**Covariate-effect analysis**

Covariate-effect analysis is the analysis of the effects of a set of covariables (covariates) on a set of response variables. For example, the yield of a set of genotypes measured in different environments can be treated as response variables, whereas the genetic values of other traits such as yield components, plant height, maturity, etc.,can be treated as covariables (or explanatory variables), which are used to "explain" the responses of the genotypes to the environments (Yan and Tinker, 2005, Crop Science). Genetic markers can also be regarded as explanatory "traits" in such analysis (Yan and Tinker 2005, Molecular Breeding).

Covariate-effect analysis is also referred to as variable-by-variable biplot analysis, and it involves the following steps:

- Calculate the effects (linear correlation or regression coefficient) of each explanatory variable on each of the response variables, which resultsin an explanatory variable by response variable (variable-by-variable or VBV) two-way table of "covariate effects", which is referred to as the "covariate-effect table" or VBV table.

- Remove explanatory variables that have no significant effects on any of the response variables at a user-specified significance level, resulting in a reduced covariate-effect table.

- Generate a covariate-effect biplot based on the reduced covariate-effect table.

- Study the covariate-effect patterns in the biplot, which includes

1) similarities among the explanatory variables in their effects on the response variables,

2) similarities among the response variables in their response to the explanatory variables, and

3) interactions between the two sets of variables.

See QTL identification based on phenotypic data from multiple environments for a full example of this type of analysis.