Topic
Studentized range
About: Studentized range is a research topic. Over the lifetime, 799 publications have been published within this topic receiving 44307 citations.
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Papers
TL;DR: In this article, the authors proposed a new estimation method by incorporating the sample size and compared the estimators of the sample mean and standard deviation under all three scenarios and presented some suggestions on which scenario is preferred in real-world applications.
Abstract: In systematic reviews and meta-analysis, researchers often pool the results of the sample mean and standard deviation from a set of similar clinical trials. A number of the trials, however, reported the study using the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, one may have to estimate the sample mean and standard deviation for such trials. In this paper, we propose to improve the existing literature in several directions. First, we show that the sample standard deviation estimation in Hozo et al.’s method (BMC Med Res Methodol 5:13, 2005) has some serious limitations and is always less satisfactory in practice. Inspired by this, we propose a new estimation method by incorporating the sample size. Second, we systematically study the sample mean and standard deviation estimation problem under several other interesting settings where the interquartile range is also available for the trials. We demonstrate the performance of the proposed methods through simulation studies for the three frequently encountered scenarios, respectively. For the first two scenarios, our method greatly improves existing methods and provides a nearly unbiased estimate of the true sample standard deviation for normal data and a slightly biased estimate for skewed data. For the third scenario, our method still performs very well for both normal data and skewed data. Furthermore, we compare the estimators of the sample mean and standard deviation under all three scenarios and present some suggestions on which scenario is preferred in real-world applications. In this paper, we discuss different approximation methods in the estimation of the sample mean and standard deviation and propose some new estimation methods to improve the existing literature. We conclude our work with a summary table (an Excel spread sheet including all formulas) that serves as a comprehensive guidance for performing meta-analysis in different situations.
8,014 citations
Book•
19 Jan 2007
TL;DR: This handbook provides you with everything you need to know about parametric and nonparametric statistical procedures, and helps you choose the best test for your data, interpret the results, and better evaluate the research of others.
Abstract: With more than 500 pages of new material, the Handbook of Parametric and Nonparametric Statistical Procedures, Fourth Edition carries on the esteemed tradition of the previous editions, providing up-to-date, in-depth coverage of now more than 160 statistical procedures. The book also discusses both theoretical and practical statistical topics, such as experimental design, experimental control, and statistical analysis. New to the Fourth Edition Multivariate statistics including matrix algebra, multiple regression, Hotellings T2, MANOVA, MANCOVA, discriminant function analysis, canonical correlation, logistic regression, and principal components/factor analysis Clinical trials, survival analysis, tests of equivalence, analysis of censored data, and analytical procedures for crossover design Regression diagnostics that include the Durbin-Watson test Log-linear analysis of contingency tables, Mantel-Haenszel analysis of multiple 2 2 contingency tables, trend analysis, and analysis of variance for a Latin square design Levene and Brown-Forsythe tests for evaluating homogeneity of variance, the Jarque-Bera test of normality, and the extreme studentized deviate test for identifying outliers Confidence intervals for computing the population median and the difference between two population medians The relationship between exponential and Poisson distribution Eliminating the need to search across numerous books, this handbook provides you with everything you need to know about parametric and nonparametric statistical procedures. It helps you choose the best test for your data, interpret the results, and better evaluate the research of others.
5,387 citations
Book•
1 Jan 1966
TL;DR: In this article, the authors presented a case of two means regression method for the family error rate, which was used to estimate the probability of a family having a nonzero family error.
