Modeling and Multivariate Methods - SAS

522 Fitting Partial Least Squares Models Chapter 21
Statistical Details
T 2 Plot
The T 2 value for the i th observation is computed as follows:
p n
T2
i
( n – 1) t2 ij
⁄ t2
= kj
j = 1
k = 1
where t ij = X score for the i th row and j th extracted factor, p = number of extracted factors, and n = number
of observations used to train the model. If validation is not used, n = total number of observations.
The control limit for the T 2 Plot is computed as follows:
((n-1) 2 /n)*BetaQuantile(0.95, p/2, (n-p-1)/2)
where p = number of extracted factors, and n = number of observations used to train the model. If
validation is not used, n = total number of observations.
van der Voet T 2
The van der Voet T 2 test helps determine whether a model with a specified number of extracted factors
differs significantly from a proposed optimum model. The test is a randomization test based on the null
hypothesis that the squared residuals for both models have the same distribution. Intuitively, one can think
of the null hypothesis as stating that both models have the same predictive ability.
The test statistic is
C i
=
( R2 ijk ,
– R2
opt,
jk
)
jk
where R i, jk
is the jth predicted residual for response k for the model with i extracted factors, and R opt,
jk
is
the corresponding quantity for the model based on the proposed optimum number of factors, opt. The
significance level is obtained by comparing C i with the distribution of values that results from randomly
exchanging R2 i, jk
and R2
opt,
jk
. A Monte Carlo sample of such values is simulated and the significance level
is approximated as the proportion of simulated critical values that are greater than C i .
Confidence Ellipse for Scatter Scores Plots
The Scatter Scores Plots option produces scatterplots with a confidence ellipse. The coordinates of the top,
bottom, left, and right extremes of the ellipse are as follows:
For a scatterplot of score i versus score j:
• the top and bottom extremes are +/-sqrt(var(score i)*z)
• the left and right extremes are +/-sqrt(var(score j)*z)
where z = ((n-1)*(n-1)/n)*BetaQuantile(0.95, 1, (n-3)/2).