DEPARTMENT OF STATISTICS AND COMPUTER SCIENCE NON-PARAMETRIC AND CATEGORICAL DATA ANALYSIS - ST XXXXXXXXXX/2011 TUTORIAL: XXXXXXXXXXa) Describe the purpose of the 3 components of a GLM. Define the...

DEPARTMENT OF STATISTICS AND COMPUTER SCIENCE NON-PARAMETRIC AND CATEGORICAL DATA ANALYSIS - ST309-2010/2011 TUTORIAL:04 01. (a) Describe the purpose of the 3 components of a GLM. Define the identity link, and explain why it is not often used with the binomial random component. (b) Define natural exponential family of distributions. Show that the following distributions can be written in this form: Exponential, Gamma. (c) Explain what is meant by the canonical link function, and identify the canonical link functions for the above two distributions. (d) Define the deviance and state how the parameters of a GLM may be estimated. (e) Describe how a suitable model may be chosen by using an analysis of deviance and by examining the significance of the parameters. (f) Define the Pearson residuals and describe how they can be used to assess the goodness of fits of GLMs. 02. For the population of subjects having , has a distribution, (a) Using Bayes theorem, show that satisfies the logistic regression model with (b) Suppose that is a with . Show that the logistic model holds with a quadratic term. 03. Sudden death is an important, lethal cardiovascular endpoint. Most previous studies of risk factors for sudden death have focused on men. Looking at this issue for women is important as well. For this purpose, data were used from the Framingham Heart Study. Several potential risk factors, such as age, blood pressure and cigarette smoking are of interest and need to be controlled for simultaneously. Therefore a multiple logistic regression was fitted to these data as shown below. The response is 2-year incidence of sudden death in females without prior coronary heart disease. Risk Factor Regression Coefficient Standard Error Constant -15.3 Blood Pressure (mm Hg) 0.0019 0.0070 Weight (% of study mean) -0.0060 0.0100 Cholesterol (mg/100 mL) 0.0056 0.0029 Glucose (mg/100 mL) 0.0066 0.0038 Smoking (cigarettes/day) 0.0069 0.0199 Hematocrit (%) 0.1110 0.0490 Vital capacity (centiliters) -0.0098 0.0036 Age (years) 0.0686 0.0225 (a) Assess the statistical significance of the individual risk factors and explain the practical implications of your findings. (b) Give brief interpretations of the age and vital capacity coefficients. (c) Compute the odds ratios relating the additional risk of sudden death associated with (i) A 100-centiliter decrease in vital capacity and (ii) An additional year of age after adjusting for the other risk factors. (d) Provide 95% confidence intervals for the odds ratios in part (c) (e) Predict the probability of sudden death for a 50 year old woman with systolic blood pressure of 120 mmHg, a relative weight of 100% a cholesterol level of 250 mg/100mL, a glucose level of 100 mg/100mL, a hematocrit of 40%, and a vital capacity of 450centiliters who smokes 10 cigarettes per day