Require R Studio Experimental Design Worksheet

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Homework 6 Stat 158, Fall 2023 Due November 9, 2023 by 11:59 PM PDT on Gradescope Please remember to keep homework assignments structured and legible, to use a new page for each problem, and to upload a .pdf with separate pages instead of one long page. Include all R code written in a code appendix at the end of the homework, and link pages with relevant code to each question when you upload to Gradescope. 1. (30 points) The model for a split-plot design is given as yijk = µ + αi + bj(i) + δk + (αδ)ik + ejk(i) where αi is the effect due to the treatment applied at the block level (e.g. diabetic or not), bj(i) is the random effect for the block j in the ith group (e.g. the jth dog in group i), and δk is the effect due to the treatment applied within each block (e.g. method of infusion/inject). We assume that 2 2 bj(i) ∼ N (0, σblock ), ejk(i) ∼ N (0, σerr ) and that all of the bj(i) and ejk(i) terms are independent of each other. The mean sum-of-squares of the blocking factor is given by I I J J K K 1 XXX 1 XXX 2 b̂j(i) = M Sblock = (ȳij· − ȳi·· )2 . IJ − I i=1 j=1 k=1 IJ − I i=1 j=1 k=1 Prove the following result given in the handout: 2 2 + σerr . E(M Sblock ) = Kσblock 2. Analyze the data from the bridge experiment from class (available on bCourses as file bridge strength.txt). (a) (5 points) Visualize the data and describe what you see. (b) (20 points) Perform randomization inference, using F-statistics extracted from a random effects fit using the nlme package. Include crumpling, orientation, and their interaction as fixed effects. Report and interpret the results of your tests. 1 3. We are planning an experiment on the quality of video tape and have purchased 24 tapes, four tapes from each of six types1 . The six types of tape were 1) brand A high cost, 2) brand A low cost, 3) brand B high cost, 4) brand B low cost, 5) brand C high cost, 6) brand D high cost. Each tape will be recorded with a series of standard test patterns, replayed 10 times, and then replayed an eleventh time into a device that measures the distortion on the tape. The distortion measure is the response, and the tapes will be recorded and replayed in random order. Previous similar tests had an error variance of about .25. (a) (10 points) What is the power when testing at the .01 level if the high cost tapes have an average one unit different from the low cost tapes? (b) (15 points) How many tapes per brand would we need to obtain a 95% confidence interval of no wider than 2 for the mean difference between tapes of brand A and tapes of brand B? 4. We decide to alter the experiment in the previous question in two ways. First, in addition to comparing the six types we will measure the effect of storage temperature on video quality by selecting half the tapes of each type at random to be stored at a higher temperature than the others. In addition, to speed up the running of the experiment, we will use two different distortion-measuring devices concurrently. Since we expect that there may be some variation in measurement across devices, we will assign half the tapes to each device at random, ensuring that the number of tapes in each brand/temperature combination is balanced across devices. (a) (5 points) Identify the design of the experiment, including both the type and the number of treatment factors at each level. Justify your answer. (b) (15 points) Repeat the analysis from question 3(a) for the new design. Comment on any differences between the results and give intuition for why they do or do not differ. 1 Adapted from Oehlert problem 7.5 2 Run Sheet # Crumpled Orientation Coins at Collaps 1 1 N L 18 2 1 N S 35 3 2 Y L 40 4 2 Y S 42 5 3 Y S 51 6 3 Y L 42 7 4 N S 24 8 4 N L 18 9 5 N S 20 10 5 N L 13 11 6 Y S 33 12 6 Y L 39 13 7 Y L 28 14 7 Y S 66 15 8 N S 33 16 8 N L 13
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