Estimating Interaction Effects Using Multiple Regression

1
Estimating Interaction Effects Using Multiple
Regression

• Herman Aguinis, Ph.D.
• Mehalchin Term Professor of Management
• The Business School
• University of Colorado at Denver
• www.cudenver.edu/haguinis

2
Overview

• What is an Interaction Effect?
• The So What Question Importance of Interaction
  Effects for Theory and Practice
• Estimating Interaction Effects Using Moderated
  Multiple Regression (MMR)
• Problems with MMR
• Aguinis, Beaty, Boik, Pierce (2005, J. of
  Applied Psychology)
• The Now What Question Addressing problems with
  MMR
• Some Conclusions

3
What is an Interaction Effect?

• The relationship between X and Y depends on Z
  (i.e., a moderator)
• X Y X Y
• Z Z
• Other terms used
• Population control variable (Gaylord Carroll,
  1948) Subgrouping variable (Frederiksen
  Melville, 1954) Predictability variable
  (Ghiselli, 1956) Referent variable (Toops,
  1959) Modifier variable (Grooms Endler,
  1960)Homologizer variable (Johnson, 1966)

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Importance of Interaction Effects Theory

• Going beyond main effects
• We typically say it depends
• More complex models
• If we want to know how well we are doing in the
  biological, psychological, and social sciences,
  an index that will serve us well is how far we
  have advanced in our understanding of the
  moderator variables of our field (Hall
  Rosenthal, 1991, p. 447)

5
Importance of Interaction Effects Practice

• For example, personnel selection
• Test bias The relationship between a test and a
  criterion depends on gender or ethnicity
• No bias exists if the regression equations
  relating the test and the criterion are
  indistinguishable for the groups in question
  (Standards, 1999, p. 79)
• In other words, the X-Y relationship differs
  depending on the value of Z (e.g., 1 Female, 0
  Male)

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Illustration of Gender as a Moderator in
Personnel Selection
Women
Ywomen
Common line
Ycommon
Ymen
Job Performance
Men
X
Test Scores
7
Importance of Interaction Effects Practice

• Management in General
• Does an intervention work similarly well for, for
  example, Cantonese and American employees working
  in Hong Kong? (categorical moderator)
• Example Performance management system regarding
  teaching at university in Hong Kong. Would the
  same evaluation methods lead to employee (i.e.,
  faculty) satisfaction depending on the national
  origin of faculty members?

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Estimating Interaction Effects

• Moderated Multiple Regression (MMR)
• Y a b1 X b2 Z b3 XZ,
• where Y criterion (continuous variable)
• X predictor (typically continuous)
• Z moderator (continuous or
  categorical)
• XZ product term carrying information
  about the moderating effect (i.e., interaction
  between X and Z)

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Statistical Significance Test

• Y a b1 X b2 Z
• Y a b1 X b2 Z b3 XZ
• Ho ?1 ?2
• Ho ß3 0 (using a t-statistic)

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Estimating Interaction Effects Using Moderated
Multiple Regression (MMR)

• For example
• Personnel selection Y measure of performance,
  X test score, Z gender
• Additional research areas training, turnover,
  performance appraisal, return on investment,
  mentoring, self-efficacy, job satisfaction,
  organizational commitment, and career
  development, among others

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Interpreting Interactions(Z is continuous)

• Y a b1 X b2 Z b3 XZ,
• b3 2 means that a one-unit change in X (Z)
  increases the slope of Y on Z (Y on X) by 2 points

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Interpreting Interactions(Z is binary, dummy
coded)

• Y a b1 X b2 Z b3 XZ,
• b3 estimated difference between the slope of Y
  on X between the group coded as 1 and the group
  coded as 0.
• b2 estimated difference between X scores for a
  member in group coded as 1 and a member in group
  coded as 0 assuming the scores on Y are 0.
• b1 estimated X score for members of the group
  coded as 1 assuming the scores on Y are 0.
• a mean score on X for members of group coded as
  0.

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Pervasive Use of MMR in the Organizational
Sciences

• Recent review MMR was used in over 600 attempts
  to detect moderating effects of categorical
  variables in AMJ, JAP, and PP between 1977-1998
  (Aguinis, Beaty, Boik, Pierce, 2005, JAP)

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Selected Research on MMR

• Aguinis (2004, Regression Analysis for
  Categorical Moderators, Guilford Press)
• Aguinis, Beaty, Boik, and Pierce (2005, J. of
  Applied Psychology)
• Aguinis, Boik, and Pierce (2001, Organizational
  Research Methods)
• Aguinis, Petersen, and Pierce (1999,
  Organizational Research Methods)
• Aguinis and Pierce (1998, Organizational Research
  Methods)
• Aguinis and Pierce (1998, Ed. Psychological
  Measurement)
• Aguinis and Stone-Romero (1997, J. of Applied
  Psychology)
• Aguinis, Bommer, and Pierce (1996, Ed.
  Psychological Measurement)
• Aguinis (1995, J. of Management)

