Split-plot Designs. Bruce A Craig. Department of Statistics Purdue University. STAT 514 Topic 21 1

Transcription

1 Split-plot Designs Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 21 1

2 Randomization Defines the Design Want to study the effect of oven temp (3 levels) and amount of baking soda (4 levels) on the consistency of a 6-inch chocolate chip cookie. [Design 1] Factorial: Each of the 12 combinations of temp and baking soda is replicated three times. You mix up cookie dough and then cook it 36 times. [Design 2] Split plot: Four batches of dough are created, each with a different amount of baking soda. Oven is heated to specific temp and the four doughs are put in the oven at the same time. Replicate this process three times at each oven temp. This means we make 36 batches of dough but only run 9 cooking trials. STAT 514 Topic 21 2

3 Randomization Defines the Design Design 2 is different from Design 1 because of a randomization restriction. Instead of randomly assigning temp to each batch of dough, it is instead randomly assigned to a group of four batches. In other words, the experimental unit of each factor is different. EU for amount of baking soda batch of dough EU for oven temperature group of four dough batches This kind of design is often used because it is easier to implement. For this experiment, nine cooking trials is far more manageable than 36 cooking trials. STAT 514 Topic 21 3

4 Split-plot Design Arose in agriculture Whole plot - Large field Subplot - Smaller sections of field - Want to study 4 fertilizers and 6 corn varieties - Spreader covers 15 foot wide section and planter covers 5 foot wide section - Spread fertilizer on 15x10 foot section (whole plot) - Plant seed in 5x5 foot sections (subplot) for a total of 6 subplots per whole plot Very useful in other areas (done out of convenience) Engineering - certain settings fixed for a group of runs Repeated measures - subject split into time sections STAT 514 Topic 21 4

5 Split-Plot Design For the cookie study, the experimental unit for oven temp is the group (or sheet) of four cookies. Since the four cookies within a sheet are randomly assigned amounts of baking soda, the experimental unit for baking soda is still the individual batch of dough. The larger experimental unit (cookie sheet) is divided or split into smaller experimental units (cookies). Whole plot: Batch of four cookies Subplot : Individual cookies The whole plots are always divided into smaller entities called subplots. The key for proper analysis is determining the whole plot and subplot factors and their experimental units STAT 514 Topic 21 5

6 Split Plot Structure Different from nested model because factors are crossed Different from factorial model because of randomization Information collected from two levels or strata Each level has its own experimental design Whole plot EUs serve as blocks at subplot level Can often consider split-plot consisting of a) RCBD in whole plot and RCBD in subplot b) CRD in whole plot and RCBD in subplot More power for subplot trt factor and interaction Should use this design only for practical reasons as the factorial design, if feasible, is overall more powerful STAT 514 Topic 21 6

7 EMS - CRD in Whole Plot Fixed A and B (r replicates of each level A) Whole plot EUs are these replicates Source of Degrees of Expected Variation Freedom Mean Square A a 1 rbφ A +bσr 2 +σ2 Rep(A) a(r 1) bσr 2 +σ2 B b 1 arφ B +σ 2 AB (a 1)(b 1) rφ AB +σ 2 Error a(b 1)(r 1) σ 2 STAT 514 Topic 21 7

8 EMS - RCBD in Whole Plot Fixed A and B treatment factors r random blocks contain similar whole plot EUs These whole plots EUs serve as blocks for subplot factor Source of Degrees of Expected Variation Freedom Mean Square Blk r 1 abσr 2 +(bσ2 RA )+σ2 A a 1 rbφ A +bσra 2 +σ2 Blk*A (a 1)(r 1) bσra 2 +σ2 B b 1 arφ B +σ 2 AB (a 1)(b 1) rφ AB +σ 2 Error a(b 1)(r 1) σ 2 Sometimes blocking interactions not pooled (Page 622) STAT 514 Topic 21 8

9 Example: Soybean Yields Interested in the effect of soybean varieties and fertilizers on the yield (bushels per subplot unit). Fertilizers were randomly applied to acres within each farm, varieties then randomly applied to subunits of each acre. Consider fertilizers and varieties as fixed. Farm, as a block, is considered random. Whole plot testing similar if block random or fixed factors. In subplot, if block fixed, all interactions with block are pooled into error. If it is random, this may or may not be done. If it is not done, there are other tests that may be of interest (see page 622). STAT 514 Topic 21 9

10 Soybean Yields - Data and Layout Farm Fertilizer Fertilizer Fertilizer Variety 1 2 Variety 2 1 Variety STAT 514 Topic 21 10

11 data new; infile "soy.dat"; input farm fert var resp; SAS Programs proc glm plots=all; **Pooling; class farm fert var; model resp=farm fert farm*fert var fert*var; random farm farm*fert / test; proc glm plots=all; **No pooling; class farm fert var; model resp=farm fert farm*fert var farm*var fert*var; random farm farm*fert farm*var / test; STAT 514 Topic 21 11

