Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. 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.     

Low dose risk estimation via simultaneous statistical inferences

Summary.  The paper develops and studies simultaneous confidence bounds that are useful for making low dose inferences in quantitative risk analysis. Application is intended for risk assessment studies where human, animal or ecological data are used to set safe low dose levels of a toxic agent, but where study information is limited to high dose levels of the agent. Methods are derived for estimating simultaneous, one-sided, upper confidence limits on risk for end points measured on a continuous scale. From the simultaneous confidence bounds, lower confidence limits on the dose that is associated with a particular risk (often referred to as a bench-mark dose ) are calculated. An important feature of the simultaneous construction is that any inferences that are based on inverting the simultaneous confidence bounds apply automatically to inverse bounds on the bench-mark dose.     

Simultaneous confidence bands for nonlinear regression models

The maximization and minimization procedure for constructing confidence bands about general regression models is explained. Then, using an existing confidence region about the parameters of a nonlinear regression model and the maximization and minimization procedure, a generally conservative simultaneous confidence band is constructed about the model. Two examples are given, and some problems with the procedure are discussed     

Confidence bands for the laplace distribution

Based on a random sample from the Laplace population with unknown shape and scale parameters, one- and two-sided confidence bands on the entire cumulative distribution function and simultaneous confidence intervals for the interval probabilities under the distribution are constructed using Kolmogorov–Smirnov type statistics. Small sample and asymptotic percentiles of the relevant statistics are provided.     

Empirical Likelihood Inference for the Cox Model with Time-dependent Coefficients via Local Partial Likelihood

Abstract.  The Cox model with time-dependent coefficients has been studied by a number of authors recently. In this paper, we develop empirical likelihood (EL) pointwise confidence regions for the time-dependent regression coefficients via local partial likelihood smoothing. The EL simultaneous confidence bands for a linear combination of the coefficients are also derived based on the strong approximation methods. The EL ratio is formulated through the local partial log-likelihood for the regression coefficient functions. Our numerical studies indicate that the EL pointwise/simultaneous confidence regions/bands have satisfactory finite sample performances. Compared with the confidence regions derived directly based on the asymptotic normal distribution of the local constant estimator, the EL confidence regions are overall tighter and can better capture the curvature of the underlying regression coefficient functions. Two data sets, the gastric cancer data and the Mayo Clinic primary biliary cirrhosis data, are analysed using the proposed method.     

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

Exact simultaneous confidence bands (SCBs) for a polynomial regression model are available only in some special situations. In this paper, simultaneous confidence levels for both hyperbolic and constant width bands for a polynomial function over a given interval are expressed as multidimensional integrals. The dimension of these integrals is equal to the degree of the polynomial. Hence the values can be calculated quickly and accurately via numerical quadrature provided that the degree of the polynomial is small (e.g. 2 or 3). This allows the construction of exact SCBs for quadratic and cubic regression functions over any given interval and for any given design matrix. Quadratic and cubic regressions are frequently used to characterise dose response relationships in addition to many other applications. Comparison between the hyperbolic and constant width bands under both the average width and minimum volume confidence set criteria shows that the constant width band can be much less efficient than the hyperbolic band. For hyperbolic bands, comparison between the exact critical constant and conservative or approximate critical constants indicates that the exact critical constant can be substantially smaller than the conservative or approximate critical constants. Numerical examples from a dose response study are used to illustrate the methods.     

Simultaneous confidence bands for growth incidence curves in weighted sup-norm metrics

ABSTRACT

The so-called growth incidence curve (GIC) is a popular way to evaluate the distributional pattern of economic growth and pro-poorness of growth in development economics. The log-transformation of the the GIC is related to the sum of empirical quantile processes which allows for constructions of simultaneous confidence bands for the GIC. However, standard constructions of these bands tend to be too wide at the extreme points 0 and 1 because the estimator of the quantile function can be very volatile at the extreme points. In order to construct simultaneous confidence bands which are narrower at the ends, we consider the convergence of quantile processes with weight functions. In particular, we investigate the asymptotic convergence under specific weighted sup-norm metrics and compare different kinds of qualified weight functions. This implies simultaneous confidence bands that are narrower at the boundaries 0 and 1. We show in simulations that these bands have a more regular shape. Finally, we evaluate real data from Uganda with the improved confidence bands.     

Adjusted Confidence Bands in Nonparametric Regression

Suppose we have {(x _i , y _i )} i = 1, 2,…, n, a sequence of independent observations. We wish to find approximate 1 ? α simultaneous confidence bands for the regression curve. Many previous confidence bands in the literature have practical difficulties. In this article, the local linear smoother is used to estimate the regression curve. The bias of the estimator is considered. Different methods of constructing confidence bands are discussed. Finally, a possible method incorporating logistic regression in an innovative way is proposed to construct the bands for random designs. Simulations are used to study the performance or properties of the methods. The procedure for constructing confidence bands is entirely data-driven. The advantage of the proposed method is that it is simple to use and can be applied to random designs. It can be considered as a practically useful and efficient method.     

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−α$1-\alpha$ 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 Tolerance Limits and a Unified Approach to some Nonparametric Theory

In this paper nonparametric simultaneous tolerance limits are developed using rectangle probabilities for uniform order statistics. Consideration is given to the handling of censored data, and some comparisons are made with the parametric normal theory. The nonparametric regional estimation techniques of (i) confidence bands for a distribution function, (ii) simultaneous confidence intervals for quantiles and (iii) simultaneous tolerance limits are unified. A Bayesian approach is also discussed.     

Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182–213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.     

Exact simultaneous confidence segments for all contrast comparisons

This paper provides an exact method to construct simultaneous confidence bands for all contrasts of several regression lines over a restricted explanatory variable. Due to the lack of exact methods in the literature, currently existing approaches consist mainly of simulation based approaches. Using confidence bands for regression analysis occurs ubiquitously in practice, for example, inference on the shelf-life or stability of a drug, on the reliability of an engineering system over time, on the environmental impact of a fertilizer in a field over time, to list just a few. The new method enhances currently existing approaches that are based on simulations.     

Confidence bands for the logistic and probit regression models over intervals

This article presents methods for the construction of two-sided and one-sided simultaneous hyperbolic bands for the logistic and probit regression models when the predictor variable is restricted to a given interval. The bands are constructed based on the asymptotic properties of the maximum likelihood estimators. Past articles have considered building two-sided asymptotic confidence bands for the logistic model, such as Piegorsch and Casella (1988 Piegorsch, W.W., Casella, G. (1988). Confidence bands for logistic regression with restricted predictor variables. Biometrics 44:739–750.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). However, the confidence bands given by Piegorsch and Casella are conservative under a single interval restriction, and it is shown in this article that their bands can be sharpened using the methods proposed here. Furthermore, no method has yet appeared in the literature for constructing one-sided confidence bands for the logistic model, and no work has been done for building confidence bands for the probit model, over a limited range of the predictor variable. This article provides methods for computing critical points in these areas.     

Tests for comparison of competing risks under the additive risk model

It is of interest that researchers study competing risks in which subjects may fail from any one of k causes. Comparing any two competing risks with covariate effects is very important in medical studies. In this paper, we develop tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels under the additive risk model by a weighted difference of estimates of cumulative cause-specific hazard rates. Motivated by McKeague et al. (2001), we construct simultaneous confidence bands for the difference of two conditional cumulative incidence functions as a useful graphical tool. In addition, we conduct a simulation study, and the simulation result shows that the proposed procedure has a good finite sample performance. A melanoma data set in clinical trial is used for the purpose of illustration.     

Simultaneous Confidence Intervals for the Population Cell Means,for Two-by-Two Factorial Data,that Utilize Uncertain Prior Information

Consider a two-by-two factorial experiment with more than one replicate. Suppose that we have uncertain prior information that the two-factor interaction is zero. We describe new simultaneous frequentist confidence intervals for the four population cell means, with simultaneous confidence coefficient 1 ? α, that utilize this prior information in the following sense. These simultaneous confidence intervals define a cube with expected volume that (a) is relatively small when the two-factor interaction is zero and (b) has maximum value that is not too large. Also, these intervals coincide with the standard simultaneous confidence intervals obtained by Tukey’s method, with simultaneous confidence coefficient 1 ? α, when the data strongly contradict the prior information that the two-factor interaction is zero. We illustrate the application of these new simultaneous confidence intervals to a real data set.     

Confidence Bands for Comparison of Transition Probabilities in a Markov Chain Model

We consider graphs, confidence procedures and tests that can be used to compare transition probabilities in a Markov chain model with intensities specified by a Cox proportional hazard model. Under assumptions of this model, the regression coefficients provide information about the relative risks of covariates in one–step transitions, however, they cannot in general be used to to assess whether or not the covariates have a beneficial or detrimental effect on the endpoint events. To alleviate this problem, we consider graphical tests based on confidence procedures for a generalized Q–Q plot and for the difference between transition probabilities. The procedures are illustrated using data of the International Bone Marrow Transplant Registry.     

Multiple comparisons with a control for exponential location parameters under heteroscedasticity

In this paper, a new design-oriented two-stage two-sided simultaneous confidence intervals, for comparing several exponential populations with control population in terms of location parameters under heteroscedasticity, are proposed. If there is a prior information that the location parameter of k exponential populations are not less than the location parameter of control population, one-sided simultaneous confidence intervals provide more inferential sensitivity than two-sided simultaneous confidence intervals. But the two-sided simultaneous confidence intervals have advantages over the one-sided simultaneous confidence intervals as they provide both lower and upper bounds for the parameters of interest. The proposed design-oriented two-stage two-sided simultaneous confidence intervals provide the benefits of both the two-stage one-sided and two-sided simultaneous confidence intervals. When the additional sample at the second stage may not be available due to the experimental budget shortage or other factors in an experiment, one-stage two-sided confidence intervals are proposed, which combine the advantages of one-stage one-sided and two-sided simultaneous confidence intervals. The critical constants are obtained using the techniques given in Lam [9,10]. These critical constant are compared with the critical constants obtained by Bonferroni inequality techniques and found that critical constant obtained by Lam [9,10] are less conservative than critical constants computed from the Bonferroni inequality technique. Implementation of the proposed simultaneous confidence intervals is demonstrated by a numerical example.     

Multiple comparisons of matched proportions

It is well known (see, e.g., Scheffé (1959)) that if confidence intervals are desired for several treatment comparisons of interest, especially after a preliminary test of significance, then the appropriate technique is to consider simultaneous confidence intervals with a certain joint confidence coefficient. Goodman (1964) derived such simultaneous confidence intervals for contrasts among several multinomial populations, each with the same number, say J, of classes. The special case involving simultaneous confidence intervals for contrasts among several binomial populations on the basis of independent samples follows simply by taking J=2. This paper now deals with the problem of construction of simultaneous confidence intervals among probabilities of ‘success’ on the basis of matched samples.