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

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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  [7] and Glass's  [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

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Suppose the random values in t groups represented by random variables   have means   and variances  , respectively. A contrast variable   is defined by

 

where the  's are a set of coefficients representing a comparison of interest and satisfy  . The SMCV of contrast variable  , denoted by  , is defined as[1]

 

where   is the covariance of   and  . When   are independent,

 

Classifying rule for the strength of group comparisons

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The population value (denoted by   ) 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    
Very strong    
Strong    
Fairly strong    
Large Moderate    
Fairly moderate    
Medium Fairly weak    
Weak    
Very weak    
Small Extremely weak    
No effect  

Statistical estimation and inference

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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

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Consider an independent sample of size  ,

 

from the   group  .  's are independent. Let  ,

 
 

and

 

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

 

and

 

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

 

where  .

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

 

where  

Matched samples

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In matched contrast analysis, assume that there are   independent samples   from   groups ( 's), where  . Then the   observed value of a contrast   is  .

Let   and   be the sample mean and sample variance of the contrast variable  , respectively. Under normality assumptions, the UMVUE estimate of SMCV is[1]

 

where  

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

 

See also

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References

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  1. ^ a b c d e f g h i j k 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.
  2. ^ a b c d e f g 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.
  3. ^ a b 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.
  4. ^ Rosenthal R, Rosnow RL, Rubin DB (2000). Contrasts and Effect Sizes in Behavioral Research. Cambridge University Press. ISBN 0-521-65980-9.
  5. ^ Huberty CJ (2002). "A history of effect size indices". Educational and Psychological Measurement. 62: 227–40. doi:10.1177/0013164402062002002.
  6. ^ Kirk RE (1996). "Practical significance: A concept whose time has come". Educational and Psychological Measurement. 56: 746–59. doi:10.1177/0013164496056005002.
  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.
  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. PMID 15137887.