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In [[statistics]], a '''contrast variable''' is a [[linear combination]] of [[random variable]]s in which the sum of the coefficients is zero<ref name=ZhangPharmacogenomics2009>{{cite journal |author=Zhang XHD
|title= A method for effectively comparing gene effects in multiple conditions in RNAi and expression-profiling research
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|year=2009 |month= |pmid= |doi=10.2217/14622416.10.3.345 |url=}}</ref>. Each variable may represent random values in one of multiple groups involved in a comparison. Associated with a contrast variable are two terms: the standardized mean of contrast variable ([[SMCV]]) and [[c+-probability|c<sup>+</sup>-probability]]. The SMCV is the ratio of [[mean]] to [[standard deviation]] of a contrast variable and the c<sup>+</sup>-probability is the probability that a contrast variable obtains a positive value.
==Background==
Traditional contrast analysis tackles the question of whether a linear combination of group means is exactly zero using significance testing. However, in reality, the key question of interest is usually how far a linear combination of values in multiple groups involved in a comparison is away from zero in a distribution level. <ref name=ZhangBook2011>{{cite book
|title= Contrast variable potentially providing a consistent interpretation to effect sizes▼
|journal=Journal of Biometrics & Biostatistics |volume=1 |issue= |pages=108 ▼
|year=2010 |month= |pmid= |doi= doi:10.4172/2155-6180.1000108 ▼
|journal=Pharmacogenomics |volume=11 |issue= |pages=199–213 ▼
|year=2010 |month= |pmid= |doi=10.2217/PGS.09.136 |url=}}</ref>▼
|author= Zhang XHD
|year=2011
Line 21 ⟶ 11:
|publisher =Cambridge University Press
|url=
|isbn=978-0-521-73444-8}}</ref> Therefore, to effectively compare groups, we need additional analysis to incorporate information in a distribution level. In addition, the value of a traditional contrast has issues in capturing data variability. The p-value from classical t-test of testing a traditional contrast can capture data variability; however, it is affected by both sample size and the strength of a comparison. <ref name=Kirk1996>{{cite journal |author=Kirk RE
|isbn=978-0-521-73444-8}}</ref>▼
|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=}}</ref>
To address the issues of traditional contrast analysis, various effect sizes have been proposed. <ref name=Huberty2002>{{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=}}</ref> Many of them may fall into two categories: probabilistic indices for comparing groups in a distribution level<ref name=Owen1964>{{cite journal |author=Owen DB, Graswell KJ, Hanson DL
|title=Nonparametric upper confidence bounds for P(Y < X) and confidence limits for P(Y<X) when X and Y are normal
|journal=Journal of American Statistical Association |volume=59 |issue= |pages=906–24
|year=1964 |month= |pmid= |doi= |url=}}</ref>
<ref name=Church1970>{{cite journal |author=Church JD, Harris B
|title=The estimation of reliability from stress-strength relationships
|year=1970 |month= |pmid= |doi= |url=}}</ref>
<ref name=Downton1973>{{cite journal |author=Downton F
|title=The estimation of Pr(Y < X) in normal case
|journal=Technometrics |volume=15 |issue= |pages=551–8
|year=1973 |month= |pmid= |doi= |url=}}</ref>
<ref name=Reiser1986>{{cite journal |author=Reiser B, Guttman I
|title=Statistical inference for of Pr(Y-less-thaqn-X) - normal case
|journal=Technometrics |volume=28 |issue= |pages=253–7
|year=1986 |month= |pmid= |doi= |url=}}</ref>
<ref name=Acion2006>{{cite journal |author=Acion L, Peterson JJ, Temple S, Arndt S
|title=Probabilistic index: an intuitive non-parametric approach to measuring the size of treatment effects
|journal=Statistics in Medicine |volume=25 |issue= |pages=591–602
<ref name=Stine2001>{{cite journal |author=Stine RA, Heyse JF
|title=Non-parametric estimates of overlap
|journal=Statistics in Medicine |volume=20 |issue= |pages=215–36
|year=2001 |month= |pmid= 11169598
|doi= |url=}}</ref> and metrics for capturing both mean and variability, which includes various effect sizes similar to standardized mean differences<ref name=RosenthaletalBook2000>{{cite book
|author= Rosenthal R, Rosnow RL, Rubin DB
|year=2000
|title= Contrasts and Effect Sizes in Behavioral Research
|publisher =Cambridge University Press
<ref name=ZhangGenomics2007>{{cite journal |author=Zhang XHD
|title=A pair of new statistical parameters for quality control in RNA interference [[high-throughput screening]] assays
|journal=Genomics |volume=89 |issue= |pages=552–61
|year=2007 |month= |pmid= |doi=10.1016/j.ygeno.2006.12.014 |url=}}</ref>
<ref name=Cohen1962>{{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= |url=}}</ref>
<ref name=Glass1976>{{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=}}</ref>. However, different effect size measures are suitable for different types of data, and the interpretations of effect sizes are generally arbitrary and remain problematic even for the same effect size measure. <ref name=Onwuegbuzie2003>{{cite journal |author= Onwuegbuzie AJ, Levin JR
|title=Without supporting statistical evidence, where would reported measures of substantive importance lead? To no good effect
|journal= Journal of Modern Applied Statistical Methods |volume=65 |issue= |pages=133–51
|year=2003 |month= |pmid= |doi= |url=}}</ref> The contrast variable, along with SMCV and c<sup>+</sup>-probability, provides potential solutions to these issues.
