An exact test for comparing two predictive values in small‐size clinical trials

Positive and negative predictive values describe the performance of a diagnostic test. There are several methods to test the equality of predictive values in paired designs. However, these methods were premised on large sample theory, and they may not be suitable for small‐size clinical trials because of inflation of the type 1 error rate. In this study, we propose an exact test to control the type 1 error rate strictly for conducting a small‐size clinical trial that investigates the equality of predictive values in paired designs. In addition, we execute simulation studies to evaluate the performance of the proposed exact test and existing methods in small‐size clinical trials. The proposed test can calculate the exact P value, and as a result of simulations, the empirical type 1 error rate for the proposed test did not exceed the significance level regardless of the setting, and the empirical power for the proposed test is not much different from the other methods based on large‐sample theory. Therefore, it is considered that the proposed exact test is useful when the type 1 error rate needs to be controlled strictly.     

Bayesian inference on extreme value distribution using upper record values

In this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.     

Rejoinder

Abstract

In this article several formulae for the approximation of the critical values for tests on the actual values of the process capability indices CPL, CPU, and C_pk are provided. These formulae are based on different approximations of the percentiles of the noncentral t distribution and their performance is evaluated by comparing the values assessed through them from the exact critical values, for several significance levels, test values, and sample sizes. As supported by the obtained results, some of the presented techniques constitute valuable tools in situations where the exact critical values of the tests are not available, since one may approximate them readily and rather accurately through them.     

On C pk Assessment in the Presence of Tool Wear

Process capability indices are routinely used in manufacturing industries for process monitoring. A basic assumption while using process capability indices is that there are no assignable causes of variation present. However, when variation due to an assignable cause is present and is tolerated, the conventional methods of capability measurement become inaccurate. In this article, we suggest an estimate of C_pk assuming that the process capability changes dynamically. We obtain an exact form of the sampling distribution in the presence of a systematic assignable cause. We discuss the problem of testing whether a given process is capable. The critical values for different sample sizes are obtained based on the sampling distribution. An example involving tool wear problem is presented.     

An exact Kolmogorov–Smirnov test for the Poisson distribution with unknown mean

We develop an exact Kolmogorov–Smirnov goodness-of-fit test for the Poisson distribution with an unknown mean. This test is conditional, with the test statistic being the maximum absolute difference between the empirical distribution function and its conditional expectation given the sample total. Exact critical values are obtained using a new algorithm. We explore properties of the test, and we illustrate it with three examples. The new test seems to be the first exact Poisson goodness-of-fit test for which critical values are available without simulation or exhaustive enumeration.     

Computation of the poisson-inverse gaussian distribution

Recursion relations suitable for rapid computation are derived for the probabilities of the compound Poisson distribution when the compounder is the inverse-Gaussian distribution. Series representation of the probabilities are given. Asymptotic results as well as approximations for probabilities, compared with the exact values, are investigated.     

A Permutation Test for Planar Regression

Oja (1987) presents some distribution-free tests applicable in the presence of covariates when treatment values are randomly assigned. The formulas and calculations are cumbersome, however, and implementation of the tests relies on using a x² approximation to the exact null distribution. In this paper a re-formulation of his test statistic is given which has the advantages of ease of calculation, explicit formulas for permutation moments, and allowing a Beta distribution to be fitted to the exact null distribution.     

The Exact Null Distribution of Bartlett's Criterion

In this paper the exact null distribution of Bartlett's criterion for testing the homogeneity of variances in normal samples with unequal sizes is derived. The most general form of the density function is obtained by using contour integration. The expression for the cumulative distribution, being a series in simple algebraic functions, seems quite tractable for computation of the exact critical values. In the special case of equal sample sizes, some indication of the relation of the work of others to our series expansions has also been given.     

Statistical inference about the shape parameter of the exponential power distribution

After being proposed by Smith & Bain (1975), the exponential power distribution has been discussed by many authors. This paper proposes a simple exact statistical test for the shape parameter of an exponential power distribution, as well as an exact confidence interval for the same parameter. Necessary critical values of the test are given. The method provided in this paper can be used for type II censored data. Comparing this method to the existing approaches, this method requires less calculation or less tables, and is easier to use in practice.     

Some Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation

In this article, we point out some interesting relations between the exact test and the score test for a binomial proportion p. Based on the properties of the tests, we propose some approximate as well as exact methods of computing sample sizes required for the tests to attain a specified power. Sample sizes required for the tests are tabulated for various values of p to attain a power of 0.80 at level 0.05. We also propose approximate and exact methods of computing sample sizes needed to construct confidence intervals with a given precision. Using the proposed exact methods, sample sizes required to construct 95% confidence intervals with various precisions are tabulated for p = .05(.05).5. The approximate methods for computing sample sizes for score confidence intervals are very satisfactory and the results coincide with those of the exact methods for many cases.     

