ANCOVA
Encyclopedia
In statistics, analysis of covariance (ANCOVA) is a general linear model
with a continuous outcome variable (quantitative, scaled) and two or more predictor variables where at least one is continuous (quantitative, scaled) and at least one is categorical (nominal, non-scaled). ANCOVA is a merger of ANOVA and regression
for continuous variables. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative predictors (covariate
s) account. The inclusion of covariates can increase statistical power
because it accounts for some of the variability.
assumes that the errors are normally distributed and homoscedastic
.
Since ANCOVA is a method based on linear regression, the relationship of the dependent variable to the independent variable
(s) must be linear in the parameters.
Simplifying assumption (not necessary to run ANCOVA): homogeneity of regression which says that the relationship between the covariate and the dependent variable should be similar across all groups of the independent variable.
by accounting for some of the variance in the dependent variable and thus increasing the ratio of variance explained by the independent variables, adding a covariate into ANOVA also reduces the degrees of freedom
. Accordingly, adding a covariate which accounts for very little variance in the dependent variable might actually reduce power.
General linear model
The general linear model is a statistical linear model.It may be written aswhere Y is a matrix with series of multivariate measurements, X is a matrix that might be a design matrix, B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors or...
with a continuous outcome variable (quantitative, scaled) and two or more predictor variables where at least one is continuous (quantitative, scaled) and at least one is categorical (nominal, non-scaled). ANCOVA is a merger of ANOVA and regression
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
for continuous variables. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative predictors (covariate
Covariate
In statistics, a covariate is a variable that is possibly predictive of the outcome under study. A covariate may be of direct interest or it may be a confounding or interacting variable....
s) account. The inclusion of covariates can increase statistical power
Statistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
because it accounts for some of the variability.
Assumptions
Like any statistical procedure, the interpretation of ANCOVA depends on certain assumptions about the data entered into the model. For instance, the F-testF-test
An F-test is any statistical test in which the test statistic has an F-distribution under the null hypothesis.It is most often used when comparing statistical models that have been fit to a data set, in order to identify the model that best fits the population from which the data were sampled. ...
assumes that the errors are normally distributed and homoscedastic
Homoscedasticity
In statistics, a sequence or a vector of random variables is homoscedastic if all random variables in the sequence or vector have the same finite variance. This is also known as homogeneity of variance. The complementary notion is called heteroscedasticity...
.
Since ANCOVA is a method based on linear regression, the relationship of the dependent variable to the independent variable
Independent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
(s) must be linear in the parameters.
Simplifying assumption (not necessary to run ANCOVA): homogeneity of regression which says that the relationship between the covariate and the dependent variable should be similar across all groups of the independent variable.
Power considerations
While the inclusion of a covariate into an ANOVA generally increases statistical powerStatistical power
The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...
by accounting for some of the variance in the dependent variable and thus increasing the ratio of variance explained by the independent variables, adding a covariate into ANOVA also reduces the degrees of freedom
Degrees of freedom (statistics)
In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
. Accordingly, adding a covariate which accounts for very little variance in the dependent variable might actually reduce power.
External links
- Examples of all ANOVA and ANCOVA models with up to three treatment factors, including randomized block, split plot, repeated measures, and Latin squares
- One-Way Analysis of Covariance for Independent Samples
- Use of covariates in randomized controlled trials by G.J.P. Van Breukelen and K.R.A. Van Dijk (2007)