The Analysis of Covariance In an analysis of covariance, quantitative variables (interval or ordinal level) are included in the model in addition to the factors. In our problems, we will use a single covariate. The rationale for adding a covariate is to increase the opportunity to find statistical significance for the factors. Recalling that each independent variable is only credited with the variance in the dependent variable that it uniquely explains, the inclusion of a covariate independent variable reduces the variance that is to be explained by the factors. With less variance to explain, the chances that the factor will explain a significant portion of the variance increases. There are several reasons why a covariate (also called a concomitant variable or a control variable) is included in the analysis. In experimental designs, the rationale is usually an effort to account for the effect of a variable that does affect the dependent variable, but could not be accounted for in the experimental design. In observational studies, the covariate is used to control for variables that rival the independent variable of interest. In order for a covariate to be useful, it should have a reasonable correlation with the dependent variable. A reasonable correlation can be translated into a linear relationship, so analysis of covariance has an additional assumption of a linear relationship between the dependent variable and the covariate. In addition, the covariate should not have a significant interaction with the factors. This implies that the slopes of the regression lines representing the relationship between the dependent variable and the covariate are similar for all of the cells represented by all combinations of factors. Thus, an analysis of covariance must also satisfy the assumption of homogeneous regression slopes. If the assumption of homogeneous regression slopes cannot be satisfied, the model including the covariate should not be interpreted, because the relationships between the factors and the dependent variable change with different scores of the covariate. The covariate does not have the same meaning or play the same role for all cells in the analysis. Once the assumption of homogeneous regression slopes has been met, the covariate interaction term is removed from the model, and the model becomes a full factorial model plus the covariate. The covariate relationship may or may not be interpreted, since it may not be a predictor of interest. If the covariate is not interpreted, the findings for the analysis of covariance parallel those of factorial analysis of variance. It is required to meet the assumptions of normality and homogeneity of variance. The main effects for the factors are not interpreted if a significant interaction is found for the factors. Level of Measurement Requirement In an analysis of covariance, the level of measurement for the independent variables can be any level that defines groups (dichotomous, nominal, ordinal, or grouped interval) and the dependent variable and covariate are required to be interval level. If the dependent variable or covariate is ordinal level, we will follow the common convention of treating ordinal variables as interval level, but we should note the use of an ordinal variable in the discussion of our findings.If the level of measurement requirement (along with the sample size requirement) is satisfied, the check box “The level of measurement requirement and the sample size requirement are satisfied” should be marked. If the level of measurement requirement is not satisfied, the correct answer to the problem is “Inappropriate application of the statistic.” All other answers should be unmarked when the answer about level of measurement and sample size is “Inappropriate application of the statistic.” Sample Size Requirement I have imposed a minimum sample size requirement of 5 cases per cell for these problems. The cells are the possible combinations of categories for the two factors. If the factor one contained 2 categories and the factor two contained three categories, the total number of cells would be 6, as shown in the following table: Factor two Factor one Category A Category B Category C Category 1 Cell 1 Cell 2 Cell 3 Category 2 Cell 4 Cell 5 Cell 6 If the sample size requirement (along with the level of measurement requirement) is satisfied, the check box “The level of measurement requirement and the sample size requirement are satisfied” should be marked. If the sample size requirement is not satisfied, the correct answer to the problem is “Inappropriate application of the statistic.” All other answers should be unmarked when the answer is “Inappropriate application of the statistic” due to level of measurement or sample size. The Assumption of Normality Analysis of covariance assumes that the dependent variable and the covariate are both normally distributed, but there is general consensus that violations of this assumption do not seriously affect the probabilities needed for statistical decision making. The problems evaluate normality based on the criteria that the skewness and kurtosis of the dependent variable fall within the range from -1.0 to +1.0. If the dependent variable or the covariate satisfies these criteria for skewness and kurtosis, the check box “The skewness and kurtosis of both the independent variable and the covariate satisfy the assumption of normality” should be marked. If the criteria for normality are not satisfied, the check box should remain unmarked and we should consider including a statement about the violation of this assumption in the discussion of our results. In these problems we will not test transformations or consider removing outliers to improve the normality of the variables. The Assumption of Linearity If the covariate is to improve the analysis, it should be selected because it has a relationship to the dependent variable. We will use the SPSS test of linearity for thisassumption. The test of linearity tests for the presence of a linear relationship and a non-linear relationship. Statistical significance for the linear relationship satisfies the assumption. If the relationship is not linear, we should consider including a statement about the violation of this assumption in the discussion of our results. It is very possible to find a statistically significant linear relationship between the covariate and the dependent