Presentation on theme: "Latin Square Designs (§15.4)"— Presentation transcript:
1Latin Square Designs (§15.4) Lecture ObjectiveIntroduce basic experimental designs that account for two orthogonal sources of extraneous variation.TerminologySquare designOrthogonal blocksRandomizations
2ExamplesA 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.
3Latin Square DesignA 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.
4Cold 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).
5highNitrogen LevellowField LayoutnonedripoverMoisturenoneovernonedripdripoverMoisture and Soil Nitrogen are two sources of extraneous variation that we wish to simultaneously control for.CANALNitrogen LevelhighlownonedripoverWhich design will best allow us to account for both soil moisture and nitrogen gradients?MoisturedripovernoneovernonedripCANAL
6Advantages 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.
7Constructing 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.
9Randomization Get a random ordering of the rows. 1 2 3 4 replaced by Reorder the rows according to randomization.
10Randomization Get a random ordering of the columns. 1 2 3 4 replaced byReorder the columns according to randomization.Two Blocking Factors = Two Randomizations= Two Constraints on Randomization
11Latin 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).
14Experimental ErrorExperimental 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…
15Latin 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.
16Latin Square ExampleThe 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?”
17Latin Square in SAS Data strawb; input row column irrig $ weight @@; datalines;1 1 drip over none 602 1 none drip over 313 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 SASSum ofSource DF Squares Mean Square F Value Pr > FModelErrorCorrected TotalR-Square Coeff Var Root MSE weight MeanSource DF Type I SS Mean Square F Value Pr > FrowcolumnirrigSource DF Type III SS Mean Square F Value Pr > F
18Latin Square in SPSS Input Data Analyze > General Linear Model > UnivariateNote: You must use a custom model andonly ask for main effects.
20Latin Square in Minitab Stat > ANOVA > General Linear Model
21MTB: ANOVA and Sums of Squares General Linear Model: weight versus row, column, irrigFactor Type Levels Valuesrow fixed , 2, 3column fixed , 2, 3irrig fixed drip, none, overAnalysis of Variance for weight, using Adjusted SS for TestsSource DF Seq SS Adj SS Adj MS F ProwcolumnirrigErrorTotalS = R-Sq = 78.27% R-Sq(adj) = 13.07%
22Latin 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 TableResponse: weightDf Sum Sq Mean Sq F value Pr(>F)factor(row)factor(column)factor(irrig)Residuals