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Interactive Effects of Chemicals in Wistar Rats’ Diets. (60 points) An experiment wasperformed (Groten et al. 1997) to find out interactive effects on nine chemicals in Wistar Rats’diets. The data can be found in: “http://www.stat.sinica.edu.tw/fredphoa/teaching/data/ASAT.txt”.
The response is the concentration of aspartate aminotransferase (ASAT) in rats’ blood. Thenine chemicals in this study are given below: Given the defining relations of this design is I = ABK = AEG = ADF = ACH = BCDG.
(a) This data is raw and each row stands for a trial. Rewrite the data into a standard plan-ning matrix of a 2n−k fractional factorial designs where − stands for chemical absence and +stands for chemical presence in the trial. What is n and k in this experiment?(b) Write down all 31 defining relations in its defining contrast subgroup.
Hint: Build a full model and obtain the model matrix via the R command “model.matrix”.
Then the defining relations are the column names that the sum of column is not zero.
(c) How many clear effects in this design? How many strongly clear effects in this design?(d) Write down the wordlength pattern.
(e) Find the resolution and projectivity of this design.
(f) How many effects can be estimated and what are they? Are there any assumptions behindthese estimations?(g) An expert suggests that no interaction effects are important. Use half-normal probabilityplot to figure out which effects are important to the response.
(h) Another expert suggests that the interactions DE and DF are probably important. Usehalf-normal probability plot to figure out which effects are important to the response.
(i) Suggest a potential problem in this analysis.
Resolving the Ambiguities of Aliasing via Optimal Design Approach. (30 points)We continue to investigate in the toxicological experiment. If you perform the correct anal-ysis, effect aliasing causes some troubles. Therefore, we decide to resolve this ambiguity viaOptimal Design Approach. We assume that main effects C, D, E, F and interaction DE areimportant.
(a) Prepare the augmented model matrix (not the design matrix). For the ? entries, simplyput 0. Provide the R code how you get this matrix. Name the matrix AM.
(b) Prepare all possible combinations of factor level setting of trial 17. List each combination in a row and you should get a 32 × 5 matrix. Provide the R code how you get this matrix.
Name the matrix PP.
(c) Find the factor level settings of rows 17 and 18 such that the augmented model matrixis D-optimal. To obtain it, please follow the steps below. First, obtain D-efficiencies for allcombinations of rows 17 and 18:> Deff=matrix(0,nrow(PP),nrow(PP)); Di=as.matrix(AM)> for(i in 1:(nrow(PP)-1)) { for(j in (i+1):nrow(PP)) {+ Di[17,3:7]=PP[i,]+ Di[18,3:7]=PP[j,]+ Deff[i,j]=det(t(Di) % ∗ % Di) }}Then look for the maximum D-efficiencies in the “Deff” matrix and report it. There shouldbe 24 combinations that return the maximum D-efficiency, which means there should be 24different D-optimal designs. Report the factor level setting of the first combination.
Investigation to 26−2 FFD. (10 points) Table 5A.2 suggests a 26−2 design of resolution 4with generator 5 = 123 and 6 = 124. Is it possible to have a 26−2 of resolution 5? If yes,please write down the defining contrast subgroup. If no, please provide the reason.
Hint: Think about the definition of Resolution 5 design and the degree of freedom of thedesign.
Design Table - A Deeper Look (Bonus 20 points) You may wonder how the textbookcomes up with the designs listed in the design tables. In this question, let’s construct minimumaberration 2n−2 (quarter-fraction) designs. Let n − 2 = 3m + r, where 0 ≤ r < 3. Define:B1 = 12 · · · (2m)(n − 1), B2 = (m + 1)(m + 2) · · · (3m)n for r = 0;B1 = 12 · · · (2m + 1)(n − 1), B2 = (m + 1)(m + 2) · · · (3m + 1)n for r = 1;B1 = 12 · · · (2m + 1)(n − 1), B2 = (m + 1)(m + 2) · · · (3m + 2)n for r = 2;(a) Prove that the 2n−2 design D with the defining relation I = B1 = B2 = B1B2, where B1and B2 are given above, has the maximum resolution 2n/3 , where x denotes the integerpart of x.
It is not simple as in (a) to prove that the designs constructed above have minimum aberration,but let’s have faith on this result by now.
(b) The textbook provides these quarter-fraction designs up to 128 runs in Table 5A. Usingthe above rules, construct the minimum aberration 210−2 designs (256 runs) and find theresolution of this design.

Source: http://www.stat.sinica.edu.tw/fredphoa/teaching/2013F_NCU/HW5.pdf