Standardized mean of a contrast variable

In statistics, the standardized mean of a contrast variable (SMCV or SMC), is a parameter assessing effect size. The SMCV is defined as mean divided by the standard deviation of a contrast variable.^[1][2] The SMCV was first proposed for one-way ANOVA cases ^[2] and was then extended to multi-factor ANOVA cases .^[3]

Background

Consistent interpretations for the strength of group comparison, as represented by a contrast, are important.^[4][5]

When there are only two groups involved in a comparison, SMCV is the same as SSMD. SSMD belongs to a popular type of effect-size measure called "standardized mean differences"^[6] which includes Cohen's $d$^[7] and Glass's ${\displaystyle \delta .}$^[8] In ANOVA, a similar parameter for measuring the strength of group comparison is standardized effect size (SES).^[9] One issue with SES is that its values are incomparable for contrasts with different coefficients. SMCV does not have such an issue.

Concept

Suppose the random values in t groups represented by random variables ${\displaystyle G_{1},G_{2},\ldots ,G_{t}}$ have means ${\displaystyle \mu _{1},\mu _{2},\ldots ,\mu _{t}}$ and variances ${\displaystyle \sigma _{1}^{2},\sigma _{2}^{2},\ldots ,\sigma _{t}^{2}}$, respectively. A contrast variable $V$ is defined by

${\displaystyle V=\sum _{i=1}^{t}c_{i}G_{i},}$

where the $c_{i}$'s are a set of coefficients representing a comparison of interest and satisfy ${\displaystyle \sum _{i=1}^{t}c_{i}=0}$. The SMCV of contrast variable $V$, denoted by $\lambda$, is defined as^[1]

${\displaystyle \lambda ={\frac {\operatorname {E} (V)}{\operatorname {stdev} (V)}}={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {{\text{Var}}(\sum _{i=1}^{t}c_{i}G_{i})}}}={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {\sum _{i=1}^{t}c_{i}^{2}\sigma _{i}^{2}+2\sum _{i=1}^{t}\sum _{j=i}c_{i}c_{j}\sigma _{ij}}}}}$

where ${\displaystyle \sigma _{ij}}$ is the covariance of ${\displaystyle G_{i}}$ and ${\displaystyle G_{j}}$. When ${\displaystyle G_{1},G_{2},\ldots ,G_{t}}$ are independent,

${\displaystyle \lambda ={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {\sum _{i=1}^{t}c_{i}^{2}\sigma _{i}^{2}}}}.}$

Classifying rule for the strength of group comparisons

The population value (denoted by $\lambda$ ) of SMCV can be used to classify the strength of a comparison represented by a contrast variable, as shown in the following table.^[1][2] This classifying rule has a probabilistic basis due to the link between SMCV and c⁺-probability.^[1]

Effect type	Effect subtype	Thresholds for negative SMCV	Thresholds for positive SMCV
Extra large	Extremely strong	${\displaystyle \lambda \leq -5}$	${\displaystyle \lambda \geq 5}$
	Very strong	${\displaystyle -5<\lambda \leq -3}$	${\displaystyle 5>\lambda \geq 3}$
	Strong	${\displaystyle -3<\lambda \leq -2}$	${\displaystyle 3>\lambda \geq 2}$
	Fairly strong	${\displaystyle -2<\lambda \leq -1.645}$	${\displaystyle 2>\lambda \geq 1.645}$
Large	Moderate	${\displaystyle -1.645<\lambda \leq -1.28}$	${\displaystyle 1.645>\lambda \geq 1.28}$
	Fairly moderate	${\displaystyle -1.28<\lambda \leq -1}$	${\displaystyle 1.28>\lambda \geq 1}$
Medium	Fairly weak	${\displaystyle -1<\lambda \leq -0.75}$	${\displaystyle 1>\lambda \geq 0.75}$
	Weak	${\displaystyle -0.75<\lambda <-0.5}$	${\displaystyle 0.75>\lambda >0.5}$
	Very weak	${\displaystyle -0.5\leq \lambda <-0.25}$	${\displaystyle 0.5\geq \lambda >0.25}$
Small	Extremely weak	${\displaystyle -0.25\leq \lambda <0}$	${\displaystyle 0.25\geq \lambda >0}$
	No effect	${\displaystyle \lambda =0}$

Statistical estimation and inference

The estimation and inference of SMCV presented below is for one-factor experiments.^[1][2] Estimation and inference of SMCV for multi-factor experiments has also been discussed.^[1][3]

The estimation of SMCV relies on how samples are obtained in a study. When the groups are correlated, it is usually difficult to estimate the covariance among groups. In such a case, a good strategy is to obtain matched or paired samples (or subjects) and to conduct contrast analysis based on the matched samples. A simple example of matched contrast analysis is the analysis of paired difference of drug effects after and before taking a drug in the same patients. By contrast, another strategy is to not match or pair the samples and to conduct contrast analysis based on the unmatched or unpaired samples. A simple example of unmatched contrast analysis is the comparison of efficacy between a new drug taken by some patients and a standard drug taken by other patients. Methods of estimation for SMCV and c⁺-probability in matched contrast analysis may differ from those used in unmatched contrast analysis.

