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From |
Joseph Coveney <jcoveney@bigplanet.com> |

To |
Statalist <statalist@hsphsun2.harvard.edu> |

Subject |
Re: st: bivariate random effects meta-analysis of diagnostic test |

Date |
Fri, 30 Apr 2004 18:39:30 +0900 |

Ben Dwamena asked earlier this month about using bivariate random effects modeling in meta-analysis of accuracy indexes of diagnostic tests. >I was interested in using bivariate random effects regression to >meta-analyze simultaneously sensitivity and specificity ( logit >transforms) as correlated heterogeneous outcomes versus a number of >covariates such as study quality, sample size, prevance of disease and >other clinical/methodologic characteristics. This has been done by other >investigators using Proc Mixed in SAS. How may this be done with Stata? >Gllamm? mvreg?. Thanks I had posted some possibilities with Stata, acknowledging not knowing what other investigators had done with SAS's PROC MIX in this regard. As a follow-up, Gerben ter Riet provided partial citations to two literature articles for the method, and posted some control language that his colleagues use to implement bivariate random effects meta-analysis using PROC MIX. I tracked down the articles and finally received photocopies of them yesterday. The full citations are: H. C. van Houwelingen & K. H. Zwinderman, A bivariate approach to meta-analysis. _Statistics in Medicine_ 12:2273-84, 1993. H. C. van Houwelingen, L. R. Arends & T. Stijnen, Tutorial in biostatistics. Advanced methods in meta-analysis: multivariate approach and meta-regression. _Statistics in Medicine_ 21:589-624, 2002. There is good news and bad news. First, the bad news. The suggestions that I posted are not for what Ben's other investigators meant by "bivariate random effects meta-analysis." As a rough analogy, bivariate random effects meta-analysis is to what I posted as MANOVA is to repeated-measures ANOVA. The good news is that -gllamm- can indeed handily perform bivariate random effects meta-analysis. The do-files below show how, and reproduce the results in each of the two articles that Gerben cites. Convergence is rather rapid as far as -gllamm- analyses go. A few notes: First, the first do-file below follows the method that van Houwelingen and his colleagues use in the first article above, with the exception that the do-file doesn't add 0.5 to both successes (failures) and trials when there are zero successes (failures). -gllamm- doesn't seem bothered by the occasional clinical study with a zero in one or two cells of the fourfold table. So, without the kludge of adding small increments "to avoid degeneracy" (from the first article, p. 2274), the results from the first do-file below occasionally differ in the third decimal place from the values given in Table III (p. 2281) of the first article. Second, the method that van Houwelingen and his colleagues use in the second article is, as Ben and Gerben describe, a logit transformation followed by linear mixed-effects regression (using PROC MIXED) with the residual variance fixed at the Woolf-formula value of the log odds. The second do-file below implements this in Stata, again, using -gllamm-. Realize, however, that this method is not the same as that in their first article, which was by maximizing the full likelihood. (They wrote GAUSS code to do this maximum likelihood estimation for the first article.) If you use the first article's approach on the second article's example dataset, then the results will differ from the results published in the second article using the PROC MIXED approach. I'm not sure why van Houwelingen and his colleagues chose to use PROC MIXED and the fixed, "known"-residual-variance expedient for the second article, instead of, say, PROC NLMIXED (which would be like -gllamm-), or perhaps the GLIMMIX macro (with a caveat about potential for bias in some circumstances with binomial data). Their first article's approach (with -gllamm-) seems more elegant and and would seem more accurate than that using PROC MIXED. Third, the SAS control language that Gerben posted indicates that his colleagues at his institution use PROC MIX's default restricted maximum likelihood (REML) estimation method instead of maximum likelihood. In the second article above, van Houwelingen in contrast deliberately uses maximum likelihood with PROC MIX in order to be able to estimate likelihood-ratio confidence intervals around the parameter estimates, admonishing that REML won't allow this. (One of the advantages that van Houwelingen and his colleagues mention for the bivariate random effects approach using maximum likelihood method in the first place was that it is able to avoid using Wald confidence intervals for parameter estimates; likelihood ratio confidence intervals accommodate the variance