doe course part 14

8/19/2019 DOE Course Part 14

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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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L. M. Lye DOE Course $

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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L. M. Lye DOE Course /

% !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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% 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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"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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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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"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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L. M. Lye DOE Course 1$

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L. M. Lye DOE Course 1/

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&divided and eac! of t!e t(o

levels of factor and applied to t!e su&divided 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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Data and Analyses

A B Observations

-1 -1 $.9 $.5 $.0

-1 1 $.> $.0 $.9

1 -1 $./ $./ $.5

1 1 $.> $.> $.:


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#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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L. M. Lye DOE Course 1>

% 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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L. M. Lye DOE Course $

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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L. M. Lye DOE Course $1

"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.

%

doe course part 14

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