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.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Robustness of subset selection procedures

The robustness (and the number of non-best populations selected) of 11 subset selection procedures is investigated by means of simulation experiments. If the underlying distributions differ only in their location parameter, the subset selection procedures are robust for symmetric distributions or distributions with negative skewness. With increasing positive skewness and increasing number of populations the considered parametric procedures fail in robustness slightly. This non-robustness is more serious in the case of unequal variances. Non-parametric subset selection rules show then an increasing non-robustness with increasing sample size.     

A Comparison of Bayesian Models of Heteroscedasticity in Nested Normal Data

We consider the fitting of a Bayesian model to grouped data in which observations are assumed normally distributed around group means that are themselves normally distributed, and consider several alternatives for accommodating the possibility of heteroscedasticity within the data. We consider the case where the underlying distribution of the variances is unknown, and investigate several candidate prior distributions for those variances. In each case, the parameters of the candidate priors (the hyperparameters) are themselves given uninformative priors (hyperpriors). The most mathematically convenient model for the group variances is to assign them inverse gamma distributed priors, the inverse gamma distribution being the conjugate prior distribution for the unknown variance of a normal population. We demonstrate that for a wide class of underlying distributions of the group variances, a model that assigns the variances an inverse gamma-distributed prior displays favorable goodness-of-fit properties relative to other candidate priors, and hence may be used as standard for modeling such data. This allows us to take advantage of the elegant mathematical property of prior conjugacy in a wide variety of contexts without compromising model fitness. We test our findings on nine real world publicly available datasets from different domains, and on a wide range of artificially generated datasets.     

Robustness of the t and U tests under combined assumption violations

SUMMARY When the assumptions of parametric statistical tests for the difference between two means are violated, it is commonly advised that non-parametric tests are a more robust substitute. The history of the investigation of this issue is summarized. The robustness of the t -test was evaluated, by repeated computer testing for differences between samples from two populations of equal means but non-normal distributions and with different variances and sample sizes. Two common alternatives to t -Welch's approximate t and the Mann-Whitney U -test-were evaluated in the same way. The t -test is sufficiently robust for use in all likely cases, except when skew is severe or when population variances and sample sizes both differ. The Welch test satisfactorily addressed the latter problem, but was itself sensitive to departures from normality. Contrary to its popular reputation, the U -test showed a dramatic 'lack of robustness' in many cases-largely because it is sensitive to population differences other than between means, so it is not properly a 'non-parametric analogue' of the t -test, as it is too often described.     

Estimating kurtosis and confidence intervals for the variance under nonnormality

Exact confidence intervals for variances rely on normal distribution assumptions. Alternatively, large-sample confidence intervals for the variance can be attained if one estimates the kurtosis of the underlying distribution. The method used to estimate the kurtosis has a direct impact on the performance of the interval and thus the quality of statistical inferences. In this paper the author considers a number of kurtosis estimators combined with large-sample theory to construct approximate confidence intervals for the variance. In addition, a nonparametric bootstrap resampling procedure is used to build bootstrap confidence intervals for the variance. Simulated coverage probabilities using different confidence interval methods are computed for a variety of sample sizes and distributions. A modification to a conventional estimator of the kurtosis, in conjunction with adjustments to the mean and variance of the asymptotic distribution of a function of the sample variance, improves the resulting coverage values for leptokurtically distributed populations.     

Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.     

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.     

Improved estimators for the common mean and ordered means of two normal distributions with ordered variances

We first consider the problem of estimating the common mean of two normal distributions with unknown ordered variances. We give a broad class of estimators which includes the estimators proposed by Nair (1982) and Elfessi et al. (1992) and show that the estimators stochastically dominate the estimators which do not take into account the order restriction on variances, including the one given by Graybill and Deal (1959). Then we propose a broad class of individual estimators of two ordered means when unknown variances are ordered. We show that in estimating the mean with larger variance, estimators which do not take into account the order restriction on variances are stochastically dominated by the proposed class of estimators which take into account both order restrictions. However, in estimating the mean with smaller variance, similar improvement is not possible even in terms of mean squared error. We also show a domination result in the simultaneous estimation problem of two ordered means. Further, improving upon the unbiased estimators of the two means is discussed.     

Proportion estimation in mixtures with covariates

Linear maps of a single unclassified observation are used to estimate the mixing proportion in a mixture of two populations with homogeneous variances in the presence of covariates. with complete knowledge of the parameters of the individual populations, the linear map for which the estimator is unbiased and has minimum variance amongst all similar estimators can be determined. Plug-in estimator based on independent training samples from the component populations can be constructed and is asymptotically equivalent to Cochran's classification statistic V* for covariate classification; see Memon and Okamoto (1970). Under normality assumptions, asymptotic expansion of the distribution of the plug-in estimator is available. In the absence of covariates, our estimator reduces to that suggested by Walker (1980) who has investigated the problem based on information on large unclassified samples from a mixture of two populations with heterogeneous variances. In contrast, distribution of Walker's estimator seems intractable in moderate sample sizes even with normality assumption.     

