Sample Size Estimation And Meta-analysis Methods For Youden Index:Accounting For The Association Between Sensitivity And Specificity

Background:Methodological research for the evaluation in diagnostic test remains hot in biostatistics fields. Youden index, as the linear combination of sensitivity and specificity, is one of the comprehensive measurements of diagnostic test accuracy. However, the association between sensitivity and specificity had not been considered for original Youden index, and paired Youden index was unavailable.In 2005, my supervisor and me proposed the statistical inference method for Youden index accounting for the association between sensitivity and speicificity (hereinafter referred to as:the improved Youden inde) and the paired statistical inference method which solved the two long remained question in the application of Youden index. As proved by mathematical derivation and simulation, new methods yield better type Ⅰ error, higher power and more conformable to practice. However, the relative methods for sample size estimation and meta-analysis, which with no doubt are of great importance and practical meaning in the evaluation of diagnostic tests, remain absent.Object:1. To establish sample size estimation method for Youden index based on the previews result for one sample, two independent samples and paired sample design diagnostic test.2. To establish combined effect size for Youden index under fixed effect model and random effect model for one sample, two independent samples and paired sample design diagnostic test.Method:This research has been conducted under the condition that sensitivity and specificity are associated. Sensitivity and specificity are all binomial distributed variable. One sample, two independent samples and paired sample design are taken into consideration.3. Based on previews research, sample size estimation methods for three different designs were obtained by mathematical derivations according to the fundamentals of sample size estimation with approximate normal test. Monte Carlo simulations were conducted to study the test power and statistical properties of the proposed methods. The influence of the association caused by paired design to the estimation of sample size has also been discussed.4. Based on previews research, the combined effect size for Youden index has been proposed under fixed and random effect model. For fixed effect model, Mantel Haenszel method, Least Square method and Maximum Likelihood method were applied to conduct the estimator of the combined effect size of Youden index. For random effect model, DerSimonian-Laird method was conducted to estimate the combined effect size of Youden index. Monte Carlo simulations were conducted to discuss the coverage probability of (1-α)% to the combined Youden index estimated by different methods and different models.Result:1. Sample size estimation for Youden indexFor single Youden index (J), let Sen represent the sensitivity, Spe the specificity, Var(*) the variance, n1 the sample size for diseased group, n0 the sample size for non-diseased group, and n0/n1=r, N the total sample size. Let a represents the significant level,β the type Ⅱ error, and 1-β the power. Two sides test is applied. (Symbols and assumptions denoted in the same way for sample size determination parts)Based on the previews research, there is: Var(J)= Sen2Var(Spe)+Spe2Var(Sen),Then, based on the relationship between the sample size and the variance, type Ⅰ/Ⅱ errors under approximate normal test, the estimated sample size for diseased group can be expressed as:Then, the sample size for non-diseased group can be calculated based on the denotations above.For two independent sample diagnostic test, let DJ。 be the difference between two independent Youden indexes. For ith diagnostic test, Ji represent the Youden index, ni1 the sample size for diseased group, n,o the sample size for non-diseased group, Ni the total sample size. Let n10=r1n11,n21=r2n11,n20=r3n11.Based on the previews research, there is:Then, based on the relationship between the sample size and the variance, type Ⅰ/Ⅱ errors under approximate normal test, the estimated sample size for diseased group can be expressed as:Then, the sample size for non-diseased group can be calculated based on the denotations above.For paired diagnostic test, let DJbe the difference between two independent Youden indexes, n1 the sample size for diseased group, n0 the sample size for non-diseased group, and n0/n1=r, N the total sample size. For ith diagnostic test, Ji represent the Youden index.Based on the previews research, there is:Then, based on the relationship between the sample size and the variance, type Ⅰ/Ⅱ errors under approximate normal test, there is:The estimated sample size for diseased group can be expressed as:Then, the sample size for non-diseased group can be calculated based on the denotations above.Monte Carlo simulation indicated the sample size estimation for Youden index accounting for the association between sensitivity and specificity for three different designs yields stable type Ⅱ error, the power stables around the set level. The established method can satisfy the practical requirement in diagnostic test.2. Meta-analysis methods for Youden indexFixed effect model applied with homogeneity situations. Three weight methods were applied to estimate the combined effect size for Youden index.(1). Based on the Mantel Haenszel method (MH method)a. For single Youden index, let WMHi,represent the weight for the ith diagnostic test. ni1 the sample size for diseased group, ni0 the sample size for non-diseased group, Seni the sensitivity, Spei the specificity, Ji the Youden index, and JMH the combined effect size. Then the weights should be as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:b. For two independent samples diagnostic test, let WMHi, represent the weight for the ith diagnostic test. For the first diagnostic test, let ni1 represent the sample size for diseased group, ni0 the sample size for non-diseased group, Seni1 the sensitivity, Spei1 the specificity, Ji1 the Youden index; For another, mi1 the sample size for diseased group,mi0 the sample size for non-diseased group, Seni2 the sensitivity, Spei2 the specificity, Ji2 the Youden index. let Dj be the difference between two independent Youden indexes and DMH represent its combined effect size, Then the weights can be expressed as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:c. For paired diagnostic test, let WMHi, represent the weight for the ith diagnostic test, ni1 the sample size for diseased