Heteroskedasticity in logistic regression

Question

I'd like to solve the heteroskedasticity in logistic regression. In my problem, I have two numeric and 23 dummies variables. I tried to transform the two numerical variables using log, min-max normalization and standard normal transformation but the model continues presenting this phenomenon. How to solve this problem?

My R output

Call:
glm(formula = TURMA_PROFICIENTE ~ ., family = "binomial", data = treinamento3, 
    model = T)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.5633  -0.6633  -0.4702  -0.2725   3.2180  

Coefficients:
                                Estimate Std. Error z value Pr(>|z|)    
(Intercept)                   -11.468260   0.234033 -49.003  < 2e-16 ***
MODA_ID_DEPENDENCIA_ADM_TURMA   0.207687   0.029116   7.133 9.82e-13 ***
TAMANHO_TURMA                   0.025761   0.002113  12.191  < 2e-16 ***
PERC_ALUNOS_GOSTAM_MT           0.855038   0.092606   9.233  < 2e-16 ***
TX_RESP_Q001B                   0.294212   0.029333  10.030  < 2e-16 ***
TX_RESP_Q004S_EM                0.204347   0.087208   2.343 0.019119 *  
TX_RESP_Q005                    0.139776   0.012944  10.798  < 2e-16 ***
TX_RESP_Q008                    0.073287   0.014984   4.891 1.00e-06 ***
TX_RESP_Q010                    0.032345   0.006231   5.191 2.09e-07 ***
TX_RESP_Q018                    0.057162   0.020725   2.758 0.005815 ** 
TX_RESP_Q020                    0.042434   0.017486   2.427 0.015233 *  
TX_RESP_Q022C                   0.133927   0.031147   4.300 1.71e-05 ***
TX_RESP_Q028                    0.026202   0.014779   1.773 0.076234 .  
TX_RESP_Q048                    0.188193   0.022012   8.549  < 2e-16 ***
TX_RESP_Q052                    0.239548   0.015695  15.263  < 2e-16 ***
TX_RESP_Q054                    0.031970   0.011816   2.706 0.006814 ** 
TX_RESP_Q060                    0.036555   0.016207   2.255 0.024106 *  
TX_RESP_Q074                    0.166943   0.032754   5.097 3.45e-07 ***
TX_RESP_Q075                    0.121384   0.033159   3.661 0.000252 ***
TX_RESP_Q095                    0.206870   0.023490   8.807  < 2e-16 ***
TX_RESP_Q096                    0.328982   0.016370  20.097  < 2e-16 ***
TX_RESP_Q098                    0.117467   0.033336   3.524 0.000426 ***
TX_RESP_Q099                    0.203174   0.013005  15.622  < 2e-16 ***
TX_RESP_Q106                    0.469938   0.022099  21.265  < 2e-16 ***
TX_RESP_Q108                    0.047157   0.015743   2.995 0.002740 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 39156  on 42108  degrees of freedom
Residual deviance: 34932  on 42084  degrees of freedom
AIC: 34982

Number of Fisher Scoring iterations: 5

Breush-Pagan test

 bptest(fit3)

    studentized Breusch-Pagan test

data:  fit3
BP = 3559.6, df = 24, p-value < 2.2e-16

My plot of the fitted_values vs residuals

Answer 1

Logistic regression is for a binary response variable. It should be distributed as a Bernoulli or, more generally, a binomial. For either of those, the variance is a function of the mean:

\begin{align} \newcommand{\Var}{{\rm Var}} \text{Bernoulli: }\quad \Var(Y) &= \quad\!\pi(1-\pi) \\ \text{Binomial: }\quad \Var(Y) &= N\pi(1-\pi) \end{align}

where $\pi$ is the parameter that controls the behavior of the distribution, namely the probability of 'success' (or the mean of a vector of $0$s and $1$s).

Thus, if the variables have any association with the response at all, even if not significant, then the variance also has to change as a function of the variables. That is, you expect to have heteroscedasticity. Homoscedasticity is not an assumption of logistic regression the way it is with linear regression (OLS).