Power & Sample Size Calculation

Transcription

1 Chapter 7 Power & Sample Size Calculation Yibi Huang Chapter 7 Section 10.3 Power & Sample Size Calculation for CRDs Power & Sample Size for Factorial Designs Chapter 7-1

2 Power & Sample Size Calculation When proposing an experiment (applying for funding etc), nowadays one needs to show that the proposed sample size (i.e. the number of experiment units) is neither so small that scientifically interesting effects will be swamped by random variation (i.e., has enough power to reject a false H 0 ) nor larger than necessary, wasting resources (time & money) Chapter 7-2

3 Errors and Power in Hypothesis Testing A Type I error occurs when H 0 is true but is rejected. A Type II error occurs when H 0 is false but is accepted. The (significance) level of a test is the chance of making a Type I error, i.e., the chance to reject a H 0 when it is true. The power of the test is the the chance of rejecting H 0 when H a is true: power = 1 P(making type II error H 0 is false) = 1 β = P(correctly reject H 0 H 0 is false) A good test should have small test-level and large power. H 0 is true H 0 is false H a is rejected H 0 is rejected (H 0 is accepted) (H a is accepted) α = P(Type I error) β = P(Type II error) Chapter 7-3

4 Power Calculation for ANOVA For a CRD design with g treatments, recall the means model: y ij = µ i + ε ij and the effects model: y ij = µ + α i + ε ij, for i = 1,..., g, and j = 1,..., n i. The two models are related via the relationship µ i = µ + α i. For CRDs, we use the means model more often. However, for power and sample size calculation, we need to use the effects model and µ must be defined as µ = 1 g n i µ i = 1 g n i µ i. N N i=1 j=1 which implies that g i=1 n iα i = 0. Here N = g i=1 n i is the total number of experimental units. Chapter 7-4 i=1

5 Recall the H 0 and H a for the ANOVA F test are H 0 : µ 1 = = µ g v.s. H a : µ i s not all equal, in terms of the means model, or equivalently in terms of the effects model, they are H 0 : α 1 = = α g = 0 v.s. H a : Not all α i s are 0. Recall we reject H 0 at level α if the test statistic F = MS Trt MSE exceeds the critical value F α,g 1,N g. So Power = P(Reject H 0 H a is true) = P(F > F α, g 1, N g ). To find P(F > F α, g 1, N g ), we must know the distribution of F. What is the distribution of F under H 0? F g 1,N g. And under H a? Chapter 7-5

6 For F = MS Trt MSE = SS Trt/(g 1), recall in Ch3 we ve shown that SSE/(N g) no matter under H 0 or H a, it is always true that SSE σ 2 χ2 N g, and E(MSE) = σ2 For the numerator, SS Trt /σ 2 has a χ 2 g 1 distribution under H 0. But under H a, it has a non-central Chi-square distribution χ 2 g 1,δ with non-central parameter Moreover, δ = g i=1 n iα 2 i σ 2. E(SS Trt ) = (g 1)σ 2 + g n iα 2 i=1 i }{{} δσ { 2 (g 1)σ 2 under H 0 = (g 1 + δ)σ 2 under H a Chapter 7-6

7 Non-Central F -Distribution Under H a, it can be shown that F = MS Trt MSE has a non-central F -distribution on degrees of freedom g 1 and N g, with non-centrality parameter δ, denoted as g i=1 F F g 1,N g,δ where δ = n iαi 2 σ 2. Recall that α i = µ i µ, and µ = 1 N g n i µ i = 1 N i=1 j=1 g n i µ i. Non-central F -distribution is also right skewed, and the greater the non-centrality parameter δ, the further away the peak of the distribution from 0. Chapter 7-7 i=1

8 Example 1: Power Calculation (1) g = 5 treatment groups group sizes: n 1 = n 2 = n 3 = 5, n 4 = 6, n 5 = 4 assume σ = 0.8 desired significance level α = 0.05 find the power of the test when H a is true with µ 1 = 1.6, µ 2 = 0.6, µ 3 = 2, µ 4 = 0, µ 5 = 1. Solution. The grand mean µ is g i=1 µ = n iµ i = N = = 1 The α i s are thus α i = µ i µ = µ i 1: α 1 = 0.6, α 2 = 0.4, α 3 = 1, α 4 = 1, α 5 = 0. Chapter 7-8

