“Standardized mean of a contrast variable”的意思、由来-开放百科全书

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]

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

^[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

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]

Classifying rule for the strength of group comparisons

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]

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

When the

groups have unequal variance, the maximal likelihood estimate (MLE) and method-of-moment estimate (MM) of SMCV (

) are, respectively^[1]^[2]

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]

Matched samples

In matched contrast analysis, assume that there are

independent samples

from

groups (

's), where

. Then

Let

and

be the sample mean and sample variance of the contrast variable

, respectively. Under normality assumptions, the UMVUE estimate of SMCV is^[1]

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

See also

References

1. ^¹²³⁴⁵⁶⁷⁸⁹¹⁰{{cite book |author= Zhang XHD|year=2011|title= Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research |publisher =Cambridge University Press|url= |isbn=978-0-521-73444-8}}
2. ^¹²³⁴⁵⁶{{cite journal |author=Zhang XHD|title= A method for effectively comparing gene effects in multiple conditions in RNAi and expression-profiling research |journal=Pharmacogenomics |volume=10 |issue= |pages=345–58 |year=2009 |month= |pmid=20397965 |doi=10.2217/14622416.10.3.345 |url=}}
3. ^¹{{cite journal |author=Zhang XHD|title= Assessing the size of gene or RNAi effects in multifactor high-throughput experiments |journal=Pharmacogenomics |volume=11 |issue= |pages=199–213 |year=2010 |month= |pmid= 20136359|doi=10.2217/PGS.09.136 |url=}}
4. ^{{cite book |vauthors=Rosenthal R, Rosnow RL, Rubin DB |year=2000|title= Contrasts and Effect Sizes in Behavioral Research|publisher =Cambridge University Press|url= |isbn=0-521-65980-9}}
5. ^{{cite journal |author=Huberty CJ|title= A history of effect size indices |journal=Educational and Psychological Measurement |volume=62 |issue= |pages=227–40 |year=2002 |month= |pmid= |doi=10.1177/0013164402062002002 |url=}}
6. ^{{cite journal |author=Kirk RE|title= Practical significance: A concept whose time has come |journal=Educational and Psychological Measurement |volume=56 |issue= |pages=746–59 |year=1996 |month= |pmid= |doi= 10.1177/0013164496056005002 |url=}}
7. ^{{cite journal |author=Cohen J|title= The statistical power of abnormal-social psychological research: A review|journal= Journal of Abnormal and Social Psychology |volume=65 |issue= |pages=145–53 |year=1962 |month= |pmid= 13880271 |doi= 10.1037/h0045186|url=}}
8. ^{{cite journal |author=Glass GV|title= Primary, secondary, and meta-analysis of research |journal= Educational Researcher |volume=5 |issue= |pages=3–8 |year=1976 |month= |pmid= |doi= 10.3102/0013189X005010003 |url=}}
9. ^{{cite journal |author=Steiger JH|title= Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis|journal= Psychological Methods |volume=9 |issue= |pages=164–82 |year=2004 |month= |pmid= |doi= 10.1037/1082-989x.9.2.164|url=}}