group, ni0 the sample size for non-diseased group. For the first diagnostic test, let Seni1 represent the sensitivity, Spei1 the specificity, Ji1 the Youden index; for another, Seni2 the sensitivity, Spei2 the specificity, Ji2 the Youden index. let DJ be the difference between paired Youden indexes and DMH represent its combined effect size. Then the weights can be expressed as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:(2). Based on Least square method (Inverse Variance method)a. For single Youden index, let WLSi, represent the weight for the ith diagnostic test, ni1 the sample size for diseased group, ni0 the sample size for non-diseased group, Seni the sensitivity, Spei the specificity,Ji the Youden index, and JMH the combined effect size. Then the weights can be expressed as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:b. For two independent samples diagnostic test, let WLSi, represent the weight for the ith diagnostic test. For the first diagnostic test, let ni1 represent the sample size for diseased group, ni0 the sample size for non-diseased group, Seni1 the sensitivity, Spei1 the specificity, Ji1 the Youden index; For another, mi1 the sample size for diseased group, mi0 the sample size for non-diseased group, Seni2 the sensitivity, Spei2 the specificity, Ji2 the Youden index. let DJ be the difference between two independent Youden indexes and DLS, represent its combined effect size, Then the weights can be expressed as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:c. For paired diagnostic test, let WLSsi, represent the weight for the ith diagnostic test, ni1 the sample size for diseased group, ni0 the sample size for non-diseased group. For the first diagnostic test, let Seni1 represent the sensitivity, Spei1 the specificity, Ji1 the Youden index; For another, Seni2 represents the sensitivity, Spei2 the specificity, Ji2 the Youden index. let DJ be the difference between paired Youden indexes and DLS, represent its combined effect size. Then the weights should be as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:(3). Maximum Likelihood Method (ML method)There is no explicit for the estimation of the combined effect size and variance for Youden index obtained by Maximum Likelihood method.a. For single Youden index, let Seni represent sensitivity, Spei the specificity, Ji the Youden index, and JML the combined effect size, for the ith diagnostic test. Then the Log-likelihood equation can be obtained as:After partial derivative transformation applied there are:The estimation of combined effect size can be obtained by solving the above equation set. The estimation of the combined variance can be obtained through the inverse of the Fisher information matrix of the Log-likelihood equation.For two independent and paired sample Youden index, let Seni represent sensitivity, Spei the specificity, Ji the Youden index, and JML the combined effect size, for the ith diagnostic test. Let DJ be the difference between paired Youden indexes and DML, represent its combined effect size. Then the Log-likelihood equation can be obtained as:After partial derivative transformation applied there are:The estimation of combined effect size can be obtained by solving the above equation set. The estimation of the combined variance can be obtained through the inverse of the Fisher information matrix of the Log-likelihood equation.Random effect model applied with heterogeneity situations. DerSimonian-Laird method (DL method) was applied to estimate the combined effect size for Youden index.a. For single Youden index, let WLsi, represent the weight for the ith diagnostic test. nit the sample size for diseased group obtained by Least Square method, ni0 the sample size for non-diseased group, Seni the sensitivity, Spei the specificity, Ji the Youden index, and JDL the combined effect size. Denote τ2 as the heterogeneity variance.Then the weights with DerSimonian-Laird method under random effect model is:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:b. For two independent samples diagnostic test, let WLsi, represent the weight for the ith diagnostic test under FEM with Least Square method. For the first diagnostic test, let Seni1 represent the sensitivity, Spei1 the specificity, Ji1 the Youden index; For another, Seni2 represents the sensitivity, Spei2 the specificity, Ji2 the Youden index. let Dj be the difference between two independent Youden indexes and DDs, represent its combined effect size, Then the weights W*i could be expressed as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:c. For paired diagnostic test, let WLsi, represent the weight for the ith diagnostic test under FEM with Least Square method, For the first diagnostic test, let Seni1 represent the sensitivity, Spei1 the specificity, Ji1 the Youden index; for another, Seni2 the sensitivity, Spei2 the specificity, Ji2 the Youden index, let DJi represent the difference between paired Youden indexes and DDS represent its combined effect size. Then the weights Wi* should be as:Then the combined Youden index is:The combined variance would be:The (1-α)% confidence interval would be:Monte Carlo simulation indicated under the fixe effect model, the coverage probability of the 95% confidence interval of combined Youden index obtained by Mantel Haenszel was stabled around the set level, it yielded the best performance. Its performance was slightly better than that obtained by Maximum Likelihood method whose performance was also acceptable especially when number of included studies is small than 5. CI obtained by Least Square method yielded the worst performance, its trend along with the increasing of number of included studies was unreasonable.Under random effect model, coverage probability of the 95% CI of the combined Youden index obtained DerSimonianDerSimonian-Laird method was also unsatisfied; the performance of it was the same as Least Square method under the fixed effect model.Conclusion:This research proposed the sample size estimation and the estimation of the combined Youden index in meta-analysis accounting for the association between sensitivity and specificity towards three kinds of design for diagnostic test. Results show that all three methods for sample size estimation have perfect statistical performance, and can meet the requirement of practical use. When under fixed effect model, combined effect size estimated through either Maximum Likelihood or Mantel Haenszel method is recommended. When under random effect model, the combined effect size for Youden index propose in this research is not satisfied by now.