Distribution-free Tolerance Intervals Research Articles

Distribution-free tolerance intervals with nomination samples: Applications to mercury contamination in fish

Tolerance intervals are enclosure intervals which will cover a fixed portion of the population distribution with a specified confidence. These intervals are widely used in clinical, environmental, biological and industrial applications, including quality control and environmental monitoring, to help determine limits for detection or assessment monitoring. In many of these applications the measurement of the variable of interest is costly and/or destructive but a small number of sampling units can be ranked easily by using expert-opinion knowledge or inexpensive and easily obtained measurements from these units. In this paper, we construct tolerance intervals based on the expensive measurements that are obtained using randomized nomination sampling (RNS) with the help of inexpensive auxiliary information. We study the performance of our proposed RNS-based tolerance intervals based on the corresponding coverage probabilities and the necessary sample size for their existence with those based on simple random sampling (SRS). The efficiency of the constructed RNS-based tolerance intervals compared to their SRS counterparts is discussed. We investigate the performance of RNS-based tolerance intervals for different values of the design parameters and various population shapes. We find the values of the design parameters which improve RNS over SRS. The RNS design in presence of ranking error is discussed and a new method for estimating ranking error probabilities is proposed. Theoretical results are augmented with numerical evaluations and a case study based on a fish mercury level dataset.

Synthesis of Regional Wildlife and Vegetation Field Studies to Guide Management of Standing and Down Dead Trees

Abstract We used novel methods for combining information from wildlife and vegetation field studies to develop guidelines for managing dead wood for wildlife and biodiversity. The DecAID Decayed Wood Advisor presents data on wildlife use of standing and down dead trees (snags and down wood) and summaries of regional vegetation plot data depicting dead wood conditions, for forests across the Pacific Northwest United States. We combined data on wildlife use by snag diameter and density and by down wood diameter and cover, across studies, using parametric techniques of meta-analysis. We calculated tolerance intervals, which represent the percentage of each species' population that uses particular sizes or amounts of snags and down wood, and rank-ordered the species into cumulative species curves. We combined data on snags and down wood from > 16,000 field plots from three regional forest inventories and calculated distribution-free tolerance intervals compatible with those compiled for wildlife to facilitate integrated analysis. We illustrate our methods using an example for one vegetation condition. The statistical summaries in DecAID use a probabilistic approach, which works well in a risk analysis and management framework, rather than a deterministic approach. Our methods may prove useful to others faced with similar problems of combining information across studies in other regions or for other data types.

Safe composition levels of transgenic crops assessed via a clinical medicine model.

Re-use of this article is permitted in accordance with the Creative Commons Deed, Attribution 2.5, which does not permit commercial exploitation.Substantial equivalence has become established as a foundation concept in the safety evaluation of transgenic crops. In the case of a food and feed crop, no single variety is considered the standard for safety or nutrition, so the substantial equivalence of transgenic crops is investigated relative to the array of commercial crop varieties with a history of safe consumption. Although used extensively in clinical medicine to compare new generic drugs with brand-name drugs, equivalence limits are shown to be a poor model for comparing transgenic crops with an array of reference crop varieties. We suggest an alternate model, also analogous to that used in clinical medicine, where reference intervals are constructed for a healthy heterogeneous population. Specifically, we advocate the use of distribution-free tolerance intervals calculated across a large amount of publicly available compositional data such as is found in the International Life Sciences Institute Crop Composition Database.

Order statistics and probabilistic robust control

Order statistics theory is applied in this paper to probabilistic robust control theory to compute the minimum sample size needed to come up with a reliable estimate of an uncertain quantity under continuity assumption of the related probability distribution. Also, the concept of distribution-free tolerance intervals is applied to estimate the range of an uncertain quantity and extract the information about its distribution. To overcome the limitations imposed by the continuity assumption in the existing order statistics theory, we have derived a cumulative distribution function of the order statistics without the continuity assumption and developed an inequality showing that this distribution has an upper bound which equals to the corresponding distribution when the continuity assumption is satisfied. By applying this inequality, we investigate the minimum computational effort needed to come up with an reliable estimate for the upper bound (or the lower bound) and the range of a quantity. We also give conditions, which are much weaker than the absolute continuity assumption, for the existence of such minimum sample size. Furthermore, the issue of making tradeoff between performance level and risk is addressed and a guideline for making this kind of tradeoff is established. This guideline can be applied in general without continuity assumption.

The assessment of individual and population bioequivalence.

We develop a statistical methodology for the assessment of individual bioequivalence when a crossover design is invoked. The location parameters for our model consist of population-averaged parameters for formulation, population-averaged parameters for nuisance effects (sequence, period, carryover, etc.), and subject-specific parameters for formulation. We do not impose any distributional assumptions other than the existence of first- and second-order moments. We derive unbiased estimators for all of the parameters in the model and construct subject-specific bioequivalence scores which exclude the effects of the nuisance parameters. We assess individual bioequivalence by constructing distribution-free tolerance intervals based on the sample of bioequivalence scores.

Extension to the use of tolerance intervals for the assessment of individual bioequivalence

For the determination of bioequivalence, researchers have recently shifted their emphasis from average bioequivalence alone to average and individual bioequivalence. Existing methods for assessing average bioequivalence were first developed for the standard 2 x 2 crossover design, but these methods are easily generalized to the two-treatment, p-period crossover designs (e.g., TRR, RTT, and TTRR, RRTT, TRRT, RTTR). With respect to individual bioequivalence, Westlake (1,2) implemented the use of parametric and distribution-free tolerance intervals for assessing individual bioequivalence. Anderson and Hauck (3) described what they call the test of individual equivalence ratios (TIER) for the same purpose. Note that these methods have been applied and/or developed only for the standard 2 x 2 crossover design. The present work extends the method of using parametric tolerance intervals for assessing individual bioequivalence.

