Stat 601 The Design of Experiments

Transcription

1 Stat 601 The Design of Experiments Yuqing Xu Department of Statistics University of Wisconsin Madison, WI 53706, USA October 28, 2016 Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

2 Logistics Exam on next Wed. Bring calculator and tables (t, F and Studentized Range). Office Hours adjustment: (One time adjustment) Mon: 12:15-1:15pm, Wed MSC 6740 MSC 6740: Sixth floor, on the East side of MSC, take lift No.13 Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

3 RCBD with Random Block Effect Statistic Models for RCBD with Random Effect Y ij = µ + b i + t j + ɛ ij where i = 1,..., b, j = 1,..., t, ɛ ij N(0, σ 2 ) and b i N(0, σ 2 β ). Why Random When we are trying to learn about variation in the treatment effects: we want to design an experiment and look at the variability that arises in sample. Random effects models are also used in subsampling situations. Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

4 ANOVA RCBD with Fixed Effect Source DF SS MS E[MS] b Block b-1 i=1 t(y i. y.. ) 2 SSB (b 1) σ 2 + t t Treatment t-1 j=1 b(y.j y.. ) 2 SSTr t 1 σ 2 + b Error (b-1)(t-1) i j (y ij y i. y.j + y.. ) 2 SSE df σ 2 Total n-1 S total = i j (y ij y.. ) 2 i b2 i b 1 j t2 j t 1 RCBD with Random Effect Source DF SS MS E[MS] b Block b-1 i=1 t(y i. y.. ) 2 SSB (b 1) σ 2 + tσb 2 t Treatment t-1 j=1 b(y.j y.. ) 2 SSTr t 1 σ 2 + b Error (b-1)(t-1) i j (y ij y i. y.j + y.. ) 2 SSE df σ 2 Total n-1 S total = i j (y ij y.. ) 2 j t2 j t 1 Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

5 RCBD with Random Block Effect Final Question Y ij = µ + b i + t j + ɛ ij where i = 1,..., b, j = 1,..., t, ɛ ij N(0, σ 2 ) and b i N(0, σβ 2). How to estimate σ 2? How about σβ 2? Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

6 Model Diagnostics This is far too rough but important 1 What if independence is not satisfied? 2 What if homoscedasticity or normality is not satisfied? 3 What if you have CRD but fitted RCBD? or you have RCBD but fitted CRD? 4 What kind of transformation should we use? 5 After transformation, what value is the most important one? 3 You will lose power of the test. 4 Monotone function. 5 Median. Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

7 Model Diagnostics This is far too rough but important Median/quantiles will not be changed by monotone transformation Suppose we did transformation: ỹ jk = log y jk Then median( µ j ) = log[median(µ j )] CI for median( µ j ) = log[ci for median(µ j )] This is not always true for mean. Think about Jensen s inequality: E(logy) log(ey) Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

8 Review of Selected Topics in CRD Example 1 Four different brands of pillows were tested by consumers. The rates on a scale of 1(inferior) to 7(superior) is summarized below: Pillow brand Rating A 1,3,5,7,2,3,4 B 7,6,7,7,6 C 1,2,3,2,3,2,1 D 4,3,4,1,5 Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

9 Example 1 Solution: Source DF SS MSS F Between Tr t-1 S T = t j=1 n j(y j. y.. ) 2 S T t 1 Within Tr n-t S E = j k (y jk y j. ) 2 S E n t Total n-1 S total = j k (y jk y.. ) 2 S T /(t 1) S E /(n t) Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

10 Example 1 Questions (b) Based on this study, manufacturers of pillow B want to claim their pillow is significantly better than brand C. Test this contrast. (c)how large must the difference δ = ȳ B ȳ C be in order to declare significant at the 0.05 level by Scheffe s method? Is the observed difference significant? (d)how large must the difference δ = ȳ B ȳ C be in order to declare significant at the 0.05 level by Tukey s method? Is the observed difference significant? (e)how large must the difference δ = ȳ B ȳ C be in order to declare significant at the 0.05 level by Bonferroni s method? Is the observed difference significant? (consider m = ( 4 2), FWER=0.05) Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

11 Example 1 (b) Solution: Hypothesis: H 0 : µ 2 = µ 3, H 1 : µ 2 > µ 3 Contrast: C = (0, 1, 1, 0) t-statistic: (c)(d)(e) Solution: Sheffe: ( µ ˆ B µ ˆ C ) > j c j ˆµ j j c jµ j ˆσ2 j c2 j /n j Tukey: ( µ ˆ B µ ˆ C ) > q t,n t (α)ˆσ Bonferroni: t n t (t 1)F t 1,n t (α) ˆσ 2 (n 1 B (n 1 B + n 1 C )/2 ( µ ˆ B µ ˆ C ) > ˆσ2 (n 1 B + n 1 C ) + n 1 C )t n t( α 2m ) Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

12 Example 2 Example 2 We are planning an experiment on the quality of video tape and have purchased 24 tapes, four tapes from each of six types. 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 C low 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. Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

13 Example 2 Question How large should the sample size have been to have a 95% brand A versus brand B confidence interval of no wider than 2? Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

14 Example 3 HW6 Q1 A chemist wishes to test the effect of four chemical agents on the strength of a particular type of cloth. Because there might be variability from one bolt to another, the chemist decides to use a randomized block design, with the bolts of cloth considered as blocks. She selects five bolts and applies all four chemicals in random order to each bolt. The resulting tensile strengths are given below. Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

15 Example 3 Questions (a) Analyze the data and state your conclusions. (b) Suppose that the observations for chemical type 2 and bolt 3 and chemical type 4 and bolt 4 are missing. i. Estimate the missing values by differentiating the residual sum of squares with respect to them, equating the results to zero, and solving the equations. Analyze the data using these two estimates of the missing values. ii. Another way to estimate the missing values is to set their residuals to zero. Does this method give the same estimates as the previous one? Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16

16 Example 3 HW3 Q1 (b): SSE = RSS = i Let y 32 = x 1, y 44 = x 2. Then (y ij y i. y.j + y.. ) 2 j y 3. = (74 + x )/4 y.2 = ( x )/5 y 4. = ( x 2 )/4 y.4 = ( x )/5 y.. = (... + x x )/20 i. Find solutions to SSE x 1 = 0, SSE x 2 = 0. ii. Find x 1, x 2 such that y 32 y 3. y.2 + y.. = 0 and y 44 y 4. y.4 + y.. = 0 Yuqing Xu (UW-Madison) Stat 601 Week 8 October 28, / 16