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I generated binomial data showing a logistic correlation to a predictor. I analysed such dataset with a generalised linear model which assumes a binomial distribution of residuals and a "logit" link function, but the residuals of such a model still fail to respect the model assumptions: the quantile-quantile plot shows that the residuals' distribution departs from the theoretical one, and residuals seem heteroscedastic. This happens regardless of sample size. Why is that?

Here is my example, simulated and analysed in R. To give the data some realism: let's imagine that I am interested in the probability of bird predation under different degrees of vegetation cover (expressed as percentage). For each level of vegetation cover I tracked 15 birds and recorded if they had been killed by a predator by the end of the study (data coded as 1 or 0).

# Generate data:
set.seed(666)
predictor <- c(0, 20, 40, 50, 70, 90)
y = 4 - 0.2 * predictor # linear combination of the variables
pr = 1/(1+exp(-y)) # pass y through an inv-logit function to get probability

# build dataset:

df <- data.frame(vegetation.cover = rep(predictor, 15), prob = rep(pr, 15))
df$predation.01 <- rbinom(length(df[,1]), 1, pr)
    
# df$predation.01 refers to the probability of a bird to be killed by a predator. 
# To model predation.01 vs vegetation.cover:

glm.01 <- glm(predation.01 ~ vegetation.cover,
    data= df,
    family=binomial(link="logit")
    )

These are the model diagnostic plots:

Diagnostic plots for glm.01 fitted to a dataset with replication n=15 for each value of the predictor

For comparison, these are the diagnostic plots for the same model fitted on a dataset with a replication of 1500 for each value of the predictor:

Diagnostic plots for glm.01 fitted to a dataset with replication n=1500 for each value of the predictor

Here are the data together with the model predictions for n=15:

data and predictions by glm.01 fitted to the dataset with replication n=15 for each value of the predictor

I added a pinch of artificial noise to the observations to avoid a perfect overlap of all the 0s and 1s in an attempt to aid the interpretation of the graph.

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  • $\begingroup$ See: stats.stackexchange.com/questions/121490/… and stats.stackexchange.com/questions/307221/… $\endgroup$ Commented Jun 1, 2023 at 16:11
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    $\begingroup$ I suggest adding quantile residuals as implemented in the DHARMa package. $\endgroup$ Commented Jun 1, 2023 at 16:28
  • $\begingroup$ You are using extraordinarily small probabilities at cover values greater than 40. A quick check suggests that at cover = 40 the probability is less than 0.02, and for the higher values it becomes increasingly infinitesimal (less than $10^{-6}$ at cover = 90). $\endgroup$
    – EdM
    Commented Jun 1, 2023 at 17:04

1 Answer 1

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It may help you to read my answer to Interpretation of plot (glm.model). The plots that R is generating are not specific to a logistic regression. They are the plots for a standard OLS regression. For example, the qq-plot is not of the residuals vs a binomial distribution. It shows them relative to a normal distribution. However, there is no theoretical reason that the residuals from a logistic regression 'should' be normally distributed. On the other hand, the residuals from a logistic regression very much should be heteroscedastic, because the variance is a function of the mean. Your model seems fine, as far as I can tell.

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  • $\begingroup$ Thanks! What confuses me is that if I generate y values with an exponential correlation to x values and a Poisson distribution, the diagnostic plots for the corresponding glm(y~x, family=binomial(link="logit")) model indicate normality and homoscedasticity of the residuals. How comes the diagnostic plots show normalisation and homogenization of residuals for a Poisson GLM fitted to Poisson-distributed data, but the same doesn't happen for a binomial GLM fitted to binomial data? $\endgroup$ Commented Jun 4, 2023 at 12:17
  • $\begingroup$ @MarcoPlebani, they aren't actually normal, but they're close. The Poisson is much closer to a normal than a binomial. $\endgroup$ Commented Jun 4, 2023 at 12:23

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