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The interpretation of glass evidence is crucial to the accurate appraisal of the evidential value of the analytical results. Frequently, the first step in this assessment is a comparison between the analytical results of the recovered and control glass fragments to determine whether they could have come from the same source of glass. A number of different methods are available depending on the type of analytical test performed. The second step is to determine the evidential value of these findings. This involves consideration of a number of factors, including the amount of matching and nonmatching glass found, the frequency of the matching glass, the probability of glass being transferred given the particular case scenario, and the amount of background glass found on a random person. The assessment of the evidential value can be made in an essentially subjective manner or by using a more formal framework, such as the calculation of the likelihood ratio (LR). Methods to report these findings are also presented.

A transformation of the Welch statistic to compare means is proposed to correct skewness and kurtosis of parent populations. The results show that this transformation seems to improve the performance of the test in heavy-tailed distributions more than other transformations focused only on skewness. The proposed test outperforms the Welch test in asymmetric heavy-tailed distributions with high heteroscedasticity and it behaves better than the Johnson’s transformation trimmed mean Welch test in normal, near-normal and light-tailed distributions. It may also be a better option when some of the distributions are heavy-tailed and some light-tailed.

Researchers are generally required to report and interpret effect sizes and associated confidence intervals. When comparing two independent groups, the most commonly used estimator of effect size is Cohen’s ds where sample mean difference is divided by the pooled standard deviation. However, computing the pooled error term is not valid when both groups do not share common population variances. Furthermore, the assumption of equal population variances is unlikely in many psychological fields. Consequently, researchers shift to the use of Welch’s t-test over Student’s t-test in the context of hypothesis testing. Meanwhile, the question which effect size to report when equal variances are not assumed remains open. Based on Monte Carlo simulations, we compare Hedges’ gs (i.e. Cohen’s ds with correction for bias) to Glass’s gs, Shieh’s gs and Hedges’ g_s^*. Comparisons are made under normality as well as under realistic deviations from the assumptions of normality and equal variances. Although it is not directly related with Welch’s t-test (unlike Shieh’s gs), we recommend the use of Hedges’ g_s^* because it shows better properties than all other estimators. Practical recommendations, R package and Shiny App in order to compute effect size estimators and confidence intervals are provided.

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., SCOPUS: ar.j, info:eu-repo/semantics/published

This article proposes a useful extension of the ANOVA F-test for examining mean differences for evaluating the comparability of standardized mean effects. The equivalence procedure presumably suffers from the same disadvantage as the traditional ANOVA F-statistic regarding the violations of homogeneous variance assumption. In view of the absence of vital clarification for theory development and supportive technique, this article provides a critical exposition of the extended Welch test for the equivalence of standardized means. To enhance the usefulness of equivalence testing, the theoretical properties and practical implications of the Welch-type procedure are demonstrated. Moreover, the corresponding power and sample size algorithms are described for advance planning of equivalence studies. Despite the approximate nature, the accuracy of Type I error rate, statistical power, and sample size calculations of the suggested equivalence test is justified under a wide range of model configurations. The proposed procedures are demonstrated with the data of a clinical trial regarding the comparative study of four antihypertensive treatments. A complete set of computer codes is presented to calculate the corresponding critical values, p-values, power levels, and sample sizes for data analysis and design planning of equivalence studies.

The parametric Welch $t$-test and the non-parametric Wilcoxon-Mann-Whitney test are the most commonly used two independent sample means tests. More recent testing approaches include the non-parametric, empirical likelihood and exponential empirical likelihood. However, the applicability of these non-parametric likelihood testing procedures is limited partially because of their tendency to inflate the type I error in small sized samples. In order to circumvent the type I error problem, we propose simple calibrations using the $t$ distribution and bootstrapping. The two non-parametric likelihood testing procedures, with and without those calibrations, are then compared against the Wilcoxon-Mann-Whitney test and the Welch $t$-test. The comparisons are implemented via extensive Monte Carlo simulations on the grounds of type I error and power in small/medium sized samples generated from various non-normal populations. The simulation studies clearly demonstrate that a) the $t$ calibration improves the type I error of the empirical likelihood, b) bootstrap calibration improves the type I error of both non-parametric likelihoods, c) the Welch $t$-test with or without bootstrap calibration attains the type I error and produces similar levels of power with the former testing procedures, and d) the Wilcoxon-Mann-Whitney test produces inflated type I error while the computation of an exact p-value is not feasible in the presence of ties with discrete data. Further, an application to real gene expression data illustrates the computational high cost and thus the impracticality of the non parametric likelihoods. Overall, the Welch t-test, which is highly computationally efficient and readily interpretable, is shown to be the best method when testing equality of two population means., Accepted for publication in the Journal of Statistical Computation and Simulation

