Estimation Of Combined Weighted Youden Index In Meta-analysis

Background:In recent years, Evidence Based Medicine (EBM) has been developing rapidly. The results based on meta-analysis have been regarded as most strong evidence to guide clinical practice. Now the meta-analysis evaluating indicators used in the diagnosis test fields, such as combined sensitivity and specificity, SROC curve and so on, can not evaluate the overall effect of the diagnostic test yet. The combined sensitivity and specificity have traditionally been used in the meta-analysis of diagnostic test data. While whether the sensitivity or the specificity only reflects one side of the diagnostic test. And the summary ROC curve which considers the influence of cut-points most used to find the optimum cut-point in meta-analysis of a diagnostic test. While in the real clinical practice we can only choose one cut-point for a certain diagnotisis test. So the sensitivity and specificity information at other cut-points are useless. Youden Index, a combined index, has been proposed for a long time and it has been widely used in the diagnostic test evaluation. While the methods for estimation the Youden Index in meta-analysis are still blank.The Youden index is linear combination of sensitivity and specificity. It considers the sensitivity and specificity have the equal weights in one diagnostic test. In clinical practice, different situations dictate different importance of sensitivity and specificity. Li and Chen proposed the weighted Youden Index to make the application of Youden Index more adaptively. The weighted Youden index was defined as Jw=2[w·SEN+(1-w)SPE]-1, where0≤w≤1.The methods of estimation for weighted Youden index in meta-analysis are also an important issue to study.Objective:In this thesis, we want to establish the methods to estmate the combined Youden index and weighted Youden index in both fixed effect model and random effect model and apply them in meta-analysis. And we also want to establish the methods that could use use to compare two weighted Youden Indexes.Simulation study will be used to compare the statistical characteristics of different estimation methods.Methods:Two assumptions are made for all the estimation methods. Firstly all the studies included are independent. Secondly the sensitivity and specificity both follow a binomial distribution in each study and the sensitivity and specificity are independent.We consider both fixed effect model and random effect model to establish the estimating methods of the combined Youden Index and weighted Youden index. In the fixed effect model we use the max-likelihood estimation, Mantel-Haenszl estimation and weighted least square estimation methods to estimate the combined Youden Index and weighted Youden index. In the random effect model, we establish the weighted least square method to estimate the combined Youden Index and combined weighted Youen Index.We also establish the methods to estimate the difference between two weighted Youden Indexes for comparing two diagnostic tests. We also consider both fixed effect model and random effect model. In the fixed effect model we use the max-likelihood estimation, Mantel-Haenszl estimation and weighted least square estimation methods to estimate the difference between two weighted Youden indexes. And give the test statistic to test the difference between two weighted Youden indexes. In the random effect model, we establish the weighted least square method to do the estimation.We use Monte Carlo simulation to compare the different methods characteristics of the estimation.Results:1. Estimation of combined weighted Youden index.①Fixed effect modelMax-likelihood estimation:According to max-likelihood estimation idea we first build the likelihood function and log-likelihood function. The Newton-Raphson iterative method is used to obtain the estimation of combined Youden index Jw. The varicance is estimated as where Then the two-sided (1-α)%confidence interval of Jw is Jw±Zα2(?). When the weight of weighted Youden index equals0.5, we can obtain the estimation of combined Youden index J.The varicance of J is estimated as where The two-sided(1-α)%confidence interval of J is J±Zα2(?).Mantel-Haenszel estimation:The estimaTion of combined weighted YoudenAnd THe two-sided(1-α)%confidence interval of is JwMH±Zα2(?). When the weight of weighted Youden index equals0.5,the weighted Youden index will simplify as Youden index.The estimation of combined Youden index is two-sided (1-α)%confidence interval of JMH isWeighted least square estimation:The estimation of combined weighted Youden index is where the Wi is estimatied as The variance of Jw/s is And the two-sided (1-α)%confidence interval of Jw/s is When the weight of weighted Youden index equals0.5, the weighted Youden index will simplify as Youden index. The