# How to back-transform glmer() cbind() proportion data

I have fit a binomial GLMM (glmer) to a response variable that is a proportion.

How my data looks:

ind_id   total_off  total_on  total  OFT1  MIS1  sex  litterid
122      50         40        90     2.33  -1.32  F    5643
134      10         20        30     -1.00 -0.22  M    2345
145      1          10        11     3.19   0.89  F    8743


Where:

• ind_id: individual identity
• total_off: count of # of times off territory
• total_on: count of # of times on territory
• total: sum of # of times off and on territory
• OFT1: continuous behaviour score
• MIS1: continuous behaviour score
• sex: categorical sex (M or F)
• litterid: identity of litter

Model:

library(lme4)
m.1<-glmer(cbind(total_off, total_on) ~ OFT1 + MIS1 + sex + (1|litterid), family=binomial, data=new)
summary(m.1)


Fixed effects output:

            Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.08019    0.19200   0.418    0.676
OFT1        -0.07030    0.06919  -1.016    0.310
MIS1         0.05392    0.04753   1.135    0.257
sexM         0.19664    0.19385   1.014    0.310


Based on chapter 16 in The R Book, I can back-transform the estimates with the following equation:

1/(1+1/(exp(-0.07030)))


But, the examples Crawley goes over are for categorical explanatory variables with a glm() model (no random effect), not glmer() model (random effect).

I have two questions:

1. For starters, should my cbind() actually be cbind(total_off, total) instead of cbind(total_off, total_on)?
2. How can I back-transform glmer model estimates? Once back transformed, how are they interpreted?

For continuous explanatory variables, such as OFT1, I want to be able to say something like: for every 1 unit increase in OFT1 score, an individual spends X% more/less time off their territory. For categorical explanatory variables, such as sex, I want to be able to say: females spend X% more/less time off their natal territory compared to males.