Latin Square Design. Design of Experiments - Montgomery Section 4-2

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1 Latin Square Design Design of Experiments - Montgomery Section 4-2

2 Latin Square Design Can be used when goal is to block on two nuisance factors Constructed so blocking factors orthogonal to treatment Must have same the number of blocks and treatments y ijk = µ + α i + τ j + β k + ǫ ijk α i - ith Block 1 effect (row) τ j - jth treatment effect β k - kth Block 2 effect (column) ǫ ijk N(0, σ 2 ) i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., p Completely additive model (no interaction) Latin Square 2

3 Latin Square Design Design commonly represented in p p grid There are now two randomization restrictions One trt per row (row = blocking factor 1) One trt per column (column = blocking factor 2) Can randomly shuffle rows, columns, and treatments of standard square to get other variations of layout Standard square has treatment levels written alphabetically in the first row and first column. Other cells are filled using the same alphabetic order Latin Square 3

4 Layout Examples C B A B A C A C B D B C A B A C C D A B A C B B A D C C B A A C B D Latin Square 4

5 Proc PLAN title Latin Square Design ; proc plan seed=12; factors rows=4 ordered cols=4 ordered / NOPRINT; treatments tmts=4 cyclic; output out=g rows cvals=( Day 1 Day 2 Day 3 Day 4 ) random cols cvals=( Lab 1 Lab 2 Lab 3 Lab 4 ) random tmts nvals=( ) random; proc tabulate; class rows cols; var tmts; table rows, cols*(tmts*f=6.) / rts=8; run; Latin Square 5

6 Proc PLAN Output cols Lab 1 Lab 2 Lab 3 Lab 4 tmts tmts tmts tmts Sum Sum Sum Sum rows Day Day Day Day Latin Square 6

7 Partitioning the SS Rewrite observation as: y ijk = y... + (y i.. y... ) + (y.j. y... ) + (y..k y... ) + (y ijk y i.. y.j. y..k + 2y.. ) = ˆµ + ˆα i + ˆτ j + ˆβk + ˆǫ ijk p Partition SS T into (y i.. y... ) 2 + p (y.j. y... ) 2 + p (y..k y... ) 2 + ˆǫ 2 ijk SS Row + SS Treatment + SS Col + SS E Under H 0, all SS/σ 2 independent chi-squared RVs Usual F-test analysis Caution testing column and row effects Latin Square 7

8 Analysis of Variance Table Source of Sum of Degrees of Mean F 0 Variation Squares Freedom Square Rows SS Row p 1 MS Row Treatment SS Treatment p 1 MS Treatment F 0 Column SS Column p 1 MS Column Error SS E (p 2)(p 1) MS E Total SS T p 2 1 If F 0 > F α,p 1,(p 2)(p 1) then reject H 0 Latin Square 8

9 Comparisons of Treatments Multiple Comparisons/Contrasts Similar procedures as before with CRD Replace n with p in all standard error formulas Degrees of freedom error is (p 2)(p 1) y.j. = µ + ᾱ. + τ j + β. + ǫ.j. Var(y.j. ) = σ 2 /p y.j. y.j. = τ j τ j + ǫ.j. ǫ.j. Var(y.j. y.j.) = σ 2 /p + σ 2 /p Latin Square 9

10 Missing Values When missing (and not due to treatment) Design unbalanced Orthogonality lost Order of fit important Procedures 1 Regression approach Use Type III sum of squares 2 Estimate missing value Choose value to minimize SS E Take derivative and set equal to zero y ijk = p(y i.. +y.j. +y..k ) 2y... (p 2)(p 1) Latin Square 10

11 Using SAS Consider experiment to investigate the effect of 4 diets on milk production. There are 4 cows. Each lactation period the cows receive a different diet. Assume there is a washout period so previous diet does not affect future results. Will therefore block on lactation period and cow. data new; input cow period trt cards; ; proc glm; class cow trt period; model resp=trt period cow; means trt/ lines tukey; lsmeans trt / adjust=tukey lines; output out=new1 r=res p=pred; symbol1 v=circle; proc gplot; plot res*pred; plot res*cow; plot res*trt; plot res*period; proc univariate noprint; histogram res / normal (L=1 mu=0 sigma=est) kernel (L=2); run; Latin Square 11

12 ANOVA Table Dependent Variable: resp Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total Source DF Type I SS Mean Square F Value Pr > F trt period <.0001 cow Source DF Type III SS Mean Square F Value Pr > F trt period <.0001 cow Latin Square 12

