This article was downloaded by: [University of York]On: 02 December 2014, At: 06:54Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Quality EngineeringPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lqen20

Split-Split-Plot Experimental Design in a High-Throughput ReactorFlor Castillo aa The Dow Chemical Company , Freeport, TexasPublished online: 15 Sep 2010.

To cite this article: Flor Castillo (2010) Split-Split-Plot Experimental Design in a High-Throughput Reactor, QualityEngineering, 22:4, 328-335, DOI: 10.1080/08982112.2010.495101

To link to this article: http://dx.doi.org/10.1080/08982112.2010.495101

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Split-Split-Plot Experimental Designin a High-Throughput Reactor

Flor Castillo

The Dow Chemical Company,

Freeport, Texas

ABSTRACT In the last few years, high-throughput (HT) reactors

have received significant attention due to the potential for fast material

development. Split-plot experimental design plays a critical role in this

type of application given randomization restrictions often imposed by

equipment constraints. A case study in a parallel polymerization reactor is

presented.

KEYWORDS combinatorial experimentation, high-throughput experimental

design, randomization restrictions, split-split-plot

INTRODUCTION

Experimental design techniques have played a critical role in industry

from product development to process optimization. Of particular impor-

tance are the strip and split-plot designs because of the restrictions on

randomization frequently encountered in industrial experimentation

where limitations associated with equipment, material availability, time,

and cost often preclude the execution of a completely randomized

design. Starting from the late 1990s split-plot experiments have received

a significant increase in research interest. Many authors have contributed

to the development of this field, increasing awareness of this rather com-

mon situation (Bingham and Sitter 2001, 2003; Bisgaard 2000; Bisgaard

and Steinberg 1997; Goos and Vandebroek 2001, 2003, 2004; Kowalski

2002; Kowalski et al. 2007; Kulahci 2007; Vining and Kowalski 2008;

Vining et al. 2005).

Applications of strip and split-plot designs in industrial settings have

been recently presented in the literature (Box et al. 2005; Montgomery

2008; Paniagua-Quinones and Box 2008; Vivacqua and Bisgaard 2009).

However, the number of applications is relatively small considering that

in industrial experimentation, restrictions on randomization are the rule

rather than the exception. In this article we present a split-split-plot

situation in a high-throughput parallel polymerization reactor (PPR): an

array of several parallel miniature reactors capable of completing a

number of experiments at once.Address correspondence to FlorCastillo, 2301 N. Brazosport Blvd.,B1605, Freeport, TX 77541. E-mail:Facastillo@dow.com

Quality Engineering, 22:328–335, 2010Copyright # Taylor & Francis Group, LLCISSN: 0898-2112 print=1532-4222 onlineDOI: 10.1080/08982112.2010.495101

328

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

The restrictions on randomization are due to

equipment configuration in which microreactors

or wells1 are organized in columns or modules2

and variables describing the synthesis components

are easy to change from well to well, but process

variables such as pressure or temperature are difficult

to change from well to well but easy to change

from module to module. Split-plot designs in high-

throughput systems tend to be large experiments

because they are likely to include two or more whole-

plot and subplot factors. In addition, multiple replica-

tions of the whole plots and subplots are possible

given the speed of the reaction and the capabilities

of the equipment.

THE STUDY

The high-throughput reactor used in this study is

a PPR-483 manufactured by Symyx Technologies. A

diagram of the reactor is shown in Figure 1. It is laid

out as six modules (columns 1–6) housing eight

wells (chambers) each. Each reactor module is

essentially a contiguous steel block into which the

wells, each approximately 8.5 cm deep and 1.7 cm

wide, are machined. Each reactor well is heated by

means of two cylindrical heater elements, which

are aligned with the bore of the well. Individual wells

are only partially thermally decoupled from their

neighbors by approximately 4-mm-wide slits that

are machined into the reactor block. Process con-

ditions such as pressure (P) and temperature (T)

can be kept constant throughout the module but can-

not be varied from well to well within the module.

Synthesis factors, such as chemical composition,

can be individually changed across all wells, because

all wells have individual compound injector ports.

