The confidence limits are computed using Fieller’s theorem (Fieller, 1954), 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.
where the parenthesized subscript (i) indicates that the ith component is omitted. A confidence interval can be formed from the relation
where t is the t value for the specified confidence level.
an interval of the form 1, φ2), where φ1 < φ2
Note: The Fit Y by X logistic platform and the Fit Model Nominal Logistic personalities use t values when computing confidence intervals for inverse prediction. The Fit Model Generalized Linear Model personality, as well as PROC PROBIT in SAS/STAT, use z values, which give different results.