Replicated Latin Square and Crossover Designs
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1 Replicated Latin Square and Crossover Designs
2 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 E, power Methods of replication Use the same row and column blocks Use new row blocks but same column blocks Use the same row blocks and new column blocks Use new row and column blocks Degrees of freedom depend on what is new /randomized Often include additional block - replicate effect Replicated Latin Squares 2
3 Replicate the Square Same row/column blocks used in additional squares Usually includes replicate (e.g., time) effect y ijkl = µ + α i + τ j + β k + δ l + ǫ ijkl i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., p l = 1, 2,..., n Replicated Latin Squares 3
4 Source of Sum of Degrees of Mean F Variation Squares Freedom Square Rows SS Row p 1 Columns SS Column p 1 Replicate SS Replicate n 1 Treatment SS Treatment p 1 MS Treatment F 0 Error SS E (p 1)(n(p + 1) 3) MS E Total SS T np 2 1 Replicated Latin Squares 4
5 Replicate the Rows (or columns) Different rows (columns) in new square Row(column) effects nested within square Same column(row) effects y ijkl = µ + α i(l) + τ j + β k + δ l + ǫ ijkl i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., p l = 1, 2,..., n Replicated Latin Squares 5
6 Source of Sum of Degrees of Mean F Variation Squares Freedom Square Rows SS Row n(p 1) Columns SS Column p 1 Replicate SS Replicate n 1 Treatment SS Treatment p 1 MS Treatment F 0 Error SS E (p 1)(np 2) MS E Total SS T np 2 1 Replicated Latin Squares 6
7 Latin Rectangle Replicate rows(columns) but not multiple squares np separate rows (n integer) y ijk = µ + α i + τ j + β k + ǫ ijk i = 1, 2,..., np j = 1, 2,..., p k = 1, 2,..., p Replicated Latin Squares 7
8 Source of Sum of Degrees of Mean F Variation Squares Freedom Square Rows SS Row np 1 Columns SS Column p 1 Treatment SS Treatment p 1 MS Treatment F 0 Error SS E (p 1)(np 2) MS E Total SS T np 2 1 Replicated Latin Squares 8
9 Replicated Rows and Columns Have completely separate squares Row and column effect nested within square y ijkl = µ + α i(l) + τ j + β k(l) + δ l + ǫ ijkl i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., p l = 1, 2,..., n Replicated Latin Squares 9
10 Source of Sum of Degrees of Mean F Variation Squares Freedom Square Rows SS Row n(p 1) Columns SS Column n(p 1) Replicate SS Replicate n 1 Treatment SS Treatment p 1 MS Treatment F 0 Error SS E (p 1)(n(p 1) 1) MS E Total SS T np 2 1 Replicated Latin Squares 10
11 Graeco-Latin Square Design Described in Section 4.3 Superimposes two Latin Squares onto each other Allows blocking on three factors Exists for all p 3 except p 6 Degrees of freedom error is (p 3)(p 1) p = 4 df E =3 p = 5 df E =8 Replicated Latin Squares 11
12 Crossover Design A commonly-use within-subject design Considers s subjects as blocks Each subject undergoes p treatments run over p periods Can consider incomplete block structure (# periods < # trts) Used in drug comparisons/physiology experiments Delay between periods to remove residual effect Residual effect also called carryover effect Replicated Latin Squares 12
13 Crossover Design Used because one anticipates high level of variability between subjects block on subject to remove it Subject (S k ) is serving as its own control Commonly used for 2, 3, or 4 periods Period (P i ) is typically considered a blocking factor too Potential drawbacks: Subsequent use / carryover effect Replicated Latin Squares 13
14 Analysis of a Crossover Design Another variation of a repeated measures design Linear model approach similar to that of Latin Rectangle y ijk = µ + P i + τ j + S k + ǫ ijk Assumes no residual effects, subjects ǫ s can be correlated Consider 2 2 experiment with n subjects per group (order of treatments). Using model with ǫ N(0, sigma 2 ), the difference in trts for the two groups can be written Subjects who received Trt 1 first : diff 1k = (τ 1 τ 2 ) + (P 1 P 2 ) + (ǫ 11k ǫ 22k ) Subjects who received Trt 2 first : diff 2k = (τ 2 τ 1 ) + (P 1 P 2 ) + (ǫ 21k ǫ 12k ) Subject effects cancel out. Only within-subject variability left. Thus diff 1. diff 2. estimates 2(τ 1 τ 2 ) with standard error 4ˆσ 2 /n This result is equivalent to fitting the linear model above Replicated Latin Squares 14
