Analysis of covariance (ANCOVA) is a more sophisticated form of analysis of variance. It accounts for a shared variable between populations that may be explaining variation. For example, three groups may receive three treatments. Variations between group survivals may be driven by age differences; therefore, controlling for the effect of age by including it as a variable could improve the model's explanatory power.
Things You'll Need
- ANCOVA Formulas
- Statistical Software
- Statistical Calculator
Determine the independent and dependent variables, including the covariate.
In the above example, the two dependent variables are the covariate, age and which treatment the group received. The covariate should be continuous. Beyond avoiding empty cells, the benefit of this will become clear in the following steps.
Determine the linear regression for each group. In our example, age is an independent variable, survival time is an independent variable and each group has its own line of regression. Age has therefore been accounted for.
Reject the null hypothesis (that the treatments are the same, i.e., that the treatment coefficient is zero) if the difference between slopes is statistically significant.
Determine whether the intercepts are the same if the slopes were not found to be statistically different.
Reject the null hypothesis if the intercepts are significantly different. If the three lines of regression have the same slope but their intercepts are significantly different, then their parallel nature means they won't meet anywhere and the treatments are different.
Tips & Warnings
- ANCOVA is a blend of discrete (i.e., categorical) and continuous variables. In the example above, age is the continuous variable and whether one is in a group or not is a (binary) discrete variable. SAS handles such regression problems easily, sidestepping the above complication.
- If bias can be removed by better randomizing the populations, this would be preferable to compensating for such bias mathematically. In the above example, age should be distributed randomly before the experiment is begun, if possible.
- The assumptions of the ANCOVA model must be met by the data to a significant extent for the model's predictions to be valid. These assumptions, as in ANOVA (analysis of variance), are that the variance of the errors are not a function of the independent variables, that the errors are normally distributed and that the relation between the independent and dependent variables is linear.
- Two-way ANOVA is more appropriate if the covariate (time, in the above example) is categorical/discrete, thus avoiding the problem of empty cells and therefore a need for more test subjects.
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SAS Code for ANCOVA: Intuitive Explanation; Richard Suen: Introduction to Optimal Control Theory; US Dept. of Justice Glossary; Resources. Statistics.com: Statistical Glossary;
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