Anova: Components of variation in population (notation in Doncaster and Davey)


I am reading Doncaster and Davey's 2007 book. The book lists the appropriate ANOVA/ANCOVA models for combinations of nested and crossed factors. While I think the book to be a great practical asset, I am confused by the authors' notation.

For each model they provide an anova table listing the sources of variation and the construction of the F ratio for each effect/factor. The tables also have column titled 'Components of variation estimated in the population' (See attached image for a simple example).

My problem is that I do not understand how to read this column. For example, I have attached their table for the simple one-way ANOVA case. In this table, the components of variation estimated in the population read

  • S'(A), for the residual variance
  • S'(A) + A, for the explained variance.

In this, S'(A) denotes that the subjects are a random factor nested in the factor A.

This notation seems to indicate that the variance at the level of the factor is the sum of (1) the residual variance and (2) the "variance associated with the factor" (which I would think to be the between group variance?). However, that would be the total variance - and that's obviously wrong.

Hence, I am not sure how to interpret this notation of the 'Components of variation estimated in the population'. Why do Doncaster and Davey note the between group variation as S'(A) + A in this model?

My problem is not that I do not know how the calculate the sum of squares and/or mean sum of squares for this anova table. I do know how to calculate the sum of squares. It's just the notation that is bothering me. As the examples in the book become more complex it would help me to know how to read this column.

Any help would be greatly appreciated. Also, it would be great to hear from other people who have read the book and their thoughts on it.