15 Issue of Residual Effects But what if there are residual effects. This alters the overall effect in the second period. Considering r 1 and r 2 the residuals effects, the difference can be written Trt 1 first : diff 1k = (τ 1 (τ 2 + r 1 )) + (P 1 P 2 ) + (ǫ 11k ǫ 22k ) Trt 2 first : diff 2k = (τ 2 (τ 1 + r 2 )) + (P 1 P 2 ) + (ǫ 21k ǫ 12k ) Thus diff 1 diff 2 estimates 2(τ 1 τ 2 )+(r 2 r 1 ). Mean difference no longer estimates just difference in treatments (confounded with difference in residual effects). Can test for residual effect by looking at sums instead of differences. Consider subject effects random so subject variability incorporated into error (δ ijk = ǫ ijk +S k ) Trt 1 first : sum 1k = 2µ + (τ 1 + τ 2 + r 1 ) + (P 1 + P 2 ) + δ 1k Trt 2 first : sum 2k = 2µ + (τ 2 + τ 1 + r 2 ) + (P 1 + P 2 ) + δ 2k Thus sum 1 sum 2 estimates (r 1 r 2 ). Can check to see if different from zero. Unfortunately this is a low power test because it incorporates between-subject variability, which is often larger than within-subject variability. Replicated Latin Squares 15
16 Modeling Residual Effects Can attempt to include residual effects in model Need p > 2 if subjects considered fixed effect Not orthogonal so fit order important (Type III SS if using OLS) First-order residual effects model y ijk = µ + P i + τ j + S k + r ij + ǫ ijk i = 1, 2,..., p j = 1, 2,..., p k = 1, 2,..., np where r ij only occurs when i 1 and j references the trt used in the previous period. Replicated Latin Squares 16
17 Crossover Analysis when p = 2 Will analyze the following data set using Differences and sums Proc GLM - fixed effects linear model Proc Mixed - subjects considered random effects Need to create design column(s) for residual effects data trt2cross; input subj period trt resid cards; ; Replicated Latin Squares 17
18 Demonstration Using SAS **Calculate differences and sums by hand and enter into data set; data trt2cross_diff; input subj order ydiff cards; ; proc ttest; var ydiff; class order; run; ***Test if trts differ data trt2cross_sum; input subj order cards; ; proc ttest; var ysum; class order; run; ***Test if res effects differ Replicated Latin Squares 18
19 SAS Output Variable: ydiff order N Mean Std Dev Std Err Minimum Maximum Diff (1-2) Method Variances DF t Value Pr > t Pooled Equal ***Trt Satterthwaite Unequal Variable: ysum order N Mean Std Dev Std Err Minimum Maximum Diff (1-2) Method Variances DF t Value Pr > t Pooled Equal ***Residual Satterthwaite Unequal Replicated Latin Squares 19
20 Demonstration Using SAS ***Get trt*period means***; proc sort; by trt period; proc means; var y; by trt period; run; ***Fit using fixed effects linear model***; proc glm; class subj trt period; model y = subj trt period; lsmeans trt / lines; ***Try to include residual effects***; proc glm; class subj trt period; model y = resid period subj trt; lsmeans trt / lines; run; Replicated Latin Squares 20
21 SAS Output ***Summary of the Proc Means Output*** Trt1 Period Trt2 Period > Trt Period Trt1 Period > Trt Period Trt2 Period > GrandMean 34.2 ***Using latin rectangle linear model**** Estimated treatment difference is = 3.2 ***Using first-order residual effects model**** Period 1 Trt2 - Trt1 = = 4.2 Period 2 Trt2 - Trt1 = = 2.2 ** **Includes possible residuals effects **Therefore, estimated treatment difference is 4.2 **(r1 - r2) = 2*r1 = -2 --> r1 = -1 Replicated Latin Squares 21
22 GLM Output - No Residual Effects 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 subj trt *** period Source DF Type III SS Mean Square F Value Pr > F subj trt *** period Parameter Estimate Std Error t Value Pr > t trt B *** period B Replicated Latin Squares 22
23 GLM Output - Residual Effects 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 resid period subj trt Source DF Type III SS Mean Square F Value Pr > F resid period subj trt *************CONFOUNDING OF EFFECTS PRESENT********************; Replicated Latin Squares 23
24 Demonstration Using SAS proc mixed; class subj trt period; model y = period trt / s ddfm=kr; random subj; lsmeans trt; run; proc mixed; class subj trt period; model y = resid period trt / s ddfm=kr; ***With residual effect; random subj; lsmeans trt; Replicated Latin Squares 24
