split-plot designs martin arvidsson

Chalmers University of Technology

Split-plot designs

Martin Arvidsson


A simple test performed at Cochlear BAS to evaluate a new supplier of components

• The objective of the test was to evaluate whether washers from a new supplier could be used• Altogether 120 transducers where produced, 60 with washersordinary used and 60 with washers from a new potential supplier

• The order in which the 120 transducers was produced was randomised


Details of the improvement work

The objective of the project is to improve the production yield of the transducers

• The transducers are made up by a rather large number of components• The assembly process of transducers include a ratherlarge number of operations• The assembly process requires that measurement equipment work satisfactory


Individual value plot

Washer from new supplierOrdinary used

1200

1100

1000

900

800

700

Type of washer

Individual Value Plot


Individual value plot – two outliers removed

Washer from new supplierOrdinary used

800

790

780

770

760

750

740

Type of washer

Individual Value Plot


Time series plot to investigate whether the process was stable during the test

10896847260483624121

800

790

780

770

760

750

740

Index

Time Series Plot


Histogram of the”populations”

790780770760750740

0,04

0,03

0,02

0,01

0,00

Densi

ty

760,1 9,643 59769,9 11,82 59

Mean StDev N

Ordinary usedWasher from new supplier

Type of washer

Normal Histogram


Complete randomisation

• Randomisation of run order

• Resetting of all factor levels between each experiment


Randomizing

• Problem: Systematic dependence between the experiments.

• Solution: Make the experiments in random order.

order Exp.nr

A B C Y

8

5

1

2

4

7

6

3

1

2

3

4

5

6

7

8

-

+

-

+

-

+

-

+

-

-

+

+

-

-

+

+

-

-

-

-

+

+

+

+

53.8

51.8

47.4

47.8

50.6

51.8

48.2

48.6


Resetting of factor levels

4

8

1

i

i

contrasts

24)(

2

8

1

i

i

VarcontrastsVar

Exp. A B C y ε

1 - - - 53.8 εA1+εB1+εC1+ε1 2 + - - 51.8 εA2+εB2+εC2+ε2 3 - + - 47.4 εA3+εB3+εC3+ε3 4 + + - 47.8 εA4+εB4+εC4+ε4 5 - - + 50.6 εA5+εB5+εC5+ε5 6 + - + 51.8 εA6+εB6+εC6+ε6 7 - + + 48.2 εA7+εB7+εC7+ε7 8 + + + 48.6 εA8+εB8+εC8+ε8


If factors are not reset between each experiment, contrasts will have unequal variance!

Exp. A B C y ε

1 - - - 53.8 εA1+εB1+εC1+ε1 2 + - - 51.8 εA2+εB2+εC1+ε2 3 - + - 47.4 εA3+εB3+εC1+ε3 4 + + - 47.8 εA4+εB4+εC1+ε4 5 - - + 50.6 εA5+εB5+εC2+ε5 6 + - + 51.8 εA6+εB6+εC2+ε6 7 - + + 48.2 εA7+εB7+εC2+ε7 8 + + + 48.6 εA8+εB8+εC2+ε8

4

8

1

i

iBiAi

Acontrast

4

42

1

8

1

j

Cji

iBiAi

Ccontrast

2

222 BA

AcontrastVar

2222

22 CBA

CcontrastVar

Responses arenot independent!


• Four different process conditions• Eight batches of raw material

Split-plot designs: A Composite Material Example

Manufacturing process of composite material

y – bending strength response variable

A – curing temperatureB – pressureC – holding time

control factors (process variables)

D – proportion of hardenerE – thermo-plastic contentF – proportion of epoxyG – material ageingH – process type

noise factors

y = f (A,B,C,D,E,F,G,H)?


