How to change the parameters of the binomial distribution used in a binomial glmer?

#1
I wish to assess how being part of a majority helps in emerging as a leader of an animal group.

Say I have 10 cases in which I assessed whether the leader came from the majority or the minority.

Leader <- c(1,1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0)
Case <- as.factor(c(1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10))
Majority <- as.factor(c("Maj","Maj","Maj","Maj","Maj","Maj","Maj","Maj","Maj","Maj",
"Min","Min","Min","Min","Min","Min","Min","Min","Min","Min"))
leadMaj <- data.frame(Leader,Case,Majority)

binomial.glmer <- glmer(Leader ~ Majority + (1|Case),
family = binomial, data = leadMaj)
summary(binomial.glmer)

The outcome says that being from the minority drastically decreases the probability to lead the group

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) [glmerMod]
Family: binomial ( logit )
Formula: Leader ~ Majority + (1 | Case)
Data: leadMaj

AIC BIC logLik deviance df.resid
26 29 -10 20 17

Scaled residuals:
Min 1Q Median 3Q Max
-2.0 -0.5 0.0 0.5 2.0

Random effects:
Groups Name Variance Std.Dev.
Case (Intercept) 0 0
Number of obs: 20, groups: Case, 10

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.3863 0.7906 1.754 0.0795 .
MajorityMin -2.7726 1.1180 -2.480 0.0131 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr)
MajorityMin -0.707

However, the groups were composed of 8 individuals in the majority and 2 individuals in the minority. We can see that in 80% of the cases the majority led, which is what is expected.

So the question is: how can I include the binomial distribution with p=0.8, and not p=0.5?
 

ledzep

Point Mass at Zero
#2
Re: How to change the parameters of the binomial distribution used in a binomial glme

First thing regarding the output, your variance of random effect is zero. Hence, there is no need to specify Case as a random effect. You can simply run a glm.

Code:
binomial.glm<- glm(Leader ~ Majority , 
			family = binomial, data = leadMaj)
However, the groups were composed of 8 individuals in the majority and 2 individuals in the minority. We can see that in 80% of the cases the majority led, which is what is expected.
So the question is: how can I include the binomial distribution with p=0.8, and not p=0.5?
Your model accounts the data pretty well. Just look at the predicted values.

Code:
# Compute the predicted response
pred<-predict(binomial.glm, type="response")

#Examine the histogram of predicted probabilities
hist(pred)

# You see two peaks here: one at 0.2 and other at 0.8- as observed in your dataset.
I hope this helps. Sorry, I didnt get your question very well.