Approximate sample size formulas for the two‐sample trimmed mean test with unequal variances

Luh, Wei-Ming; Guo, Jiin Huarng

doi:10.1348/000711006x100491

Cited by 20 publications

(11 citation statements)

References 28 publications

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“…Alternatives exist, known as the "modern robust statistics" (Wilcox, Granger, & Clark, 2013). For example, data sets with low kurtosis (i.e., a distribution flatter than the normal distribution) should be analyzed with the two-sample trimmed t-test for unequal population variances, also called Yuen's t-test (Luh & Guo, 2007;Yuen, 1974). However, analyses in a later section will show that the normality assumption is not very important for Welch's t-test and that there are good reasons to, in general, prefer Welch's t-test over Yuen's t-test. With respect to the assumption of homogeneity of variance, if the test of the equality of variance is non-significant and the assumption of equal variances cannot be rejected, homoscedastic methods such as the Student's t-test should be used (Wilcox et al, 2013).…”

Section: Limitations Of Two-step Proceduresmentioning

confidence: 99%

Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test

Delacre¹,

Lakens²,

Leys

2017

International Review of Social Psychology

679

383

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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. Keywords:Welch's t-test; Student's t-test; homogeneity of variance; Levene's test; Homoscedasticity; statistical power; type 1 error; type 2 error Independent sample t-tests are commonly used in the psychological literature to statistically test differences between means. There are different types of t-tests, such as Student's t-test, Welch's t-test, Yuen's t-test, and a bootstrapped t-test. These variations differ in the underlying assumptions about whether data is normally distributed and whether variances in both groups are equal (see, e.g., Rasch, Kubinger, & Moder, 2011;Yuen, 1974). Student's t-test is the default method to compare two groups in psychology. The alternatives that are available are considerably less often reported. This is surprising, since Welch's t-test is often the preferred choice and is available in practically all statistical software packages.In this article, we will review the differences between Welch's t-test, Student's t-test, and Yuen's t-test, and we suggest that Welch's t-test is a better default for the social sciences than Student's and Yuen's t-tests. We do not include the bootstrapped t-test because it is known to fail in specific situations, such as when there are unequal sample sizes and standard deviations differ moderately (Hayes & Cai, 2007).When performing a t-test, several software packages (i.e., R and Minitab) present Welch's t-test by default. Users can request Student's t-test, but only after explicitly stating that the assumption of equal variances is met. Student's t-test is a parametric test, which means it relies on assumptions about the data that are analyzed. Parametric tests are believed to be more powerful than non-parametric tests (i.e., tests that do not require assumptions about the population parameters; Sheskin, 2003). However, Student's t-test is generally only more powerful when the data are normally distributed (the assumption of normality) and the variances are equal in both groups (homoscedasticity; the assumption of homogeneity of variance; Carroll & Schneider, 1985;Erceg-Hurn & Mirosevich, 2008).When sample sizes are equal between groups, Student's t-test is robust to violations of the assump...

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Section: Limitations Of Two-step Proceduresmentioning

confidence: 99%

Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test

Delacre¹,

Lakens²,

Leys

2017

International Review of Social Psychology

679

383

View full text Add to dashboard Cite

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“…Moreover, when variances are heterogeneous, Luh and Guo (2007) developed a sample size formula for trimmed means. When variance is unknown, researchers suggest Stein's two-stage sampling procedure (Desu & Raghavarao, 1990;Luh & Olejnik, 1994).…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Sample Size Determination for One-and Two-Sample Trimmed Mean Tests

Luh

Olejnik²,

Guo

2008

The Journal of Experimental Education

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Formulas to determine the necessary sample sizes for parametric tests of group comparisons are available from several sources and appropriate when population distributions are normal. However, in the context of nonnormal population distributions, researchers recommend Yuen's trimmed mean test, but formulas to determine sample sizes have not been developed. The authors address this omission and provide statistically rigorous formulas of sample size determination for 1-and 2-sample Yuen's trimmed mean tests. Computer simulations for a wide range of nonnormal distribution shapes evaluated the adequacy of the proposed formulas. The results show that the proposed formulas control Type I error satisfactorily and achieve desired power, whereas the traditional formulas resulted in oversampling units for nonnormal distributions.

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“…Luh and Guo (2007) have already developed approximate sample size formulas for Yuen's trimmed t test. Therefore, in this paper, random samples of size n 1 and n 2 will be assigned to groups 1 and 2, respectively, along with the anticipated unequal variances () and a different cost per subject for each group.…”

Section: Introductionmentioning

confidence: 99%

Optimum sample size allocation to minimize cost or maximize power for the two‐sample trimmed mean test

Guo¹,

Luh²

2009

Brit J Math & Statis

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When planning a study, sample size determination is one of the most important tasks facing the researcher. The size will depend on the purpose of the study, the cost limitations, and the nature of the data. By specifying the standard deviation ratio and/or the sample size ratio, the present study considers the problem of heterogeneous variances and non-normality for Yuen's two-group test and develops sample size formulas to minimize the total cost or maximize the power of the test. For a given power, the sample size allocation ratio can be manipulated so that the proposed formulas can minimize the total cost, the total sample size, or the sum of total sample size and total cost. On the other hand, for a given total cost, the optimum sample size allocation ratio can maximize the statistical power of the test. After the sample size is determined, the present simulation applies Yuen's test to the sample generated, and then the procedure is validated in terms of Type I errors and power. Simulation results show that the proposed formulas can control Type I errors and achieve the desired power under the various conditions specified. Finally, the implications for determining sample sizes in experimental studies and future research are discussed.

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Approximate sample size formulas for the two‐sample trimmed mean test with unequal variances

Cited by 20 publications

References 28 publications

Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test

Why Psychologists Should by Default Use Welch’s t-test Instead of Student’s t-test

Sample Size Determination for One-and Two-Sample Trimmed Mean Tests

Optimum sample size allocation to minimize cost or maximize power for the two‐sample trimmed mean test

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