Prior Covariance Matrix for Fixed Effects.
1 
formula 

data 

intercept 
prior standard deviation for the intercept 
scale 
prior standard deviation for regression parameters 
singular.ok 
logical: if 
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 tdistribution 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$V
can be defined for the regression parameters without input standardisation that corresponds to a diagonal prior D for the regression parameters had the inputs been standardised. The diagonal elements of D are set to scale^2
except the first which is set to intercept^2
. With logistic regression D=pi^2/3+v gives a prior that is approximately flat on the probability scale, where v is the total variance due to the random effects. For probit regression it is 1+v.
prior covariance matrix
Jarrod Hadfield j.hadfield@ed.ac.uk
Gelman, A. et al. (2008) The Annals of Appled Statistics 2 4 13601383
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  dat<data.frame(y=c(0,0,1,1), x=gl(2,2))
# data with complete separation
#####################
# probit regression #
#####################
prior1<list(
B=list(mu=c(0,0), V=gelman.prior(~x, data=dat, scale=sqrt(1+1))),
R=list(V=1,fix=1))
m1<MCMCglmm(y~x, prior=prior1, data=dat, family="ordinal", verbose=FALSE)
c2<1
p1<pnorm(m1$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))

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