Clinical biochemists often wish to compare two groups of measurements. In order to do so, they must be familiar with Student's t-test. This article provides guidance for the use of the Welch t-test, with subtle but important differences and validity in a broader range of settings: advises the use of the Welch t-test rather than Student's t-test.

Urban trees are generally considered to be a public asset and are an important part of a city’s heritage. The aim of this work is to analyse the influence of season on the economic appraisal of various trees in Madrid. Photographs were taken of 43 individual tree specimens in summer and winter. The survey was designed to compare differences of opinion in the economic assessment of trees. The trees were assessed by eight valuation methods used worldwide. A total of 78 agroforestry engineering students answered a written survey, and the variables considered were: percentage of students who always evaluated the tree equally (%0), percentage of students who assigned more value to the summer photograph (%S), and percentage of students who assigned more value to the winter photograph (%W). The results were analysed by the statistical test of equal proportions and ANOVA to detect differences according to tree type (evergreen or deciduous), species, and other groupings made by the authors in previous works. W and S percentages are similar. The ANOVA analysis rejects the equality of percentages of S and W between groups. The Welch test rejects the equality of the percentage of S, W, and O between species.

Reference (Rasch, Kubinger and Moder (2011b). Stat. Pap. 52, 219–231.) [4] showed that in case that nothing is known about the two variances it is better to use the approximate Welch test instead of the two-sample t-test for comparing means of two continuous distributions with existing first two moments. An analogue approach for the triangular sequential t test is not possible because it is based on the first two derivatives of the underlying likelihood functions. Extensive simulations have been done and are reported in this chapter. It is shown that the two-sample triangular sequential t test in most interesting cases holds the type I and type II risks when variances are unequal.

In this research, we propose simultaneous confidence intervals for all pairwise multiple comparisons in a two-way unbalanced design with unequal variances, using a parametric bootstrap approach. Simulation results show that Type 1 error of the multiple comparison test is close to the nominal level even for small samples. They also show that the proposed method outperforms Tukey–Kramer procedure when variances are heteroscedastic and group sizes are unequal.

An alternative to the Satterswaite-Welch degrees of freedom (df) is developed in this paper based on unbiased estimating equations. The method enables us to evaluate the impact of sample variation due to using estimated df instead of the true df by computer intensive method. It is shown by simulation that the alternative df is close to the Welch df and the method developed could be also used for evaluating the impact of sample variation due to using estimated df in the Welch test.

There are two well-known versions of the t-test for comparing means from unpaired data: Student's t-test and Welch's t-test. While Welch's t-test does not assume homoscedasticity (i.e., equal variances), nit involves approximations. A classical textbook recommendation would be to use Student's t-test if either the two sample sizes are similar or the two sample variances are similar, and to use Welch's t-test only when both of the above conditions are violated. However, a more recent recommendation seems to be to use Welch's t-test unconditionally. Using past data from both TREC and NTCIR, the present study demonstrates that the latter advice should not be followed blindly in the context of IR system evaluation. More specifically, our results suggest that if the sample sizes differ substantially and if the larger sample has a substantially larger variance,Welch's t-test may not be reliable.

In order to objectively evaluate the systematic differences in the area of annual disaster relief before and after the scientific implementation of artificial flood control in the Akesu area of Xinjiang, the effect of artificial flood control in the area was analyzed. This paper uses the data of the annual disaster area of the Akesu area from 1978 to 2013, and starts the year of artificial flood control operation in 1996 as a science. Using the statistical methods such as sequence test, unpaired rank sum test and Welch test, artificial prevention of science The systematic differences in the annual disaster area of each 18a before and after the operation were analyzed. The results show that the hail disasters in the Akesu area are significantly reduced after the artificial anti-mite operation. The non-parametric unpaired rank sum test has a significance level of 0.05, the parametric Welch test has a significance level of 0.01, and the average annual disaster area is reduced by 15063 hm2. The reduction rate was 43.14%. Combined with agricultural economic data, the annual average reduction of disaster relief losses is 281.09 million yuan, the annual input-output ratio is 1:6, and the statistical significance level reaches 0.1. Therefore, after the scientific implementation of artificial flood control operations in the Akesu area, the hail disaster losses are significantly reduced, and the social and economic benefits of disaster reduction are significant.

