Robust Tests for the Equality of Variances
01 Jun 1974-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 69, Iss: 346, pp 364-367
TL;DR: In this paper, alternative formulations of Levene's test statistic for equality of variances are found to be robust under nonnormality, using more robust estimators of central location in place of the mean.
Abstract: Alternative formulations of Levene's test statistic for equality of variances are found to be robust under nonnormality. These statistics use more robust estimators of central location in place of the mean. They are compared with the unmodified Levene's statistic, a jackknife procedure, and a χ2 test suggested by Layard which are all found to be less robust under nonnormality.
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TL;DR: In this paper, distance-based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed, relying on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes.
Abstract: The traditional likelihood-based test for differences in multivariate dispersions is known to be sensitive to nonnormality. It is also impossible to use when the number of variables exceeds the number of observations. Many biological and ecological data sets have many variables, are highly skewed, and are zero-inflated. The traditional test and even some more robust alternatives are also unreasonable in many contexts where measures of dispersion based on a non-Euclidean dissimilarity would be more appropriate. Distance-based tests of homogeneity of multivariate dispersions, which can be based on any dissimilarity measure of choice, are proposed here. They rely on the rotational invariance of either the multivariate centroid or the spatial median to obtain measures of spread using principal coordinate axes. The tests are straightforward multivariate extensions of Levene's test, with P-values obtained either using the traditional F-distribution or using permutation of either least-squares or LAD residuals. Examples illustrate the utility of the approach, including the analysis of stabilizing selection in sparrows, biodiversity of New Zealand fish assemblages, and the response of Indonesian reef corals to an El Nino. Monte Carlo simulations from the real data sets show that the distance-based tests are robust and powerful for relevant alternative hypotheses of real differences in spread.
2,255 citations
TL;DR: The statistical literature on tests to compare treatments after the analysis of variance is reviewed, and the use of these tests in ecology is examined, and particular strategies are recommended.
Abstract: The statistical literature on tests to compare treatments after the analysis of variance is reviewed, and the use of these tests in ecology is examined. Monte Carlo simulations on normal and lognormal data indicate that many of the tests commonly used are inappropriate or inefficient. Particular tests are recommended for unplanned multiple comparisons on the basis of controlling experimentwise type I error rate and providing maximum power. These include tests for parametric and nonparametric cases, equal and unequal sample sizes, homogeneous and heterogeneous variances, non-independent means (repeated measures or adjusted means), and comparing treatments to a control. Formulae and a worked example are provided. The problem of violations of assumptions, especially variance heterogeneity, was investigated using simulations, and particular strategies are recommended. The advantages and use of planned comparisons in ecology are discussed, and the philosophy of hypothesis testing with unplanned multiple comparisons is consid- ered in relation to confidence intervals and statistical estimation.
1,841 citations
TL;DR: The role of kurtosis in testing univariate and multivariate normality, as a measure of departures from normality; in issues of robustness, outliers, and bimodality; and in generalized tests and estimators, as well as limitations of and alternatives to the Kurtosis measure [32,.
Abstract: For symmetric unimodal distributions, positive kurtosis indicates heavy tails and peakedness relative to the normal distribution, whereas negative kurtosis indicates light tails and flatness. Many textbooks, however, describe or illustrate kurtosis incompletely or incorrectly. In this article, kurtosis is illustrated with well-known distributions, and aspects of its interpretation and misinterpretation are discussed. The role of kurtosis in testing univariate and multivariate normality; as a measure of departures from normality; in issues of robustness, outliers, and bimodality; in generalized tests and estimators, as well as limitations of and alternatives to the kurtosis measure [32, are discussed.
1,527 citations
TL;DR: A model in which M. tuberculosis arrests growth, decreases its respiration rate and is resistant to isoniazid, rifampicin and metronidazole is established, which is generated a model with which to search for agents active against persistent M.culosis.
