# Nested ANOVA evaluation in R

#### rogojel

##### TS Contributor
hi,
I have an experiment to check factors that influence the results of a
measurement which is done like this:

Several pieces of substrate are selected (a kind of paper), on each
substrate a left and a right side area is defined and in each area five
measurement points going from top to bottom. Then the slips are painted with a machine and various quality measures of the painting captured in each point.

So, each substrate receives the same treatment (except for random variation
in the painting process of course) and the results are measured at each
measurement point.

The objective of the exeriment is to investigate whether there are
systematic diferences in the results, due to differences in the
substrate, the side which is measured (i.e. left or right) or the
position of the measurement points (from top to bottom).

So, my questions are:

1. This design needs to be analysed with a nested ANOVA, right?
2. If using R the error term shuld be Error(Substrate/Side/Points)?

and most importantly

3. I do not get a p-value for the highest level (Substrate). I checked
in the Crowley book and the example he uses for split-plot designs also
does not have a p-value for the highest level, however this is not
commented in any way in the book. If I do a nested ANOVA with Minitab
(same data) I get a p-value for the Substrate variable, so my question
is: why do I not get a p value with the evaluation of the nested design
in R ?

#### Jake

1. This design needs to be analysed with a nested ANOVA, right?
That is not obvious to me. Could you not view this as a fully crossed experiment? For example, if you are interested in investigating statements like "across all the substrate, measurements on the left side tended to be higher than measurements on the right side," then in that case we would treat Side as a fixed factor crossed with the random Substrates. If Side is nested in Substrate (and hence is random since it is nested in a random factor) then the test of Side only tests whether there is significant variation due to Side in any direction (i.e., rather than in a consistent direction). But maybe it is this latter question that you are interested in? (Similar questions arise for the Points factor.)

2. If using R the error term shuld be Error(Substrate/Side/Points)?
If the nested model is really what you want, then yes, this looks right.

I checked
in the Crowley book and the example he uses for split-plot designs...
But if this is really a hierarchical/nested design as you say then it's not a split-plot, right...? Maybe I'm missing something.

why do I not get a p value with the evaluation of the nested design
in R ?
Can you provide a reproducible example?

#### rogojel

##### TS Contributor
Hi Jake,
I guess I am quite confused about some of the finer points, so I would go into the details for each of your questions:

That is not obvious to me. Could you not view this as a fully crossed experiment? For example, if you are interested in investigating statements like "across all the substrate, measurements on the left side tended to be higher than measurements on the right side," then in that case we would treat Side as a fixed factor crossed with the random Substrates. If Side is nested in Substrate (and hence is random since it is nested in a random factor) then the test of Side only tests whether there is significant variation due to Side in any direction (i.e., rather than in a consistent direction). But maybe it is this latter question that you are interested in? (Similar questions arise for the Points factor.)
Welll, this is where I have my doubts. The data can be analysed as if it was a normal balanced ANOVA but in fact the Substrates do contain the Sides and the Sides the Points, so would the balanced approach not be more appropriate? The p-values are quite different though the interpretation does not change.

But if this is really a hierarchical/nested design as you say then it's not a split-plot, right...? Maybe I'm missing something.
Nope, it is myself who is confused or it possibly comes from Crowley's R bok he discusses nested designs by using the example of a split-plot design. As far as I can think of it the difference would be that the treatments are applied randomly to each of the sub-plots in a split-plot design right? In my case the treatments are definitely not random, but the error structure should still be applicable if I understood it correctly. Is this right?

#### rogojel

##### TS Contributor
I just found a nice explanation here: http://udel.edu/~mcdonald/statnested.html

" Think of it this way: if you dissected out the muscles, labeled the tubes "A" and "B," then forgot which was right and which was left, it wouldn't matter if you were doing a nested anova; it would be a disaster if you were doing a two-way anova. "
So, by this logic, as I am not interested in determining WHAT is the difference between the left and the right side, just in the fact whether there is a difference or not it seems that the nested approach would be better.

What do you think?