MCMCglmm (version 2.36)

gelman.prior: Prior Covariance Matrix for Fixed Effects.

Description

Prior Covariance Matrix for Fixed Effects.

Usage

gelman.prior(formula, data, scale=1, intercept=scale, singular.ok=FALSE)

Value

prior covariance matrix

Arguments

formula

formula for the fixed effects.

data

data.frame.

intercept

prior standard deviation for the intercept

scale

prior standard deviation for regression parameters

singular.ok

logical: if FALSE linear dependencies in the fixed effects are removed. if TRUE they are left in an estimated, although all information comes form the prior

Author

Jarrod Hadfield

Details

Gelman et al. (2008) suggest that the input variables of a categorical regression are standardised and that the associated regression parameters are assumed independent in the prior. Gelman et al. (2008) recommend a scaled t-distribution with a single degree of freedom (scaled Cauchy) and a scale of 10 for the intercept and 2.5 for the regression parameters. If the degree of freedom is infinity (i.e. a normal distribution) then a prior covariance matrix B$V can be defined for the regression parameters without input standardisation that corresponds to a diagonal prior \({\bf D}\) for the regression parameters had the inputs been standardised. The diagonal elements of \({\bf D}\) are set to scale^2 except the first which is set to intercept^2. With logistic regression \(D=\pi^{2}/3+\sigma^{2}\) gives a prior that is approximately flat on the probability scale, where \(\sigma^{2}\) is the total variance due to the random effects. For probit regression it is \(D=1+\sigma^{2}\).

References

Gelman, A. et al. (2008) The Annals of Appled Statistics 2 4 1360-1383

Examples

dat<-data.frame(y=c(0,0,1,1), x=gl(2,2))
# data with complete separation

#####################
# probit regression #
#####################

prior1<-list(
  B=list(mu=c(0,0), V=gelman.prior(~x, data=dat, scale=sqrt(1+1))), 
  R=list(V=1,fix=1))

m1<-MCMCglmm(y~x, prior=prior1, data=dat, family="ordinal", verbose=FALSE)

c2<-1
p1<-pnorm(m1$Sol[,1]/sqrt(1+c2)) # marginal probability when x=1

#######################
# logistic regression #
#######################

prior2<-list(B=list(mu=c(0,0), V=gelman.prior(~x, data=dat, scale=sqrt(pi^2/3+1))),
             R=list(V=1,fix=1))

m2<-MCMCglmm(y~x, prior=prior2, data=dat, family="categorical", verbose=FALSE)

c2 <- (16 * sqrt(3)/(15 * pi))^2
p2<-plogis(m2$Sol[,1]/sqrt(1+c2)) # marginal probability when x=1

plot(mcmc.list(p1,p2))