Checking linearity in logistic regression

Question

I am conducting logistic regression, and I am a bit confused about the linearity check? I have 18 independent variables, among them, 13 are continuous, and 5 are categorical variables. In this scenario,

1. Do I need to check for violation of linearity assumption?
2. I tried myself to check the assumption by creating an interaction term log(IV)*IV of a continuous variable. When I regressed (only with continuous variables), I found 4 variables were found significant (that means the violation of the assumption).
3. Therefore, If I need to check and treat the model accordingly, how I can treat those variables? I hope to get your advice. Thank you

Answer 1

1. Yes

2. I'm not sure how this assesses the assumption.

3. One way to assess the linearity assumption is to check the deviance residuals. If the outcome is 0/1 you will have to group the variables in an intelligent way so that the outcome is binomial rather than bernoulli.

Here is an example. In the following code, I purposefully create a non-linear logistic regression. I fit a model with only a linear term and evaluate the deviance residuals. The residuals display a non-linear pattern where they should look like a cloud around 0. This is evidence the fit is poor.

library(tidyverse)
library(broom)

set.seed(0)

N = 10000
x = rnorm(N, 0, 2)
eta = -0.4 + 0.2*x - 0.1*x^2
p = plogis(eta)
y = rbinom(N,1,p)

d = tibble(x, y) %>% 
    # bucket the continuous predictor into deciles
    mutate(q = ntile(x, 10)) %>% 
    group_by(q) %>% 
    # n- how many onbservations in the bucket, y - how many successes
    summarise(n = n(), u = sum(y))

model = glm(cbind(u, n-u)~q, data = d, family=binomial())


d$fitted = predict(model)
d$resid = resid(model, type = 'deviance')

d %>% 
  ggplot(aes(fitted, resid))+
  geom_point()

See how the residuals look U shaped? That means the linearity assumption is likely incorrect. If you have good reasons to include an interaction or higher order terms, you may do so. Else, you could use splines. Here is an example

library(splines)
model = glm(cbind(u, n-u)~bs(q), data = d, family=binomial())


d$fitted = predict(model)
d$resid = resid(model, type = 'deviance')

d %>% 
  ggplot(aes(fitted, resid))+
  geom_point()

]

Note how the deviance residuals are clustered around 0 now, with no discernible pattern.