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Donald W Zimmerman - One of the best experts on this subject based on the ideXlab platform.
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Consequences of choosing samples in hypothesis testing to ensure Homogeneity of Variance.
The British journal of mathematical and statistical psychology, 2013Co-Authors: Donald W ZimmermanAbstract:The two-sample t-test/">Student t test of location was performed on random samples of scores and on rank-transformed scores from normal and non-normal population distributions with unequal Variances. The same test also was performed on scores that had been explicitly selected to have nearly equal sample Variances. The desired Homogeneity of Variance was brought about by repeatedly rejecting pairs of samples having a ratio of standard deviations that exceeded a predetermined cut-off value of 1.1, 1.2, or 1.3, while retaining pairs with ratios less than the cut-off value. Despite this forced conformity with the assumption of equal Variances, the tests on the selected samples were no more robust than tests on unselected samples, and in most cases substantially less robust. Under conditions where sample sizes were unequal, so that Type I error rates were inflated and power curves were atypical, the selection procedure produced still greater inflation and distortion of the power curves.
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A Note on Homogeneity of Variance of Scores and Ranks
The Journal of Experimental Education, 1996Co-Authors: Donald W ZimmermanAbstract:When any two or more sets of scores with unequal Variances are com bined and ranked together as one set, the corresponding sets of ranks inherit the unequal Variances. This fact is well known in the theory of nonparametric statistics, but in practice researchers and applied statisticians frequently overlook its implica tions. Because of this property, familiar nonparametric rank tests cannot overcome effects of heterogeneous Variances of treatment groups in statistical significance test ing. A simulation study demonstrates explicitly that transformation of scores to ranks reduces Variance heterogeneity, although not enough to prevent gross distor tion of the probabilities of type I and type II errors of statistical significance tests, including the t test, the Wilcoxon-Mann-Whitney test, and the van der Waerden, or normal scores, test. The present note also focuses attention on an aspect of the prob lem that is neglected in the literature: The equivalence of various nonparametric tests and their parametric counterparts performed on ranks, or the rank transfor mation concept, provides a rationale for the influence of unequal Variances on test statistics calculated from ranks. MY PURPOSE in the present note is to call attention to a simple property of ranks that researchers and applied statisticians sometimes overlook when using statistical methods based on ranks. This property, noted by Pratt (1964) and Zaremba (1965), and later by many other authors, has implications for the prob lem of Homogeneity of Variance in statistical significance testing. It reveals that substitution of nonparametric methods for parametric tests such as t and F to overcome violation of the assumption, a procedure widely recommended in introductory textbooks, does not accomplish what is intended. Although some textbook authors have become aware of this problem in recent years, they have missed its connection with the rank transformation concept, or the equivalence of various nonparametric tests with parametric counterparts per formed on ranks replacing scores. In the present note, I emphasize that this con
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rank transformations and the power of the student t test and welch t test for non normal populations with unequal Variances
Canadian Journal of Experimental Psychology, 1993Co-Authors: Donald W Zimmerman, Bruno D ZumboAbstract:Abstract Classical studies have disclosed that parametric significance tests such as t and F are robust under violation of Homogeneity of Variance, provided sample sizes are equal. But relatively little is known about effects of unequal Variances on nonparametric counterparts of the tests or about non - normality combined with unequal Variances. In the present computer simulation study, the t-test/">Student t test and the Welch version of the t test (the t' test) were performed first on the initial sample values and then on ranks of the sample values. Unequal Variances together with unequal N's markedly altered