Standardized mean of a contrast variable explained

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 the strictly standardized mean difference (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 \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

G _{1,} G _{2,} \ldots, G _{t} have means \mu _{1,} \mu _{2,} \ldots, \mu _{t} and variances \sigma ^{2 ,} _{1} \sigma ^{2 ,} _{2} \ldots, \sigma ^{2} _{t} , respectively. A contrast variable V is defined byV = \sum ^{t} _{i =1} c _{i} G _{i} , where the c _{i} 's are a set of coefficients representing a comparison of interest and satisfy \sum ^{t} _{i =1} c _{i} = 0 . The SMCV of contrast variable V , denoted by λ , is defined as^[1] λ = (\operatorname{E)/((V)}{\operatorname{stdev}(V)}) = (\sum ^{t} _{i =1} c _{i} \mu _{i})/(\sqrt{ Var \left(\sum ^{t} _{i =1} c _{i} G _{i\right)}) } = \frac

where

\sigma _{ij} is the covariance of G _{i} and G _{j} . When G _{1,} G _{2,} \ldots, G _{t} are independent,λ = (\sum ^{t} _{i =1} c _{i} \mu _{i})/(\sqrt{\sum ^{t} _{i =1} c ^{2} _{i} \sigma ^{2} _{i}) }.

Classifying rule for the strength of group comparisons

The population value (denoted by

λ ) 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	λ \le -5	λ \ge 5
	Very strong	-5 < λ \le -3	5 > λ \ge 3
	Strong	-3 < λ \le -2	3 > λ \ge 2
	Fairly strong	-2 < λ \le -1 . 645	2 > λ \ge 1 . 645
Large	Moderate	-1 . 645 < λ \le -1 . 28	1 . 645 > λ \ge 1 . 28
	Fairly moderate	-1 . 28 < λ \le -1	1 . 28 > λ \ge 1
Medium	Fairly weak	-1 < λ \le -0 . 75	1 > λ \ge 0 . 75
	Weak	-0 . 75 < λ < -0 . 5	0 . 75 > λ > 0 . 5
	Very weak	-0 . 5 \le λ < -0 . 25	0 . 5 \ge λ > 0 . 25
Small	Extremely weak	-0 . 25 \le λ < 0	0 . 25 \ge λ > 0
	No effect	λ = 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} ,Y _{i} = \left(Y _{i1} , Y _{i2} , \ldots, Y _{i n _{i}} \right)

from the

i ^{th} (i =1 , 2 , \ldots, t) group G _{i} . Y _{i} 's are independent. Let \bar{Y} _{i} = (1)/(n _{i}) \sum ^{n _{i}} _{j =1} Y _{ij} ,s ^{2} _{i} = (1)/(n _{i-1}) \sum ^{n _{i}} _{j =1} \left(Y _{ij} - \bar{Y} ^{2 ,} _{i\right)} N = \sum ^{t} _{i =1} n _{i} and MSE = (1)/(N - t) \sum ^{t} _{i =1} \left(n _{i} - 1 \right)s ^{2 .} _{i}

When the

t groups have unequal variance, the maximal likelihood estimate (MLE) and method-of-moment estimate (MM) of SMCV (λ ) are, respectively^{[1] [2]} \hat{ λ } _{MLE} = (\sum ^{t} _{i =1} c _{i} \bar{Y)/(_{i}{\sqrt{\sum}) ^{t} ^{i =1} (n _{i} -1)/(n _{i}) c ^{2} _{i} s ^{2} _{i} }} and\hat{ λ } _{MM} = (\sum ^{t} _{i =1} c _{i} \bar{Y)/(_{i}{\sqrt{\sum}) ^{t} ^{i =1} c ^{2} _{i} s ^{2} _{i} }}.

When the

t groups have equal variance, under normality assumption, the uniformly minimal variance unbiased estimate (UMVUE) of SMCV (λ ) is^{[1] [2]} \hat{ λ } _{UMVUE} = \sqrt (K)/(N - t) (\sum ^{t} _{i =1} c _{i} \bar{Y)/(_{i}{\sqrt{\sum}) ^{t} ^{i =1} MSE c ^{2 }}} _{i}

where

K = (2 \left(\Gamma\left( (N - t)/(2) \right)\right) ^{2})/(\left(\Gamma\left( (N - t -1)/(\right)\right) ^{2} 2)) .

The confidence interval of SMCV can be made using the following non-central t-distribution:^{[1] [2]}

T = (\sum ^{t} _{i =1} c _{i} \bar{Y)/(_{i}{\sqrt{\sum}) ^{t} ^{i =1} MSE c ^{2 /n} _{i}}} \sim noncentral t(N - t, b λ )

where

b = \sqrt{ (\sum ^{t} _{i =1} c ^{2} _{i})/(\sum ^{t} _{i =1} c ^{2 /n} _{i}) }.

Matched samples

In matched contrast analysis, assume that there are

n independent samples \left(Y _{1j} , Y _{2j} , … , Y _{tj} \right) from t groups (G _{i} 's), where i = 1 , 2 , … , t; j = 1 , 2 , … , n . Then the j ^{th} observed value of a contrast V = \sum ^{t} _{i =1} c _{i} G _{i} is v _{j} = \sum ^{t} _{i =1} c _{i} Y _{i} .

Let

\bar{V} and s ^{2} _{V} be the sample mean and sample variance of the contrast variable V , respectively. Under normality assumptions, the UMVUE estimate of SMCV is^[1] \hat{ λ } _{UMVUE} = \sqrt (K)/(n -1) (\bar{V)/() }

where

K = (2 \left(\Gamma\left( (n -1)/(\right)\right) ^{2} 2))/(\left(\Gamma\left( (n -2)/(\right)\right) ^{2} 2)) .

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

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

See also

• Dual-flashlight plot

Notes and References

1. Book: Zhang XHD. 2011. Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research . Cambridge University Press. 978-0-521-73444-8.
2. Zhang XHD. A method for effectively comparing gene effects in multiple conditions in RNAi and expression-profiling research . Pharmacogenomics . 10 . 345–58 . 2009 . 20397965 . 10.2217/14622416.10.3.345 .
3. Zhang XHD. Assessing the size of gene or RNAi effects in multifactor high-throughput experiments . Pharmacogenomics . 11 . 199–213 . 2010 . 20136359. 10.2217/PGS.09.136 .
4. Book: Rosenthal R, Rosnow RL, Rubin DB . 2000 . Contrasts and Effect Sizes in Behavioral Research . Cambridge University Press . 0-521-65980-9.
5. Huberty CJ . A history of effect size indices . Educational and Psychological Measurement . 62 . 227–40 . 2002 . 10.1177/0013164402062002002 .
6. Kirk RE . Practical significance: A concept whose time has come . Educational and Psychological Measurement . 56 . 746–59 . 1996 . 10.1177/0013164496056005002 .
7. Cohen J . The statistical power of abnormal-social psychological research: A review . Journal of Abnormal and Social Psychology . 65 . 145–53 . 1962 . 13880271 . 10.1037/h0045186 .
8. Glass GV . Primary, secondary, and meta-analysis of research . Educational Researcher . 5 . 3–8 . 1976 . 10.3102/0013189X005010003 .
9. Steiger JH . Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis . Psychological Methods . 9 . 164–82 . 2004 . 15137887. 10.1037/1082-989x.9.2.164 .