*! version 4.07 David Fisher 15sep2023
help metan_model

Title

model_spec -- Specify models and methods for meta-analytic pooling of aggregate (summary) data with metan,
including estimation of heterogeneity statistics

Syntax

model_spec is:

model( model [ \ model [ \ model [...]]] ) [ options_het ]

where model is either:

model_name [ , options_model
options_test options_label ]

or model is:

user_spec [ , options_label ]

where user_spec is ES lci uci, representing a pooled effect size estimate and 95% confidence limits
which are supplied manually rather than estimated from the data.

model_name

Note: references for particular models may be found in Fisher 2015
if not specifically referenced below.

Methods applicable to two-group comparison of binary outcomes only
mhaenszel | fixed Mantel-Haenszel model (default)

peto common-effect pooling of Peto odds ratios

Tau-squared estimators for standard inverse-variance random-effects model
ivcommon | fe | fixedi Common (aka "fixed") effect inverse-variance (default unless two-group comparison of binary outcomes)

random | re | dlaird [, pooled] DerSimonian-Laird estimator

bdl | dlb Bootstrap DerSimonian-Laird estimator

hedges Hedges estimator aka "Cochran ANOVA-like" aka "variance component" estimator

mpaule | ebayes [, pooled] Mandel-Paule aka "empirical Bayes" estimator

pmm "Median-unbiased" estimator (Viechtbauer 2021)

mle Maximum likelihood (ML) estimator

reml [, pooled] Restricted maximum likelihood (REML) estimator

hmakambi Hartung-Makambi estimator (Hartung and Makambi 2003)

b0 bp Rukhin B0 and BP estimators

sj2s [, init(model_name)] Sidik-Jonkman two-step estimator

dk2s [, init(model_name)] DerSimonian-Kacker two-step estimator (DerSimonian and Kacker 2007)

sa [, isq(real) tausq(real)] Sensitivity analysis with user-defined I-squared or tau-squared

Non-standard models, or modifications to standard models
hksj [, truncate(one | zovert)] Hartung-Knapp-Sidik-Jonkman (HKSJ) variance correction to DerSimonian-Laird estimator

pl [, bartlett skovgaard] Estimation using profile likelihood

kroger [, eim oim] Kenward-Roger variance-corrected REML model (Morris et al 2018)

btweedie Biggerstaff-Tweedie approximate Gamma model

hcopas Henmi-Copas approximate Gamma model (Henmi and Copas 2010)

mult Multiplicative heterogeneity model (Thompson and Sharp 1999)

ivhet "Inverse-variance heterogeneity" (IVhet) model (Doi et al 2015a)

qe [, qwt(varname)] Quality Effects model (Doi et al 2015b)

options_model

wgt(varname) specify a variable containing user-defined weights, as described under options_main,
but applicable to a specific model

hksj Hartung-Knapp-Sidik-Jonkman (HKSJ) variance correction, applicable to any standard tau-squared estimator

robust Sidik-Jonkman robust (sandwich-like) variance estimator (Sidik and Jonkman 2006)

Model-specific options (see model_name)
init(model_name) initial estimate of tau-squared for two-step estimators sj2s and dk2s

isq(real) tausq(real) user-defined I-squared or tau-squared values for use with sensitivity analysis sa

pooled heterogeneity estimation will be pooled across subgroups (Borenstein et al 2009); for use with option by()
and only with models dlaird, mpaule, mult, or reml

truncate(one | zovert) optional truncation of the Hartung-Knapp-Sidik-Jonkman correction factor

bartlett skovgaard Bartlett's (Huizenga et al 2011)
or Skovgaard's (Guolo 2012) corrections to the likelihood, for use with the profile-likelihood model pl

eim oim use expected (default) or observed information matrix to compute degrees of freedom for Kenward-Roger model kroger
(Morris et al 2018)

qwt(varname) variable containing quality information (model qe only; see Doi et al 2015b)

