The extent of heterogeneity in a meta-analysis partly determines the difficulty in drawing overall conclusions. This extent may be measured by estimating a between-study variance, but interpretation is then specific to a particular treatment effect metric. A test for the existence of heterogeneity exists, but depends on the number of studies in the meta-analysis. We develop measures of the impact of heterogeneity on a meta-analysis, from mathematical criteria, that are independent of the number of studies and the treatment effect metric. We derive and propose three suitable statistics: H is the square root of the chi2 heterogeneity statistic divided by its degrees of freedom; R is the ratio of the standard error of the underlying mean from a random effects meta-analysis to the standard error of a fixed effect meta-analytic estimate, and I2 is a transformation of (H) that describes the proportion of total variation in study estimates that is due to heterogeneity. We discuss interpretation, interval estimates and other properties of these measures and examine them in five example data sets showing different amounts of heterogeneity. We conclude that H and I2, which can usually be calculated for published meta-analyses, are particularly useful summaries of the impact of heterogeneity. One or both should be presented in published meta-analyses in preference to the test for heterogeneity.

Quantifying heterogeneity in a meta‐analysis

Additional co-authors: TJ Heaton, AG Hogg, KA Hughen, KF Kaiser, B Kromer, SW Manning, RW Reimer, DA Richards, JR Southon, S Talamo, CSM Turney, J van der Plicht, CE Weyhenmeyer

/pdf/intcal13-and-marine13-radiocarbon-age-calibration-curves-0-4f3vsbzfqx.pdf

IntCal13 and Marine13 radiocarbon age calibration curves 0-50,000 years cal BP

The metafor package provides functions for conducting meta-analyses in R. The package includes functions for fitting the meta-analytic fixed- and random-effects models and allows for the inclusion of moderators variables (study-level covariates) in these models. Meta-regression analyses with continuous and categorical moderators can be conducted in this way. Functions for the Mantel-Haenszel and Peto's one-step method for meta-analyses of 2 x 2 table data are also available. Finally, the package provides various plot functions (for example, for forest, funnel, and radial plots) and functions for assessing the model fit, for obtaining case diagnostics, and for tests of publication bias.

Conducting Meta-Analyses in R with the metafor Package

Luminescence dating of quartz using an improved single aliquot regenerative-dose protocol

Background: The number of Mendelian randomization analyses including large numbers of genetic variants is rapidly increasing. This is due to the proliferation of genome-wide association studies, and the desire to obtain more precise estimates of causal effects. However, some genetic variants may not be valid instrumental variables, in particular due to them having more than one proximal phenotypic correlate (pleiotropy).

Methods: We view Mendelian randomization with multiple instruments as a meta-analysis, and show that bias caused by pleiotropy can be regarded as analogous to small study bias. Causal estimates using each instrument can be displayed visually by a funnel plot to assess potential asymmetry. Egger regression, a tool to detect small study bias in meta-analysis, can be adapted to test for bias from pleiotropy, and the slope coefficient from Egger regression provides an estimate of the causal effect. Under the assumption that the association of each genetic variant with the exposure is independent of the pleiotropic effect of the variant (not via the exposure), Egger’s test gives a valid test of the null causal hypothesis and a consistent causal effect estimate even when all the genetic variants are invalid instrumental variables.

Results: We illustrate the use of this approach by re-analysing two published Mendelian randomization studies of the causal effect of height on lung function, and the causal effect of blood pressure on coronary artery disease risk. The conservative nature of this approach is illustrated with these examples.

Conclusions: An adaption of Egger regression (which we call MR-Egger) can detect some violations of the standard instrumental variable assumptions, and provide an effect estimate which is not subject to these violations. The approach provides a sensitivity analysis for the robustness of the findings from a Mendelian randomization investigation.

/pdf/mendelian-randomization-with-invalid-instruments-effect-1pqi6rglyj.pdf

Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression

The analysis of projected fission track lengths

Statistical models for mixed ages

I comment on some aspects of fission-track observational biases and orientation effects arising in recent work by Donelick, Ketcham, and Carlson, and their relation to earlier work by myself and colleagues. Isotropic and anisotropic random line segment models provide a sound basis for understanding variation in fission-track measurements, and some useful mathematical formulae are available. The effects of orientation bias can be avoided, without introducing arbitrary assumptions, by modeling confined track length conditional on angle to the c axis.

Some remarks on fission-track observational biases and crystallographic orientation effects

On the mis-use of mathematics: a comment on "How confident are we about the chronology of the transition between Howieson's Poort and Still Bay?" by Guérin et al (2013).

R.F. Galbraith

Papers

The analysis of projected fission track lengths

Statistical models for mixed ages

Some remarks on fission-track observational biases and crystallographic orientation effects

On the mis-use of mathematics: a comment on "How confident are we about the chronology of the transition between Howieson's Poort and Still Bay?" by Guérin et al (2013).

Statistical properties of semi-tracks in fission track analysis