Contrast variable: Difference between revisions

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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

|journal=Pharmacogenomics |volume=10 |issue= |pages=345–58

|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⁺-probability]]. The SMCV is the ratio of [[mean]] to [[standard deviation]] of a contrast variable and the c⁺-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

|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}}</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

|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

|journal=Technometrics |volume=12 |issue= |pages=49–54

|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

|year=2006 |month= |pmid= |doi= 10.1002/sim.2256 |url=}}</ref>

<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

|url= |isbn=0-521-65980-9}}</ref>

<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⁺-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⁺-probability is <math>\text{Pr}(V > 0)</math>.

==Discussions==

The c⁺-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⁺-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⁺-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⁺-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

|year=2010 |month= |pmid= |doi=10.4172/2155-6180.1000108

|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==

* [[Effect size]]

* [[SSMD]]

* [[SMCV]]

* [[c+-probability|c⁺-probability]]

* [[Contrast (statistics)]]

* [[Dual-flashlight plot]]

==References==

{{reflist}}

{{DEFAULTSORT:Contrast variable}}

[[Category:Effect size]]

[[Category:Analysis of variance]]