Besides independency, normal distribution and homoscedasticity, why don't we also need the sample sizes among groups to be somewhat equal?
Imagine we have a factor with three conditions. Let's say sample sizes are n_1=20, n_2=100, n_3=100 and all mentioned requirements are fulfilled. Calculating the variance within groups and between groups combines the variance of all samples now. Accordingly, degrees of freedom are df_within = p-1 = 2 and df_between = n-p = 217. Don't we in average overestimate the degrees of freedom for n_1 and underestimate it for n_2 and n_3 then? Or is that the very reason why we can't actually say something about individual groups when calculating an ANOVA?
Thanks in advance,
Steffen
One assumption of ANOVA is that the variance in all the groups is equal. Under this assumption, we can use all the data to estimate the (one!) variance, independent of any group membership. We have all the data (N) and need only k (=the number of groups) group means to estimate the residuals (and, thus, the residual variance), what leaves N-k degrees of freedom (on which this one variance is estimated).
It's a different story when we don't want to assume equality of variances. There are some alternatives like Welsh ANOVA, weighted and robust regression, and bootstrap. However, the practical differences are often rather small (the key problem is that variances are really hard to estimate, and it is much much more difficult and error prone to estimate k variances from less data for each rather than one using all the data; the k group-wise estimated variances can easily be more wrong than using the common variance as an estimate for each group).
The equations for a one-way ANOVA accounts for the different sample sizes; I would be more concerned about the fact that the sample sizes are small relatively speaking - in general, a sample size of at least 30 is preferred.
The construction of the variances are illustrated in the attached layout for a one-way ANOVA.
One assumption of ANOVA is that the variance in all the groups is equal. Under this assumption, we can use all the data to estimate the (one!) variance, independent of any group membership. We have all the data (N) and need only k (=the number of groups) group means to estimate the residuals (and, thus, the residual variance), what leaves N-k degrees of freedom (on which this one variance is estimated).
It's a different story when we don't want to assume equality of variances. There are some alternatives like Welsh ANOVA, weighted and robust regression, and bootstrap. However, the practical differences are often rather small (the key problem is that variances are really hard to estimate, and it is much much more difficult and error prone to estimate k variances from less data for each rather than one using all the data; the k group-wise estimated variances can easily be more wrong than using the common variance as an estimate for each group).
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