SIZE CHARACTERISTICS OF TESTS FOR SAMPLE SELECTION BIAS: A MONTE CARLO COMPARISON AND EMPIRICAL EXAMPLE

The t-test of an individual coefficient is used widely in models of qualitative choice. However, it is well known that the t-test can yield misleading results when the sample size is small. This paper provides some experimental evidence on the finite sample properties of the t-test in models with sample selection biases, through a comparison of the t-test with the likelihood ratio and Lagrange multiplier tests, which are asymptotically equivalent to the squared t-test. The finite sample problems with the t-test are shown to be alarming, and much more serious than in models such as binary choice models. An empirical example is also presented to highlight the differences in the calculated test statistics.     

A TWO STAGE PROCEDURE FOR ESTIMATING THE SIZE OF A TRUNCATED EXPONENTIAL SAMPLE

Let X₁, …, X_N be i.i.d. exponential random variables with unknown scale parameter θ. If one can observe only those X_i in (0, T₀), an estimate of N based on the J observations obtained has a variance which explodes as θ→θC. Using θC based on the observations in (0, T₀) T is computed and all X_i in (0, ) are observed. An estimate of N based on all observations in (0, ) has a bounded variance where the bound can be adjusted by proper choice of .     

SAMPLE SIZE DETERMINATION FOR MULTIPLE MANY-ONE AND PAIRWISE COMPARISONS OF PROPORTIONS USING THE ARCSIN TRANSFORMATION

Our paper provides sample size formulas for multiple comparisons of k treatments with a control as well as with pairwise comparisons of k treatments when observing proportions. The test statistics we regard use the arcsin transformation of the proportions. We determine sample sizes with preassigned all-pairs, any-pair and per-pair power considering the control either of the comparisonwise or of the familywise type I error rate.     

AN EXACT TEST FOR HAZARD SIMILARITY

An exact test is developed for hazard similarity and in particular for exponentiality. This test is distinct from more common goodness-of-fit tests such as the Kolmogorov–Smirnov goodness-of-fit test, as it does not require full specification of the null distribution. This test is obtained through a characterization of hazard-similar distributions and a generalization of Fisher's test for association.     

A BAYESIAN METHOD FOR THE CHOICE OF THE SAMPLE SIZE IN EQUIVALENCE TRIALS

In this paper we consider a Bayesian predictive approach to sample size determination in equivalence trials. Equivalence experiments are conducted to show that the unknown difference between two parameters is small. For instance, in clinical practice this kind of experiment aims to determine whether the effects of two medical interventions are therapeutically similar. We declare an experiment successful if an interval estimate of the effects‐difference is included in a set of values of the parameter of interest indicating a negligible difference between treatment effects (equivalence interval). We derive two alternative criteria for the selection of the optimal sample size, one based on the predictive expectation of the interval limits and the other based on the predictive probability that these limits fall in the equivalence interval. Moreover, for both criteria we derive a robust version with respect to the choice of the prior distribution. Numerical results are provided and an application is illustrated when the normal model with conjugate prior distributions is assumed.     

A TEST FOR THE NONPARAMETRIC TWO SAMPLE SCALE PROBLEM

We propose a distribution-free test for the nonparametric two sample scale problem. Unlike the other tests for this problem, we do not assume that the two distribution functions have a common median. We assume that they have a common quantile of order a (not necessarily 1/2). The test statistic is a modification of the Sukhatme statistic for the scale problem and the Wilcoxon-Mann-Whitney statistic for stochastic dominance. It is shown that the new test is uniformly more efficient (in the Pitman sense) than the Sukhatme test and has very good efficiency when compared to the Mood test.     

VARIANCE ESTIMATION FOR SAMPLE AUTOCOVARIANCES: DIRECT AND RESAMPLING APPROACHES

The usual covariance estimates for data _n-1 from a stationary zero-mean stochastic process {X_t} are the sample covariances Both direct and resampling approaches are used to estimate the variance of the sample covariances. This paper compares the performance of these variance estimates. Using a direct approach, we show that a consistent windowed periodogram estimate for the spectrum is more effective than using the periodogram itself. A frequency domain bootstrap for time series is proposed and analyzed, and we introduce a frequency domain version of the jackknife that is shown to be asymptotically unbiased and consistent for Gaussian processes. Monte Carlo techniques show that the time domain jackknife and subseries method cannot be recommended. For a Gaussian underlying series a direct approach using a smoothed periodogram is best; for a non-Gaussian series the frequency domain bootstrap appears preferable. For small samples, the bootstraps are dangerous: both the direct approach and frequency domain jackknife are better.     