Modified Baumgartner statistics for the two-sample and multisample problems: a numerical comparison

Various non-parametric rank tests based on the Baumgartner statistic have been proposed for testing the location, scale and location–scale parameters. The modified Baumgartner statistics are not suitable for the scale shifts for a two-sample problem. Two modified Baumgartner statistics are proposed by changing the weight function. The suggested statistics are extended to the multisample problem. Some exact critical values of the suggested test statistics are evaluated. Simulations are used to investigate the power of the modified Baumgartner statistics.     

On the Gini Mean Difference Test for Circular Data

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributions under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency.     

A representation in beta series for the exact density of some random volume contents

Consider r independent and identically distributed random points in a unit n-ball of which p are in the interior and rp are on the surface. These r points, via their convex hull, generate an r-simplex. This article deals with the exact density of the r-content when the points are uniformly distributed. The exact density of the r-content is obtained for the general values of the parameters r, n and p. A representation of the density is given as a mixture of beta type-1 densities so that one can evaluate various types of probabilities by using incom-plete beta tables.     

Tables of two-sided tolerance factors for a normal distribution

Given a random sample of size N from a normal distribution, we consider tolerance intervals of the form X ? ks to X + ks, where X is the sample mean and s is the sample standard deviation. The value of k is chosen so that the interval covers a given proportion P of the population with confidence γ. Exact values of k, computed from numerical integration, are given for N = 2(1)100; P = 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.999; and γ = 0.5, 0.75, 0.90, 0.95, 0.975, 0.99, 0.995. The exact values are compared with the values obtained from an approximation developed by Wald and Wolfowitz (1946).     

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.     

Tests and limits for the common odds ratio of several 2 × 2 contingency tables: methods in analogy with the Mantel-Haenszel procedure

For a postulated common odds ratio for several 2 × 2 contingency tables one may, by conditioning on the marginals of the seperate tables, determine the exact expectation and variance of the entry in a particular cell of each table, hence for the total of such cells across all tables. This makes it feasible to determine limiting values, via single-degree-of-freedom, continuity-corrected chi-square tests on the common odds ratio–one determines lower and upper limits corresponding to just barely significant chi-square values. The Mantel-Haenszel approach can be viewed as a special application of this, but directed specifically to the case of unity for the odds ratio, for which the expectation and variance formulas are particularly simple. Computation of exact expectations and variances may be feasible only for 2 × 2 tables of limited size, but asymptotic formulas can be applied in other instances.Illustration is given for a particular set of four 2 × 2 tables in which both exact limits and limits by the proposed method could be applied, the two methods giving reasonably good agreement. Both procedures are directed at the distribution of the total over the designated cells, the proposed method treating that distribution as being asymptotically normal. Especially good agreement of proposed with exact limits could be anticipated in more asymptotic situations (overall, not for individual tables) but in practice this may not be demonstrable as the computation of exact limits is then unfeasible.     

Record values and the geometric distribution

Suppose {X_n, n≥1} is a sequence of independent and identically distributed discrete random variables having the common distribution function F(x). The exact distribution of the n-th record value is given under the assumption that F(x) has the geometric distribution. Various properties of the record values and some new characterizations of the geometric distribution are presented.     

Exact Tests for 2 × 2 Contingency Tables

Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.     

Designing historical control studies with survival endpoints using exact statistical inference

Historical control trials compare an experimental treatment with a previously conducted control treatment. By assigning all recruited samples to the experimental arm, historical control trials can better identify promising treatments in early phase trials compared with randomized control trials. Existing designs of historical control trials with survival endpoints are based on asymptotic normal distribution. However, it remains unclear whether the asymptotic distribution of the test statistic is close enough to the true distribution given relatively small sample sizes in early phase trials. In this article, we address this question by introducing an exact design approach for exponentially distributed survival endpoints, and compare it with an asymptotic design in both real examples and simulation examples. Simulation results show that the asymptotic test could lead to bias in the sample size estimation. We conclude the proposed exact design should be used in the design of historical control trials.     

A new sampling plan under the exponential distribution

This article considers a sampling plan when the quality characteristic follows the exponential distribution. We provide the exact approach and propose an approximated approach. In the proposed approximation, a new statistic combined with a Weibull transformation is used for the normal approximation. The plan parameters are obtained through the two-point approach at the acceptable quality level (AQL) and the limiting quality level (LQL). The tables for plan parameters are reported according to various values of the AQL and the LQL when the producer's and the consumer's risks are given. A real example is given to illustrate the proposed approximation approach.