Options for iteration, replication or numerical integration
itol(#) tolerance for iteration convergence (with mpaule, mle, reml, pl, kroger, btweedie or hcopas)

maxiter(#) maximum number of iterations (as above)

maxtausq(#) upper bound of search for tau-squared confidence limits; may need to be raised in extreme cases (as above)

difficult technique(algorithm_spec) likelihood maximisation options; see help maximize

quadpts(#) number of quadrature points to use in numerical integration (with bt or hc; see help for integrate)

reps(#) number of replications for Bootstrap DerSimonian-Laird estimator (with bdl)

options_test

z t chi2 cmh specify test statistic for significance of pooled result

options_het

Homogeneity test statistic
cochranq Cochran's Q statistic (default, unless Mantel-Haenszel)

breslow Breslow-Day test for homogeneity of odds ratios (Mantel-Haenszel only)

tarone Breslow-Day-Tarone test for homogeneity of odds ratios
(Mantel-Haenszel only; preferred to breslow, see e.g. Breslow 1996)

Heterogeneity confidence intervals
hlevel(#) set confidence level for reporting confidence intervals for heterogeneity; default is hlevel(95)

isqparam define I-squared parametrically, based on variance considerations; see below

testbased use alternative test-based confidence interval for H (Higgins and Thompson 2002)

options_label

label(string) specify alternative display label for a model

extralabel(string) optional further description, for display within the forest plot only

nomodellabels suppress all model labels in output and forest plot

Description

model_spec specifies method(s) for meta-analytic pooling with metan.
If no model_spec is supplied, the default for two-group comparison of binary outcomes is mhaenszel;
otherwise the default is ivcommon.

model_spec allows the results of multiple different pooling methods (or variations of methods) to be displayed simultaneously.
Such methods include alternative estimators of the between-study heterogeneity (tau-squared),
methods which correct post hoc the variance of the pooled estimate, or those which use alternative weighting systems
in order to improve statistical performance in the suspected presence of publication bias; see model_name.

Furthermore, the user_spec syntax allows pooled effect estimates from sources external to metan
- such as WinBUGS (Lunn et al 2000), or one-stage modelling - to be displayed alongside the usual metan output on-screen and in the forest plot.
Note that:

If displayed study estimates and non-user-defined pooled effect estimates are exponentiated
(either via explicit use of eform, or by default e.g. with binary data),
then the elements of user_spec will also be exponentiated prior to display.

If user_spec is specified as the first model element, then wgt(varname) must also be supplied.

user_spec cannot be used to specify subgroup effect estimates.
Therefore, by() may not be used if user_spec is specified as the first model element.
Similarly, if user_spec is specified other than as the first model element,
it will be ignored when printing subgroup estimates and tests.

All previous syntax of metan v3.04 for Stata v9 relating to model specification continues to be supported,
including options first(), firststats(), second() and secondstats().
(These are implemented as special cases of model_spec with options_label.)

Another, more flexible, way of incorporating user-supplied estimates and/or text is to use forestplot "results sets"; see forestplot.

Underneath the table of study effects, metan reports a homogeneity test, typically based on Cochran's Q statistic;
plus the following heterogeneity statistics defined in terms of Q and its degrees of freedom Q_df as follows:

H = sqrt(Q / Q_df), and I^2 = 100% * (Q - Q_df) / Q

Following Hedges and Pigott (2001), when the null hypothesis of homogeneity is not true,
the homogeneity statistic (e.g. Cochran's Q) may be considered approximately to follow a non-central chi-square distribution
if under a common-effect model, or a Gamma distribution (incorporating the DerSimonian-Laird estimator of tau-squared)
if under a random-effects model.
metan uses these distributions to construct confidence intervals, based on observed data, for the heterogeneity statistics H and I-squared.

Note that some minor variation in the way metan presents heterogeneity statistics may result depending on the model(s) specified.
If more than one model is specified, heterogeneity statistics are presented consistent with the first model only.

Click here to return to the main metan help page.

Options

options_model

hksj applies the Hartung-Knapp-Sidik-Jonkman (HKSJ) variance correction to any standard tau-squared estimator;
that is, in the absense of other variance correction or re-weighting mechanisms.
hksj may instead be supplied as model_name, in which case the DerSimonian-Laird estimator is implied.

By default, the correction factor is untruncated.
This means that, if its value is less than one, the confidence interval around the pooled effect may be less conservative
than under a model without the HKSJ correction being applied.
In that case, a warning message is printed underneath the results table, recommending that the truncate() option is used.

truncate(one) truncates the HKSJ correction factor at one (Jackson et al 2017).

truncate(zovert) applies a less conservative truncation (van Aert et al 2019)
under which the correction factor is compared to the ratio of the z-based to the t-based critical values.

init(model_name) specifies an initial estimate of tau-squared for "two-step" estimation models sj2s and dk2s.

