Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 685
Power Calculations
Computations for the Power
To calculate power, first get the critical value for the central F by solving for F C in the equation
α = 1–
ProbF[ F C
, dfH,
n – dfR – 1]
Then obtain the probability of getting this critical value or higher
nδ 2
Power = 1–
ProbF F C
, dfH, n – dfR – 1,
--------
σ 2
nδ 2
The last argument to ProbF is the noncentrality value λ --------
SS(H)
= , which can be estimated as ------------- for
retrospective power.
σ 2
σˆ 2
Computations for Adjusted Power
The adjusted power is a function a noncentrality estimate that has been adjusted to remove positive bias in
the original estimate (Wright and O’Brien 1988). Unfortunately, the adjustment to the noncentrality
estimate can lead to negative values. Negative values are handled by letting the adjusted noncentrality
estimate be
λˆ λˆ ( N – dfR– 1 – 2)
adj = Max 0,
-------------------------------------------- – dfH
N– dfR – 1
where N is the actual sample size, dfH is the degrees of freedom for the hypothesis and dfR is the degrees of
freedom for regression in the whole model. F s is a F-value calculated from data. F C is the critical value for
the F-test.
The adjusted power is
Power adj = 1–
ProbF[ F C
, dfH, n – dfR – 1,
λˆ adj]
.
Confidence limits for the noncentrality parameter are constructed according to Dwass (1955) as
Lower CL for λ = dfH( Max[ 0,
F s
– F C
]) 2
Upper CL for λ =
dfH( F s
– F C
) 2
Power confidence limits are obtained by substituting confidence limits for λ into
Power = 1 - ProbF[F C , dfH, n - dfR - 1, λ]