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] The standardized mean of a contrast variable, along with c⁺-probability , can provide a consistent interpretation of the strength of a comparison.^[6] 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"^[7] which includes Cohen's $d$ ^[8] and Glass's $\delta.$ ^[9] In ANOVA, a similar parameter for measuring the strength of group comparison is standardized effect size (SES).^[10] 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 $G_1, G_2, \ldots, G_t$ have means $\mu_1, \mu_2, \ldots, \mu_t$ and variances $\sigma_1^2, \sigma_2^2, \ldots, \sigma_t^2$ , respectively. A contrast variable $V$ is defined by

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 $\sum_{i=1}^t c_i = 0$ . The SMCV of contrast variable $V$ , denoted by $\lambda$ , is defined as^[1]

\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 $\sigma_{ij}$ is the covariance of $G_{i}$ and $G_{j}$ . When $G_1, G_2, \ldots, G_t$ are independent,

\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	$\lambda \le -5$	$\lambda \ge 5$
	Very strong	$-5 < \lambda \le -3$	$5 > \lambda \ge 3$
	Strong	$-3 < \lambda \le -2$	$3 > \lambda \ge 2$
	Fairly strong	$-2 < \lambda \le -1.645$	$2 > \lambda \ge 1.645$
Large	Moderate	$-1.645 < \lambda \le -1.28$	$1.645 > \lambda \ge 1.28$
Large	Fairly moderate	$-1.28 < \lambda \le -1$	$1.28 > \lambda \ge 1$
Medium	Fairly weak	$-1 < \lambda \le -0.75$	$1 > \lambda \ge 0.75$
	Weak	$-0.75 < \lambda < -0.5$	$0.75 > \lambda > 0.5$
	Very weak	$-0.5 \le \lambda < -0.25$	$0.5 \ge \lambda > 0.25$
Small	Extremely weak	$-0.25 \le \lambda < 0$	$0.25 \ge \lambda > 0$
Small	No effect	$\lambda = 0$

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]^[6]

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 $n_i$ ,

Y_i=(Y_{i1}, Y_{i2}, \ldots, Y_{i n_i})

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

s_i^2 = \frac{1}{n_i-1} \sum_{j=1}^{n_i} (Y_{ij}-\bar{Y}_i)^2,

N = \sum_{i=1}^t n_i

and

\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]

\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

\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]

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

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 $b=\sqrt{\frac{\sum_{i=1}^t c_i^2}{\sum_{i=1}^t c_i^2/n_i}}.$

Matched samples

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

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

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

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

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

References

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