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 $\delta .$ ^[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 $\sigma _{1}^{2},\sigma _{2}^{2},\ldots ,\sigma _{t}^{2}$ , 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	$\lambda \leq -5$	$\lambda \geq 5$
	Very strong	$-5<\lambda \leq -3$	$5>\lambda \geq 3$
	Strong	$-3<\lambda \leq -2$	$3>\lambda \geq 2$
	Fairly strong	$-2<\lambda \leq -1.645$	$2>\lambda \geq 1.645$
Large	Moderate	$-1.645<\lambda \leq -1.28$	$1.645>\lambda \geq 1.28$
Large	Fairly moderate	$-1.28<\lambda \leq -1$	$1.28>\lambda \geq 1$
Medium	Fairly weak	$-1<\lambda \leq -0.75$	$1>\lambda \geq 0.75$
	Weak	$-0.75<\lambda <-0.5$	$0.75>\lambda >0.5$
	Very weak	$-0.5\leq \lambda <-0.25$	$0.5\geq \lambda >0.25$
Small	Extremely weak	$-0.25\leq \lambda <0$	$0.25\geq \lambda >0$
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]

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_{in_{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

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

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[ZhangBook2011-1] 1 2 3 4 5 6 7 8 9 10 11 Zhang XHD (2011). Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research. Cambridge University Press. ISBN 978-0-521-73444-8.

[ZhangPharmacogenomics2009-2] 1 2 3 4 5 6 7 Zhang XHD (2009). "A method for effectively comparing gene effects in multiple conditions in RNAi and expression-profiling research". Pharmacogenomics. 10: 345–58. doi:10.2217/14622416.10.3.345. PMID 20397965.

[ZhangPharmacogenomics2010-3] 1 2 Zhang XHD (2010). "Assessing the size of gene or RNAi effects in multifactor high-throughput experiments". Pharmacogenomics. 11: 199–213. doi:10.2217/PGS.09.136. PMID 20136359.

[RosenthaletalBook2000-4] Rosenthal R, Rosnow RL, Rubin DB (2000). Contrasts and Effect Sizes in Behavioral Research. Cambridge University Press. ISBN 0-521-65980-9.

[Huberty2002-5] Huberty CJ (2002). "A history of effect size indices". Educational and Psychological Measurement. 62: 227–40. doi:10.1177/0013164402062002002.

[Kirk1996-6] Kirk RE (1996). "Practical significance: A concept whose time has come". Educational and Psychological Measurement. 56: 746–59. doi:10.1177/0013164496056005002.

[Cohen1962-7] Cohen J (1962). "The statistical power of abnormal-social psychological research: A review". Journal of Abnormal and Social Psychology. 65: 145–53. doi:10.1037/h0045186. PMID 13880271.

[Glass1976-8] Glass GV (1976). "Primary, secondary, and meta-analysis of research". Educational Researcher. 5: 3–8. doi:10.3102/0013189X005010003.

[9] Steiger JH (2004). "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. doi:10.1037/1082-989x.9.2.164.