Split-plotDesignsbacraig/notes514/topic21a.pdf · Whole plot: Batch of four cookies Subplot : Individual cookies The whole plots are always divided into smaller entities called subplots

Split-plot Designs

Bruce A Craig

Department of StatisticsPurdue University

STAT 514 Topic 21 1

Randomization Defines the Design

Want to study the effect of oven temp (3 levels) andamount of baking soda (4 levels) on the consistency of a6-inch chocolate chip cookie.

[Design 1] Factorial: Each of the 12 combinations of temp andbaking soda is replicated three times. You mix up cookie doughand then cook it 36 times.

[Design 2] Split plot: Four batches of dough are created, eachwith a different amount of baking soda. Oven is heated to specifictemp and the four doughs are put in the oven at the same time.Replicate this process three times at each oven temp. This meanswe make 36 batches of dough but only run 9 cooking trials.

STAT 514 Topic 21 2

Randomization Defines the Design

Design 2 is different from Design 1 because of a randomizationrestriction. Instead of randomly assigning temp to each batchof dough, it is instead randomly assigned to a group of fourbatches. In other words, the experimental unit of each factoris 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 toimplement. For this experiment, nine cooking trials is far moremanageable than 36 cooking trials.

STAT 514 Topic 21 3

Split-plot Design

Arose in agriculture

Whole plot - Large fieldSubplot - Smaller sections of field

- Want to study 4 fertilizers and 6 corn varieties- Spreader covers 15 foot wide section and planter covers5 foot wide section

- Spread fertilizer on 15x10 foot section (whole plot)- Plant seed in 5x5 foot sections (subplot) for a total of 6subplots per whole plot

Very useful in other areas (done out of convenience)

Engineering - certain settings fixed for a group of runsRepeated measures - subject “split” into time sections

STAT 514 Topic 21 4

Split-Plot Design

For the cookie study, the experimental unit for oven temp is thegroup (or sheet) of four cookies. Since the four cookies within asheet are randomly assigned amounts of baking soda, theexperimental unit for baking soda is still the individual batch ofdough.The larger experimental unit (cookie sheet) is divided or split intosmaller experimental units (cookies).

Whole plot: Batch of four cookies

Subplot : Individual cookies

The whole plots are always divided into smaller entities calledsubplots. The key for proper analysis is determining the whole plotand subplot factors and their experimental units

STAT 514 Topic 21 5

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 subplotb) 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 thefactorial design, if feasible, is overall more powerful

STAT 514 Topic 21 6

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 ExpectedVariation Freedom Mean Square

A a− 1 rbφA + bσ2R+ σ2

Rep(A) a(r − 1) bσ2R+ σ2

B b − 1 arφB + σ2

AB (a− 1)(b − 1) rφAB + σ2

Error a(b − 1)(r − 1) σ2

STAT 514 Topic 21 7

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


Blk r − 1 abσ2R+ (bσ2

RA) + σ2

A a− 1 rbφA + bσ2RA

+ σ2

Blk*A (a− 1)(r − 1) bσ2RA

+ σ2

B b − 1 arφB + σ2

AB (a− 1)(b − 1) rφAB + σ2

Error a(b − 1)(r − 1) σ2

Sometimes blocking interactions not pooled (Page 622)

STAT 514 Topic 21 8

Example: Soybean Yields

Interested in the effect of soybean varieties and fertilizers onthe yield (bushels per subplot unit). Fertilizers were randomlyapplied to acres within each farm, varieties then randomlyapplied to subunits of each acre. Consider fertilizers andvarieties as fixed. Farm, as a block, is considered random.Whole plot testing similar if block random or fixed factors. Insubplot, if block fixed, all interactions with block are pooledinto error. If it is random, this may or may not be done. If it isnot done, there are other tests that may be of interest (seepage 622).

