Interpreting intercept in logistic regression when there is more than one categorical variable

Question

mydata <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
#create a binary variable just for the purpose of experimentation
mydata$bin = rbinom(nrow(mydata), p = 0.7, size = 1)
mydata$rank <- factor(mydata$rank)
mydata$bin <- factor(mydata$bin)
#Run the model with just bin
my.mod <- glm(admit ~ bin, data = mydata, family = "binomial")
summary(my.mod)

Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.8434294 0.1966085 -4.28989 0.000017876
bin1 0.1121304 0.2347628 0.47763 0.63291

The intercept is simply logodds of admit == 1 when bin1 = 0

Check:

x = subset(mydata, bin == "0")
mean(x$admit)
log(0.3008130081/(1-0.3008130081))

Result:
-0.843429383 Tallied

Next, do the same with rank variable alone in the model

my.mod <- glm(admit ~ rank, data = mydata, family = "binomial")
summary(my.mod)

Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.1643031 0.2569384 0.63946 0.522521
rank2 -0.7500300 0.3079693 -2.43540 0.014875
rank3 -1.3646980 0.3353867 -4.06903 0.000047210
rank4 -1.6867296 0.4093073 -4.12094 0.000037733

The intercept is simply logodds of admit == 1 when rank2 = 0 & rank3 = 0 & rank4 = 0; in other words rank == 1

Check:

x = subset(mydata, rank == "1" )
mean(x$admit)
log(0.5409836066/(1-0.5409836066))

Result:
0.1643030515 Tallied

Next, I add both the categorical variables in the model

my.mod <- glm(admit ~ rank + bin, data = mydata, family = "binomial")
summary(my.mod)

Coefficients: Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.23245129 0.32715773 0.71052 0.477383
rank2 -0.75673758 0.30869909 -2.45138 0.014231
rank3 -1.38523274 0.34119770 -4.05991 0.000049091
rank4 -1.69956417 0.41127752 -4.13240 0.000035899
bin1 -0.08308446 0.24667503 -0.33682 0.736255

Question: How to make sense of this intercept like above 2 examples? I don't think in this case intercept equals logodds of admit == 1 when (bin = 0 and rank = 1).

Check:

 x = subset(mydata, rank == "1" & bin == "0" )
 mean(x$admit)
 log(0.6363636364/(1-0.6363636364))

Result: 0.5596157881 Not tallied!

Answer 1

So, you have two binary predictors, say $X$ and $Z$, and the logistic model $$\DeclareMathOperator{\P}{\mathbb{P}} \P(Y=1 \mid X=x) = \frac{e^{\beta_0 + \beta_1 x}}{1+e^{\beta_0 + \beta_1 x}} $$ setting $X=0$ you can see that $\beta_0$ is the log odds of probability that $Y=1$ given $X=0$. Now adding the second predictor in the model: $$ \P(Y=1 \mid X=x,Z=z) = \frac{e^{\beta_0 + \beta_1 x+\beta_2 z}}{1+e^{\beta_0 + \beta_1 x + \beta_2 z}} $$ and you can see that in this model $\beta_0$ is the log odds of the probability that $Y=1$ given both $X=0$ and $Z=0$ (in your code example you look at the conditioning $X=0, Z=1$).