Description

Fixed effect and random effects meta-analysis based on estimates (e.g. log hazard ratios) and their standard errors. The inverse variance method is used for pooling.

Usage

Arguments

Details

This function provides the generic inverse variance method for meta-analysis which requires treatment estimates and their standard errors (Borenstein et al., 2010). The method is useful, e.g., for pooling of survival data (using log hazard ratio and standard errors as input). Arguments TE and seTE can be used to provide treatment estimates and standard errors directly. However, it is possible to derive these quantities from other information.

Default settings are utilised for several arguments (assignments using gs function). These defaults can be changed for the current R session using the settings.meta function.

Furthermore, R function update.meta can be used to rerun a meta-analysis with different settings.

Approximate treatment estimates

Missing treatment estimates can be derived from

1. confidence limits provided by arguments lower and upper;

2. median, interquartile range and range (arguments median, q1, q3, min, and max);

3. median and interquartile range (arguments median, q1 and q3);

4. median and range (arguments median, min and max).

For confidence limits, the treatment estimate is defined as the center of the confidence interval (on the log scale for relative effect measures like the odds ratio or hazard ratio).

If the treatment effect is a mean it can be approximated from sample size, median, interquartile range and range. By default, methods described in Luo et al. (2018) are utilized (argument method.mean = "Luo"):

• equation (7) if sample size, median and range are available,

• equation (11) if sample size, median and interquartile range are available,

• equation (15) if sample size, median, range and interquartile range are available.

Instead the methods described in Wan et al. (2014) are used if argument method.mean = "Wan"):

• equation (2) if sample size, median and range are available,

• equation (14) if sample size, median and interquartile range are available,

• equation (10) if sample size, median, range and interquartile range are available.

By default, missing treatment estimates are replaced successively using these method, i.e., confidence limits are utilised before interquartile ranges. Argument approx.TE can be used to overwrite this default for each individual study:

• Use treatment estimate directly (entry "" in argument approx.TE);

• confidence limits ("ci" in argument approx.TE);

• median, interquartile range and range ("iqr.range");

• median and interquartile range ("iqr");

• median and range ("range").

Approximate standard errors

Missing standard errors can be derived from

1. p-value provided by arguments pval and (optional) df;

2. confidence limits (arguments lower, upper, and (optional) df);

3. sample size, median, interquartile range and range (arguments n.e and / or n.c, median, q1, q3, min, and max);

4. sample size, median and interquartile range (arguments n.e and / or n.c, median, q1 and q3);

5. sample size, median and range (arguments n.e and / or n.c, median, min and max).

For p-values and confidence limits, calculations are either based on the standard normal or t distribution if argument df is provided. Furthermore, argument level.ci can be used to provide the level of the confidence interval.

Wan et al. (2014) describe methods to estimate the standard deviation (and thus the standard error by deviding the standard deviation with the square root of the sample size) from the sample size, median and additional statistics. Shi et al. (2020) provide an improved estimate of the standard deviation if the interquartile range and range are available in addition to the sample size and median. Accordingly, equation (11) in Shi et al. (2020) is the default (argument method.sd = "Shi"), if the median, interquartile range and range are provided (arguments median, q1, q3, min and max). The method by Wan et al. (2014) is used if argument method.sd = "Wan" and, depending on the sample size, either equation (12) or (13) is used. If only the interquartile range or range is available, equations (15) / (16) and (7) / (9) in Wan et al. (2014) are used, respectively. The sample size of individual studies must be provided with arguments n.e and / or n.c. The total sample size is calculated as n.e + n.c if both arguments are provided.

By default, missing standard errors are replaced successively using these method, e.g., p-value before confidence limits before interquartile range and range. Argument approx.seTE can be used to overwrite this default for each individual study:

• Use standard error directly (entry "" in argument approx.seTE);

• p-value ("pval" in argument approx.seTE);

• confidence limits ("ci");

• median, interquartile range and range ("iqr.range");

• median and interquartile range ("iqr");

• median and range ("range").

Confidence intervals for individual studies

For the mean difference (argument sm = "MD"), the confidence interval for individual studies can be based on the

• standard normal distribution (method.ci = "z"), or

• t-distribution (method.ci = "t").

By default, the first method is used if argument df is missing and the second method otherwise.

Note, this choice does not affect the results of the fixed effect and random effects meta-analysis.

