Contrast variable

In statistics, a contrast variable is a linear combination of random variables in which the sum of the coefficients is zero. 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. 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. 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.

To address the issues of traditional contrast analysis, various effect sizes have been proposed. Many of them may fall into two categories: probabilistic indices for comparing groups in a distribution level and metrics for capturing both mean and variability, which includes various effect sizes similar to standardized mean differences . 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. 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 $G_{1},G_{2},\cdots ,G_{t}$ have means $\mu _{1},\mu _{2},\cdots ,\mu _{t}$ and variances $\sigma _{1}^{2},\sigma _{2}^{2},\cdots ,\sigma _{t}^{2}$, respectively. Then a traditional contrast $L$ is $L=\sum _{i=1}^{t}c_{i}\mu _{i}$ where $c_{i}$'s are a set of coefficients representing a comparison of interest and satisfy $\sum _{i=1}^{t}c_{i}=0$. Traditional contrast analysis focuses on testing ${\text{H}}_{0}:L=0$, ${\text{H}}_{0}:\leq 0$ or ${\text{H}}_{0}:L\geq 0$. Correspondingly, a contrast variable $V$ is defined as a linear combination of the random variables, i.e., $V=\sum _{i=1}^{t}c_{i}G_{i}$. Thus, the traditional contrast equals the mean of the contrast variable $V$, that is, $L={\text{E}}(V)$. The SMCV of contrast variable $V$, denoted by $\lambda$, is defined as

$\lambda ={\frac {\mu _{V}}{\sigma _{V}}}={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {{\text{Var}}(\sum _{i=1}^{t}c_{i}G_{i})}}}={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {\sum _{i=1}^{t}c_{i}^{2}\sigma _{i}^{2}+2\sum _{i=1}^{t}\sum _{j=i}c_{i}c_{j}\sigma _{ij}}}}$

where $\sigma _{ij}$ is the covariance of $G_{i}$ and $G_{j}$. When $G_{1},G_{2},\cdots ,G_{t}$ are independent,

$\lambda ={\frac {\sum _{i=1}^{t}c_{i}\mu _{i}}{\sqrt {\sum _{i=1}^{t}c_{i}^{2}\sigma _{i}^{2}}}}$. The c⁺-probability is ${\text{Pr}}(V>0)$.

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 $d$ and Glass's $\Delta$) incorporating both mean and variance of groups. There is a link between SMCV and c⁺-probability . 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. 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.

See also

• Effect size
• SSMD
• SMCV
• c⁺-probability
• Contrast (statistics)
• Dual-flashlight plot