How to compute Deviance Statistic for a simple Logistic Regression Model in the case that any $n_i = y_i$?

Question

I am working on example 7.3.1 from the Second Edition of the book An Introduction to Generalized Linear Models in section 7.3 Dose response models. This example fits a simple logistic regression model on the following data:

This seems easy enough. However, I am having an issue with the Deviance Statistic calculated for this example. The following is my R code that will reproduce a Deviance Statistic $D=11.23$ just like this example in the book has.

#original data
#copied in by row
( df <-  data.frame( 
  Trial = 1:8,
  Dose = c(1.6907, 1.7242, 1.7552, 1.7842, 1.8113, 1.8369, 1.8610, 1.8839),
  Yes = c(6, 13, 18, 28, 52, 53, 61, 60),
  No = c(59, 60, 62, 56, 63, 59, 62, 60)- c(6, 13, 18, 28, 52, 53, 61, 60),
  Total = c(59, 60, 62, 56, 63, 59, 62, 60)
) )

#Logistic Regression Model
mle_beet <- glm(cbind(Yes, No)~Dose, family=binomial(logit), data=df)
mle_beet$deviance
##

Section 5.6.1 of this same book derives the Deviance Statistic for the Binomial Model to be:

$D = 2\sum^{N}_{i=1}y_{i}[ log_{e}(\frac{y_i}{\hat{y_i}})+(n_i - y_i)log_{e}(\frac{n_i - y_i}{n_i - \hat{y_i}}) ]$

However, looking closely at the given data, it can be seen that for the last row, the number of beetles killed is the same as the total number of beetles ( $n_{8}=y_{8}$ ). This means that the very last part in the sum for D is:

$ y_{8}log_{e}(\frac{y_8}{\hat{y_8}})+(n_8 - y_8)log_{e}(\frac{n_8 - y_8}{n_8 - \hat{y_8}}) = 60log_{e}(\frac{60}{\hat{y_8}})+(0)log_{e}(\frac{0}{n_8 - \hat{y_8}})$

In particular, this value contains:

$0log_{e}(0)=0(-\infty)=$ undefined

Here is the R code that agrees with this:

sum( 2*(df$Yes*(log(df$Yes/(mle_beet$fitted.values*df$Total))) + (df$Total-df$Yes)*
log((df$Total-df$Yes)/(df$Total-mle_beet$fitted.values*df$Total) ) ) )

My question is: What is the mathematical reasoning for computing the Deviance Statistic when $n_i=y_i$? What do the book and R do in the background to obtain $D=11.23$?

(Note that the book likely didn't use R to get this value, but the two agree)

Thank you!

EDIT: See the accepted answer and its comments for a great explanation.

If you happen to be computing the Deviance through the formula in R (you likely shouldn't since mle_beet$deviance shows this for you), you can replace -Inf or Nan in each vector that results from an individual operation. The following works for this example:

x <- df$Yes*(log(df$Yes/(mle_beet$fitted.values*df$Total))) 
x[is.na(x) | x==-Inf ] <- 0 #only in a case $n_i = y_i$ 
y <- (df$Total-df$Yes)*
    log((df$Total-df$Yes)/(df$Total-mle_beet$fitted.values*df$Total) ) ) 
y[is.na(y) | y==-Inf ] <- 0 #only in a case $n_i = y_i$ 

sum(x+y)*2 #the deviance

Answer 1

When calculating the binomial deviance we use the convention that $0 \log(0) = \lim_{x \downarrow 0} x \log(x) = 0$.

To understand why this convention makes sense here it's helpful for simplicity to think in terms of the Bernoulli likelihood function $\prod_{i=1}^{n} p_i^{y_i} (1 - p_i)^{1 - y_i}$ (the general binomial case will just be an extension of the main idea here), where $y_i$ is either zero or one.

Now when we talk about the saturated model (the likelihood of which is part of the deviance statistic) we mean the model where $\hat{p}_i = y_i$ for each $i$. But clearly the likelihood should equal one here (our model says that the data are certain to arise as they did), which means we should treat $0^0$ as itself being equal to one, and for the sake of consistency we should therefore treat the logarithm of $0^0$ as being equal to zero.