Simultaneous confidence bands in linear regression analysis

A simultaneous confidence band provides useful information on the plausible range of anunknown regression model. For a simple linear regression model, the most frequentlyquoted bands in the statistical literature include the two-segment band, the three-segmentband and the hyperbolic band, and for a multiple linear regression model, the most com-mon bands in the statistical literature include the hyperbolic band and the constant widthband. The optimality criteria for confidence bands include the Average Width criterionconsidered by Gafarian (1964) and Naiman (1984) among others, and the Minimum AreaConfidence Set (MACS) criterion of Liu and Hayter (2007). A concise review of theconstruction of two-sided simultaneous confidence bands in simple and multiple linear re-gressions and their comparison under the two mentioned optimality criteria is provided inthe thesis. Two families of confidence bands, the inner-hyperbolic bands and the outerhyperbolicbands, which include the hyperbolic and three-segment bands as special cases,are introduced for a simple linear regression. Under the MACS criterion, the best con-fidence band within each family is found by numerical search and compared with thehyperbolic band, the best three-segment band and with each other. The inner-hyperbolicfamily of confidence bands, which include the hyperbolic and constant-width bands asspecial cases, is also constructed for a multiple linear regression model over an ellipsoidalcovariate region and the best band within the family is found by numerical search. Fora multiple linear regression model over a rectangular covariate region (i.e. the predictorvariables are constrained in intervals), no method of constructing exact simultaneous con-fidence bands has been published so far. A method to construct exact two-sided hyperbolicand constant width bands over a rectangular covariate region and compare between themis provided in this thesis when there are up to three predictor variables. A simulationmethod similar to the ones used by Liu et al. (2005a) and Liu et al. (2005b) is alsoprovided for the calculation of the average width and the minimum volume of confidenceset when there are more than three predictor variables. The methods used in this thesisare illustrated with numerical examples and the Matlab programs used are available uponrequest

Comparison of hyperbolic and constant width simultaneous confidence bands in multiple linear regression under MVCS criterion

A simultaneous confidence band provides useful information on the plausible range of the unknown regression model, and different confidence bands can often be constructed for the same regression model. For a simple regression line, Liu and Hayter (2007) propose use of the area of the confidence set corresponding to a confidence band as an optimality criterion in comparison of confidence bands; the smaller the area of the confidence set, the better the corresponding confidence band. This minimum area confidence set (MACS) criterion can begeneralized to a minimum volume confidence set (MVCS) criterion in the study of confidence bands for a multiple linear regression model. In this paper hyperbolic and constant width confidence bands for a multiple linear regression model over a particular ellipsoidal region of the predictor variables are compared under the MVCS criterion. It is observed that whether one band is better than the other depends on the magnitude of one particular angle that determines the size of the predictor variable region. When the angle and so the size of the predictor variable region is small, the constant width band is better than the hyperbolic band but only marginally. When the angle and so the size of the predictor variable region is large the hyperbolic band can be substantially better than the constant width band

Optimal simultaneous confidence bands in multiple linear regression with predictor variables constrained in an ellipsoidal region

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model function, just as a confidence interval gives the plausible range of an unknown parameter. For a multiple linear regression model, confidence bands of different shapes, such as the hyperbolic band and the constant width band, can be constructed and the predictor variable region over which a confidence band is constructed can take various forms. One interesting but unsolved problem is to find the optimal (shape) confidence band over an ellipsoidal region ? E under the Minimum Volume Confidence Set (MVCS) criterion of Liu and Hayter (2007) and Liu et al. (2009). This problem is challenging as it involves optimization over an unknown function that determines the shape of the confidence band over ? E . As a step towards solving this difficult problem, in this paper, we introduce a family of confidence bands over ? E , called the inner-hyperbolic bands, which includes the hyperbolic and constant-width bands as special cases. We then search for the optimal confidence band within this family under the MVCS criterion. The conclusion from this study is that the hyperbolic band is not optimal even within this family of inner-hyperbolic bands and so cannot be the overall optimal band. On the other hand, the constant width band can be optimal within the family of inner-hyperbolic bands when the region ? E is small and so might be the overall optimal band

Optimal simultaneous confidence bands in simple linear regression

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model. For simple linear regression models, the most frequently quoted bands in the statistical literature include the hyperbolic band and the three-segment bands. One interesting question is whether one can construct confidence bands better than the hyperbolic and three-segment bands. The optimality criteria for confidence bands include the average width criterion considered by Gafarian (1964) and Naiman (1984) among others, and the minimum area confidence set (MACS) criterion of Liu and Hayter (2007). In this paper, two families of exact 1?? confidence bands, the inner-hyperbolic bands and the outer-hyperbolic bands, which include the hyperbolic and three-segment bands as special cases, are introduced in simple linear regression. Under the MACS criterion, the best confidence band within each family is found by numerical search and compared with the hyperbolic band, the best three-segment band and with each other. The methodologies are illustrated with a numerical example and the Matlab programs used are available upon request

A ray method of confidence band construction for multiple linear regression models

This paper addresses the problem of confidence band construction for a standard multiple linear regression model. A “ray” method of construction is developed which generalizes the method of Graybill and Bowden [1967. Linear segment confidence bands for simple linear regression models. J. Amer. Statist. Assoc. 62, 403–408] for a simple linear regression model to a multiple linear regression model. By choosing suitable directions for the rays this method requires only critical points from t-distributions so that the confidence bands are easy to construct. Both one-sided and two-sided confidence bands can be constructed using this method. An illustration of the new method is provided

Simultaneous confidence bands for linear regression with covariates constrained in intervals

The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc.24, 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k-dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k-dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed-effect or random-effect generalized linear regression models

Exact simultaneous confidence intervals for a finite set of contrasts of three, four or five generally correlated normal means

The construction of a set of simultaneous confidence intervals for any finite number of contrasts of pp generally correlated normal means is considered. It is shown that the simultaneous confidence level can be expressed as a (p?2)(p?2)-dimensional integral for a general p?3p?3. This expression allows one to compute quickly and accurately, by using numerical quadrature, the required critical constants and multiplicity adjusted pp-values for at least p=3p=3, 4 and 5, involving only one-, two- and three-dimensional integrals, respectively. Real data examples from a drug stability study and a dose response study are used to illustrate the metho