the probability of Type I and Type II errors for normal and for eight kinds of non - normal distributions, including mixed - normal, exponential, lognormal, and Cauchy distributions. Substitution of the Welch t' test for the t-test/">Student t test eliminated effects of unequal Variances, but not effects of non - normality. The t test on ranks, which is equivalent to the Mann - Whitney - Wilcoxon test, was more powerful than the t-test/">Student t test for several non - normal distributions, but exhibited a substantial power loss when Variances were unequal. The Welch t' test in conjunction with the rank transformation simultaneously counteracted effects of both non - normality and unequal Variances. Resume Des etudes classiques ont revele que des tests d'hypothese parametriques comme les tests t et F sont rigoureux dans les cas ou l'homogeneite de la Variance est perturbee, pourvu que les echantillons aient la me@me taille. Mais on en sait relativement peu au sujet des effets des Variances inegales sur les versions non parametriques des tests ou au sujet de la non - normalite combinee a des Variances inegales. Dans la presente etude de simulation par ordinateur, le test de Student et la version Welch du test t ont ete appliques d'abord aux valeurs initiales de l'echantillon, puis aux rangs des valeurs. Les Variances inegales jointes aux N inegaux modifiaient nettement la probabilite des erreurs de type I et de type II dans le cas des distributions normales et de huit genres de distributions non normales, dont les distributions mixtes - normales, exponentielles, normales logarithmiques, et des distributions de Cauchy. Le remplacement du test de Student par le test t de Welch a elimine les effets des Variances inegales, mais non ceux de la non - normalite. Le test t effectue sur les rangs, qui equivaut au test de Mann - Whitney - Wilcoxon, etait plus rigoureux que le test de Student pour plusieurs distributions non normales, mais il perdait considerablement de pouvoir lorsque les Variances etaient inegales. Le test t de Welch joint a la transformation en rangs neutralise simultanement les effets de la non - normalite et des Variances inegales.It is well known that parametric significance tests such as t and F are based on an assumption of equality of Variances in treatment groups, or "Homogeneity of Variance," as it is known. For a long time, researchers have been concerned about how violation of this assumption affects statistical tests (see, for example, Box, 1953, Glass, Peckham, & Sanders, 1972, Scheffe, 1959). As a result of numerous simulation studies, as well as theoretical investigations, there is now general agreement that the t and F tests are robust under violation of Homogeneity of Variance, provided sample sizes are equal, although some exceptions have been found recently (Tomarkin & Serlin, 1986).When sample sizes are unequal, the Type I error probabilities of the tests are decidedly influenced by unequal Variances (see, for example, Boneau, 1960, Box, 1953, Games & Howell, 1976, Hsu, 1938, Kohr & Games, 1977, Ramsey, 1980, Rogan & Keselman, 1977, Scheffe, 1959). It has been found that, when the larger Variance is associated with the larger sample size, there is a depression of the Type I error probability, and when the larger Variance is associated with the smaller sample size, there is a spurious elevation of that probability. …
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parametric alternatives to the student t test under violation of normality and Homogeneity of Variance
Perceptual and Motor Skills, 1992Co-Authors: Donald W Zimmerman, Bruno D ZumboAbstract:Introductory statistics textbooks in psychology, education, and social sciences have contributed to the belief that nonparametric tests, such as the Wilcoxon-Mann-Whitney test, are effective against violations of both normality and Homogeneity of Variance. The present paper emphasizes that, although rank methods often are useful when samples are obtained from heavy-tailed, nonnormal distributions, they are influenced by unequal Variances just like parametric tests. Computer programs are now available to perform modified t tests based on unequal sample Variances, in which degrees of freedom and critical values are altered from sample to sample. These procedures, although neglected for many years because they are computationally complex, are far more effective than nonparametric methods in protecting against violation of Homogeneity of Variance.
Bruno D Zumbo - One of the best experts on this subject based on the ideXlab platform.