9 Example 1: Power Calculation (2) The non-centrality parameter is g i=1 δ = n iαi 2 σ 2 = ( 0.4) ( 1) = = So F = MS Trt MSE { F g 1, N g = F 4, 25 5 under H 0 F g 1, N g,δ = F 4, 25 5, under H a H 0 : F F 4,20 H a : F F 4,20, Chapter 7-9

10 Example 1: Power Calculation (3) The critical value to reject H 0 keeping the significance level at α = 0.05 is F 0.05,4, > qf(0.05,4,20, lower.tail=f) [1] When H 0 is true, the chance that H 0 is rejected is only 0.05 (the red shaded area.) Accept H 0 Reject H 0 F 0.05,4,20 Area = Reminder: Don t confuse the following two: in most STAT books, F α,df 1,df 2 means the area of the upper tail is α. in R, qf(alpha, df1, df2) means the the area of the lower Chapter 7-10

11 Example 1: Power Calculation (4) If H a is true and µ 1 = 1.6, µ 2 = 0.6, µ 3 = 2, µ 4 = 0, µ 5 = 1, we know then F F 4,20,δ= The power to reject H 0 is the area under the density of F 4,20,δ=21.25 beyond the critical value (the blue shaded area,) which is > pf(qf(.95,4,20),4,20, ncp=21.25, lower.tail=f) [1] In the R codes, ncp means the non-centrality parameter Accept H 0 Reject H 0 Power F 0.05,4, Chapter 7-11

12 Example 1: Power Calculation (5) Remark The power of a test is not a single value, but a function of the parameters in H a. If parameter µ i s change their values, the power of the test also changes. It makes no sense to talk about the power of a test without specifying the parameters in H a Chapter 7-12

13 Power Of A Test Is Affected By... The larger the non-centrality parameter the greater the power. δ = The power of a test will increase if g i=1 n iα 2 i σ 2, the number of replicate n i per treatment increases the magnitude of treatment effects α i s increase (under the constraint i n iα i = 0) the size of noise σ 2 decreases. Chapter 7-13

14 Example 2: Sample Size Calculation (1) g = 5 treatment groups of equal sample size n i = n for all i assume σ = 0.8 desired significance level α = 0.05 Assuming the design is balanced that all groups have identical sample size n, what is the minimal sample size n per treatment to have power 0.95 when α 1 = 0.5, α 2 = 0.5, α 3 = 1, α 4 = 1, α 5 = 0? Solution. The non-centrality parameter is g i=1 δ = n iαi 2 σ 2 = n [ ( 0.5) ( 1) ] = n 2.5 = 3.9n 0.82 So F = MS Trt MSE { F g 1, N g = F 4, 5n 5 under H 0 F g 1, N g,δ = F 4, 5n 5, 3.9n under H a Chapter 7-14

15 Recall the critical value F for rejecting H 0 at level α = 0.05 is F = F α, g 1, N g = F 0.05, 4, 5n 5 which can be find in R via the command > F.crit = qf(alpha, g-1, N-g, lower.tail=f) # syntax > F.crit = qf(0.05, 4, 5*n-5, lower.tail=f) # sub-in the values By definition, Power = P(reject H 0 H a is true) = P(the non central F statistic F ) = P(F (g 1, N g, δ) F ) = P(F (4, 5n 5, 3.9n) F ) which can be found in R via the command > pf(f.crit, g-1, N-g, ncp=delta, lower.tail=f) # syntax > pf(f.crit, 4, 5*n-5, ncp=3.9*n, lower.tail=f) # sub-in values In the R codes, ncp means the non-centrality parameter. Chapter 7-15

16 Now we find the R code to find the power of the ANOVA F -test when n is known. Let s plug in different values of n and see what is the smallest n to make power > F.crit = qf(alpha, g-1, N-g, lower.tail=f) > pf(f.crit, g-1, N-g, ncp=delta, lower.tail=f) # syntax > n = 5 > F.crit = qf(0.05, 4, 5*n-5, lower.tail=f) > pf(f.crit, 4, 5*n-5, ncp=3.9*n, lower.tail=f) [1] # less than 0.95, not high enough > n = 6 > F.crit = qf(0.05, 4, 5*n-5, lower.tail=f) > pf(f.crit, 4, 5*n-5, ncp=3.9*n, lower.tail=f) [1] # greater than 0.95, bingo! So we need 6 replicates in each of the 5 treatment groups to ensure a power of 0.95 when α 1 = 0.5, α 2 = 0.5, α 3 = 1, α 4 = 1, α 5 = 0. Remark: Again, we have to specify the H a fully to to find the appropriate sample size. Chapter 7-16

