Simultaneous Tolerance Intervals Research Articles

Multiple-Use Calibration for All Future Values and Exact Two-Sided Simultaneous Tolerance Intervals in Linear Regression

Multiple-use calibration using regression is an important statistical tool. Confidence sets for the x-values associated with all future y-values should guarantee a key property, which can be satisfied by simultaneous tolerance intervals (STI’s), and so multiple-use calibration requires the construction of STI’s. In this article, exact two-sided STI’s have been constructed for polynomial regression over any given covariate interval. There is a misconception that two-sided pointwise tolerance intervals (PTI’s) can be employed for multiple-use calibration. This article shows that the confidence sets based on the two-sided PTI’s do not satisfy the key property and so should not be used. Real-world data examples are given in this article for illustration. Supplementary materials for this article are available online.

Simultaneous tolerance intervals for linear regression models using an adjusted product set method

Pointwise tolerance intervals for linear regression models are well established in the literature, however, there is no gold standard for a procedure to construct simultaneous tolerance intervals for linear regression models. The existing methods vary considerably with respect to their finite sample performance in terms of coverage probabilities. Moreover, the numerical investigations into their coverage performance have only been conducted under very limited settings. In this work, we investigate the coverage performance for the major procedures available for the construction of simultaneous tolerance intervals for linear regression models, with increasing complexity in the structure of the models considered. We further show that an adjustment made to the confidence level for a product-set-based method provides coverages close to nominal for the majority of models considered, and this adjusted method is nearly always better than each of the other existing methods. Simultaneous tolerance intervals are then constructed and interpreted for a chemical oxygen demand regression model and two models for the strength of polymer-modified cement mortar.

Statistical calibration for infinite many future values in linear regression: simultaneous or pointwise tolerance intervals or what else?

AbstractStatistical calibration using regression is a useful statistical tool with many applications. For confidence sets for x-values associated with infinitely many future y-values, there is a consensus in the statistical literature that the confidence sets constructed should guarantee a key property. While it is well known that the confidence sets based on the simultaneous tolerance intervals (STIs) guarantee this key property conservatively, it is desirable to construct confidence sets that satisfy this property exactly. Also, there is a misconception that the confidence sets based on the pointwise tolerance intervals (PTIs) also guarantee this property. This paper constructs the weighted simultaneous tolerance intervals (WSTIs) so that the confidence sets based on the WSTIs satisfy this property exactly if the future observations have the x-values distributed according to a known specific distribution F(⋅). Through the lens of the WSTIs, convincing counter examples are also provided to demonstrate that the confidence sets based on the PTIs do not guarantee the key property in general and so should not be used. The WSTIs have been applied to real data examples to show that the WSTIs can produce more accurate calibration intervals than STIs and PTIs.

Rectangular tolerance regions and multivariate normal reference regions in laboratory medicine.

Reference intervals are widely used in the interpretation of results of biochemical and physiological tests of patients. When there are multiple biochemical analytes measured from each subject, a multivariate reference region is needed. Because of their greater specificity against false positives, such reference regions are more desirable than separate univariate reference intervals that disregard the cross-correlations between variables. Traditionally, under multivariate normality, reference regions have been constructed as ellipsoidal regions. This approach suffers from a major drawback: it cannot detect component-wise extreme observations. In the present work, procedures are developed to construct rectangular reference regions in the multivariate normal setup. The construction is based on the criteria for tolerance intervals. The problems addressed include the computation of a rectangular tolerance region and simultaneous tolerance intervals. Also addressed is the computation of mixed reference intervals that include both two-sided and one-sided limits, simultaneously. A parametric bootstrap approach is used in the computations, and the accuracy of the proposed methodology is assessed using estimated coverage probabilities. The problem of sample size determination is also addressed, and the results are illustrated using examples that call for the computation of referenceregions.

Construction of simultaneous tolerance intervals for several normal distributions

Methods for computing simultaneous tolerance limits and tolerance intervals (TIs) for several normal populations with a common unknown variance are proposed. We propose numerical methods that determine the confidence coefficient so that the simultaneous TIs include at least specified proportions of the populations with the intended confidence level. The methods are exact and as simple as the existing ones. They are also applicable when sample sizes are unequal and content levels of simultaneous TIs are unequal. Algorithms and R functions are provided to compute factors for constructing simultaneous one-sided tolerance limits, tolerance intervals and equal-tailed tolerance intervals. The methods are illustrated using an example with a real data set.

