{smcl}
{* 31oct2008}{...}
{cmd:help metareg}{right: ({browse "http://www.stata-journal.com/article.html?article=up0023":SJ8-4: sbe23_1})}
{hline}

{title:Title}

{p2colset 5 16 18 2}{...}
{p2col :{hi:metareg} {hline 2}}Meta-analysis regression (revised){p_end}
{p2colreset}{...}


{title:Syntax}

{p 8 15 2}
{cmd: metareg}
{it:{help varname:depvar}} [{it:{help varlist:indepvars}}]
{ifin} {cmd:wsse(}{it:varname}{cmd:)}
[, {cmdab:ef:orm}
{cmdab:g:raph}
{cmdab:ra:ndomsize}
{cmdab:nocons:tant}
{cmd:mm} {cmd:reml} {cmd:eb}
{cmdab:k:napphartung} {cmd:z}
{cmdab:tau:2test}
{cmdab:l:evel(}{it:#}{cmd:)}
{cmdab:perm:ute(}{it:#} [{cmd:,} {cmdab:u:nivariable} {cmdab:d:etail} 
{cmdab:j:oint(}{it:varlist1} [{cmd:|} {it:varlist2} ...]{cmd:)}]{cmd:)}
{cmd:log}
{it:maximize_options}]

{pstd}
{cmd:by} can be used with {cmd:metareg}; see {manhelp by D}.


{title:Description}

{pstd}
{cmd:metareg} performs random-effects meta-regression
using aggregate-level data.

{pstd}
From a more abstract perspective, it extends {helpb vwls}
by estimating an extra additive component of variance tau2:

{p 8 17 2}
y_i = a + B*x_i + u_i + e_i

{pstd} where a is a constant, u_i is a normal error term with known standard deviations
wsse_i that may vary across units, and e_i is a normal error with variance
tau2 to be estimated, assumed equal across units.  This is a similar model to
those fit by the {helpb xt} commands, except that the within-unit data have
been summarized by an effect estimate and its standard error for each unit i.


{title:Options}

{phang}{cmd:wsse(}{it:varname}{cmd:)} specifies the variable
containing the standard error of {it:depvar} within each study
({ul:w}ithin-{ul:s}tudy {ul:s}tandard {ul:e}rror).  All values of {it:varname}
must be greater than zero. {cmd:wsse()} is required.

{phang}{cmd:eform} indicates to output the exponentiated form of the
coefficients and to suppress reporting of the constant.  This option may
be useful when {it:depvar} is the logarithm of a ratio measure, such as a log
odds-ratio or a log risk-ratio.

{phang}{cmd:graph} requests a line graph of fitted values
plotted against the first covariate in {it:indepvars},
together with the estimates from each study represented by circles.
By default, the circle sizes depend on the precision of each estimate
(the inverse of its within-study variance),
which is the weight given to each study in the fixed-effects model.

{phang}{cmd:randomsize} is for use with the {cmd:graph} option.  It
specifies that the size of the circles will depend on the weights in the
random-effects model rather than the precision of each estimate.  These
random-effects weights depend on the estimate of tau2.

{pstd} The remaining options will mainly be of interest to more advanced users:

{phang}{cmd:noconstant} suppresses the constant term (intercept).
This is rarely appropriate in meta-regression. 

{pstd} The {cmd:mm}, {cmd:reml}, and {cmd:eb} options are alternatives 
that specify the method of estimation 
of the additive (between-study) component of variance tau2.

{phang}{cmd:mm} specifies the use of method of moments to estimate the
additive (between-study) component of variance tau2; this is a generalization
of the DerSimonian and Laird (1986) method commonly used for random-effects
meta-analysis.  For speed, this is the default when the {cmd:permute()} option
is specified, because it is the only noniterative method.

