# For a logistic regression of a 2 by 2 table using glm in R, is using cbind or using a full data matrix for the response the correct method?

For a 2 by 2 table that looks like:

Response       No Response
Treatment Given                 25                60
Treatment Not Given             55                43

We may fit a logistic regression by introducing the table as a binomial relationthrough cbind:

glm(formula = cbind(c(25, 60),c(55, 43)) ~ as.factor(c(1, 0)), family = binomial())

It seems from reading through related posts on this topic that another option exists, by using:

DAT <- cbind(c(rep(1, 80), rep(0, 103)), c(rep(1, 25), rep(0,55), rep(1, 60), rep(0, 43)))

with model:

glm(DAT[,2]~DAT[,1], family = binomial())

Now, the FIRST MODEL outputs:

Call:  glm(formula = cbind(c(25, 60), c(55, 43)) ~ as.factor(c(1, 0)),
family = binomial())

Coefficients:
(Intercept)  as.factor(c(1, 0))1
0.3331              -1.1216

Degrees of Freedom: 1 Total (i.e. Null);  0 Residual
Null Deviance:      13.42
Residual Deviance: -7.55e-15    AIC: 13.75

while the SECOND MODEL outputs:

Call:  glm(formula = DAT[, 2] ~ DAT[, 1], family = binomial())

Coefficients:
(Intercept)     DAT[, 1]
0.3331      -1.1216

Degrees of Freedom: 182 Total (i.e. Null);  181 Residual
Null Deviance:      252.8
Residual Deviance: 239.3    AIC: 243.3

The coefficient estimates and p-values are the SAME, but they differ on the degree of freedom and Null/Residual deviance. The question is: are these two models the same or are they actually different? Which one is the correct one? Thanks!