Split-plotDesignsbacraig/notes514/topic21a.pdf · Whole plot: Batch of four cookies Subplot :...
Transcript of Split-plotDesignsbacraig/notes514/topic21a.pdf · Whole plot: Batch of four cookies Subplot :...
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
Source of Degrees of ExpectedVariation Freedom Mean Square
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;
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
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
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
STAT 514 Topic 21 13
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 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
STAT 514 Topic 21 14
SAS Output - SP Interactions
Dependent Variable: resp
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 13 35.19166667 2.70705128 81.21 0.0003
Error 4 0.13333333 0.03333333
Cor Total 17 35.32500000
Source DF Type III SS Mean Square F Value Pr > F
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
STAT 514 Topic 21 15
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 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
STAT 514 Topic 21 16
SAS Output - WP Analysis Only
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 3 9.90277778 3.30092593 457.05 0.0022
Error 2 0.01444444 0.00722222
Cor Total 5 9.91722222
Source DF Type III SS Mean Square F Value Pr > F
farm 2 9.62111111 4.81055556 666.08 0.0015
fert 1 0.28166667 0.28166667 39.00 0.0247
**** Same results *****
STAT 514 Topic 21 17
SAS Programs
data new; infile "soy.dat";
input farm fert var resp;
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
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***
STAT 514 Topic 21 19
Using Proc Mixedproc mixed plots=all; **pooling;
class fert var farm;
model resp=fert|var / ddfm=kr;
random farm farm*fert;
Cov Parm Estimate
FARM 2.40077740
FERT*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***
STAT 514 Topic 21 20
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 2.40166
FERT*FARM -0.00222
Residual 0.02833
Tests of Fixed Effects
Source NDF DDF Type III F Pr > F
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***
STAT 514 Topic 21 21
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
STAT 514 Topic 21 22
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 ExpectedVariation Freedom Mean Square
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
STAT 514 Topic 21 23
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
STAT 514 Topic 21 24
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.
Source of Degrees of ExpectedVariation Freedom Mean Square
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
STAT 514 Topic 21 25
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
STAT 514 Topic 21 26
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)
STAT 514 Topic 21 27
Strip Plot ANOVA table
Source of Degrees of ExpectedVariation Freedom Mean Square
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
STAT 514 Topic 21 28
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.
STAT 514 Topic 21 29
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
STAT 514 Topic 21 30
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;
STAT 514 Topic 21 31
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
STAT 514 Topic 21 32
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
STAT 514 Topic 21 33
STAT 514 Topic 21 34
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
STAT 514 Topic 21 35
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
STAT 514 Topic 21 36