Split-Plot Design: Statistical Approach for Analyzing Experiments with Factors, Exams of Design

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Split-plot Designs

Bruce A Craig

Department of Statistics Purdue University

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.

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

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

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 rbφA + bσ R^2 + σ^2

Rep(A) a(r − 1) bσ R^2 + σ^2

B b − 1 ar φB + σ^2

AB (a − 1)(b − 1) r φAB + σ^2

Error a(b − 1)(r − 1) σ^2

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 abσ R^2 + (bσ^2 RA) + σ^2

A a − 1 rbφA + bσ^2 RA + σ^2

Blk*A (a − 1)(r − 1) bσ^2 RA + σ^2

B b − 1 ar φB + σ^2

AB (a − 1)(b − 1) r φAB + σ^2

Error a(b − 1)(r − 1) σ^2

Sometimes blocking interactions not pooled (Page 622)

Soybean Yields - Data and Layout

Farm

Fertilizer Fertilizer Fertilizer

Variety 1 2 Variety 2 1 Variety 1 2

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 farmfert var fertvar; random farm farm*fert / test;

proc glm plots=all; No pooling; class farm fert var; model resp=farm fert farmfert var farmvar fertvar; random farm farmfert farm*var / test;

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 <. Error 8 0.22666667 0. Cor Total 17 35.

Source DF Type III SS Mean Square F Value Pr > F farm 2 28.86333333 14.43166667 509.35 <. fert 1 0.84500000 0.84500000 29.82 0. farmfert 2 0.04333333 0.02166667 0.76 0.4967* var 2 5.34333333 2.67166667 94.29 <.0001* fertvar 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

Source DF Type III SS Mean Square F Value Pr > F farm 2 28.863333 14.431667 666.08 0. fert 1 0.845000 0.845000 39.00 0. MS(farm*fert) 2 0.043333 0.

farmfert 2 0.043333 0.021667 0.76 0. var 2 5.343333 2.671667 94.29 <. fertvar 2 0.003333 0.001667 0.06 0. MS(Error) 8 0.226667 0.

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. MS(farm*fert) 2 0.043333 0.

farmfert 2 0.043333 0.021667 0.65 0. farmvar 4 0.093333 0.023333 0.70 0. fert*var 2 0.003333 0.001667 0.05 0. MS(Error) 4 0.133333 0.

var 2 5.343333 2.671667 114.50 0. MS(farm*var) 4 0.093333 0.

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. Error 2 0.01444444 0. Cor Total 5 9.

Source DF Type III SS Mean Square F Value Pr > F farm 2 9.62111111 4.81055556 666.08 0. fert 1 0.28166667 0.28166667 39.00 0.

**** Same results *****

Using Proc Mixed

proc mixed plots=all; no pooling; class fert var farm; model resp=fert|var / ddfm=kr; random farm farmfert farmvar; Cov Parm Estimate FARM 2. FERTFARM 0. VARFARM 0. Residual 0.

Tests of Fixed Effects Source NDF DDF Type III F Pr > F FERT 1 10 31.30 0. VAR 2 10 98.95 <. FERT*VAR 2 10 0.06 0.

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; *pooling; class fert var farm; model resp=fert|var / ddfm=kr; random farm farmfert;

Cov Parm Estimate FARM 2. FERT*FARM 0. Residual 0.

Tests of Fixed Effects Source NDF DDF Type III F Pr > F FERT 1 10 31.30 0. VAR 2 10 98.95 <. FERT*VAR 2 10 0.06 0.

ddfm=kr is causing pooling of WP and SP errors Need to remove ddfm=kr or use the nobound option