Stat 710: Mathematical Statistics Lecture 41

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1 Stat 710: Mathematical Statistics Lecture 41 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

2 Lecture 41: One-way ANOVA and Tukey s method Example 7.27 (Multiple comparison in one-way ANOVA models) Consider the one-way ANOVA model X ij = N(µ i,σ 2 ), j = 1,...,n i,i = 1,...,m. If the hypothesis H 0 : µ 1 = = µ m is rejected, one typically would like to compare µ i s. One way to compare µ i s is to consider simultaneous confidence intervals for µ i µ j, 1 i < j m. Since X ij s are independently normal, the sample means X i are independently normal N(µ i,σ 2 /n i ), i = 1,...,m, respectively, and they are independent of SSR = m n i i=1 j=1 (X ij X i ) 2. Consequently, ( X i X j )/ v ij has the t-distribution t n m, 1 i < j m, where v ij = (n 1 i + n 1 j )SSR/(n m). Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

3 Example 7.27 (continued) For each (i,j), a confidence interval for µ i µ j with confidence coefficient 1 α is C ij,α (X) = [ X i X j t n m,α/2 vij, X i X j + t n m,α/2 vij ], where t n m,α is the (1 α)th quantile of the t-distribution t n m. One can show that C ij,α (X) is actually UMAU (exercise). Bonferroni s level 1 α simultaneous confidence intervals for µ i µ j, 1 i < j m, are C ij,α (X), 1 i < j m, where α = 2α/[m(m 1)]. When m is large, these confidence intervals are very conservative in the sense that the confidence coefficient of these intervals may be much larger than the nominal level 1 α and these intervals may be too wide to be useful. If the normality assumption is removed, then C ij,α (X) is 1 α asymptotically correct as min{n 1,...,n m } and max{n 1,...,n m }/min{n 1,...,n m } c <. Therefore, C ij,α (X), 1 i < j m, are simultaneous confidence intervals with asymptotic confidence level 1 α. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

4 Example 7.27 (continued) For each (i,j), a confidence interval for µ i µ j with confidence coefficient 1 α is C ij,α (X) = [ X i X j t n m,α/2 vij, X i X j + t n m,α/2 vij ], where t n m,α is the (1 α)th quantile of the t-distribution t n m. One can show that C ij,α (X) is actually UMAU (exercise). Bonferroni s level 1 α simultaneous confidence intervals for µ i µ j, 1 i < j m, are C ij,α (X), 1 i < j m, where α = 2α/[m(m 1)]. When m is large, these confidence intervals are very conservative in the sense that the confidence coefficient of these intervals may be much larger than the nominal level 1 α and these intervals may be too wide to be useful. If the normality assumption is removed, then C ij,α (X) is 1 α asymptotically correct as min{n 1,...,n m } and max{n 1,...,n m }/min{n 1,...,n m } c <. Therefore, C ij,α (X), 1 i < j m, are simultaneous confidence intervals with asymptotic confidence level 1 α. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

5 Example 7.27 (continued) For each (i,j), a confidence interval for µ i µ j with confidence coefficient 1 α is C ij,α (X) = [ X i X j t n m,α/2 vij, X i X j + t n m,α/2 vij ], where t n m,α is the (1 α)th quantile of the t-distribution t n m. One can show that C ij,α (X) is actually UMAU (exercise). Bonferroni s level 1 α simultaneous confidence intervals for µ i µ j, 1 i < j m, are C ij,α (X), 1 i < j m, where α = 2α/[m(m 1)]. When m is large, these confidence intervals are very conservative in the sense that the confidence coefficient of these intervals may be much larger than the nominal level 1 α and these intervals may be too wide to be useful. If the normality assumption is removed, then C ij,α (X) is 1 α asymptotically correct as min{n 1,...,n m } and max{n 1,...,n m }/min{n 1,...,n m } c <. Therefore, C ij,α (X), 1 i < j m, are simultaneous confidence intervals with asymptotic confidence level 1 α. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

6 Example 7.27 (continued) We can also use Scheffé s method to obtain simultaneous confidence intervals for µ i µ j, 1 i < j m. Scheffé s intervals have confidence coefficient 1 α for t τ Lβ, t T, where β = (µ 1,..., µ m ) and L = Scheffé s intervals are also too conservative. In fact, they are often more conservative than Bonferroni s intervals. Tukey s method in one-way ANOVA models Consider the one-way ANOVA model in Example Tukey s method introduced next produces simultaneous confidence intervals for all nonzero contrasts (including the differences µ i µ j, 1 i < j m) with confidence coefficient 1 α. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

7 Example 7.27 (continued) We can also use Scheffé s method to obtain simultaneous confidence intervals for µ i µ j, 1 i < j m. Scheffé s intervals have confidence coefficient 1 α for t τ Lβ, t T, where β = (µ 1,..., µ m ) and L = Scheffé s intervals are also too conservative. In fact, they are often more conservative than Bonferroni s intervals. Tukey s method in one-way ANOVA models Consider the one-way ANOVA model in Example Tukey s method introduced next produces simultaneous confidence intervals for all nonzero contrasts (including the differences µ i µ j, 1 i < j m) with confidence coefficient 1 α. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

8 Studentized range Let σ 2 = SSR/(n m), where SSR is given in Example The studentized range is defined to be R st = max 1 i<j m ( X i µ i ) ( X j µ j ). σ The distribution of R st does not depend on any unknown parameter. Theorem 7.11 Assume the one-way ANOVA model in Example Let q α be the (1 α)th quantile of the studentized range R st. Then Tukey s intervals [c τ β qα σc +, c τ β + qα σc + ], c R m {0},c τ J = 0, are simultaneous confidence intervals for c τ β, c R m {0}, c τ J = 0, with confidence coefficient 1 α, where c + is the sum of all positive components of c, β = (µ 1,..., µ m ), β = ( X 1,..., X m ), and J is the m-vector of ones. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

