stats.
4. (a) In a study done about 50 years ago, doctors studied the effect on the body weight (in Coefficients: ounce) of a new born baby, of: the mother's age; weight (in pounds); height (in inches); smoking habit; length of pregnancy (or "gestation" ); and order of birth (or "parity"). Estimate Std. Error z value Pr(> =[) Here the mother's smoking habit is coded as 1 if she smokes, and ( otherwise. The Intercept -3.6393 1.7380 -7.039 2.0175 * *X parity parameter is coded as 0 if the baby is the mother's first born, and is 1 otherwise. 1.2822 0.01082 2145 0.0032 * * Output of linearly regressing body weight on the covariates mentioned above, is given 1.4232 0.0482 6.645 70-5 * * * in the table below. Signif. codes: 0" * * *" 0.001 "* *" 0.01 " *"0.05" . " 0.1 " " 1 Estimate Standard Error t-value Pr(3 t) intercept -80.41 14.35 -5.60 0.0000 (Dispersion parameter for binomial family is taken to be 1) gestation 0.44 0.03 15.26 0.0000 parity 3.33 1.13 -2.95 0.0033 Null deviance: 273.52 on 199 degrees of freedom age -0.01 0.09 0.10 0.9170 Residual deviance: 254.85 on 197 degrees of freedom height 1.15 0.21 5.63 0.0000 Interpret the effect of X2 on Y using the output provided by explaining the weight 0.05 0.03 1.99 0.0471 meaning of the estimated coefficient in the context of the study. Calculate the smoke -8.40 0.95 8.81 0.0000 95% confidence interval on X2- What does this confidence interval imply for the i. Write the equation of the model in which the response - namely new-born's weight association between this covariate and the response? (5 marks is regressed linearly against the covariates contained in the results tabulated is. The final lines of the output provide the null and full deviance. State the mathe- above. [4marks] matical definitions of the null and the full deviances. Use the outputs to carry out a hypothesis test at a 1% significance level by stating explicitly the null/ alternative hypotheses, and the test statistic. (5 marks] ii. Interpret what the "Estimate's for covariates "gestation" and "age" suggest for changes in this response, for a given change in the respective covariate. (2marks] [Total marks: 10] iil. For a new-born with weight 120, gestation = 284, parity=0, age-27, height-62, weight =100, smoke-0, what is the residual? (2marks] [Total marks: 20] iv. Unbiased sample variance of residuals for the given data set is 249.28, while the unbiased sample variance of the body weights of the new-borns observed in this survey is 332.57. What is the coefficient of determination of this model? Note that the data set has 1236 data points in it. (2marks] (Total marks: 10) (b) In a new model, effect of only the covariates: gestation (X1) and mother's age (X2), is examined, on the response variable Z that takes the value "healthy" or "no-healthy". depending on whether new-born body weight exceeds 6, or not. Here "Healthy" is coded as I and "not healthy" as 0. Doctors use a subset with 200 data points, from the old 1236-sized sample, to construct estimates under this model. i. The following provides the output of the analysis on the data. Call : glm(Y ~ X1 + Xz, family - binomial, data = subset - newborn.dat) Deviance Residuals: Min 1Q Median 3Q Max -1.6211 -0.5699 - -0.2309 0.3954 2.8751