13 Multiple Comparisons Tukey s Studentized Range (HSD) Test for resp Alpha 0.05 Error Degrees of Freedom 6 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Mean N trt A A B B Latin Square 13

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17 Using Proc Mixed Often at least one block is considered random Example: 4 cows randomly chosen from large herd Want the inference to extend to the herd Sometimes time is one of the blocking factors Example: Each cow measured over 4 lactation periods This is an example of a crossover design Could observations closer together be more correlated? Proc Mixed can incorporate both concepts into model Latin Square 17

18 Random Effects Similar approach and results as with the RCBD Must be careful if using Proc GLM Standard error for a mean: Proc GLM incorrect Standard error for a contrast: Proc GLM correct GLM very limited in terms of handling repeated measures Latin Square 18

19 Comparison of Approaches proc glm; class cow trt period; model resp=cow trt period; random cow; lsmeans trt / stderr tdiff lines; proc mixed; class cow trt period; model resp=trt period / ddfm=kr; ***recommended df adjustment option random cow; lsmeans trt/ diff; run; ****F test for treatment is the same for both models ***** Latin Square 19

20 GLM Selected Output Standard LSMEAN trt resp LSMEAN Error Pr > t Number < < < < resp LSMEAN LSMEAN trt Number A A A B B B Latin Square 20

21 MIXED Selected Output Least Squares Means Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 trt <.0001 trt <.0001 trt <.0001 Differences of Least Squares Means Effect trt _trt Estimate Std Error DF t Value Pr > t trt trt trt trt trt trt Latin Square 21

22 Correlated Errors Random effects induce correlations among observations Errors from same EU may also be correlated Errors closer in time may be more similar Can incorporate various correlation structures Given p obs per EU, consider a p p covariance matrix Main diagonal contains the variances Off-diagonal elements represent covariances Latin Square 22

23 Covariance Matrices Uncorrelated errors (for p = 4 obs/cow) σ σ σ 2 0 = σ σ st order autoregressive (for p = 4 obs/cow) σ 2 1 ρ ρ 2 ρ 3 ρ 1 ρ ρ 2 ρ 2 ρ 1 ρ ρ 3 ρ 2 ρ 1 Latin Square 23

24 Covariance Matrices Compound symmetry (for p = 4 obs/cow) σ 2 + σ 1 σ 1 σ 1 σ 1 σ 1 σ 2 + σ 1 σ 1 σ 1 σ 1 σ 1 σ 2 + σ 1 σ 1 σ 1 σ 1 σ 1 σ 2 + σ 1 Toeplitz (for p = 4 obs/cow) σ 2 σ 1 σ 2 σ 3 σ 1 σ 2 σ 1 σ 2 σ 2 σ 1 σ 2 σ 1 σ 3 σ 2 σ 1 σ 2 Latin Square 24

25 Covariance Matrices Banded Toeplitz d = 2(for p = 4 obs/cow) σ 2 σ σ 1 σ 2 σ σ 1 σ 2 σ σ 1 σ 2 Unstructured (for p = 4 obs/cow) σ1 2 σ 12 σ 13 σ 14 σ 12 σ2 2 σ 23 σ 24 σ 13 σ 23 σ3 2 σ 34 σ 14 σ 24 σ 34 σ4 2 Latin Square 25

26 Mixed Model Can be expressed Y = Xβ + Zδ + e X and Z describe the treatment and design structure β is a vector of the fixed effect parameters δ is a vector of the random effect parameters Typically assume δ and e are multivariate Normal (MVN) Mean 0 Covariance matrices G and R uncorrelated Latin Square 26

27 Conditional/Marginal Models Conditional model of Y δ is MVN E(Y δ) = Xβ + Zδ Cov(Y δ) = R Marginal model of Y is MVN E(Y ) = Xβ Cov(Y ) = ZGZ + R Can specify either model in MIXED. Random statement needed for conditional model. When dists Normal, both models give identical estimates, unless covariances negative Latin Square 27

28 SAS Code /* Fit a simple correlation structure for residuals */ /* Assuming cow is fixed - matches glm output on Slide 20 */ proc mixed covtest cl; class cow trt period; model resp=trt period cow / ddfm=kr outp=diag; repeated period / subject=cow type=simple r=1; lsmeans trt / diff; run; /* Assuming cow is random - matches mixed output on Slide 21 */ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; random intercept / subject=cow v=1; repeated period / subject=cow type=simple r=1; lsmeans trt / diff; run; Latin Square 28

