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help for umeta and umeta_postestimation
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Title

umeta - U-statistics-based random-effects meta-analyses

Description

The umeta command performs U-statistics-based random-effects
meta-analysis on a dataset of univariate, bivariate or trivariate point
estimates, sampling variances, and for bivariate or trivariate data,
within-study correlations or covariances.  The methodology is described
in Ma and Mazumdar (2011).

For each outcome, umeta calculates the overall effect and a confidence
interval for the effect. The command also displays the between-study
variance (or alternatively between-study standard deviation),
between-study correlation(s) for bivariate or trivariate data and
inconsistency (I-squared) statistics.

umeta Syntax

umeta yvar* svar* [wsvar*] [if] [in] [, covvar(string) level(#) predint
tscale(logit|log|asin) noestimates bssd zci i2]

where the data are arranged with one line per study: the point estimates
are held in variables yvar*, the sampling variances are held in svar*,
and within-study correlations (or covariances) for 2 or 3 outcomes are
held in variable wsvar*.

For univariate data, yvar* is yvar and svar* is svar

For bivariate data, yvar* is yvar1 yvar2, svar* is svar1 svar2 and wsvar*
is wsvar12

For trivariate data, yvar* represents yvar1 yvar2 yvar3, svar* is svar1
svar2 svar3 and wsvar* is wsvar12, wsvar13 wsvar23

For any unreported outcomes, umeta sets the outcome and its variance at 0
and 1E12, respectively.

Options for umeta

covvar(string) For bivariate or trivariate data analysis, you must
specify covvar(rho) or covvar(cov) depending on whether you are using
within-study correlation(s) or covariance(s).

level(#) specifies the significance level for probability intervals.

predint displays outcome-specific mean estimates with the probability
interval of the approximate predictive distribution of a future
trial, based on the extent of heterogeneity. No method has been
developed as yet for multivariate predictive distribution.

tscale(logit|log|asin) transformation of estimates to original scale, if
data was transformed prior to analysis.

bssd reports the between-study standard deviations with confidence
intervals (calculated as a function of inconsistency statistic and
typical within-study variance as by White(2009)) instead of the
default between-study variances.

noestimates prevents display of mean estimates, between-study variances
(or standard deviations) and correlation(s)

zci uses z-statistics instead of default t-statistics for confidence
interval calculation. This is overriden if option predint specified.

i2 reports I-squared statistic for each outcome, together with confidence
intervals as is described in White(2009).

umeta, typed without specifying varlist, redisplays the latest estimation
results.  All the output options listed above may be used

by...: or statsby...: may be used with umeta to perform subgroup
analyses; see help by or statsby.

Remarks

Multivariate meta-analysis is used to synthesize multiple outcomes
simultaneously taking into account the correlation between the outcomes
(Riley(2009)). Likelihood based approaches, in particular, Restricted
Maximum Likelihood (REML) method is commonly utilized in this context.
REML assumes a multivariate normal distribution for the random-effects
model. This assumption is difficult to verify, especially for
meta-analysis with small number of component studies. Use of REML also
requires iterative estimation between parameters, needing moderately high
computation time, especially when the dimension of outcomes is large
(White(2009)).  Jackson, White and Thompson(2010) have developed a
multivariate method of moments (MMM) which has been shown to perform
equally well to REML.

Ma and Mazumdar recently proposed a new method for multivariate
meta-analysis based on the theory of U-statistic.  The motivation for
using U-statistic stems from the fact that it provides a a robust,
nonparametric and noniterative approach.  Additionally, the asymptotic
behavior of the related statistics and their estimates are easy to derive
being based on theorems already available for U-statistics.

Since the between-study variance matrix for the random-effects
meta-analysis model involves second order moments, U-statistic
formulation is especially beneficial. It is easily applied to estimate
the variance matrix components and to develop their joint asymptotic
distribution for related inference. Because the U-statistic-based method
does not depend on parametric distributional assumptions for both random
effects and sampling errors, it provides robust inference irrespective of
the data distribution

For a detailed description of the u-statistic methodology, see Ma and
Mazumdar (2011).

By convention, the within-study variances are assumed known and replaced
by their sample estimates. Thus imprecision in within-study variance
estimates may affect the estimation of pooled effect size especially when
the size of within-study variation is relatively large.

This program does not assume that variables need log, logit or arcsin or
other transformation(s).  However, if study-level outcome data are
available as odds ratios, risk ratios or proportions, the user may choose
to log-, logit-or arcsin-transform them first. Then tscale option may be
used to change back to the original scale for reporting if so desired.

