Latin Square Designs ( § 15.4)

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 over none 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 Moisture Which design will best allow us to account for both soil moisture and nitrogen gradients? drip over none over drip none CANAL

6. Advantages and Disadvantages • Advantages: • 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 throught, 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 2 1 4 3 Reorder the rows according to randomization.

10. Randomization Get a random ordering of the columns. 1 2 3 4 replaced by 4 2 3 1 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. Data strawb; input row column irrig $ weight @@; datalines; 1 1 drip 51 1 2 over 119 1 3 none 60 2 1 none 98 2 2 drip 43 2 3 over 31 3 1 over 99 3 2 none 87 3 3 drip 49 ; run; procglm; 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 6 5840.000000 973.333333 1.20 0.5205 Error 2 1621.555556 810.777778 Corrected Total 8 7461.555556 R-Square Coeff Var Root MSE weight Mean 0.782679 40.23037 28.47416 70.77778 Source DF Type I SS Mean Square F Value Pr > F row 2 817.555556 408.777778 0.50 0.6648 column 2 2616.222222 1308.111111 1.61 0.3826 irrig 2 2406.222222 1203.111111 1.48 0.4026 Source DF Type III SS Mean Square F Value Pr > F row 2 817.555556 408.777778 0.50 0.6648 column 2 2616.222222 1308.111111 1.61 0.3826 irrig 2 2406.222222 1203.111111 1.48 0.4026

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

19. SPSS Ouptut

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

21. MTB: ANOVA and Sums of Squares General Linear Model: weight versus row, column, irrig Factor Type Levels Values row fixed 3 1, 2, 3 column fixed 3 1, 2, 3 irrig fixed 3 drip, none, over Analysis of Variance for weight, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P row 2 817.6 817.6 408.8 0.50 0.665 column 2 2616.2 2616.2 1308.1 1.61 0.383 irrig 2 2406.2 2406.2 1203.1 1.48 0.403 Error 2 1621.6 1621.6 810.8 Total 8 7461.6 S = 28.4742 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) 2 817.56 408.78 0.5042 0.6648 factor(column) 2 2616.22 1308.11 1.6134 0.3826 factor(irrig) 2 2406.22 1203.11 1.4839 0.4026 Residuals 2 1621.56 810.78

23. Medical Example of a Latin Square

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