1. PSY 626: Bayesian Statistics for Psychological Science
5/8/2018
Another example
Greg Francis
PSY 626: Bayesian Statistics for Psychological Science
Fall 2016
Purdue University
PSY200 Cognitive Psychology

2. Facial feedback
Is your emotional state influenced by your facial muscles?
If I ask you to smile, you may report feeling happier
But this could just be because you guess I want you to report feeling happier
Or because intentional smiling is associated with feeling happier
You can ask people to use smiling muscles without them realizing it

3. Facial feedback Within subjects design (n=21)
Judge “happiness” in a piece of abstract art while
Holding a pen in your teeth (smiling)
Holding a pen in your lips (frowning/pouting)
No pen
11 trials for each condition
Different art on each trial

4. Facial feedback Within subjects design (n=21)
Judge “happiness” in a piece of abstract art while
Holding a pen in your teeth (smiling)
Holding a pen in your lips (frowning/pouting)
No pen
11 trials for each condition
Different art on each trial

5. Facial feedback Within subjects design (n=21)
Judge “happiness” in a piece of abstract art while
Holding a pen in your teeth (smiling)
Holding a pen in your lips (frowning/pouting)
No pen
11 trials for each condition
Different art on each trial

6. Data
The HappinessRating is a number between 0 (no happiness) to 100 (lots of happiness)
The facial feedback hypothesis suggests that the mean HappinessRating values should be larger when the pen is held in the teeth and lower when the pen is held in the lips
File FacialFeedback.csv contains all the data

7. Loading data # load full data file
FFdata<-read.csv(file="FacialFeedback.csv",header=TRUE,stringsAsFactors=FALSE)
# load the rethinking library
library(rethinking)
# Set up dummy variables
FFdata$PenInTeeth <- ifelse(FFdata$Condition =="PenInTeeth", 1, 0)
FFdata$NoPen <- ifelse(FFdata$Condition =="NoPen", 1, 0)
FFdata$PenInLips <- ifelse(FFdata$Condition =="PenInLips", 1, 0)

8. Set up model
Dummy variables set up different values for mu (to be a1, a2, or a3)
FFmodel1 <- map(
alist( HappinessRating ~ dnorm(mu, sigma),
mu <- a1*PenInTeeth + a2*NoPen + a3*PenInLips,
a1 ~ dnorm(50, 100),
a2 ~ dnorm(50, 100),
a3 ~ dnorm(50, 100),
sigma ~ dunif(0, 50)
), data= FFdata )

9. Model Results Maximum a posteriori (MAP) model fit Formula:
HappinessRating ~ dnorm(mu, sigma)
mu <- a1 * PenInTeeth + a2 * NoPen + a3 * PenInLips
a1 ~ dnorm(50, 100)
a2 ~ dnorm(50, 100)
a3 ~ dnorm(50, 100)
sigma ~ dunif(0, 50)
MAP values:
a a a3 sigma
Log-likelihood:

10. Alternative model set up
# load full data file
FFdata<-read.csv(file="FacialFeedback.csv",header=TRUE,stringsAsFactors=FALSE)
# load the rethinking library
library(rethinking)
# Dummy variable to indicate condition
FFdata$ConditionIndex <-0*FFdata$Trial
FFdata$ConditionIndex[FFdata$Condition =="PenInTeeth"] =1
FFdata$ConditionIndex[FFdata$Condition =="NoPen"] =2
FFdata$ConditionIndex[FFdata$Condition =="PenInLips"] =3

11. Alternative model set up
FFmodel2 <- map(
alist( HappinessRating ~ dnorm(mu, sigma),
mu <- a[ConditionIndex],
a[ConditionIndex] ~ dnorm(50, 100),
sigma ~ dunif(0, 50)
), data= FFdata )
Mathematically, this is the same as the previous model, but we use ConditionIndex to automatically indicate different a[ ] values (means) for the different conditions
Each a[ ] value has the same prior applied to it

12. Alternative model set up
Maximum a posteriori (MAP) model fit
Formula:
HappinessRating ~ dnorm(mu, sigma)
mu <- a[ConditionIndex]
a[ConditionIndex] ~ dnorm(50, 100)
sigma ~ dunif(0, 50)
MAP values:
a[1] a[2] a[3] sigma
Log-likelihood:
Previous (equivalent model)
a a a3 sigma