For the Sidik-Jonkman two-step estimator sj2s, the default initial estimate
is the mean dispersion of effect sizes from their unweighted mean (Sidik and Jonkman 2005);
any standard tau-squared estimator model_name may instead be used.

For the DerSimonian-Kacker two-step estimator dk2s, the default initial model_name is hedges,
with the single alternative of dlaird.

isq(real) or tausq(real) impose a user-specified amount of heterogeneity for use with the sensitivity-analysis model sa.
At most one of these options may be used.
If neither is used, a value for I-squared of 80% is taken as the default.
A random-effects model is then fitted, with tau-squared constrained accordingly.
Accompanying heterogeneity statistics are calculated based on tau-squared and sigma-squared using formulae presented in Higgins and Thompson (2002).

isq(real) constrains the value of I-squared, and takes values between 0 and 100.
A value for tau-squared is back-derived in order for the random-effects model to be fitted.

tausq(real) constrains the value of tau-squared, and takes values of zero or above.

pooled is only for use with trial-level subgroups; see by().
It requests that the between-study heterogeneity variance is pooled across subgroups, and that the resulting estimate is used for all subgroups
rather than being estimated separately for each subgroup as by default.
Because a pooled estimate of heterogeneity variance is equivalent to a meta-regression on the set of subgroup categories,
a call is made to the user-written meta-regression package metareg (Harbord and Higgins 2008).
Hence, only the random-effects models implemented by metareg are allowed with pooled; that is: dlaird, mpaule, or reml.
(The multiplicative heterogeneity model mult may also be used, via a call to regress rather than to metareg.)

bartlett and skovgaard are options for use with the profile-likelihood model pl.
These apply small-sample corrections to the likelihood, designed to improve performance when the number of studies is small.

bartlett implements Bartlett's correction (Huizenga et al 2011).
This is applied to the likelihood-ratio statistic,
and therefore a chi-squared test statistic is presented by default for the pooled estimate,
displayed as "LR chi2" to differentiate from a standard (non-likelihood based) chi-squared statistic.

skovgaard implements Skovgaard's correction (Guolo 2012).
This is applied to the signed log-likelihood ratio statistic,
and therefore a z-based test statistic is presented, displayed as "LL z" to differentiate from a standard Wald-type statistic.

options_test

options_test can be z, t or chi2 (or cmh; see below).
These options specify a distribution for testing the significance of the pooled result.
The default is usually z, but is t with hksj or robust, and is chi2 with mhaenszel, peto or pl.
Any of z, t or chi2 may be specified to override these defaults.

Additionally, with Mantel-Haenszel odds ratios only, the Cochran-Mantel-Haenszel test statistic (McDonald 2014)
may be requested with option cmh.

options_het

cochranq, breslow and tarone are used to over-ride the default choice of homogeneity statistic.
The default is usually cochranq; that is, the standard Cochran's Q statistic
using common-effect inverse-variance weighting and pooled effect size (Deeks et al 2001).
When using alternative common-effect models mhaenszel or peto, the default homogeneity statistic changes
to incorporate the study weighting and pooled effect size derived from the specified model;
in the Display Window and forest plot the description becomes "Mantel-Haenszel Q" or "Peto Q" as appropriate.

breslow and tarone are applicable only with Mantel-Haenszel odds ratios.
breslow specifies the Breslow-Day homogeneity statistic (Breslow and Day 1980).
Although widely used, this statistic is unnecessarily approximate;
tarone instead specifies the superior Breslow-Day-Tarone statistic (Breslow 1996).

testbased requests that an alternative, test-based method of constructing confidence intervals for H and I-squared is used
(Higgins and Thompson 2002), instead of the default intervals
based on an approximate distribution for Q under the assumption that the null hypothesis of homogeneity is false.

isqparam requests that metan presents heterogeneity statistics based upon variance considerations.
That is, I-squared is to be defined parametrically based on the within-study variance tau-squared as estimated from a random-effects model,
rather than on a homogeneity statistic calculated directly from the data.
In the notation of Higgins and Thompson (2002): I^2 = tau^2 / (tau^2 + sigma^2)
instead of the default I^2 = (Q - Q_df) / Q.
Confidence intervals for I-squared may be derived similarly, using model-derived confidence limits for tau-squared.
(Estimates and confidence intervals for H may also be derived in this way; these are not shown in the Display Window
but are returned within the matrix r(hetstats).)

