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I have to analyse the results of an experiment, say, a dependent variable Y, in relation to the different treatments applied. However, I am not sure about the model to use with this experimental design.

There are 40 plots of land, organized as 4 rows * 8 columns

01 02 03 04 05 06 07 08 09 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

It seems similar (but different) to split plot experimental designs. Actually, The land is divided into two whole plots => 2 different levels (of the whole plot factor ?) are applied (e.g. levels A1 and A2 from factor A).

Edit (2014/10/15): levels of factor A are controlled levels of fertility. I don't know if this information changes anything, as I am not interested in assessing the effect of A over Y. I though maybe A can be considered a blocking factor (some kind of controlled fertility gradient) ?

_____A1_____ | _____A2_____
01 02 03 04 05 | 06 07 08 09 10
11 12 13 14 15 | 16 17 18 19 20
21 22 23 24 25 | 26 27 28 29 30
31 32 33 34 35 | 36 37 38 39 40

Whole plot factor is crossed with a blocking factor B (with 4 levels B1 .. B4)

_____A1_____ | ______A2____
01 02 03 04 05 | 06 07 08 09 10 |B1
--------------------- ----------------------
11 12 13 14 15 | 16 17 18 19 20 |B2
--------------------- ----------------------
21 22 23 24 25 | 26 27 28 29 30 |B3
--------------------- ----------------------
31 32 33 34 35 | 36 37 38 39 40 |B4

The factor of interest, to finish, is factor C with 5 independent levels C1 .. C5. Each of the 5 plots in each combination of A and B is randomly assigned to one of the 5 levels of factor C.

______A1_____ | ______A2_____
C1 C2 C3 C4 C5 | C4 C1 C3 C2 C5 |B1
----------------------- ------------------------
C5 C1 C4 C3 C2 | C5 C2 C1 C3 C4 |B2
----------------------- ------------------------
C4 C3 C1 C2 C5 | C1 C3 C2 C4 C5 |B3
----------------------- ------------------------
C2 C3 C5 C4 C1 | C3 C4 C5 C1 C2 |B4

THIS is the real experimental design that I have problems understanding. As I said earlier, a measure yi of a dependent variable Y is associated with each i=1 .. 40 plot.

My question is : How to compare Y among levels of factor C ?

Thanks for your kind help.

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Do you use R? If so, I suggest you use lmer in the lme4 package, assuming that Y is a continuous outcome variable. In your experimental design, A, B, and C are all crossed: every level of one factor happens in every combination of the other two factors. For example, all levels of C, C1 ~ C5, happen all combinations of A and B, A1B1, A1B2, ..., A2B4. The lmer function is design to handle data obtained from such crossed designs.

I assume that C a fixed effect and A and B are both random effects. In this case we are considering B as a random sample from a population. Then the code for analyzing your data would be

library(lme4)
library(lmerTest)
summary(lmer(Y ~ C + (1|A) + (1|B), data=your_df, RELM=FALSE)).

If you treat B as a fixed blocking factor, then the code becomes

summary(lmer(Y ~ B + C + (1|A), data=your_df, RELM=FALSE)).

I believe treating B as a random effect is preferable, especially if you want to generalize the results beyond those 4 blocks.

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  • $\begingroup$ How to take into account the fact that each level of factor A were not randomly assigned to the left or right part of each block ? This is the thing that makes this experimental design slightly different than a split-plot experimental design, right ? $\endgroup$
    – Rodolphe
    Commented Oct 16, 2014 at 12:34
  • $\begingroup$ Including a variable as a random factor does not necessary imply that the sample is taken randomly (even though that is the definition). For example, mice used in an experiment are not necessarily randomly selected from a larger population. So probably you can still consider A as a random factor, especially because you are not interested in its effect per se. Of course you can include it as a fixed effect but then It would be hard to justify generalizing the results beyond the sample you have. $\endgroup$
    – Masato Nakazawa
    Commented Oct 16, 2014 at 12:47
  • $\begingroup$ If I consider factor A as random, what do you think of the following model : Y ~ C + (1|A:B) ? Is it different than the first one you proposed ( Y ~ C + (1|A) + (1|B) ) ? I mean, would the conclusions I could draw from each model be different ? $\endgroup$
    – Rodolphe
    Commented Oct 16, 2014 at 13:32
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    $\begingroup$ (Comment 2) (1|A:B) implies that the models fit 8 intercepts for each combination of A and B and computes 1 $\sigma$, while (1|A) + (1|B) implies that the model fits 2 intercepts for each level of A and 4 intercepts for each level of B, giving 2 $\sigma$'s. I do not know how or whether these models change your conclusions. On one hand, (1|A:B) seems to ignore the crossed nature of your design. On the other hand, if A and B interact, (1|A:B) may be able to reduce residual variance, hence smaller standard errors for the test of the C fixed effect. $\endgroup$
    – Masato Nakazawa
    Commented Oct 16, 2014 at 14:47

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