Statistical parameter

As a statistical parameter, SSMD (denoted as ${\displaystyle \beta }$) is defined as the ratio of mean to standard deviation of the difference of two random values respectively from two groups. Assume that one group with random values has mean ${\displaystyle \mu _{1}}$ and variance ${\displaystyle \sigma _{1}^{2}}$ and another group has mean ${\displaystyle \mu _{2}}$ and variance ${\displaystyle \sigma _{2}^{2}}$. The covariance between the two groups is ${\displaystyle \sigma _{12}.}$ Then, the SSMD for the comparison of these two groups is defined as^[1]

${\displaystyle \beta ={\frac {\mu _{1}-\mu _{2}}{\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}-2\sigma _{12}}}}.}$

If the two groups are independent,

${\displaystyle \beta ={\frac {\mu _{1}-\mu _{2}}{\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}}}}.}$

If the two independent groups have equal variances ${\displaystyle \sigma ^{2}}$,

${\displaystyle \beta ={\frac {\mu _{1}-\mu _{2}}{{\sqrt {2}}\sigma }}.}$

In the situation where the two groups are correlated, a commonly used strategy to avoid the calculation of ${\displaystyle \sigma _{12}}$ is first to obtain paired observations from the two groups and then to estimate SSMD based on the paired observations. Based on a paired difference ${\displaystyle D}$ with population mean ${\displaystyle \mu _{D}}$ and ${\displaystyle \sigma _{D}^{2}}$, SSMD is

${\displaystyle \beta ={\frac {\mu _{D}}{\sigma _{D}}}.}$

Statistical estimation

In the situation where the two groups are independent, Zhang XHD ^[1] derived the maximum-likelihood estimate (MLE) and method-of-moment (MM) estimate of SSMD. Assume that groups 1 and 2 have sample mean ${\displaystyle {\bar {X}}_{1},{\bar {X}}_{2}}$, and sample variances ${\displaystyle s_{1}^{2},s_{2}^{2}}$. The MM estimate of SSMD is then^[1]

${\displaystyle {\hat {\beta }}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\sqrt {s_{1}^{2}+s_{2}^{2}}}}.}$

When the two groups have normal distributions with equal variance, the uniformly minimal variance unbiased estimate (UMVUE) of SSMD is,^[10]

${\displaystyle {\hat {\beta }}={\frac {{\bar {X}}_{1}-{\bar {X}}_{2}}{\sqrt {{\frac {2}{K}}((n_{1}-1)s_{1}^{2}+(n_{2}-1)s_{2}^{2})}}},}$

where ${\displaystyle n_{1},n_{2}}$ are the sample sizes in the two groups and ${\displaystyle K\approx n_{1}+n_{2}-3.48}$.^[3]

In the situation where the two groups are correlated, based on a paired difference with a sample size ${\displaystyle n}$, sample mean ${\displaystyle {\bar {D}}}$ and sample variance ${\displaystyle s_{D}^{2}}$, the MM estimate of SSMD is

${\displaystyle {\hat {\beta }}={\frac {\bar {D}}{s_{D}}}.}$

The UMVUE estimate of SSMD is ^[22]

${\displaystyle {\hat {\beta }}={\frac {\Gamma ({\frac {n-1}{2}})}{\Gamma ({\frac {n-2}{2}})}}{\sqrt {\frac {2}{n-1}}}{\frac {\bar {D}}{s_{D}}}.}$

SSMD looks similar to t-statistic and Cohen's d, but they are different with one another as illustrated in.^[3]

Quality control

The number of wells for the positive and negative controls in a plate in the 384-well or 1536-well platform is normally designed to be reasonably large .^[24] Assume that the positive and negative controls in a plate have sample mean ${\displaystyle {\bar {X}}_{P},{\bar {X}}_{N}}$, sample variances ${\displaystyle s_{P}^{2},s_{N}^{2}}$, and sample sizes ${\displaystyle n_{P},n_{N}}$. Usually, the assumption that the controls have equal variance in a plate holds. In such a case, The SSMD for assessing quality in that plate is estimated as ^[10]

