When dealing with an imbalanced dataset in the context of applying a logistic regression (Logit) model on heart disease data, there are several steps you can take to address the imbalance and evaluate the performance of your model. Let's go through your approach and address your questions:

Approach A: Logistic regression with the original imbalanced dataset

1. Splitting data: Splitting the data into training and test sets is a good practice to evaluate your model's performance.

2. Fitting the model and making predictions: Fit a logistic regression classifier on the training data and use the trained model to make predictions on the test set.

3. Confusion matrix and metrics: Calculate the confusion matrix (CM) using the predicted values from step 2 and the actual values in the test set. From the confusion matrix, you can compute metrics such as recall and precision.

4. Performance measure: You mentioned choosing the Area Under the Precision-Recall Curve (AUC-PR) as your performance measure. It captures the trade-off between precision and recall, which is relevant for imbalanced datasets.

Approach B: Applying SMOTE or random oversampling to address imbalance

1. Splitting data: Similar to approach A, split the data into training and test sets.

2. Oversampling: Apply either SMOTE or random oversampling techniques to balance the dataset. These techniques generate synthetic or random samples to increase the representation of the minority class.

3. Fitting the model and making predictions: Fit a logistic regression classifier on the balanced training data and make predictions on the test set.

4. Confusion matrix and metrics: Calculate the confusion matrix and evaluate metrics such as recall and precision using the predicted values and the true values in the test set.

5. Performance measure: Continue using the AUC-PR as your performance measure to assess the model's performance.

Now let's address your additional questions:

- Threshold tuning: Threshold tuning is applicable in both cases A and B. You can optimize the threshold value to balance precision and recall based on your specific requirements. You can use metrics like the precision-recall curve or the receiver operating characteristic (ROC) curve to determine the best threshold value.

- Precision and recall metrics: Yes, you can calculate precision and recall metrics after threshold tuning to assess the performance of your model at the chosen threshold value.

- Pseudo R2: Pseudo R2 measures, such as McFadden's R2 or Nagelkerke's R2, are often used to assess the goodness-of-fit of logistic regression models. They provide information about how well the model explains the variation in the data. Checking these coefficients can be helpful in understanding the overall fit of the model.

Remember that addressing class imbalance is crucial, and oversampling techniques like SMOTE or random oversampling can help improve the performance of your model on the minority class. Additionally, evaluating multiple metrics and considering the specific requirements of your problem is important for a comprehensive assessment of your model's performance.

If you have further questions or need more specific guidance, please don't hesitate to ask.