Experimental designD E F G H-1 -1 -1 1 -1 20751 -1 -1 1 1 2117-1 1 -1 -1 1 22211 1 -1 -1 -1 2227

-1 -1 1 -1 1 22011 -1 1 -1 -1 2179-1 1 1 1 -1 19881 1 1 1 1 1858-1 -1 -1 1 -1 18291 -1 -1 1 1 1978-1 1 -1 -1 1 21111 1 -1 -1 -1 2205

-1 -1 1 -1 1 2127

A B C 1 -1 1 -1 -1 2106-1 -1 1 -1 1 1 1 -1 1870

1 -1 -1 1 1 1 1 1 1879-1 1 -1 -1 -1 -1 1 -1 22451 1 1 1 -1 -1 1 1 2242

-1 1 -1 -1 1 22451 1 -1 -1 -1 2258

-1 -1 1 -1 1 22061 -1 1 -1 -1 2207-1 1 1 1 -1 20531 1 1 1 1 2188-1 -1 -1 1 -1 22191 -1 -1 1 1 2145-1 1 -1 -1 1 21741 1 -1 -1 -1 2265

-1 -1 1 -1 1 22411 -1 1 -1 -1 2187-1 1 1 1 -1 22081 1 1 1 1 2181

Process variables (control factors)A Curing temperatureB PressureC Holding time

Incoming material (noise factors)D Proportion of hardenerE Thermo-plastic contentF Proportion of epoxyG Material agingH Type of process

Process

Product


1 I EFG DEFH ABC ABCEFG ABCDEFH ABCDGH DGH 2 A AEFG ADEFH BC BCEFG BCDEFH BCDGH ADGH 3 B BEFG BDEFH AC ACEFG ACDEFH ACDGH BDGH 4 D DEFG EFH ABCD ABCDEFG ABCEFH ABCGH GH 5 E FG DFH ABCE ABCFG ABCDFH ABCDEGH DEGH 6 F EG DEH ABCF ABCEG ABCDEH ABCDFGH DFGH 7 AB ABEFG ABDEFH C CEFG CDEFH CDGH ABDGH 8 AD ADEFG AEFH BCD BCDEFG BCEFH BCGH AGH 9 AE AFG ADFH BCE BCFG BCDFH BCDEGH ADEGH 10 AF AEG ADEH BCF BCEG BCDEH BCDFGH ADFGH 11 BD BDEFG BEFH ACD ACDEFG ACEFH ACGH BGH 12 BE BFG BDFH ACE ACFG ACDFH ACDEGH BDEGH 13 BF BEG BDEH ACF ACEG ACDEH ACDFGH BDGH 14 DE DFG FH ABCDE ABCDFG ABCFH ABCEGH EGH 15 DF DEG EH ABCDF ABCDEG ABCEH ABCFGH FGH 16 EF G DH ABCEF ABCG ABCDH ABCDEFGH DEFGH 17 ABD ABDEFG ABEFH CD CDEFG CEFH CGH ABGH 18 ABE ABFG ABDFH CE CFG CDFH CDEGH ABDEGH 19 ABF ABEG ABDEH CF CEG CDEH CDFGH ABDFGH 20 ADE ADFG AFH BCDE BCDFG BCFH BCEGH AEGH 21 ADF ADEG AEH BCDF BCDEG BCEH BCFGH AFGH 22 AEF AG ADH BCEF BCG BCDH BCDEFGH ADEFGH 23 BDE BDFG BFH ACDE ACDFG ACFH ACEGH BEGH 24 BDF BDEG BEH ACDF ACDEG ACEH ACFGH BFGH 25 BEF BG BDH ACEF ACG ACDH ACDEFGH BDEFGH 26 DEF DG H ABCDEF ABCDG ABCH ABCEFGH EFGH 27 ABDE ABDFG ABFH CDE CDFG CFH CEGH ABEGH 28 ABDF ABDEG ABEH CDF CDEG CEH CFGH ABFGH 29 ABEF ABG ABDH CEF CG CDH CDEFGH ABDEFGH 30 ADEF ADG AH BCDEF BCDG BCH BCEFGH AEFGH 31 BDEF BDG BH ACDEF ACDG ACH ACEFGH BEFGH 32 ABDEF ABDG ABH CDEF CDG CH CEFGH ABFEGH

Confounding pattern


1 I 2 A -49,0625 3 B 143,3125 4 D 46,0625 5 E 13,0625 6 F -23,3125 7 AB -54,8125 8 AD -130,313 9 AE -0,4375 10 AF 10,8125 11 BD -26,6875 12 BE 7,8125 13 BF 30,9375 14 DE 38,9375 15 DF -2,8125 16 EF 8,3125 17 ABD 14,5625 18 ABE -34,0625 19 ABF -6,9375 20 ADE 4,8125 21 ADF 10,0625 22 AEF -17,4375 23 BDE -8,0625 24 BDF 31,9375 25 BEF 30,1875 26 DEF 92,5625 27 ABDE -4,8125 28 ABDF -10,5625 29 ABEF 17,3125 30 ADEF -6,1875 31 BDEF -2,0625 32 ABDEF -25,8125

Contrasts!


