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L. M. Lye DOE Course 1
Design and Analysis of
Multi-Factored ExperimentsAdvanced Designs
-ard to C!ange Factors-
"plit-#lot Design and Analysis
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ard-to-C!ange Factors
% Assume t!at a factor can &e varied ' (it! great difficulty'in an experimental setup )suc! as a pilot plant*' alt!oug! itcannot &e freely varied during normal operatingconditions.
% Assume furt!er t!at eac! of t(o factors !as t(o levels andt!e design is to !ave a factorial structure' and it isimperative t!at t!e num&er of c!anges of t!e !ard-to-c!ange factor &e minimi+ed.
% ,e can minimi+e t!e num&er of level c!anges of one
factor simply &y eeping t!e level constant in pairs ofconsecutive runs. !at is' eit!er t!e !ig! level is used onconsecutive runs and t!en t!e lo( level on t!e next t(oruns' or t!e reverse.
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% !is means (e t!at (e !ave restricted randomi+ation' as
t!ere are 0 possi&le run orders of t!at one factor (it!outany restrictions' &ut (it! t!e restriction' t!ere are only $
possi&le run orders ) ' - -* or )- -' *
% 2estricted randomi+ation increases t!e lieli!ood t!at
extraneous factors )i.e. factors not included in t!e design*
could affect t!e conclusions t!at are dra(n from t!e
analysis.
% Furt!ermore' t!is (ill also cause &ias in t!e statistics t!at
are used to assess significance. i.e. normal A3O4A &ased
on a completely randomi+ed design may give t!e (rong
conclusions.
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L. M. Lye DOE Course 5
% Alt!oug! !ard-to-c!ange factors !ave not &een discussedextensively in text&oos' it is safe to assume t!at suc!
factors occur very fre6uently in practice.% "ometimes t!ere may &e no !ard-to-c!ange factors at all in
t!e experiments' &ut t!e experimenters or tec!nician (!o(ants to save time may not !ave follo(ed t!e randomi+ed
design as prescri&ed &y t!e experimental design.% ence it is very important for t!e analyst performing t!estatistical analysis to no( exactly !o( t!e experiments(ere performed. ,ere t!e runs randomi+ed as prescri&edor (ere t!e runs made 7convenient8 to save time.
% o( t!e experiments (ere carried out can !ave seriousconse6uences on t!e results. "ignificant effects may turnout to &e insignificant or vice versa if is not properlyanaly+ed. !e soft(are (ill not no( unless you tell it.
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L. M. Lye DOE Course 9
"plit-#lot Design (it! ard-to-C!ange Factors
% For example' all of you no( a $/ full factorial design. Most(ould c!oose to run t!e : treatment com&inations in acompletely randomi+ed order as given say &y Design-Expert.
% ;nfortunately' limitations involving time' cost' material' andexperimental e6uipment can mae it inefficient and' at times'impossi&le to run a completely randomi+ed design.
%
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L. M. Lye DOE Course 0
2ecogni+ing a "plit-#lot Design
% "plit-plot experiments &egan in t!e agricultural industry &ecause one factor in t!e experiment usually a fertili+er or
irrigation met!od can only &e applied to large sections of
land called “whole plots”.
% !e factor associated (it! t!is is t!erefore called a (!ole plot factor.
% ,it!in t!e (!ole plot' anot!er factor' suc! as seed variety'
is applied to smaller sections of t!e land' (!ic! is o&tained
&y splitting t!e larger section of land into subplotssubplots. !is
factor is t!erefore referred to as t!e su&plot factor.
% !ese same experimental situations are also common in
industrial settings.
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L. M. Lye DOE Course >
/ Main C!aracteristics of "plit-#lot Designs
% !e levels of all t!e factors are not randomly determinedand reset for eac! run.
= Did you !old a factor at a particular level and t!e run all t!e
com&inations of t!e ot!er factors?
% !e si+e of t!e experimental unit is not t!e same for all
experimental factors.
= Did you apply one factor to a larger unit or group of units
involving com&inations of t!e ot!er factors?
% !ere is a restriction on t!e random assignment of t!e
treatment com&inations to t!e experimental units. =
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L. M. Lye DOE Course :
Effect of restricted randomi+ation on
statistical analysis
% Consider a very simple example of $ factors eac! at $
levels.
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L. M. Lye DOE Course B
% 3otice t!at t!is se6uence of runs could !ave of course risen
from t!e completely randomi+ed experiment' &ut t!e data
(ould still !ave to &e analy+ed differently &ecause of t!e
restricted randomi+ation in t!e second case.
% !at is' t!ere are only : possi&le se6uences of treatment
com&inations (it! t!e restriction' (!ereas t!ere are $5
possi&le se6uences (it!out t!e restriction.
% Anot!er ey point@ ,it! complete randomi+ation' eac! run is
completely reset' (!ereas' (it! restricted randomi+ation' t!e
!ard-to-c!ange factor (as not reset.
%
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L. M. Lye DOE Course 1
"plit-#lot Designs !ave $ error terms
%2ecall t!at in a $
-
design' eac! effect is estimated (it! t!esame precision. i.e. t!ey !ave t!e same standard error.