If multiple models are requested (see model_spec) alongside isqparam, then potentially
there will be multiple sets of heterogeneity statistics, each associated with a model-derived estimate of tau-squared.
This information is displayed in an additional table in the Results Window; and if a forest plot is presented,
then a set of heterogeneity information is displayed in brackets alongside the name of each individual model.

options_label

label(label_string) specifies an alternative display label for a particular model.
For example, the default label for the DerSimonian-Laird random-effects model is "DL",
but the simpler label "Random" might be preferred if no other random-effects models are used.

label("") will suppress the label for a particular model.
To suppress all model labels, specify nomodellabels as a main option to metan.

If user_spec is used without label(label_string), the default label_string is "User".

extralabel(extra_string) optionally specifies further descriptive text
to appear on the forest plot in place of the standard heterogeneity text; extra_string is not displayed in the Results Window.
This option is designed for use alongside user_spec, for example to display externally-derived parameters related to study heterogeneity.

Authors

Original authors:
Michael J Bradburn, Jonathan J Deeks, Douglas G Altman.
Centre for Statistics in Medicine, University of Oxford, Oxford, UK

metan v3.04 for Stata v9:
Ross J Harris, Roger M Harbord, Jonathan A C Sterne.
Department of Social Medicine, University of Bristol, Bristol, UK

metan v4.00 and later (including current version):
David Fisher, MRC Clinical Trials Unit at UCL, London, UK.

Email d.fisher@ucl.ac.uk

Acknowledgments

Thanks to Patrick Royston (MRC Clinical Trials Unit at UCL, London, UK) for suggestions of improvements to the code and help file.

Thanks to Vince Wiggins, Kit Baum and Jeff Pitblado of Statacorp who offered advice and helped facilitate the version 9 update.

Thanks to Julian Higgins and Jonathan A C Sterne (University of Bristol, Bristol, UK) who offered advice and helped facilitate this latest update,
and thanks to Daniel Klein (Universität Kassel, Germany) for assistance with testing under older Stata versions.

The "click to run" element of the examples in this document is handled using an idea originally developed by Robert Picard.

References

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Chichester: Wiley.

Breslow NE, Day NE. 1980. Statistical Methods in Cancer Research: Vol. I - The Analysis of Case-Control Studies.
Lyon: International Agency for Research on Cancer.

Breslow NE. 1996.
Statistics in epidemiology: The case-control study.
Journal of the American Statistical Association 91: 14-28

Deeks JJ, Altman DG, Bradburn MJ. 2001.
Statistical methods for examining heterogeneity and combining results from several studies in meta-analysis.
In Systematic Reviews in Health Care: Meta-analysis in Context, ed. Egger M, Davey Smith G, Altman DG, 2nd ed., 285-312. London: BMJ Books.

DerSimonian R, Kacker R. 2007.
Random-effects model for meta-analysis of clinical trials: An update.
Contemporary Clinical Trials 28: 105-114. doi: 10.1016/j.cct.2006.04.004

Doi SAR, Barendregt JJ, Khan S, Thalib L, Williams GM. 2015a.
Advances in the meta-analysis of heterogeneous clinical trials I: The inverse variance heterogeneity model.
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Doi SAR, Barendregt JJ, Khan S, Thalib L, Williams GM. 2015b.
Advances in the meta-analysis of heterogeneous clinical trials II: The quality effects model.
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Fisher DJ. 2015.
Two-stage individual participant data meta-analysis and generalized forest plots.
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Guolo A. 2012.
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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?
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Lunn DJ, Thomas A, Best N, Spiegelhalter D. 2000.
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McDonald JH. 2014.
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Morris TP, Fisher DJ, Kenward MG, Carpenter JR. 2018.
Meta-analysis of quantitative individual patient data: two stage or not two stage?
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Sidik K, Jonkman JN. 2005.
Simple heterogeneity variance estimation for meta-analysis.
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Sidik K, Jonkman JN. 2006.
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van Aert RCM, Jackson D. 2019.
A new justification of the Hartung-Knapp method for random-effects meta-analysis
based on weighted least squares regression.
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Viechtbauer W. 2007.
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