Overlooking Stimulus Variance

Jake WestfallUniversity of Colorado Boulder

Charles M. Judd David A. KennyUniversity of Colorado Boulder University of Connecticut

Cornfield & Tukey (1956):“The two spans of the bridge of inference”

My actual samples

50 University of Colorado undergraduates;40 positively/negatively valenced English adjectives

Ultimate targets of generalization

My actual samples

All healthy, Western adults; All non-neutral visual stimuli

50 University of Colorado undergraduates;40 positively/negatively valenced English adjectives

Ultimate targets of generalization

My actual samples

All healthy, Western adults; All non-neutral visual stimuli

All CU undergraduates takingPsych 101 in Spring 2014;All short, common, stronglyvalenced English adjectives

50 University of Colorado undergraduates;40 positively/negatively valenced English adjectives

All potentially sampled participants/stimuli

Ultimate targets of generalization

My actual samples

All healthy, Western adults; All non-neutral visual stimuli

“Subject-matter span”

“Statistical span”

50 University of Colorado undergraduates;40 positively/negatively valenced English adjectives

All potentially sampled participants/stimuli

Difficulties crossing the statistical span• 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

• 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!

• Modern statistical procedures solve the statistical problem of stimulus sampling

• These linear mixed models with crossed random effects are easy to apply and are already widely available in major statistical packages– R, SAS, SPSS, Stata, etc.

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

effects models, hierarchical linear models, etc.• Most 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

ParticipantMeans5.86

7.09

-1.09

-4.53

Stimulus Means: -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

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)

Conclusion• Stimulus variation is a generalizability issue• The conclusions we draw in the Discussion sections

of our papers ought to be in line with the assumptions of the statistical methods we use

• Mixed models with crossed random effects allow us to generalize across both participants and stimuli

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.