A Bayesian non‐inferiority test for two independent binomial proportions

In drug development, non‐inferiority tests are often employed to determine the difference between two independent binomial proportions. Many test statistics for non‐inferiority are based on the frequentist framework. However, research on non‐inferiority in the Bayesian framework is limited. In this paper, we suggest a new Bayesian index τ = P(π₁ > π₂ ? Δ₀ | X₁,X₂), where X₁ and X₂ denote binomial random variables for trials n₁ and n₂, and parameters π₁ and π₂, respectively, and the non‐inferiority margin is Δ₀ > 0. We show two calculation methods for τ, an approximate method that uses normal approximation and an exact method that uses an exact posterior PDF. We compare the approximate probability with the exact probability for τ. Finally, we present the results of actual clinical trials to show the utility of index τ. Copyright © 2013 John Wiley & Sons, Ltd.     

TEST FOR TREATMENT EFFECT BASED ON BINARY DATA WITH RANDOM SAMPLE SIZES

The problem of testing for treatment effect based on binary response data is considered, assuming that the sample size for each experimental unit and treatment combination is random. It is assumed that the sample size follows a distribution that belongs to a parametric family. The uniformly most powerful unbiased tests, which are equivalent to the likelihood ratio tests, are obtained when the probability of the sample size being zero is positive. For the situation where the sample sizes are always positive, the likelihood ratio tests are derived. These test procedures, which are unconditional on the random sample sizes, are useful even when the random sample sizes are not observed. Some examples are presented as illustration.     

LOWER BOUNDS FOR THE NORMAL APPROXIMATION OF A SUM OF INDEPENDENT RANDOM VARIABLES

The rates of convergence to the normal distribution are investigated for a sum of independent random variables. Using Stein's method, we derive a lower bound of the uniform distance between two distributions of independent sum and normal.     

EXISTENCE OF A SEN—YATES—GRUNDY FORM FOR A DISPERSION

The condition of fixed sample size is essential for the existence of a Sen-Yates-Grundy form variance and its design unbiased estimator, in the problem of estimating the mean, variance and covariance of a finite population.     

THE CONTROL CHART FOR INDIVIDUAL OBSERVATIONS FROM A MULTIVARIATE NON-NORMAL DISTRIBUTION

The Hotelling's T²statistic has been used in constructing a multivariate control chart for individual observations. In Phase II operations, the distribution of the T²statistic is related to the F distribution provided the underlying population is multivariate normal. Thus, the upper control limit (UCL) is proportional to a percentile of the F distribution. However, if the process data show sufficient evidence of a marked departure from multivariate normality, the UCL based on the F distribution may be very inaccurate. In such situations, it will usually be helpful to determine the UCL based on the percentile of the estimated distribution for T². In this paper, we use a kernel smoothing technique to estimate the distribution of the T²statistic as well as of the UCL of the T²chart, when the process data are taken from a multivariate non-normal distribution. Through simulations, we examine the sample size requirement and the in-control average run length of the T²control chart for sample observations taken from a multivariate exponential distribution. The paper focuses on the Phase II situation with individual observations.     

A comparison of two approaches for power and sample size calculations in logistic regression models

Whittemore (1981) proposed an approach for calculating the sample size needed to test hypotheses with specified significance and power against a given alternative for logistic regression with small response probability. Based on the distribution of covariate, which could be either discrete or continuous, this approach first provides a simple closed-form approximation to the asymptotic covariance matrix of the maximum likelihood estimates, and then uses it to calculate the sample size needed to test a hypothesis about the parameter. Self et al. (1992) described a general approach for power and sample size calculations within the framework of generalized linear models, which include logistic regression as a special case. Their approach is based on an approximation to the distribution of the likelihood ratio statistic. Unlike the Whittemore approach, their approach is not limited to situations of small response probability. However, it is restricted to models with a finite number of covariate configurations. This study compares these two approaches to see how accurate they would be for the calculations of power and sample size in logistic regression models with various response probabilities and covariate distributions. The results indicate that the Whittemore approach has a slight advantage in achieving the nominal power only for one case with small response probability. It is outperformed for all other cases with larger response probabilities. In general, the approach proposed in Self et al. (1992) is recommended for all values of the response probability. However, its extension for logistic regression models with an infinite number of covariate configurations involves an arbitrary decision for categorization and leads to a discrete approximation. As shown in this paper, the examined discrete approximations appear to be sufficiently accurate for practical purpose.     

