Strictly Standardised Mean Difference (SSMD)

SSMD is a measure of effects size and can be used for both quality control like Z-prime, as well as hit-selection. It was initially prosed by Zhang.

\[ \beta = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1^2 + \sigma_2^2 - 2\sigma_{12}}} \]

def ssmd(x: np.ndarray, y: np.ndarray) -> float:
    mu_x = np.mean(x)
    mu_y = np.mean(y)
    sigma_x = np.var(x)
    sigma_y = np.var(y)
    cov_xy = np.cov(x, y)[0][1]
    return (mu_x - mu_y) / np.sqrt(sigma_x + sigma_y - 2 * cov_xy)

The \(\beta\) value returned from SSMD can be compared against a table to get an idea of the effect size.

\(\beta\)	description
0	no effect
\(0 < \beta \le 0.25\)	extremely weak
\(0.25 < \beta \le 0.5\)	very week
\(0.5 < \beta \le 0.75\)	weak
\(0.75 < \beta \le 1\)	fairly weak
\(1 < \beta \le 1.28\)	fairly moderate
\(1.28 < \beta \le 1.654\)	moderate
\(1.654 < \beta \le 2\)	fairly strong
\(2 < \beta \le 3\)	strong
\(3 < \beta \le 5\)	very strong
\(\beta > 5\)	extremely strong