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Latin Square -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

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Latin Square -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.

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Latin Square -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 t 2 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.

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Latin Square -4 Cold Protection of Strawberries Three different irrigation methods (treatment levels) are used on strawberries : 1.drip, 2.overhead sprinkler, 3.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).

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

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Latin Square -6 Advantages and Disadvantages Advantages: Allows for control of two extraneous sources of variation. Analysis is quite simple. Disadvantages: Requires t 2 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.

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Latin Square -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.

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Latin Square -8 Basic Square

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Latin Square -9 Randomization Get a random ordering of the rows replaced by Reorder the rows according to randomization.

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Latin Square -10 Randomization Get a random ordering of the columns replaced by Reorder the columns according to randomization. Two Blocking Factors = Two Randomizations = Two Constraints on Randomization

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Latin Square -11 Latin Square Linear Model: A Three-Way AOV t=number of treatments, rows and columns. y ij(k) =observation on the unit in the i th row, j th column given the k th treatment. The indicator k is in parenthesis to remind us that specifying i and j effectively determines the treatment k. =the general mean common to all experimental units. i =the effect of level i of the row blocking factor. Usually assumed N(0, 2 ), a random effect. j =the effect of level j of the column blocking factor. Usually assumed N(0, 2 ), a random effect. k =the effect of level k of treatment factor, a fixed effect. ij(k) =component of random variation associated with observation ij(k). Usually assumed N(0, 2 ).

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Latin Square -12 Latin Square Analysis of Variance

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Latin Square -13 Sums of Squares

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Latin Square -14 Experimental Error

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Latin Square -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- ) th percentile of the F distribution with degrees of freedom specified above.

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Latin Square -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?”

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Latin Square -17 Latin Square in SAS Data strawb; input row column irrig $ weight datalines; 1 1 drip over none none drip over over none drip 49 ; run; proc glm; class row column irrig; model weight = row column irrig; title 'Strawberry Irrigation Latin Square Exp'; run; 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 row column irrig

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Latin Square -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.

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Latin Square -19 SPSS Ouptut

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

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Latin Square -21 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 column irrig Error Total S = R-Sq = 78.27% R-Sq(adj) = 13.07% MTB: ANOVA and Sums of Squares

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Latin Square -22 > 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 Latin Square with R

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Latin Square -23 Medical Example of a Latin Square

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

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