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Methodology Monte Carlo Simulations

• Research question Does MMR do a good job at
  estimating moderating effects?
• Difficulty We dont know the population
• Solution Monte Carlo methodology
• Create a population
• Generate random samples
• Perform MMR analyses on samples
• Compare population versus samples
• Assess of hits and misses

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Problems with MMR

• We dont find moderators
• If we find them, they are small
• Why should we care?
• Theory Failure to find support for correct
  hypotheses (derailment of theory advancement
  process model misspecification)
• Practice Erroneous decision making (e.g., over
  and under prediction of performance,
  implementation of ineffective interventions)
• Ethical implications
• Legal implications

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Some Culprits for Erroneous Estimation of
Moderating Effects

• Small total sample size
• Unequal sample size across moderator-based groups
  
• Range restriction (i.e., truncation) in predictor
  variable X
• Scale coarseness
• Violation of homogeneity of error variance
  assumption
• Unreliability of measurement
• Artificial dichotomization/polichotomization of
  continuous variables
• Interactive effects

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Unequal Sample Size Across Moderator-based
Subgroups

• Applies to categorical moderators (e.g., gender,
  national origin)
• In many research situations, n1 ? n2
• Two studies examined this issue (Aguinis
  Stone-Romero, 1997 Stone, Alliger, and Aguinis,
  1994) (see also Aguinis, 1995)
• Conclusion n1 needs to be (.3 n2) or larger to
  detect medium moderating effects

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Truncation in Predictor X

• Non-random sampling
• Pervasive in field settings (systematic in
  personnel selection/test validation research,
  X,Y X gt x)
• Aguinis and Stone-Romero (1997) (categorical
  moderator) McClelland and Judd, 1993 (continuous
  moderator)
• Truncation has a dramatic impact on power
• N 300, medium moderating effect, power .81
• Same conditions, truncation .80, power .51
• Conclusion Even mild levels of truncation can
  have a substantial detrimental effect on power

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Violation of Homogeneity of Error Variance
Assumption

• Applies to categorical moderators
• Error variance Variance in Y that remains after
  predicting Y from X is equal across subgroups
  (e.g., women, men)
• Distinct from homoscedasticity assumption

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Regression of Homoscedastic Data
Total Sample Women Men
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Regression for Subgroups
Women
Men
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Artificial polichotomization of continuous
variables

• Median split and other common methods for
  simplifying the data before conducting ANOVAs
• Cohen (1983) showed this practice is
  inappropriate
• In the context of MMR, some have used a median
  split procedure on continuous predictor Z and
  compared correlations across groups
• MMR always performs better than comparing
  artificially-created subgroups (Stone-Romero
  Anderson, 1994)
• Conclusion Do not polichotomize truly continuous
  predictors

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Interactions Among Artifacts

• Concurrent manipulation of truncation, N, n1 and
  n2, and moderating effect magnitude (Aguinis
  Stone-Romero, JAP, 1997) .
• Results Methodological artifacts have
  interactive effects on power.
• Even if conditions conducive to high power are
  favorable regarding one factor (e.g., N),
  conditions unfavorable regarding other factors
  (e.g., truncation) will lead to low power.
• Conclusion Relying on a single strategy (e.g.,
  increase N) to improve power will not be
  successful if other methodological and
  statistical artifacts

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Aguinis, Beaty, Boik, Pierce (2005, JAP)

• Q1 What is the size of observed moderating
  effects of categorical variables in published
  research?
• Q2 What would the size of moderating effects of
  categorical variables be in published research
  under conditions of perfect reliability?
• Q3 What is the a priori power of MMR to detect
  moderating effects of categorical variables in
  published research?
• Q4 Do MMR tests reported in published research
  have sufficient statistical power to detect
  moderating effects conventionally defined as
  small, medium, and large?

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Method

• Review of all articles published from 1969 to
  1998 in Academy of Management Journal (AMJ),
  Journal of Applied Psychology (JAP), and
  Personnel Psychology (PP)
• Criteria for study inclusion
• At least one MMR analysis
• The MMR analysis included a continuous criterion
  Y, a continuous predictor X, and a categorical
  moderator Z

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Effect Size and Power Computation

• Total of 636 MMR analyses
• Moderator sample sizes for 507 (79.72)
• Moderator group sample sizes and
  predictor-criterion rs for 261 (41.04)
• Effect sizes and power computation based on 261
  MMR analyses for which ns and rs were available.
  We used SD information when available, and
  assumed homogeneity or error variance when this
  information was not available

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Results (I)

• Frequency of MMR Use over Time

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Q1 Size of Observed Effects (I)

• Effect size metric
• Median f 2 .002,
• Mean (SD) .009 (.025)
• 95 CI .0089 to .0091
• 25th percentile .0004
• 75th percentile .0053
• Effect size values over time r(261) .15, p lt
  .05

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Q1 Size of Observed Effects (II)

