Computation of simultaneous confidence bands is described for simple linear regressions where the band is constructed to be asymmetric about the predictor mean. Both two-sided and one-sided bands are constructed. The bands represent extensions of a class of symmetric confidence bands due to Bowden, 1970. J. Amer. Statist. Assoc. 65, 413–421. An example illustrates the computations, and a WWW-based applet for computing the bands is described.
In many experimental situations, the response variable, Y, is observed in tandem with a non-stochastic predictor variable, x. Often, the mean response is assumed linear in . A common model for this simple linear regression setting assumes that the observations are sampled from a normal parent distribution: . In this case, least-squares estimators b=[b0,b1]′ of the unknown parameter vector β=[β0,β1]′ correspond to those achieved under maximum likelihood, and exact inferences on the effects of x or on the mean response are readily available ( Neter et al., 1996).
Herein we direct attention at construction of simultaneous 1−αconfidence bands for the underlying mean β0+β1x over all values of x in some relevant restriction set . A confidence band is best written as a set of parameters. , where hα is a critical point that gives the band 1−α coverage, S2 is the usual mean square error estimator of σ2 and is the band shape function. In cases where the population variance is known, one simply replaces S2 with the known value of σ2 and modifies the value of hα appropriately (see Section 2, below).
To simplify this notation, let . Then, the band's confidence region becomes
equation(1.1)
If one-sided bands are desired, the confidence region may be written simply as
equation(1.2)
where kα is a one-sided critical point that again gives the band 1−α coverage. (The display gives upper one-sided bands; lower one-sided bands follow by reversing the set of inequalities and changing the sign on kα.)
In practice, an interval restriction on x is common: , where A and B>0 are predetermined constants. Interval restrictions have been considered by many authors; see, e.g., Wynn and Bloomfield (1971), Naiman (1983), or Stewart (1991) among many others. In some applications the predictor variable may be a positive quantity such as dose, mass, time, a dietary supplement, etc. In such cases, it is natural to restrict x to be positive, i.e., where A=0 and B becomes some known upper bound on the possible value of x. Below, we operate under this assumption and fix the restriction set as . (Notice, however, that setting A=0 does not sacrifice any generality: when interest exists over the more general interval, simply transform the predictor variable to x′=x−A and then work on the scale we use herein.)
The most widely recognized band is the Working–Hotelling–Scheffé construction (Working and Hotelling, 1929; Scheffé, 1953) which in a simplified form is This is in fact a member of a larger class of bands, the Bowden p-family , where p≥1 is a shape parameter that determines the bands’ degree of incline near ( Bowden, 1970).
