Scheffé s Method. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

Transcription

1 Scheffé s Method opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

2 Scheffé s Method: Suppose where Let y = Xβ + ε, ε N(0, σ 2 I). c 1β,..., c qβ be q estimable functions, where c 1,..., c q are linearly independent. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

3 Let C = c 1. c q and define θ = θ 1. θ q = c 1 β. c qβ = Cβ. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

4 Let W = C(X X) C with diagonal elements denoted w 11,..., w qq. Then Var(Cˆβ) = σ 2 W and Var(ˆθ j ) = Var(c j ˆβ) = σ 2 w jj j = 1,..., q. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

5 For any q 1 vector u and any k R, let L(u, k) denote the interval [u ˆθ k ˆσ 2 u Wu, u ˆθ + k ˆσ 2 u Wu]. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

6 We want to find k P(u θ L(u, k) u R q ) = 1 α. Thus, we seek simultaneously coverage probability 1 α for an infinite set of intervals. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

7 Show that u θ L(u, k) u R q (Cˆβ Cβ) (C(X X) C ) 1 (Cˆβ Cβ) qˆσ 2 k2 q. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

8 First, note that u θ L(u, k) u R q u θ L(u, k) u R q \ {0} 0 θ = 0 [0, 0] = L(0, k). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

9 u θ L(u, k) u ˆθ k ˆσ 2 u Wu u θ u ˆθ + k ˆσ 2 u Wu k ˆσ 2 u Wu u θ u ˆθ k ˆσ 2 u Wu k u θ u ˆθ ˆσ 2 u Wu k u θ u ˆθ ˆσ 2 u Wu k [u (θ ˆθ)] 2 ˆσ 2 u Wu k 2. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

10 u θ L(u, k) u R q \ {0} [u (θ ˆθ)] 2 ˆσ 2 u k 2 u R q \ {0} Wu [u (θ max ˆθ)] 2 u 0 ˆσ 2 u k 2 Wu (ˆθ θ) W 1 (ˆθ θ) ˆσ 2 k 2 (by C-S generalization in previous notes) Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

11 (ˆθ θ) W 1 (ˆθ θ) qˆσ 2 k2 q (Cˆβ Cβ) (C(X X) C ) 1 (Cˆβ Cβ) qˆσ 2 k2 q. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

12 Thus, we have P(u θ L(u, k) u R q ) [ (Cˆβ Cβ) (C(X X) C ) 1 (Cˆβ Cβ) = P qˆσ 2 ] k2. q What shall we choose for k to make this probability equal to 1 α? Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

13 Thus, if (Cˆβ Cβ) (C(X X) C ) 1 (Cˆβ Cβ) qˆσ 2 F q,n r. k = qf q,n r,α, then k 2 q = F q,n r,α so that the simultaneous coverage probability is 1 α. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

14 Example: Suppose an experiment was conducted using a completely randomized design with 10 subjects in each of 4 treatment groups. The treatment groups were defined by the combinations of levels from 2 factors: diet (1 or 2) and exercise program (1 or 2). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

15 The model y ijk = µ ij + ε ijk was fit the response data y ijk = measure of overall health for diet i, exercise program j, subject k (i = 1, 2; j = 1, 2; k = 1,..., 10). The parameter µ ij represents the mean response for diet i, exercise program j (i = 1, 2; j = 1, 2). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

16 The ε ijk terms are assumed to be iid N(0, σ 2 ). A summary of the data is as follows: ȳ 11 = 9 ȳ 12 = 7 ȳ 21 = 8 ȳ 22 = 3 ˆσ 2 = 5. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

17 Suppose we want to construct a set of confidence intervals using a method that gives simultaneous coverage probability at least 95%. Suppose the confidence intervals will be used to address the following questions: opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

18 1. Diet main effect? 2. Exercise program main effect? 3. Diet-by-exercise program interaction? 4. Difference between diet 1 and diet 2 under exercise program 1? 5. Difference between exercise program 1 and 2 under diet 1? 6. Diet 1, exercise program 1 vs. mean of other treatments? Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

19 What estimable function of β = µ 11 µ 12 µ 21 µ 22 is of interest in each of the questions 1 through 6, respectively? opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

20 1. 2. µ 11 +µ 12 2 µ 21+µ 22 2 = µ 11+µ 12 µ 21 µ 22 2 µ 11 +µ 21 2 µ 12+µ 22 2 = µ 11 µ 12 +µ 21 µ (µ 11 µ 12 ) (µ 21 µ 22 ) = µ 11 µ 12 µ 21 + µ µ 11 µ µ 11 µ µ 11 (µ 12+µ 21 +µ 22 ) 3 = 3µ 11 µ 12 µ 21 µ opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

21 Compute an estimate and standard error for each estimable function of interest. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

22 X = I 1, X X = 10 I ȳ 11 9 ˆβ = (X X) X ȳ 12 7 y = = ȳ ȳ Var(c ˆβ) = σ 2 c (X X) c = σ2 10 c c ˆσ se(c 2 5 ˆβ) = 10 c c = 10 c c = c c/2. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

23 1. c 1 ˆβ = 2.5 se 1 = 1/2 2. c 2 ˆβ = 3.5 se 2 = 1/2 3. c 3 ˆβ = 3 se 3 = 2 4. c 4 ˆβ = 2 se 4 = 1 5. c 5 ˆβ = 1 se 5 = 1 6. c 6 ˆβ = 3 se 6 = 2/3. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

24 Determine appropriate intervals to address questions 1 through 6. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

25 Each interval is of the form c j ˆβ ± kse j. How shall we choose k? opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

26 If we use the Bonferroni approach, then k = t 40 4, 0.05 (2)(6) This approach would be legitimate if we were interested in these 6 intervals, and only these 6 intervals, prior to observing the data. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

27 Alternatively, we can consider Scheffé intervals, k = qf q,40 4,0.05. What is the value of q in our situation? Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

28 Recall that Scheffé intervals have the property P(u θ L(u, k) u R q ) = 1 α, where θ = Cβ for some q p C of rank q. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

29 If we choose q p C so that c 1β,..., c qβ {u θ : u R q }, then we will have Scheffé intervals that give simultaneous coverage at least 1 α for c 1 β,..., c 6 β. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

30 Because qf q,n r,α is an increasing function of q, we would like to pick q no larger than necessary. opyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

31 The matrix 2c 1 2c 2 c 3 c 4 c 5 3c = has rank 3. To see this, note that each row is a contrast vector (sums to zero) and is thus in N ( 1 ), which has dim Thus, rank 3. The last 3 rows are LI, so the rank is exactly 3. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

32 Thus, dim(span{c 1,..., c 6 }) = 3. We can take q = 3 to obtain k = 3F 3,40 4, The resulting intervals c j ˆβ j ± 2.93se j j = 1,..., 6 have simultaneous coverage at least 95%. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

33 We could also include additional intervals for other c β without changing k or losing the guarantee of simultaneous coverage probability at least 95%, as long as c C(C ). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

34 We have not specified C explicitly, but we can choose C the rows of C form a basis for the set of all contrasts; i.e., N ( ). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

35 This Scheffé method for all possible contrasts allows us to construct as many intervals as we wish and still have simultaneous coverage probability at least 1 α, provided each interval is for a c β 1 c = 0. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

36 We can even examine the data to decide which contrasts appear to be of most interest. When using the Bonferroni method, intervals of interest must be preplanned before observing the data. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37

37 If the contrasts in this example were preplanned, Bonferroni would be preferred over Scheffé because the Bonferroni intervals are narrower. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 37