• F(2, 258) 4.97, p .008, ?2 .04
• Tukey HSD tests AMJ gt JAP and PP gt JAP

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Q1 Size of Observed Effects (III)

• F(2, 258) 8.71, p lt .001, ?2 .06
• Tukey HSD tests Other gt Ethnicity

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Q1 Size of Observed Effects (IV)

• t(259) -.226, p ns
• t(259) -0.95, p ns

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Q2 Construct-level Effects (I)

• Median f 2 .003
• Increase of .001 over median observed effect size
• Mean (SD) .017
• Increase of .008 over mean observed effect size

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Q3 Statistical Power (I)
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Q3 Statistical Power (II)
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Q4 Power to Detect Small, Medium, and Large
Effects

• Small f 2 (.02) mean power .84 72 of tests
  would have a power of .80 or higher
• Medium f 2 (.15) mean power .98
• Large f 2 (.35) mean power 1.0

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Some Conclusions

• We expected effect size to be small, but not so
  small (i.e., median of .002)
• Computation of construct-level effect sizes did
  not improve things by much (i.e., median of .003)
• More encouraging results
• None of the 95 CIs around the mean effect size
  for the various comparisons included zero
• Effect sizes have increased over time
• Given the observed sample sizes, mean power is
  sufficient to detect effects .02
• 72 of studies had sufficient power to detect an
  effect .02

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Some Implications

• Are theories in dozens of research domains
  incorrect in hypothesizing moderators?
• Are hundreds of researchers in dozens of
  disparate domains wrong and population moderating
  effects so small?
• Could be, but.. more likely, methodological
  artifacts decrease the observed effect sizes
  substantially vis-à-vis their population
  counterparts
• More attention needs to be paid to design and
  analysis issues that decrease observed effect
  sizes
• Conventional definitions of effect size (f 2) for
  moderators should probably be revised

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The Now What Question

• Before data are collected
• Larger sample size
• More reliable measures
• Avoid truncated samples
• Use non-coarse scales (e.g., program by Aguinis,
  Bommer, Pierce, 1996, Ed. Psych. Measurement)
• Equalize sample size across moderator-based
  subgroups
• Use computer programs in the public domain to
  estimate sample size needed for desired power
  level
• Gather information on research design trade-offs
• Easier said that done!

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Tools to Improve Moderating Effect Estimation
(Aguinis, 2004)

• Scale coarseness
• Aguinis, Bommer, and Pierce (1996, Educational
  Psychological Measurement)
• Homogeneity of error variance
• Aguinis, Petersen, and Pierce (1999,
  Organizational Research Methods)
• Power estimation and research design trade-offs
• Aguinis, Pierce, and Stone-Romero (1994,
  Educational Psychological Measurement)
• Aguinis and Pierce (1998, Educational
  Psychological Measurement)
• Aguinis, Boik, and Pierce (2001, Organizational
  Research Methods)

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Assessment of Assumption Compliance

• DeShon and Alexanders (1996) 1.5 rule of thumb
• Bartletts homogeneity test
• M
• k number of sub-groups
• nk number of observations in each sub-group
• s2 sub-group variance on the criterion
• v degrees of freedom from which s2 is based

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Homogeneity is not Met... Now What?

• Use alternatives to MMR
• Alexander and colleagues' normalized-t
  approximation
• OR James's second-order approximation

where
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Program ALTMMR

• Calculates
• Error variance ratio (highest if more than 2
  subgroups)
• Bartletts M
• Jamess J
• Alexanders A
• Uses sample descriptive data
• nk , sx , sy , rxy
• User sets p .05 or .01 (for all but Jamess
  statistic)

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Program ALTMMR

• Described in detail in Aguinis (2004)
• Available at www.cudenver.edu/haguinis/ (click
  on MMR icon on left side of page)
• Executable on-line or locally

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Power Estimation

• Program POWER
• Aguinis, Pierce, and Stone-Romero (1994, Ed.
  Psych. Measurement)
• Program MMRPWR
• Aguinis and Pierce (1998, Ed. Psych.
  Measurement)
• Program MMRPOWER
• Aguinis, Boik, and Pierce (2001, Organizational
  Research Methods)

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Program MMRPOWER

• Problems/Challenges regarding POWER and MMRPWR
  programs
• Based on extrapolation from simulations Range of
  values is limited
• Absence of factors known to affect power of MMR
  (e.g., unreliability)
• Theoretical approximation to power

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Program MMRPOWER

• Described in detail in Aguinis (2004)
• Available at www.cudenver.edu/haguinis/ (click
  on MMR icon on left side of page)
• Executable on-line or locally

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Some Conclusions

• Observed moderating effects are very small
• MMR is a low power test for detecting effect
  sizes as typically observed
• Researchers are not aware of problems with MMR
• Implications for theory and practice
• User-friendly programs are available and allow
  researchers to improve moderating effect
  estimation
• Using these tools will allow researchers to make
  more informed decisions regarding the operation
  of moderating effects