Covariance models

Models containing some quantitative (categorical) and some qualitative explanatory variables, where the chief explanatory variables of interest are qualitative and the quantitative variables are introduced primarily to reduce the variance of the error terms. [Models in which all explanatory variables are qualitative are called analysis of variance -ANOVA- models.] Analysis of covariance -ANCOVA- combines features of ANOVA and regression. It augments the ANOVA model containing the factor effects with one or more additional quantitative variables that are related to the response variable. The intention is to make the analysis more precise by reducing the variance of the error terms. Each continuous quantitative variable added to the ANOVA model is called a concomitant variable (and sometimes also covariates). If qualitative variables are added to an ANOVA model to reduce error variance, the model remains to be ANOVA. By adding extra variables, the results is said to be controlled or adjusted for the additional variables (like age or sex). Apart from the above use of the term, analysis of covariance is more generally used for almost any analysis assessing the relationship between a response variable and a number of explanatory variables. In a multiple regression model, additional variables which are known not to have any effect on the response variable, such as age and sex, are sometimes added to the model to adjust the response for these variables (age and sex in this case). Such variables are called confounders (or covariates). When the response is normally distributed, this is the preferred method over a simple t-test when the two groups compared differ, say, in their age and sex distribution (or any other confounding variable). The result is then controlled (or adjusted) for age and sex. When such adjustments are made, the regression coefficient for the significant effect variable will be (most probably) different from the one obtained from a univariate model involving only that variable (say the effect of a disease on pulse rate as compared to healthy controls).