==Concepts==
Suppose the random values in t groups represented by random variables <math>G_1, G_2, \cdots, G_t </math> have means <math>\mu_1, \mu_2, \cdots, \mu_t </math> and variances <math>\sigma_1^2, \sigma_2^2, \cdots, \sigma_t^2 </math>, respectively. Then a traditional contrast <math>L</math> is <math>L = \sum_{i=1}^t c_i \mu_i </math> where <math>c_i</math>'s are a set of coefficients representing a comparison of interest and satisfy <math>\sum_{i=1}^t c_i = 0</math>. Traditional contrast analysis focuses on testing <math>\text{H}_0: L=0</math>, <math>\text{H}_0: \le 0</math> or <math>\text{H}_0: L \ge 0</math>. Correspondingly, a contrast variable <math>V</math> is defined as a linear combination of the random variables, i.e., <math> V = \sum_{i=1}^t c_i G_i </math>. Thus, the traditional contrast equals the mean of the contrast variable <math>V</math>, that is, <math>L = \text{E}(V)</math>. The SMCV of contrast variable <math>V</math>, denoted by <math>\lambda</math>, is defined as<ref name="ZhangBook2011"/>
:<math>\lambda = \frac{\mu_V}{\sigma_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} }} </math>
where <math> \sigma_{ij}</math> is the covariance of <math>G_{i}</math> and <math>G_{j}</math>. When <math>G_1, G_2, \cdots, G_t </math> are independent,
:<math>\lambda = \frac{\sum_{i=1}^t c_i \mu_i}{\sqrt{\sum_{i=1}^t c_i^2 \sigma_i^2 }} </math>. The c<sup>+</sup>-probability is <math>\text{Pr}(V > 0)</math>.
==Discussions==
The c<sup>+</sup>-probability is a probabilistic index accounting for distributions of compared groups whereas SMCV is an extended variant of standardized mean difference (such as Cohen's <math>d</math><ref name="Cohen1962"/> and Glass's <math>\Delta</math><ref name="Glass1976"/>) incorporating both mean and variance of groups. There is a link between SMCV and c<sup>+</sup>-probability<ref name="ZhangPharmacogenomics2009"/> <ref name="ZhangBook2011"/>. Thus, standardized mean difference and probabilistic index are now integrated to effectively assess the strength of a comparison. In addition, the concepts of SMCV and c<sup>+</sup>-probability are applicable to not only the comparison of two groups but also the comparison of more than two groups. Based on contrast variable, [[SMCV]] along with [[c<sup>+</sup>-probability]] may provide a consistent interpretation to the strength of a comparison.<ref name=ZhangJBiometBiostat2010>{{cite journal |author=Zhang XHD
▲|title= Contrast variable potentially providing a consistent interpretation to effect sizes
▲|journal=Journal of Biometrics & Biostatistics |volume=1 |issue= |pages=108
|url= http://www.omicsonline.org/2155-6180/2155-6180-1-108.php}}</ref>
<ref name="ZhangBook2011"/> Based on the concept of contrast variable, a traditional contrast (i.e., mean of a contrast variable) and an effect size (i.e., SMCV) are now two characteristics of the same random variable (i.e., a contrast variable); subsequently, they are integrated for any comparison in contrast analysis. <ref name="ZhangBook2011"/>
==See also==
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