STAT 514 Topic 21 9

Soybean Yields - Data and Layout

Farm1 2 3

Fertilizer Fertilizer FertilizerVariety 1 2 Variety 2 1 Variety 1 2

1 10.6 10.9 2 11.9 11.5 3 9.5 9.82 11.4 11.7 3 12.6 12.1 1 8.1 8.23 11.8 12.4 1 11.6 10.8 2 8.7 9.3

STAT 514 Topic 21 10

SAS Programs

data new; infile "soy.dat";

input farm fert var resp;

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;


model resp=farm fert farm*fert var farm*var fert*var;

random farm farm*fert farm*var / test;


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;


SAS Output - Pooled SP Interactions

Dependent Variable: resp

Sum of

Source DF Squares Mean Square F Value Pr > F

Model 9 35.09833333 3.89981481 137.64 <.0001

Error 8 0.22666667 0.02833333

Cor Total 17 35.32500000

Source DF Type III SS Mean Square F Value Pr > F

farm 2 28.86333333 14.43166667 509.35 <.0001

fert 1 0.84500000 0.84500000 29.82 0.0006

farm*fert 2 0.04333333 0.02166667 0.76 0.4967**

var 2 5.34333333 2.67166667 94.29 <.0001*

fert*var 2 0.00333333 0.00166667 0.06 0.9433*

*Correct F-test

**Necessary to keep in model to maintain SP structure


SAS Output - Pooled SP Interactions

Tests of Hypotheses for Mixed Model Analysis of Variance


farm 2 28.863333 14.431667 666.08 0.0015

fert 1 0.845000 0.845000 39.00 0.0247

MS(farm*fert) 2 0.043333 0.021667

farm*fert 2 0.043333 0.021667 0.76 0.4967

var 2 5.343333 2.671667 94.29 <.0001

fert*var 2 0.003333 0.001667 0.06 0.9433

MS(Error) 8 0.226667 0.028333


SAS Output - SP Interactions

Dependent Variable: resp

Sum of


Model 13 35.19166667 2.70705128 81.21 0.0003

Error 4 0.13333333 0.03333333

Cor Total 17 35.32500000


farm 2 28.86333333 14.43166667 432.95 <.0001

fert 1 0.84500000 0.84500000 25.35 0.0073

farm*fert 2 0.04333333 0.02166667 0.65 0.5696**

var 2 5.34333333 2.67166667 80.15 0.0006

farm*var 4 0.09333333 0.02333333 0.70 0.6310*

fert*var 2 0.00333333 0.00166667 0.05 0.9518*

*Correct F-test

**Necessary to keep in model to maintain SP structure


SAS Output - SP Interactions

Tests of Hypotheses for Mixed Model Analysis of Variance


fert 1 0.845000 0.845000 39.00 0.0247

MS(farm*fert) 2 0.043333 0.021667

farm*fert 2 0.043333 0.021667 0.65 0.5696

farm*var 4 0.093333 0.023333 0.70 0.6310

fert*var 2 0.003333 0.001667 0.05 0.9518

MS(Error) 4 0.133333 0.033333

var 2 5.343333 2.671667 114.50 0.0003

MS(farm*var) 4 0.093333 0.023333


SAS Output - WP Analysis Only

Sum of


Model 3 9.90277778 3.30092593 457.05 0.0022

Error 2 0.01444444 0.00722222

Cor Total 5 9.91722222


farm 2 9.62111111 4.81055556 666.08 0.0015

fert 1 0.28166667 0.28166667 39.00 0.0247

**** Same results *****


SAS Programs

data new; infile "soy.dat";

input farm fert var resp;

proc mixed plots=all; **Pooling;


model resp= fert var fert*var / ddfm=kr;

random farm farm*fert;

proc mixed plots=all; **No pooling;


model resp=fert var fert*var / ddfm=kr;

random farm farm*fert farm*var;


Using Proc Mixedproc 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 2.40077740

FERT*FARM 0.00000000

VAR*FARM 0.00000000

Residual 0.02700000

Tests of Fixed Effects

Source NDF DDF Type III F Pr > F

FERT 1 10 31.30 0.0002

VAR 2 10 98.95 <.0001

FERT*VAR 2 10 0.06 0.9405

***ddfm=kr is causing pooling of WP and SP errors***

***Need to remove ddfm=kr or use the nobound option***


Using Proc Mixedproc mixed plots=all; **pooling;




