Thesensitivity index ordiscriminability index ordetectability index is a dimensionlessstatistic used insignal detection theory. A higher index indicates that the signal can be more readily detected.

The discriminability index is the separation between the means of two distributions (typically the signal and the noise distributions), in units of thestandard deviation.

For twounivariate distributions${\displaystyle a}$ and${\displaystyle b}$ with the same standard deviation, it is denoted by${\displaystyle d^{\prime }}$ ('dee-prime'):

${\displaystyle d^{\prime }=\frac{\left|{\mu }_a-{\mu }_b\right|}{\sigma }}$.

In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix${\displaystyle \mathbf{\Sigma }}$, (whose symmetric square-root, the standard deviation matrix, is${\displaystyle \mathbf{S}}$), this generalizes to theMahalanobis distance between the two distributions:

${\displaystyle d^{\prime }=\sqrt{({\mathit{\mu }}_a-{\mathit{\mu }}_b{)}^{\prime }{\mathbf{\Sigma }}^{-1}({\mathit{\mu }}_a-{\mathit{\mu }}_b)}=\Vert {\mathbf{S}}^{-1}({\mathit{\mu }}_a-{\mathit{\mu }}_b)\Vert =\Vert {\mathit{\mu }}_a-{\mathit{\mu }}_b\Vert /{\sigma }_{\mathit{\mu }}}$,

where${\displaystyle {\sigma }_{\mathit{\mu }}=1/\Vert {\mathbf{S}}^{-1}\mathit{\mu }\Vert }$ is the 1d slice of the sd along the unit vector${\displaystyle \mathit{\mu }}$ through the means, i.e. the${\displaystyle d^{\prime }}$ equals the${\displaystyle d^{\prime }}$ along the 1d slice through the means.^[1]

For two bivariate distributions with equal variance-covariance, this is given by:

${\displaystyle {d^{\prime }}^2=\frac{1}{1-{\rho }^2}\left({d^{\prime }}_x^2+{d^{\prime }}_y^2-2\rho {d^{\prime }}_x{d^{\prime }}_y\right)}$,

where${\displaystyle \rho }$ is the correlation coefficient, and here${\displaystyle d_x^{\prime }=\frac{{{\mu }_b}_x-{{\mu }_a}_x}{{\sigma }_x}}$ and${\displaystyle d_y^{\prime }=\frac{{{\mu }_b}_y-{{\mu }_a}_y}{{\sigma }_y}}$, i.e. including the signs of the mean differences instead of the absolute.^[1]

${\displaystyle d^{\prime }}$ is also estimated as${\displaystyle Z(\text{hit rate})-Z(\text{false alarm rate})}$.^[2]: 8

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${\displaystyle d^{\prime }}$ for equal variance/covariance.

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${\displaystyle e_b}$ by an ideal observer, or its complement, the optimal accuracy${\displaystyle a_b}$:

${\displaystyle d_b^{\prime }=-2Z\left(\text{Bayes error rate }{e}_b\right)=2Z\left(\text{best accuracy rate }{a}_b\right)}$,^[1]

where${\displaystyle Z}$ is the inversecumulative distribution function of the standard normal. The Bayes discriminability between univariate or multivariate normal distributions can be numerically computed^[1] (Matlab code), and may also be used as an approximation when the distributions are close to normal.

${\displaystyle d_b^{\prime }}$ is a positive-definite statistical distance measure that is free of assumptions about the distributions, like theKullback–Leibler divergence${\displaystyle D_{\text{KL}}}$.${\displaystyle D_{\text{KL}}(a,b)}$ is asymmetric, whereas${\displaystyle d_b^{\prime }(a,b)}$ is symmetric for the two distributions. However,${\displaystyle d_b^{\prime }}$ does not satisfy thetriangle inequality, so it is not a full metric.^[1]

In particular, for a yes/no task between two univariate normal distributions with means${\displaystyle {\mu }_a,{\mu }_b}$ and variances${\displaystyle v_a>v_b}$, the Bayes-optimal classification accuracies are:^[1]

,

where${\displaystyle {\chi }^{\prime 2}}$ denotes thenon-central chi-squared distribution,${\displaystyle \lambda ={\left(\frac{{\mu }_a-{\mu }_b}{v_a-v_b}\right)}^2}$, and${\displaystyle c=\lambda +\frac{\ln v_a-\ln v_b}{v_a-v_b}}$. The Bayes discriminability${\displaystyle d_b^{\prime }=2Z\left(\frac{p\left(A|a\right)+p\left(B|b\right)}{2}\right).}$

${\displaystyle d_b^{\prime }}$ can also be computed from theROC 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.^[1]

For a two-interval task between these distributions, the optimal accuracy is${\displaystyle a_b=p\left({\tilde{\chi }}_{\mathit{w},\mathit{k},\mathit{\lambda },0,0}^2>0\right)}$ (${\displaystyle {\tilde{\chi }}^2}$ denotes thegeneralized chi-squared distribution), where.^[1] The Bayes discriminability${\displaystyle d_b^{\prime }=2Z\left(a_b\right)}$.

