Why do I get different results when I want to find the AIC of my model in Python and R?

I have a dataset and I want to study the relationship between the age and the probability of using a credit card. So I decided to apply weighted logistic regression.

Python code:

#Imports

import numpy as np

import pandas as pd

import statsmodels.formula.api as smf

import statsmodels.api as sm

#Let's create a dataframe of our data

use = np.array([5,7,11,20,47,49,54,53,76])

total = np.array([30,33,35,40,67,61,63,59,80])

age = np.arange(16, 25)

p = use / total

df = pd.DataFrame({"Use": use, "Total": total, "Age": age, "P": p})

#Let's create our model

logit_model = smf.glm(formula = "P ~ Age", data = df,
                  family = sm.families.Binomial(link=sm.families.links.Logit()),
                  var_weights = total).fit()

logit_model.summary()

Output

             Generalized Linear Model Regression Results
==============================================================================
Dep. Variable:                      P   No. Observations:                    9
Model:                            GLM   Df Residuals:                        7
Model Family:                Binomial   Df Model:                            1
Link Function:                  Logit   Scale:                          1.0000
Method:                          IRLS   Log-Likelihood:                -147.07
Date:                Mon, 08 May 2023   Deviance:                       1.9469
Time:                        10:22:00   Pearson chi2:                     1.95
No. Iterations:                     5   Pseudo R-squ. (CS):              1.000
Covariance Type:            nonrobust
==============================================================================
             coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    -11.3464      1.172     -9.678      0.000     -13.644      -9.049
Age            0.5996      0.059     10.231      0.000       0.485       0.715
==============================================================================

In this model, I want to calculate the AIC.

logit_model.aic

Output

298.1385764743574

(In the above model I used the argument var_weights as Josef suggested in this thread)

Let's do the same in R.

Rcode:

use = c(5,7,11,20,47,49,54,53,76)

total = c(30,33,35,40,67,61,63,59,80)

age = seq(16,24)

p = use/total

logit_temp = glm(p~age,family = binomial, weights = total)

logit_temp

Output

Call:  glm(formula = p ~ age, family = binomial, weights = total)

Coefficients:
(Intercept)          age  
   -11.3464       0.5996  

Degrees of Freedom: 8 Total (i.e. Null);  7 Residual
Null Deviance:	    156.4 
Residual Deviance: 1.947 	AIC: 40.12

As you can see now, the AIC of the model I created with R is very different from the AIC I found with Python. What should I change in order to have the same results?

The difference reduces to a difference in the log-likelihoods: both packages appear to (correctly) compute AIC as (-2*loglikelihood + 2*df).

The help for statsmodels GLM function says:

> WARNING: Loglikelihood and deviance are not valid in models where scale is equal to 1 (i.e., Binomial, NegativeBinomial, and Poisson). If variance weights are specified, then results such as loglike and deviance are based on a quasi-likelihood interpretation. The loglikelihood is not correctly specified in this case, and statistics based on it, such AIC or likelihood ratio tests, are not appropriate.

I'm not quite sure what to do about that ...

(I spent a little while digging down to try to reproduce the log-likelihood of -147.07 from Python by computing a quasi-likelihood, but so far without success)

I'm not sure (don't remember) how aic is defined with var_weights in statsmodels GLM.

update
var_weights are treated as scale or dispersion factor in statsmodels GLM. It does not assume that there is a likelihood interpretation of var_weights, and so we only have QMLE and not MLE. (see Ben Bolker's answer).
Parameter estimates and standard errors are the same in both cases and do not directly rely on the MLE/QMLE distinction. However, loglikelihood and statistics based on it, like aic, bic, will not correspond to a correctly specified likelihood in QMLE.

However, the example is just a standard binomial model.

In this case we can specify (success, failure) counts directly as dependent variable. This produces then the same estimate and same aic as the R version.
The results are interpreted as standard maximum likelihood estimate in this case.

fail = total - use
df["fail"] = fail
logit_model2 = smf.glm(formula = "use + fail ~ Age", data = df,
                       family = m.families.Binomial(link=sm.families.links.Logit()),
                       ).fit()
​
print(logit_model2.summary())
                 Generalized Linear Model Regression Results                  
==============================================================================
Dep. Variable:        ['use', 'fail']   No. Observations:                    9
Model:                            GLM   Df Residuals:                        7
Model Family:                Binomial   Df Model:                            1
Link Function:                  Logit   Scale:                          1.0000
Method:                          IRLS   Log-Likelihood:                -18.059
Date:                Mon, 08 May 2023   Deviance:                       1.9469
Time:                        11:46:25   Pearson chi2:                     1.95
No. Iterations:                     5   Pseudo R-squ. (CS):              1.000
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept    -11.3464      1.172     -9.678      0.000     -13.644      -9.049
Age            0.5996      0.059     10.231      0.000       0.485       0.715
==============================================================================

logit_model.aic, logit_model2.aic
(298.1385764743575, 40.11730698160575)

As background.
This notebook https://www.statsmodels.org/dev/examples/notebooks/generated/glm_weights.html illustrates some of the properties of var and freq weights and their equivalent standard model when the MLE assumption holds.
This relies to a large extent on averaging and aggregation properties of the linear exponential families in GLM.

I would like to provide an answer which is similar to Josef's answer for someone who wants to use sm.GLM() function instead of smf.glm()

#Imports

import numpy as np

import pandas as pd

import statsmodels.api as sm

#Let's create a dataframe of our data

use = np.array([5,7,11,20,47,49,54,53,76])

total = np.array([30,33,35,40,67,61,63,59,80])

age = np.arange(16, 25)

fail = total - use

df = pd.DataFrame({"Use": use, "Total": total, "Age": age, "Fail": fail})

endog_star = df[["Use","Fail"]] #our response variable in form [success, failure]

exog_star = df["Age"] #our independent variable

exog_star = sm.add_constant(exog_star) #we have to include a constant because by default
#constant is not included

#Let's create our model

logit_model = sm.GLM(
    endog = endog_star,
    exog = exog_star,
    family = sm.families.Binomial(link=sm.families.links.Logit())).fit()

logit_model.summary()

logit_model.aic

Output:

                 Generalized Linear Model Regression Results
==============================================================================
Dep. Variable:        ['Use', 'fail']   No. Observations:                    9
Model:                            GLM   Df Residuals:                        7
Model Family:                Binomial   Df Model:                            1
Link Function:                  Logit   Scale:                          1.0000
Method:                          IRLS   Log-Likelihood:                -18.059
Date:                Fri, 12 May 2023   Deviance:                       1.9469
Time:                        12:53:23   Pearson chi2:                     1.95
No. Iterations:                     5   Pseudo R-squ. (CS):              1.000
Covariance Type:            nonrobust
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
const        -11.3464      1.172     -9.678      0.000     -13.644      -9.049
Age            0.5996      0.059     10.231      0.000       0.485       0.715
==============================================================================

40.11730698160573