Cov Parm Estimate

FARM 2.40077740

FERT*FARM 0.00000000

Residual 0.02700000



FERT 1 10 31.30 0.0002

VAR 2 10 98.95 <.0001

FERT*VAR 2 10 0.06 0.9405

***ddfm=kr is causing pooling of WP and SP errors***

***Need to remove ddfm=kr or use the nobound option***


Using Proc Mixed

proc mixed plots=all nobound;




Cov Parm Estimate

FARM 2.40166

FERT*FARM -0.00222

Residual 0.02833



FERT 1 2 39.00 0.0247

VAR 2 8 94.29 <.0001

FERT*VAR 2 8 0.06 0.9433

***Results same as GLM pooled SP interactions***


Whole Plot/Subplot Experiments

Can have more than one factor in whole plot or subplot

Common whole plot designs

CRDRCBDFactorial (k factors)BIB

Subplot

RCBDBIBBlocked Factorial Design

Analysis of Covariance

Covariate linear with response in subplot and whole plot


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


A a− 1 bcφA + cσ2WP

+ σ2

B b − 1 acφB + cσ2WP

+ σ2

AB (a− 1)(b − 1) cφAB + cσ2WP

+ σ2

Rep1(AB) 0 cσ2WP

+ σ2

C c − 1 abφC + σ2

AC (a− 1)(c − 1) bφAC + σ2

BC (b − 1)(c − 1) aφ2BC

+ σ2

ABC (a− 1)(b − 1)(c − 1) σ2ABC

+ σ2

Rep(ABC) 0 σ2


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 factorIf σ2

WP> σ2

SP, understate subplot factor

Should pool errors separately

Need to maintain the design structure


Example: Pooling in Split Plot

Consider A (fixed) and B (random) in whole plot, C fixedfactor in subplot. Pool B, AB with Rep(AB) and pool BC,ABC with error. Other combinations alter design.


A a− 1 bcnφA + ncσ2AB

+ cσ2WP

+ σ2

B b − 1 acnσ2B+ cσ2

WP+ σ2

AB (a− 1)(b − 1) cnσ2AB

+ cσ2WP

+ σ2

Rep(AB) ab(n − 1) cσ2WP

+ σ2

C c − 1 abnφC + anσ2BC

+ σ2

AC (a− 1)(c − 1) bnφAC + nσ2ABC

+ σ2

BC (b − 1)(c − 1) anσ2BC

+ σ2

ABC (a− 1)(b − 1)(c − 1) nσ2ABC

+ σ2

Error ab(c − 1)(n − 1) σ2


Extensions of Split-Plot DesignCan further split subplot units into sub-subplotsKnown as Split-Split Plot Design

CRD with 2 RCBDsThree RCBDsSource of Degrees of ExpectedVariation Freedom Mean Square

Blk r − 1 abcσ2R+ σ2

A a− 1 bcrφA + bcσ2AR

+ σ2

Blk*A (a− 1)(r − 1) bcσ2AR

+ σ2

B b − 1 acrφB + acσ2BR

+ σ2

Blk*B (b − 1)(r − 1) acσ2BR

+ σ2

AB (a− 1)(b − 1) crφAB + cσ2ABR

+ σ2

Blk*AB (a− 1)(b − 1)(r − 1) cσ2ABR

+ σ2

C c − 1 abrφC + abσ2CR

+ σ2

Blk*C (c − 1)(r − 1) abσ2CR

+ σ2

AC (a− 1)(c − 1) brφAC + bσ2ACR

+ σ2

Blk*AC (a− 1)(c − 1)(r − 1) bσ2ACR

+ σ2

BC (b − 1)(c − 1) arφBC + aσ2BCR

+ σ2

Blk*BC (b − 1)(c − 1)(r − 1) aσ2BCR

+ σ2

ABC (a− 1)(b − 1)(c − 1) rφABC + σ2ABCR

+ σ2

Blk*ABC (a− 1)(b − 1)(c − 1)(r − 1) σ2ABCR

+ σ2


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

RowsColumnsRow*Column (cell)