Liquid reactant injection is done by two robot

injectors. Robot 1 injects modules 1 to 3, and robot

2 injects modules 4 to 6. Both robots operate

simultaneously and independently of each other.

The process variables were two levels of tempera-

ture (T) and two levels of pressure (P) and were

treated as the difficult to change factors allocated to

columns in the reactor. The synthesis variables were

two types of catalyst (C) and two levels of catalyst

concentration (D). These were treated as the

easy-to-change factors. They were allocated to the

wells or chambers inside each module using two

replicates because there were eight wells in each

module. Finally, only four out of the six modules

were available for the experiment. The final experi-

mental setup is seen in Figure 1.

The four modules of the reactor are capable of

making 32 runs per library (block) and four libraries

were planned. The wells allocated to the different

combinations of catalyst type (C) and catalyst concen-

tration (D) were randomized for the different libraries.

For confidentiality reasons the response variable

is referred to as Y. The data are shown in Table 1.4

Note from Table 1 that the design also blocks the

effect of the robots so that no single level of tempera-

ture (T) or pressure (P) is associated with the same

robot. The design was initially planed as a 22� 22

split-plot design with (P) and (T) as the whole plot

factors, (C) and (D) as the subplot factors (with

two replicates at the subplot level), and four replica-

tions of the whole design (blocks). Unfortunately, it

was not run as such, because the experimenter made

some changes at the time of the execution, running

each block with a different purge: type of chemical

used to terminate the reaction, thereby introducing

an additional four-level fix variable (type of purge)

and inadvertently executing the experiments as a

split-split-plot design: Once a particular purge is

selected (whole plot), the combinations of tempera-

ture and pressure were set in the different modules

(subplots), and the levels of catalyst type and cata-

lysts concentration (sub-subplot) were randomly

allocated to the different wells within the modules.

Situations like this are frequently encountered in

industrial experimentation. The experimenter, often

unaware of the potential implications, modifies the

execution of the experiments in response to limita-

tions of time, cost, or materials supply. Indeed, this

is frequently what generates split-plot in industrial

situations.

1Here, the designation well is used for individual reaction orsynthesis chambers that are chemically isolated from other cham-bers and form the basic experimental unit of high-throughputexperimentation.2A module is a block of wells tied together by engineeringconstraints, such as a common pressure control unit or a commonheater block.

3PPR is a registered trademark of Symyx Technologies, Santa Clara,California.

4T¼ temperature, P¼pressure, C¼ catalyst type, D¼ catalystconcentration.

329 Split-Split-Plot

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

Adjustments to prescribed designs happen

more often than consultants would think and hope.

Sometimes the experimenter can change a perfectly

good design so much that analysis of the data

becomes quite challenging if not impossible.

Thus, it is critical for consultants to verify and

monitor the experimental runs to determine the

way the experiments were actually executed so that

the proper analysis can be carried out.

Considering the changes introduced in the parallel

polymerization reactor (PPR) example, there are two

randomization restrictions. The first one is the purge

(block), and the second one is temperature (T) and

pressure (P). These restrictions generate a split-split-

plot design with no replications at the whole plot

level. The analysis of this design requires consider-

ation of whole-plots, split-plot, and split-split-plot

random errors (three error terms).

FIGURE 1 Experimental setup.

