Confidence Bands in Linear Regression with Constraints on the Independent Variables

Halperin, Max; Gurian, Joan

doi:10.2307/2283893

Cited by 22 publications

(4 citation statements)

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“…For an overview of various finite-interval bands and their computational details, see Miller (1981) and Naiman (1987a). Many authors have discussed finite-interval bands of hyperbolic width over [a,b] for multiple linear regression; in particular, see Bohrer (1967Bohrer ( , 1969Bohrer ( , 1973, Halperin and Gurian (1968), Bohrer and Francis (1972a,b), Wynn and Bloomfield (1971), Wynn (1971), Casella and Strawderman (1980), and Knafl, Sacks, and Ylvisaker (1985).…”

Section: Other Considerationsmentioning

confidence: 99%

The graphical advantages of finite interval confidence band procedures

Stewart

1991

Communications in Statistics - Theory and Methods

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When presented as graphical illustrations, regression surface confidence bands for linear statistical models quickly convey detailed information about analysis results. A taut confidence band is a compact set of cl~rves which are estimation candidates for the unobservable, fixed regression curve. The bounds of the band are usually plotted with the estimated regression curve and may be overlaid by a scatter-plot of the data to provide an integrated visual impression. Finite-interval confidence bands offer the advantages of clearer interpretation and improved efficiency and avoid visual ambiguities inherent t o infinite-interval bands. The definitive characteristic of a finite-interval confidence band is that it is only necessary to plot it over a finite interval in order to visually communicate all its information. In contrast, visual representations of infiniteinterval bands are not fully informative and can be misleading. When an infinite-interval band is plotted, and therefore truncated, substantial information given by its asymptotic behavior is lost. Many curves that are clearly within the plotted portion of the infiniteCopyright @ 1991 by Marcel Dekker, Inc. Downloaded by [New York University] at 16:52 09 January 2015 3976STEWART interval confidence band eventually cross a boundary. In practice, a finite-interval band can always be easily obtained from any infinite-interval band. This article focuses on interpretational considerations of symmetric confidence bands as graphical devices.

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Section: Other Considerationsmentioning

confidence: 99%

The graphical advantages of finite interval confidence band procedures

Stewart

1991

Communications in Statistics - Theory and Methods

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“…A considerable amount of work has been done on improving the earlier work by restricting the predictor variables to some specified regions. For example, Halperin & Gurain [6] and Liu & Lin [7] constructed two-sided hyperbolic confidence bands over an ellipsoidal region. Wynn & Bloomfield [8] and Uusipaikka [9] developed exact two-sided hyperbolic bands when the predictor variable is restricted to an interval or union of intervals.…”

Section: Introductionmentioning

confidence: 99%

“…Wynn & Bloomfield [8] and Uusipaikka [9] developed exact two-sided hyperbolic bands when the predictor variable is restricted to an interval or union of intervals. Casella and Strawderman [10] were able to build an exact two-sided band over a region that was more general than the region considered by Halperin & Gurain [6] and Liu & Lin [7]. Seppanen and Uusipaikka [2] further studied the region investigated by Casella and Strawderman [10], and developed an exact two-sided confidence band over a region that is even more general than the one considered by Casella and Strawderman.…”

Section: Introductionmentioning

confidence: 99%

Construction of simultaneous confidence bands for multiple logistic regression models over restricted regions

Kerns

2016

Statistics

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This article presents methods for constructing an asymptotic hyperbolic band under the multiple logistic regression model when the predictor variables are restricted to a specific region X . Scheffé's method yields unnecessarily wide, and hence conservative, bands if the predictor variables can be restricted to a certain region. Piegorsch and Casella [1] developed a procedure to build an asymptotic confidence band for the multiple logistic regression model over particular regions. Those regions are shown to be special cases of the region X , which was first investigated by Seppanen and Uusipaikka [2] in the multiple linear regression context. This article also provides methods for constructing conservative confidence bands when the restricted region is not of the specified form. Particularly, rectangular restricted regions, which are commonly encountered in practice, are considered. Two examples are given to illustrate the proposed methodology, and one example shows that the proposed procedure outperforms the method given by Piegorsch and Casella [1].

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“…S-method bands on '(x) have been restricted to the positive orthant {xi: xi~O, i=l, ••• ,K} by Bohrer (1967), and then to ellipsoidal regions for x by Halperin and Gurian (1968). Casella and Strawderman (1980) later derived restricted bands over a wide class of regions for x.…”

Section: Introductionmentioning

confidence: 99%

Confidence Bands for Logistic Regression with Restricted Predictor Variables

Piegorsch

Casella

1988

Biometrics

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Confidence bands are constructed for the logistic response function when there is an interval restriction on each of the predictor variables. Scheffe's Smethod is employed. Specific details are given for the case of one predictor variable, along with details for a fixed-width alternative to the S-method bands. In the one-predictor case, Monte Carlo results suggest that both bands are conservative for small sample sizes, such as N=25. By N=200 the S-method's coverage probabilities are seen to attain their nominal levels while the fixedwidth bands remain conservative. The procedures are exemplified with data from a short-term mutagenicity experiment.

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Confidence Bands in Linear Regression with Constraints on the Independent Variables

Cited by 22 publications

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The graphical advantages of finite interval confidence band procedures

The graphical advantages of finite interval confidence band procedures

Construction of simultaneous confidence bands for multiple logistic regression models over restricted regions

Confidence Bands for Logistic Regression with Restricted Predictor Variables

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