Null deviance

Topic of Paired comparisons, and how far a fit is from the data, part of Generalised linear models

Null deviance

Type	Topic
Taught in	Paired comparisons, and how far a fit is from the data
Part	Generalised linear models
Questions using it	2

The deviance of the intercept-only model. The drop from the null deviance to the residual deviance measures how much the covariates explain, and is tested on the difference in degrees of freedom.

In Paired comparisons, and how far a fit is from the data

From Paired comparisons, and how far a fit is from the data

Sex drops out, as we already found. Cholesterol alone, three links:

chol_file <- "data/cholest.csv"
if (!file.exists(chol_file)) {
  stop("Could not find data/cholest.csv. Set the working directory to the folder that contains it.")
}
chol <- read.csv(chol_file)
for (lnk in c("logit", "probit", "cloglog")) {
  fit <- glm(disease ~ cholesterol, data = chol, family = binomial(link = lnk))
  cat(lnk, "null", fit$null.deviance, "on", fit$df.null, "df,",
      "residual", fit$deviance, "on", fit$df.residual, "df,",
      "AIC", AIC(fit), "\n")
}

binomial logit:    null 137.63 on 99 df,  residual 114.04 on 98 df,  AIC 118.04
binomial probit:   null 137.63 on 99 df,  residual 114.12 on 98 df,  AIC 118.12
binomial cloglog:  null 137.63 on 99 df,  residual 111.90 on 98 df,  AIC 115.90

The null deviance does not depend on the link. It is the intercept-only model, one probability for everyone. Logit and probit are not separable on this sample.

Figure 1: The same cholesterol data under three links. The logit and the probit sit on each other. Complementary log-log stays a little lower through the middle and then climbs more steeply through the high-cholesterol cases.

In Figure the solid and dashed curves sit on each other. Complementary log-log has the smaller deviance and the smaller AIC. It rises more slowly and then more steeply, and it picks up the high-cholesterol ones a little better. We cannot test that improvement. The models are not nested. There is no constraint on the logit coefficients that turns the fitted curve into a probit.

Used in

Taught in Paired comparisons, and how far a fit is from the data
Also used in Logistic regression
Questions that use it (2) — Q34 · Question 19
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In Paired comparisons, and how far a fit is from the data

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