]+ [(b2 + b5Z) + (a1 + a3Z)(b1 + b4Z)]X+ eY + (b1 + b4Z)eM

• [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)] = simple intercept• [(b2 + b5Z) + (a1 + a3Z)(b1 + b4Z)] = simple slope


simpledirecteffect

simplefirst

stage

simplesecondstage


Simple Effects for theTotal Effect Moderation Model

Direct Effect Moderation Model


Y = b0 + b1M + b2X + b3Z + b4XZ + eY

X

M

Y

a1 b1

b2

Z

b4

First Stage and Direct Effect Moderation Model


Y = b0 + b1M + b2X + b3Z + b4XZ + eY

X

M

Y

a1 b1

b2

Z

a3

b4

Second Stage and Direct Effect Moderation Model


Y = b0 + b1M + b2X + b3Z + b4MZ + b5XZ + eY

X

M

Y

a1 b1

b2

Z

b4

b5

Integrating and Extending Moderated Mediation and Mediated Moderation

• Moderated mediation is captured by models in which the first stage, second stage, or both stages of the mediated effect are moderated.

• Mediated moderated corresponds to the first stage moderation model.§ The moderating effect impacts the mediator variable,

which in turn influences the dependent variable.§ A moderating effect on the mediator variable is

estimated by the following equation:M = a0 + a1X + a2Z + a3XZ + eM

§ The equation relating X and M to Y can include Z and XZ if mediation is partial rather than complete, yielding the first stage and direct effect moderation model.

Model Estimation and Interpretation

• Equations are estimated using OLS regression, thereby incorporating its usual assumptions.

• Individual coefficients in each equation are tested using the reported standard errors.

• Products of coefficients are tested using bias-corrected confidence intervals derived from the bootstrap.

• Simple paths and effects are computed by substituting meaningful values of Z into the reduced form equations.

Empirical Example

• Sample: 1,307 employed adults• Measures:§ X = family role characteristics (intimacy, feedback,

segmentation)§ Y = commitment to family§ M = satisfaction with family§ Z = gender, family centrality§ All measures except gender had multiple items and

exhibited reliabilities ranging from .79 to .91.• Results are illustrated for feedback (X) and

centrality (Z).

Piecemeal Approach

• Moderation:

• Mediation:

• Conclusion: Centrality moderates the effect of feedback on commitment; satisfaction partially mediates the effect of feedback on commitment.

X Z XZ R2

Commitment 0.26** 0.39** -0.13** .49**

M X R2

Commitment 0.42** .30**

Satisfaction 0.72** .44**

Commitment 0.31** 0.20** .41**

Subgroup Approach

• Conclusion: Satisfaction partially mediates the effect of feedback on commitment for people who report either low or high centrality.

Low HighCentrality Centrality

M X M XCommitment 0.45** .20**

Satisfaction 0.77** .56**

Commitment 0.37** 0.16** .12** .13**

Moderated Causal Steps Approach

• Conclusion: Satisfaction partially mediates the effect of the interaction between feedback and centrality on commitment.

M X Z XZ R2

Commitment 0.26** 0.39** -0.13** .49**

Satisfaction 0.65** 0.17** -0.05** .46**

Commitment 0.24** 0.11** 0.35** -0.11** .56**

Moderated Path Analyses

• Simple direct effect: 0.12 – 0.05Z• Simple first stage: 0.65 – 0.05Z• Simple second stage: 0.22 – 0.09Z

M X Z MZ XZ R2

Satisfaction 0.65** 0.17** -0.05** .46**

Commitment 0.22** 0.12** 0.33** -0.09** -0.05** .57**

X

M

Y

.70** .30**

.17**

Low Centrality

High Centrality

X

M

Y

.60** .13**

.07**


• Effect decomposition for low centrality:

• Effect decomposition for high centrality:

• Note: All effects differed across levels of centrality (p < .05). Products of regression coefficients were compared with the bootstrap using the bias-corrected percentile method.

pMX pYM pYX pMXpYM pMXpYM + pYX

0.60** 0.13** 0.07** 0.08** 0.15**

pMX pYM pYX pMXpYM pMXpYM + pYX

0.70** 0.30** 0.17** 0.21** 0.38**


Bootstrap Estimates:Indirect Effect for Low Centrality

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Bootstrap Estimates:Total Effect for Low Centrality

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Bootstrap Estimates:Indirect Effect for High Centrality

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Bootstrap Estimates:Total Effect for High Centrality

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Moderation Excluding Mediating Variable

1 2 3 4 5 6 7Feedback

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Low Centrality: High Centrality:

First Stage Moderation

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Second Stage Moderation

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Indirect Effect Moderation

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Direct Effect Moderation

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Total Effect Moderation

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Substantive Conclusions fromModerated Path Analysis

• Centrality moderated the path from feedback to satisfaction and the paths from feedback and satisfaction to commitment.

• Simple effects showed that these three paths and their associated indirect and total effects were more positive when centrality was low.