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rank transformations and the power of the student t test and welch t test for non normal populations with unequal Variances
Canadian Journal of Experimental Psychology, 1993Co-Authors: Donald W Zimmerman, Bruno D ZumboAbstract:Abstract Classical studies have disclosed that parametric significance tests such as t and F are robust under violation of Homogeneity of Variance, provided sample sizes are equal. But relatively little is known about effects of unequal Variances on nonparametric counterparts of the tests or about non - normality combined with unequal Variances. In the present computer simulation study, the t-test/">Student t test and the Welch version of the t test (the t' test) were performed first on the initial sample values and then on ranks of the sample values. Unequal Variances together with unequal N's markedly altered the probability of Type I and Type II errors for normal and for eight kinds of non - normal distributions, including mixed - normal, exponential, lognormal, and Cauchy distributions. Substitution of the Welch t' test for the t-test/">Student t test eliminated effects of unequal Variances, but not effects of non - normality. The t test on ranks, which is equivalent to the Mann - Whitney - Wilcoxon test, was more powerful than the t-test/">Student t test for several non - normal distributions, but exhibited a substantial power loss when Variances were unequal. The Welch t' test in conjunction with the rank transformation simultaneously counteracted effects of both non - normality and unequal Variances. Resume Des etudes classiques ont revele que des tests d'hypothese parametriques comme les tests t et F sont rigoureux dans les cas ou l'homogeneite de la Variance est perturbee, pourvu que les echantillons aient la me@me taille. Mais on en sait relativement peu au sujet des effets des Variances inegales sur les versions non parametriques des tests ou au sujet de la non - normalite combinee a des Variances inegales. Dans la presente etude de simulation par ordinateur, le test de Student et la version Welch du test t ont ete appliques d'abord aux valeurs initiales de l'echantillon, puis aux rangs des valeurs. Les Variances inegales jointes aux N inegaux modifiaient nettement la probabilite des erreurs de type I et de type II dans le cas des distributions normales et de huit genres de distributions non normales, dont les distributions mixtes - normales, exponentielles, normales logarithmiques, et des distributions de Cauchy. Le remplacement du test de Student par le test t de Welch a elimine les effets des Variances inegales, mais non ceux de la non - normalite. Le test t effectue sur les rangs, qui equivaut au test de Mann - Whitney - Wilcoxon, etait plus rigoureux que le test de Student pour plusieurs distributions non normales, mais il perdait considerablement de pouvoir lorsque les Variances etaient inegales. Le test t de Welch joint a la transformation en rangs neutralise simultanement les effets de la non - normalite et des Variances inegales.It is well known that parametric significance tests such as t and F are based on an assumption of equality of Variances in treatment groups, or "Homogeneity of Variance," as it is known. For a long time, researchers have been concerned about how violation of this assumption affects statistical tests (see, for example, Box, 1953, Glass, Peckham, & Sanders, 1972, Scheffe, 1959). As a result of numerous simulation studies, as well as theoretical investigations, there is now general agreement that the t and F tests are robust under violation of Homogeneity of Variance, provided sample sizes are equal, although some exceptions have been found recently (Tomarkin & Serlin, 1986).When sample sizes are unequal, the Type I error probabilities of the tests are decidedly influenced by unequal Variances (see, for example, Boneau, 1960, Box, 1953, Games & Howell, 1976, Hsu, 1938, Kohr & Games, 1977, Ramsey, 1980, Rogan & Keselman, 1977, Scheffe, 1959). It has been found that, when the larger Variance is associated with the larger sample size, there is a depression of the Type I error probability, and when the larger Variance is associated with the smaller sample size, there is a spurious elevation of that probability. …
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parametric alternatives to the student t test under violation of normality and Homogeneity of Variance
Perceptual and Motor Skills, 1992Co-Authors: Donald W Zimmerman, Bruno D ZumboAbstract:Introductory statistics textbooks in psychology, education, and social sciences have contributed to the belief that nonparametric tests, such as the Wilcoxon-Mann-Whitney test, are effective against violations of both normality and Homogeneity of Variance. The present paper emphasizes that, although rank methods often are useful when samples are obtained from heavy-tailed, nonnormal distributions, they are influenced by unequal Variances just like parametric tests. Computer programs are now available to perform modified t tests based on unequal sample Variances, in which degrees of freedom and critical values are altered from sample to sample. These procedures, although neglected for many years because they are computationally complex, are far more effective than nonparametric methods in protecting against violation of Homogeneity of Variance.