12 SAS Programs ****** Doing just the whole plot analysis ***** ***** Averaging out Variety ***** proc sort data=new; by farm fert; proc means NOPRINT; var resp; by farm fert; output out=new1 mean=resp1; proc glm data=new; class farm fert; model resp1=farm fert; run; STAT 514 Topic 21 12

13 SAS Output - Pooled SP Interactions Dependent Variable: resp Sum of Source DF Squares Mean Square F Value Pr > F Model <.0001 Error Cor Total Source DF Type III SS Mean Square F Value Pr > F farm <.0001 fert farm*fert ** var <.0001* fert*var * *Correct F-test **Necessary to keep in model to maintain SP structure STAT 514 Topic 21 13

14 SAS Output - Pooled SP Interactions Tests of Hypotheses for Mixed Model Analysis of Variance Source DF Type III SS Mean Square F Value Pr > F farm fert MS(farm*fert) farm*fert var <.0001 fert*var MS(Error) STAT 514 Topic 21 14

15 SAS Output - SP Interactions Dependent Variable: resp Sum of Source DF Squares Mean Square F Value Pr > F Model Error Cor Total Source DF Type III SS Mean Square F Value Pr > F farm <.0001 fert farm*fert ** var farm*var * fert*var * *Correct F-test **Necessary to keep in model to maintain SP structure STAT 514 Topic 21 15

16 SAS Output - SP Interactions Tests of Hypotheses for Mixed Model Analysis of Variance Source DF Type III SS Mean Square F Value Pr > F fert MS(farm*fert) farm*fert farm*var fert*var MS(Error) var MS(farm*var) STAT 514 Topic 21 16

17 SAS Output - WP Analysis Only Sum of Source DF Squares Mean Square F Value Pr > F Model Error Cor Total Source DF Type III SS Mean Square F Value Pr > F farm fert **** Same results ***** STAT 514 Topic 21 17

18 data new; infile "soy.dat"; input farm fert var resp; SAS Programs proc mixed plots=all; **Pooling; class farm fert var; model resp= fert var fert*var / ddfm=kr; random farm farm*fert; proc mixed plots=all; **No pooling; class farm fert var; model resp=fert var fert*var / ddfm=kr; random farm farm*fert farm*var; STAT 514 Topic 21 18

19 Using Proc Mixed proc mixed plots=all; **no pooling; class fert var farm; model resp=fert var / ddfm=kr; random farm farm*fert farm*var; Cov Parm Estimate FARM FERT*FARM VAR*FARM Residual Tests of Fixed Effects Source NDF DDF Type III F Pr > F FERT VAR <.0001 FERT*VAR ***ddfm=kr is causing pooling of WP and SP errors*** ***Need to remove ddfm=kr or use the nobound option*** STAT 514 Topic 21 19

20 Using Proc Mixed proc mixed plots=all; **pooling; class fert var farm; model resp=fert var / ddfm=kr; random farm farm*fert; Cov Parm Estimate FARM FERT*FARM Residual Tests of Fixed Effects Source NDF DDF Type III F Pr > F FERT VAR <.0001 FERT*VAR ***ddfm=kr is causing pooling of WP and SP errors*** ***Need to remove ddfm=kr or use the nobound option*** STAT 514 Topic 21 20

21 Using Proc Mixed proc mixed plots=all nobound; class fert var farm; model resp=fert var / ddfm=kr; random farm farm*fert; Cov Parm Estimate FARM FERT*FARM Residual Tests of Fixed Effects Source NDF DDF Type III F Pr > F FERT VAR <.0001 FERT*VAR ***Results same as GLM pooled SP interactions*** STAT 514 Topic 21 21

22 Whole Plot/Subplot Experiments Can have more than one factor in whole plot or subplot Common whole plot designs CRD RCBD Factorial (k factors) BIB Subplot RCBD BIB Blocked Factorial Design Analysis of Covariance Covariate linear with response in subplot and whole plot STAT 514 Topic 21 22

23 EMS Calculation Caution Must include whole plot EU in EMS table Otherwise may be misled and test all over subplot error Consider single replicate of factorial in WP Source of Degrees of Expected Variation Freedom Mean Square A a 1 bcφ A +cσwp 2 +σ2 B b 1 acφ B +cσwp 2 +σ2 AB (a 1)(b 1) cφ AB +cσwp 2 +σ2 Rep1(AB) 0 cσwp 2 +σ2 C c 1 abφ C +σ 2 AC (a 1)(c 1) bφ AC +σ 2 BC (b 1)(c 1) aφ 2 BC +σ2 ABC (a 1)(b 1)(c 1) σabc 2 +σ2 Rep(ABC) 0 σ 2 STAT 514 Topic 21 23

24 Pooling in Split Plot Have two layers so we can t simply pool all errors If we did, this would commonly result in Overstating significance of the whole plot factor If σwp 2 > σ2 SP, understate subplot factor Should pool errors separately Need to maintain the design structure STAT 514 Topic 21 24

