Power Analysis n Many of you have seen

Power Analysis n Many of you have seen OCC’s – First specify test size a, s 2, m in a CRD – Compute : – Compute

Power Analysis-Noncentrality parameter n Compute a summary measure of Ha: n OCC curves will depend on a, and the numerator and denominator df

Power Analysis—Using tables n Select the appropriate OCC curve f on horizontal axis intersects appropriate OCC n Read b on vertical axis; compute 1 -b n Find where vertical line drawn from

Power Analysis—Sample Size Analysis n OCC’s can be used for sample size analysis, but they are awkward n The curves are computed from the distribution of the F statistic under HA

Derivation of OCC’s n Recall that

Properties of Mean Squares n Regardless of the true state of nature, where

Non-centrality parameter n A non-central c 2 rv is often introduced as a sum of independent squared N(li, s 2) rv’s; its noncentrality parameter would be: n In our case, the normal components are not independent.

Non-central F n We say that F has a noncentral F distribution with noncentrality parameter d 2 n A non-central F rv is based on a ratio of independent non-central c 2 and central c 2 rv’s

Derivation of Power

Power formulas n For the balanced case, we have:

Computer Code n SAS example n R code S 02<-rep(seq(1, 5), each=14) n<-rep(seq(2, 15), 5) ncvec<-n*s 02

R Computer Code Power<-1 -pf(qf(. 95, a-1, n*a-a), a 1, n*a-a, ncvec) Contour(seq(2, 15), seq(1, 5), matrix (power, ncol=5), xlab=“Sample Size”, ylab=“Effect”)

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

Non-central F for contrasts n We will test Ho: L-Lo=0 vs. HA: L-Lo≠ 0 – Typically, Lo=0 n Regardless of the true state of nature,

Non-central F: balanced case n For the balanced case,

SAS code for contrasts n To adapt the SAS program to contrasts, note that the coefficient of n in the noncentrality parameter has changed – Loop on L (not s 02) and calculate the remaining terms in the noncentrality parameter in the loop – This ensures that L is output – Remember to change numerator df