Within this research the travel patterns of passenger car drivers are investigated in relation to their age, as recorded from a national travel survey conducted in Greece. For the needs of this research drivers are classified into three mutually exclusive age groups: young (18–24), adult (25–64) and elderly (65+). The mobility of these three driver groups—expressed in terms of average annual distance driven—is then investigated with respect to parameters involving driver characteristics (experience, gender), vehicle characteristics (engine size) and the nature of the trip (time of trip and road type). Several patterns are analysed, and the dependence of the various travel patterns on driver age is tested through analysis of variance (ANOVA), mainly by the use of the Welch test. The results show that driver experience is not an influencing factor for driver mobility, whereas driver gender and vehicle characteristics, such as engine size, are. Furthermore, the results illustrate dependences between driver age and driving at night and during the weekends, highlighting the relationship between driver age and trip purpose. The distribution of mobility across the different road types does not reveal any interactions with driver age; however, specific trends are detected when distance driven is also segregated into weekday and weekend categories, indicating a relationship between driver age and mobility in relation to type of day.

Recently, CRESSIE and WHITFORD (1986) showed that Welch's test of H0 :μ1 = μ2 can be biased, under nonnormality, where μ1 and μ2 are the means of two independent treatment groups. They suggested, therefore, that a two-sample analog of Johnson's test be used instead. One goal in this paper is to examine the extent to which a two-sample analog of Johnson's test improves upon Welch's technique in terms of Type I errors and power when sample sizes are small or moderately large. Several alternative procedures are also considered including an additional modification of Johnson's procedure, a procedure suggested by DUNNETT (1982) that uses Tiku's modified maximum likelihood estimate with 10% trimming, two versions of Efron's bootstrap, and a test recently proposed by WILCOX (1989). The paper also describes situations where Welch's procedure is not robust in terms of Type I errors. This is important because based on results currently available, Welch's procedure is thought to be nonrobust in terms of power, but robust in terms of Type I errors.

For the two-sample problem with location and/or scale alternatives, as well as different shapes, several statistical tests are presented, such as of Kolmogorov-Smirnov and Cramer-von Mises type for the general alternative, and such as of Lepage type for location and scale alternatives. We compare these tests with the t-test and other location tests, such as the Welch test, and also the Levene test for scale. It turns out that there is, of course, no clear winner among the tests but, for symmetric distributions with the same shape, tests of Lepage type are the best ones whereas, for different shapes, Cramer-von Mises type tests are preferred. For extremely right-skewed distributions, a modification of the Kolmogorov-Smirnov test should be applied.

Behrens-Fisher problem has a lot of test, two of them are Cohran-Cox test and Welch test. This study will examine the stability of both tests by comparing thesignificance levels of both tests to a specific (given) significance level. The result shows that the Welch test is more stable than the Cochran-Cox test which means that theoretically Welch test is better than Cochran-Cox test.

Guenther derived a simple formula for the accurate computation of equal sample sizes when the two-sample t-test with pooled variance estimate is used. In the present paper, the Guenther formula is generalized to the case of unequal sample sizes. In addition, the case of unequal variances is considered.