Estimation of between-study variance

The following methods are available to estimate the between-study variance τ^2.

Historically, the DerSimonian-Laird method was the de facto standard to estimate the between-study variance τ^2 and is still the default in many software packages including Review Manager 5 (RevMan 5) and R package meta. However, its role has been challenged and especially the Paule-Mandel and REML estimators have been recommended (Veroniki et al., 2016). Accordingly, the following R command can be used to use the Paule-Mandel estimator in all meta-analyses of the R session: settings.meta(method.tau = "PM")

Confidence interval for the between-study variance

The following methods to calculate a confidence interval for τ^2 and τ are available.

The first three methods have been recommended by Veroniki et al. (2016). By default, the Jackson method is used for the DerSimonian-Laird estimator of τ^2 and the Q-profile method for all other estimators of τ^2. The Profile-Likelihood method is the only method available for the three-level meta-analysis model. No confidence intervals for τ^2 and τ are calculated if method.tau.ci = "".

Hartung-Knapp method

Hartung and Knapp (2001a,b) proposed an alternative method for random effects meta-analysis based on a refined variance estimator for the treatment estimate. Simulation studies (Hartung and Knapp, 2001a,b; IntHout et al., 2014; Langan et al., 2019) show improved coverage probabilities compared to the classic random effects method. However, in rare settings with very homogeneous treatment estimates, the Hartung-Knapp (HK) variance estimate can be arbitrarily small resulting in a very narrow confidence interval (Knapp and Hartung, 2003; Wiksten et al., 2016). In such cases, an ad hoc variance correction has been proposed by utilising the variance estimate from the classic random effects model with the HK method (Knapp and Hartung, 2003; IQWiQ, 2020). An alternative approach is to use the wider confidence interval of classic fixed or random effects meta-analysis and the HK method (Wiksten et al., 2016; Jackson et al., 2017).

Argument adhoc.hakn can be used to choose the ad hoc method:

Prediction interval

A prediction interval for the treatment effect of a new study (Higgins et al., 2009) is calculated if arguments prediction and comb.random are TRUE. Note, the definition of prediction intervals varies in the literature. This function implements equation (12) of Higgins et al., (2009) which proposed a t distribution with K-2 degrees of freedom where K corresponds to the number of studies in the meta-analysis.

Subgroup analysis

Argument byvar can be used to conduct subgroup analysis for a categorical covariate. The metareg function can be used instead for more than one categorical covariate or continuous covariates.

Specify the null hypothesis of test for an overall effect

Argument null.effect can be used to specify the (treatment) effect under the null hypothesis in a test for an overall effect.

By default (null.effect = 0), the null hypothesis corresponds to "no difference" (which is obvious for absolute effect measures like the mean difference (sm = "MD") or standardised mean difference (sm = "SMD")). For relative effect measures, e.g., risk ratio (sm = "RR") or odds ratio (sm = "OR"), the null effect is defined on the log scale, i.e., ln(RR) = 0 or ln(OR) = 0 which is equivalent to testing RR = 1 or OR = 1.

Use of argument null.effect is especially useful for summary measures without a "natural" null effect, i.e., in situations without a second (treatment) group. For example, an overall proportion of 50% could be tested in the meta-analysis of single proportions with argument null.effect = 0.5.

Note, all tests for an overall effect are two-sided with the alternative hypothesis that the effect is unequal to null.effect.

Exclusion of studies from meta-analysis

Arguments subset and exclude can be used to exclude studies from the meta-analysis. Studies are removed completely from the meta-analysis using argument subset, while excluded studies are shown in printouts and forest plots using argument exclude (see Examples). Meta-analysis results are the same for both arguments.

Presentation of meta-analysis results

Internally, both fixed effect and random effects models are calculated regardless of values choosen for arguments comb.fixed and comb.random. Accordingly, the estimate for the random effects model can be extracted from component TE.random of an object of class "meta" even if argument comb.random = FALSE. However, all functions in R package meta will adequately consider the values for comb.fixed and comb.random. For example, functions print.meta and forest.meta will not show results for the random effects model if comb.random = FALSE.

Argument pscale can be used to rescale single proportions or risk differences, e.g. pscale = 1000 means that proportions are expressed as events per 1000 observations. This is useful in situations with (very) low event probabilities.