Abstract: 1 Introduction.- 1 Case of two means.- 2 Error rates.- 2.1 Probability of a nonzero family error rate.- 2.2 Expected family error rate.- 2.3 Allocation of error.- 3 Basic techniques.- 3.1 Repeated normal statistics.- 3.2 Maximum modulus (Tukey).- 3.3 Bonferroni normal statistics.- 3.4 ?2 projections (Scheffe).- 3.5 Allocation.- 3.6 Multiple modulus tests (Duncan).- 3.7 Least significant difference test (Fisher).- 4 p-mean significance levels.- 5 Families.- 2 Normal Univariate Techniques.- 1 Studentized range (Tukey).- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 F projections (Scheffe)48.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 Bonferroni t statistics.- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 Studentized maximum modulus.- 4.1 Method.- 4.2 Applications.- 4.3 Comparison.- 4.4 Derivation.- 4.5 Distributions and tables.- 5 Many-one t statistics76.- 5.1 Method.- 5.2 Applications.- 5.3 Comparison.- 5.4 Derivation.- 5.5 Distributions and tables.- 6 Multiple range tests (Duncan).- 6.1 Method.- 6.2 Applications.- 6.3 Comparison.- 6.4 Derivation.- 6.5 Distributions and tables.- 7 Least significant difference test (Fisher).- 7.1 Method.- 7.2 Applications.- 7.3 Comparison.- 7.4 Derivation.- 7.5 Distributions and tables.- 8 Other techniques.- 8.1 Tukey's gap-straggler-variance test.- 8.2 Shortcut methods.- 8.3 Multiple F tests.- 8.4 Two-sample confidence intervals of predetermined length.- 8.5 An improved Bonferroni inequality.- 9 Power.- 10 Robustness.- 3 Regression Techniques.- 1 Regression surface confidence bands.- 1.1 Method.- 1.2 Comparison.- 1.3 Derivation.- 2 Prediction.- 2.1 Method.- 2.2 Comparison.- 2.3 Derivation.- 3 Discrimination.- 3.1 Method.- 3.2 Comparison.- 3.3 Derivation.- 4 Other techniques.- 4.1 Linear confidence bands.- 4.2 Tolerance intervals.- 4.3 Unlimited discrimination intervals.- 4 Nonparametric Techniques.- 1 Many-one sign statistics (Steel).- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 k-sample sign statistics.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 Many-one rank statistics (Steel).- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 k-sample rank statistics.- 4.1 Method.- 4.2 Applications.- 4.3 Comparison.- 4.4 Derivation.- 4.5 Distributions and tables.- 5 Signed-rank statistics.- 6 Kruskal-Wallis rank statistics (Nemenyi).- 6.1 Method.- 6.2 Applications.- 6.3 Comparison.- 6.4 Derivation.- 6.5 Distributions and tables.- 7 Friedman rank statistics (Nemenyi).- 7.1 Method.- 7.2 Applications.- 7.3 Comparison.- 7.4 Derivation.- 7.5 Distributions and tables.- 8 Other techniques.- 8.1 Permutation tests.- 8.2 Median tests (Nemenyi).- 8.3 Kolmogorov-Smirnov statistics.- 5 Multivariate Techniques.- 1 Single population covariance scalar unknown.- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 Single population covariance matrix unknown.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 k populations covariance matrix unknown.- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 Other techniques.- 4.1 Variances known covariances unknown.- 4.2 Variance-covariance intervals.- 4.3 Two-sample confidence intervals of predetermined length.- 6 Miscellaneous Techniques.- 1 Outlier detection.- 2 Multinomial populations.- 2.1 Single population.- 2.2 Several populations.- 2.3 Cross-product ratios.- 2.4 Logistic response curves.- 3 Equality of variances.- 4 Periodogram analysis.- 5 Alternative approaches: selection, ranking, slippage.- A Strong Law For The Expected Error Rate.- B TABLES.- I Percentage points of the studentized range.- II Percentage points of the Bonferroni t statistic.- III Percentage points of the studentized maximum modulus.- IV Percentage points of the many-one t statistics.- V Percentage points of the Duncan multiple range test.- VI Percentage points of the many-one sign statistics.- VIII Percentage points of the many-one rank statistics.- IX Percentage points of the k-sample rank statistics.- Developments in Multiple Comparisons 1966-).- 3.5 Allocation.- 3.6 Multiple modulus tests (Duncan).- 3.7 Least significant difference test (Fisher).- 4 p-mean significance levels.- 5 Families.- 2 Normal Univariate Techniques.- 1 Studentized range (Tukey).- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 F projections (Scheffe)48.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 Bonferroni t statistics.- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 Studentized maximum modulus.- 4.1 Method.- 4.2 Applications.- 4.3 Comparison.- 4.4 Derivation.- 4.5 Distributions and tables.- 5 Many-one t statistics76.- 5.1 Method.- 5.2 Applications.- 5.3 Comparison.- 5.4 Derivation.- 5.5 Distributions and tables.- 6 Multiple range tests (Duncan).- 6.1 Method.- 6.2 Applications.- 6.3 Comparison.- 6.4 Derivation.- 6.5 Distributions and tables.- 7 Least significant difference test (Fisher).- 7.1 Method.- 7.2 Applications.- 7.3 Comparison.- 7.4 Derivation.- 7.5 Distributions and tables.- 8 Other techniques.- 8.1 Tukey's gap-straggler-variance test.- 8.2 Shortcut methods.- 8.3 Multiple F tests.- 8.4 Two-sample confidence intervals of predetermined length.- 8.5 An improved Bonferroni inequality.- 9 Power.- 10 Robustness.- 3 Regression Techniques.- 1 Regression surface confidence bands.- 1.1 Method.- 1.2 Comparison.- 1.3 Derivation.- 2 Prediction.- 2.1 Method.- 2.2 Comparison.- 2.3 Derivation.- 3 Discrimination.- 3.1 Method.- 3.2 Comparison.- 3.3 Derivation.- 4 Other techniques.- 4.1 Linear confidence bands.- 4.2 Tolerance intervals.- 4.3 Unlimited discrimination intervals.- 4 Nonparametric Techniques.- 1 Many-one sign statistics (Steel).- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 k-sample sign statistics.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 Many-one rank statistics (Steel).- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 k-sample rank statistics.- 4.1 Method.- 4.2 Applications.- 4.3 Comparison.- 4.4 Derivation.- 4.5 Distributions and tables.- 5 Signed-rank statistics.- 6 Kruskal-Wallis rank statistics (Nemenyi).- 6.1 Method.- 6.2 Applications.- 6.3 Comparison.- 6.4 Derivation.- 6.5 Distributions and tables.- 7 Friedman rank statistics (Nemenyi).- 7.1 Method.- 7.2 Applications.- 7.3 Comparison.- 7.4 Derivation.- 7.5 Distributions and tables.- 8 Other techniques.- 8.1 Permutation tests.- 8.2 Median tests (Nemenyi).- 8.3 Kolmogorov-Smirnov statistics.- 5 Multivariate Techniques.- 1 Single population covariance scalar unknown.- 1.1 Method.- 1.2 Applications.- 1.3 Comparison.- 1.4 Derivation.- 1.5 Distributions and tables.- 2 Single population covariance matrix unknown.- 2.1 Method.- 2.2 Applications.- 2.3 Comparison.- 2.4 Derivation.- 2.5 Distributions and tables.- 3 k populations covariance matrix unknown.- 3.1 Method.- 3.2 Applications.- 3.3 Comparison.- 3.4 Derivation.- 3.5 Distributions and tables.- 4 Other techniques.- 4.1 Variances known covariances unknown.- 4.2 Variance-covariance intervals.- 4.3 Two-sample confidence intervals of predetermined length.- 6 Miscellaneous Techniques.- 1 Outlier detection.- 2 Multinomial populations.- 2.1 Single population.- 2.2 Several populations.- 2.3 Cross-product ratios.- 2.4 Logistic response curves.- 3 Equality of variances.- 4 Periodogram analysis.- 5 Alternative approaches: selection, ranking, slippage.- A Strong Law For The Expected Error Rate.- B TABLES.- I Percentage points of the studentized range.- II Percentage points of the Bonferroni t statistic.- III Percentage points of the studentized maximum modulus.- IV Percentage points of the many-one t statistics.- V Percentage points of the Duncan multiple range test.- VI Percentage points of the many-one sign statistics.- VIII Percentage points of the many-one rank statistics.- IX Percentage points of the k-sample rank statistics.- Developments in Multiple Comparisons 1966-1976.- 1 Introduction.- 2 Papers of special interest.- 2.1 Probability inequalities.- 2.2 Methods for unbalanced ANOVA.- 2.3 Conditional confidence levels.- 2.4 Empirical Bayes approach.- 2.5 Confidence bands in regression.- 3 References.- 4 Bibliography 1966-1976.- 4.1 Survey articles.- 4.2 Probability inequalities.- 4.3 Tables.- 4.4 Normal multifactor methods.- 4.5 Regression.- 4.6 Categorical data.- 4.7 Nonparametric techniques.- 4.8 Multivariate methods.- 4.9 Miscellaneous.- 4.10 Pre-1966 articles missed in [6].- 4.11 Late additions.- 5 List of journals scanned.- Addendum New Table of the Studentized Maximum Modulus.- Table IIIA Percentage points of the studentized maximum modulus.- Author Index.
4,763 citations
TL;DR: In this paper, the authors considered the possibility of picking in advance a number (say m) of linear contrasts among k means, and then estimating these m linear contrasts by confidence intervals based on a Student t statistic, in such a way that the overall confidence level for the m intervals is greater than or equal to a preassigned value.
Abstract: Methods for constructing simultaneous confidence intervals for all possible linear contrasts among several means of normally distributed variables have been given by Scheffe and Tukey. In this paper the possibility is considered of picking in advance a number (say m) of linear contrasts among k means, and then estimating these m linear contrasts by confidence intervals based on a Student t statistic, in such a way that the overall confidence level for the m intervals is greater than or equal to a preassigned value. It is found that for some values of k, and for m not too large, intervals obtained in this way are shorter than those using the F distribution or the Studentized range. When this is so, the experimenter may be willing to select the linear combinations in advance which he wishes to estimate in order to have m shorter intervals instead of an infinite number of longer intervals.
4,466 citations
Abstract: In many fields of research, one is faced with the task of comparing the effects of treatments which have been replicated unequally. This happens for a number of reasons. In an experiment on animals, some may get sick and have to be removed from the experiment. In some experiments, the amount of material available for certain treatments may not be as much as for other treatments. If the experimenter has specified orthogonal contrasts that he is interested in before he runs the experiment, one can test the various treatment effects by an F-test after the treatment sum of squares has been partitioned into individual degrees of freedom for each orthogonal contrast. If the experimenter has not specified orthogonal contrasts, one is faced with the problem of deciding which treatments are significantly different. Several writers, including Duncan, Keuls, Newman, and Tukey, have developed multiple range tests to show differences among treatments that have been replicated the same number of times and when nothing was specified concerning the treatments. Duncan [1] compares the above methods and gives citations. This extension to unequal numbers of replications will be exemplified with reference to Duncan's "New Multiple Range Test," but is applicable to any of the above writers' tests; all one has to do is use their tabled ranges. In Duncan's test for an equal number of replications, the difference between any two ranked means is significant if the difference exceeds a shortest significant range. This shortest significant range is designated by R, and is obtained by multiplying the standard error of a mean, s,, by a given value, zn2, obtained from a table of significant studentized ranges which Duncan has tabled for both the 5% and 1% test. In Duncan's terminology, n2 is the degrees of freedom of the error mean square and p = 1, 2, * *, t is the number of means concerned. Consider an experiment with five treatments, A, B. C, D, and E, each replicated n times. Suppose on ranking the means from low to high one obtains
3,259 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 8 |
| 2024 | 8 |
| 2023 | 15 |
| 2022 | 19 |
| 2021 | 11 |
| 2020 | 11 |