29 Simple - Fixed effects Estimated R Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Cov Standard Z Parm Subject Estimate Error Value Pr > Z Alpha Lower Upper period cow Fit Statistics -2 Res Log Likelihood 26.9 AIC (Smaller is Better) 28.9 BIC (Smaller is Better) 28.3 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt Latin Square 29

30 Simple - Random effects Estimated V Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr > Z Alpha Lower Upper Intercept cow period cow Fit Statistics -2 Res Log Likelihood 41.3 AIC (Smaller is Better) 45.3 BIC (Smaller is Better) 44.1 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt Latin Square 30

31 SAS Code /* Fit Compound Symmetry - matches mixed output on Slide 21 */ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; repeated period / subject=cow type=cs r=1; lsmeans trt / diff; run; /* Fit an Unstructured correlation structure */ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; repeated period / type=un subject=cow; lsmeans trt / diff; run; Latin Square 31

32 Compound Symmetry Estimated R Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr > Z Alpha Lower Upper CS cow Residual Fit Statistics -2 Res Log Likelihood 41.3 AIC (Smaller is Better) 45.3 BIC (Smaller is Better) 44.1 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt Latin Square 32

33 Unstructured WARNING: Stopped because of too many likelihood evaluations. Covariance Parameter Values At Last Iteration Cov Parm Subject Estimate UN(1,1) cow UN(2,1) cow UN(2,2) cow UN(3,1) cow UN(3,2) cow UN(3,3) cow UN(4,1) cow UN(4,2) cow UN(4,3) cow UN(4,4) cow **** SOME COVARIANCE STRUCTURES ARE TOO COMPLEX FOR THE DATA **** Latin Square 33

34 SAS Code /* Fit an AR(1) correlation structure for residuals*/ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; repeated period / type=ar(1) subject=cow r=1; lsmeans trt / diff; run; /* Fit an AR(1) correlation structure for residuals*/ /* Add cow random effect */ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; random intercept / subject=cow v=1; repeated period / type=ar(1) subject=cow r=1; lsmeans trt / diff; run; Latin Square 34

35 AR(1) Estimated R Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Alpha Lower Upper AR(1) cow < Residual Fit Statistics -2 Res Log Likelihood 42.0 AIC (Smaller is Better) 46.0 BIC (Smaller is Better) 44.8 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt trt Latin Square 35

36 AR(1) with random cow effect Estimated V Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Alpha Lower Upper Intercept cow AR(1) cow Residual Fit Statistics -2 Res Log Likelihood 41.2 AIC (Smaller is Better) 47.2 BIC (Smaller is Better) 45.3 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt trt Latin Square 36

37 SAS Code /* Fit an TOEP(2) correlation structure for residuals*/ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; repeated period / type=toep(2) subject=cow r=1; lsmeans trt / diff; run; /* Fit an Heterogeneous CS correlation structure for residuals*/ proc mixed covtest cl; class cow trt period; model resp=trt period / ddfm=kr outp=diag; repeated period / type=csh subject=cow r=1; lsmeans trt / diff; run; Latin Square 37

38 Banded TOEP(2) Estimated R Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Standard Z Cov Parm Subject Estimate Error Value Pr Z Alpha Lower Upper TOEP(2) cow Residual Fit Statistics -2 Res Log Likelihood 45.0 AIC (Smaller is Better) 49.0 BIC (Smaller is Better) 47.8 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt trt Latin Square 38

39 CSH Estimated R Matrix for cow 1 Row Col1 Col2 Col3 Col Covariance Parameter Estimates Cov Standard Z Parm Subject Estimate Error Value Pr Z Alpha Lower Upper Var(1) cow Var(2) cow Var(3) cow Var(4) cow CSH cow < Fit Statistics -2 Res Log Likelihood 34.8 AIC (Smaller is Better) 44.8 BIC (Smaller is Better) 41.7 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F trt Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 Effect trt _trt Estimate Std Error DF t Value Pr > t trt trt Latin Square 39

40 Summary When comparing models with a model selection criterion, REML fitting can only be used when fixed effects structure does not change. If different models with different fixed effects were compared, would have to use ML estimation, even though variance/covariance parameters are biased This is because REML uses a transformed set of data that takes into account loss of DF due to estimating fixed effects From the results here, heterogenous compound symmetry is the best choice provided residuals look ok Latin Square 40

Replicated Latin Square and Crossover Designs

Replicated Latin Square and Crossover Designs Replicated Latin Square Latin Square Design small df E, low power If 3 treatments 2 df error If 4 treatments 6 df error Can use replication to increase df