probability in mixed multinomial logistic regression

Question

I am using MCMCglmm package in R for my multilevel multinomial logistic regression model. I have a level-1 binary outcome 'Sex', which was coded as 1,2, and a level-1 three category unordered multinomial outcome 'mathach', which was coded as 0,1,2. I also have level-1 continous predictor 'SES'. The ID variable is 'School'. The reproducible code using R dataset MathAchieve and MathAchSchool from nlme package is as below:

library(MCMCglmm)
library(nlme)
data(MathAchieve,package='nlme')
data(MathAchSchool,package='nlme')
dat=merge(MathAchSchool,MathAchieve,by='School')
dat$mathach[dat$MathAch<5]=0
dat$mathach[dat$MathAch>=5 & dat$MathAch<15]=1
dat$mathach[dat$MathAch>15]=2
dat$mathach=as.factor(dat$mathach)
str(dat)

set.seed(9689724)
m1=MCMCglmm(mathach~SES,random=~School+SES,data=dat,rcov=~us(trait):units,family='categorical',verbose=F) 
summary(m1)


 Iterations = 3001:12991
 Thinning interval  = 10
 Sample size  = 1000 

 DIC: 9944.844 

 G-structure:  ~School

       post.mean l-95% CI u-95% CI eff.samp
School    0.9069   0.6566    1.219    25.86

               ~SES

    post.mean  l-95% CI u-95% CI eff.samp
SES 0.0007462 2.728e-06 0.004624    13.33

 R-structure:  ~us(trait):units

                                    post.mean l-95% CI u-95% CI eff.samp
traitmathach.1:traitmathach.1.units   36.6564 12.44909  63.0278    1.016
traitmathach.2:traitmathach.1.units    0.2105 -0.79940   1.1571    3.582
traitmathach.1:traitmathach.2.units    0.2105 -0.79940   1.1571    3.582
traitmathach.2:traitmathach.2.units    0.1275  0.07205   0.2024   11.354

 Location effects: mathach ~ SES 

            post.mean l-95% CI u-95% CI eff.samp  pMCMC    
(Intercept)    1.1074   0.8905   1.3358    12.12 <0.001 ***
SES            1.0324   0.8863   1.1513    12.21 <0.001 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

My questions are:

1. The regression coeffecient of SES in the multinomial logistic regression is 1.0324, but which category of 'mathach' does the coefficient refer to? how to interpret it?

2. If I want to make predictions with individual having SES=0.5 and come from School 1224, how do to do that?

Answer 1

The regression coeffecient of SES in the multinomial logistic regression is 1.0324, but which category of 'mathach' does the coefficient refer to? how to interpret it?

Both, because your model has SES as both a random and fixed effect.

If I want to make predictions with individual having SES=0.5 and come from School 1224, how do to do that?

You would call the predict() function, passing the fitted model object and a data frame with SES of 0.5 and SchoolID of 1224