Sensitivity index

The sensitivity index or discriminability index or detectability index d′ (pronounced 'dee-prime') is a statistic used in signal detection theory. It provides the separation between the means of the signal and the noise distributions, in units of the standard deviation of the signal or noise distribution. When both distributions are univariate normal distributions with the same standard deviation,

${\displaystyle d'={\frac {|\mu _{\mathrm {s} }-\mu _{\mathrm {n} }|}{\sigma }}}$.

Classification error probability ${\displaystyle p_{e}}$, and effective and approximate discriminability indices of two univariate normal distributions with unequal variances (top), and of two bivariate normal distributions with unequal covariance matrices (bottom, ellipses are 1 sd error ellipses). The classification boundary is in black. These are computed by numerical methods[1].

When the distributions are multivariate normal distributions with the same covariance matrix, ${\displaystyle d'}$ is the Mahalanobis distance between them:

${\displaystyle d'={\sqrt {({\boldsymbol {\mu }}_{s}-{\boldsymbol {\mu }}_{n})^{T}\mathbf {\Sigma } ^{-1}({\boldsymbol {\mu }}_{s}-{\boldsymbol {\mu }}_{n})}}.}$

When they have different standard deviations ${\displaystyle \sigma _{s}}$ and ${\displaystyle \sigma _{n}}$ (or different covariance matrices ${\displaystyle \mathbf {\Sigma } _{s}}$ and ${\displaystyle \mathbf {\Sigma } _{n}}$ in more than one dimension), the effective ${\displaystyle d'}$ can be defined as the normalized mean separation of two equal-variance normals that corresponds to the same classification error rate. This does not have a closed-form expression, but can be computed using numerical methods[1] (Matlab code). When variances are unequal, a common approximation is to take them to be equal to their average, which leads to[2]:

${\displaystyle d'\approx {\frac {|\mu _{s}-\mu _{n}|}{\sqrt {{\frac {1}{2}}\left(\sigma _{s}^{2}+\sigma _{n}^{2}\right)}}}.}$

in higher dimensions, it is the Mahalanobis distance with the average covariance matrix.

d′ can also be estimated from the observed hit rate and false-alarm rate, as follows:[2]^:7

d′ = Z(hit rate) − Z(false alarm rate),

where function Z(p), p ∈ [0,1], is the inverse of the cumulative distribution function of the Gaussian distribution.

d′ can be related to the area under the receiver operating characteristic curve, or AUC, via:[3]

${\displaystyle d'={\sqrt {2}}Z({\mbox{AUC}}).}$

d′ is a dimensionless statistic. A higher d′ indicates that the signal can be more readily detected.