• Plots of simple effects showed that commitment was greater for high centrality respondents and lower but more strongly related to feedback for low centrality respondents.

Methodological Conclusions

• Moderated path analysis offers a straightforward and useful approach for combining moderation and mediation.

• Moderated path analysis avoids the ambiguities and limitations of current approaches.

• The basic model illustrated here can be extended to include additional independent, dependent, mediating, and moderating variables.

• The logic of moderated path analysis can be applied to curvilinearity and structural equation modeling with latent variables.

Sample SPSS Syntax for the Bootstrap

* CNLR syntax to produce bootstrap estimates for Equation 5.SET RNG=MT MTINDEX=54321 /*Merseene Twister random number generator, seed set at 54321*/.MODEL PROGRAM a05=.04 aX5=.81 aZ5=-.05 aXZ5=-.14 /*Starting values*/.COMPUTE PRED = a05 + aX5*fbkc + aZ5*gen + aXZ5*fbkcgen /*Coefficients and independent variables*/.CNLR satc /*CNLR procedure, dependent variable*//OUTFILE=FBKGEN05.SAV /*File for bootstrap coefficient estimates*//BOOTSTRAP=1000 /*Number of bootstrap samples*/.

* CNLR syntax to produce bootstrap estimates for Equation 20.SET RNG=MT MTINDEX=54321 /*Merseene Twister random number generator, seed set at 54321*/.MODEL PROGRAM b020=-.03 bX20=.28 bM20=.31 bZ20=.06 bXZ20=-.13 bMZ20=-.01 /*Starting values*/.COMPUTE PRED = b020 + bX20*fbkc + bM20*satc + bZ20*gen + bXZ20*fbkcgen + bMZ20*satcgen

/*Coefficients and independent variables*/.CNLR comc /*CNLR procedure, dependent variable*//OUTFILE=FBKGEN20.SAV /*File for bootstrap coefficient estimates*//BOOTSTRAP=1000 /*Number of bootstrap samples*/.

Appendix:Why I Like the Bootstrap

(and Why You Should Too)

Jeffrey R. EdwardsUniversity of North Carolina

What is the Bootstrap?

• The bootstrap is a resampling procedure used to derive standard errors and construct confidence intervals.

• The bootstrap treats the sample as if it were the population and draws random samples with replacement from the original sample.

• The bootstrap samples are used to estimate parameters repeatedly, and these estimates are used to derive the sampling distributions of the parameters estimated from the original sample.

Why Should I Like the Bootstrap?

• The bootstrap can be applied to virtually any statistic.

• The bootstrap is useful when statistical theory cannot be used to derive a standard error or confidence interval.

• The bootstrap has been incorporated into popular statistical software packages, such as SAS, SYSTAT, and LISREL.

When Would I Use the Bootstrap?

• Comparing R2s from nonnested regression equations.

• Testing indirect effects in path analysis.• Testing regression and path coefficients

based on disattenuated correlations.• Evaluating features of response surfaces.• Testing parameters in structural equation

modeling with nonnormal data.

How Do I Use the Bootstrap?

• Estimate the parameters of interest using the full sample.

• Reestimate the parameters using bootstrap samples of size n.– 500 replications may work if the parameters are

normally distributed.– 10,000 samples may be needed if parameter

distributions are nonnormal and/or contain outliers.• Use the bootstrap estimates to compute

standard errors and construct confidence intervals (Excel is well suited to this task).

R2s From Non-Nested Equations

• Y = b0 + b1S + b2N + b3S2 + b4SN + b5N2

+ b6O + b7I + b8O2 + b9OI + b10I2 + e• Question: Is the ∆R2 for the S and N terms

greater than the ∆R2 for the O and I terms?• Problem: The typical F-test only applies to

nested equations.• Solution: Use bootstrap samples to

estimate full and reduced equations and compare R2s.

Using the MGLH Module in SYSTATOUTPUT=BOOT.OUTMODEL Y=CONSTANT+S+N+S*S+S*N+N*N,+O+I+O*O+O*I+I*IESTIMATERSEED=54321ESTIMATE/SAMPLE=BOOT(1000)MODEL Y=CONSTANT+S+N+S*S+S*N+N*NESTIMATERSEED=54321ESTIMATE/SAMPLE=BOOT(1000)MODEL Y=CONSTANT+O+I+O*O+O*I+I*IESTIMATERSEED=54321ESTIMATE/SAMPLE=BOOT(1000)OUTPUT

Using the Output

• The output file will contain 3003 sets of regression results, consisting of estimates from the full sample and 1000 bootstrap samples for each of the three equations.

• SYSTAT can save coefficients into a nice neat file, but in this case, we want R2s. These can be extracted using a text editor with macro capabilities.

• The extracted values can be copied into Excel.

What Should I Do with the Estimates?

• Compute descriptive statistics (e.g., mean, minimum, maximum, standard deviation).

• Construct confidence intervals using the normal approximation (e.g., the parameter estimate +/- 1.96 times the standard deviation of the bootstrap estimates).

• Construct confidence intervals using the percentile feature of Excel, with and without bias correction.

