Factorial ANOVA assumptions and test and ANOVA and regression

In this page, i'm going to talk about assumptions and tests in terms of factorial ANOVA and the link between ANOVA and multiple regression.

 

1. Assumptions : the assumptions are the same as in one way ANOVA.

 1-1) assumptions

• The observation should be independent
• The response variables should be normally distributed in each group
• The variances need to be homegeneous or the same. With unequal sample sizes, the largest variance is no more than twice the size of the smallest variance.

 1-2) hypotheses : we specify statistical hypotheses for each of the main and interaction effects. For each main effect, the null hypothesis states that the relevant marginal population means are equal. The alternative hypothesis states that at least one population mean differs from the rest. The interaction effect specifies that the difference between means on one factor are the same for each level of the other factor. The alternative hypothesis is that the differences are unequal. For each effect, the test testing F equals the between group variance divided by the within group variance. The test statistic follows an f distribution with 2 degrees of freedom.

 

2. ANOVA and regression : It's about the link between ANOVA and multiple regression. the between and within group variances are expressed in terms of sums of squares and mean squares, which we also use in multiple regression. In fact, multiple regression and ANOVA are technically the same. 

 2-1) the meaning of regression coefficients : the regression coefficient represents the difference in the population mean of the case included minus the population mean of the case not included.