25 Mixed Output - No Residual Effects Cov Parm Estimate subj Residual Solution for Fixed Effects Effect Estimate Std Error DF t Value Pr > t Intercept <.0001 period trt *** Type 3 Tests of Fixed Effects Effect DF DF F Value Pr > F period trt *** Least Squares Means Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 trt <.0001 Replicated Latin Squares 25
26 Mixed Output - Residual Effects Cov Parm Estimate subj Residual Solution for Fixed Effects Effect Estimate Std Error DF t Value Pr > t Intercept <.0001 resid *** period trt Least Squares Means Effect trt Estimate Std Error DF t Value Pr > t trt <.0001 trt <.0001 Replicated Latin Squares 26
27 SAS Code - p = 4 data new; input cow period trt if period=1 then resid=0; else resid=a; resid1=0; resid2=0; resid3=0; if resid=1 then resid1=1; if resid=4 then resid1=-1; if resid=2 then resid2=1; if resid=4 then resid2=-1; if resid=3 then resid3=1; if resid=4 then resid3=-1; a=trt; retain a; cards; ; proc print; run; Replicated Latin Squares 27
28 SAS Code proc glm; class cow period trt; model resp=cow period trt resid1 resid2 resid3 / solution; lsmeans trt / stderr pdiff cl lines; Obs cow period trt resp resid resid1 resid2 resid Replicated Latin Squares 28
29 SAS Output 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 cow period trt resid *** Sum these resid *** together to resid *** get SS(resid) Source DF Type III SS Mean Square F Value Pr > F cow period trt *** Want to look at resid trt after adjusting resid for all else resid Tests of importance based off of SAS results above Source DF Type III SS Mean Square F Value Pr > F trt resid Replicated Latin Squares 29
30 Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 cow B cow B cow B cow B... period B period B period B period B... trt B trt B trt B trt B... resid resid resid Residual Effect of A increases response 0.75 units Residual Effect of B increases response 1.25 units Residual Effect of C decreases response 1.25 units Residual Effect of D decreases response 0.75 units LSMEAN trt 1 = ( ( )) = = intercept + trt effect LSMEAN trt 2 = ( ( )) = = intercept + trt effect Replicated Latin Squares 30
31 Least Squares Means Standard LSMEAN trt resp LSMEAN Error Pr > t Number < < < < resp LSMEAN LSMEAN trt Number A A A A A A A Least Squares Means for Effect trt Difference Between 95% Confidence Limits for i j Means LSMean(i)-LSMean(j) Replicated Latin Squares 31
32 SAS Code ***Provides same results as Proc GLM output; proc mixed; class cow period trt; model resp=period trt cow resid1 resid2 resid3 / solution; lsmeans trt / cl; contrast resid eff resid1 1, resid2 1, resid3 1; run; ***Slightly different because cows random effects; proc mixed; class cow period trt; model resp=period trt resid1 resid2 resid3 / solution; random cow; lsmeans trt / cl; contrast resid eff resid1 1, resid2 1, resid3 1; run; Replicated Latin Squares 32
33 Designs Balanced For Residual Effects Consider the following two Latin squares Suppose row=period and column=subject D C B A C D A B B A D C A B C D D C B A C A D B B D A C A B C D (Left) C D twice, A D once, B D never (Right) Each trt follows each other trt once Right square balanced for residual effects If p even, can be balanced using p subjects If p odd, need multiple of 2p subjects Replicated Latin Squares 33
34 Alternative Notation Consider a 2 2 crossover design with n subjects Subject n-1 Order 1 Period 1 Subject(order) n-2 Trt 1 Trt 1 Error n-2 Trt*Order 1 Error n-2 Period confounded with Trt*Order When subjects are random, don t expect Order differences so when significant, suggests there may be carryover effects. In fact, this is the same test as the difference in sums or including a residual effect. Replicated Latin Squares 34
35 Advantages Benefits and Drawbacks Subjects serve as their own control to reduce error May be able to get more volunteers if subjects particular to a treatment Drawbacks Subjects must be available for longer periods of time Carryover issue...are subjects the same at the start of each period? Crossover design avoided in comparative clinical studies Replicated Latin Squares 35
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