Sensitivity index
The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected. Definition. The discriminability index is the separation between the means of two distributions (typically the signal and the noise distributions), in units of the standard deviation. Equal variances/covariances. For two univariate distributions formula_1 and formula_2 with the same standard deviation, it is denoted by formula_3 ('dee-prime'): In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix formula_5, (whose symmetric square-root, the  standard deviation matrix, is formula_6), this generalizes to the Mahalanobis distance between the two distributions: where formula_8 is the 1d slice of the sd along the unit vector formula_9 through the means, i.e. the formula_3 equals the formula_3 along the 1d slice through the means. This is also estimated as formula_12 . Unequal variances/covariances. When the two distributions have different standard deviations (or in general dimensions, different covariance matrices), there exist several contending indices, all of which reduce to formula_3 for equal variance/covariance. Bayes discriminability index. This is the maximum (Bayes-optimal) discriminability index for two distributions, based on  the amount of their overlap, i.e. the optimal (Bayes) error of classification formula_14 by an ideal observer, or its complement, the optimal accuracy formula_15: where formula_17 is the inverse cumulative distribution function of the standard normal. The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed (Matlab code), and may also be used as an approximation when the distributions are close to normal. formula_18 is a positive-definite statistical distance measure that is free of assumptions about the distributions, like the Kullback-Leibler divergence formula_19. formula_20 is asymmetric, whereas formula_21 is symmetric for the two distributions. However , formula_18 does not satisfy the triangle inequality, so it is not a full metric. In particular, for a yes/no task between two univariate normal distributions with means formula_23 and variances formula_24, the Bayes-optimal classification accuracies are: where formula_26 denotes the non-central chi-squared distribution, formula_27, and formula_28. The Bayes discriminability formula_29 formula_18 can also be computed from the ROC curve of a yes/no task between two univariate normal distributions with a single shifting criterion. It can also be computed from the ROC curve of any two distributions (in any number of variables) with a shifting likelihood -ratio, by locating the point on the ROC curve that is farthest from the diagonal. For a two-interval task between these distributions, the optimal accuracy is formula_31 (formula_32 denotes the generalized chi-squared distribution), where formula_33. The Bayes discriminability formula_34. RMS sd discriminability index. A common approximate (i.e. sub-optimal) discriminability index that has a closed-form is to take the average of the variances, i.e. the rms of the two standard deviations: formula_35 (also denoted by formula_36). It is formula_37 times the formula_38-score of the area under the receiver operating characteristic curve (AUC) of a single-criterion  observer. This index is extended to general dimensions as the Mahalanobis distance using the pooled covariance, i.e. with formula_39 as the common sd matrix. Average sd discriminability index. Another index is formula_40, extended to general dimensions using formula_41 as the common sd matrix. Comparison of the indices. It has been shown that for two univariate normal distributions, formula_42, and for multivariate normal distributions, formula_43 still. Thus, formula_44 and formula_45 underestimate the maximum discriminability formula_18 of univariate normal distributions. formula_44 can underestimate formula_18 by a maximum of approximately 30%. At the limit of high discriminability for univariate normal distributions, formula_45 converges  to formula_18. These results often hold true in higher dimensions, but not always. Simpson and Fitter promoted formula_44 as the best index, particularly for two-interval tasks, but Das and Geisler have shown that formula_18 is the optimal discriminability in all cases, and formula_45 is often a better closed-form approximation than formula_44, even for two-interval tasks. The approximate index formula_55, which uses the geometric mean of the sd's, is less than formula_18 at small discriminability, but greater at large discriminability.