TABLE 1 Data for the PPR Case Study

C �1 �1 1 1 �1 �1 1 1

D �1 1 �1 1 �1 1 �1 1

Block Robot T P

1 1 �1 �1 25.3 43.3 10.1 26.1 35.6 46.2 16.3 25.0

1 1 1 1 22.9 60.1 25.6 37.7 22.1 54.2 26.4 47.8

1 2 �1 1 49.3 45.9 15.3 41.2 18.8 43.8 16.8 33.1

1 2 1 �1 37.9 63.8 18.2 55.9 30.7 52.4 27 42.0

2 1 �1 �1 25.9 44.1 11.4 28.8 31.4 48.4 18.3 28.5

2 1 1 1 22.4 58.2 25.9 38.8 22.9 54.2 22 43.7

2 2 �1 1 20.3 51.1 16.2 38.5 22.9 48.8 16.6 35.4

2 2 1 �1 36.5 63.9 21.7 50.4 31.3 54.3 28.8 38.3

3 2 �1 �1 26.7 42.1 7.5 26.4 31.6 47.9 9.8 25.6

3 2 1 1 21.1 55.6 23.8 37.1 20.6 52.4 21.4 43.7

3 1 �1 1 21.2 51.6 17.4 37.4 21.2 47.3 16.5 35.1

3 1 1 �1 32.8 62.9 17.9 46.2 29.6 55.1 25.8 38.5

4 2 �1 �1 23.8 42.3 6.4 25.2 28.5 45.8 10.2 22.3

4 2 1 1 22.9 51.1 24.1 37.5 18.8 50.1 21.8 43.2

4 1 �1 1 16.9 50.7 17.2 37.4 18.6 47.2 16.2 34.3

4 1 1 �1 35.1 63.3 16.9 46.9 28.7 56.6 27.2 37.1

F. Castillo 330

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

The following is the general formof themodel taking

into account that, given the changes, it is important to

know the effect of the purge as well as the interactions.

Yijkr ¼ lþ Rr þWk þ hkr þ Sj þ ðWSÞkjþ ejkr þ SSi þ ðWSSÞki þ ðSxSSÞji þ eijkr

where Yijkr is the response of the rth replicate of the kth

level of the whole-plot factor (W) and the jth level of

the subplot factor (S) and the ith level of the

sub-subplot factor (SS); l is the overall mean; Rr is

the random effect of the rth replicate, with Rr �Nð0; r2r Þ; Wk is the fixed effect of the kth level of (W);

(Sj) is the fixed effect of the jth level of (S); (WS)kj is

the interaction effect of the kth level of (W) with the

jth level of the subplot factor (S); (SS) is the fixed effect

of the ith level of the sub-subplot factor (SS); (WSS)ki is

the interaction effect of the kth level of (W) with the ith

level of the subplot factor (SS); (SxSS)ji is the interac-

tion effect of the jth level of (S) with the ith level of

the sub-subplot factor (SS); Hkr is the whole plot error;

ejkr is the subplot error; eijkr is the sub-subplot error;

Hkr � Nð0;r2wÞ;ejkr � Nð0;r2s Þ eijkr � Nð0;r2e Þ; and

Hkr, ejkr, and eijkr are independent.

The analysis of variance (ANOVA) table is given in

Table 2.5

Experiments with more than two factors in the sub-

plot and more than two factors in the sub-subplot are

large experiments. In this particular application, it was

important to detect third-order interactions between

the process factors and the synthesis factors because

of their significance in kinetics reactionmodeling. Thus,

we assume that fourth- and higher order interactions

between the whole plot, subplot, and sub-subplot

factors are negligible. These interactions are pooled

with r2e to form the sub-subplot error (Table 2).

Given that there are no replicates at the

whole-plot level, there is no exact test for the effect

of the purge (B). Also, the test for the subplot factors

and interactions between subplot and whole-plot

factors cannot be obtained without the assumption

that the interaction (BPT) was negligible. The BPT

interaction will be used as the subplot error.

The subplot main effects (T), (P) and interactions

among subplot factors and between the whole-plot

(B) and subplot factors are tested against the subplot

error, whereas the sub-subplot factors (C) and (D)

and interactions involving sub-subplot factors are

tested against the sub-subplot error.

Table 3 shows the results. The computations were

performed using SYSTAT (SPSS Inc., Chicago, IL).

At a¼ 5% significance (P), (PT), (C), (D), (CD),

(PC), (PD), (TC), and (TD) are significant effects.

The interactions with the purge (B) turned out

not to be significant. This was an important result,

indicating that the different purges did not interfere

with any of the process or synthesis variables.

