31 Circadian Pattern of NSTEMIs Versus STEMIs
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Quartile
Bonferroni correction
Medical record
Chi-square test
The difference in statistical power between the original Bonferroni and five modified Bonferroni procedures that control the overall Type I error rate is examined in the context of a correlation matrix where multiple null hypotheses, H 0 : ρ ij = 0 for all i ≠ j, are tested. Using 50 real correlation matrices reported in educational and psychological journals, a difference in the number of hypotheses rejected of less than 4% was observed among the procedures. When simulated data were used, very small differences were found among the six procedures in detecting at least one true relationship, but in detecting all true relationships the power of the modified Bonferroni procedures exceeded that of the original Bonferroni procedure by at least .18 and by as much as .55 when all null hypotheses were false. The power difference decreased as the number of true relationships decreased. Power differences obtained for the average power were of a much smaller magnitude but still favored the modified Bonferroni procedures. For the five modified Bonferroni procedures, power differences less than .05 were typically observed; the Holm procedure had the lowest power, and the Rom procedure had the highest.
Bonferroni correction
Statistical power
Multiple comparisons problem
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Bonferroni correction
Multiple comparisons problem
p-value
False Discovery Rate
Nominal level
Statistical power
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A modification of the Bonferroni procedure for testing multiple hypotheses is presented. The method, based on the ordered p-values of the individual tests, is less conservative than the classical Bonferroni procedure but is still simple to apply. A simulation study shows that the probability of a type I error of the procedure does not exceed the nominal significance level, α, for a variety of multivariate normal and multivariate gamma test statistics. For independent tests the procedure has type I error probability equal to α. The method appears particularly advantageous over the classical Bonferroni procedure when several highly-correlated test statistics are involved.
Bonferroni correction
Multiple comparisons problem
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Abstract Ecological research often involves multiple statistical tests. It is common practice to employ the Bonferroni technique or its more advanced sequential variant for such multiple tests. Indeed, Moran (Oikos, 100, 2003, 403) found that 13% of ecological papers apply this technique. The seminal paper by Rice (Evolution, 43, 1989, 223) that introduced this technique to the ecological community, is cited to date over 12 000 times. However, these techniques are conservative and some null hypotheses that should be rejected are not. Using order statistics we find that significant results are correlated even when the data consist of independent events. The Bonferroni methods assume independent significant results which results in Type II error with their application. We propose a simple approach, which we term the correlated Bonferroni technique, to rectify this shortcoming, which reduces rejection of significant results. Ecologists may be able to confirm the significance of their results while they are unable to confirm it using the original Bonferroni technique. Researchers may revisit their projects and find that significant results were mistakenly ignored. We provide an Excel file (see supplement) that researchers can easily use. We illustrate the correlated Bonferroni technique with an example.
Bonferroni correction
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Null (SQL)
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In a given study, if many related outcomes are tested for statistical significance, one or more outcomes may emerge significant at the P < 0.05 level not because they are truly significant in the population but because of chance. The larger the number of statistical tests performed, the greater the risk that some of the significant findings are significant because of chance. There are many ways to protect against such false positive or Type 1 errors. The simplest way is to set a more stringent threshold for statistical significance than P < 0.05. This can be done using either the Bonferroni or the Hochberg correction. Using the Bonferroni correction, 0.05 is divided by the number of statistical tests being performed and the result is set as the critical P value for statistical significance. Using the Hochberg correction, the P values obtained from the different statistical tests are arranged in descending order of magnitude, and each P value is assessed for significance against progressively more stringent levels for significance. The Bonferroni and Hochberg procedures are explained with the help of examples.
Bonferroni correction
Multiple comparisons problem
p-value
Statistical Analysis
Statistical power
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Bonferroni correction procedures are commonly used for performing multiple hypothesis tests in linear data problems. Moreover, several improved Bonferroni type procedures have been proposed and shown that they attain a type-I error rate (α) better than the classical Bonferroni approach, but all of the studies are considered for linear data problems. The circular data analysis is a developing field of statistics that lacks similar studies, and also lacks computer programs to implement Bonferroni procedures. The aim of this study is to perform a comparative study of improved Bonferroni procedures and also to provide a computer program in R that performs classical and improved Bonferroni procedures for circular data problems.
Bonferroni correction
Multiple comparisons problem
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When the differences of average levels among several groups are statistically significant, the multiple comparisons should be carried out. Using α=0.05 directly as significance level in comparison between each pair may result in too high typeⅠ error probability. Bonferroni method is one of the most common methods for multiple comparisons, which could reduce the accumulated typeⅠ error probability by adjusting α level. In this report, the fundamental principle of Bonferroni method is explained and examples are presented in some specific situations. This article is supposed to guide researchers to utilize Bonferroni method correctly in multiple comparisons.
Bonferroni correction
Multiple comparisons problem
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Multiple hypothesis testing is commonly used in genome research such as genome-wide studies and gene expression data analysis (Lin, 2005). The widely used Bonferroni procedure controls the family-wise error rate (FWER) for multiple hypothesis testing, but has limited statistical power as the number of hypotheses tested increases. The power of multiple testing procedures can be increased by using weighted p-values (Genovese et al., 2006). The weights for the p-values can be estimated by using certain prior information. Wasserman and Roeder (2006) described a weighted Bonferroni procedure, which incorporates weighted p-values into the Bonferroni procedure, and Rubin et al. (2006) and Wasserman and Roeder (2006) estimated the optimal weights that maximize the power of the weighted Bonferroni procedure under the assumption that the means of the test statistics in the multiple testing are known (these weights are called optimal Bonferroni weights). This weighted Bonferroni procedure controls FWER and can have higher power than the Bonferroni procedure, especially when the optimal Bonferroni weights are used. To further improve the power of the weighted Bonferroni procedure, first we propose a weighted Šidák procedure that incorporates weighted p-values into the Šidák procedure, and then we estimate the optimal weights that maximize the average power of the weighted Šidák procedure under the assumption that the means of the test statistics in the multiple testing are known (these weights are called optimal Šidák weights). This weighted Šidák procedure can have higher power than the weighted Bonferroni procedure. Second, we develop a generalized sequential (GS) Šidák procedure that incorporates weighted p-values into the sequential Šidák procedure (Scherrer, 1984). This GS Šidák procedure is an extension of and has higher power than the GS Bonferroni procedure of Holm (1979). Finally, under the assumption that the means of the test statistics in the multiple testing are known, we incorporate the optimal Šidák weights and the optimal Bonferroni weights into the GS Šidák procedure and the GS Bonferroni procedure, respectively. Theoretical proof and/or simulation studies show that the GS Šidák procedure can have higher power than the GS Bonferroni procedure when their corresponding optimal weights are used, and that both of these GS procedures can have much higher power than the weighted Šidák and the weighted Bonferroni procedures. All proposed procedures control the FWER well and are useful when prior information is available to estimate the weights.
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