${\displaystyle {\hat {\beta }}={\frac {{\bar {X}}_{P}-{\bar {X}}_{N}}{\sqrt {{\frac {2}{K}}((n_{P}-1)s_{P}^{2}+(n_{N}-1)s_{N}^{2})}}},}$

where ${\displaystyle K\approx n_{P}+n_{N}-3.48}$. When the assumption of equal variance does not hold, the SSMD for assessing quality in that plate is estimated as ^[1]

${\displaystyle {\hat {\beta }}={\frac {{\bar {X}}_{P}-{\bar {X}}_{N}}{\sqrt {s_{P}^{2}+s_{N}^{2}}}}.}$

If there are clearly outliers in the controls, the SSMD can be estimated as ^[23]

${\displaystyle {\hat {\beta }}={\frac {{\tilde {X}}_{P}-{\tilde {X}}_{N}}{1.4826{\sqrt {{\tilde {s}}_{P}^{2}+{\tilde {s}}_{N}^{2}}}}},}$

where ${\displaystyle {\tilde {X}}_{P},{\tilde {X}}_{N},{\tilde {s}}_{P},{\tilde {s}}_{N}}$ are the medians and median absolute deviations in the positive and negative controls, respectively.

The Z-factor based QC criterion is popularly used in HTS assays. However, it has been demonstrated that this QC criterion is most suitable for an assay with very or extremely strong positive controls.^[10] In an RNAi HTS assay, a strong or moderate positive control is usually more instructive than a very or extremely strong positive control because the effectiveness of this control is more similar to the hits of interest. In addition, the positive controls in the two HTS experiments theoretically have different sizes of effects. Consequently, the QC thresholds for the moderate control should be different from those for the strong control in these two experiments. Furthermore, it is common that two or more positive controls are adopted in a single experiment.^[11] Applying the same Z-factor-based QC criteria to both controls leads to inconsistent results as illustrated in the literatures.^[10][11]

The SSMD-based QC criteria listed in the following table^[20] take into account the effect size of a positive control in an HTS assay where the positive control (such as an inhibition control) theoretically has values less than the negative reference.

Quality Type	A: Moderate Control	B: Strong Control	C: Very Strong Control	D: Extremely Strong Control
Excellent	${\displaystyle \beta \leq -2}$	${\displaystyle \beta \leq -3}$	${\displaystyle \beta \leq -5}$	${\displaystyle \beta \leq -7}$
Good	${\displaystyle -2<\beta \leq -1}$	${\displaystyle -3<\beta \leq -2}$	${\displaystyle -5<\beta \leq -3}$	${\displaystyle -7<\beta \leq -5}$
Inferior	${\displaystyle -1<\beta \leq -0.5}$	${\displaystyle -2<\beta \leq -1}$	${\displaystyle -3<\beta \leq -2}$	${\displaystyle -5<\beta \leq -3}$
Poor	${\displaystyle \beta >-0.5}$	${\displaystyle \beta >-1}$	${\displaystyle \beta >-2}$	${\displaystyle \beta >-3}$

In application, if the effect size of a positive control is known biologically, adopt the corresponding criterion based on this table. Otherwise, the following strategy should help to determine which QC criterion should be applied: (i) in many small molecule HTS assay with one positive control, usually criterion D (and occasionally criterion C) should be adopted because this control usually has very or extremely strong effects; (ii) for RNAi HTS assays in which cell viability is the measured response, criterion D should be adopted for the controls without cells (namely, the wells with no cells added) or background controls; (iii) in a viral assay in which the amount of viruses in host cells is the interest, criterion C is usually used, and criterion D is occasionally used for the positive control consisting of siRNA from the virus.^[20]

Similar SSMD-based QC criteria can be constructed for an HTS assay where the positive control (such as an activation control) theoretically has values greater than the negative reference. More details about how to apply SSMD-based QC criteria in HTS experiments can be found in a book.^[20]