TABLE 2 ANOVA Table Split-Split-Plot Design

df Term

Expected

mean squares

Whole plot 0 Replicates (R) r2e þ 128r2

R

3 Block (B)—purge r2e þ 32r2

W þ 32ø2ðBÞ

0 RB r2e þ 32r2

W

Subplot 1 T r2e þ r2

S þ 64ø2ðTÞ

1 P r2e þ r2

S þ 64ø2ðPÞ

1 TP r2e þ r2

S þ 32ø2ðPTÞ

3 BT r2e þ r2

S þ 16ø2ðBTÞ

3 BP r2e þ r2

S þ 16ø2ðBPÞ

3 BPT r2e þ r2

S þ 8ø2ðBPTÞ

0 R� Subplot-

interactions

r2e þ r2

S

Sub-subplot 1 Subplot

replicates (F)

r2e þ 64r2

F

1 C r2e þ 64ø2

ðCÞ1 D r2

e þ 64ø2ðDÞ

1 CD r2e þ 32ø2

ðCDÞ1 TC r2

e þ 32ø2ðTCÞ

1 TD r2e þ 32ø2

ðTDÞ1 PC r2

e þ 32ø2ðPCÞ

1 PD r2e þ 32ø2

ðPDÞ3 BC r2

e þ 16ø2ðBCÞ

3 BD r2e þ 16ø2

ðBDÞ3 BTD r2

e þ 8ø2ðBDÞ

3 BTC r2e þ 8ø2

ðBTCÞ3 BPC r2

e þ 8ø2ðBPCÞ

3 BCD r2e þ 8ø2

ðBCDÞ3 BPD r2

e þ 8ø2ðBPDÞ

1 TCD r2e þ 16ø2

ðTCDÞ1 PCD r2

e þ 16ø2ðPCDÞ

1 TPC r2e þ 16ø2

ðTPCÞ1 TPD r2

e þ 16ø2ðTPDÞ

79 Error higher order

interaction with

sub-subplot

factors

r2e (Sub-subplot

error)

5Ø2 is used to represent a fixed effect that is the sum squares of themodel component associated with the particular factor divided byits degrees of freedom: ø2ðCÞ is the fixed effect for catalyst type (C).

331 Split-Split-Plot

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

Based on the analysis, two significant third-order

interactions important to the kinetic modelers were

identified: (PCD) and (TCD).

Of particular interest were the interactions between

subplot and sub-subplot factors. Figure 2 shows the

interaction between pressure (P) and the sub-subplot

factors catalyst type (C) and concentration (D).

Increasing the pressure increases the response (Y)

for catalyst 1 and catalyst 2. However, overall

catalyst 1 has higher response than catalyst 2. At

the same time, increasing the pressure increases the

response for both levels of concentration; however,

the higher concentration has a higher response.

Figure 3 shows the interaction between tempera-

ture (T) and the sub-subplot factors catalyst type

(C) and concentration (D).

Increasing temperature (T) increases the response

when catalyst 2 is used but has an opposite effect

when catalyst 1 is used instead. However, in general,

catalyst 1 has a higher response than catalyst 2. FIGURE 2 Pressure sub-subplot interactions.

