Cochrane Reviews have recently started including the quantity I 2 to help readers assess the consistency of the results of studies in meta-analyses. What does this new quantity mean, and why is assessment of heterogeneity so important to clinical practice? 

Systematic reviews and meta-analyses can provide convincing and reliable evidence relevant to many aspects of medicine and health care.1 Their value is especially clear when the results of the studies they include show clinically important effects of similar magnitude. However, the conclusions are less clear when the included studies have differing results. In an attempt to establish whether studies are consistent, reports of meta-analyses commonly present a statistical test of heterogeneity. The test seeks to determine whether there are genuine differences underlying the results of the studies (heterogeneity), or whether the variation in findings is compatible with chance alone (homogeneity). However, the test is susceptible to the number of trials included in the meta-analysis. We have developed a new quantity, I 2, which we believe gives a better measure of the consistency between trials in a meta-analysis.

Assessment of the consistency of effects across studies is an essential part of meta-analysis. Unless we know how consistent the results of studies are, we cannot determine the generalisability of the findings of the meta-analysis. Indeed, several hierarchical systems for grading evidence state that the results of studies must be consistent or homogeneous to obtain the highest grading.2–4

Tests for heterogeneity are commonly used to decide on methods for combining studies and for concluding consistency or inconsistency of findings.5 6 But what does the test achieve in practice, and how should the resulting P values be interpreted?

A test for heterogeneity examines the null hypothesis that all studies are evaluating the same effect. The usual test statistic …

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Measuring inconsistency in meta-analyses

High-throughput sequencing assays such as RNA-Seq, ChIP-Seq or barcode counting provide quantitative readouts in the form of count data. To infer differential signal in such data correctly and with good statistical power, estimation of data variability throughout the dynamic range and a suitable error model are required. We propose a method based on the negative binomial distribution, with variance and mean linked by local regression and present an implementation, DESeq, as an R/Bioconductor package.

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Differential expression analysis for sequence count data.

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

1 Compositional data: some challenging problems.- 1.1 Introduction.- 1.2 Geochemical compositions of rocks.- 1.3 Sediments at different depths.- 1.4 Ternary diagrams.- 1.5 Partial analyses and subcompositions.- 1.6 Supervisory behaviour.- 1.7 Household budget surveys.- 1.8 Steroid metabolite patterns in adults and children.- 1.9 Activity patterns of a statistician.- 1.10 Calibration of white-cell compositions.- 1.11 Fruit evaluation.- 1.12 Firework mixtures.- 1.13 Clam ecology.- 1.14 Bibliographic notes.- Problems.- 2 The simplex as sample space.- 2.1 Choice of sample space.- 2.2 Compositions and simplexes.- 2.3 Spaces, vectors, matrices.- 2.4 Bases and compositions.- 2.5 Subcompositions.- 2.6 Amalgamations.- 2.7 Partitions.- 2.8 Perturbations.- 2.9 Geometrical representations of compositional data.- 2.10 Bibliographic notes.- Problems.- 3 The special difficulties of compositional data analysis.- 3.1 Introduction.- 3.2 High dimensionality.- 3.3 Absence of an interpretable covariance structure.- 3.4 Difficulty of parametric modelling.- 3.5 The mixture variation difficulty.- 3.6 Bibliographic notes.- Problems.- 4 Covariance structure.- 4.1 Fundamentals.- 4.2 Specification of the covariance structure.- 4.3 The compositional variation array.- 4.4 Recovery of the compositional variation array from the crude mean vector and covariance matrix.- 4.5 Subcompositional analysis.- 4.6 Matrix specifications of covariance structures.- 4.7 Some important elementary matrices.- 4.8 Relationships between the matrix specifications.- 4.9 Estimated matrices for hongite compositions.- 4.10 Logratios and logcontrasts.- 4.11 Covariance structure of a basis.- 4.12 Commentary.- 4.13 Bibliographic notes.- Problems.- 5 Properties of matrix covariance specifications.- 5.1 Logratio notation.- 5.2 Logcontrast variances and covariances.- 5.3 Permutations.- 5.4 Properties of P and QP matrices.- 5.5 Permutation invariants involving ?.- 5.6 Covariance matrix inverses.- 5.7 Subcompositions.- 5.8 Equivalence of characteristics of ?, ?, ?.- 5.9 Logratio-uncorrelated compositions.- 5.10 Isotropic covariance structures.- 5.11 Bibliographic notes.- Problems.- 6 Logistic normal distributions on the simplex.- 6.1 Introduction.- 6.2 The additive logistic normal class.- 6.3 Density function.- 6.4 Moment properties.- 6.5 Composition of a lognormal basis.- 6.6 Class-preserving properties.- 6.7 Conditional subcompositional properties.- 6.8 Perturbation properties.- 6.9 A central limit theorem.- 6.10 A characterization by logcontrasts.- 6.11 Relationships with the Dirichlet class.- 6.12 Potential for statistical analysis.- 6.13 The multiplicative logistic normal class.- 6.14 Partitioned logistic normal classes.- 6.15 Some notation.- 6.16 Bibliographic notes.- Problems.- 7 Logratio analysis of compositions.- 7.1 Introduction.- 7.2 Estimation of ? and ?.- 7.3 Validation: tests of logistic normality.- 7.4 Hypothesis testing strategy and techniques.- 7.5 Testing hypotheses about ? and ?.- 7.6 Logratio linear modelling.- 7.7 Testing logratio linear hypotheses.- 7.8 Further aspects of logratio linear modelling.- 7.9 An application of logratio linear modelling.- 7.10 Predictive distributions, atypicality indices and outliers.- 7.11 Statistical discrimination.- 7.12 Conditional compositional modelling.- 7.13 Bibliographic notes.- Problems.- 8 Dimension-reducing techniques.- 8.1 Introduction.- 8.2 Crude principal component analysis.- 8.3 Logcontrast principal component analysis.- 8.4 Applications of logcontrast principal component analysis.- 8.5 Subcompositional analysis.- 8.6 Applications of subcompositional analysis.- 8.7 Canonical component analysis.- 8.8 Bibliographic notes.- Problems.- 9 Bases and compositions.- 9.1 Fundamentals.- 9.2 Covariance relationships.- 9.3 Principal and canonical component comparisons.- 9.4 Distributional relationships.- 9.5 Compositional invariance.- 9.6 An application to household budget analysis.- 9.7 An application to clinical biochemistry.- 9.8 Reappraisal of an early shape and size analysis.- 9.9 Bibliographic notes.- Problems.- 10 Subcompositions and partitions.- 10.1 Introduction.- 10.2 Complete subcompositional independence.- 10.3 Partitions of order 1.- 10.4 Ordered sequences of partitions.- 10.5 Caveat.- 10.6 Partitions of higher order.- 10.7 Bibliographic notes.- Problems.- 11 Irregular compositional data.- 11.1 Introduction.- 11.2 Modelling imprecision in compositions.- 11.3 Analysis of sources of imprecision.- 11.4 Imprecision and tests of independence.- 11.5 Rounded or trace zeros.- 11.6 Essential zeros.- 11.7 Missing components.- 11.8 Bibliographic notes.- Problems.- 12 Compositions in a covariate role.- 12.1 Introduction.- 12.2 Calibration.- 12.3 A before-and-after treatment problem.- 12.4 Experiments with mixtures.- 12.5 An application to firework mixtures.- 12.6 Classification from compositions.- 12.7 An application to geological classification.- 12.8 Bibliographic notes.- Problems.- 13 Further distributions on the simplex.- 13.1 Some generalizations of the Dirichlet class.- 13.2 Some generalizations of the logistic normal classes.- 13.3 Recapitulation.- 13.4 The Ad(?,B) class.- 13.5 Maximum likelihood estimation.- 13.6 Neutrality and partition independence.- 13.7 Subcompositional independence.- 13.8 A generalized lognormal gamma distribution with compositional in variance.- 13.9 Discussion.- 13.10 Bibliographic notes.- Problems.- 14 Miscellaneous problems.- 14.1 Introduction.- 14.2 Multi-way compositions.- 14.3 Multi-stage compositions.- 14.4 Multiple compositions.- 14.5 Kernel density estimation for compositional data.- 14.6 Compositional stochastic processes.- 14.7 Relation to Bayesian statistical analysis.- 14.8 Compositional and directional data.- Problems.- Appendices.- A Algebraic properties of elementary matrices.- B Bibliography.- C Computer software for compositional data analysis.- D Data sets.- Author index.