The probability interval of the approximate predictive distribution of a
future trial, is based on the extent of heterogeneity. This incorporates
uncertainty in the location and spread of the random effects distribution
using the formula t(df) x sqrt(se2 + tau2) where t is the t-distribution
with n-2 degrees of freedom, se2 is the squared standard error and tau2
the heterogeneity statistic and n is the number of observations(studies).
This is applied to each outcome separately.  For further information see
Higgins, Thompson and Spiegelhalter(2009)

I-squared formulated by Higgins and Thompson (2002), describes the
percentage of total variation across studies that is attributable to
heterogeneity rather than chance and measures impact of heterogeneity.  .
Negative values of I-squared are made equal to 0 so that I-squared lies
between 0% and 100%.  A value of 0% indicates no observed heterogeneity,
and values greater than 50% may be considered substantial heterogeneity.
The main advantage of I-squared is that it does not inherently depend on
the number of the studies in the meta-analysis

Examples

Example 1: Univariate Data

. use umeta_example1, clear

. list yvar svar, clean noobs
. umeta yvar svar

Example 2: Bivariate logit-transformed Data, No within-study correlation

. use umeta_example2, clear

. list yvar* svar* rho*, clean noobs

. umeta yvar* svar* rho*, p

. umeta yvar* svar* rho*, z bssd p tscale(logit)

Example 3: Bivariate Outcomes with missing Data

. use umeta_example3, clear

. list yvar* svar* rho*, clean noobs

. umeta yvar* svar* rho*

. umeta yvar* svar* rho*, pred

. umeta, noest i2 z q

Example 4: Trivariate Outcomes with Zero within-study covariance matrix

. use umeta_example4, clear

. list yvar* svar* rho*, clean noobs

. umeta yvar* svar* rho*

. umeta, noest i2 z q

Example 5: Trivariate Outcomes with within-study correlations

. use umeta_example5, clear

. list yvar* svar* rho*, clean noobs

. umeta yvar* svar* rho*, pred

Saved results

umeta saves the following in e():

Scalars
e(N)                    number of observations
e(dims)                 number of outcomes for meta-analysis
e(df_r)                 degrees of freedom for meta-analysis estimation
e(Qdf)                  degrees of freedom for homogeneity testing

Macros
e(cmd)                  umeta
e(cmdline)              command as typed
e(properties)           b V
e(yvars)                names of study-specific outcome variables
(point estimates)
e(svars)                names of study-specific sampling variances
e(predict)              program used to implement predict

Matrices
e(b)                    coefficient vector
e(V)                    variance-covariance matrix of the estimators
e(Isqmat)               matrix of outcome-specific I^2 values
e(Qmat)                 matrix of outcome-specific heterogeneity
statistic
e(Vtyp)                 typical within-study variance
e(Sigma)                between-study variance-covariance matrix
e(svars)                matrix of study-specific sampling variances
e(rho)                  matrix of between-study correlation
e(yvars)                matrix of study-specific point estimates

Functions
e(sample)               marks estimation sample

Authors
Ben A. Dwamena, Department of Radiology, Division of Nuclear Medicine,
University of Michigan Medical School, Ann Arbor, Michigan

Yan Ma, Hospital for Special Surgery, Weill Medical College of Cornell
University, New York, New York

programming problems:
bdwamena@umich.edu.

u-statistic-based questions:
yam2007@med.cornell.edu.

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Title

umeta postestimation --   Postestimation tools for umeta

Description

umeta is programmed as an Stata estimation command and so supports many of
the commands listed under help estcom and postest.  The following standard
postestimation commands may be particularly useful:

Command         Description
-------------------------------------------------------------------------
estat           VCE and estimation sample summary. See help estat
estimates       Cataloging estimation results. See help estimates
lincom          Point estimates, standard errors, testing, and inference
for linear combinations of coefficients. See lincom
nlcom           Point estimates, standard errors, testing, and inference
for nonlinear combinations of coefficients. See nlcom
predict         predictions, residuals, influence statistics, and other
diagnostic measures
test            Wald tests of linear hypotheses. See help test
testnl          Wald tests of non-linear hypotheses. See help testnl
-------------------------------------------------------------------------

predict Syntax

The syntax of predict following umeta is

syntax 1:

predict [type] newvarname [if exp] [in range] [, statistic ]

syntax 2:

predict newvarname [if exp] [in range] [, statistic show(string)]

-------------------------------------------------------------------------
statistic      Description
-------------------------------------------------------------------------
fixed        prediction of fixed-effects; the default
stfixed      standard error of the fixed-effects prediction
fitted       prediction including random effects
stfit        standard error of fitted
stdf         standard error of the forecast
reffects     predicted random effects
reses        standard error of predicted random effects
rstandard    standardized predicted random effects
lev          leverage (diagonal elements of projection matrix)
cooksd       Cook's influence measure
-------------------------------------------------------------------------

These statistics are available both in and out of sample; type "predict ...
if e(sample) ..." if wanted only for the estimation sample.