Statistical calibration and exact one-sided simultaneous tolerance intervals for polynomial regression

Statistical calibration using linear regression is a useful statistical tool having many applications. Calibration for infinitely many future y-values requires the construction of simultaneous tolerance intervals (STI’s). As calibration often involves only two variables x and y and polynomial regression is probably the most frequently used model for relating y with x, construction of STI’s for polynomial regression plays a key role in statistical calibration for infinitely many future y-values. The only exact STI’s published in the statistical literature are provided by Mee et al. (1991) and Odeh and Mee (1990). But they are for a multiple linear regression model, in which the covariates are assumed to have no functional relationships. When applied to polynomial regression, the resultant STI’s are conservative. In this paper, one-sided exact STI’s have been constructed for a polynomial regression model over any given interval. The available computer program allows the exact methods developed in this paper to be implemented easily. Real examples are given for illustration.

Calibration and Simultaneous Tolerance Intervals for Regression

Simultaneous calibration (or discrimination) intervals in regression were proposed by Lieberman, Miller, and Hamilton (1967) and by Scheffe (1973). Those procedures enable one to construct confidence intervals for the unobserved values of the independent variable corresponding to an unlimited sequence of observations of the dependent variable in a regression model. These calibration intervals are conservative in that they are obtained from simultaneous tolerance intervals for which the actual confidence level exceeds the nominal level. Furthermore, all other existing simultaneous tolerance intervals in regression are likewise conservative. In this article, we propose simultaneous tolerance intervals that are narrower than previous intervals. Given the tables of factors included in this article, they are also simple to construct and use in straightline calibration applications.

Calibration and Simultaneous Tolerance Intervals for Regression

Simultaneous calibration (or discrimination) intervals in regression were proposed by Lieberman, Miller, and Hamilton (1967) and by Scheffe (1973). Those procedures enable one to construct confidence intervals for the unobserved values of the independent variable corresponding to an unlimited sequence of observations of the dependent variable in a regression model. These calibration intervals are conservative in that they are obtained from simultaneous tolerance intervals for which the actual confidence level exceeds the nominal level. Furthermore, all other existing simultaneous tolerance intervals in regression are likewise conservative. In this article, we propose simultaneous tolerance intervals that are narrower than previous intervals. Given the tables of factors included in this article, they are also simple to construct and use in straightline calibration applications.

Simultaneous Tolerance Intervals for Normal Populations With, Common Variance

This article proposes simultaneous tolerance limits for L normally distributed populations with a common variance, assuming that independent random samples are available from the populations. Tables of factors are provided for both one-sided limits and two-sided intervals for the case of equal sample sizes. Approximations and formulas for exact computations are discussed for applications in which the sample sizes differ. The related problem of simultaneous prediction intervals for the L populations is also discussed as a special case of the simultaneous prediction intervals proposed by Hahn (1972). Citations are provided for tables of factors for both one- and two-sided simultaneous prediction intervals.

Simultaneous Tolerance Intervals for Normal Populations with Common Variance

This article proposes simultaneous tolerance limits for L normally distributed populations with a common variance, assuming that independent random samples are available from the populations. Tables of factors are provided for both one-sided limits and two-sided intervals for the case of equal sample sizes. Approximations and formulas for exact computations are discussed for applications in which the sample sizes differ. The related problem of simultaneous prediction intervals for the L populations is also discussed as a special case of the simultaneous prediction intervals proposed by Hahn (1972). Citations are provided for tables of factors for both one- and two-sided simultaneous prediction intervals.

Simultaneous tolerance intervals in the random one-way model with covariates

Simultaneous tolerance intervals developed by Limam and Thomas (19881, for the normal regression model, are generalized to the random one-way model with covariates. Simultaneous tolerance intervals for unit means are developed for the balanced model. A simulation study is used to estimate the exact confidence of the tolerance intervals for models with one covariate.

Tolerance regions for random vectors and best linear predictors

In the case that vectors X and Y have a joint multivariate normal distribution, tolerance regions are found for the best linear predictor of Y using X if samples are used to estimate the regression coeffierante. Tolerance regions are also found for Y. In addition, simultaneous tolerance intervals for all linear functions of Y or of the best linear predictor of Y using X are found.