Jeffrey R. Edwards University of North Carolina Lisa S. Lambert Georgia State University

Research that Examines Moderation 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 Combining Moderation 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 = b 0 + b1X + e Y 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 Integrating Mediation 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 a1 X

M

b2

b1 Y

• 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

First Stage Moderation Model Z a3 a1 X

M

b2

b1 Y

• Equations for the model: M = a0 + a1X + a2Z + a3XZ + eM Y = b0 + b1M + b2X + eY

Reduced Form Equation for the First 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

Simple Effects for the First Stage Moderation Model • [b0 + (a0 + a2Z)b1] = simple intercept • [b2 + (a1 + a3Z)b1] = simple slope simple direct effect

simple simple first second stage stage simple indirect effect

• b1eM + eY = simple residual

Second Stage Moderation Model Z b4 a1 X

M

b2

b1 Y

• Equations for the model: M = a0 + a1X + eM Y = b0 + b1M + b2X + b3Z + b4MZ + eY

Reduced Form Equation for the Second 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

Simple Effects for the Second Stage Moderation Model • [(b0 + b3Z) + a0(b1 + b4Z)] = simple intercept • [b2 + a1(b1 + b4Z)] = simple slope simple simple simple direct first second effect stage stage simple indirect effect

• eY + (b1 + b4Z)eM = simple residual

First and Second Stage Moderation Model Z a3 a1 X

b4 M

b2

b1 Y

• Equations for the model: M = a0 + a1X + a2Z + a3XZ + eM Y = b0 + b1M + b2X + b3Z + b4MZ + eY

Reduced Form Equation for the First 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

Simple Effects for the First and Second Stage Moderation Model • [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)] = simple intercept • [b2 + (a1 + a3Z)(b1 + b4Z)] = simple slope simple direct effect

simple first stage

simple second stage

simple indirect effect

• eY + (b1 + b4Z)eM = simple residual

Total Effect Moderation Model Z a3 a1 X

b4 M b5 b2

b1 Y

• Equations for the model: M = a0 + a1X + a2Z + a3XZ + eM Y = b0 + b1M + b2X + b3Z + b4MZ + b5XZ + eY

Reduced Form Equation for the Total 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

Simple Effects for the Total Effect Moderation Model • [(b0 + b3Z) + (a0 + a2Z)(b1 + b4Z)] = simple intercept • [(b2 + b5Z) + (a1 + a3Z)(b1 + b4Z)] = simple slope simple direct effect

simple first stage

simple second stage

simple indirect effect

• eY + (b1 + b4Z)eM = simple residual

Direct Effect Moderation Model Z

a1 X

M b4 b2

b1 Y

• Equations for the model: M = a0 + a1X + eM Y = b0 + b1M + b2X + b3Z + b4XZ + eY

First Stage and Direct Effect Moderation Model Z a3 a1 X

M b4 b2

b1 Y

• Equations for the model: M = a0 + a1X + a2Z + a3XZ + eM Y = b0 + b1M + b2X + b3Z + b4XZ + eY

Second Stage and Direct Effect Moderation Model Z b4 a1 X

M b5 b2

b1 Y

• Equations for the model: M = a0 + a1X + eM Y = b0 + b1M + b2X + b3Z + b4MZ + b5XZ + eY

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 biascorrected 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: Commitment

• Mediation: Commitment Satisfaction Commitment

X 0.26**

Z 0.39**

XZ -0.13**

M

X 0.42** 0.72** 0.20**

R2 .30** .44** .41**

0.31**

R2 .49**

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

Subgroup Approach

Commitment Satisfaction Commitment

Low Centrality M X 0.45** 0.77** 0.37** 0.16**

High Centrality M X .20** .56** .12** .13**

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

Moderated Causal Steps Approach

M Commitment Satisfaction Commitment

0.24**

X 0.26** 0.65** 0.11**

Z 0.39** 0.17** 0.35**

XZ -0.13** -0.05** -0.11**

R2 .49** .46** .56**

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

Moderated Path Analyses

M Satisfaction Commitment

0.22**

X Z MZ XZ R2 0.65** 0.17** -0.05** .46** 0.12** 0.33** -0.09** -0.05** .57**

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

Moderated Path Analyses Low Centrality .70** X

M

.17**

.30** Y

High Centrality .60** X

M

.07**

.13** Y

Moderated Path Analyses • Effect decomposition for low centrality: pMX 0.70**

pYM 0.30**

pYX 0.17**

pMXpYM 0.21**

pMXpYM + pYX 0.38**

• Effect decomposition for high centrality: pMX 0.60**

pYM 0.13**

pYX 0.07**

pMXpYM 0.08**

pMXpYM + pYX 0.15**

• 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.

