Treating Stimuli as a Random Factor in Social Psychology:
A New and Comprehensive Solutionto a Pervasive but Largely Ignored Problem
Jacob WestfallUniversity of Colorado Boulder
Charles M. Judd David A. KennyUniversity of Colorado Boulder University of Connecticut
What to do about replicability?• Mandatory reporting of all DVs, studies, etc.?• Journals or journal sections devoted to straight
replication attempts?• Pre-registration of studies?
• Many of the proposed solutions involve large-scale institutional changes, restructuring incentives, etc.
• These are good ideas worthy of discussing, but surely not quick or easy to implement
One way to increase replicability:Treat stimuli as random
• Failure to account for uncertainty associated with stimulus sampling (i.e., treating stimuli as fixed rather than random) leads to biased, overconfident estimates of effects (Clark, 1973; Coleman, 1964)
• The pervasive failure to model stimulus as a random factor is probably responsible for many failures to replicate when future studies use different stimulus samples
Doing the correct analysis is easy!
• Recently developed statistical methods solve the statistical problem of stimulus sampling
• These mixed models with crossed random effects are easy to apply and are already widely available in major statistical packages (R, SAS, SPSS, Stata, etc.)
Outline of rest of talk1. The problem– Illustrative design and typical RM-ANOVA analyses– Estimated type 1 error rates
2. The solution– Introducing mixed models with crossed random
effects for participants and stimuli– Applications of mixed model analyses to actual
datasets
Illustrative Design• Participants crossed with Stimuli
– Each Participant responds to each Stimulus • Stimuli nested under Condition
– Each Stimulus always in either Condition A or Condition B• Participants crossed with Condition
– Participants make responses under both Conditions
Sample of hypothetical dataset:
5 4 6 7 3 8 8 7 9 5 6 5
4 4 7 8 4 6 9 6 7 4 5 6
5 3 6 7 4 5 7 5 8 3 4 5
Typical repeated measures analyses (RM-ANOVA)
MBlack MWhite Difference
5.5 6.67 1.17
5.5 6.17 0.67
5.0 5.33 0.33
5 4 6 7 3 8 8 7 9 5 6 5
4 4 7 8 4 6 9 6 7 4 5 6
5 3 6 7 4 5 7 5 8 3 4 5
How variable are the stimulus ratings around each of the participant means? The variance is lost due to the aggregation
“By-participant analysis”
Typical repeated measures analyses (RM-ANOVA)
5 4 6 7 3 8 8 7 9 5 6 5
4 4 7 8 4 6 9 6 7 4 5 6
5 3 6 7 4 5 7 5 8 3 4 5
4.00 3.67 6.33 7.33 3.67 6.33 8.00 6.00 8.00 4.00 5.00 5.33
Sample 1 v.s. Sample 2
“By-stimulus analysis”
Simulation of type 1 error rates for typical RM-ANOVA analyses
• Design is the same as previously discussed• Draw random samples of participants and stimuli– Variance components = 4, Error variance = 16
• Number of participants {10, 30, 50, 70, 90}∈• Number of stimuli {10, 30, 50, 70, 90}∈• Conducted both by-participant and by-stimulus
analysis on each simulated dataset• True Condition effect = 0
Type 1 error rate simulation results• The exact simulated error rates depend on the
variance components, which although realistic, were ultimately arbitrary
• The main points to take away here are:1. The standard analyses will virtually always show
some degree of positive bias2. In some (entirely realistic) cases, this bias can be
extreme3. The degree of bias depends in a predictable way on
the design of the experiment (e.g., the sample sizes)
The old solution: Quasi-F statistics• Although quasi-Fs successfully address the
statistical problem, they suffer from a variety of limitations– Require complete orthogonal design (balanced factors)– No missing data– No continuous covariates– A different quasi-F must be derived (often laboriously)
for each new experimental design – Not widely implemented in major statistical packages
The new solution: Mixed models• Known variously as:– Mixed-effects models, multilevel models, random
effect models, hierarchical linear models, etc.• Most social psychologists familiar with mixed
models for hierarchical random factors– E.g., students nested in classrooms
• Less well known is that mixed models can also easily accommodate designs with crossed random factors– E.g., participants crossed with stimuli
Grand mean = 100
MeanA = -5 MeanB = 5
ParticipantIntercepts5.86
7.09
-1.09
-4.53
Stim. Intercepts: -2.84 -9.19 -1.16 18.17
ParticipantSlopes3.02
-9.09
3.15
-1.38
Everything else = residual error
The linear mixed-effects modelwith crossed random effects
Fixed effects Random effects
The linear mixed-effects modelwith crossed random effects
Intercept Slope
6 parameters
Fitting mixed models is easy: Sample syntaxlibrary(lme4)model <- lmer(y ~ c + (1 | j) + (c | i))
proc mixed covtest;class i j;model y=c/solution;random intercept c/sub=i type=un;random intercept/sub=j;run;
MIXED y WITH c /FIXED=c /PRINT=SOLUTION TESTCOV /RANDOM=INTERCEPT c | SUBJECT(i) COVTYPE(UN) /RANDOM=INTERCEPT | SUBJECT(j).