-------------------------------------------------------------------------
show           Description
-------------------------------------------------------------------------
clean        force table format with no divider or separator lines
table        force table format
abbreviate(#)
abbreviate variable names to # characters; default is
ab(8)
noobs        do not list observation numbers
divider      draw divider lines between columns
separator(#) draw a separator line every # lines; default is
separator(5)
-------------------------------------------------------------------------

Options for predict

fixed calculates the linear prediction for the fixed portion of the model.

stfixed calculates the outcome-specific standard error of the fixed-portion
linear prediction

stfitted calculates the outcome-specific standard error of the prediction
including random effects.

fitted calculates the outcome-specific prediction including random effects,
Xb[i] + u[i], also known as the empirical Bayes estimates of the
effects in each study.

stdf calculates the outcome-specific standard error of the forecast.  This
gives the standard deviation of the predicted distribution of the true
value of depvar in a future study
stdf^2 = stdp^2 + tau2.

reffects calculates the outcome-specific best linear unbiased predictions
(BLUPs) of the random effects, also known as the posterior mean or
empirical Bayes estimates of the random effects, or as shrunken
residuals.

reses calculates the outcome-specific standard error of predicted random
effects.

rstandard calculates the outcome-specific 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.

lev calculates the study-specific leverages

cooksd calculates the study-specific Cook's influence statistic.

Remarks

Similar to other types of data, it is not uncommon to observe extreme
effect size values when conducting a meta-analysis.  As the main
objective of a meta-analysis is to provide a reasonable summary of the
effect sizes of a body of empirical studies, the presence of such
outliers may distort the conclusions of a meta-analysis.  Moreover, if
the conclusions of a meta-analysis hinge on the data of only one or two
influential studies, then the robustness of the conclusions are called
into question.  Researchers, therefore, generally agree that the effect
sizes should be examined for potential outliers and influential cases
when conducting a meta-analysis.

The most thorough treatment of outlier diagnostics in the context of
meta-analysis to date can be found in the classic book by Hedges and
Olkin, who devoted a whole chapter to diagnostic procedures for effect
size data.  However, the methods developed by Hedges and Olkin(1985) are
only applicable to fixed-effects models.  Given that random- and
mixed-effects models are gaining popularity in the meta-analytic context,
corresponding methods for outlier and influential case diagnostics need
to be developed.

Viechtbauer and Cheung(2010) have introduced several outlier and
influence diagnostic procedures for the random- and mixed-effects model
in meta-analysis.  These procedures are logical extensions of the
standard outlier and case-deletion influence diagnostics for regular
regression models as in Demidenko and Stukel(2005) and take both sampling
variability and between-study heterogeneity into account. The proposed
measures provide a simple framework for evaluating the potential impact
of outliers or influential cases in meta-analysis.

Examples

. use umeta_example5, clear

. umeta yvar* svar* rho*

. predict lev, lev show(clean)

. predict cook, cooksd show(clean)

. predict fit, fit

. predict fix

. predict reff, reff show(clean noobs)

. predict res, res

. predict rst, rst

. predict stpred, stfit

. predict double stdf, stdf

Author
Ben A. Dwamena, Department of Radiology, Division of Nuclear Medicine,
University of Michigan Medical School, Ann Arbor, Michigan.
bdwamena@umich.edu.

References

Demidenko, E., T. A. Stukel. 2005 Influence analysis for linear
mixed-effects models Statistics in Medicine 24: 893–909

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

Hedges LV, I. Olkin. 1985.  Statistical Methods for Meta-Analysis

Higgins, J. P. T., and S. G. Thompson.  2002.  Quantifying heterogeneity
in a meta-analysis.  Statistics in Medicine 21: 1539-1558.

Higgins, J. P. T., S. G. Thompson, and D. J. Spiegelhalter.  2009.  A
re-evaluation of random-effects meta-analysis.  Journal of the Royal
Statistical Society, Series A 172: 137-159.

Jackson, D., I. R. White, and S. G. Thompson. 2010.  Extending
DerSimonian and Laird's methodology to perform multivariate random
effects meta-analyses.  Statistics in Medicine 29: 1282-1297.

Ma, Y., and M. Mazumdar. 2011. Multivariate meta-analysis:  a robust
approach based on the theory of U-Statistic.  Statistics in Medicine
30: 2911-2929.

Riley, R. D.  2009.  Multivariate meta-analysis: The effect of ignoring
within-study correlation.  Journal of the Royal Statistical Society,
Series A 172: 789-811.

Viechtbauer, W., M. W.-L. Cheung. 2010.  Outlier and influence
diagnostics for meta-analysis.  Research Synthesis Methods 1:
112-125.

White, I. R.  2009.  Multivariate random-effects meta-analysis.  Stata
Journal 9: 40-56.

Also see

Help: mvmeta (if installed)

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