17 But σ 2 is Unknown... As σ 2 is usually unknown, here are a few ways to make a guess. Make a small-sample pilot study to get an estimate of σ. Based on prior studies or knowledge about the experimental units, can you think of a range of plausible values for σ? If so, choose the biggest one. You could repeat the sample size calculations for various levels of σ to see how it affects the needed sample size. Chapter 7-17

18 How to Specify the H a? As the power of a test depends on the alternative hypothesis H a, that is, the α i s, one might has to try several sets of α i s to find the appropriate sample size. But how many H a s we have to try? Here is a useful trick. 1. Suppose we would be interested if any two means differed by D or more. 2. The smallest value of δ in this case is when two means differ by exactly, D, and the other g 2 means are halfway between. 3. So try α 1 = D/2, α 2 = D/2, and α i = 0 for all other τ. Assuming equal sample sizes, the non-centrality parameter is δ = i nαi 2 σ 2 = n(d2 /4 + D 2 /4) σ 2 = nd2 2σ 2. Chapter 7-18

19 Power and Sample Size Calculation for Complete Block Designs For complete block designs y ij = µ + α i + β j + ε ij y ijk = µ + α i + β j + γ k + ε ijk y ijkl = µ + α i + β j + γ k + δ l + ε ijkl RCBD Latin Square Graeco-Latin Square where α i s are the treatment effect, and β j, γ k, δ l are effects of blocking factors, when H a is true, treatments have different effects, the ANOVA F -statistic also has a non-central F -distribution F = MS trt MSE { F g 1, df of MSE under H 0 under H a F g 1, df of MSE,δ where the non-centrality parameter δ is δ = r i α2 i σ 2, where r = # of replicates per treatment Note that the df changes from N g to the df of MSE. Chapter 7-19

20 Power and Sample Size Calculation for Balanced Factorial Designs Section 10.3 in Oehlert s book We may ignore factorial structure and compute power and sample size for the overall null hypothesis of no model effects. We will address methods for balanced data only because most factorial experiments are designed to be balanced, and simple formulae Power for balanced data for noncentrality parameters exist only for balanced data. For factorial data, we usually test H 0 about main effects or interactions in addition to the overall H 0 of no model effects. Chapter 7-20

21 Non-Centrality Parameter for Balanced Factorial Designs Ex, consider a 3-way factorial design y ijkl = µ + α i + β j + γ k + αβ ij + βγ jk + αγ ij + αβγ ijk + ε ijkl to calculate the power of the F -test of a main effect or an interaction effect under H a, the non-centrality parameter can be calculated as follows: Square the factorial effects (using the zero-sum constraint) and sum them, Multiply this sum by the total number of data in the design divided by the number of levels in the effect, and Divide that product by the error variance. Chapter 7-21

22 E.g., consider a a b c factorial design with r replicates per treatment y ijkl = µ + α i + β j + γ k + αβ ij + βγ jk + αγ ij + αβγ ijk + ε ijkl To test about the B main effects H 0 : β 1 = = β b = 0 v.s. H a : not all β j are 0 the ANOVA F -statistic for B main-effect is F = MS B MSE { F b 1, df of MSE under H 0 under H a F b 1, df of MSE,δ where the non-centrality parameter δ is ( N ) / ( δ = b j β2 j σ 2 = rac ) / j β2 j σ 2. Here N = abcr is the total number of observations. Chapter 7-22

23 E.g., consider a a b c factorial design with r replicates per treatment y ijkl = µ + α i + β j + γ k + αβ ij + βγ jk + αγ ij + αβγ ijk + ε ijkl To test about the AB interactions H 0 : all αβ ij = 0 v.s. H a : not all αβ ij = 0 the ANOVA F -statistic for AB interactions F = MS AB MSE { F (a 1)(b 1), df of MSE under H 0 under H a F (a 1)(b 1), df of MSE,δ where the non-centrality parameter δ is ( N ) / ( δ = ab (αβ ij) 2 σ 2 = rc (αβ ij) 2) / σ 2. ij ij Here N = abcr is the total number of observations. Chapter 7-23

24 Sample Size Should Be The Last Thing To Consider In practice, sample size calculation is the last thing to do in designing an experiment. Consider the followings before calculating the required sample size. asking the right question, choosing an appropriate response, thinking about what factors need to be blocked on, choosing a right design, setting up the right procedure to recruit subjects, and so on, All the above are more important than doing the right sample size calculation! Sometimes, choosing a better design can substantially reduces the required sample size. Chapter 7-24