Unmatched samples

Consider an independent sample of size $n_{i}$,

${\displaystyle Y_{i}=(Y_{i1},Y_{i2},\ldots ,Y_{in_{i}})}$

from the ${\displaystyle i^{\text{th}}(i=1,2,\ldots ,t)}$ group $G_i$. $Y_{i}$'s are independent. Let ${\displaystyle {\bar {Y}}_{i}={\frac {1}{n_{i}}}\sum _{j=1}^{n_{i}}Y_{ij}}$,

${\displaystyle s_{i}^{2}={\frac {1}{n_{i}-1}}\sum _{j=1}^{n_{i}}(Y_{ij}-{\bar {Y}}_{i})^{2},}$

${\displaystyle N=\sum _{i=1}^{t}n_{i}}$

and

${\displaystyle {\text{MSE }}={\frac {1}{N-t}}\sum _{i=1}^{t}(n_{i}-1)s_{i}^{2}.}$

When the $t$ groups have unequal variance, the maximal likelihood estimate (MLE) and method-of-moment estimate (MM) of SMCV ($\lambda$) are, respectively^[1][2]

${\displaystyle {\hat {\lambda }}_{\text{MLE }}={\frac {\sum _{i=1}^{t}c_{i}{\bar {Y}}_{i}}{\sqrt {\sum _{i=1}^{t}{\frac {n_{i}-1}{n_{i}}}c_{i}^{2}s_{i}^{2}}}}}$

and

${\displaystyle {\hat {\lambda }}_{\text{MM}}={\frac {\sum _{i=1}^{t}c_{i}{\bar {Y}}_{i}}{\sqrt {\sum _{i=1}^{t}c_{i}^{2}s_{i}^{2}}}}.}$

When the $t$ groups have equal variance, under normality assumption, the uniformly minimal variance unbiased estimate (UMVUE) of SMCV ($\lambda$) is^[1][2]

${\displaystyle {\hat {\lambda }}_{\text{UMVUE}}={\sqrt {\frac {K}{N-t}}}{\frac {\sum _{i=1}^{t}c_{i}{\bar {Y}}_{i}}{\sqrt {\sum _{i=1}^{t}{\text{MSE }}c_{i}^{2}}}}}$

where ${\displaystyle K={\frac {2(\Gamma ({\frac {N-t}{2}}))^{2}}{(\Gamma ({\frac {N-t-1}{2}}))^{2}}}}$. The confidence interval of SMCV can be made using the following non-central t-distribution:^[1][2]

${\displaystyle T={\frac {\sum _{i=1}^{t}c_{i}{\bar {Y}}_{i}}{\sqrt {\sum _{i=1}^{t}{\text{MSE }}c_{i}^{2}/n_{i}}}}\sim {\text{noncentral }}t(N-t,b\lambda )}$

where ${\displaystyle b={\sqrt {\frac {\sum _{i=1}^{t}c_{i}^{2}}{\sum _{i=1}^{t}c_{i}^{2}/n_{i}}}}.}$

Matched samples

In matched contrast analysis, assume that there are $n$ independent samples ${\displaystyle (Y_{1j},Y_{2j},\cdots ,Y_{tj})}$ from $t$ groups ($G_i$'s), where ${\displaystyle i=1,2,\cdots ,t;j=1,2,\cdots ,n}$. Then the $j^\text{th}$ observed value of a contrast ${\displaystyle V=\sum _{i=1}^{t}c_{i}G_{i}}$ is ${\displaystyle v_{j}=\sum _{i=1}^{t}c_{i}Y_{i}}$. Let ${\bar {V}}$ and ${\displaystyle s_{V}^{2}}$ be the sample mean and sample variance of the contrast variable $V$, respectively. Under normality assumptions, the UMVUE estimate of SMCV is^[1]

${\displaystyle {\hat {\lambda }}_{\text{UMVUE}}={\sqrt {\frac {K}{n-1}}}{\frac {\bar {V}}{s_{V}}}}$

where ${\displaystyle K={\frac {2(\Gamma ({\frac {n-1}{2}}))^{2}}{(\Gamma ({\frac {n-2}{2}}))^{2}}}.}$

A confidence interval for SMCV can be made using the following non-central t-distribution:^[1]

${\displaystyle T={\frac {\bar {V}}{s_{V}/{\sqrt {n}}}}\sim {\text{noncentral }}t(n-1,{\sqrt {n}}\lambda ).}$

See also

• Dual-flashlight plot

References

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