# Assumptions of the Factorial ANOVA

The **factorial ANOVA **has a several assumptions that need to be fulfilled – (1) interval data of the dependent variable, (2) normality, (3) homoscedasticity, and (4) no multicollinearity. Furthermore similar to all tests that are based on variation (e.g. t-test, regression analysis, and correlation analyses) the quality of results is stronger when the sample contains a lot of variation – i.e., the variation is unrestricted and not truncated.

Firstly, the factorial ANOVA requires the dependent variable in the analysis to be of metric measurement level (that is ratio or interval data) the independent variables can be nominal or better. If the independent variables are not nominal or ordinal they need to be grouped first before the factorial ANOVA can be done.

Secondly, the factorial analysis of variance assumes that the dependent variable approximates a multivariate normal distribution. The assumption needs can be verified by checking graphically (either a histogram with normal distribution curve, or with a Q-Q-Plot) or tested with a goodness of fit test against normal distribution (Chi-Square or Kolmogorov-Smirnov test, the later being preferable for interval or ratio scaled data).

Some statisticians argue that the limit theorem implies that large random samples automatically approximate normal distribution. Small, non-normal samples can be increased in size by boot-strapping.

However if the observations are not completely random, e.g., when a specific subset of the general population has been chosen for the analysis, increasing the sample size might not fix the violation of multivariate normality. In these cases it is best to apply a non-linear transformation, e.g., log transformation, to the data. The transformation would be correctly described as transforming the scores into an index. For example, we would transform our murder rate per 100,000 inhabitants into a murder index, because the log-transformation of the murder rate would not easily make sense numerically.

Thirdly, the factorial ANOVA assumes homoscedasticity of error variances, which means that the error variances of all data points of the dependent variable are equal or homogenous throughout the sample. In simpler terms this means that the variability in the measurement error should be constant along the scale and not increase or decrease with larger values. The Levene's Test addresses this assumption.

The factorial ANOVA requires the observations to be mutually independent from each other (e.g., no repeated measurements) and that the independent variables are independent from each other. Since the factorial ANOVA includes two or more independent variables it is important that the factorial ANOVA model contains little or no Multicollinearity. Multicollinearity occurs when the independent variables are intercorrelated and not independent from each other.

In other terms the factorial ANOVA should not have any between-factor effects. If multicollinearity occurs the problem can be corrected by conducting a factor analysis. The factor analysis will extract factors that group the variables. After extraction the factor solution should be rotated orthogonally, e.g., with the varimax method. An orthogonal rotation ensures that the resulting factors are independent (orthogonal = 90° angle between the vectors of the factors and correlation between factors is defined as their cosines and cos(90°) = 0).

Generally as with all analyses minimal measurement error is needed because low reliability in data results in low reliability of analyses.

And like most statistical analysis, the higher the variation within the sample the better the results of the factorial ANOVA. Restricted or truncated variance, e.g., because of biased sampling, results in lower F-values, which increases the p-values.

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