Summary:
Provides an overview of the topics covered in the module and their significance in statistical analysis.
Learning Objectives:
Understand the scope and importance of the module on ANOVA, MANOVA, ANCOVA, T-test, Correlation, and Regression Analysis.
Gain an appreciation for the role of these statistical techniques in research and data analysis.
Identify the specific statistical methods covered and their applications in various research contexts.
Although Generalized Linear Models (GLM) allows for easy execution of regression models, in practice, univariate GLM is primarily used for analysis of variance (ANOVA) and analysis of covariance (ANCOVA) models (Rutherford, 2011:1-2). On the other hand, multivariate GLM is primarily utilized for multiple analysis of variance (MANOVA) and multiple analysis of covariance (MANCOVA) models (Huberty & Petoskey, 2000). In SPSS, multivariate GLM is a separate module, while in SAS, it is implemented within PROC GLM using the MANOVA statement.
ANOVA is a statistical technique used to investigate the main and interaction effects of categorical independent variables (referred to as "factors") on a continuous dependent variable (West etal., 1996). It examines whether the means of the groups formed by different values of the independent variable(s) differ significantly. ANOVA allows for the identification of both main effects (the direct effect of an independent variable on the dependent variable) and interaction effects (the combined effect of two or more independent variables). In contrast to regression models, which require the explicit addition of interaction terms, ANOVA inherently detects interaction effects (Jaccard, 1998). In the case of multiple dependent variables, multivariate GLM implements MANOVA, which can also incorporate control variables as covariates (MANCOVA).
The key statistic in ANOVA is the F-test, which evaluates whether the differences in group means are significant enough to suggest that they did not occur by chance (Tian etal., 2018: 61) If the group means do not differ significantly, it implies that the independent variable(s) did not have a significant effect on the dependent variable. However, if the F-test indicates a significant relationship between the independent variable(s) and the dependent variable, multiple comparison tests can be conducted to determine which specific values of the independent variable(s) contribute the most to this relationship.
It is important to note that ANOVA tests the null hypothesis that group means are equal, not that variances are equal. However, ANOVA assumes relative homogeneity of variances, meaning that the groups formed by the independent variable(s) have similar variances on the dependent variable. Homogeneity of variances can be assessed using tests such as Levene's test (Levene, 1960). Similar to regression, ANOVA is a parametric procedure that assumes the multivariate normality of the dependent variable for each value category of the independent variable(s) (Dattalo, 2013: 14).
ANCOVA, on the other hand, is used to test the main and interaction effects of categorical variables on a continuous dependent variable while controlling for the effects of selected continuous variables that covary with the dependent variable (Ankarali etal, 2018: 283). These covariates, also known as control variables, can be used to predict the dependent variable through regression analysis. ANCOVA then performs an ANOVA on the residuals (the predicted minus actual dependent variables) to determine if the factors are still significantly related to the dependent variable after accounting for the variation explained by the covariates. ANCOVA serves three purposes: 1) in quasi-experimental designs, it helps remove the effects of variables that modify the relationship between categorical independent variables and the interval dependent variable; 2) in experimental designs, it controls for factors that cannot be randomized but can be measured on an interval scale; and 3) in regression models, it accommodates the presence of both categorical and interval independent variables.
All three purposes of ANCOVA aim to reduce the error term in the model. ANCOVA can be seen as a type of "what if" analysis, examining what would happen if all cases had equal scores on the covariates, allowing for the isolation of the effects of factors beyond the influence of the covariates. The use of ANCOVA is applicable in various ANOVA designs, and the same assumptions regarding homogeneity of variances and multivariate normality still apply.
It is important to distinguish GLM from other types of models, such as generalized linear models (GZLM) that incorporate nonlinear link functions, linear mixed models (LMM) that handle multilevel data, and generalized linear mixed models (GLMM) that combine nonlinear link functions with LMM. SPSS also offers analysis of variance components (VC), which is a subset of LMM and serves similar functions as ANOVA under GLM. A comparison between GLM, LMM, and VC, along with data illustrations, can be found in the section on linear mixed models. While both GLM and LMM allow for the inclusion of random effects in models, LMM is generally preferred when random effects are present, as explained in the comparison.
