Analysis of Scientific Data - Semester One, 2021 Week 10 Tutorial Solutions Linear Regression Question 1 Lung function data for 25 cystic fibrosis patients in a study by O'Neill et al. (1983) was presented in Douglas Altman's Practical Statistics for Medical Research (1991). The data contains a number of variables recorded for each patient and here we are interested in the relationship between two of these, weight (weight, kg) and pemax (maximum expiratory pressure, cm of H2O). a) Write down the linear model and its assumptions to study the relationship between the response variable, pemax, and the explanatory variable, weight. Y = Bo + B1X + U where, Y = pemax; X = weight Assume that observed pemax comes from a normal distribution. That is, Y~N(Bo + B1X,0) U~N(0,0); o is a constant. E(U) = 0, Var(U) = G2 b) The R output for the linear model is presented below. Compute the values of A and C, and then give a value for B. Coefficients : Estimate Std. Error t value Pr(> |t | ) (Intercept) weight 63.5456 1.1867 A 5.003 4.63e-05 0.3009 3.944 B Residual standard error: 26.38 on C degrees of freedom C = n-2 = 23 A = se(b0) Then, tstat = bo- 0 A 5.003 = 63.5456-0 A B= 2 * P(t23 ? 3.944) Using R: B = 2 * (1 - pt(3.944, df = 23) A = 63.5456 A = 12.70 5.003 B = 0.000646 c) Based on the linear model, what is the estimated mean maximum expiratory pressure for patients with a weight of 50 kg? Estimated SLR equation is Ý = 63.5456 + 1.1867X Substitute X = 50. Ý= 63.5456 + 1.1867 * 50 gives Ý = 122.88 cmH20 d) The Pearson correlation coefficient, r, between the variables pemax and weight is 0.635. Describe the strength and the direction of the relationship between pemax and weight. Moderately strong positive association between pemax and weight. e) Conduct a hypothesis test to determine whether there is any evidence of a linear association between pemax and weight. What do you conclude? Ho: B1 = 0 Vs H1: B1 0 From the table in part (b), p-value = 0.000646 p-value = 0.000646 < 0.01, We have strong evidence to conclude that there is a linear relationship between pemax and weight.
Question 2 Polychlorinated biphenyls (PCBs) were once used in industry but were banned in the 1970s because of concerns about their toxicity. Despite the ban, PCBs can still be detected in most people because they are persistent in the environment. A team of researchers recorded the amount of PCBs detected in maternal milk from mothers who had eaten fish from a particular lake considered to be contaminated with PCBs. They subsequently administered an IQ test to the children when they were 11 years old. The results are shown in the following scatter plot along with the least-squares line fitting a linear relationship between the two variables: 0 o 120 O 0 110 Full Scale IQ =105 100