Argument irscale can be used to rescale single rates or rate differences, e.g. irscale = 1000 means that rates are expressed as events per 1000 time units, e.g. person-years. This is useful in situations with (very) low rates. Argument irunit can be used to specify the time unit used in individual studies (default: "person-years"). This information is printed in summaries and forest plots if argument irscale is not equal to 1.

Default settings for comb.fixed, comb.random, pscale, irscale, irunit and several other arguments can be set for the whole R session using settings.meta.

Value

An object of class c("metagen", "meta") with corresponding print, summary, and forest functions. The object is a list containing the following components:

Note

R function rma.uni from R package metafor (Viechtbauer 2010) is called internally to estimate the between-study variance τ^2.

Author(s)

Guido Schwarzer sc@imbi.uni-freiburg.de

References

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Hedges LV & Olkin I (1985): Statistical methods for meta-analysis. San Diego, CA: Academic Press

Higgins JPT, Thompson SG, Spiegelhalter DJ (2009): A re-evaluation of random-effects meta-analysis. Journal of the Royal Statistical Society: Series A, 172, 137–59

Hunter JE & Schmidt FL (2015): Methods of Meta-Analysis: Correcting Error and Bias in Research Findings (Third edition). Thousand Oaks, CA: Sage

Hartung J, Knapp G (2001a): On tests of the overall treatment effect in meta-analysis with normally distributed responses. Statistics in Medicine, 20, 1771–82

Hartung J, Knapp G (2001b): A refined method for the meta-analysis of controlled clinical trials with binary outcome. Statistics in Medicine, 20, 3875–89

IntHout J, Ioannidis JPA, Borm GF (2014): The Hartung-Knapp-Sidik-Jonkman method for random effects meta-analysis is straightforward and considerably outperforms the standard DerSimonian-Laird method. BMC Medical Research Methodology, 14, 25

IQWiG (2020): General Methods: Version 6.0. https://www.iqwig.de/en/about-us/methods/methods-paper/

Jackson D (2013): Confidence intervals for the between-study variance in random effects meta-analysis using generalised Cochran heterogeneity statistics. Research Synthesis Methods, 4, 220–229

Jackson D, Law M, Rücker G, Schwarzer G (2017): The Hartung-Knapp modification for random-effects meta-analysis: A useful refinement but are there any residual concerns? Statistics in Medicine, 36, 3923–34

Knapp G & Hartung J (2003): Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22, 2693–710

Langan D, Higgins JPT, Jackson D, Bowden J, Veroniki AA, Kontopantelis E, et al. (2019): A comparison of heterogeneity variance estimators in simulated random-effects meta-analyses. Research Synthesis Methods, 10, 83–98

Luo D, Wan X, Liu J, Tong T (2018): Optimally estimating the sample mean from the sample size, median, mid-range, and/or mid-quartile range. Statistical Methods in Medical Research, 27, 1785–805

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Review Manager (RevMan) [Computer program]. Version 5.4. The Cochrane Collaboration, 2020

Shi J, Luo D, Weng H, Zeng X-T, Lin L, Chu H, et al. (2020): Optimally estimating the sample standard deviation from the five-number summary. Research Synthesis Methods.

Sidik K & Jonkman JN (2005): Simple heterogeneity variance estimation for meta-analysis. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54, 367–84

Veroniki AA, Jackson D, Viechtbauer W, Bender R, Bowden J, Knapp G, et al. (2016): Methods to estimate the between-study variance and its uncertainty in meta-analysis. Research Synthesis Methods, 7, 55–79

Viechtbauer W (2005): Bias and efficiency of meta-analytic variance estimators in the random-effects model. Journal of Educational and Behavioral Statistics, 30, 261–93

Viechtbauer W (2007): Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26, 37–52

Viechtbauer W (2010): Conducting Meta-Analyses in R with the metafor Package. Journal of Statistical Software, 36, 1–48

Van den Noortgate W, López-López JA, Marín-Martínez F, Sánchez-Meca J (2013): Three-level meta-analysis of dependent effect sizes. Behavior Research Methods, 45, 576–94

Wan X, Wang W, Liu J, Tong T (2014): Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range. BMC Medical Research Methodology, 14, 135

Wiksten A, Rücker G, Schwarzer G (2016): Hartung-Knapp method is not always conservative compared with fixed-effect meta-analysis. Statistics in Medicine, 35, 2503–15

See Also

update.meta, metabin, metacont, print.meta, settings.meta

Examples