% !is does not !appen (it! a split-plot design as su&plotfactors are generally estimated (it! greater precision )smallererrors* t!an are (!ole plot factors.
% !is is &ecause t!ere is greater !omogeneity among su&plotst!an are t!e (!ole plots' especially if t!e (!ole plots arelarge.
% E.g. "maller pieces cut from a s!eet of ply(ood are more
!omogeneous t!an &et(een $ different s!eets of ply(ood.i.e. pieces (it!in a s!eet !as less varia&ility t!an &et(een $s!eets of ply(ood.
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Example
% Assume t!at factor A is a !ard-to-c!ange factor and factor is not !ard to c!ange' (it! t!e experiment suc! t!at
material )e.g. a &oard* is divided into t(o pieces and t!e
t(o levels of factor A applied to t!e t(o pieces' one level
to eac! piece.% !en t!e pieces are furt!er su÷d and eac! of t!e t(o
levels of factor and applied to t!e su÷d pieces.
!ree pieces of t!e original lengt! )e.g. t!ree full &oards*
are used.
% !e data (ill &e analy+ed assuming a fully randomi+ed
design lie a regular $$ design' and t!en correctly using a
split-plot design.
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L. M. Lye DOE Course 15
Data and Analyses
A B Observations
-1 -1 $.9 $.5 $.0
-1 1 $.> $.0 $.9
1 -1 $./ $./ $.5
1 1 $.> $.> $.:
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L. M. Lye DOE Course 10
#roper statistical analysis@ split-plot analysis
%eneral &inear 'odel" # versus A$ B$
Factor ype Levels 4alues
A fixed $ -1' 1
,#)A* random 0 1' $' /' 5' 9' 0 need to set up t!is column
fixed $ -1' 1
Analysis of 4ariance for H' using "e6uential "" for ests
"ource DF "e6 "" "e6 M" F #
A 1 .:// .:// .15 .>$9
,#)A* 5 .$//// .9:// I I
,# error term 1 .1:>9 .1:>9 $9. .>
AI 1 .0>9 .0>9 B. .5 / times !ig!er t!an C2D
Error 5 ./ .>9 su&plot error term
otal 11 ./B10>
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% A some(!at different picture emerges (!en t!e data are
analy+ed correctly.
% !e p-value for A is more t!an / times t!an of for C2D.% !e difference in t!e conclusions dra(n (it! t!e (rong
analysis and t!e conclusions made (it! t!e proper analysis
can &e muc! greater t!an t!e difference in t!is example.
% As illustrated &y #otcner and Go(alsi )$5*' asignificant main effect in t!e complete randomi+ation
analysis can &ecome a non-significant (!ole-plot main
effect (!en t!e split-plot analysis is performed.
% And' a non-significant main effect in t!e completerandomi+ation analysis can &ecome a significant su&plot
main effect (!en t!e split plot analysis is performed.
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L. M. Lye DOE Course 1:
"everal !ard-to-c!ange factors
% "ometimes t!ere may &e instances (!ere t!ere are several
!ard-to-c!ange factors and one or more easy to c!angefactors.
% For example' 5 of t!e factors )A' ' C' and D* may &e
!ard-to-c!ange (!ereas E may &e easy to c!ange. Or (e
may !ave / !ard-to-c!ange factors and say 0 easy toc!ange factors' etc.
%
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L. M. Lye DOE Course 1B
Dividing into ,!ole #lots and "u&plots
% Lets consider $ examples@
% Example 1@ 9 factors )A' ' C' D' E*. A' ' C' D are !ard-to-c!ange factors' and E is easy to c!ange.
% ,!ole plot group@ A' ' C' D and interactions involving onlyt!ese factors.
% "u&plot group@ E' and all interactions involving E only. E.g.AE' E' CDE' etc.
% Example $@ B factors )A' ' C' D' E' F' ' ' J*. A' ' C are!ard-to-c!ange' and D' E' F' ' ' J are easy to c!ange.
% ,!ole plot group@ A' ' C' and all interactions involving onlyt!ese / factors
% "u&plot group@ D' E' F' ' ' J and all interactions involvingt!ese factors. E.g. AD' DE' etc' &ut not A' C' or AC.
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Effects and alf-normal plots
% !e effects of eac! factor and its interaction are determined
in exactly t!e same (ay as in regular factorial design.
% Once t!e (!ole plot group and su&plot group !ave &eendecided' a !alf-normal plot of effects are used to determinedt!e significant effects for eac! group.
% ence' t(o !alf-normal plots are constructed.% !e significant effects from &ot! groups are t!en com&ined
to give t!e final model and prediction e6uation.
%
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"ummary% ,!en it is not convenient or not economical to do a
completely randomi+e experiment due to one or more!ard-to-c!ange factors' (e !ave a restricted randomi+ationcase.
% A common and often used approac! is a split plotexperiment (!ic! !as a (!ole plot group of effects and asu&plot group of effects leading to t(o error terms in t!eA3O4A or t(o !alf-normal plots for t!e unreplicated case.
%