25 Example: Pooling in Split Plot Consider A (fixed) and B (random) in whole plot, C fixed factor in subplot. Pool B, AB with Rep(AB) and pool BC, ABC with error. Other combinations alter design. Source of Degrees of Expected Variation Freedom Mean Square A a 1 bcnφ A +ncσab 2 +cσ2 WP +σ2 B b 1 acnσb 2 +cσ2 WP +σ2 AB (a 1)(b 1) cnσab 2 +cσ2 WP +σ2 Rep(AB) ab(n 1) cσwp 2 +σ2 C c 1 abnφ C +anσbc 2 +σ2 AC (a 1)(c 1) bnφ AC +nσabc 2 +σ2 BC (b 1)(c 1) anσbc 2 +σ2 ABC (a 1)(b 1)(c 1) nσabc 2 +σ2 Error ab(c 1)(n 1) σ 2 STAT 514 Topic 21 25

26 Extensions of Split-Plot Design Can further split subplot units into sub-subplots Known as Split-Split Plot Design CRD with 2 RCBDs Three RCBDs Source of Degrees of Expected Variation Freedom Mean Square Blk r 1 A a 1 abcσr 2 +σ2 bcrφ A +bcσar 2 +σ2 Blk*A (a 1)(r 1) B b 1 bcσar 2 +σ2 acrφ B +acσbr 2 +σ2 Blk*B (b 1)(r 1) AB (a 1)(b 1) acσbr 2 +σ2 crφ AB +cσabr 2 +σ2 Blk*AB (a 1)(b 1)(r 1) C c 1 cσabr 2 +σ2 abrφ C +abσcr 2 +σ2 Blk*C (c 1)(r 1) AC (a 1)(c 1) abσcr 2 +σ2 brφ AC +bσacr 2 +σ2 Blk*AC (a 1)(c 1)(r 1) BC (b 1)(c 1) bσacr 2 +σ2 arφ BC +aσbcr 2 +σ2 Blk*BC (b 1)(c 1)(r 1) ABC (a 1)(b 1)(c 1) aσbcr 2 +σ2 rφ ABC +σabcr 2 +σ2 Blk*ABC (a 1)(b 1)(c 1)(r 1) σabcr 2 +σ2 STAT 514 Topic 21 26

27 Strip Plot/Criss Cross Design Criss-Cross or Strip-Plot Design Two-factor treatment structure Both treatments require large EUs Arrange EUs in blocks (rectangles of size a b) Each block : whole plot rows and whole plot columns Three levels of information Rows Columns Row*Column (cell) STAT 514 Topic 21 27

28 Strip Plot ANOVA table Source of Degrees of Expected Variation Freedom Mean Square Blk r 1 abσr 2 +σ2 A a 1 brφ A +bσar 2 +σ2 Blk*A (a 1)(r 1) bσar 2 +σ2 B b 1 arφ B +aσbr 2 +σ2 aσbr 2 +σ2 rφ AB +σabr 2 +σ2 Blk*AB (a 1)(b 1)(r 1) σabr 2 +σ2 Blk*B (b 1)(r 1) AB (a 1)(b 1) Blk*AB would be the error term in most analyses STAT 514 Topic 21 28

29 Example of Strip Plot / Split Plot Investigating the long term effects of pasture composition for different patterns of grazing. Response is the percent of area covered by principal grass. Considered three factors: Length of time grazing (3, 9, 18 days) (SP)ring grazing cycles (2 with long gap or 4 with short gap) (S)ummer grazing cycles (2 with long gap or 4 with short gap) Experiment set up in a 3 3 Latin Square design for grazing time. Each of the nine whole plots split using a criss-cross design for the two grazing cycle factors. STAT 514 Topic 21 29

30 Data and Layout S SP SP SP 18 S 9 S S S S SP 9 SP 3 SP S S SP SP 3 SP 18 S STAT 514 Topic 21 30

31 data new; input row column time sp sum resp; cards; *** time*row*column serves as WP error *** Have three diff subplot errors ; proc mixed plots=all nobound; class row column time sp sum; model resp= time sp sum; random row column time*row*column time*row*column*sp time*row*column*sum; lsmeans sum; lsmeans sp*time / adjust=tukey; run; STAT 514 Topic 21 31

32 Cov Parm Estimate row **Was a Latin Square column really necessary? row*column*time row*column*time*sp row*column*time*sum Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F time sp time*sp sum time*sum sp*sum time*sp*sum STAT 514 Topic 21 32

33 Least Squares Means Standard Effect time sp sum Estimate Error DF t Value Pr > t sum <.0001 sum <.0001 time*sp <.0001 time*sp time*sp <.0001 time*sp time*sp time*sp STAT 514 Topic 21 33

34 STAT 514 Topic 21 34

35 Conclusions Significant main effect for summer grazing cycle. Larger percent of principal grass when one uses 4 cycles with a short gap Significant interaction between spring grazing and length of time grazing As the length increases, the difference between the 2 and 4 cycles decreases. In all cases, the larger percent occurs when 2 cycles are used with a long gap STAT 514 Topic 21 35

36 Background Reading The split-plot design : Montgomery Section 14.4 Split-plot design with multiple trt factors : Montgomery Section Split-split plot design : Montgomery Section Strip split-plot design : Montgomery Section STAT 514 Topic 21 36