Modeling and Multivariate Methods - SAS

684 Statistical Details Appendix A
Power Calculations
Solving for Lb, a scalar, given an F for some α-level, like 0.05, and using a t as the square root of a
one-degree-of-freedom F, making it properly two-sided, gives
( Lb) LSV = t α ⁄ 2
s LX'X ( ) – 1 L'
For L testing some β i , this is
b i
LSV
= t α ⁄ 2
s (( X'X) – 1 ) ii
which can be written with respect to the standard error as
b i
LSV
=
t α ⁄ 2
stderr( b i
)
If you have a simple model of two means where the parameter of interest measures the difference between
the means, this formula is the same as the LSD, least significant difference, from the literature
LSD = t α ⁄ 2
s
-----
1 1
+ -----
n 1
n 2
In the JMP Fit Model platform, the parameter for a nominal effect measures the difference to the average of
the levels, not to the other mean. So, the LSV for the parameter is half the LSD for the differences of the
means.
Computations for the LSN
The LSN solves for n in the equation:
nδ
α = 1 – ProbF --------------- 2
, dfH,
n – dfR – 1
dfHσ 2
where
ProbF is the central F-distribution function
dfH is the degrees of freedom for the hypothesis
dfR is the degrees of freedom for regression in the whole model
σ 2 is the error variance
δ 2 is the squared effect size, which can be estimated by
SS(H) -------------
n
When planning a study, the LSN serves as the lower bound.