Improved tests for homogeneity of variances

Equality of variances is one of the key assumptions of analysis of variances (ANOVA). There are several testing procedures available to validate this assumption, but it is rare to find a test procedure which controls the type I error rate while providing high statistical power. In this article, we introduce a bootstrap test based on the ratio of mean absolute deviances (RMD). We also propose a two-stage testing procedure where we first quantify the skewness of the distributions and then choose an appropriate test for homogeneity of variances. The performances of these test procedures are studied via a simulation study.     

Testing equality of variances for several normal populations

In this study, we develop a test based on computational approach for the equality of variances of several normal populations. The proposed method is numerically compared with the existing methods. The numeric results demonstrate that the proposed method performs very well in terms of type I error rate and power of test. Furthermore we study the robustness of the tests by using simulation study when the underlying data are from t, exponential and uniform distributions. Finally we analyze a real dataset that motivated our study using the proposed test.     

A MONTE CARLO COMPARISON OF ELLIPTICAL-THEORY AND OTHER TESTS FOR EQUALITY OF COVARIANCE MATRICES

A Monte Carlo study of the size and power of tests of equality of two covariance matrices is carried out. Tests based upon normality assumptions, elliptical distribution assumptions as well as distribution-free tests are compared. Samples are generated from normal, elliptical and non-elliptical populations. The elliptical-theory tests, in particular, have poor size properties for both elliptical distributions with moderate sample sizes and for non-elliptical distributions.     

Sensitivity of a bayesian inference to prior assumptions

The sensitivity of-a Bayesian inference to prior assumptions is examined by Monte Carlo simulation for the beta-binomial conjugate family of distributions. Results for the effect on a Bayesian probability interval of the binomial parameter indicate that the Bayesian inference is for the most part quite sensitive to misspecification of the prior distribution. The magnitude of the sensitivity depends primarily on the difference of assigned means and variances from the respective means and variances of the actually-sampled prior distributions. The effect of a disparity in form between the assigned prior and actually-sampled distributions was less important for the cases tested.     

Testing homoosenetty of t normal means against ordered alternatives in 2 groups having uncommon and unknown variance

Given two series of normal populations with variances unknown and common within the series but possibly differ between the series, the likelihood ratio test of simultaneous homogeneity of means against sinpie order alternative is considered. Sole approxirations Par the null distribution of thetest statistic are also proposed.     

Anomalies in the analysis of calibrated data

This study examines the effects of calibration errors on model assumptions and data-analytic tools in direct calibration assays. These effects encompass induced dependencies, inflated variances, and heteroscedasticity among the calibrated measurements, whose distributions arise as mixtures. These anomalies adversely affect conventional inferences, including the inconsistency of sample means; the underestimation of measurement variance; and the distributions of sample means, sample variances, and student's t as mixtures. Inferences in comparative experiments remain largely intact, although error mean squares continue to underestimate the measurement variances. These anomalies are masked in practice, as conventional diagnostics cannot discern the irregularities induced through calibration. Case studies illustrate the principal issues.     

TWO STAGE SELECTION PROBLEM FOR NORMAL POPULATIONS WITH UNEQUAL VARIANCES

The paper discusses the Dudewicz-Dalal modification of a Stein-type two sample procedure for the goal of selecting the population with the largest mean from k normal populations with unknown variances. Largest k values are obtained such that a procedure based on sample means is preferred to the Dudewicz-Dalal procedure. The more general goal of choosing those populations with the t largest means is also considered.     

LOGISTIC REGRESSION COMPARED TO NORMAL DISCRIMINATION FOR NON-NORMAL POPULATIONS'

An investigation is undertaken of the logistic regression procedure for estimating the posterior probability of an object belonging to one of two populations. The asymptotic bias and mean square error associated with the procedure are derived for univariate populations whose distributions satisfy the general Day-Kerridge model for which the logistic form is valid for the posterior probability. These properties are compared with those of the normal discrimination method based on the classical assumption of normal populations with common variances. The asymptotic relative efficiency of logistic regression is considered on the basis of asymptotic mean square error.     

A note on dealing with missing standard errors in meta-analyses of continuous outcome measures in WinBUGS

A meta-analysis of a continuous outcome measure may involve missing standard errors. This is not a problem depending on assumptions made about the population standard deviation. Multiple imputation can be used to impute missing values while allowing for uncertainty in the imputation. Markov chain Monte Carlo simulation is a multiple imputation technique for generating posterior predictive distributions for missing data. We present an example of imputing missing variances using WinBUGS. The example highlights the importance of checking model assumptions, whether for missing or observed data.