How Latin squares remove variation in experiment design? (2-way blocking)

Question

I read that latin squares are used to design of experiments when there are 2 sources of nuisance (factors). In that setting, every treatment appears exactly once in each row (factor 1) and column (factor 2). While I think I understand why it is important, I would like to get an explanation with a specific example. My understanding is that it allows to clearly identify a source of any influence, e.g. if one of the factors influences the treatments, it will be shown clearly irrespective of the other factor.

Answer 1

Here is an example from an older edition of Ott & Longnecker: Statistical Methods and Data Analysis. I suppose data are fake, but I believe the example suffices to answer your question. (Printouts are from Minitab statistical software, but are generic in format.)

Suppose an oil company tested four different blends of gasoline for fuel efficiency according to a Latin square design in order to control for the effects of four different drivers and four different models of car. The data are given in the printout below.

Obs MPG Driver Model Blend

 1   15.5       1      1      4
 2   33.9       1      2      2
 3   13.2       1      3      3
 4   29.1       1      4      1
 5   16.3       2      1      2
 6   26.6       2      2      3
 7   19.4       2      3      1
 8   22.8       2      4      4
 9   10.8       3      1      3
10   31.1       3      2      1
11   17.1       3      3      4
12   30.3       3      4      2
13   14.7       4      1      1
14   34.0       4      2      4
15   19.7       4      3      2
16   21.6       4      4      3

The ANOVA Table is as follows:

Analysis of Variance for MPG     

 Source     DF     Seq SS     Adj SS     Adj MS       F      P
 Driver      3      5.897      5.897      1.966    0.50  0.699
 Model       3    736.912    736.912    245.637   61.90  0.000
 Blend       3    108.982    108.982     36.327    9.15  0.012
 Error       6     23.809     23.809      3.968
 Total      15    875.599  

This model is othogonal so that SS's for Driver, Model, Blend, and Error are independent random variables. Thus 'Seq SS' and 'Adj SS' are the same. F-tests are exact.

From the point of view of the oil company, the Blend effect is the crucial one. Drivers, and Models of cars are introduced to explain variability If 'Model' had been omitted from the model, then (roughly speaking) SS Model would be added to SS Error, making SS Error much larger. This would decrease the F-statistic for Blend, and increase the P-value. (This is a now a block design in which Blend is the 'main' effect, and Drivers are blocks.)

If 'Driver' had also been omitted, and we had picked car models and drivers for each observation at random, we might have gotten an ANOVA table like the one below, in which it is not possible to see any difference among blends. (Of course, we will never know, because this hypothetical completely randomized design was not used to obtain our observations.)

SOURCE     DF        SS        MS        F        p
Blend       3     109.0      36.3     0.57    0.646
ERROR      12     766.6      63.9
TOTAL      15     875.6 

A design in which we control for some possible sources of variablility (such as car models and drivers) may decrease MS(Error) so that the power for detecting differences among the levels of the main effect, if there are any differences to be detected.

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Notes: (a) Block designs are widely used. It is often easy enough to identify one likely major source of variability and make it a block. If there are four levels of the main effect, then there may be almost any desired number of levels of the block effect.

(b) By contrast Latin Squares are used less often because of the considerable constraints of such a design. If there are $k$ levels of the main effect, then there must also be exactly $k$ levels of the two "blocking" effects. In practice, it may be difficult to schedule the taking of observations so that each one is taken with the appropriate levels of each factor.

(c) For certain values of $k$ it may even be possible to use a Greco-Latin square, in which one designates levels of second and third blocking effects by using Greek and Latin letters, respectively, in a $k \times k$ data table.

(d) Partially balanced incomplete block designs offer some flexibility in the numbers of blocking effects and their levels. Some books on the design of experiments include tables showing the structures and models of such designs. Generally, the rows of the ANOVA table for such models are not independent, so it is necessary to use adjusted values of SS's and MS's to get approximate F-tests. An advantage of Latin square designs is that they are orthogonal so that exact F-tests are possible.