Contrast (statistics)

In statistics, particularly analysis of variance, a contrast is a linear combination of two or more factor level means (averages) whose coefficients add up to zero.^[1][2] A simple contrast is the difference between two means. A contrast may be any of: the set of coefficients used to specify a comparison; the specific value of the linear combination obtained for a given study or experiment; the random quantity defined by applying the linear combination to treament effects when these are themselves considered as random variables. In the last context here, the term contrast variable is sometimes used.

Background

Contrasts are sometimes used to compare mixed effects. A common example can be the difference between two test scores — one at the beginning of the semester and one at its end. Note that we are not interested in one of these scores by itself, but only in the contrast (in this case — the difference). Since this is a linear combination of independent variables, its variance will match accordingly, as the weighted sum of the variances; in this case both weights are one. This "blending" of two variables into one might be useful in many cases such as ANOVA, regression, or even as descriptive statistics in its own right.

Another example would be comparing 5 standard treatments to a new treatment, hence giving each old treatment a weight of 1/5, and the new sixth treatment a weight of −1. If this new linear combination has a mean zero, this will mean that the old treatments are not different from the new on average.

The usual results for linear combinations of independent random variables mean that the variance of a contrast is equal to the weighted sum of the variances.^[1] If two contrasts are orthogonal, estimates created by using such contrasts will be uncorrelated. This has implications in ANOVA and regression where in certain designed experiments the design is such as to ensure that estimates for a number of different parameters, each related to a different contrast, will be uncorrelated. Thus, if orthogonal contrasts are available, it is possible to summarise the results of a statistical analysis in the form of a simple analysis of variance table, in such a way that it contains the results for different test statistics relating to different contrasts, each of which are statistically independent.

A recent development in statistical analysis is the standardized mean of a contrast variable. This makes a comparison between the size of the differences between groups, as measured by a contrast and the accuracy with which that contrast can be measured by a given study or experiment.^[3].

Types of constrast

• Orthogonal contrasts are a set of contrasts in which, for any distinct pair, the sum of the cross-products of the coefficients is zero^[4]
• Orthonormal contrasts are orthogonal constrasts which satify the additional condition that, for each contrast, the sum squares of the coefficients add up to one^[4]

References

1. ^ ^{a b} NIST/SEMATECH e-Handbook of Statistical Methods
2. ^ Everitt, B.S. (2002) Cambridge Dictionary of Statistics, CUP, ISBN 0-521-81099-x (Entry for "Contrast"
3. ^ Zhang XHD (2011). Optimal High-Throughput Screening: Practical Experimental Design and Data Analysis for Genome-scale RNAi Research. Cambridge University Press. ISBN 978-0-521-73444-8. 
4. ^ ^{a b} Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x (entry for "Orthogonal contrasts")

External links

• Examples of orthogonal contrasts for analysis of variance