# How should I analyse this experiment with two blocking factors

In this experiment, a machine is used to compare the wear resistance of 4 different types of rubber-coated fabric (variable mat coded A, B, C and D). To do this, the machine is run 4 times (variable run, values 1,2,3,4 - these are labels not numbers). The machine has 4 different (independent) surfaces that mechanically rub the fabric, slowly removing the rubber coat (variable sur values 1,2,3, and 4 - these are labels, not values). The response variable, wear is the weight loss of the fabric samples after a fixed time on the machine. All runs of the machine were of the same length. This is the data (all of it).

As mentioned at the top, the research question is to compare the wear resistance of the 4 materials.

  wear run sur mat
1  218   1   1   B
2  227   2   1   D
3  274   3   1   A
4  230   4   1   C
5  236   1   2   D
6  241   2   2   B
7  273   3   2   C
8  270   4   2   A
9  268   1   3   A
10 229   2   3   C
11 226   3   3   B
12 225   4   3   D
13 235   1   4   C
14 251   2   4   A
15 234   3   4   D
16 195   4   4   B



Is this another type of Randomized Complete Block Design. Should I analyze it the same way as suggested in the answer to my previous question ? That is, just a straight forward linear regression:

PROC GLM data=mydata;
CLASS MAT RUN SUR;
MODEL WEAR = MAT RUN SUR;
RUN;

• can you explain how the sur and mat factors were randomised, and what hypotheses you are interested in testing? Nov 16, 2023 at 12:01
• This looks like 16 independent samples? There is only one machine, run 16 times? Please answer the other comment about how randomized? If the goal is to compare the materials, and the different machine surfaces is seen as a noise factor, then: To me it seems like this asks for a regression analysis maybe a mixed model with wear ~ 1 + mat + (1 | run) + (1 | sur) (using R notation) You also need to clarify your research question! Nov 16, 2023 at 12:08

Is this another type of Randomized Complete Block Design?

It is very closely related but this seems to be a Latin Square design, not a RCB design, though we could say that it's an extension of the RCB design. If we look at a table of the two blocking factors and see which treatment (materials) are applied, we obtain this:

/* Remove response column, as we don't want the response variable in the table */
PROC SQL;
CREATE TABLE dt_selected AS
SELECT * EXCEPT WEAR
FROM dt;
QUIT;

PROC TRANSPOSE DATA=dt_selected OUT=dt_wide;
BY run; /* ID variable */
ID sur; /* Time variable */
VAR mat; /* Variables to transpose */
PREFIX=mat_; /* Prefix for new columns
RUN;


which results in:

  run mat_1 mat_2 mat_3 mat_4
1     B     D     A     C
2     D     B     C     A
3     A     C     B     D
4     C     A     D     B


This shows each run as a row, and each column contains the treatments (mat) for that run, and is labelled mat_X where X is the surface used for that run. Notice that the columns appear to be randomised, as per the RCB design, that is, each in column (blocking factor run), each treatment (mat = A, B, C or D) appears exactly once. But in addition, in each row (blocking factor sur, the treatments also appear exactly once. That is one of the defining characteristics of a Latin Square (another characteristic is that both blocking factors should have the same number of levels (in your case 4 each). This is in contrast to the RCB design where the rows do not necessarily have every treatment exactly once (1 or more treatments can appear more than once, and 1 or more treatments may be absent, from the rows). A Latin Square is a design in which two gradients are controlled with crossed blocks, but in each intersection there is only one treatment level.

Model: We can write the model as follows, where we refer to the above Latin Square layout with the runs (in rows) indexed by $$j$$, and the surfaces (in columns) indexed by $$k$$:

$$y_{ijk} = \mu + t_i + b_j + c_k + \epsilon_{ijk}$$

where:
$$y_{ijk}$$ is the response from the experimental unit in row $$i$$ (the $$i$$th, column $$j$$ of the Latin Square above, that received treatment (material) $$k$$
$$\mu$$ is the overall mean response, taken over all treatments and all blocks,
$$t_i$$ is the difference between $$\mu$$ and the mean response for treatment (material) $$i$$ ie, the treatment effect (which we are interested in),
$$b_j$$ is the difference between $$\mu$$ and the mean response for row $$j$$, ie the blocking effect of the machine runs (which we are not interested in), and
$$c_k$$ is the difference between $$\mu$$ and the mean response for column $$k$$, ie the blocking effect of the machine surfaces (which we are not interested in), and
$$\epsilon_{ijk}$$ is the residual error for the $$i$$th material in the $$j$$th run block using the $$k$$th surface block.

To fit such a model, in SAS we can use:

PROC GLM data=dt;
CLASS run sur mat;
MODEL wear = run sur mat;
RUN;


where we instruct SAS to compute all the pairwise comparisons for the treatment effect, as per the research question with the LSMEANS statement.

Note that, as with the RCB design, if the Latin Square design had more levels of the blocking factors, then we could use a mixed effects model with crossed random effects, and such a model would be fitted with:

PROC MIXED data=dt;
CLASS run sur mat;
MODEL wear = mat;
RANDOM run sur / TYPE=VC;

where we use the TYPE=VC option (Variance Components) to specify that these random effects are additive and independent. Note that PROC MIXED also supports TYPE=CS (Compound Symmetry), UN (Unstructured) and AR(1) (Autoregressive of order 1) among others.