Abstract: The search for new TB drugs that rapidly and effectively sterilize the tissues and are thus able to shorten the duration of chemotherapy from the current 6 months has been hampered by a lack of understanding of the metabolism of the bacterium when in a 'persistent' or latent form. Little is known about the condition in which the bacilli survive, although laboratory models have shown that Mycobacterium tuberculosis can exist in a non-growing, drug-resistant state that may mimic persistence in vivo. Using nutrient starvation, we have established a model in which M. tuberculosis arrests growth, decreases its respiration rate and is resistant to isoniazid, rifampicin and metronidazole. We have used microarray and proteome analysis to investigate the response of M. tuberculosis to nutrient starvation. Proteome analysis of 6-week-starved cultures revealed the induction of several proteins. Microarray analysis enabled us to monitor gene expression during adaptation to nutrient starvation and confirmed the changes seen at the protein level. This has provided evidence for slowdown of the transcription apparatus, energy metabolism, lipid biosynthesis and cell division in addition to induction of the stringent response and several other genes that may play a role in maintaining long-term survival within the host. Thus, we have generated a model with which we can search for agents active against persistent M. tuberculosis and revealed a number of potential targets expressed under these conditions.
1,361 citations
01 Jan 2011
TL;DR: In this paper, a compact, hands-on and step-by-step introduction to quantitative market research techniques is presented and the most important techniques and shows how to translate theoretical choices into SPSS and how to analyze the output.
Abstract: ▶ Compact, hands-on and step-by-step introduction to quantitative market research techniques ▶ Presents the most important techniques and shows how to translate theoretical choices into SPSS and how to analyze the output ▶ Range of education elements such as learning objectives, keywords, self-assessment tests, case studies ▶ Innovative supplementary online concept, including mobile tags, sample datasets and additional cases
911 citations
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1,165 citations
"Robust Tests for the Equality of Va..." refers background or methods in this paper
...The statistics least sensitive to nonnormality were of the same form: each group is randomly divided into subgroups; the standard deviation of each subgroup is calculated; and a one-way ANOVA is calculated between groups using the logarithms of the subgroup variances as data points [2, 3]....
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...Unfortunately, the common F-ratio and Bartlett's test are very sensitive to the assumption that the underlying populations are from a Gaussian distribution [3, 12]....
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TL;DR: In this article, the authors compared four statistics which may be used to test the equality of population means with respect to their robustness under heteroscedasticity, their power, and the overlap of their critical regions.
Abstract: Four statistics which may be used to test the equality of population means are com-pared with respect to their robustness under heteroscedasticity, their power, and the overlap of their critical regions. The four are: the ANOVA F-statistic; a modified F which has the same numerator as the ANOVA but an altered denominator; and two similar statistics proposed by Welch and James which differ primarily in their approximations for their critical values. The critical values proposed by Welch are a better approximation for small sample sizes than that proposed by James. Both Welch's statistic and the modified F are robust under the inequality of variances. The choice between them depends upon the magnitude of the means and their standard errors. When the population variances are equal, the critical region of the modified F more closely approximates that of the ANOVA than does Welch's.
600 citations
"Robust Tests for the Equality of Va..." refers background in this paper
...This may be due to the lack of robustness of the ANOVA when the within group variances are unequal [5]....
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TL;DR: A selective review on robust statistics, centering on estimates of location, but extending into other estimation and testing problems, can be found in this paper, where three important classes of estimates are singled out and some basic heuristic tools for assessing properties of robust estimates (or test statistics) are discussed.
Abstract: This is a selective review on robust statistics, centering on estimates of location, but extending into other estimation and testing problems. After some historical remarks, several possible concepts of robustness are critically reviewed. Three important classes of estimates are singled out and some basic heuristic tools for assessing properties of robust estimates (or test statistics) are discussed: influence curve, jackknifing. Then we give some asymptotic and finite sample minimax results for estimation and testing. The material is complemented by miscellaneous remarks on: computational aspects; other estimates; scale, regression, time series and other estimation problems; some tentative practical recommendations.
557 citations