Christophe Leys - One of the best experts on this subject based on the ideXlab platform.
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why psychologists should by default use welch s t test instead of student s t test
International Review of Social Psychology, 2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract: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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Additional file to "Why Psychologists Should by Default Use Welch's t-test Instead of Student's t-test." (in press for the International Review of Social Psychology)
2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract:When comparing two independent groups, researchers in Psychology commonly use Student’s t-test. Assumptions of normality and of 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 leads to invalid statistical inferences. Moreover, we argue that the assumption of equal Variances will seldom hold in psychological research and that choosing between Student’s t-test or 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 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 (in press for the International Review of Social Psychology).
2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract:When comparing two independent groups, researchers in Psychology commonly use Student’s t-test. Assumptions of normality and of 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 leads to invalid statistical inferences. Moreover, we argue that the assumption of equal Variances will seldom hold in psychological research and that choosing between Student’s t-test or 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 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.
Gero Stefan Michael Kinzinger - One of the best experts on this subject based on the ideXlab platform.
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Soft tissue profile changes after Functional Mandibular Advancer or Herbst appliance treatment in class II patients
Clinical Oral Investigations, 2018Co-Authors: Jan Hourfar, Jorg Alexander Lisson, Ulrich Gross, Linda Frye, Gero Stefan Michael KinzingerAbstract:Objective The objective of the present study is to compare the effects on soft tissue profile in class II patients after treatment with either “Functional Mandibular Advancer” (FMA) or Herbst appliance. Materials and methods The study included n = 42 patients treated with either FMA ( n = 21) or Herbst appliance ( n = 21) by the same experienced orthodontist. The treatment followed a single-step advancement protocol. Lateral cephalograms were analyzed through a set of customized measurements. The actual therapeutic effect was calculated using data from a growth survey. After testing for normal distribution and Homogeneity of Variance, data were analyzed by one-sample Student’s t tests and independent Student’s t tests. Statistical significance was set at p
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Soft tissue profile changes after Functional Mandibular Advancer or Herbst appliance treatment in class II patients.
Clinical Oral Investigations, 2017Co-Authors: Jan Hourfar, Jorg Alexander Lisson, Ulrich Gross, Linda Frye, Gero Stefan Michael KinzingerAbstract:The objective of the present study is to compare the effects on soft tissue profile in class II patients after treatment with either “Functional Mandibular Advancer” (FMA) or Herbst appliance. The study included n = 42 patients treated with either FMA (n = 21) or Herbst appliance (n = 21) by the same experienced orthodontist. The treatment followed a single-step advancement protocol. Lateral cephalograms were analyzed through a set of customized measurements. The actual therapeutic effect was calculated using data from a growth survey. After testing for normal distribution and Homogeneity of Variance, data were analyzed by one-sample Student’s t tests and independent Student’s t tests. Statistical significance was set at p
Marie Delacre - One of the best experts on this subject based on the ideXlab platform.
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why psychologists should by default use welch s t test instead of student s t test
International Review of Social Psychology, 2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract: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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Additional file to "Why Psychologists Should by Default Use Welch's t-test Instead of Student's t-test." (in press for the International Review of Social Psychology)
2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract:When comparing two independent groups, researchers in Psychology commonly use Student’s t-test. Assumptions of normality and of 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 leads to invalid statistical inferences. Moreover, we argue that the assumption of equal Variances will seldom hold in psychological research and that choosing between Student’s t-test or 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 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 (in press for the International Review of Social Psychology).
2017Co-Authors: Marie Delacre, Daniel Lakens, Christophe LeysAbstract:When comparing two independent groups, researchers in Psychology commonly use Student’s t-test. Assumptions of normality and of 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 leads to invalid statistical inferences. Moreover, we argue that the assumption of equal Variances will seldom hold in psychological research and that choosing between Student’s t-test or 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 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.