# What is the right name for the variant of the Kolmogorov-Smirnov statistic that retains the sign of the difference?

The two-sample Kolmogorov-Smirnov statistic is normally defined as

$D = \max_x |A(x) - B(x)|.$

I would like to compute a variant that retains the sign of the difference between the distribution functions:

$D^\prime = A(x) - B(x) \quad\text{where}\quad x = \arg\max_x |A(x) - B(x)|.$

What is the right name of $D^\prime$?

Note that $D^\prime$ is different than $D^+ = \max_x [A(x) - B(x)]$ and $D^- = \max_x [B(x) - A(x)],$ which Coberly and Lewis describe as “one-sided” KS-statistics.

For context, the $D^\prime$ as defined above is used by Perlman et al. to profile drug-treated cultured cells. (See page 3–4 of the supplement.) They compute $D^\prime$ independently for each image feature of the sample, then standardize the $D^\prime$-values (using the mean and standard deviation of the same measurement on mock-treated samples). Correlated profiles (of standardized $D^\prime$-values) indicate drugs that have similar biological effects.

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$D^{+}$ and $D^{-}$ are common names for the one-sided statistics (cf springerlink.com/content/y460k248m8w1822j ). In what context does your $D'$ appear? –  whuber Mar 14 '11 at 18:29
@whuber: Added context and comparison to $D^+$ and $D^-$ to the question. –  Vebjorn Ljosa Mar 14 '11 at 18:46
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## 1 Answer

It looks like a variant of Kuiper's test to me, although Kuiper's V = D+ + DD'.

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