{phang}{cmd:reml} specifies the use of residual maximum likelihood (REML) to estimate the additive
(between-study) component of variance tau2.  This is the default unless the
{cmd:permute()} option is specified.  This revised version uses
Stata's maximum likelihood facilities to maximize the REML
log likelihood.  It will therefore not give identical results to the previous
version of {cmd:metareg}, which used an approximate iterative method.

{phang}{cmd:eb} specifies the use of the "empirical Bayes" method
to estimate tau2 (Morris 1983).

{phang}{cmd:knapphartung} makes a modification to the variance of the
estimated coefficients suggested by Knapp and Hartung (2003), accompanied by
the use of a t distribution in place of the standard normal distribution when
calculating p-values and confidence intervals.  This is the default unless the
{cmd:permute()} option is specified.

{phang}{cmd:z} requests that the {cmd:knapphartung} modification not be
applied and that the standard normal distribution be used to calculate
p-values and confidence intervals.  This is the default when the
{cmd:permute()} option is specified with a fixed-effects model.

{phang}{cmd:tau2test} adds to the output two tests of tau2 = 0.  The first is
based on the residual heterogeneity statistic, Q_res. The second (not
available if the {cmd:mm} option is also
specified) is a likelihood-ratio test based on the REML log likelihood. These
are two tests of the same null hypothesis (the fixed-effects model with tau2
= 0), but the alternative hypotheses are different, as are the distributions of
the test statistics under the null, so close agreement of the two tests is not
guaranteed.  Both tests are typically of little interest because it is more
helpful to quantify heterogeneity than to test for it.

{phang} {opt level(#)} specifies the confidence level, as a percentage, for
confidence intervals.  The default is {cmd:level(95)} or as set by 
{helpb set level}.

{phang} {cmd:permute(}...{cmd:)} calculates p-values by using a Monte Carlo permutation
test. See {help metareg##permute:Option for permutation test} for more information about the option.

{phang}{cmd:log} requests the display of the iteration log during estimation of
tau2.  This is ignored if the {cmd:mm} option is specified, because this uses a
noniterative method.

{phang}{it:maximize_options} are ignored unless estimation of tau2 is by REML.
These options control the maximization process; see {helpb maximize}.  You
should never need to specify them; they are supported only in case problems in
the REML estimation of tau2 are ever reported or suspected.


{marker permute}{...}
{title:Option for permutation test}

{pstd} The {cmd:permute()} option calculates p-values by using a Monte
Carlo permutation test, as recommended by Higgins and
Thompson (2004).  To address multiple testing, {cmd:permute()} also calculates
p-values for the most- to least-significant covariates, as the same
authors also recommend.

{pstd}
The syntax of {cmd:permute()} is

{p 8 15 2}{cmd:permute(}{it:#}
[{cmd:,}
	{cmd:univariable}
	{cmd:detail}
	{cmd:joint(}{it:varlist1} [{cmd:|} {it:varlist2} ...]{cmd:)}]{cmd:)}

{pstd}
where {it:#} is required and specifies the number of random permutations to
perform.  Larger values give more precise p-values but take longer.

{pstd}
There are three suboptions:

{phang}{cmd:univariable} indicates that p-values should be calculated for a
series of single covariate meta-regressions of each covariate in {it:varlist}
separately, instead of a multiple meta-regression of all covariates in
{it:varlist} simultaneously.

{phang}{cmd:detail} requests more detailed output in the style given by
{helpb permute}.

{phang}{cmd:joint(}{it:varlist1} [{cmd:|} {it:varlist2} ...]{cmd:)}
specifies that a permutation p-value should also be computed for a joint
test of the variables in each {help varlist}.

{pstd}The {cmd:eform}, {cmd:level()}, and {cmd:z} options have
no effect when the {cmd:permute()} option is specified.