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:${\displaystyle d_a^{\prime }=\left|{\mu }_a-{\mu }_b\right|/{\sigma }_{\text{rms}}}$^[3] (also denoted by${\displaystyle d_a}$). It is${\displaystyle \sqrt{2}}$ times the${\displaystyle z}$-score of the area under thereceiver 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${\displaystyle {\mathbf{S}}_{\text{rms}}={\left[\left({\mathbf{\Sigma }}_a+{\mathbf{\Sigma }}_b\right)/2\right]}^{\frac{1}{2}}}$ as the common sd matrix.^[1]

Another index is${\displaystyle d_e^{\prime }=\left|{\mu }_a-{\mu }_b\right|/{\sigma }_{\text{avg}}}$, extended to general dimensions using${\displaystyle {\mathbf{S}}_{\text{avg}}=\left({\mathbf{S}}_a+{\mathbf{S}}_b\right)/2}$ as the common sd matrix.^[1]

It has been shown that for two univariate normal distributions,${\displaystyle d_a^{\prime }\le d_e^{\prime }\le d_b^{\prime }}$, and for multivariate normal distributions,${\displaystyle d_a^{\prime }\le d_e^{\prime }}$ still.^[1]

Thus,${\displaystyle d_a^{\prime }}$ and${\displaystyle d_e^{\prime }}$ underestimate the maximum discriminability${\displaystyle d_b^{\prime }}$ of univariate normal distributions.${\displaystyle d_a^{\prime }}$ can underestimate${\displaystyle d_b^{\prime }}$ by a maximum of approximately 30%. At the limit of high discriminability for univariate normal distributions,${\displaystyle d_e^{\prime }}$ converges to${\displaystyle d_b^{\prime }}$. These results often hold true in higher dimensions, but not always.^[1] Simpson and Fitter^[3] promoted${\displaystyle d_a^{\prime }}$ as the best index, particularly for two-interval tasks, but Das and Geisler^[1] have shown that${\displaystyle d_b^{\prime }}$ is the optimal discriminability in all cases, and${\displaystyle d_e^{\prime }}$ is often a better closed-form approximation than${\displaystyle d_a^{\prime }}$, even for two-interval tasks.

The approximate index${\displaystyle d_{gm}^{\prime }}$, which uses thegeometric mean of the sd's, is less than${\displaystyle d_b^{\prime }}$ at small discriminability, but greater at large discriminability.^[1]

In general, the contribution to the total discriminability by each dimension or feature may be measured using the amount by which the discriminability drops when that dimension is removed. If the total Bayes discriminability is${\displaystyle d^{\prime }}$ and the Bayes discriminability with dimension${\displaystyle i}$ removed is${\displaystyle d_{-i}^{\prime }}$, we can define the contribution of dimension${\displaystyle i}$ as${\displaystyle \sqrt{d^{\prime 2}-{d_{-i}^{\prime }}^2}}$. This is the same as the individual discriminability of dimension${\displaystyle i}$ when the covariance matrices are equal and diagonal, but in the other cases, this measure more accurately reflects the contribution of a dimension than its individual discriminability.^[1]

We may sometimes want to scale the discriminability of two data distributions by moving them closer or farther apart. One such case is when we are modeling a detection or classification task, and the model performance exceeds that of the subject or observed data. In that case, we can move the model variable distributions closer together so that it matches the observed performance, while also predicting which specific data points should start overlapping and be misclassified.

There are several ways of doing this. One is to compute the mean vector andcovariance matrix of the two distributions, then effect a linear transformation to interpolate the mean and sd matrix (square root of the covariance matrix) of one of the distributions towards the other.^[1]

Another way that is by computing the decision variables of the data points (log likelihood ratio that a point belongs to one distribution vs another) under a multinormal model, then moving these decision variables closer together or farther apart.^[1]

• Receiver operating characteristic (ROC)
• Summary statistics
• Effect size

• Wickens, Thomas D. (2001).Elementary Signal Detection Theory. OUP USA. ch. 2, p. 20.ISBN 0-19-509250-3.

• Interactive signal detection theory tutorial including calculation ofd′.

• Detection theory
• Signal processing
• Summary statistics

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