components' contributions.) Likelihood-ratio confidence intervals are estimated by profiling the likelihood. Although I haven't shown it below, -gllamm-'s -eval from()- options enable this. Last, one of the motivations of the bivariate random effects meta-analysis for van Houwelingen and his colleagues was to allow the log odds of the control treatment group and of the experimental treatment group to be individually estimated. This enables examination of the baseline risk ("underlying risk") among studies independently of the treatment effect. The merits and demerits of this are discussed, among other places, in an online article accessible without charge to anyone who's interested at http://bmj.bmjjournals.com/cgi/content/full/313/7059/735 . (van Houwelingen's approach is broached in the article.) Joseph Coveney ---------------------------------------------------------------------------- clear set more off /* The dataset is from R. Collins & M. Langman, Treatment with histamine H1 antagonists in acute upper gastrointestinal hemorrhage. _New England Journal of Medicine_ 313:660-66, 1985, cited in H. C. van Houwelingen & K. H. Zwinderman, A bivariate approach to meta-analysis. _Statistics in Medicine_ 12:2273-84, 1993. */ input trial n_T X_T n_C X_C 1 259 50 260 51 2 153 44 132 30 3 106 16 107 15 4 78 14 80 21 5 51 16 54 15 6 56 11 53 12 7 50 8 50 16 8 40 9 48 11 9 33 12 36 10 10 46 5 47 12 11 31 6 29 12 12 33 6 39 7 13 36 11 26 5 14 21 10 19 8 15 20 7 20 8 16 45 9 43 3 17 34 3 31 13 18 24 4 24 5 19 14 2 15 6 20 15 2 14 3 21 18 5 12 1 22 10 1 9 4 23 10 3 11 1 24 18 0 19 3 25 14 0 14 0 end compress reshape long n_ X_, i(trial) j(trt) string label define Group 0 C 1 T encode trt, generate(grp) label(Group) drop trt rename grp trt rensfix _ /* Actual model fitting begins here. */ quietly tabulate trt, generate(grp) eq grp1: grp1 eq grp2: grp2 gllamm X grp1 grp2, nocons i(trial) nrf(2) eqs(grp1 grp2) /// family(binomial) denom(n) link(logit) adapt trace allc exit ---------------------------------------------------------------------------- clear set more off /* The dataset is from G. A. Colditz, F. B. Brewer, C. S. Berkey, E. M. Wilson, E. Burdick, H. V. Fineberg & F. Mosteller, Efficacy of BCG vaccine in the prevention of tuberculosis. _Journal of the American Medical Association_ 271:698-702, 1994, cited in H. C. van Houwelingen, L. R. Arends and T. Stijnen, Tutorial in Biostatistics. Advanced methods in meta-analysis: multivariate approach and meta-regression. _Statistics in Medicine_ 21:589-624, 2002. */ input trial vd vwd nvd nvwd 1 4 119 11 128 2 6 300 29 274 3 3 228 11 209 4 62 13536 248 12619 5 33 5036 47 5761 6 180 1361 372 1079 7 8 2537 10 619 8 505 87886 499 87892 9 29 7470 45 7232 10 17 1699 65 1600 11 186 50448 141 27197 12 5 2493 3 2338 13 27 16886 29 17825 end compress /* This is in the spirit of the SAS data step in the article; a Stata implementation ab initio would be briefer, -reshape-d long first. */ generate float lno1 = ln(vd/ vwd) generate float lno0 = ln(nvd / nvwd) generate float var1 = 1/vd + 1/vwd generate float var0 = 1/nvd + 1/nvwd /* The article is confusing in that it describes that the standard errors are stored in the PARMSDATA file (p. 602), but what's stored are really variance estimates. For stability, -gllamm- parametrizes the residual variance as logarithms of the standard error, so . . . */ generate float lsd1 = ln(sqrt(var1)) generate float lsd0 = ln(sqrt(var0)) reshape long lno var lsd, i(trial) j(trt) /* The two random effects, one for each treatment. */ quietly tabulate trt, generate(grp) eq grp1: grp1 eq grp2: grp2 /* Setting up the fixed ("known") residual (first-level) variances. */ egen int cell = group(trial trt) quietly tabulate cell, generate(dum) eq wgt: dum1-dum26 forvalues i = 1/26 { local dummy = lsd[`i'] constraint define `i' [lns1]dum`i' = `dummy' } macro drop dummy /* First-pass estimation. -gllamm- doesn't apply constraints in preliminary estimation of the fixed effects (if -gllamm- decides to use itself for this), so don't allow it to iterate here and get lost trying to estimate as many or more parameters as there are data. */ gllamm lno grp1 grp2, nocons i(trial) nrf(2) eqs(grp1 grp2) /// s(wgt) constraint(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 /// 17 18 19 20 21 22 23 24 25 26) noest matrix A = M_initf /* Fit the model here. */ gllamm lno grp1 grp2, nocons i(trial) nrf(2) eqs(grp1 grp2) /// s(wgt) constraint(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 /// 17 18 19 20 21 22 23 24 25 26) from(A) long adapt trace allc exit ---------------------------------------------------------------------------- * * For searches and help try: * http://www.stata.com/support/faqs/res/findit.html * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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