TABLE 3 ANOVA Table Results

Source df Sum of squares Mean square F ratio p-Value

Whole Plot Replicates (R) 0

Block (B) 3 169.4 56.5

Subplot Pressure (P) 1 2102.5 2102.5 721.79 0.0000

Temperature (T) 1 6.1 6.1 2.09 0.2440

BP 3 6.2 2.1 0.71 0.6090

BT 3 12.1 4.0 1.39 0.3970

PT 1 526.6 526.6 180.77 0.0010

BPT (Subplot error) 3 8.7 2.9

Sub-subplot Sub-subplot Replicates (F) 1 36.7 36.7 1.75 0.1902

C 1 4166.8 4166.8 198.25 0.0000

D 1 15000.9 15000.9 713.71 0.0000

C�D 1 360.3 360.3 17.14 0.0001

B�C 3 4.9 1.6 0.08 0.9720

B�D 3 30.9 10.3 0.49 0.6904

P�C 1 198.7 198.7 9.46 0.0029

P�D 1 247.0 247.0 11.75 0.0010

T�C 1 488.9 488.9 23.26 0.0000

T�D 1 131.8 131.8 6.27 0.0143

B�P�C 3 20.2 6.7 0.32 0.8106

B�P�D 3 73.6 24.5 1.17 0.3275

B�T�C 3 42.7 14.2 0.68 0.5681

B�T�D 3 22.5 7.5 0.36 0.7848

B�C�D 3 67.2 22.4 1.07 0.3685

P�T�C 1 9.7 9.7 0.46 0.4987

P�T�D 1 66.2 66.2 3.15 0.0799

P�C�D 1 92.5 92.5 4.40 0.0391

T�C�D 1 88.9 88.9 4.23 0.0430

sub-subplot error 79 1660.4 21.0

Total 127 25642.5 .

F. Castillo 332

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

Likewise, increasing temperature (T) increases the

response at the higher catalyst concentration but has

the opposite effect at lower catalyst concentrations.

In general, higher concentration produces higher

response than lower concentration.

Split-Plot Situation

Had the experiments been executed as initially

prescribed, that is, with the block variable as the rep-

lication not associated with the purge, the situation

would have been that of a split-plot design. One

level of randomization would have applied: Tem-

perature (T) and pressure (P), which would have

been treated as the whole plot factors. The catalyst

type (C) and concentration (D) would have been

the subplot factors and the block(s) would corre-

spond to replicates of the whole design. This would

result in a mixed model with factors (P), (T), (C), (D)

as fixed factors and the blocks (B) as a random fac-

tor. In this situation the interaction with the block

variable (the replicates) would not have been of

particular significance to the researcher.

The model considered would be

Yijk ¼ lþ Bk þWj þ hjk þ Si þ ðWSÞji þ eijk

where Yijk is the response of the kth block or repli-

cate of the jth level of the whole plot factor W and

the ith level of the subplot factor S. l is the overall

mean; Bk is the random effect of the kth block,

with Bk�N(0, rB);Wj is the fixed effect of the jth level

ofW; Si is the fixed effect of the ith level of S; (WS)ji is

the interaction effect of the jth level of (W) with the

ith level of the subplot factor (S); Hjk is the whole-

plot error; Hjk � Nð0;r2wÞ, eijk is the subplot error,

eijk � Nð0;r2e Þ; and Hjk and eijk are independent.

Table 4 shows the ANOVA table for this situation

under the restricted model. See Montgomery (2008)

for details on the restricted and unrestricted mixed

model mean squares.

In this situation, the whole-plot error could be

found by pooling the (BP), (BT), and (BPT) interac-

tions, resulting in 9 degrees of freedom for the

whole-plot error. The subplot error can be found

by pooling the fourth- and higher order interactions

among whole-plot and subplot factors along with

(BC), (BD) with re. The variance of the whole plot

is represented by r2H, and the variance of the subplot

error is represented by r2e .In this case the whole-plot factor (P), (T), and the

interaction (PT) are tested against the whole-plot

error. The subplot factors and interactions involving

TABLE 4 ANOVA Restricted Model

Term df

Expected mean

squares

Replicates (B) 3 r2e þ 32r2

B

Whole plot T 1 r2e þ r2

h þ 64ø2ðTÞ

P 1 r2e þ r2

h þ 64ø2ðPÞ

TP 1 r2e þ r2

h þ 32ø2ðPTÞ

Whole plot

error B�W

interactions

9 r2e þ r2

h

Sub-subplot Subplot

replicates (F)

1 r2e þ 64r2

F

C 1 r2e þ 64fø2

ðCÞD 1 r2

e þ 64ø2ðDÞ

CD 1 r2e þ 32ø2

ðCDÞTC 1 r2

e þ 32ø2ðTCÞ

TD 1 r2e þ 32ø2

ðTDÞPC 1 r2

e þ 32ø2ðPCÞ

PD 1 r2e þ 32ø2

ðPDÞPTC 1 r2

e þ 16ø2ðPTCÞ

PTD 1 r2e þ 16ø2

ðPTDÞPCD 1 r2

e þ 16fø2ðPCDÞ

TCD 1 r2e þ 16ø2

ðTCDÞError 100 r2

e ðSubplot errorÞ

FIGURE 3 Temperature sub-subplot interactions.

