berkeley <- as.data.frame(UCBAdmissions) 
head(berkeley)
##      Admit Gender Dept Freq
## 1 Admitted   Male    A  512
## 2 Rejected   Male    A  313
## 3 Admitted Female    A   89
## 4 Rejected Female    A   19
## 5 Admitted   Male    B  353
## 6 Rejected   Male    B  207

Logit model corresponding to the Log Linear Model

berk.logit2 <- glm(Admit == "Admitted" ~ Dept + Gender,data = berkeley, weights = Freq, family = "binomial") 
summary(berk.logit2)
## 
## Call:
## glm(formula = Admit == "Admitted" ~ Dept + Gender, family = "binomial", 
##     data = berkeley, weights = Freq)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -25.3424  -13.0584   -0.1631   16.0167   21.3199  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   0.58205    0.06899   8.436   <2e-16 ***
## DeptB        -0.04340    0.10984  -0.395    0.693    
## DeptC        -1.26260    0.10663 -11.841   <2e-16 ***
## DeptD        -1.29461    0.10582 -12.234   <2e-16 ***
## DeptE        -1.73931    0.12611 -13.792   <2e-16 ***
## DeptF        -3.30648    0.16998 -19.452   <2e-16 ***
## GenderFemale  0.09987    0.08085   1.235    0.217    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 6044.3  on 23  degrees of freedom
## Residual deviance: 5187.5  on 17  degrees of freedom
## AIC: 5201.5
## 
## Number of Fisher Scoring iterations: 6

The summary of the logit model shows that the coefficients for departments decline as we go from Dept. A to Dept. F. GenderFemale coefficient [exp(0:0999) = 1.105] indicates the women more likely to be admitted. They have around 10.5% advantage over men.

Logit model allowing 1 df term for Dept. A

berkeley <- within(berkeley,
                   dept1AG <- (Dept=='A')*(Gender=='Female'))
berk.logit3 <- glm(Admit=="Admitted" ~ Dept + Gender + dept1AG,
                   data=berkeley, weights=Freq, family="binomial")

summary(berk.logit3)
## 
## Call:
## glm(formula = Admit == "Admitted" ~ Dept + Gender + dept1AG, 
##     family = "binomial", data = berkeley, weights = Freq)
## 
## Deviance Residuals: 
##      Min        1Q    Median        3Q       Max  
## -24.6314  -13.1867    0.0142   15.8566   22.1022  
## 
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept)   0.49212    0.07175   6.859 6.94e-12 ***
## DeptB         0.05206    0.11187   0.465    0.642    
## DeptC        -1.08802    0.11425  -9.523  < 2e-16 ***
## DeptD        -1.14250    0.11157 -10.240  < 2e-16 ***
## DeptE        -1.56102    0.13272 -11.762  < 2e-16 ***
## DeptF        -3.15321    0.17336 -18.189  < 2e-16 ***
## GenderFemale -0.03069    0.08676  -0.354    0.724    
## dept1AG       1.08277    0.27666   3.914 9.09e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 6044.3  on 23  degrees of freedom
## Residual deviance: 5169.8  on 16  degrees of freedom
## AIC: 5185.8
## 
## Number of Fisher Scoring iterations: 6

Computing likelihood-ratio tests of the terms in a model

library(car)
Anova(berk.logit2)
## Analysis of Deviance Table (Type II tests)
## 
## Response: Admit == "Admitted"
##        LR Chisq Df Pr(>Chisq)    
## Dept     763.40  5     <2e-16 ***
## Gender     1.53  1     0.2159    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(berk.logit3)
## Analysis of Deviance Table (Type II tests)
## 
## Response: Admit == "Admitted"
##         LR Chisq Df Pr(>Chisq)    
## Dept      646.72  5  < 2.2e-16 ***
## Gender      0.13  1     0.7236    
## dept1AG    17.65  1  2.658e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1