13. Prediction of means 89% HPDI derived from the posterior distributions
# Define a sequence of ConditionIndex to compute predicted means
Condition.seq <-seq(from=1, to=3, by=1)
# use link to compute mu for each sample from posterior and for each value in Condition.seq
mu<-link(FFmodel1, data=data.frame(ConditionIndex=Condition.seq))
mu.mean <- apply(mu, 2, mean)
mu.HPDI <-apply(mu, 2, HPDI, prob=0.89)
# Plot the MAP line (same as abline done previously from the linear regression coefficients)
dev.new()
plot(HappinessRating ~ConditionIndex, data=FFdata, xlab="Condition", xaxt="n")
axis(side=1, at=c(1, 2, 3), labels=c("Pen In Teeth", "No Pen", "Pen In Lips"))
lines(Condition.seq, mu.mean, col=col.alpha("green",1))
shade(mu.HPDI, Condition.seq, col=col.alpha("green",0.4))

14. Prediction of ratings
# generate many sample HappinessRatings scores for Condition.seq using the model
sim.Happiness <- sim(FFmodel2, data=list(ConditionIndex =Condition.seq))
# Idenitfy limits of middle 89% of samples values for each NumberDistractors (PI is a function from the rethinking library)
Happiness.PI <- apply(sim.Happiness, 2, PI, prob=0.89)
# Plot with shading for prediction uncertainty of individual scores
dev.new()
plot(HappinessRating ~ConditionIndex, data=FFdata, xlab="Condition", xaxt="n")
axis(side=1, at=c(1, 2, 3), labels=c("Pen In Teeth", "No Pen", "Pen In Lips"))
lines(Condition.seq, mu.mean) # MAP line for means
shade(mu.HPDI, Condition.seq) # shaded HPDI for estimates of means
shade(Happiness.PI, Condition.seq) # shaded prediction interval for simulated Happiness values

15. Null model FFmodel <- map(
alist( HappinessRating ~ dnorm(mu, sigma),
mu <- a,
a ~ dnorm(50, 50),
sigma ~ dunif(0, 100)
), data= FFdata )

16. Null model Maximum a posteriori (MAP) model fit Formula:
HappinessRating ~ dnorm(mu, sigma)
mu <- a
a ~ dnorm(50, 50)
sigma ~ dunif(0, 100)
MAP values:
a sigma
Log-likelihood:

17. More flexible model Different means and sd’s for each condition
FFmodel3 <- map(
alist( HappinessRating ~ dnorm(mu, sigma),
mu <- a[ConditionIndex],
a[ConditionIndex] ~ dnorm(50, 50),
sigma[ConditionIndex] ~ dunif(0, 100)
), data= FFdata )

18. More flexible model Maximum a posteriori (MAP) model fit Formula:
HappinessRating ~ dnorm(mu, sigma)
mu <- a[ConditionIndex]
a[ConditionIndex] ~ dnorm(50, 50)
sigma[ConditionIndex] ~ dunif(0, 100)
MAP values:
a[1] a[2] a[3] sigma[1] sigma[2] sigma[3]
Log-likelihood:

19. Comparing models
plot(coeftab(FFmodel, FFmodel2, FFmodel3))

20. Comparing models print(compare(FFmodel, FFmodel2, FFmodel3))
WAIC pWAIC dWAIC weight SE dSE
FFmodel NA
FFmodel
FFmodel
Null model (FFmodel) and different means with same sd (FFmodel2) are nearly the same

21. Model uncertainty
When you think about predicting means, you should really consider not just a single model, but the uncertainty across the models
The Akaike weights estimate this uncertainty
Take the prediction for each model and weight it by the Akaike weight
FF.ensemble <- ensemble(FFmodel, FFmodel2, FFmodel3, data=data.frame(ConditionIndex=Condition.seq))
dev.new()
plot(HappinessRating ~ConditionIndex, data=FFdata, xlab="Condition", xaxt="n")
axis(side=1, at=c(1, 2, 3), labels=c("Pen In Teeth", "No Pen", "Pen In Lips"))
lines(Condition.seq, mu.mean) # MAP line for means
shade(mu.HPDI, Condition.seq) # shaded HPDI for estimates of means
shade(Happiness.PI, Condition.seq) # shaped prediction interval for simulated Happiness values
mu.PI <- apply(FF.ensemble$link, 2, PI)
shade(mu.PI, Condition.seq, col=col.alpha("green",0.2))

22. What does it all mean?
Relative to the variability in the HappinessRatings, the differences in means are pretty small
There’s not much difference between the null model and a model with different means
With equal sd, they are expected do about equally well at predicting future data
What do you do?
Gather more data
Develop a better model
Give up
Use what you’ve got

23. What are the limitations?
How would you improve these models?