What are some of the methods to check and improve the fit of a logistic regression model?

To make a confusion matrix, you have to first binarize your predictions by applying some threshold. A threshold of 0.5 is often applied by default, which may or may not be appropriate for your use case. Consider to what extent your model's predictions are (or should be) calibrated, and what precision/recall trade-off is appropriate for your problem. You may want to think through the costs to your application of different kinds of error. You might even choose your threshold on the basis of some aspect of the resulting confusion matrix.

As rightly pointed out by Aaron Schumacher, you need to binarize predictions before building a confusion matrix. With that, you lose a lot of information! Say you have a threshold of 0.5. Is a prediction of 0.44 the same as a prediction of 0.01? Communicating uncertainty within a company and outwards to customers is hard, but I think we should persevere. So, please use other techniques as well, not just confusion matrices and ROC/AUC.

To check and improve the fit of a logistic regression model, methods include analyzing the confusion matrix to evaluate the model's predictive accuracy by comparing actual versus predicted classifications. This aids in identifying the rates of true positives, true negatives, false positives, and false negatives. Additionally, one can use metrics derived from the confusion matrix such as precision, recall, F1-score, and accuracy to assess model performance. To improve fit, consider techniques like feature selection, regularization (e.g., L1, L2 penalties), and interaction terms to address overfitting or underfitting, enhancing the model's predictive capability.

To evaluate and enhance the fit of a logistic regression model, the ROC curve (Receiver Operating Characteristic curve) and AUC (Area Under the Curve) are pivotal. The ROC curve plots the true positive rate against the false positive rate at various threshold settings, providing insights into the model's ability to distinguish between classes. AUC, quantifying the entire two-dimensional area underneath the entire ROC curve, offers a single measure of the model's performance. A higher AUC indicates better model performance. Improving the model might involve adjusting the decision threshold, feature engineering, or model tuning to maximize the AUC, thereby enhancing the model's discriminative ability.

The Hosmer-Lemeshow test is a statistical test used to evaluate the goodness-of-fit of a logistic regression model. It compares the observed event rates in groups of observations to the predicted probabilities of the event to assess whether the model adequately fits the data. To improve model fit if the Hosmer-Lemeshow test indicates a lack of fit, consider revisiting model assumptions, incorporating interaction terms, or using more flexible modeling approaches like spline techniques for continuous variables. Adjusting the model based on test results can help achieve a better alignment between predicted probabilities and observed outcomes, enhancing the model's predictive accuracy.

It is important to note that the Hosmer-Lemeshow test is a low-power test. This means that even if the test produced a high p-value, there could still be a lack of fit to the model. On the other hand, if the null hypothesis was rejected, then we can be highly confident something was wrong with the model fit! Also, the test is appropriate only when the set of predictors are unique (or approximately) across observations. If multiple observations are outputting the exact same fitted probabilities, the assumption that the test statistic comes from a Chi-squared distribution will not hold.

Residual analysis in logistic regression involves examining the differences between observed outcomes and the probabilities predicted by the model. It's a crucial method for checking the model's fit. By analyzing patterns in the residuals, you can identify model misspecifications, such as non-linearity, omitted variables, or interactions that have not been considered. To improve the model's fit, you may need to transform variables, add or remove predictors, or include interaction terms. Residual plots against predicted values or individual predictors can help visually diagnose issues, guiding adjustments to enhance the model's predictive performance.

Not a good idea! The suggested scatter plot (predictions versus residuals) will generate an "X", with large residuals (~1 in abs value) occurring when prediction ~ 0 and ~1, and smaller residuals (abs(r)<0.5) when prediction ~ 0.5. I recommend an analysis of quantile residuals (Dunn and Smyth, JCGS 1996) instead.

Few use cases would help. For me I prefer practical examples rather then theoretical examples. For example any real-time project where these methods have been used.