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?