333 Split-Split-Plot

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

TABLE 5 Created Data

C �1 �1 1 1 �1 �1 1 1

D �1 1 �1 1 �1 1 �1 1

Block Robot T P

1 1 �1 �1 �82.58 �39.55 �39.55 �87.5 �71.68 �54 �35.95 �73.59

1 1 1 1 208.2 93.85 128.5 118 144.3 78 132.6 123.3

1 2 �1 1 81.71 �6.14 �10.71 �42.1 36.12 �10 27.5 �7.6

1 2 1 �1 69.05 75.01 �13.75 31.2 �3.06 �14 �26.68 �44.8

2 1 �1 �1 32.29 67.18 16.95 48.4 109.6 93 131.5 65.8

2 1 1 1 25.54 �32.7 �12.08 38.4 25.5 �25 �0.42 25.71

2 2 �1 1 65.18 23.21 �17.74 6.14 43.77 19 5.97 �10.2

2 2 1 �1 14.3 25.04 �10.98 42.4 38.79 3 31.48 �6.3

3 2 �1 �1 32.59 64.32 19.21 49.8 109.8 92 129.7 61.9

3 2 1 1 24.53 �32.78 �11.64 38.3 10.58 �25 �0.94 21.5

3 1 �1 1 64.24 23.36 �15.88 2.68 43.57 17 5.82 �9.1

3 1 1 �1 16.18 23.93 �9.35 43.8 37.09 4 28.92 �8.4

4 2 �1 �1 30.66 65.15 18.6 50.6 111.8 93 131.4 63.2

4 2 1 1 24.92 �35.24 �12.82 38.9 13.4 �25 �1.3 24.4

4 1 �1 1 64.72 22.7 �16.3 5.82 43.76 17 6.81 �9.5

4 1 1 �1 15.69 23.11 �10.64 44.7 36.97 3 29.42 �7.0

TABLE 6 ANOVA Table Created Data

Source df Sum of squares Mean square F ratio p-Value

Whole Plot Replicates (R) 0

Block (B) 3 1644.03 548.01

Subplot Pressure (P) 1 0.62 0.62 0.22 0.666 (0.996)

Temperature (T) 1 1.34 1.34 0.48 0.539 (0.994)

BP 3 105701.35 35233.78 12606.26 0.000

BT 3 106014.69 35338.23 12643.63 0.000

PT 1 13068.59 13068.59 4675.80 0.000 (0.475)

BPT (Subplot error) 3 8.38 2.79

Sub-subplot Sub-subplot Replicates (F) 1 1329.67 1329.67 1.84 0.178

C 1 6183.28 6183.28 8.57 0.004

D 1 3479.03 3479.03 4.82 0.031

C�D 1 6555.41 6555.41 9.09 0.003

B�C 3 1460.87 486.96 0.68 0.570

B�D 3 986.88 328.96 0.46 0.714

P�C 1 0.79 0.79 0.00 0.974

P�D 1 3342.35 3342.35 4.64 0.034

T�C 1 4254.80 4254.80 5.90 0.017

T�D 1 91.77 91.77 0.13 0.722

B�P�C 3 407.10 135.70 0.19 0.904

B�P�D 3 4178.03 1392.68 1.93 0.131

B�T�C 3 1361.71 453.90 0.63 0.598

B�T�D 3 176.27 58.76 0.08 0.970

B�C�D 3 51.35 17.12 0.02 0.995

P�T�C 1 5933.14 5933.14 8.23 0.005

P�T�D 1 106.69 106.69 0.15 0.702

P�C�D 1 8373.15 8373.15 11.61 0.001

T�C�D 1 4743.87 4743.87 6.58 0.012

sub-subplot error 79 56967.49 721.11

Total 127 336295.48

F. Castillo 334

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14

at least one subplot factor are tested against the sub-

plot error. In this case, the split-plot and split-split-

plot analyses produce essentially the same results;

however, this is not always the case. As an example,

consider the data presented in Table 5. In this table,

values of the response have been artificially created

considering the same configuration of the reactors

presented in Figure 1.