{title:Syntax of predict}

{p 4 4}
The syntax of {helpb predict} following {cmd:metareg} is

{p 8 15 2}{cmd:predict} [{it:type}] {it:newvar} {ifin} [{cmd:,} {it:statistic}]

{pstd}
where {it:statistic} is

{p 8 25}{cmd:xb}{space 11}fitted values; the default{p_end}
{p 8 25}{cmd:stdp}{space 9}standard error of the prediction{p_end}
{p 8 25}{cmd:stdf}{space 9}standard error of the forecast{p_end}
{p 8 25}{cmd:u}{space 12}predicted random effects{p_end}
{p 8 25}{cmdab:usta:ndard}{space 4}standardized predicted random effects{p_end}
{p 8 25}{cmd:xbu}{space 10}prediction including random effects{p_end}
{p 8 25}{cmd:stdxbu}{space 7}standard error of xbu{p_end}
{p 8 25}{cmdab:h:at}{space 10}leverage (diagonal elements of hat matrix)

{pstd}
These statistics are available both in and out of sample;
type {cmd:predict ... if e(sample) ...} if wanted only for the estimation sample.


{title:Options for predict}

{phang}{cmd:xb}, the default, calculates the linear prediction, x_i*b, that is, the fitted values excluding the random effects.

{phang}{cmd:stdp} calculates the standard error of the prediction
(the standard error of the fitted values excluding the random effects).

{phang}{cmd:stdf} calculates the standard error of the forecast.
This gives the standard deviation of the predicted distribution
of the true value of {it:depvar} in a future study,
with the covariates given by {it:varlist}.
{break}stdf^2 = stdp^2 + tau2.

{phang}{cmd:u} calculates the predicted random effects, u_i.  These are the
best linear unbiased predictions of the random effects, also known as
the empirical Bayes (or posterior mean) estimates of the random effects,
or as shrunken residuals.

{phang}{cmd:ustandard} calculates the standardized predicted random effects,
i.e., the predicted random effects, u_i,
divided by their (unconditional) standard errors.
These may be useful for diagnostics and model checking.

{phang}{cmd:xbu} calculates the prediction including random effects, a +
B*x_i + u_i, also known as the empirical Bayes estimates of the effects for
each study.

{phang}{cmd:stdxbu} calculates the standard error of the prediction including
random effects.

{phang}{cmd:hat} calculates the leverages (the diagonal elements of the
projection hat matrix).


{title:Saved results}

{pstd}When the {cmd:permute()} option is not specified, {cmd:metareg} saves
the following in {cmd:e()}:

{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Scalars}{p_end}
{synopt:{cmd:e(N)}}number of observations{p_end}
{synopt:{cmd:e(df_m)}}model degrees of freedom{p_end}
{synopt:{cmd:e(df_Q)}}degrees of freedom for test of Q=0{p_end}
{synopt:{cmd:e(df_r)}}residual degrees of freedom (if t tests used){p_end}
{synopt:{cmd:e(remll)}}REML log likelihood{p_end}
{synopt:{cmd:e(chi2_c)}}chi^2 for comparison test{p_end}
{synopt:{cmd:e(F)}}model F statistic{p_end}
{synopt:{cmd:e(tau2)}}estimate of tau2{p_end}
{synopt:{cmd:e(Q)}}Cochran's Q{p_end}
{synopt:{cmd:e(I2)}}I-squared{p_end}
{synopt:{cmd:e(q_KH)}}Knapp-Hartung variance modification factor{p_end}
{synopt:{cmd:e(remll_c)}}REML log likelihood, comparison model{p_end}
{synopt:{cmd:e(tau2_0)}}tau2, constant-only model{p_end}
{synopt:{cmd:e(chi2)}}model chi^2{p_end}

{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Macros}{p_end}
{synopt:{cmd:e(cmd)}}{cmd:metareg}{p_end}
{synopt:{cmd:e(predict)}}program used to implement {cmd:predict}{p_end}
{synopt:{cmd:e(wsse)}}name of {cmd:wsse()} variable{p_end}
{synopt:{cmd:e(depvar)}}name of dependent variable{p_end}
{synopt:{cmd:e(method)}}{cmd:REML}, {cmd:Method of moments}, or 
            {cmd:Empirical Bayes}{p_end}
{synopt:{cmd:e(properties)}}{cmd:b V}{p_end}

