A monte carlo study on the robustness of the two-sample sequential t test

The robustness of the two-sample sequentla1 t test was studied against departures from normality and equality of variances The effect of skewness and kurtosis of the underlying distribution on the test 1s relatively mild but the effect of heteroscedasticity serious.     

Coefficients of lee-gurland two-sample test on normal means

The Behrens-Fisher problem in comparing means of two normal populations is revisited Lee and Gurland (1975) suggested a solution to the problem and provided the set of coefficients required in computing critical values for the case α=005, where α is the nominal level of significance This solution, called the Lee-Guiland Test in this article, has proven to be practical as far as calculation is involved, and more importantly, it maintains the actual size very close to α= 0.05 for possible values of the ratio of population variances This merit has not been attained by most of the Behrens-Fisher solutions in the literature. In this article, the coefficients for other values of α, namely 0 025, 0 01 and 0.005 are provided for wider applications of the test Moreover, careful and detailed comparisons are made in terms of size and power with the other practical solution:the Welch's Approximate t I est Due to a possible drawback of the Welch's Approximate t I est in controlling the actual size, especially for small a and small sample sizes, the Lee-Gurland lest presents itself as a slightly better' alternative in testing equality of two normal population means I he coefficients mentioned above are also fitted by the functions of the reciprocals of the degrees of freedom, so that the substantial amount of table-looking can be avoided Some discussions are also made in regarding the recent “Welch vs Gosset” argument: Should the Student's t Test be dispensed off’from the routine use in testing the equality of two normal means?.     

A sample size selection procedure for pairwise comparisons of means with unequal variances

A sample size selection procedure for paired comparisons of means is presented which controls the half width of the confidence intervals while allowing for unequal variances of treatment means.     

Contributions to two-sample statistics

When testing the equality of the means from two independent normally distributed populations given that the variances of the two populations are unknown but assumed equal, the classical Student's two-sample t-test is recommended. If the underlying population distributions are normal with unequal and unknown variances, either Welch's t-statistic or Satterthwaite's approximate F test is suggested. However, Welch's procedure is non-robust under most non-normal distributions. There is a variable tolerance level around the strict assumptions of data independence, homogeneity of variances, and identical and normal distributions. Few textbooks offer alternatives when one or more of the underlying assumptions are not defensible. While there are more than a few non-parametric (rank) procedures that provide alternatives to Student's t-test, we restrict this review to the promising alternatives to Student's two-sample t-test in non-normal models.     

The two-sample t test versus satterthwaite's approximate f test

A comparison between the two-sample t test and Satterthwaite's approximate F test is made, assuming the choice between these two tests is based on a preliminary test on the variances. Exact formulas for the sizes and powers of the tests are derived. Sizes and powers are then calculated and compared for several situations.     

Non-Gaussian limit distributions for U-statistics based on trimmed and Winsorized samples

Conditions ensuring the asymptotic normality of U-statistics based on either trimmed samples or Winsorized samples are well known [P. Janssen, R. Serfling, and N. Veraverbeke, Asymptotic normality of U-statistics based on trimmed samples, J. Statist. Plann. Inference 16 (1987), pp. 63–74; U-statistics on Winsorized and trimmed samples, Statist. Probab. Lett. 9 (1990), pp. 439–447]. However, the class of U-statistics has a much richer family of limiting distributions. This paper complements known results by providing general limit theorems for U-statistics based on trimmed or Winsorized samples where the limiting distribution is given in terms of multiple Ito–Wiener stochastic integrals.     

An adaptive two-sample location-scale test of lepage type for symmetric distributions

For the two-sample location and scale problem we propose an adaptive test which is based on so called Lepage type tests. The well known test of Lepage (1971) is a combination of the Wilcoxon test for location alternatives and the Ansari-Bradley test for scale alternatives and it behaves well for symmetric and medium-tailed distributions. For the cae of short-, medium- and long-tailed distributions we replace the Wilcoxon test and the .Ansari-Bradley test by suitable other two-sample tests for location and scale, respectively, in oder to get higher power than the classical Lepage test for such distribotions. These tests here are called Lepage type tests. in practice, however, we generally have no clear idea about the distribution having generated our data. Thus, an adaptive test should be applied which takes the the given data set inio consideration. The proposed adaptive test is based on the concept of Hogg (1974), i.e., first, to classify the unknown symmetric distribution function with respect to a measure for tailweight and second, to apply an appropriate Lepage type test for this classified type of distribution. We compare the adaptive test with the three Lepage type tests in the adaptive scheme and with the classical Lepage test as well as with other parametric and nonparametric tests. The power comparison is carried out via Monte Carlo simulation. It is shown that the adaptive test is the best one for the broad class of distributions considered.     