ON THE LIKELIHOOD RATIO TEST FOR A SINGLE MODEL AGAINST THE MIXTURE OF TWO KNOWN DENSITIES

In this paper, we consider the likelihood ratio test for a single model against the mixture of two known densities. We research the accuracy of the limiting null distribution obtained by Titterington 1-2 Titterington, D. M. 1981. Contribution to the discussion of paper by M. Aitkin, D. Amderson and J. Hinde, J. R.. Statist. Soc. A, 144: 459 Titterington, D. M., Smith, A. F.M. and Makov, U. E. 1985. Statistical Analysis of Finite Mixture Distribution New York: Wiley.  .After that, we improve the limiting null distribution and confirm its usefulness by the simulation study.     

ON A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD RATIO CRITERIA

In this paper the exact distribution of a statistic with general moment function has been derived in a gamma series and also in a beta series. It is shown that the Box expansion can be obtained from the gamma series by collecting terms of the same order.     

Tests for noninferiority trials with binomial endpoints: A guide to modern and quasi‐exact methods for biomedical researchers

Applied statisticians and pharmaceutical researchers are frequently involved in the design and analysis of clinical trials where at least one of the outcomes is binary. Treatments are judged by the probability of a positive binary response. A typical example is the noninferiority trial, where it is tested whether a new experimental treatment is practically not inferior to an active comparator with a prespecified margin δ. Except for the special case of δ = 0, no exact conditional test is available although approximate conditional methods (also called second‐order methods) can be applied. However, in some situations, the approximation can be poor and the logical argument for approximate conditioning is not compelling. The alternative is to consider an unconditional approach. Standard methods like the pooled z‐test are already unconditional although approximate. In this article, we review and illustrate unconditional methods with a heavy emphasis on modern methods that can deliver exact, or near exact, results. For noninferiority trials based on either rate difference or rate ratio, our recommendation is to use the so‐called E‐procedure, based on either the score or likelihood ratio statistic. This test is effectively exact, computationally efficient, and respects monotonicity constraints in practice. We support our assertions with a numerical study, and we illustrate the concepts developed in theory with a clinical example in pulmonary oncology; R code to conduct all these analyses is available from the authors.     

A Five-Decision Testing Procedure to Infer the Value of a Unidimensional Parameter

ABSTRACT

A statistical test can be seen as a procedure to produce a decision based on observed data, where some decisions consist of rejecting a hypothesis (yielding a significant result) and some do not, and where one controls the probability to make a wrong rejection at some prespecified significance level. Whereas traditional hypothesis testing involves only two possible decisions (to reject or not a null hypothesis), Kaiser’s directional two-sided test as well as the more recently introduced testing procedure of Jones and Tukey, each equivalent to running two one-sided tests, involve three possible decisions to infer the value of a unidimensional parameter. The latter procedure assumes that a point null hypothesis is impossible (e.g., that two treatments cannot have exactly the same effect), allowing a gain of statistical power. There are, however, situations where a point hypothesis is indeed plausible, for example, when considering hypotheses derived from Einstein’s theories. In this article, we introduce a five-decision rule testing procedure, equivalent to running a traditional two-sided test in addition to two one-sided tests, which combines the advantages of the testing procedures of Kaiser (no assumption on a point hypothesis being impossible) and Jones and Tukey (higher power), allowing for a nonnegligible (typically 20%) reduction of the sample size needed to reach a given statistical power to get a significant result, compared to the traditional approach.     

Inferences on the Means of Two Log-Normal Distributions: A Computational Approach Test

In this article, we propose a novel approach for testing the equality of two log-normal populations using a computational approach test (CAT) that does not require explicit knowledge of the sampling distribution of the test statistic. Simulation studies demonstrate that the proposed approach can perform hypothesis testing with satisfying actual size even at small sample sizes. Overall, it is superior to other existing methods. Also, a CAT is proposed for testing about reliability of two log-normal populations when the means are the same. Simulations show that the actual size of this new approach is close to nominal level and better than the score test. At the end, the proposed methods are illustrated using two examples.