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.

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

The following data on diagnosis of coronary heart disease (where 0 indicating absence and 1 indicating presence), serum cholesterol (in mg/dl), resting blood pressure (in mmHg) and weight (in 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

 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.

Kaplan and Meier method

Poisson regression

Polytomous logistic models

Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/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.

Explain the assumptions underlying multiple linear regression model.

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.

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

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 loglog  .

(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.

Seven successive observations on a stationary time-series are as follows:

12, 14, 13, 10, 15, 12, 15

(a) Calculate auto-covariances C₀, C₁, C₂, C₃ and C₄.

(b) Calculate auto-correlation coefficients r₁, r₂, r₃ and r₄.

(c) Plot the correlogram.

The following table represents the sales (in thousands) of mobile sets of a shop for 16 quarters over four years:

(a) Compute the seasonal indices for four quarters by Simple average method. (b) Obtain deseasonlised values.

A researcher is interested in developing a linear model for the electricity consumption of a household having an AC (1.5 ton) so that she can predict the electricity consumption. For this purpose, she selects 25 houses and records the electricity consumption (in kWh), size of house (in square feet) and AC hours for one month during summers. The results obtained are:

SS(B₀) = 12526.08, SS(B₀,B₁) = 17908.47, SS(B₀,B₂) = 17125.23, SS(B₀, B₁, B₂) = 18079.0,

Build a regression model by selecting appropriate regressors in the model using the Stepwise Selection method.

A firm wants to know whether there is any linear relationship between the sales (X) and its yearly revenue (Y). The records for 10 years were examined and the following results were obtained:

(a) Fit a regression line taking Y as the dependent variable and X as the independent variable.

(b) Test whether the sales have any effect on revenue at 5% level of significance. (c) Comment on the goodness of fit of the regression line.

Using the graphical method to minimise the time required to process Job 1 and Job 2 on five machines A, B, C, D and E, find the minimum elapsed times an idle times to complete both jobs.

In a railway marshalling yard, goods trains arrive at a rate of 36 trains per day. Assuming that the inter-arrival and service time distributions both follow exponential distribution with an average of 30 minutes, calculate the following:

(i) Traffic intensity

(ii) The mean queue length

(iii) Probability that the queue size exceeds

Four professors are capable of teaching any one of four different courses. Class preparation time in hours for different topics varies from professor to professor and is given in the table below:

Each professor is assigned only one course. Determine an assignment schedule so as to minimise the total course preparation time for all courses.

A company has three production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100s) per week of a product, respectively. These units are to be shipped to four warehouses D₁, D₂, D₃ and D₄ with requirement of 5, 6, 7 and 14 units (in 100s) per week, respectively. The transportation costs (in Rs) per unit between factories to warehouses are given in the table below:

Obtain optimal solution by the MODI method.

Use the penalty (Big M) method to solve the following LP problem:

Minimise Z 5x₁+ 3x₂

Subject to the constraints: