Showing papers by "Ingram Olkin published in 1985"

Statistical Methods for Meta-Analysis

Abstract: Preface. Introduction. Data Sets. Tests of Statistical Significance of Combined Results. Vote-Counting Methods. Estimation of a Single Effect Size: Parametric and Nonparametric Methods. Parametric Estimation of Effect Size from a Series of Experiments. Fitting Parametric Fixed Effect Models to Effect Sizes: Categorical Methods. Fitting Parametric Fixed Effect Models to Effect Sizes: General Linear Models. Random Effects Models for Effect Sizes. Multivariate Models for Effect Sizes. Combining Estimates of Correlation Coefficients. Diagnostic Procedures for Research Synthesis Models. Clustering Estimates of Effect Magnitude. Estimation of Effect Size When Not All Study Outcomes Are Observed. Meta-Analysis in the Physical and Biological Sciences. Appendix. References. Index.

Estimation of a Single Effect Size: Parametric and Nonparametric Methods

Abstract: This chapter focuses on the study of parametric and nonparametric methods for estimating the effect size (standardized mean difference) from a single experiment. It is important to recognize that estimating and interpreting a common effect size is based on the belief that the population effect size is actually the same across studies. Otherwise, estimating a mean effect may obscure important differences between the studies. The chapter discusses several alternative point estimators of the effect size δ from a single two-group experiment. These estimators are based on the sample standardized mean difference but differ by multiplicative constants that depend on the sample sizes involved. Although the estimates have identical large sample properties, they generally differ in terms of small sample properties. The statistical properties of estimators of effect size depend on the model for the observations in the experiment. A convenient and often realistic model is to assume that the observations are independently normally distributed within groups of the experiment.

A Family of Bivariate Distributions Generated by the Bivariate Bernoulli Distribution

Abstract: If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

Parametric Estimation of Effect Size from a Series of Experiments

Abstract: This chapter discusses the parametric estimation of effect size from a series of experiments. It presents two methods for obtaining optimal combinations of estimates of effect size from a series of studies: (1) a direct weighted linear combination of estimators from different studies and (2) a maximum likelihood estimator. Both the estimators have the same asymptotic distribution and, hence, are asymptotically equivalent. The chapter highlights two other methods that involve a transformation of the effect size estimates. Statistical properties of procedures for combining results from a series of experiments depend on the structural model for the results of the experiments. There are several alternative methods for estimating the effect size from a large series of studies, each of which has a small sample size. One method is based on weighted linear combinations of estimators. The second method is analogous to maximum likelihood and is based on a suggestion of Neyman and Scott.

Multivariate Distributions of Directional Wind Speeds

Abstract: The probability of failure of a structure or structural element subjected to wind forces depends, in large part, on the distribution of extreme wind speeds acting on the structure. In the past, distributions of extreme wind speeds were based on extreme wind data without regard to wind direction, and probabilities of failure were computed accordingly. A method is presented herein for calculating failure probabilities by using directional wind speed data as obtained from weather station records. The method takes advantage of the weak correlations found among wind speeds from different directions and the property of extreme value random variables that zero correlation implies statistical independence. The method is applicable to any type of structure, including structures exhibiting aerodynamic amplification or aeroelastic effects. In addition, it is shown that, in practice, the necessary distributions can be estimated almost as accurately from data obtained from readily available published documents as from...

Inequalities for the trace function

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Diagnostic Procedures for Research Synthesis Models

Abstract: This chapter describes a few methods for discovering potential sources of poor fit to fixed effects models. These diagnostic procedures provide methods for recognizing one or more estimates that deviate greatly from their expected values if the model were correct. These procedures often point to studies that differ from others in ways that are remediable; for example, they may represent mistakes in coding or calculation. Sometimes the diagnostic procedures point to sets of studies that differ in a collective way that suggests a new explanatory variable. Sometimes a study is an outlier that cannot be explained by an obvious characteristic of the study. The analysis of data containing a few observations that are outliers is a complicated task. It invariably requires the use of good judgment and decisions that are, in some sense, compromises. There are two extreme positions on dealing with outliers: (1) data are sacred, and no datum point (study) should ever be set aside for any reason and (2) data should be tested for outliers, and data points (studies) that fail to conform to the hypothesized model should be removed.

Vote-Counting Methods

Abstract: Publisher Summary This chapter discusses vote-counting methods. The vote-counting methods described in this chapter differ from the vote-counting or box-score techniques that are sometimes used in research reviews. The conventional vote-counting or box-score review attempts to draw inferences about the existence of treatment effects by sorting studies into categories according to the outcome of tests of hypotheses reported in the studies. The inference procedures used in conventional vote-counting procedures are inherently flawed and are likely to be misleading. The chapter discusses statistical methods for the analysis of vote-count data that provide explicit estimates of effect magnitude parameters such as the correlation coefficient or standardized mean difference. Vote-counting methods are partially parametric in the sense that they permit inferences about scale-invariant indices of effect size. Conventional vote-counting or box-score methodology uses the outcome of the test of significance in a series of replicated studies to draw conclusions about the magnitude of the treatment effect.

Estimation of Effect Size When Not All Study Outcomes are Observed

Abstract: The properties of statistical procedures depend on the availability of unrestricted samples of effect size estimates If the effect size estimates available to the investigator are systematically biased, special statistical procedures are needed to take account of the biasing mechanism The chapter discusses the problem created by the censoring of effect size estimates corresponding to statistically nonsignificant results It presents some evidence on the existence of the sampling bias The chapter also presents a statistical model for studying the effects of the sampling bias and discusses the consequences of such bias on the estimation of effect size It highlights the maximum likelihood estimates of effect size under the model of sampling bias The chapter discusses a combination of estimates from several experiments under the model of sampling bias in addition to the ways in which the results of this chapter can be used to draw conclusions when sampling bias may exist

Tests of Statistical Significance of Combined Results

Abstract: This chapter presents tests of statistical significance of combined results. It reviews so-called omnibus statistical methods for testing the statistical significance of combined results. The procedures are called omnibus or nonparametric because they do not depend on the form of the underlying data but only on the exact significance levels commonly called p -values. A key point is that observed p-values derived from continuous test statistics have a uniform distribution under the null hypothesis irrespective of the test statistics or distribution from which they arise. Therefore, combined significance tests that depend on the underlying data only through p -values are nonparametric in the sense that they do not depend on the parametric form of the data. Omnibus tests of statistical significance can almost always be applied to data collected for the synthesis of social science research. However, they do not always provide a test of the hypothesis of interest to the research reviewer. Such tests do not support inferences about the average magnitude of effects or about consistency of results across studies.

Combining Estimates of Correlation Coefficients

Abstract: This chapter discusses the combining estimates of correlation coefficients. Correlation coefficients have been used extensively as an index of the relationship between two normally distributed variables. As the correlation coefficient is a scale-free measure of the relationship between variables, it is invariant under substitution of different but linearly equitable measures of the same construct. The correlation coefficient is, therefore, a natural candidate as an index of effect magnitude suitable for cumulation across studies. The chapter explains models in which correlations depend on a linear combination of known predictor variables. These predictor variables can be either discrete or continuous, and represent study characteristics that are likely to be related to the correlation. Often the predictor variables arise from discrete categorizations of studies such as sex or socioeconomic status of subjects. On other times, the predictors are continuous variables, such as the time elapsed between certain measurements. The chapter presents methods that are analogous to multiple linear regression, and any coding scheme used in ordinary linear regression can be used in this context.

Fitting Parametric Fixed Effect Models to Effect Sizes: Categorical Models

Abstract: This chapter presents a method for fitting models to effect sizes when the independent variables are categorical. This method is essentially an analogue to the analysis of variance for effect sizes. The method is useful when it is possible to group studies that have similar characteristics, such as experimental conditions or stimuli. It provides valid large sample tests for differences in average effect sizes between classes and also tests for the homogeneity of effect size within classes. The within-class test of homogeneity can be used as a test for the specification of the categorical model. This method is a special case of a method of fitting general linear models to effect sizes. The technique presented in this chapter is quite direct. The first step is to determine, by means of a statistical test, whether all studies share a common effect size. If the hypothesis that all the effect sizes are equal is rejected, then the experimenter breaks the series of studies into groups or a priori classes, in such a way that the effect sizes within each class are approximately equal.

Clustering Estimates of Effect Magnitude

Abstract: This chapter discusses two methods for grouping estimates of effect magnitude into homogeneous classes proposed by Hedges and Olkin One procedure decomposes a set of effect magnitude indices into disjoint or nonoverlapping classes, whereas in another procedure, the decomposition is into overlapping groups In either case, methods are provided for determining statistical significance levels of the clusters Both procedures are based on clustering theory for standard normal random variables However, correlations and standardized mean differences are generally not normally distributed except in large samples The chapter highlights the use of large sample theory for the transformation of the estimators to approximately standard normal variates It explains the clustering procedures for arbitrary unit normal variates and discusses the application of those methods to correlations and effect sizes

Fitting Parametric Fixed Effect Models to Effect Sizes: General Linear Models

Abstract: This chapter presents methods for fitting models to effect size data when the models include either continuous or discrete independent variables. These methods are analogues to multiple regression analyses for effect sizes and provide estimates of the parameters of the model, large sample tests of significance, and an explicit test of the specification of the model. Thus, it is possible to test whether a model adequately explains the observed variability in effect size estimates. Natural tests of model specification are not available in the context of usual normal theory regression analysis. The remarkable fact that model specification can be tested in research synthesis has important advantages for the research reviewer. If tests for model specification fail to reject the specification of an explanatory model for effect sizes, then the reviewer is in a strong position to assert that the results of the series of experiments are explained. The chapter presents two techniques for fitting general linear models to effect sizes from a series of independent experiments.

Multivariate Models for Effect Sizes

Abstract: This chapter discusses multivariate models for effect sizes. A key feature of multivariate procedures is that they deal with all the correlated effect sizes simultaneously. A disadvantage of these multivariate techniques is that they usually require knowledge of the correlations between variables—information that is not always available. In a few instances, such correlations may actually be available. For example, test-norming studies may provide very good estimates of correlations between subscales of psychological tests. Such estimates can be treated as known values to provide the correlations necessary to use the methods given in this chapter. The chapter presents the multivariate distribution of a vector of effect sizes derived from correlated observations. It also discusses the estimation of a common effect size from a vector of correlated estimates. The chapter explores methods for estimating effect sizes from a series of studies in which a few studies provide several correlated estimates and other studies provide a single estimate.