1. Latin Square Designs (§15.4)
Lecture Objective
Introduce basic experimental designs that account for two orthogonal sources of extraneous variation.
Terminology
Square design
Orthogonal blocks
Randomizations

2. Examples
A researcher wishes to perform a yield experiment under field conditions, but she/he knows or suspects that there are two fertility trends running perpendicular to each other across the study plots.
An animal scientists wishes to study weight gain in piglets but knows that both litter membership and initial weights significantly affect the response.
In a greenhouse, researchers know that there is variation in response due to both light differences across the building and temperature differences along the building.
An agricultural engineer wishing to test the wear of different makes of tractor tire, knows that the trial and the location of the tire on the (four wheel drive, equal tire size) tractor will significantly affect wear.

3. Latin Square Design
A class of experimental designs that allow for two sources of blocking.
Can be constructed for any number of treatments, but there is a cost. If there are t treatments, then t2 experimental units will be required.
If one of the blocking factors is left out of the design, we are left with a design that could have been obtained as a randomized block design.
Analysis of a Latin square is very similar to that of a RBD, only one more source of variation in the model.
Two restrictions on randomization.

4. Cold Protection of Strawberries
Three different irrigation methods (treatment levels) are used on strawberries:
drip,
overhead sprinkler,
no irrigation.
We wish to determine which of these is most effective in protecting strawberries from extreme cold.
All strawberries grown through plastic mulch.
Measure weight of frozen fruit (lower values indicate more protection).

5. high
Nitrogen Level
low
Field Layout
none
drip
over
Moisture
none
over
none
drip
drip
over
Moisture and Soil Nitrogen are two sources of extraneous variation that we wish to simultaneously control for.
CANAL
Nitrogen Level
high
low
none
drip
over
Which design will best allow us to account for both soil moisture and nitrogen gradients?
Moisture
drip
over
none
over
none
drip
CANAL

6. Advantages and Disadvantages
Allows for control of two extraneous sources of variation.
Analysis is quite simple.
Disadvantages:
Requires t2 experimental units to study t treatments.
Best suited for t in range: 5  t  10.
The effect of each treatment on the response must be approximately the same across rows and columns.
Implementation problems.
Missing data causes major analysis problems.

7. Constructing a Latin Square Design for t Treatments
Treatments designated by first t capital letters in the alphabet (A,B,C, etc.)
Number the levels of blocking factor 1 (call it “Rows”) as R1, R2, … Rt.
Number the levels of blocking factor 2 (call it “Columns”) as C1, C2, … Ct.
Assign the treatment letters in alphabetic order, beginning with A, to the t units in the first row.
For the second row, start with the letter B and assign treatment letters to the t-th letter then follow with A.
For rows 3 through t, simply shift the treatment letters up one at a time, placing the shifted letter in the last unit of the row.

8. Basic Square

9. Randomization Get a random ordering of the rows. 1 2 3 4 replaced by
Reorder the rows according to randomization.

10. Randomization Get a random ordering of the columns. 1 2 3 4
replaced by
Reorder the columns according to randomization.
Two Blocking Factors = Two Randomizations
= Two Constraints on Randomization

11. Latin Square Linear Model: A Three-Way AOV
t = number of treatments, rows and columns.
yij(k) = observation on the unit in the ith row, jth column given the kth treatment. The indicator k is in parenthesis to remind us that specifying i and j effectively determines the treatment k.
m = the general mean common to all experimental units.
ri = the effect of level i of the row blocking factor. Usually assumed N(0,sr2), a random effect.
j = the effect of level j of the column blocking factor. Usually assumed N(0,sn2), a random effect.
tk = the effect of level k of treatment factor, a fixed effect.
eij(k) = component of random variation associated with observation ij(k). Usually assumed N(0,se2).

12. Latin Square Analysis of Variance

13. Sums of Squares

14. Experimental Error
Experimental error = response differences between two experimental units that have experienced the same treatment. In this case though, the “replicates” for each treatment are spread across the t row and t column blocks in a specific fashion.
Even more so than with randomized block designs, the variability among treatment replicates includes the row and column block effects. In similar fashion as for RCBDs, the specific latin square layout will filter out the extraneous (row & col) sources of variability when performing comparisons of treatment means. (Show on board…)
Note that this would not have been the case if the experiment had erroneously been laid out as a CRD or RBD…

15. Latin Square Mean Squares and F Statistics
We reject the null hypothesis of no main effect if the value of the F-statistic is greater than the 100(1-a)th percentile of the F distribution with degrees of freedom specified above.

16. Latin Square Example
The strawberry irrigation cold protection study data are given below. The effectiveness of the three irrigation methods was measured by the weight of the frozen fruit, with lower weights representing more effective protection. The study question is “ Which irrigation method provided the most protection?”

17. Latin Square in SAS Data strawb; input row column irrig $ weight @@;
datalines;
1 1 drip over none 60
2 1 none drip over 31
3 1 over none drip 49
; run;
proc glm;
class row column irrig;
model weight = row column irrig;
title 'Strawberry Irrigation Latin Square Exp'; run;
Latin Square in SAS
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE weight Mean
Source DF Type I SS Mean Square F Value Pr > F
row
column
irrig
Source DF Type III SS Mean Square F Value Pr > F

18. Latin Square in SPSS Input Data
Analyze > General Linear Model > Univariate
Note: You must use a custom model and
only ask for main effects.

19. SPSS Ouptut

20. Latin Square in Minitab
Stat > ANOVA > General Linear Model

21. MTB: ANOVA and Sums of Squares
General Linear Model: weight versus row, column, irrig
Factor Type Levels Values
row fixed , 2, 3
column fixed , 2, 3
irrig fixed drip, none, over
Analysis of Variance for weight, using Adjusted SS for Tests
Source DF Seq SS Adj SS Adj MS F P
row
column
irrig
Error
Total
S = R-Sq = 78.27% R-Sq(adj) = 13.07%

22. Latin Square with R
> straw <- read.table("Data/latin_square.txt",header=TRUE)
> straw.lm <- lm(weight ~ factor(row) + factor(column) factor(irrig), data=straw)
> anova(straw.lm)
Analysis of Variance Table
Response: weight
Df Sum Sq Mean Sq F value Pr(>F)
factor(row)
factor(column)
factor(irrig)
Residuals

23. Medical Example of a Latin Square

24. Randomized, controlled, double-blinded, NICE!
Design extracts out differences due to time and patients!