estimation of combined Youden index is where the Wi is estimatied The variance of Jls is And the two-sided (1-α)%confidence interval of Jls is②Random effect modelWeighted least square estimation:In the random effect model, we use the DerSimonian and Laird’s (1986) method to estimate the heterogeneity variance. The heterogeneity variance τ2is estimated as where and The combined weighted Youden index using is estimatied as where Variance of Jw/s is estimated as And the two-sided (1-α)%confidence interval isWhen the weight of weighted Youden index equals0.5, the weighted Youden index will simplify as Youden index. The heterogeneity variance r2is estimated as where and The combined Youden index using the weighted least square method is estimatied as where Variance of Jls is estimated as And the two-sided confidence interval is3. Estimation of difference between two weighted Youden indexes.①Fixed effect modelMax-likelihood estimation:According to max-likelihood estimation idea we also first build the likelihood function and log-likelihood function. We also use the Newton-Raphson iterative method to obtain the estimation of difference between two weighted Youden indexes DJW. The varicance of DJW is estimated as var(DJw). The two-sided (1-α)%confidence interval of DJW is The Z statistic can be used to test the difference between the two weighted Youden indexes,. When the weight of weighted Youden index equals0.5, the method can be used to test the difference between two Youden indexes.Mantel-Haenszel method:The difference between two weighted Youden indexes is estimated as, where The variance of DJwMH is And the two-sided (1-α)%confidence interval of DJwMH can be expressed as The Z statistic can be used to test the difference between the two weighted Youden indexes, When the weight of weighted Youden index equals0.5, the method can be used to test the difference between two Youden indexes.Weighted least square estimation:The estimation of the difference between two weighted Youden indexes is where The variance of DJw/s is And the two-sided (1-α)%confidence interval is The Z statistic can be used to test the difference between the two weighted Youden indexes, When the weight of weighted Youden index equals0.5, the method can be used to test the difference between two Youden indexes.In the random effect model, the estimation of heterogeneity variance τ2is where and The combined difference between two weighted Youden indexes using the weighted least square method is estimatied as where Variance of DJ w/s is estimated as And the1-α confidence interval is The Z statistic can be used to test the difference between the two weighted Youden indexes, When the weight of weighted Youden index equals0.5, the method can be used to test the difference between two Youden indexes.3. Monte Carlo simulation.1) For the estimating of combined Youden index and combined weighted Youden index, we compared the max-likelihood, Mantel-Haenszel and weighted least square methods in the fixed effect model the coverage probability of95%confidence interval using Mantel-Haenszel method is the most closed to95%. Mantel-Haenszel method has the highest accuracy among the three methods. The coverage probability of max-likelihood method is a little lower than95%when the sample size is small. While the coverage probability is more closed to95%when the sample size growed. The coverage of weighted least square method is lower than the Mantel-Haenszel method and max-likelihood method.In the random effect model, we compared the weighted least square method which add the heterogeneity varicance and the Mantel-Haenszel method. Even the M-H method seems the best performance in the fixed effect model, but it is not suitable in the random effect model. The weighted least square method has higer coverage probability.2) For the estimating the difference between tow weighted Youden indexes, we compare we compared the max-likelihood, Mantel-Haenszel and weighted least square methods in the fixed effect model. The type I error and power for the Z statistic using Mantel-Haenszel and weighted least square method is better than the likelihood method.In the random effect model, we compared the weighted least square method which add the heterogeneity varicance and the Mantel-Haenszel method. The type Ⅰ error using weighted least square metho is lower than Mantel-Haenszel method. The power using weighted least square metho is lower than Mantel-Haenszel method.Conclusion:We establish three methods to estimate the combined Youden Index and weighted Youden Index in the fixed effect model and one method in the random effect model. These methods can be used in meta-analysis estimation. The Mantel-Haenszel and weighted least square method are recommended to compare two weighted Youden Indexes in meta-analysis in the fixed effect model.