05 indicates that it was significant at p= ‘ < ‘. The 3.56 is the actual F-ratio, and the p ‘ < ‘. Explanation of the above example : As with the ANOVA, the F refers to our F-ratio procedure, the 2 in brackets refers to the degrees of freedom (df) for the effect of the model (between groups) and the 20 as the degrees of freedom for the residuals in the model (within groups).Reporting example : The effect of the covariance can be reported as follows when the effect is significant: “When improving our model with the inclusion of a covariate, their remains to be a significant effect of the independent variable on the dependent variable after controlling for the effect of the covariate: i.e.However, the more effective the covariates are, the less conservative our rule needs to be! A rule of thumb is that the number of covariates should be less than (.10 x sample size) – (number of groups – 1). Limited number of covariates as too many covariates will reduce the statistical efficiency (statistical power) of the ANCOVA procedures because a degree of freedom (df) is lost for every additional covariate added in the formula.The instrument that measures the covariate should be reliable with low measurement error.Independence of the error terms / residual terms should be uncorrelated (or independent).No high multicollinearity of the covariates.ANCOVA is highly sensitive to outliers in the covariates. Important here is that the covariate is linearly related to the dependent variables and is not related to the independent variables.This means the covariates should have equal mean scores across the factor groups (so no shared variance or interaction between them). Independent variables should be orthogonal (uncorrelated) to covariates.Homogeniety of regression slopes: The dependent variable and the covariates should have same slopes ( b-coefficient ) across all factor groups which means their linear relationships should be the same for the dependent variable and the covariates in each group.MANCOVA (a multivariate ANCOVA, which means more than one dependent variable).The covariates are not our focus, but systematically affects the dependent variable and bias the results. This (hopefully) will help us to more accurately assess the effects of the independent variables ( SS M ) on the dependent variable.

ANCOVA is a parametric procedure.Īnother reason for adding a covariate to an ANOVA is to reduce the within-group error variance, so we attempt to explain a higher portion of the unexplained variance (the sum of squares of the residuals – SS R ) in terms of additional variables (covariates).

In this case we adjusted (or controlled) for the disparities in income among our sample. As example, while we may find a significant effect of age and gender on customer satisfaction scores (in a factorial ANOVA), once we add income as a covariate, we may find that the significant relationship between aforementioned variable disappears – evidence of a spurious relationships.

The covariate is also referred to as a “confounding variable” as it has a hidden effect on the tested relationship. Testing different covariates allows us to answer the “what if…” question and to identify spurious relationships. The addition of a covariate is often conducted to determines of there is an exogenous variable (the covariate) that distorts the relationship between the interval dependent variable and the categorical independent variable (referred to as a factor).

Calculate degrees of freedom ancova