Hit selection

In an HTS assay, one primary goal is to select compounds with a desired size of inhibition or activation effect. The size of the compound effect is represented by the magnitude of difference between a test compound and a negative reference group with no specific inhibition/activation effects. A compound with a desired size of effects in an HTS screen is called a hit. The process of selecting hits is called hit selection. There are two main strategies of selecting hits with large effects.^[20] One is to use certain metric(s) to rank and/or classify the compounds by their effects and then to select the largest number of potent compounds that is practical for validation assays.^{[17] [19][22]} The other strategy is to test whether a compound has effects strong enough to reach a pre-set level. In this strategy, false-negative rates (FNRs) and/or false-positive rates (FPRs) must be controlled.^{[14] [15] [16][25] [26]}

SSMD can not only rank the size of effects but also classify effects as shown in the following table based on the population value (${\displaystyle \beta }$) of SSMD.^{[20] [27]}

Effect subtype	Thresholds for negative SSMD	Thresholds for positive SSMD
Extremely strong	${\displaystyle \beta \leq -5}$	${\displaystyle \beta \geq 5}$
Very strong	${\displaystyle -5<\beta \leq -3}$	${\displaystyle 5>\beta \geq 3}$
Strong	${\displaystyle -3<\beta \leq -2}$	${\displaystyle 3>\beta \geq 2}$
Fairly strong	${\displaystyle -2<\beta \leq -1.645}$	${\displaystyle 2>\beta \geq 1.645}$
Moderate	${\displaystyle -1.645<\beta \leq -1.28}$	${\displaystyle 1.645>\beta \geq 1.28}$
Fairly moderate	${\displaystyle -1.28<\beta \leq -1}$	${\displaystyle 1.28>\beta \geq 1}$
Fairly weak	${\displaystyle -1<\beta \leq -0.75}$	${\displaystyle 1>\beta \geq 0.75}$
Weak	${\displaystyle -0.75<\beta <-0.5}$	${\displaystyle 0.75>\beta >0.5}$
Very weak	${\displaystyle -0.5\leq \beta <-0.25}$	${\displaystyle 0.5\geq \beta >0.25}$
Extremely weak	${\displaystyle -0.25\leq \beta <0}$	${\displaystyle 0.25\geq \beta >0}$
No effect	${\displaystyle \beta =0}$

The estimation of SSMD for screens without replicates differs from that for screens with replicates.^[20][23]

In a primary screen without replicates, assuming the measured value (usually on the log scale) in a well for a tested compound is ${\displaystyle X_{i}}$ and the negative reference in that plate has sample size ${\displaystyle n_{N}}$, sample mean ${\displaystyle {\bar {X}}_{N}}$, median ${\displaystyle {\tilde {X}}_{N}}$, standard deviation ${\displaystyle s_{N}}$ and median absolute deviation ${\displaystyle {\tilde {s}}_{N}}$, the SSMD for this compound is estimated as ^[20][23]

${\displaystyle {\text{SSMD}}={\frac {X_{i}-{\bar {X}}_{N}}{s_{N}{\sqrt {2(n_{N}-1)/K}}}},}$

where ${\displaystyle K\approx n_{N}-2.48}$. When there are outliers in an assay which is usually common in HTS experiments, a robust version of SSMD ^[23] can be obtained using

${\displaystyle {\text{SSMD*}}={\frac {X_{i}-{\tilde {X}}_{N}}{1.4826{\tilde {s}}_{N}{\sqrt {2(n_{N}-1)/K}}}}}$

In a confirmatory or primary screen with replicates, for the i-th test compound with ${\displaystyle n}$ replicates, we calculate the paired difference between the measured value (usually on the log scale) of the compound and the median value of a negative control in a plate, then obtain the mean ${\displaystyle {\bar {d}}_{i}}$ and variance ${\displaystyle s_{i}^{2}}$ of the paired difference across replicates. The SSMD for this compound is estimated as ^[20]

${\displaystyle {\text{SSMD}}={\frac {\Gamma ({\frac {n-1}{2}})}{\Gamma ({\frac {n-2}{2}})}}{\sqrt {\frac {2}{n-1}}}{\frac {{\bar {d}}_{i}}{s_{i}}}}$

In many cases, scientists may use both SSMD and average fold change for hit selection in HTS experiments. The dual-flashlight plot ^[28] can display both average fold change and SSMD for all test compounds in an assay and help to integrate both of them to select hits in HTS experiments ^[29] . The use of SSMD for hit selection in HTS experiments is illustrated step-by-step in ^[23]