| Research background and objectiveEver since introduced by Yerushalmy (1947), sensitivity and specificity have been the most important tools and widely used in diagnostic test evaluation. As the corner stone, nearly all of diagnostic test measures can be expressed as functions of sensitivity and specificity, for example, Youden index (1950), diagnostic accuracy/efficiency rate, diagnostic likelihood ratio, odds product/ratio, positive/negative predictive value, information content, kppa coefficient, and so on.It is well recognized that sensitivity and specificity are reversely connected depending on the choice of cut-off value:for a certain diagnostic test, an increase in sensitivity certainly leads to the decrease in specificity, and vice versa. Consequently, when comparing several diagnostic tests, conclusion could be difficult to reach, when one test has higher sensitivity while another excels in specificity. Several approaches mentioned above were constructed as one-dimensional indices that combine sensitivity and specificity. Over the past several decades, some indices such as Youden index, diagnostic accuracy/efficiency rate, odds product/ratio, kappa coefficient have been used to compare several diagnostic tests. They, to certain degree, unify the contradictory results delivered by sensitivity or specificity alone. However these methods share one limitation. When sensitivity and specificity are of different importance, they are inept to accommodate the difference. How can we balance the apples and oranges of sensitivity and specificity? In clinical practice, different situations dictate different importance of sensitivity and specificity as described by Galen and Gambino. Van den Bruel A etc.(2007) indicated that diagnostic tests evaluation should consider the technical accuracy, the diagnostic accuracy, the impact on patient outcome, and the cost-effectiveness. Therefore the weightings of sensitivity and specificity were taken into consideration. Pepe (2003) and Perkins&Schisterman (2006) proposed the method to weight sensitivity and specificity as functions of the prevalence and relative costs of misclassification for Youden index. However, both the prevalence and the relative costs could be difficult to obtain in practice, so this method is lack of practicality. Although the idea of weighting of sensitivity and specificity had been proposed, at statistical point, its inference method is not yet available now. Furthermore, the idea of weighted sensitivity and specificity can not only be applied to Youden index, but also to any other diagnostic indices in defined as the combination of sensitivity and specificity.The aim of my study is to establish three novely and practicably statistical methods based on weighted sensitivity and specificity for diagnostic test evaluation—weighted Youden index, weighted standardized diagnostic efficiency rate and weighted odds product.MethodsThree principles of constructing weighted Youden index Jw are as follows:firstly, the sum of two weights which are attached to the sensitivity and specificity should equal to1; secondly, the range of possible values of Jw is within [-1,1], which is the same as the Youden index J; finally, Jw equals to J when the sensitivity and specificity have the same weights. Then, the Jw is defined by Jw=2[w·SEN+(1-w)SPE]-1(0<, w<1). According to the central-limit theorem, we obtain the standard error of Jw, and propose a statistical inference method to compare two weighted indexes. Furthermore, we also deduce the test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions.Two principles of constructing e’are as follows:firstly, the sum of two weights which are attached to the sensitivity and specificity should equal to1; secondly, weighted standardized diagnostic efficiency rate equals to standardized diagnostic efficiency rate when the sensitivity and specificity have the same weights, that is e’=e. Then, the e’is defined by (0≤w≤1). According to the central-limit theorem, we obtain the standard error of e’, and propose a statistical inference method to compare two weighted standardized diagnostic efficiency rate. Furthermore, we also deduce the test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions.Three principles of constructing weighted odds product φw are as follows: firstly, the sum of two weights which are attached to the sensitivity and specificity should equal to1; secondly,φw equals to φ when the sensitivity and specificity have the same weights.; finally the range of possible values of φw is within [0,+∞], which is the same as the odds product φ.Then, the φw is defined by (0≤w≤1). According to the central-limit theorem, we obtain the standard error of φw and propose a statistical inference method to compare two weighted indexes. Furthermore, we also deduce the test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions.Results1、The proposed weighted Youden index Jw satisfied the above-mentioned three principles.SJW1-JW2is the standard error for the difference of the two weighted Youden indices. With the independence assumption, can be expressed asThe test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions. We classify the conditions into four scenarios:(a) If SEN1≤SEN2and SPE1≥SPE2, Z monotonously decreases when w increases (only when SEN1=SEN2and SPE1=SPE2, Z=0);(b) If SEN1<SEN2and SPE1<SPE2, Z firstly decreases and then increases as w increases. The minimum of Z is reached when the weight equals wo is as follows(c) If SEN1≥SEN2and SPE1≤SPE2, then as w increases, Z monotonously increases (only when SEN1=SEN2and SPEi=SPE2, Z=0);(d) If SEN1>SEN2and SPE1,>SPE2,as w increases, Z firstly increases and then decreases. The maximum of Z is reached when the weight equals w0.2、The proposed weighted standardized diagnostic efficiency rate e’satisfied the above-mentioned two principles.S.e1w-e2w. is the standard error for the difference of the two weighted standardized diagnostic efficiency rate indices. With the independence assumption, can be expressed asThe test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions. We classify the conditions into four scenarios:(a) If SEN1≤SEN2and SPE1≥SPE2, Z monotonously decreases when w increases (only when SEN1=SEN2and SPE1=SPE2, Z=0);(b) If SEN1<SEN2and SPE1<SPE2, Z firstly decreases and then increases as w increases. The minimum of Z is reached when the weight equals w0is as follows(c) If SEN1≥SEN2and SPE1≤SPE2, then as w increases, Z monotonously increases (only when SEN1=SEN2and SPE1=SPE2,Z=0);(d) If SEN1>SEN2and SPE1>SPE2,as w increases, Z firstly increases and then decreases. The maximum of Z is reached when the weight equals w0.3、The proposed weighted odds product.φw satisfied the above-mentioned three principles. is the standard error for the difference of the natural logarithmic of two weighted odds product indices. With the independence assumption, can be expressed asThe test statistics Z can be either a monotonously increasing/decreasing function or non-monotone function of the weight w under different conditions. We classify the conditions into four scenarios:(a) If SEN1≤SEN2and SPE1≥SPE2, Z monotonously decreases when w increases (only when SEN1=SEN2and SPE1=SPE2, Z=0);(b) If SEN1<SEN2and SPE1<SPE2, Z firstly decreases and then increases as w increases. The minimum of Z is reached when the weight equals w0is as follows(c) If SEN1≥SEN2and SPE1≤SPE2, then as w increases, Z monotonously increases (only when SEN,=SEN2and SPE1=SPE2, Z=0);(d) If SEN1>SEN2and SPE1>SPE2,as w increases, Z firstly increases and then decreases. The maximum of Z is reached when the weight equals wq.ConclusionWith the consideration of the different characteristics of three indices for diagnostic test evaluation, this research ingeniously proposed two weights, which ranging from0to1with the sum1, be added to sensitivity and specificity. When equal weights are assigned, the three new indices proposed in this study should be equal to the original ones respectively. The standard errors of the new indices are derived and relative statistical inference methods for the comparison of two diagnostic tests is proposed. Then, methods which based on weighted sensitivity and specificity are established. The weighted Youden index, weighted standardized diagnostic efficiency rate and weighted odds product will become new and powerful statistical analyzing tools for clinical diagnostic test. |
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