Q1:

1.Derive and interpret the maximal likelihood estimate of the model parameters for a logistic regression model with one dichotomous predictor.

2.Use the derived model in Q1 to construct a classification criterion for Y=0 versus Y=1, interpret the meaning of your classification criterion.

3.Consider the diagnosis of diabetes with the following loss function. If a diabetes patient is misdiagnosed, the loss is 2; if a pre-diabetes or a healthy patient is misdiagnosed, the loss is 1; correct diagnoses have no loss. Derive a classification rule that minimizes the mean prediction loss.

4.Use the loss function in Question 3, assume that the likelihood of blood glucose level and systolic blood pressure for diabetes patients, pre-diabetes patients, and healthy subjects are normally distributed (bivariate normal models) with equal covariance matrices, derive a criterion to classify healthy subjects versus diabetes patients (without caring about pre-diabetes patients).

5.Use the loss function in Question 3, assume that the likelihood of blood glucose level and blood pressure for diabetes patients, pre-diabetes patients, and healthy subjects are normally distributed (bivariate normal models) with unequal covariance matrices, derive a criterion to classify healthy subjects versus diabetes patients.