Often in the study of behavioral ecology, and more widely in science, we require to statistically test whether the central tendencies (mean or median) of 2 groups are different from each other on the basis of samples of the 2 groups. In surveying recent issues of Behavioral Ecology (Volume 16, issues 1–5), I found that, of the 130 papers, 33 (25%) used at least one statistical comparison of this sort. Three different tests were used to make this comparison: Student’s t-test (67 occasions; 26 papers), Mann–Whitney U test (43 occasions; 21 papers), and the t-test for unequal variances (9 occasions; 4 papers). My aim in this forum article is to argue for the greater use of the last of these tests. The numbers just related suggest that this test is not commonly used. In my survey, I was able to identify tests described simply as ‘‘t-tests’’ with confidence as either a Student’s t-test or an unequal variance t-test because the calculation of degrees of freedom from the 2 sample sizes is different for the 2 tests (see below). Hence, the neglect of the unequal variance t-test illustrated above is a real phenomenon and can be explained in several (nonexclusive ways) ways: 1. Authors are unaware that Student’s t-test is unreliable

The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test

When comparing two independent groups, psychology researchers commonly use Student’s t-tests. Assumptions of normality and homogeneity of variance underlie this test. More often than not, when these conditions are not met, Student’s t-test can be severely biased and lead to invalid statistical inferences. Moreover, we argue that the assumption of equal variances will seldom hold in psychological research, and choosing between Student’s t-test and Welch’s t-test based on the outcomes of a test of the equality of variances often fails to provide an appropriate answer. We show that the Welch’s t-test provides a better control of Type 1 error rates when the assumption of homogeneity of variance is not met, and it loses little robustness compared to Student’s t-test when the assumptions are met. We argue that Welch’s t-test should be used as a default strategy.

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Why psychologists should by default use Welch's t-test instead of student's t-test

A meta-analysis of the effects of texting on driving

In many applications, the underlying scientific question concerns whether the variances of $k$ samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the $F$-test to the absolute deviations of the observations from their group means. Levene's approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances. This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or "open or closed gramophones" in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

The Impact of Levene's Test of Equality of Variances on Statistical Theory and Practice

INTRODUCTION TO STATCALC Introduction of StatCalc PRELIMINARIES Random Variables and Expectations Moments and Other Functions Some Functions Relevant to Reliability Model Fitting Methods of Estimation Inference Random Number Generation Some Special Functions DISCRETE UNIFORM DISTRIBUTION Description Moments BINOMIAL DISTRIBUTION Description Moments Computing Table Values Test for the Proportion Confidence Intervals for the Proportion A Test for the Difference between Two Proportions Fisher's Exact Test Properties and Results Random Number Generation Computation of Probabilities HYPERGEOMETRIC DISTRIBUTION Description Moments Computing Table Values Point Estimation Test for the Proportion Confidence Intervals and Sample Size Calculation A Test for the Difference between Two Proportions Properties and Results Random Number Generation Computation of Probabilities POISSON DISTRIBUTION Description Moments Computing Table Values Point Estimation Test for the Mean Confidence Intervals for the Mean Test for the Ratio of Two Means Confidence Intervals for the Ratio of Two Means A Test for the Difference between Two Means Model Fitting with Examples Properties and Results Random Number Generation Computation of Probabilities GEOMETRIC DISTRIBUTION Description Moments Computing Table Values Properties and Results Random Number Generation NEGATIVE BINOMIAL DISTRIBUTION Description Moments Computing Table Values Point Estimation A Test for the Proportion Confidence Intervals for the Proportion Properties and Results Random Number Generation A Computational Method for Probabilities LOGARITHMIC SERIES DISTRIBUTION Description Moments Computing Table Values Inferences Properties and Results Random Number Generation A Computational Algorithm for Probabilities UNIFORM DISTRIBUTION Description Moments Inferences Properties and Results Random Number Generation NORMAL DISTRIBUTION Description Moments Computing Table Values One-Sample Inference Two-Sample Inference Tolerance Intervals Properties and Results Relation to Other Distributions Random Number Generation Computing the Distribution Function CHI-SQUARE DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Computing the Distribution Function F DISTRIBUTION Description Moments Computing Table Values Properties and Results Random Number Generation A Computational Method for Probabilities STUDENT'S t DISTRIBUTION Description Moments Computing Table Values Distribution of the Maximum of Several |t| Variables Properties and Results Random Number Generation A Computational Method for Probabilities EXPONENTIAL DISTRIBUTION Description Moments Computing Table Values Inferences Properties and Results Random Number Generation GAMMA DISTRIBUTION Description Moments Computing Table Values Applications with Some Examples Inferences Properties and Results Random Number Generation A Computational Method for Probabilities BETA DISTRIBUTION Description Moments Computing Table Values Inferences Applications with an Example Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL CHI-SQUARE DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL F DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function NONCENTRAL t DISTRIBUTION Description Moments Computing Table Values Applications Properties and Results Random Number Generation Evaluating the Distribution Function LAPLACE DISTRIBUTION Description Moments Computing Table Values Inferences Applications Relation to Other Distributions Random Number Generation LOGISTIC DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Applications Properties and Results Random Number Generation LOGNORMAL DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Confidence Interval and Test for the Mean Inferences for the Difference between Two Means Inferences for the Ratio of Two Means Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles PARETO DISTRIBUTION Description Moments Computing Table Value Inferences Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles WEIBULL DISTRIBUTION Description Moments Computing Table Values Applications Point Estimation Properties and Results Random Number Generation Computation of Probabilities and Percentiles EXTREME VALUE DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimators Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles CAUCHY DISTRIBUTION Description Moments Computing Table Values Inference Applications Properties and Results Random Number Generation Computation of Probabilities and Percentiles INVERSE GAUSSIAN DISTRIBUTION Description Moments Computing Table Values One-Sample Inference Two-Sample Inference Random Number Generation Computational Methods for Probabilities and Percentiles RAYLEIGH DISTRIBUTION Description Moments Computing Table Values Maximum Likelihood Estimator Relation to Other Distributions Random Number Generation BIVARIATE NORMAL DISTRIBUTION Description Computing Table Values An Example Inferences on Correlation Coefficients Inferences on the Difference between Two Correlation Coefficients Some Properties Random Number Generation A Computational Algorithm for Probabilities DISTRIBUTION OF RUNS Description Computing Table Values Examples SIGN TEST AND CONFIDENCE INTERVAL FOR THE MEDIAN Hypothesis Test for the Median Confidence Interval for the Median Computing Table Values An Example WILCOXON SIGNED-RANK TEST Description Moments and an Approximation Computing Table Values An Example WILCOXON RANK-SUM TEST Description Moments and an Approximation Mann-Whitney U Statistic Computing Table Values An Example NONPARAMETRIC TOLERANCE INTERVAL Description Computing Table Values An Example TOLERANCE FACTORS FOR A MULTIVARIATE NORMAL POPULATION Description Computing Tolerance Factors Examples DISTRIBUTION OF THE SAMPLE MULTIPLE CORRELATION COEFFICIENT Description Moments Inferences Some Results Random Number Generation A Computational Method for Probabilities Computing Table Values REFERENCES INDEX

Handbook of statistical distributions with applications

In the two-sample means test, the need for a preliminary variance test and the special emphasis given to the equal variance assumption are questioned. The conclusions can be easily used in the classroom to teach the two sample means test.

Gary R. Stevens

Papers

Homogeneity of Variance in the Two-Sample Means Test