Analysis of Covariance

Analysis of covariance (ANCOVA) is a combination of analysis of variance (ANOVA) and regression analysis because the model contains both quantitative and qualitative independent variables. The idea is to enhance the ANOVA model by adding one or more quantitative independent variables that are related to the dependent variable. These variables are called concomitant variables or covariates. Increasing the precision of the model results in reducing the error terms. Without a covariate, the error mean square may be so high that a simple ANOVA may not detect differences between treatments. Covariates can also be used to remove the effect of an extraneous variable from the dependent variable. An extraneous variable influences the outcome of an experiment but is not of particular interest to the researcher.

Consider the study of a new weight-loss medication. In a double-blind study, weight is measured on subjects who have been randomly assigned to one of two treatment groups. One group receives the new medication, and the other group receives a placebo. The researcher wants to know whether the new medication produces significant weight-loss results. Since the effect of the medication may be related to the individual's initial weight, initial weight is used as a covariate in the analysis to reduce within-treatment variability. The ability to detect differences between treatments is now strengthened.

ANCOVA is also used to explore the nature of treatment effects rather than for increasing the precision of the model. In a study of the effect of two different cognitive therapy treatments, children with behavioral problems are assessed by a mental health professional using a questionnaire. Each child is given a total problem score. Parents are asked to fill out a questionnaire to establish a socioeconomic status (SES) score for each child as well. The SES score is used as a covariate in the analysis. In this case, the relationship between total problems and SES score for each treatment is of primary concern rather than the effect of the treatments on total problem score.

ANCOVA is often used as a means of correcting for bias when treatment groups are noticeably different from each other. In the double-blind study of a new medication to reduce blood pressure, subjects are randomly assigned to one of two treatments: those treated with a placebo or those treated with the medication. Suppose it is found that by chance the initial blood pressure for subjects in one group is found to be substantially higher than that of the other group. Adding initial blood pressure as a covariate in the model helps remove that bias. Using ANCOVA for this purpose must be done with caution, however. If the covariate is related to the treatment variable, any conclusions are questionable at minimum. For instance, in a study of attitudes toward two different blood glucose monitors for diabetics, it is found that older patients tend to like one monitor while younger patients tend to like the other. With little regard, age is a covariate in an ANCOVA in an attempt to remove the bias. As it turns out, however, age is related to monitor preference. Therefore, using age as a covariate could actually lead to the wrong conclusion.

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