Sample sizes for strong two-sided distribution-free tolerance limits

Tables for minimum sample sizes and accompanying order statistics are given for strong two-sided distribution-free tolerance intervals. Both inner and outer tolerance intervals are considered. The criterion used for determining minimum sample size is to require that the tail proportions be close to their corresponding expected proportions. Examples are given to illustrate the use of the tables.

Random Walks of Ordered Elements with Applications

Abstract In this partly expository paper a simple random walk model is introduced for generating sequences of ordered elements. The model is related to Polya's urn scheme. An alternative description of the walk leads to conditionally independent steps. A basic theorem for the walk is proved and applications are given. Among other things, recursion formulas for moments of order statistics are studied and connections with exceedances and distribution-free tolerance intervals are indicated.

Two-Sided Distribution-Free Tolerance Intervals and Accompanying Sample Size Problems

Three types of two-sided distribution-free tolerance intervals are discussed. These are called beta-content, Wilks' outer, and Wilks' inner tolerance intervals. For each a minimum sample size problem is proposed and solved using easily written computer ..

Distribution-Free Tolerance Intervals for Stochastically Ordered Distributions

Consider $k$ stochastically ordered distributions with $F_{(1)} \leqq\cdots\leqq F_{(k)}$. The present paper deals with distribution-free tolerance intervals for $F_{(j)}$ based on order statistics in samples of same size from each of the $k$ distributions. Two criteria are defined for determining such intervals. These two criteria are extensions of $\beta$-expectation tolerance intervals and $\beta$-content tolerance intervals with confidence coefficient $\gamma$ used in the single population literature. A tolerance interval for the lifetime distribution of a series system is considered as an example.

Distribution-Free Confidence Intervals for Quantile Intervals

Abstract Wilks [6] introduced distribution-free confidence intervals based on order statistics for quantile intervals. For the rth and sth order statistics in a sample of size n (1 ≤ r < s ≤ n), upper and lower bounds and an exact expression for the confidence coefficient are derived. Numerical examples comparing the bounds and the exact solution are provided. A connection with distribution-free tolerance intervals is indicated.

Distribution-Free Confidence Intervals for Quantile Intervals

Abstract Wilks [6] introduced distribution-free confidence intervals based on order statistics for quantile intervals. For the rth and sth order statistics in a sample of size n (1 ≤ r < s ≤ n), upper and lower bounds and an exact expression for the confidence coefficient are derived. Numerical examples comparing the bounds and the exact solution are provided. A connection with distribution-free tolerance intervals is indicated.

Sample Size Formulas for Some Binomial Type Problems

A formula is given for approximating the minimum sample size required in some binomial type problems—distribution-free tolerance intervals satisfying one or two probability conditions and hypothesis testing for p satisfying two power conditions. The approximation requires a percentile from a chi-square table, is very easy to use and is extremely accurate.

Sample Size Formulas for Some Binomial Type Problems

A formula is given for approximating the minimum sample size required in some binomial type problems—distribution-free tolerance intervals satisfying one or two probability conditions and hypothesis testing for p satisfying two power conditions. The approximation requires a percentile from a chi-square table, is very easy to use and is extremely accurate.

Some Two-Sided Distribution-Free Tolerance Intervals of a General Nature

Abstract Nearly all of the distribution-free tolerance intervals that have been considered in the statistical literature are such that each endpoint consists of a sample order statistic or is unbounded. For two-sided intervals, many other possibilities are available. One purpose of this paper is to call attention to these additional possibilities, and to discuss their advantages and disadvantages. The principal purpose, however, is to show how to construct specialized two-sided tolerance intervals for use in investigating situations of two types. For the first situation, the probability concentration in the central part of the population is investigated. For the other, it is desired to investigate the probability concentration about a specified point. In both situations, the specialized intervals presented are not difficult to apply. For the first situation, permissible additional information can be used to obtain intervals tending to have shorter lengths than those not exploiting this extra information. ...

Some Two-Sided Distribution-Free Tolerance Intervals of a General Nature

Abstract Nearly all of the distribution-free tolerance intervals that have been considered in the statistical literature are such that each endpoint consists of a sample order statistic or is unbounded. For two-sided intervals, many other possibilities are available. One purpose of this paper is to call attention to these additional possibilities, and to discuss their advantages and disadvantages. The principal purpose, however, is to show how to construct specialized two-sided tolerance intervals for use in investigating situations of two types. For the first situation, the probability concentration in the central part of the population is investigated. For the other, it is desired to investigate the probability concentration about a specified point. In both situations, the specialized intervals presented are not difficult to apply. For the first situation, permissible additional information can be used to obtain intervals tending to have shorter lengths than those not exploiting this extra information. In the other situation, the relative emphasis placed on the part of the interval to one side of the specified point, as compared to the part on the other side, can be specified.

Distribution-Free Tolerance Intervals for Continuous Symmetrical Populations

For continuous symmetrical distributions and specified tolerance interval characteristics, distribution-free tolerance intervals are presented that require substantially smaller sample sizes than those required for arbitrary continuous populations. In the case of one-sided tolerance intervals, about twice as many sample values are required for the general situation as for the symmetrical situation. For two-sided intervals, the additional sample values required for the general situation vary from about 35 percent to about 85 percent in the cases considered, depending on the specified tolerance interval characteristics. These intervals furnish at least rough protection against all possible violations of the symmetry assumption. In addition, some specialized one-sided and two-sided tolerance intervals are developed for life-test situations involving continuous symmetrical populations. Results for continuous symmetrical populations with known central median values are also presented; the life-testing versions of these intervals are especially useful.