R
SAS
SPSS
Mixed models successfully maintain the nominal type 1 error rate (α = .05)
Applications to existing datasets1. Representative simulated dataset (for
comparison)2. Afrocentric features data (Blair et al., 2002,
2004, 2005)3. Shooter data (Correll et al., 2002, 2007)4. Psi / Retroactive priming data (Bem)– Forward-priming condition (classic evaluative
priming effect)– Reverse-priming condition (psi condition)
Comparison of effectsbetween RM-ANOVA and mixed model analyses
Dataset RM-ANOVA (by-participant) Mixed model Stimulus ICC
F ratio D.F. p F ratio D.F. p
Simulated example
30.48 (1, 29) <.001 9.11 (1, 38.52) .005 r = 0.191
Shooter data 57.89 (1, 35) <.001 3.39 (1, 48.1) .072 r = 0.317
Afrocentric features data
6.40 (1, 46) .015 4.33 (1, 51.1) .043 r = 0.113
Bem (2011)Forward-priming condition
22.18 (1, 98) <.001 14.59 (1, 46.91) .029 Targets: r = 0.349Primes: r = 0.035
Bem (2011) Reverse-priming condition
6.60 (1, 98) .012 2.34 (1, 27.58) .136 Targets: r = 0.292Primes: r = 0.0
Comparison of effectsbetween RM-ANOVA and mixed model analyses
Dataset RM-ANOVA (by-participant) Mixed model Stimulus ICC
F ratio D.F. p F ratio D.F. p
Simulated example
30.48 (1, 29) <.001 9.11 (1, 38.52) .005 r = 0.191
Shooter data 57.89 (1, 35) <.001 3.39 (1, 48.1) .072 r = 0.317
Afrocentric features data
6.40 (1, 46) .015 4.33 (1, 51.1) .043 r = 0.113
Bem (2011)Forward-priming condition
22.18 (1, 98) <.001 14.59 (1, 46.91) .029 Targets: r = 0.349Primes: r = 0.035
Bem (2011) Reverse-priming condition
6.60 (1, 98) .012 2.34 (1, 27.58) .136 Targets: r = 0.292Primes: r = 0.0
Comparison of effectsbetween RM-ANOVA and mixed model analyses
Dataset RM-ANOVA (by-participant) Mixed model Stimulus ICC
F ratio D.F. p F ratio D.F. p
Simulated example
30.48 (1, 29) <.001 9.11 (1, 38.52) .005 r = 0.191
Shooter data 57.89 (1, 35) <.001 3.39 (1, 48.1) .072 r = 0.317
Afrocentric features data
6.40 (1, 46) .015 4.33 (1, 51.1) .043 r = 0.113
Bem (2011)Forward-priming condition
22.18 (1, 98) <.001 14.59 (1, 46.91) .029 Targets: r = 0.349Primes: r = 0.035
Bem (2011) Reverse-priming condition
6.60 (1, 98) .012 2.34 (1, 27.58) .136 Targets: r = 0.292Primes: r = 0.0
Conclusion• Many failures of replication are probably due to
sampling stimuli and the failure to take that into account
• Mixed models with crossed random effects allow for generalization to future studies with different samples of both stimuli and participants
The end
Further reading:Judd, C. M., Westfall, J., & Kenny, D. A. (2012). Treating stimuli as a random factor in social psychology: A new and comprehensive solution to a pervasive but largely
ignored problem. Journal of personality and social psychology, 103(1), 54-69.