Analysis of the experiment

-3

-2

-1

0

1

2

3

-150 -100 -50 0 50 100 150 200

G

contrasts

BBG


1 I EFG DEFH ABC ABCEFG ABCDEFH ABCDGH DGH 2 A AEFG ADEFH BC BCEFG BCDEFH BCDGH ADGH 3 B BEFG BDEFH AC ACEFG ACDEFH ACDGH BDGH 4 D DEFG EFH ABCD ABCDEFG ABCEFH ABCGH GH 5 E FG DFH ABCE ABCFG ABCDFH ABCDEGH DEGH 6 F EG DEH ABCF ABCEG ABCDEH ABCDFGH DFGH 7 AB ABEFG ABDEFH C CEFG CDEFH CDGH ABDGH 8 AD ADEFG AEFH BCD BCDEFG BCEFH BCGH AGH 9 AE AFG ADFH BCE BCFG BCDFH BCDEGH ADEGH 10 AF AEG ADEH BCF BCEG BCDEH BCDFGH ADFGH 11 BD BDEFG BEFH ACD ACDEFG ACEFH ACGH BGH 12 BE BFG BDFH ACE ACFG ACDFH ACDEGH BDEGH 13 BF BEG BDEH ACF ACEG ACDEH ACDFGH BDGH 14 DE DFG FH ABCDE ABCDFG ABCFH ABCEGH EGH 15 DF DEG EH ABCDF ABCDEG ABCEH ABCFGH FGH 16 EF G DH ABCEF ABCG ABCDH ABCDEFGH DEFGH 17 ABD ABDEFG ABEFH CD CDEFG CEFH CGH ABGH 18 ABE ABFG ABDFH CE CFG CDFH CDEGH ABDEGH 19 ABF ABEG ABDEH CF CEG CDEH CDFGH ABDFGH 20 ADE ADFG AFH BCDE BCDFG BCFH BCEGH AEGH 21 ADF ADEG AEH BCDF BCDEG BCEH BCFGH AFGH 22 AEF AG ADH BCEF BCG BCDH BCDEFGH ADEFGH 23 BDE BDFG BFH ACDE ACDFG ACFH ACEGH BEGH 24 BDF BDEG BEH ACDF ACDEG ACEH ACFGH BFGH 25 BEF BG BDH ACEF ACG ACDH ACDEFGH BDEFGH 26 DEF DG H ABCDEF ABCDG ABCH ABCEFGH EFGH 27 ABDE ABDFG ABFH CDE CDFG CFH CEGH ABEGH 28 ABDF ABDEG ABEH CDF CDEG CEH CFGH ABFGH 29 ABEF ABG ABDH CEF CG CDH CDEFGH ABDEFGH 30 ADEF ADG AH BCDEF BCDG BCH BCEFGH AEFGH 31 BDEF BDG BH ACDEF ACDG ACH ACEFGH BEFGH 32 ABDEF ABDG ABH CDEF CDG CH CEFGH ABFEGH

Confounding pattern


Error structure of a Strip-Block ExperimentD E F G H-1 -1 -1 1 -1 20751 -1 -1 1 1 2117-1 1 -1 -1 1 22211 1 -1 -1 -1 2227

-1 -1 1 -1 1 22011 -1 1 -1 -1 2179-1 1 1 1 -1 19881 1 1 1 1 1858-1 -1 -1 1 -1 18291 -1 -1 1 1 1978-1 1 -1 -1 1 21111 1 -1 -1 -1 2205