Bootstrap Results

0.1 0.2 0.3 0.4R2

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0.010.02

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0.070.08

0.090.10

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Descriptive Statistics

FULL R2 R2HW R2

OI ∆R2HW ∆R2

OI ∆R2HW - ∆R2

OI

SAMPLE 0.273 0.231 0.234 0.039 0.042 -0.003

BOOTSTRAP R2R2

HW R2OI ∆R2

HW ∆R2OI ∆R2

HW - ∆R2OI

AVG 0.283 0.237 0.240 0.044 0.047 -0.003MEDIAN 0.282 0.236 0.239 0.043 0.046 -0.003

MIN 0.192 0.158 0.147 0.013 0.018 -0.068MAX 0.371 0.333 0.335 0.091 0.091 0.072STD 0.028 0.027 0.028 0.012 0.012 0.022

SKEW 0.123 0.108 0.155 0.497 0.490 0.002KURT -0.069 -0.147 0.151 0.356 0.119 -0.072

Normal Approximation

FULL R2 R2HW R2

OI ∆R2HW ∆R2

OI ∆R2HW - ∆R2

OI

SAMPLE 0.273 0.231 0.234 0.039 0.042 -0.003

BOOTSTRAP R2 R2HW R2

OI ∆R2HW ∆R2

OI ∆R2HW - ∆R2

OI

-1.645*STD 0.227 0.187 0.187 0.019 0.022 -0.038+1.645*STD 0.319 0.275 0.280 0.059 0.062 0.033

-1.96*STD 0.218 0.184 0.184 0.020 0.022 -0.045+1.96*STD 0.328 0.290 0.295 0.068 0.071 0.040-2.58*STD 0.201 0.162 0.161 0.008 0.010 -0.059

+2.58*STD 0.345 0.300 0.306 0.071 0.074 0.053

Percentile Method

FULL R2 R2HW R2

OI ∆ R2HW ∆ R2

OI ∆ R2HW - ∆ R2

OI

SAMPLE 0.273 0.231 0.234 0.039 0.042 -0.003


OI ∆ R2HW ∆ R2

OI ∆ R2HW - ∆ R2

OI

5% 0.238 0.194 0.197 0.026 0.029 -0.04095% 0.332 0.282 0.288 0.066 0.070 0.032

2.5% 0.231 0.187 0.186 0.022 0.026 -0.04797.5% 0.340 0.288 0.298 0.071 0.075 0.040

0.5% 0.212 0.171 0.168 0.018 0.020 -0.05699.5% 0.356 0.303 0.317 0.081 0.081 0.049

Bias Corrected Percentile Method

FULL R2 R2HW R2

OI ∆ R2HW ∆ R2

OI ∆ R2HW - ∆ R2

OI

SAMPLE 0.273 0.231 0.234 0.039 0.042 -0.003


OI ∆ R2HW ∆ R2

OI ∆ R2HW - ∆ R2

OI

p(Θ *< Θ ) 0.370 0.432 0.435 0.374 0.383 0.498t -8.222 -4.301 -4.111 -7.969 -7.400 -0.126

Zo -0.332 -0.171 -0.164 -0.321 -0.298 -0.005BC 5% 0.224 0.186 0.186 0.020 0.024 -0.040

BC 95% 0.312 0.273 0.276 0.055 0.059 0.032BC 2.5% 0.212 0.177 0.178 0.018 0.020 -0.047

BC 97.5% 0.321 0.282 0.288 0.060 0.065 0.039BC 0.5% 0.197 0.165 0.160 0.015 0.018 -0.056

BC 99.5% 0.338 0.297 0.306 0.071 0.075 0.048

Some Parting Thoughts

• Make sure you use the same bootstrap samples to compare estimators.

• The bootstrap should be applied to the stochastic (i.e., random) part of a model.

• For hypothesis testing, confidence intervals should be constructed under the assumption that the null hypothesis is true.

• The bootstrap can fail in some cases (e.g., when observations are not independent).

Bootstrap Syntax for SPSS 14.0.2REGRESSION/DEPENDENT satc/METHOD=ENTER intc gen intcgen .REGRESSION/DEPENDENT comc/METHOD=ENTER intc satc gen intcgen satcgen .

SET RNG=MT MTINDEX=54321 .MODEL PROGRAM a05=-.04 aX5=.82 aZ5=.07 aXZ5=-.05 .COMPUTE PRED = a05 + aX5*intc + aZ5*gen + aXZ5*intcgen .CNLR satc/OUTFILE=INTGEN05.SAV/BOOTSTRAP=1000 .SET RNG=MT MTINDEX=54321 .MODEL PROGRAM b016=-.05 bX16=.43 bM16=.25 bZ16=.09 bXZ16=-.16 bMZ16=-.01 .COMPUTE PRED = b016 + bX16*intc + bM16*satc + bZ16*gen + bXZ16*intcgen + bMZ16*satcgen .CNLR comc/OUTFILE=INTGEN16.SAV/BOOTSTRAP=1000 .

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