Comparing residual deviance to null deviance in a logistic regression model: Is percentage reduction fallacious?

Question

If I run a logistic regression model in R...for example

summary(glm(data, formula = dichotomous.outcome.variable ~ age + hpv,family = binomial(link = "logit")))

I get an output such as this

Null deviance: 142.18  on 417  degrees of freedom  
Residual deviance: 112.42  on 415  degrees of freedom
(1 observation deleted due to missingness)
AIC: 118.42
Number of Fisher Scoring iterations: 9

Is there any value to looking at (142.18-112.42)/142.18 = 0.21?

Could one say, "The intercept represents the baseline model where it simply predicts for every case, the baseline probability. So if 5% were TRUE then it simply predicts 5% chance for every case. Or put differently, since only 5% were TRUE, the baseline model could predict that every case was FALSE and it would only be wrong 5% of the cases?" Now we come along with our best model. Can we say, "Our model explains 21% of the variability"? Is there great value, little value or zero value (or maybe even dangerously contradictory value) to looking at the percent reduction of the null deviance?

Please disavow me of any misconceptions that I may harbor.

Answer 1

The intercept-only model looks like this in R:

m0 <- glm(dichotomous.outcome.variable ~ 1,family = binomial(link = "logit")), data = data) 

and stipulates that the probability of success (where success means that dichotomous.outcome.variable = 1) is constant - in particular, it does not depend on any of your measured predictor variables.

The model which includes the age and hpv predictors looks like this:

m1 <- glm(dichotomous.outcome.variable ~ age + hpv,family = binomial(link = "logit")), data = data) 

and stipulates that the probability of success depends on both age and hpv (e.g., probability of success might increase with age but decrease with hpv).

When you compare the two models, you are essentially testing two competing hypotheses:

Ho: probability of success is constant
Ha: probability of success depends on both age and hpv 

If Ho is true, a subject's age and hpv have no bearings on his/her probability of success. If Ha is true, a subject's probability of success would be influenced by their age and hpv (e.g., an old subject with a low hpv value would have a high probability of success), though the influence of age on this probability is independent of hpv and viceversa.

One way you can test these mutually exclusive hypotheses is by examining the reduction in null deviance achieved by introducing the age and hpv predictors in the intercept-only predictor. If the reduction is (statistically) significant, the data provide evidence against the null hypothesis of constant probability of success and in favour of the alternative hypothesis on non-constant probability of success which depends on age and hpv.

Thus, it is not enough to merely compute the reduction in null deviance, you must also determine if this reduction is statistically significant. See Interpreting Residual and Null Deviance in GLM R for how you might do that. Alternatively, just compare your two model fits above with the anova function:

anova(m0, m1, test = "Chisq") 

The PLoS One article Non-significant p-values? Strategies to understand and better determine the importance of effects and interactions in logistic regression by Vakhitova et al. (doi:10.1371/journal.pone.0205076) will give you a more comprehensive look at the issue discussed in my answer: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6261058/.