# When should we include the “Replicate” in an ANOVA table for Factorial Design?

When I have been learning how to do Factorial Designs ($$2^k$$ mainly) the ANOVA table would usually include the main effects, all the interaction effects if necessary, and the error. However, on one of my exams it not only included the factors and their interactions, and the error, but it also included "replication".

First off, I was never taught that we could have replication as one of our sources of variation. So I don't know how that could effect the Sum of Squares of Error. I'm just guessing here, but I guess the degrees of freedom should be $$n-1$$ if we have $$n$$ replicates, but I am only guessing that because everything else appears to be of the form $$\theta - 1$$ for some parameter $$\theta$$.

Up to this point I have never included replication in my ANOVA table. Should replication be included and if it is included how does it affect the Sums of Squares?

The exact example we had was that we had a $$3$$ factor experiment with two replicates and it gave us part of the ANOVA table, and we had to fill in the rest. If it didn't include Replication, I would know exactly how to fill in the rest of the table, but I don't know exactly why Replication would need to be checked whether or not it is significant or not.

$$\begin{array}{c|c|c|c|c|c|} \text{Source of Variation} & \text{df} & \text{SS} & \text{MS} & \text{F-Value} & \text{p-value} \\ \hline \text{Replication} & & 300 \\ \hline \text{A} & & & 120\\ \hline \text{B} & & 320 \\ \hline \text{C} & & & 60 \\ \hline \text{AB} & & &55 \\ \hline \text{AC} & &160 \\ \hline \text{BC} & &100 \\ \hline \text{ABC} & & \\ \hline \text{Error} & & \\ \hline \text{Total} & & 2000\\ \hline \end{array}$$

We are given that each factor is three levels, two replicates, and that the treatment sum of squares is $$1200$$

If this table doesn't include the row "Replicates" I know how to fill in all the missing entries. But since it is there, I don't really know how to account for that correctly. Nor do I understand why we would ever want to have that included in our ANOVA

Any help on my confusion would be greatly appreciated!

• If you can present your table, even empty, it will be helpful to understand your question. – user158565 Nov 9 '18 at 3:02
• @a_statistician I've edited the question by adding the given table. If the "Replications" row isn't included I know how to solve this problem exactly, but with it , I'm not sure how to account for that exactly. More importantly, I don't understand why it is there in the first place; it seems unnecessary to me because I don't see why we would want to see if replications is significant or not. – WaveX Nov 9 '18 at 3:13
• Without your table, I though the replication is ERROR or ABC. With your table, I do not think that Replication should be there. Without replication, you already separate the total SS into 8 parts. It is the finest separation and you cannot separate SS further, because you have 8 cells in total (assume $2^3$ design). – user158565 Nov 9 '18 at 3:29
• This is what I thought as well, which is why when I received the question it puzzled me. For this particular question do you think it's supposed to be the SSE broken down into two parts: true error and error due to replication? And then in practice, you say that it probably shouldn't be included at all, correct? – WaveX Nov 9 '18 at 3:34
• I have no way to break down SSE. I think Replication should not be in the table at all. – user158565 Nov 9 '18 at 3:40