-------------------------------------------------------------------------------help for umeta and umeta_postestimation-------------------------------------------------------------------------------

Title

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

DescriptionThe

umetacommand 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,

umetacalculates 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.

umetaSyntax

umetayvar*svar*[wsvar*] [if] [in] [,covvar(string)level(#)predinttscale(logit|log|asin)noestimatesbssdzcii2]where the data are arranged with one line per study: the point estimates are held in variables

yvar*, the sampling variances are held insvar*, and within-study correlations (or covariances) for 2 or 3 outcomes are held in variablewsvar*.For univariate data,

yvar*isyvarandsvar*issvarFor bivariate data,

yvar*isyvar1 yvar2,svar*issvar1 svar2andwsvar*iswsvar12For trivariate data,

yvar*representsyvar1 yvar2 yvar3,svar*issvar1svar2 svar3andwsvar*iswsvar12, wsvar13 wsvar23

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

Options forumeta

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

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

predintdisplays 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.

bssdreports 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.

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

zciuses z-statistics instead of default t-statistics for confidence interval calculation. This is overriden if optionpredintspecified.

i2reports 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...:orstatsby...:may be used withumetato perform subgroup analyses; see help by or statsby.

RemarksMultivariate 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

tscaleoption 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 svarExample 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 qExample 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

umetasaves the following ine():Scalars

e(N)number of observationse(dims)number of outcomes for meta-analysise(df_r)degrees of freedom for meta-analysis estimatione(Qdf)degrees of freedom for homogeneity testingMacros

e(cmd)umetae(cmdline)command as typede(properties)b Ve(yvars)names of study-specific outcome variables (point estimates)e(svars)names of study-specific sampling variancese(predict)program used to implementpredictMatrices

e(b)coefficient vectore(V)variance-covariance matrix of the estimatorse(Isqmat)matrix of outcome-specific I^2 valuese(Qmat)matrix of outcome-specific heterogeneity statistice(Vtyp)typical within-study variancee(Sigma)between-study variance-covariance matrixe(svars)matrix of study-specific sampling variancese(rho)matrix of between-study correlatione(yvars)matrix of study-specific point estimatesFunctions

e(sample)marks estimation sample

Ben A. Dwamena, Department of Radiology, Division of Nuclear Medicine, University of Michigan Medical School, Ann Arbor, MichiganAuthorsYan Ma, Hospital for Special Surgery, Weill Medical College of Cornell University, New York, New York

bdwamena@umich.edu.programming problems:

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

-------------------------------------------------------------------------------

Title

umeta postestimation-- Postestimation tools for umeta

Descriptionumeta 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 -------------------------------------------------------------------------

estatVCE and estimation sample summary. See help estatestimatesCataloging estimation results. See help estimateslincomPoint estimates, standard errors, testing, and inference for linear combinations of coefficients. See lincomnlcomPoint estimates, standard errors, testing, and inference for nonlinear combinations of coefficients. See nlcompredictpredictions, residuals, influence statistics, and other diagnostic measurestestWald tests of linear hypotheses. See help testtestnlWald tests of non-linear hypotheses. See help testnl -------------------------------------------------------------------------

predictSyntaxThe syntax of predict following

umetais

syntax 1:

predict[type]newvarname[ifexp] [inrange] [,statistic]

syntax 2:

predictnewvarname[ifexp] [inrange] [,statisticshow(string)]-------------------------------------------------------------------------

statisticDescription -------------------------------------------------------------------------fixedprediction of fixed-effects; the defaultstfixedstandard error of the fixed-effects predictionfittedprediction including random effectsstfitstandard error offittedstdfstandard error of the forecastreffectspredicted random effectsresesstandard error of predicted random effectsrstandardstandardized predicted random effectslevleverage (diagonal elements of projection matrix)cooksdCook'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.

-------------------------------------------------------------------------

showDescription -------------------------------------------------------------------------cleanforce table format with no divider or separator linestableforce table formatabbreviate(#)abbreviate variable names to#characters; default isab(8)noobsdo not list observation numbersdividerdraw divider lines between columnsseparator(#)draw a separator line every#lines; default isseparator(5)-------------------------------------------------------------------------

Options forpredict

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

stfixedcalculates the outcome-specific standard error of the fixed-portion linear prediction

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

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

stdfcalculates the outcome-specific standard error of the forecast. This gives the standard deviation of the predicted distribution of thetruevalue ofdepvarin a future study stdf^2 = stdp^2 + tau2.

reffectscalculates 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.

resescalculates the outcome-specific standard error of predicted random effects.

rstandardcalculates 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.

levcalculates the study-specific leverages

cooksdcalculates the study-specific Cook's influence statistic.

RemarksSimilar 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

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

ReferencesDemidenko, E., T. A. Stukel. 2005 Influence analysis for linear mixed-effects models

Statistics in Medicine24: 893–909DerSimonian, R., and N. Laird. 1986. Meta-analysis in clinical trials.

Controlled Clinical Trials7: 177-188.Hedges LV, I. Olkin. 1985.

Statistical Methods for Meta-AnalysisAcademic Press: New York.Higgins, J. P. T., and S. G. Thompson. 2002. Quantifying heterogeneity in a meta-analysis.

Statistics in Medicine21: 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 RoyalStatistical Society, Series A172: 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 Medicine29: 1282-1297.Ma, Y., and M. Mazumdar. 2011. Multivariate meta-analysis: a robust approach based on the theory of U-Statistic.

Statistics in Medicine30: 2911-2929.Riley, R. D. 2009. Multivariate meta-analysis: The effect of ignoring within-study correlation.

Journal of the Royal Statistical Society,Series A172: 789-811.

Viechtbauer, W., M. W.-L. Cheung. 2010. Outlier and influence diagnostics for meta-analysis.

Research Synthesis Methods1: 112-125.White, I. R. 2009. Multivariate random-effects meta-analysis.

StataJournal9: 40-56.

Also see

Help:

mvmeta(if installed)