Strip Plot ANOVA table


Blk r − 1 abσ2R+ σ2

A a− 1 brφA + bσ2AR

+ σ2

Blk*A (a− 1)(r − 1) bσ2AR

+ σ2

B b − 1 arφB + aσ2BR

+ σ2

Blk*B (b − 1)(r − 1) aσ2BR

+ σ2

AB (a − 1)(b − 1) rφAB + σ2ABR

+ σ2

Blk*AB (a − 1)(b − 1)(r − 1) σ2ABR

+ σ2

Blk*AB would be the error term in most analyses


Example of Strip Plot / Split Plot

Investigating the long term effects of pasture composition fordifferent patterns of grazing. Response is the percent of areacovered 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 thetwo grazing cycle factors.


Data and Layout

S SP SP2 4 2 4 2 4

4 12.5 26.2 4 59.2 49.9 4 55.0 27.3SP 18 S 9 S 32 33.4 44.2 2 47.6 15.8 2 35.9 18.3

S S S4 2 2 4 2 4

2 56.2 52.3 2 67.7 62.2 2 28.0 29.4SP 9 SP 3 SP 184 27.5 25.1 4 24.1 27.5 4 19.5 29.9

S S SP2 4 2 4 2 4

2 57.2 69.5 2 30.3 26.6 4 61.9 26.2SP 3 SP 18 S 94 16.9 19.5 4 11.0 17.6 2 46.5 15.4


data new;

input row column time sp sum resp;

cards;

1 1 18 4 2 12.5

1 1 18 4 4 26.2

1 1 18 2 2 33.4

. *** time*row*column serves as WP error

3 3 9 2 2 46.5 *** Have three diff subplot errors

3 3 9 4 2 15.4

;

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;


Cov Parm Estimate

row -4.4646 **Was a Latin Square

column -3.8987 really necessary?

row*column*time -1.0209

row*column*time*sp 25.1028

row*column*time*sum 15.9039

Residual 29.4556

Type 3 Tests of Fixed Effects

Num Den

Effect DF DF F Value Pr > F

time 2 2 7.81 0.1135

sp 1 6 71.52 0.0001

time*sp 2 6 5.16 0.0497

sum 1 6 11.36 0.0150

time*sum 2 6 0.66 0.5503

sp*sum 1 6 0.72 0.4276

time*sp*sum 2 6 0.88 0.4609


Least Squares Means

Standard

Effect time sp sum Estimate Error DF t Value Pr > |t|

sum 2 30.9722 1.3773 6 22.49 <.0001

sum 4 39.7667 1.3773 6 28.87 <.0001

time*sp 3 2 57.9167 3.5776 6 16.19 <.0001

time*sp 3 4 22.2667 3.5776 6 6.22 0.0008

time*sp 9 2 53.9500 3.5776 6 15.08 <.0001

time*sp 9 4 26.6500 3.5776 6 7.45 0.0003

time*sp 18 2 31.9833 3.5776 6 8.94 0.0001

time*sp 18 4 19.4500 3.5776 6 5.44 0.0016



Conclusions

Significant main effect for summer grazing cycle. Largerpercent of principal grass when one uses 4 cycles with ashort gap

Significant interaction between spring grazing and lengthof time grazing

As the length increases, the difference between the 2 and4 cycles decreases. In all cases, the larger percent occurswhen 2 cycles are used with a long gap


Background Reading

The split-plot design : Montgomery Section 14.4

Split-plot design with multiple trt factors : MontgomerySection 14.5.1

Split-split plot design : Montgomery Section 14.5.2

Strip split-plot design : Montgomery Section 14.5.3


Documents

Split-plotDesignsbacraig/notes514/topic21a.pdf · Whole plot: Batch of four cookies Subplot : Individual cookies The whole plots are always divided into smaller entities called subplots