9 Studentized range Let σ 2 = SSR/(n m), where SSR is given in Example The studentized range is defined to be R st = max 1 i<j m ( X i µ i ) ( X j µ j ). σ The distribution of R st does not depend on any unknown parameter. Theorem 7.11 Assume the one-way ANOVA model in Example Let q α be the (1 α)th quantile of the studentized range R st. Then Tukey s intervals [c τ β qα σc +, c τ β + qα σc + ], c R m {0},c τ J = 0, are simultaneous confidence intervals for c τ β, c R m {0}, c τ J = 0, with confidence coefficient 1 α, where c + is the sum of all positive components of c, β = (µ 1,..., µ m ), β = ( X 1,..., X m ), and J is the m-vector of ones. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

10 Proof Let Y i = ( X i µ i )/ σ and Y = (Y 1,...,Y m ). Then the result follows if we can show that is equivalent to max 1 i<j m Y i Y j q α (1) c τ Y q α c + for all c R m satisfying c τ J = 0,c 0. (2) Let c(i,j) = (c 1,...,c m ) with c i = 1, c j = 1, and c l = 0 for l i or l j. Then c(i,j) + = 1 and [c(i,j)] τ Y = Y i Y j. Therefore, (2) implies (1). Next, we show (1) implies (2). Let c = (c 1,...,c m ) be a vector satisfying the conditions in (2). Define c to be the sum of negative components of c. Then the result follows from Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

11 Proof Let Y i = ( X i µ i )/ σ and Y = (Y 1,...,Y m ). Then the result follows if we can show that is equivalent to max 1 i<j m Y i Y j q α (1) c τ Y q α c + for all c R m satisfying c τ J = 0,c 0. (2) Let c(i,j) = (c 1,...,c m ) with c i = 1, c j = 1, and c l = 0 for l i or l j. Then c(i,j) + = 1 and [c(i,j)] τ Y = Y i Y j. Therefore, (2) implies (1). Next, we show (1) implies (2). Let c = (c 1,...,c m ) be a vector satisfying the conditions in (2). Define c to be the sum of negative components of c. Then the result follows from Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

12 Proof (continued) c τ Y = 1 c + c + c j Y j + c c i Y i j:c j <0 i:c i >0 = 1 c + c i c j Y j i:c i >0 j:c j <0 j:c j <0 = 1 c i c j (Y j Y i ) c + i:c i >0 1 c + i:c i >0 j:c j <0 max Y j Y i 1 i<j m c i c j Y j Y i j:c j <0 ( = max 1 i<j m Y j Y i c +, 1 c + i:c i >0 c i c j Y i i:c i >0 ) c i c j j:c j <0 where the first and the last equalities follow from the fact that c = c + 0. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

13 Remarks Tukey s method works well when n i s are all equal to n 0, in which case values of n 0 q α can be found using tables or statistical software. When n i s are unequal, some modifications are suggested; see Tukey (1977) and Milliken and Johnson (1992). Example 7.29 We compare the t-type confidence intervals, Bonferroni s, Scheffé s, and Tukey s simultaneous confidence intervals for µ i µ j, 1 i < j 3, based on the following data X ij given in Mendenhall and Sincich (1995): j = i = Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

14 Remarks Tukey s method works well when n i s are all equal to n 0, in which case values of n 0 q α can be found using tables or statistical software. When n i s are unequal, some modifications are suggested; see Tukey (1977) and Milliken and Johnson (1992). Example 7.29 We compare the t-type confidence intervals, Bonferroni s, Scheffé s, and Tukey s simultaneous confidence intervals for µ i µ j, 1 i < j 3, based on the following data X ij given in Mendenhall and Sincich (1995): j = i = Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

15 Example 7.29 (continued) In this example, m = 3, n i n 0 = 10, X 1 = 229.6, X 2 = 309.8, X 3 = 427.8, and σ = Let α = For the t-type intervals, t 27,0.975 = For Bonferroni s method, α = α/3 = and t 27,0.983 = For Scheffé s method, c 0.05 = 3.35 and 2c 0.05 = From Table 13 in Mendenhall and Sincich (1995, Appendix II), n0 q 0.05 = The resulting confidence intervals are given in the following table. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

16 Example 7.29 (continued) Parameter Method µ 1 µ 2 µ 1 µ 3 µ 2 µ 3 Length t-type [ 235.2, 74.6] [ 353.1, 43.3] [ 272.8, 37.0] Bonferroni [ 273.0, 112.4] [ 390.9, 5.5] [ 310.6, 74.8] Scheffé [ 276.0, 115.4] [ 393.9, 2.5] [ 313.6, 77.8] Tukey [ 267.3, 106.7] [ 385.2, 11.2] [ 304.9, 69.1] Discussions Apparently, t-type intervals have the shortest length, but they are not simultaneous confidence intervals. Tukey s intervals in this example have the shortest length among simultaneous confidence intervals. Scheffé s intervals have the longest length. Jun Shao (UW-Madison) Stat 710, Lecture 41 May 8, / 10

Stat 710: Mathematical Statistics Lecture 40

Stat 710: Mathematical Statistics Lecture 40 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 40 May 6, 2009 1 / 11 Lecture 40: Simultaneous