Modeling and Multivariate Methods - SAS

686 Statistical Details Appendix A
Inverse Prediction with Confidence Limits
Inverse Prediction with Confidence Limits
Inverse prediction estimates a value of an independent variable from a response value. In bioassay problems,
inverse prediction with confidence limits is especially useful. In JMP, you can request inverse prediction
estimates for continuous and binary response models. If the response is continuous, you can request
confidence limits for an individual response or an expected response.
The confidence limits are computed using Fieller’s theorem, which is based on the following logic. The goal
is predicting the value of a single regressor and its confidence limits given the values of the other regressors
and the response.
Let b estimate the parameters β so that we have b distributed as N(β,V).
Let x be the regressor values of interest, with the ith value to be estimated.
Let y be the response value.
We desire a confidence region on the value of x[i] such that β'x = y with all other values of x given.
The inverse prediction is just
x[]
i
y – β'
() i x () i
= ---------------------------
β[]
i
where the parenthesized-subscript “ ( i ) ” means that the ith component is omitted. A confidence interval can
be formed from the relation:
( y–
b'x) 2 < t 2 x'Vx
with specified confidence if y
where
z 2 g+ zh+ f = 0
=
β'x . A quadratic equation results of the form
g
=
b[] i
2 – t 2 V[ ii , ]
h
=
– 2yb[] i + 2b[]b' i () i x () i – 2t 2 V[ i,
() i ]'x () i
f = y 2 – 2yb' () i x () i + ( b' () x i () i ) 2 – t 2 x () i 'V () i x () i
It is possible for the quadratic equation to have only imaginary roots, in which case the confidence interval
becomes the entire real line. In this situation, Wald intervals are used. If only one side of an interval is
imaginary, the Fieller method is still used, but the imaginary side is returned as missing.
Note: The Logistic platform in JMP uses t values when computing the confidence intervals for inverse
prediction. PROC PROBIT in SAS/STAT and the Generalized Linear Model platform in JMP use z values,
which give slightly different results.