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Question 1 [worth 30 marks] A sampling experiment with 5000 trials for J = 5 has parameters μ1 = μ2 = 48, μ3 = μ4 = μ5 =32, and σ = 15. On each trial, n = 13 random observations are drawn from each population and 95% t-based CI limits and t-statistics are calculated for each pairwise comparison (tc = t.05/2;60 = 2.00). (a) Table 1 shows the raw 95% t-based CI limits for each comparison from 5 trials. (i) Choose one trial in Table 1. What inference can be made for the 95% CIs for μ2 - μ3 and μ3 - μ5? For the same trial, what test outcomes and conclusions would follow for .05-level t-tests of H0: μ2 - μ3 = 0 and H0: μ3 - μ5 = 0? (ii) [Limit: 300 Words] For each of the 10 comparisons, state whether an inferential error (or errors) has been made across the 5 trials. If so, indicate what type of inferential error (or errors) has been made for each comparison and explain why the CI limits for this particular comparison provide an example of this type of inferential error. (b) For this sampling experiment, zero Type I errors are made on 4190 trials and zero non-coverage errors are made on 3596 trials. (i) Calculate the FWER and explain what this means. · The familywise error rate refers to the probability of making at least one Type I error on a single replication, for the set of comparisons. The familywise error rate = 1.0 - .419 = .581. That is, across the set of comparisons at least one Type I error was made on 58.1% of trials. (ii) Calculate the familywise non-coverage error rate (FWNCER) and explain what this means. · The familywise noncoverage error rate refers to the probability of making at least one non- coverage error (for the set of comparisons) on a single replication. The familywise noncoverage error rate = 1.0 - .3596 = 0.6404 That is, across the set of comparisons, at least one noncoverage error was made on 64.04% of trials. (iii) [Limit: 300 Words] Why are these two values different to each other? FWER control exerts a more stringent control over false discovery compared to false discovery rate (FDR) procedures. FWER control limits the probability of at least one false discovery, whereas FDR control limits (in a loose sense) the expected proportion of false discoveries. Thus, FDR procedures have greater power at the cost of increased rates of type I errors, i.e., rejecting null hypotheses that are actually true.[11] On the other hand, FWER control is less stringent than per-family error rate control, which limits the expected number of errors per family. Because FWER control is concerned with at least one false discovery, unlike per-family error rate control it does not treat multiple simultaneous false discoveries as any worse than one false discovery. The Bonferroni correction is often considered as merely controlling the FWER, but in fact also controls the per-family error rate Question 2 [worth 20 marks] A sampling experiment with 1000 trials for J = 4 and n = 11 has parameters μ1 = μ2 = μ3 = μ4 = 105, and σ = 10. On each trial .05-level t-tests and .05-level q* tests (Tukey tests) of H0: μj - μj’ = 0 (for j, j’ = 1, 2, 3, 4 and j ≠ j’) have been conducted for each comparison. The tables below show the number of trials for which a significant test outcome occurs for the maximal comparison for either the t-test method (Table 2) or the Tukey method (Table 3). (a) [Limit: 300 Words] What is the Type 1 error rate for a .05-level t-test of the maximal comparison and a .05-level Tukey-test of the maximal comparison? Why are these two values different? (b) [Limit: 300 Words] For either test method, explain why the Type 1 error rate for the maximal comparison is the same as the FWER. Question 3 [worth 50 marks] A psychologist evaluates whether exercising with others can encourage unfit adults to engage in exercise for longer periods of time. Eighty-four adults who wish to get fit are randomly allocated to one of four support conditions (J = 4, n = 21, N = 84): · · Group 1: Large Group (all 21 participants exercise together) · Group 2: Medium Group (7 participants exercise together) · Group 3: Small Group (3 participants exercise together) · Group 4: Control condition (participants exercise by themselves). Over a four-week period, participants keep a daily record of time spent exercising. The dependent variable is the average daily number of minutes of exercise. The psychologist considers a difference of 5 minutes to be the smallest difference of practical importance between conditions. (i) State the most efficient decision rule for this analysis (that will afford the greatest statistical power). Non-significance of data gained from the planned contrast analysis does not allow for any clear inference regarding the difference in the effectiveness of group size for encouraging unfit adults to engage in exercise for long periods of time. In comparison, through the analysis of variance, the control group clearly demonstrates minimal involvement in exercise with its visibly low Sum of Square and Mean Squares, resulting in a low F-value of 0.019. However, the impact the number of people in a group has presented to be of small value, only presenting its more significant differences in the Analysis of Variance and in its CI limits, (ii) What inference follows from test outcomes? The test outcomes suggest a directional inference of B4<>
F, 1, 2 Fc = F.05, 3, 60 = 2.76 and F = 8.99 > 2.76, hence reject H0 . F = MSB/MSE = 218.667/24.33 = 8.99 (ii) What is the SSc that you would use for tests of post-hoc contrasts? (iii) The psychologist identifies two follow-up contrasts for which a directional inference can be made from test outcomes. Identify ONE such contrast and carry out a test of H0: ψ = 0 for this contrast in the context of the post-hoc contrasts analysis, controlling FWER at .05. What directional inference can be made for this contrast? [Hint: look at the coefficients of maximal contrast.] AND (c) [Limit: 300 Words] Comment on the reasons for any differences in outcome between the analysis in Q3 (a) and (b). AND (d) (i) Calculate the 95% Tukey SCI for the maximal comparison. (ii) Why is this SCI wider than the SCI in Q3(a) for the same comparison? (iii) Verify that a post-hoc 95% SCI for the maximal comparison is wider than the 95% Tukey SCI for the same comparison. End of Test