{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Matrices}{p_end}
{synopt:{cmd:e(b)}}coefficient vector{p_end}
{synopt:{cmd:e(V)}}variance-covariance matrix of estimators{p_end}

{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Functions}{p_end}
{synopt:{cmd:e(sample)}}marks estimation sample{p_end}
{p2colreset}{...}


{title:Examples}

{phang}{cmd:. metareg logrr latitude, wsse(selogrr) eform}

{phang}{cmd:. metareg logrr latitude, wsse(selogrr) graph}

{phang}{cmd:. metareg smd abstract duration itt, wsse(sesmd) permute(10000)}

{phang}{cmd:. metareg smd abstract duration itt, wsse(sesmd) permute(1000,}
          {cmd: univariable)}

{phang}{cmd:. xi: metareg logor i.group, wsse(selogor) permute(1000, joint(i.group))}


{title:Note}

{pstd} {cmd:metareg} is programmed as a Stata estimation command and so
supports many of the commands listed under {helpb estcom} and 
{helpb postest} (except when the {cmd:permute()} option is used).  One
deliberate exception is {helpb lrtest}, which is not appropriate after
{cmd:metareg} (because the REML log likelihood cannot be used to compare models
with different fixed effects, while the method of moments is not based on a
likelihood).  For this reason, when the REML method is used, the iteration log
showing the log likelihood is suppressed by default; specify
the {cmd:log} option if you wish to see it.


{title:References}

{phang}DerSimonian, R., and N. Laird. 1986. Meta analysis in clinical trials.
{it:Controlled Clinical Trials} 7: 177-188.

{phang} Higgins, J. P. T, and S. G. Thompson. 2004. Controlling the risk of
spurious findings from meta-regression. {it:Statistics in Medicine} 23:
1663-1682.

{phang} Knapp, G., and J. Hartung. 2003. Improved tests for a random effects
meta-regression with a single covariate. {it:Statistics in Medicine} 22:
2693-2710.

{phang} Morris, C. N. 1983. Parametric empirical Bayes inference: Theory and
applications. {it:Journal of the American Statistical Association} 78: 47-55.

{phang} Sharp, S. 1998. sbe23: Meta-analysis regression. {it:Stata Technical Bulletin} 42: 16-22. Reprinted in {it:Stata Technical Bulletin Reprints}, vol. 7, pp. 148-155. College Station, TX: Stata Press.


{title:Author}

{pstd}Roger M. Harbord{p_end}
{pstd}Department of Social Medicine{p_end}
{pstd}University of Bristol, UK{p_end}
{pstd}{browse "mailto:roger.harbord@bristol.ac.uk":roger.harbord@bristol.ac.uk}{p_end}


{title:Acknowledgments}

{pstd}
This is a substantial revision of the original version of {cmd:metareg}
written by Stephen Sharp (1998), who gave his permission to release this
version under the same name and to incorporate his code.  Julian Higgins gave
advice on the permutation test.  Aijing Shang tested early versions and made
helpful suggestions.  Portions of the new code borrow ideas from official
Stata commands such as {cmd:nbreg}, and I thank StataCorp for making such code
visible to the user. 

{pstd}
A dialog box, written by Thomas J. Steichen, is available for the original
version of the {cmd:metareg} command.


{title:Also see}

{psee} 
Article: {it:Stata Journal}, volume 8, number 4: {browse "http://www.stata-journal.com/article.html?article=up0023":sbe23_1},{break}
         {it:Stata Technical Bulletin} 42: {browse "http://www.stata.com/products/stb/journals/stb42.pdf":sbe23}

{psee}
Manual:  {hi:[R] meta},{break}
{hi:[R] permute}

{psee}
Online:  {manhelp vwls R}, {manhelp permute R}, {helpb meta} (if installed),
{helpb metan} (if installed), {help meta_dialog} (if installed)
{p_end}