Evaluation of a method for setting confidence intervals on the common mean of two normal populations

Let X₁, , X₂, …, X be distributed N(µ, σ² _x), let Y₁, Y₂, …, Y"_n be distributed N(µ, σ² _y), and let X , X , … X_m, Y₁, Y₂, …, Y_n be mutually independent. In this paper a method for setting confidence intervals on the common mean µ is proposed and evaluated.     

Single-stage sampling procedure of the t best populations under heteroscedasticity

Given k( ? 3) independent normal populations with unknown means and unknown and unequal variances, a single-stage sampling procedure to select the best t out of k populations is proposed and the procedure is completely independent of the unknown means and the unknown variances. For various combinations of k and probability requirement, tables of procedure parameters are provided for practitioners.     

Small sample properties of tests on homogeneity in one—way Anova and Meta—analysis

Received: January 12, 2000; revised version: July 26, 2000     

Combinatoric classification of multivariate normal variates

Consider classifying an n × I observation vector as coming from one of two multivariate normal distributions which differ both in mean vectors and covariance matrices. A class of dis-crimination rules based upon n independent univariate discrim-inate functions is developed yielding exact misclassification probabilities when the population parameters are known. An efficient search of this class to select the procedure with minimum expected misclassification is made by employing an algorithm of the implicit enumeration type used in integer programming. The procedure is applied to the classification of male twins as either monozygotic or dizygotic.     

The folded t distribution

Measurements are frequently recorded without their algebraicsign. As a consequence the underlying distribution of measurements is replaced by a distribution of absolute measurements. When the underlying distribution is t the resulting distribution is called the “folded-t distribution”. Here we study this distribution, we find the relationship between the folded-t distribution and a special case of the folded normal distribution and we derive relationships of the folded-t distribution to other distributions pertaining to computer generation. Also tables are presented which give areas of the folded-t distribution.     

Subset selection of the best treatment via many-one tests

The method of Gupta (1956, 1965) was developed to select a subset from k normal populations that contains the best populations with given probability. This paper shows a duality between the general goal of selecting a subset for the best population and many-one tests. A population should be regarded as ‘candidate’ for the best population and thus retained in the subset if the samples from the other populations are not significantly better. Based on this ‘idea’ a general selection procedure is proposed using many-one tests for the comparison of each population against the remaining ones.     

Selecting the normal population with the smallest variance: A restricted subset selection rule

Consider k( ? 2) normal populations whose means are all known or unknown and whose variances are unknown. Let σ²_[1] ? ??? ? σ_[k]² denote the ordered variances. Our goal is to select a non empty subset of the k populations whose size is at most m(1 ? m ? k ? 1) so that the population associated with the smallest variance (called the best population) is included in the selected subset with a guaranteed minimum probability P* whenever σ²_[2]/σ_[1]² ? δ* > 1, where P* and δ* are specified in advance of the experiment. Based on samples of size n from each of the populations, we propose and investigate a procedure called R_BCP. We also derive some asymptotic results for our procedure. Some comparisons with an earlier available procedure are presented in terms of the average subset sizes for selected slippage configurations based on simulations. The results are illustrated by an example.     

Adaptive sequential procedures for comparing new treatments with a standard

In this paper we propose and study two sequential elimination procedures for selecting all new treatments better than a standard or control treatment. These procedures differ from those previously proposed in that we assume variances are unequal and unknown. Expressions for asymptotic expected sample sizes are given. Confidence intervals associated with the procedures are also discussed.     

A Retrievable Recipe for Inverse t

Although the percentage points of the Student-t distribution have been widely tabulated, a simple approximation is given and derived in this article. The approximation can be re-derived easily, since it is based on the percentage points from the Gaussian distribution, and can thus be used for applications requiring non-integer degrees of freedom (e.g., Welch's two-sample t test) and for arbitrary significance levels (e.g., for Bonferroni multiple comparison procedures). Comparisons between this approximation and others suggested in the literature indicate three-digit accuracy for even small degrees of freedom and tail areas.     

On certail methods of improving ration and regression estimators

In this paper we have examined certain techiques suggested in the literature for improving the ratio and regression methods of estimation and demonstrated that these techniques are not very profitable.     

Levels of significance of some two-sample tests wkem observations are from compound normal distributions

This paper presents an investigation of the behavior of the levels of significance of the two-sample t and its related tests and the Mann-Whitney test when the samples are randomly drawn from mixtures of two normal populations (compound normals) and when the sample sizes are small (combined sample sizes ? 15). The use of the compound normal allows for investigation when the underlying populations are unequal, nonnormal, heterogeneous in variances, unimodal or bimodal, possessing smaller than normal kurtosis or containing contamination. The exact distribution of the t and its related tests are given. However, they are not readily amenable to calculations. Most of the numerical results presented were obtained by simulations