# Analysis of Covariance (ANCOVA)

**BRIEF DESCRIPTION**

The Analysis of Covariance (ANCOVA) follows the same procedures as the ANOVA except for the addition of an

*exogenous variable*(referred to as a covariate) as an independent variable. The ANCOVA procedure is quite straightforward: It uses regression to determine if the covariate can predict the dependent variable and then does a test of differences (ANOVA) of the residuals among the groups. If there remains a significant difference among the groups, it signifies a significant difference between the dependent variable and the predictors after the effect of the covariate has been removed.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

Another 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

*factor*). Testing different covariates allows us to answer the “what if…” question and to identify spurious relationships. The covariate is also referred to as a “confounding variable” as it has a hidden effect on the tested relationship. 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. In this case we adjusted (or controlled) for the disparities in income among our sample. ANCOVA is a parametric procedure.Another 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). This (hopefully) will help us to more accurately assess the effects of the independent variables (SS_{M}) on the dependent variable. The covariates are not our focus, but systematically affects the dependent variable and bias the results.

**SIMILAR STATISTICAL PROCEDURES:**

- ANOVA (without the covariate)
- MANCOVA (a
*multivariate*ANCOVA, which means more than one dependent variable)

**CHARACTERISTICS OF THE VARIABLE(S)**

**Dependent variable**: continuous scaled data**Independent variable**: categorical data with more than two groups (referred to as factors)**Covariate**: continuous scaled data. Note that categorical covariates can be used and are referred to as “blocking variables”. These blocking variables are added to the “fixed effects” input box along with the other categorical independent variables. Note that if a categorical variable is dummy coded, it can be considered as a covariate.

**DATA ASSUMPTIONS**

The same assumptions as those for the ANOVA applies, plus the following:

- 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. - Independent variables should be orthogonal (uncorrelated) to covariates. This means the covariates should have equal mean scores across the factor groups (so no shared variance or interaction between them).
- Important here is that the covariate is linearly related to the dependent variables and is not related to the independent variables.
- No covariate outliers. ANCOVA is highly sensitive to outliers in the covariates.
- No high multicollinearity of the covariates.
- Independence of the error terms / residual terms should be uncorrelated (or independent).
- The instrument that measures the covariate should be reliable with low measurement error.
- 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. A rule of thumb is that the number of covariates should be less than (.10 x sample size) – (number of groups – 1). However, the more effective the covariates are, the less conservative our rule needs to be!

**WHERE TO FIND IN SPSS?**

ANALYZE / GEN LINEAR MODEL / UNIVARIATE

**HOW TO REPORT THE FINDINGS?**

Reporting the results of the ANCOVA is the same as for the ANOVA except that we add the effect of the covariance as well.

**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. F(2, 20) = 3.56, p‘<‘.05″.**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). The 3.56 is the actual F-ratio, and the p‘<‘.05 indicates that it was significant at p=‘<‘.05. The actual p-value can be reported in lieu of the ‘<‘.05.

Note: As with the ANOVA, if you find a significant difference among the group, conduct the appropriate *post hoc* comparisons and report as: “Post hoc comparisons using the Tukey HSD test indicated that the mean score for the “teens group” (M=3.4, SD=1.2) was significantly different than the “mature group” (M=5.2, SD=0.9). However, the “young adults” group (M= 4.5, SD=2.2) did not significantly differ from either the “teens group” or the “mature group”. You may want to also report the confidence levels and intervals e.g. (M = 4.5, 95% CI [3.90, 5.20]), p =.007.

**FURTHER COMMENTS.**

Be careful with any significance tests if you have a large sample, at which time it becomes critical to calculate the effect size (Cohen’s d).

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