Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 1 of 7 Show your work. Due date: 10/12/2021 by 11:00 pm. Question 1 A study was conducted in which m = 40 devices were randomized to be operated under 4 different sets of conditions, 10 devices per set of conditions. A response reflecting “perfor- mance level” of such devices was measured on each device at time 0, i.e., at “baseline,” right before the devices were subjected to their assigned conditions. (The smaller the performance response, the better the performance of the device.) The conditions were then implemented, and the performance each device was measured every two hours thereafter over a 12 hour period. Spaghetti plots for the devices from each of the 4 groups are shown in the following Figure 1. Figure 2 shows the sample means for each condition group at each observation time using the group number as the plotting symbol. Figure 1: Performance data for devices under 4 different conditions. The experimenters expected that all devices would show degradation of performance over the study period, in particular, they thought that this behavior would be reflected in mean performance profiles that exhibit upward “curvature” to varying degrees as the study progressed. Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 2 of 7 Figure 2: Means of performance data for devices under 4 different conditions. Let (hours) denote the time of observation for the device. denote the corresponding performance response, where = 1, 2, … , = 40 Based on the investigator’s expectation and the visual evidence, an appropriate model of the form for might be different in each group. The mean response trajectories during the study period for each group to exhibit “curvature” over time may have different features for each group. One way to write such a model is = 0,1 + 1,1 + 2,1 2 + ε for group 1 = 0,2 + 1,2 + 2,2 2 + ε for group 2 = 0,3 + 1,3 + 2,3 2 + ε for group 3 = 0,4 + 1,4 + 2,4 2 + ε for group 4 We may express these models as = + i where i is the vector of random deviations ε for device and = (0,1, 1,1, 2,1, 0,2, 1,2, 2,2, 0,3, 1,3, 2,3, 0,4, 1,4, 2,4) ′. a) Give the form of for devices in each of the 4 groups. b) The experimenter’s first question was whether the study was carried out properly in the sense that the devices were similar on average at baseline, as would be the case in a Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 3 of 7 randomized study like this. Give the null and alternative hypotheses that address this question. c) Express in (b) as = for some appropriate matrix . Give the form of . d) The next question was whether we need the quadratic coefficients at all for any of the groups or whether straight line models with possibly different slopes are adequate to describe the mean performance behavior for all groups. Prove the null and alternative hypotheses that addresses this question. e) Express your in (d) as = . Give the form of . Question 2 Consider the ultrafiltration data for low flux dialyzers (homework 1 – question 3). The experimenter considers the issue of whether the mean slope of a trajectory differs across the centers with the following assumptions: – Each dialyzer has its own straight-line trajectory with its own intercept and slope. – A common () = and a common diagonal within-unit covariance matrix = 2 for all centers. Then, the model for is: = 0 + 1 + ε = + = (0,1, 1,1, 0,2, 1,2, 0,3, 1,3) ′ ~2(, ) where 0,1, 1,1 are the mean intercept and slope for the center 1. is the appropriate matrix of 0’s and 1’s that picks the correct elements of for the dialyzer, e.g. if is from center 1, then = [ 1 0 0 0 1 0 0 0 0 0 0 0 ] Consider the output by running the following program: Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 4 of 7 a) From the output, provide the estimates of , and 2. b) Provide the matrix and the Wald (chisq) test statistics for the difference in slope obtained from the contrast statement that there is a difference in mean slope for the 3 centers. What is your conclusion c) Provide the matrix for the difference in intercepts and the Wald test statistics. What is your conclusion d) Provide the BLUP of the dialyzers #2 from center 2. e) Re-fit the model with several different choices of model for . What is your conclusion Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 5 of 7 Output: Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 6 of 7 Homework STAT 592 – Fall 2021 Last updated: 9/28/2021 Page 7 of 7