Table 6 shows the results of the analysis as a split-

split-plot situation. In parenthesis are the p-values

obtained when the data is analyzed as split-plot. In

this case, the results are not the same. Clearly, in this

case, if the data are analyzed as a split-plot when

the experiment was run as split-split-plot, the effect

of the interaction PT would have been missed. This

shows the importance of determining how the actual

experiment was run in order to perform the proper

analysis and get the correct conclusions.

CONCLUSION

A case study of split-plot experimental design

applied to a high-throughput reactor system was

presented. The equipment constraints and way the

experiments were executed introduced two levels

of randomization, resulting in a split-split-plot

design. The study illustrates the importance of moni-

toring the experiments to understand the way the

experiments are executed in order to perform a cor-

rect analysis and provide results that are physically

meaningful for the researcher.

ACKNOWLEDGMENTS

The author thanks Professor Soren Bisgaard for his

feedback and encouragement to publish the work

reported in this article.

ABOUT THE AUTHOR

Flor A. Castillo is a Lead Research Specialist

within Performance Products R&D organization of

the Dow Chemical Company. She is a member of

the American Statistical Association, a Senior

Member of the American Society of Quality, and a

recipient of the Shewell Award of the American

Society of Quality.

REFERENCES

Bingham, D., Sitter, R. R. (2001). Design issues in fractional factorialsplit-plot experiments. Journal of Quality Technology, 33:2–15.

Bingham, D., Sitter, R. R. (2003). Fractional factorial split-plot designsfor robust parameter experiments. Technometrics, 45:80–89.

Bisgaard, S. (2000). The design and analysis of 2k-p�2q-r split plotexperiments. Journal of Quality Technology, 32:39–56.

Bisgaard, S., Steinberg, D. (1997). The design and analysis of 2k-p� Sprototype experiments. Technometrics, 39:52–62.

Box, G., Hunter,W., Hunter, J. S. (2005). Statistics for Experimenters: Design,Innovation and Discovery, 2nd ed. New York: John Wiley & Sons.

Goos, P., Vandebroek, M. (2001). Optimal split-plot designs. Journal ofQuality Technology, 33:436–450.

Goos, P., Vandebroek, M. (2003). D-optimal split-plot designs with givennumbers and sizes of whole plots. Technometrics, 45:235–245.

Goos, P., Vandebroek, M. (2004). Outperforming completely randomizeddesigns. Journal of Quality Technology, 36:12–26.

Kowalski, S. M. (2002). 24 Run split-plot experiments for robustparameter design. Journal of Quality Technology, 34:399–410.

Kowalski, S. M., Parker, P. A., Vining, G. G. (2007). Tutorial: Industrialsplit-plot experiments. Quality Engineering, 19:1–15.

Kulahci, M. (2007). Split-plot experiments with unusual numbers ofsubplot runs. Quality Engineering, 19:363–371.

Montgomery, D. C. (2008). Design and Analysis of Experiments, 7th ed.New York: John Wiley & Sons.

Paniagua-Quinones, M. C., Box, G. (2008). Use of strip-stripblock designfor multi-stage processes to reduce costs of experimentation. QualityEngineering, 1:46–51.

Vining, G. G., Kowalski, S. M. (2008). Exact inference for responsesurface designs within a split-plot structure. Journal of QualityTechnology, 40:394–406.

Vining, G. G., Kowalski, S. M., Montgomery, D. C. (2005). Responsesurface designs within a split-plot structure. Journal of QualityTechnology, 37:115–129.

Vivacqua, A. C., Bisgaard, S. (2009). Post fractionated strip-block designs.Technometrics, 52:47–55.

335 Split-Split-Plot

Dow

nloa

ded

by [

Uni

vers

ity o

f Y

ork]

at 0

6:54

02

Dec

embe

r 20

14