Factorial ANOVA** **a statistical method for comparing groups with multiple factors, relies on several key assumptions to ensure accurate results. Understanding and meeting these assumptions is crucial for the validity of the analysis. Here’s a simplified and expanded explanation of these assumptions: (1) interval data of the dependent variable, (2) normality, (3) homoscedasticity, and (4) no multicollinearity.

**Metric Measurement Level:**The outcome variable (dependent variable) analyzed in factorial ANOVA needs to be measured on a metric scale, meaning it should be ratio or interval data. This type of data allows for meaningful calculations of differences and averages. The factors (independent variables) influencing this outcome can be categorical, either nominal or ordinal. If the independent variables are not already in these categories, they must be grouped accordingly before analysis.

**Normal Distribution of Data:**The data for the dependent variable should follow a normal distribution across groups. This assumption can be checked in a couple of ways:**Graphical Methods:**Using a histogram overlaid with a normal distribution curve or a Q-Q plot can visually indicate if the data approximates normality.**Statistical Tests:**Goodness-of-fit tests, such as the Chi-Square or Kolmogorov-Smirnov tests (the latter is preferred for metric data), can statistically assess normality.

It’s often argued that for large samples, the central limit theorem ensures that data will approximate a normal distribution, making this less of a concern for big datasets. For smaller or non-normal samples, techniques like bootstrapping (creating many simulated samples) can help meet this assumption.

**Equal Variances:**The variability in scores (variance) should be similar across all groups being compared. This ensures that the analysis is not biased by one group having much more variability in its data than others.

**Independence Among Factors:**The independent variables should not be too highly correlated with each other. High correlation (multicollinearity) can make it difficult to distinguish the unique impact of each factor on the outcome variable.

**Unrestricted Variation:**Like other statistical tests that rely on variance (such as t-tests, regression, and correlation analyses), factorial ANOVA produces more reliable results when there’s a wide range of data points. Limited or truncated variation can weaken the analysis. Essentially, having diverse and varied data points enriches the analysis, providing a more robust understanding of the factors at play.

In summary, ensuring that these assumptions are met before conducting a factorial ANOVA is crucial for the accuracy and reliability of its results. These prerequisites help in correctly interpreting the effects of multiple factors on an outcome variable, making factorial ANOVA a powerful tool for understanding complex relationships in data.

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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.

Check out our online course for conducting an ANOVA here.

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