1. Power Analysis Many of you have seen OCC’s Many of you have seen OCC’s –First specify test size    in a CRD –Compute : –Compute

2. Power Analysis Compute a summary measure of H a : Compute a summary measure of H a : OCC curves will depend on , and the numerator and denominator df OCC curves will depend on , and the numerator and denominator df

3. Power Analysis Select the appropriate OCC curve Select the appropriate OCC curve Find where vertical line drawn from  on horizontal axis intersects appropriate OCC Find where vertical line drawn from  on horizontal axis intersects appropriate OCC Read  on vertical axis; compute  Read  on vertical axis; compute 

4. Power Analysis OCC’s can be used for sample size analysis, but they are awkward OCC’s can be used for sample size analysis, but they are awkward The curves are computed from the distribution of the F statistic under H A The curves are computed from the distribution of the F statistic under H A

5. Derivation of OCC’s Recall that Recall that

6. Derivation of OCC’s Regardless of the true state of nature, Regardless of the true state of nature,where

7. Derivation of OCC’s A non-central  2 rv is often introduced as a sum of independent squared N( i,  2 ) rv’s; its noncentrality parameter would be: A non-central  2 rv is often introduced as a sum of independent squared N( i,  2 ) rv’s; its noncentrality parameter would be: In our case, the normal components are not independent. In our case, the normal components are not independent.

8. Derivation of OCC’s We say that F has a noncentral F distribution with noncentrality parameter  2 We say that F has a noncentral F distribution with noncentrality parameter  2 A non-central F rv is based on a ratio of independent non-central  2 and central  2 rv’s A non-central F rv is based on a ratio of independent non-central  2 and central  2 rv’s

9. Derivation of OCC’s

10. For the balanced case, we have: For the balanced case, we have:

11. Computer Code SAS example SAS example R code R codeS02<-rep(seq(1,5),rep(15,5))n<-rep(seq(2,16),5)ncvec<-n*s02

12. Computer Code Power<-1-pf(qf(.95,a-1,n*a-a),a- 1,n*a-a,ncvec) Contour(seq(2,16),seq(1,5),matri x(power,ncol=5),xlab=“Sample Size”,ylab=“Effect”)

13. Power Analysis for Contrasts Magnitude of contrasts under H A is easy for experimenters to articulate (Yandell, Edwards) Magnitude of contrasts under H A is easy for experimenters to articulate (Yandell, Edwards) We consider one df contrasts only (Yandell focuses on specific cases; our treatment will be more general) We consider one df contrasts only (Yandell focuses on specific cases; our treatment will be more general)

14. Power Analysis for Contrasts We will test H o :L-L o =0 vs. H A :L-L o ≠0 We will test H o :L-L o =0 vs. H A :L-L o ≠0 –Typically, L o =0 Regardless of the true state of nature, Regardless of the true state of nature,

15. Power Analysis for Contrasts For the balanced case, For the balanced case,

16. Power Analysis for Contrasts To adapt the SAS program to contrasts, note that the coefficient of n in the noncentrality parameter has changed To adapt the SAS program to contrasts, note that the coefficient of n in the noncentrality parameter has changed –Loop on L (not s02) and calculate s02 in the loop –This ensures that L is output rather than s02 –Remember to change numerator df