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1. State whether the following statements are True or False. Give reason in support of your answer:

(a) The value of sensitivity of the following results of a diagnostic test is \(0.85\).

(b) For the following cohort study, the relative risk for the lung cancer among smokers is \(3.5\).

(c) The logit link function is \(\log [-\log (1-\pi)]\).

(d) We define three indicator/dummy variables for a regressor variable with three categories.

(e) Left censoring occurs whenever the exact time of occurrence of an event is not known.

2. Differentiate between Chi-square tests for association and homogeneity of proportions. Also mention the assumptions of these tests.

3. A random sample of 250 patients was selected and their workout timing and diabetes status were recorded. The following table shows the workout timing and severity of diabetes:

Test at \(5 \%\) level of significance whether workout habit and diabetes are associated with to each other or not.

4.(a) Explain the assumptions underlying multiple linear regression model.

(b) Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in \(\mathrm{mg} / \mathrm{dL}\) ) and systolic blood pressure (in \(\mathrm{mm} / \mathrm{Hg}\) ) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:

(i) Fit a linear regression model using the method of least squares.

(ii) Construct the normal probability plot for the data on serum cholesterol and systolic blood pressure.

(iii) Test the significance of the fitted regression model.

5. Write a short note on the following:

(i) Polytomous logistic models

(ii) Poisson regression

(iii) Kaplan and Meier method

6. The following data on diagnosis of coronary heart disease (where 0 indicating absence and 1 indicating presence), serum cholesterol (in \(\mathrm{mg} / \mathrm{dl}\) ), resting blood pressure (in \(\mathrm{mmHg}\) ) and weight (in \(\mathrm{kg}\) ) were obtained for 80 patients to explore the relationship of coronary heart disease with cholesterol and weight.

(i) Fit a multiple logistic model for the dependence of coronary heart disease on the average serum cholesterol and weight considering \(\hat{\beta}_{0}^{0}=4.279, \hat{\beta}_{1}^{0}=-0.035\) and \(\hat{\beta}_{2}^{0}=0.172\) as the initial values of the parameters (solve only for one Iteration).

(ii) Test the significance of the fitted model using Hosmer-Lemeshow test at \(5 \%\) level of significance.

7.(a) Describe censoring and differentiate between different types of censoring with the help of examples which are not considered in Block 4 of MSTE-004.

(b) A study was conducted on 185 patients aged more than 45 years which are followed until the time of death or up to 10 years, whichever comes first. The patients have different covariates: age, gender (male/female), systolic blood pressure, smoking (yes/no), total serum cholesterol and diabetes (yes/no). The objective of this study is to determine which covariate influences the survival time. An analysis is conducted to investigate differences in all-cause mortality between men and women participating in the study. Suppose we obtain the following results after applying the Cox regression hazard model analyses:

(i) Obtain hazard ratio and interpret the results.

(ii) Find the \(99 \%\) confidence interval for the hazard ratio.

(iii) Test whether the covariates are significant or not at \(1 \%\) level of significance.