# Interpretation of Intercept in mixed model with repeated contrasts

I am running the following logistic mixed model in lme4:

AllCond_logme <- glmer(Response ~ Condition + (Condition | Pt_ID),
family = binomial(), data = lme.df,
control = glmerControl(optimizer = "bobyqa"))


where Response is a binary outcome (1 vs 0).

I computed repeated contrasts and applied them to the Condition predictor, which is a factor with 4 levels (0, 25%, 50%, 75%). The contrasts are:

1. cH01: 25% - 0
2. cH02: 50% - 25%
3. cH03: 75% - 50%

To create the contrasts, I first computed the following hypothesis matrix:

0 25% 50% 75%
Intercept 0.25 0.25 0.25 0.25
cH01 -1 1 0 0
cH02 0 -1 1 0
cH03 0 0 -1 1

where the intercept is the mean probability of Response=1 across all conditions.

I then use ginv() to convert to a contrast matrix and apply this to 'Condition'.

*Note: I now realise that this was equivalent to using the contr.sdif() function in the MASS package.

The mixed model results are as follows:

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)    -0.1197     0.1021  -1.173    0.241
ConditioncH01   1.6971     0.1587  10.692   <2e-16 ***
ConditioncH02   1.9622     0.1108  17.713   <2e-16 ***
ConditioncH03   1.5400     0.1370  11.240   <2e-16 ***


The results seem plausible except for my result for the intercept.

I believe that the intercept should represent the mean probability of Response=1 across all conditions, and the mixed model tests whether this is significantly different to 0. The probability of Response=1 should be significantly different to 0 (across all conditions, about 40% of responses are Response=1). Is my interpretation of the intercept correct, and why is it not significant?

I've also attached a plot of Response= 1 ('Tone Present' response) across the conditions as a visual aid to understand the analysis and also to demonstrate how the intercept should differ from 0.