Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 683
Power Calculations
Discriminant Analysis
The distance from an observation to the multivariate mean of the ith group is the Mahalanobis distance,
D 2 , and computed as
where
In saving discriminant columns, N is the number of observations and M is the identity matrix.
The Save Discrim command in the popup menu on the platform title bar saves discriminant scores with
their formulas as columns in the current data table. SqDist[0] is a quadratic form needed in all the distance
calculations. It is the portion of the Mahalanobis distance formula that does not vary across groups.
SqDist[i] is the Mahalanobis distance of an observation from the ith centroid. SqDist[0] and SqDist[i] are
calculated as
and
D 2 = ( y – y i )'S – 1 ( y–
y i ) = y'S – 1 y – 2y'S – 1 y i + y i 'S – 1 y i
S =
------------ E
N – 1
SqDist[0] =
SqDist[i] = SqDist[0] -2y'S – 1 y i + y i 'S – 1 y i
Assuming that each group has a multivariate normal distribution, the posterior probability that an
observation belongs to the ith group is
Prob[i] = ----------------------------
exp( Dist[i] )
Prob[ 0]
where
Prob[0] =
y'S – 1 y
e – 0.5Dist[i]
Power Calculations
The next sections give formulas for computing LSV, LSN, power, and adjusted power.
Computations for the LSV
For one-degree-freedom tests, the LSV is easy to calculate. The formula for the F-test for the hypothesis
Lβ = 0 is:
( Lb)' ( LX'X ( ) – 1 L' ) – 1
( Lb) ⁄ r
F = ------------------------------------------------------------------------
s 2