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Text S1. Statistical appendix on the calculation of 95% confidence intervals for corrected programmelevel mortality by web calculator at www.iedeasa.org.
We implemented a web calculator that directly calculates the corrected, programmelevel mortality with 95% confidence intervals (95% CI) using the two methods presented in the main paper by Egger et al (PLoS Medicine 2010): a) the tracing method and b) the meta method. The web calculator can be found at HYPERLINK "http://www.iedeasa.org"www.iedeasa.org (click on tab Tools and calculators to get to the web calculator). The user chooses the method to use and then enters the required input values.
In this statistical appendix we provide details on the calculation of the 95% CIs. These are computed using the Monte Carlo method with k=100,000 draws, thus allowing for uncertainty in the estimation of i) mortality in patients remaining in care; ii) the proportion of patients lost to followup; and iii) mortality in patients lost to followup. For i) and iii) sampling is based on the asymptotic normal distribution of log(log) transformed survival (1mortality) estimates ADDIN REFMGR.CITE Kalbfleisch2002182710The Statistical Analysis of Failure Time DataBook, Whole182710The Statistical Analysis of Failure Time DataKalbfleisch,J.D.Prentice,R.L.2002Cox modelsCox regressionMEggerMULTIVARIATEANALYSISstatistical methodologystatisticsstatisticalanalysisFAILURETimedataNot in File182nd EditionNew York, N.Y.John Wiley and Sons2[1].
Let
MNL = Mortality observed in patients retained in care (not lost to followup)
SNL = Survival observed in patients retained in care (not lost to followup)
ML = Mortality estimated in patients lost to followup
SL = Survival estimated in patients lost to followup
r = Proportion lost to followup
The steps of the Monte Carlo procedure are as follows:
Tracing method
Based on estimates of MNL with 95% CI (MNL_UB and MNL_LB for upper and lower bounds, respectively), k deviates MNL_i (i = 1,...,k) of mortality among patients remaining in care are sampled. To do this, first sample sNL_i (i = 1,...,k) from the estimated normal distribution of log(log) transformed oneyear survival. The mean of this distribution is estimated as log(log(1MNL)) and the standard deviation by log(log(1MNL_UB)) log(log(MNL_LB)) divided by 21.96. The sNL_i are then back transformed to obtain deviates of survival SNL_i = exp(exp(sNL_i)) and, hence, of mortality MNL_i = (1 SNL_i).
Sample k deviates ri (i = 1,...,k) of the proportion of lost to followup. To do this, sample deviates NL_i (i = 1,...,k) of numbers lost to followup from the binomial distribution with parameters n = NR and p = NL /NR, where NR and NL are the number of eligible patients and the number of patients lost to followup, respectively. The ri are obtained from ri = NL_i /NR.
Sample k deviates ML_i (i = 1,...,k) of oneyear mortality among patients lost to followup analogously to a) using user input for ML, ML_UB and ML_LB.
Obtain k deviates of corrected oneyear mortality MC_i =(1 ri) MNL_i + ri ML_i (i = 1,...,k).
The corrected estimate of oneyear mortality and the 95%CI lower and upper bounds which are given as output correspond to the 50 (median), 2.5 and 97.5 percentiles of the sampled deviates MC_i (i = 1,...,k).
Meta method
This procedure is identical to described for 1) Tracing method, with the exception that step c) is replaced by the following:
For each deviate ri (i = 1,...,k) obtained in b) sample a deviate ML_i of oneyear mortality among patients lost to followup using the meta regression. To do this, sample deviates mL_i from the normal distribution for the logit of mortality among patients lost to followup for a programme with proportion of lost to followup equal to ri taking into account the sampling variability of the meta regression parameters. The mean of this distribution is estimated as a + bri and the variance as (aa + (abri + (bbri2 +(2 where a and b are the constant and slope parameter, (aa, (bb and (ab are the variances and covariance of these parameters respectively, and (2 is the between programme variance from the fitted meta regression. ADDIN REFMGR.CITE Brinkhof2009198107Mortality of patients lost to followup in antiretroviral treatment programmes in resourcelimited settings: systematic review and metaanalysisJournal198107Mortality of patients lost to followup in antiretroviral treatment programmes in resourcelimited settings: systematic review and metaanalysisBrinkhof,M.W.PujadesRodriguez,M.Egger,M.2009AabstractADULTADULTSafricaAIDSandantiretroviral therapyantiretroviral treatmentArtArticlebutCAREcaribbeancauses of deathchildrenCombinedCONFIDENCEconfidence intervaldataDatabasesdeathEmbaseEnvironmental HealthEvaluationFollow UpfollowupGov'thealthheterogeneityHomeIndexesINDIALiteraturemedicineMEDLARSmeta analysismetaanalysisMETAANALYSISMethodsmonitoringMORTALITYoutcomeoutcomesPATIENTPatient CarepatientsPreventive MedicinePubMedRandom effectsresearchResearch SupportRESOURCELIMITED SETTINGSREVIEWSciencesocialStudySUBSAHARAN AFRICASupportSWITZERLANDsystematic reviewtelephonetherapyTreatmentUniversitiesUniversityWHONot in Filee5790PLoS One46Division of International and Environmental Health, Institute of Social and Preventive Medicine (ISPM), University of Bern, Bern, Switzerland. brinkhof@ispm.unibe.chPM:19495419PLoS One1[2] The ML_i are then obtained from ML_i = exp(mL_i) / (1+ exp(mL_i)).
ADDIN REFMGR.REFLIST References
1. Kalbfleisch, J. D. and Prentice, R. L. (2002) The Statistical Analysis of Failure Time Data. New York, N.Y.: John Wiley and Sons.
2. Brinkhof MW, PujadesRodriguez M, Egger M (2009) Mortality of patients lost to followup in antiretroviral treatment programmes in resourcelimited settings: systematic review and metaanalysis. PLoS One 4: e5790.
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