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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

Moderation Excluding Mediating Variable 7

Commitment

6 5 4 3 2 1 1

2

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Low Centrality:

4 5 Feedback

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High Centrality:

7

First Stage Moderation 7

Satisfaction

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6

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Commitment

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4 5 Satisfaction

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Commitment

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4 5 Feedback

6

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7

Direct Effect Moderation 7

Commitment

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4 5 Feedback

6

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7

Total Effect Moderation 7

Commitment

6 5 4 3 2 1 1

2

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4 5 Feedback

6

High Centrality:

7

Substantive Conclusions from Moderated 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. Edwards University 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 SYSTAT OUTPUT=BOOT.OUT MODEL Y=CONSTANT+S+N+S*S+S*N+N*N, +O+I+O*O+O*I+I*I ESTIMATE RSEED=54321 ESTIMATE/SAMPLE=BOOT(1000) MODEL Y=CONSTANT+S+N+S*S+S*N+N*N ESTIMATE RSEED=54321 ESTIMATE/SAMPLE=BOOT(1000) MODEL Y=CONSTANT+O+I+O*O+O*I+I*I ESTIMATE RSEED=54321 ESTIMATE/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.

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Bootstrap Results

Descriptive Statistics R2

R2HW

R2OI

0.273

0.231

0.234

BOOTSTRAP

R2

R2HW

R2OI

AVG MEDIAN MIN MAX STD SKEW KURT

0.283 0.282 0.192 0.371 0.028 0.123 -0.069

0.237 0.236 0.158 0.333 0.027 0.108 -0.147

0.240 0.239 0.147 0.335 0.028 0.155 0.151

FULL SAMPLE

∆R

2

0.039 ∆R

∆R

2

HW

2

OI

0.044 0.043 0.013 0.091 0.012 0.497 0.356

2 HW

0.042 ∆R

2

HW

∆R

OI

0.047 0.046 0.018 0.091 0.012 0.490 0.119

- ∆ R2OI -0.003

∆R

2 HW

- ∆ R2OI -0.003 -0.003 -0.068 0.072 0.022 0.002 -0.072

Normal Approximation R2

R2HW

R2OI

0.273

0.231

0.234

BOOTSTRAP

R2

R2HW

R2OI

-1.645*STD +1.645*STD -1.96*STD +1.96*STD -2.58*STD +2.58*STD

0.227 0.319 0.218 0.328 0.201 0.345

0.187 0.275 0.184 0.290 0.162 0.300

0.187 0.280 0.184 0.295 0.161 0.306

FULL SAMPLE

∆R

2

0.039 ∆R

∆R

2

HW

2

OI

0.019 0.059 0.020 0.068 0.008 0.071

2 HW

0.042 ∆R

2

HW

∆R

OI

0.022 0.062 0.022 0.071 0.010 0.074

- ∆ R2OI -0.003

∆R

2 HW

- ∆ R2OI -0.038 0.033 -0.045 0.040 -0.059 0.053

Percentile Method R2

R 2 HW

R 2OI

∆ R 2HW

0.273

0.231

0.234

0.039

R2

R 2 HW

R 2OI

∆ R 2HW

5% 95% 2.5%

0.238 0.332 0.231

0.194 0.282 0.187

0.197 0.288 0.186

0.026 0.066 0.022

0.029 0.070 0.026

-0.040 0.032 -0.047

97.5% 0.5% 99.5%

0.340 0.212 0.356

0.288 0.171 0.303

0.298 0.168 0.317

0.071 0.018 0.081

0.075 0.020 0.081

0.040 -0.056 0.049

FULL SAMPLE BOOTSTRAP

∆ R 2OI ∆ R 2HW - ∆ R 2OI 0.042

-0.003

∆ R 2OI ∆ R 2HW - ∆ R 2OI

Bias Corrected Percentile Method FULL SAMPLE BOOTSTRAP p(Θ *< Θ ) t Zo BC 5% BC 95% BC 2.5% BC 97.5% BC 0.5% BC 99.5%

R2

R 2 HW

R 2OI

∆ R 2 HW

0.273

0.231

0.234

0.039

R2

R 2 HW

R 2OI

∆ R 2 HW

∆ R 2 OI ∆ R 2HW - ∆ R 2OI

0.370 -8.222 -0.332 0.224 0.312 0.212 0.321 0.197 0.338

0.432 -4.301 -0.171 0.186 0.273 0.177 0.282 0.165 0.297

0.435 -4.111 -0.164 0.186 0.276 0.178 0.288 0.160 0.306

0.374 -7.969 -0.321 0.020 0.055 0.018 0.060 0.015 0.071

0.383 -7.400 -0.298 0.024 0.059 0.020 0.065 0.018 0.075

∆ R 2 OI ∆ R 2HW - ∆ R 2OI 0.042

-0.003

0.498 -0.126 -0.005 -0.040 0.032 -0.047 0.039 -0.056 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 14.0.2 REGRESSION /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 .