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]
InANOVA, 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.
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[edit]
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]
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.
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^ abcdefgZhang 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. PMID20397965.
^ abZhang 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. PMID20136359.
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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. PMID13880271.
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Glass GV (1976). "Primary, secondary, and meta-analysis of research". Educational Researcher. 5: 3–8. doi:10.3102/0013189X005010003.
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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. PMID15137887.