Improved Statistical Inference Method For Single Youden Index And Comparison Method Of Youden Index For Paired Design

BackgroundSince Youden index was established (Youden,1950), it has been widely accepted as one of the most important measurement in evaluating diagnostic tests for more than half a century. However, there still have two perplexed questions remained.The first one is that the combined variance of the Youden index was estimated under the assumption of the independency of sensitivity and specificity. However, the association between sensitivity and specificity is fairly high that cannot be ignored. The variance estimated based on this incorrect basement would certainly result in the decrease of power when the hypothesis test for Youden index is performed under a certain significant level. Though this problem has been found by some researchers and reported in their papers or books, the solution to this problem remains absent.The second one is that the statistical inference method of Youden index was proposed for the comparison of two diagnostic tests based on the assumption that two diagnostic tests were independent mutually without consideration of whether the design was independent or paired. However, in, most cases of clinical practice, the comparisons between two diagnostic tests are designed in a paired one. In reality, the analysis for comparisons between two diagnostic tests in the paired design are performed with two independent-sample Youden index test because of the absence of proper method. This problem would certainly cause the decrease of testing power just like independent-t test is applied to an effective paired design.For the reasons mentioned above, it is of great value and importance to solve the two problems in both methodology and clinical practice.ObjectiveThis study aims to establish the novel statistical inference methods for both single Youden index and paired Youden index based on the dependency between sensitivity and specificity. These more practical and efficient analysis tools are highly expected to benefit evaluation of diagnostic tests in clinical practice.Methods1. The establishing for standard error of single Youden index.Because of the special nature of the high negative association between sensitivity and specificity, it was hard to measure it with a certain quantity. The combined variance and standard error of single Youden index was obtained based on the idea that Cleophas used to calculate the combined standard error of overall validity with Delta method. After that, the statistical inference method for single Youden index was established according to the central limit theorems as well as the (1-α)·100%confidence interval for single Youden index.2. The characteristics of statistical inference method and standard error of single Youden index.The very nature of the combined standard error was proved through mathematical derivation, so as the test power of the statistical inference method. The power of the new method is compared with the original one to see if there is a promotion. The type I error of the method established in this study was studied by computer simulation with Monte Carlo method to see if it was consistent with the set level.3. The establishing of the standard error for paired Youden index.The combined covariance of the paired Youden index was made of six covariances between two sensitivities and specificities in a paired diagnostic test. With consideration of the association between the sensitivities and specificities of two diagnostic tests, the idea which Tang and his colleges used to estimate the confidence interval of two related rates (2000) can be used here to estimate the covariances between them. For the association between sensitivity and specificity of different and same diagnostic test, it was hard to choose a proper quantity to stand for their association. However, their covariances can be proved to be a quite small one compared with other parts of the combined standard error of paired Youden index, and they are reasonable to be neglect. With the work mentioned above, the combined variance of paired Youden index can be estimated and the statistical method for it can be established, as well as the (1-α)·100%confidence interval for paired Youden index.4. The characteristics of statistical inference method and standard error of single Youden index.The every nature of the combined standard error was proved by mathematical derivation, so as the test power of the statistical inference method. The power of the new method is compared with the original one to see if there is a promotion. The type I error of the method established in this study was studied by computer simulation with Mont Carlo method to see if it is consistent with the set level. The combination of the covariance between sensitivities and specificities between and within diagnosis was proved to be a quite small amount by mathematical derivation and mechanical proof with computer. Results1. The statistical inference method for single Youden index. For a diagnostic test, Sen represents for the sensitivity, and Spe the specificity,J is the Youden index for the diagnosis, Var(*) represents for the variance and S.E.*the standard error. The combined variance and standard error of the Youden index was obtained thorough Delta method:According to the central limit theory, the statistic of the statistical inference method for the comparison between single Youden index and0was:Furthermore, the (1-α)·100%confidence interval of single Youden index was:It can be proved that the variance and combined standard error obtained with Delta method fits the general nature of standard error and it is smaller than the original one estimated based on the independency assumption.It was proved through that the power of the statistical inference method proposed in this study is higher than the original one at a set level of significance through mathematical derivation. Besides, the stronger the association between sensitivity and specificity is, the higher the power of the new method is.Simulation research indicated that the type I error of the statistical inference method proposed in this study is stable around0.05as it was set to be.2. The statistical inference method for paired Youden index. For a paired diagnostic test, with Seni represents for the sensitivity of diagnostic test i, and Spei the specificity, Ji is the Youden index for the diagnosis i, Var(*) represents for the variance and S.E.*represents for the standard error, Ksen (Kspe)is the association coefficients between sensitivities (specificities). With kappa index as an estimation for the association of the sensitivities (specificities) of two paired diagnostic tests, the variance and standard error estimated for the deviation of paired Youden index are:According to the central limit theory, the statistic for the statistical inference method for the comparison between the deviation of two paired Youden index and0is:Furthermore, the (1-α)·100%confidence interval of Dj, the deviation of the paired Youden index, is:It can be proved that the combined standard error obtained in this study fits the very nature of general standard error. With the consideration of the association of sensitivity and specificity, the new standard error is smaller than the original one association. The covariances ignored during the estimation of the variance takes quite small amount of the combined standard error of the paired Youden index is proved. The proportion is less than1%.It can also be proved through mathematical derivation that the power of the statistical inference method proposed in this study is higher than the original one at any certain level of significance especially when the association between the two diagnostic tests is strong. It is more likely to find the difference when it really exists.Simulation research indicated that the type I error of the statistical inference method established in this study is stable around0.05as it was set to be.ConclusionThe combined variance and standard error for single Youden index and paired Youden index estimated in this research fits the general nature of variance and standard error and are closer to real situations. The statistical inference methods proposed in this study have a higher power compared with the original ones, especially when the association is strong. The type I errors for both methods are all around0.05within a normal and acceptable range as it was set to be. It is able to conclude that the statistical inference methods established by this study are reasonable and effective.