-1 -1 1 -1 1 2127

A B C 1 -1 1 -1 -1 2106-1 -1 1 -1 1 1 1 -1 1870

1 -1 -1 1 1 1 1 1 1879-1 1 -1 -1 -1 -1 1 -1 22451 1 1 1 -1 -1 1 1 2242

-1 1 -1 -1 1 22451 1 -1 -1 -1 2258

-1 -1 1 -1 1 22061 -1 1 -1 -1 2207-1 1 1 1 -1 20531 1 1 1 1 2188-1 -1 -1 1 -1 22191 -1 -1 1 1 2145-1 1 -1 -1 1 21741 1 -1 -1 -1 2265

-1 -1 1 -1 1 22411 -1 1 -1 -1 2187-1 1 1 1 -1 22081 1 1 1 1 2181

εsεs1

εs2

εw1

εw2

εw3

εw4

εw

ε32

ε1

ε


A B D AD error -1 -1 -1 1 εw1+ εs1+ ε1 -1 -1 1 -1 εw1+ εs2+ ε2 -1 -1 -1 1 εw1+ εs3+ ε3 -1 -1 1 -1 εw1+ εs4+ ε4 -1 -1 -1 1 εw1+ εs5+ ε5 -1 -1 1 -1 εw1+ εs6+ ε6 -1 -1 -1 1 εw1+ εs7+ ε7 -1 -1 1 -1 εw1+ εs8+ ε8 1 -1 -1 -1 εw2+ εs1+ ε9 1 -1 1 1 εw2+ εs2+ ε10 1 -1 -1 -1 εw2+ εs3+ ε11 1 -1 1 1 εw2+ εs4+ ε12 1 -1 -1 -1 εw2+ εs5+ ε13 1 -1 1 1 εw2+ εs6+ ε14 1 -1 -1 -1 εw2+ εs7+ ε15 1 -1 1 1 εw2+ εs8+ ε16 -1 1 -1 1 εw3+ εs1+ ε17 -1 1 1 -1 εw3+ εs2+ ε18 -1 1 -1 1 εw3+ εs3+ ε19 -1 1 1 -1 εw3+ εs4+ ε20 -1 1 -1 1 εw3+ εs5+ ε21 -1 1 1 -1 εw3+ εs6+ ε22 -1 1 -1 1 εw3+ εs7+ ε23 -1 1 1 -1 εw3+ εs8+ ε24 1 1 -1 -1 εw4+ εs1+ ε25 1 1 1 1 εw4+ εs2+ ε26 1 1 -1 -1 εw4+ εs3+ ε27 1 1 1 1 εw4+ εs4+ ε28 1 1 -1 -1 εw4+ εs5+ ε29 1 1 1 1 εw4+ εs6+ ε30 1 1 -1 -1 εw4+ εs7+ ε31 1 1 1 1 εw4+ εs8+ ε32


Variances of the contrasts

8

1

32

1

4

1

32

1

32

1

416

1

816

1

16

1

isi

jifactorsmaterial

iwi

jifactorsprocess

iinsinteractiomaterialprocess

contrasts

contrasts

contrasts

22

22

2

8

1

2

18

18

1

sfactorsmaterial

wfactorsprocess

nsinteractiomaterialprocess

contrastsVar

contrastsVar

contrastsVar


-3

-2

-1

0

1

2

3

-150 -100 -50 0 50 100 150

Identification of location effects

•B, G and BG was determined to be active based on engineering knowledge and the normal plots

Process factors Factors and interactionsassociated with incoming material

Interactions between ”process factors”and ”incoming material factors”


Model

ˆ( , ) 2132 72 65 46

2132 72 46 65

y B G B G BG

B B G

B ≈ 1.4


Conclusions

• The storage time of the incoming material (G) is causing variation in the bending strength of the composite material.

• If the pressure (B) is set at high level the bending strength is made insensitive to the storage time.


Randomisation and split-plot• View randomisation as an insurance against

unknown factors - buy as much as you can afford

• It is not always advisable to reset all factor levels between each experiment!– Can be very time consuming and expensive

– Split-plot designs allow some contrasts of interest to be estimated with great precision. This characteristic can, for example, be useful in robust design experiments

split-plot designs martin arvidsson

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