The Statistical Analysis of Compositional Data

Regression Analysis with Applications

Nous decrivons un estimateur du maximum de vraisemblance avec correction du premier ordre pour le biais sur le parametre de dispersion d'une distribution binomiale negative. Cet estimateur est compare, en terme debiais et d'efficacite, avec l'estimateur du maximum de vraisemblance etudie par Piegorsch (1990, Biometrics 46, 863-867), les estimateurs de moment et du maximum de quasi-vraisemblance etendue de Clark et Perry (1989, Biometrics 45, 309-316) et un estimateur de double quasi-vraisemblance etendue. L'estimateur du maximum de vraisemblance avec correction du biais a des proprietes de biais et d'efficacite superieures dans la plupart des cas. Pour faciliter la comparaison, nous donnons les resultats pour le modele binomial negatif a deux parametres. Cependant un exemple impliquant la regression binomiale negative est donne.

Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter.

SUMMARY Data presented as contingency tables and classified by qualitative or quantitative methods are usually analysed on the basis of a Poisson log linear model. We examine the effect of Poisson mixtures, especially as represented by the negative binomial distribution, on statistical inference. The main finding is that probabilities of rejection are increased, sometimes considerably. In the case of heterogeneous binary data, attention is also given to the problems of analysis implied by Poisson mixtures.

Inference sensitivity for Poisson mixtures

In this note we study, by simulation, small sample performance in terms of size of ten procedures for testing the homogeneity of odds ratios in K 2 x 2 contingency tables. These ten statistics are derived for 'large-stratum' settings. Our study concerns the behaviour of these statistics for 'small-stratum' settings.

Small sample performance of tests of homogeneity of odds ratios in K 2 × 2 tables

We conducted a simulation study to determine the performance of nine procedures for testing the homogeneity of odds ratios in K 2 × 2 contingency tables. We recommend Tarone's approximate score test, based on the Mantel—Haenszel estimator of the common odds ratio, for use in practice. We also recommend a non-iterative statistic developed by Gart and based on the modified Woolf estimator of the common odds ratio for very large samples in balanced or mildly unbalanced designs. We base our recommendation of a statistic on its performance in terms of size and power in comparison with the other statistics considered.

Sudhir R. Paul

Papers

Regression Analysis with Applications

Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter.

Inference sensitivity for Poisson mixtures

Small sample performance of tests of homogeneity of odds ratios in K 2 × 2 tables

A comparison of tests of homogeneity of odds ratios in K 2 x 2 tables.