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

異なる海域の放射能を比較し, 両者に有意の差があるか否かを検定するため, ウェルチの検定が利用できるかどうかを議論する。検定されるべき統計量として, ウェルチの検定に対応する量を定義し, その分布をコンピュータシミュレーションで調べたところ, t分布ではなく標準形の正規分布に近くなることがわかった。したがってウェルチの検定を採用すると, 本来棄却されるべきデータが棄却されない危険性がある。

The U.S. Environmental Protection Agency (U.S. EPA) and state agencies evaluate the toxicity of effluent and surface water samples based on statistical endpoints derived from multiconcentration tests (e.g., no observed effect concentration, EC25). The test of significant toxicity (TST) analysis is a two-sample comparison test that uses Welch's t test to compare organism responses in a sample (effluent or surface water) with responses in a control or site sample. In general, any form of t test (Welch's t included) is appropriate only if the data meet assumptions of normality and homogeneous variances. Otherwise, nonparametric tests are recommended. TST was designed to use Welch's t as the statistical test for all whole effluent toxicity (WET) test data. The authors evaluated the suitability of using Welch's t test for analyzing two-sample toxicity (WET) data, and within the TST approach, by examining the distribution and variances of data from over 2,000 WET tests and by conducting multiple simulations of WET test data. Simulated data were generated having variances and nonnormal distributions similar to observed WET test data for control and the effluent treatment groups. The authors demonstrate that (1) moderately unequal variances (similar to WET data) have little effect on coverage of the t test or Welch t test (for normally distributed data), and (2) for nonnormally distributed data (similar in distribution to WET data) TST, using Welch's t test, has close to nominal coverage on the basis of simulations with up to a ninefold difference in variance between the effluent and control groups (∼95th percentile based on observed WET test data).

In the two-sample location problem the application of thet-test depends on very restrictive assumptions such as normality and equal variances of the two random variablesX andY. If the assumptions of thet-test are not satisfied it is more appropriate to apply a robust version of thet-test, like the Welch test or the trimmedt-test, or a nonparametric test, like the Wilcoxon. But usually we have no information about the underlying distribution of the data. Therefore, an adaptive test should be applied. Some of these tests are discussed and compared with each other and with the classicalt-test under different models of nonnormality and for unequal variances. It is shown that an adaptive test behaves well over a broad class of distribution functions. Im Zweistichproben-Lageproblem hangt die Anwendung dest-Tests von sehr restriktiven Modellannahmen wie die der Normalverteilung und der Gleichheit der Varianzen der ZufallsvariablenX undY ab. Falls die Annahmen dest-Tests nicht erfullt sind, ist es besser, eine robuste Version dest-Tests wie den WelchTest bzw. den getrimmtent-Test oder einen nichtparametrischen Test wie den Wilcoxon-Test anzuwenden. Da wir aber in der Regel keine Information uber die den Daten zugrundeliegende Verteilung haben, sollte ein adaptiver Test bevorzugt werden. Einige dieser Tests werden diskutiert und miteinander sowie mit dem klassischent-Test fur verschiedene Verteilungen und ungleiche Varianzen verglichen. Es wird gezeigt, daβ ein adaptiver Test gute Eigenschaften uber eine breite Klasse von Verteilungsfunktionen hat.

Tables of optimal sample sizes are provided in this article for testing the equality of means of two normal populations when variances may be unequal. The Welch's Approximate t Test is used since it is increasingly recognized as the most practical solution to the Behrens–Fisher problem

This study was undertaken to apply a new method of controlling type I error when performing pairwise comparisons. The Benjamini and Hochberg false discovery rate (FDR) controlling procedures have proved to be very powerful tools in solving many practical problems but have not yet been applied to pairwise comparisons of refractive index of glass samples. Students t-test and Welch test (unequal variance Student's t-test) were applied to all possible pairwise comparisons. The comparisons were made on the basis of refractive index values of 72 glass fragments from different car windows and 69 different fragments from one windscreen. The type I error was controlled by the use of Benjamini and Hochberg false discovery rate (FDR) controlling procedure. To illustrate the importance of controlling type I error when using pairwise comparisons, results of pairwise comparisons with the FDR controlling procedure were compared to those comparisons made without any controlling procedure. Significantly fewer false negative results (false rejection of H0) were found during the use of FDR procedure in the comparison of refractive indices from the same windscreen than when comparing without controlling type I error. The results of application of FDR method were also compared to the results of other post-hoc tests, such as Tukey HSD test and Bonfferoni test. The FDR method has higher power than Bonferroni method and Tukey HSD method and control errors better than comparing without controlling type I error. The method of choice, for pairwise comparison of glass fragments on the basis of refractive index is Welch test with FDR adjustment. The method gave type I errors at level about 5.3% and type II error at about 3.7%.

It is known that using the F test for testing the equality of meansin a one-way ANOVA is misleading when the assumption of equal populationvariances is violated. When the variances are unknown and unequal, Welch[1] developed the so-called Welch test and Chen and Chen [2] developed thesingle-stage test for ANOVA. In this paper, Welch and single-stage tests arecompared in terms of their powers and the reject ratios of the hypothesis whenthe hypothesis is true. Simulation results indicate that the power of the singlestage test is better than the Welch test and that this test performs well in termsof the reject ratios for small number of populations and sample sizes

Six procedures which convert tests of homogeneity of variance into tests for mean equality for independent groups are compared. The tests are the analysis of variance (ANOVA) and Welch F statistics. The Welch statistics are included since it was anticipated that ANOVA would not provide a robust test when samples of unequal sizes are obtained from non-normal populations. However, the Welch tests are not found to be uniformly preferrable. In addition, a prior recommendation for Miller's jackknife procedure is not supported for the unequal sample size case. The data indicates that the current tests for variance heterogeneity are either sensitive to non-normality or, if robust, lacking in power. Therefore, these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption.

Errors are given for the departure of Bartlett's exact distri-bution for the test of homogeneity of variances for unequal sample sizes from that of equal sample sizes. Let vj + 1 denote the size of the jth sample and define a departure number where Then these errors may be expressed as functions of only three parameters: the number of populations k, the depart-ure number δ and . Accurate critical values of the test, in the general case with unequal sample sizes, are given when these errors are small.

The comparison of the unknown means of two populations with unknown variances is called the Behrens-Fisher problem, if the populations are assumed to be normal and the ratio of the variances is not known. In this paper a summary of recipes is given to solve this problem in practice, as published in the past 35 years by Banerjee, Fisher and Behrens, Pagurova, Wald and Hajek, Welch, Welch and Aspin, together with two large sample solutions and one solution often used as an approximate one without justification. The solutions are presented mainly in terms of confidence intervals for the difference of the population means. Some remarks are made concerning the lengths of these intervals and the power of the corresponding tests. The solutions in this paper are dependent on the means and the variances of samples drawn from the two populations only. All solutions discussed, except the disqualified approximate one, are robust against violations of the normality assumptions with respect to the populations and they provide, at least asymptotically, good measures for the difference of the population means if the samples are drawn from populations whatsoever with finite second moment.

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.

By using Monte Carlo studies, this paper compares the Welch test, the James test and the Brown-Forsythe test for comparing several means under heteroscedasticity. It appears that all the tests are quite robust with respect to departure from normality. The Brown-Forsythe test is at least as good as the James test and the Welch test; but the differences are so small that the choice is immaterial for practical purposes.

Numerous studies have documented the robustness of t and F to heterogeneous variances under the restricted condition of equal n’s. Likewise, the distortion of α in the presence of unequal n’s and variances has been demonstrated in both mathematical and empirical studies. Several investigations, however, have shown the Welch technique to be robust to this disturbing situation in the two group case. The present study was addressed to the k group AOV situation. Monte Carlo methods were employed to contrast several procedures with respect to a.) control over Type I errors and b.) power. Results indicate that the generalized Welch technique may be substituted for the AOV when variances are heterogeneous and n’s are unequal.

A comparative study is made of three tests, developed by James (1951), Welch (1951) and Brown & Forsythe (1974). James presented two methods of which only one is considered in this paper. It is shown that this method gives better control over the size than the other two tests. None of these methods is uniformly more powerful than the other two. In some cases the tests of James and Welch reject a false null hypothesis more often than the test of Brown & Forsythe, but there are also situations in which it is the other way around. We conclude that for implementation in a statistical software package the very complicated test of James is the most attractive. A practical disadvantage of this method can be overcome by a minor modification.

This paper examines the robustness of the Welch test , the James test as well as Tan's Anova test (to be referred as Fβ test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.

The method in Krutchkoff (1988b) for one-way ANOVA is extended to the two-way situation. The K- ANOVA application program is discussed.

The probabilities of the Type I errors of the Welch approximate-t test and the Aspin-Welch test for the Behrens-Fisher problem have been calculated for selected sets of degrees of freedom and nominal significance levels. The results show satisfactory agreement to the desired level. Thus, in practice, one can just use the usual t-table to carry out the Welch approximate t-test without much loss of accuracy.