PengProd - Pertemuan 3

8/12/2019 PengProd - Pertemuan 3

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Pengembangan ProdukNajma Annuria Fithri, S.Farm.,M.Sc., Apt.

Universitas Sriwijaya

Genap 2012/2013


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Pertemuan 3Factorial Design


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Role of DOE in Process Improvement

DOE is a formal mathematical method forsystematically planning and conductingscientific studies that change experimentalvariables together in order to determine their

effect of a given response.

DOE makes controlled changes to input

variables in order to gain maximum amountsof information on cause and effectrelationships with a minimum sample size.


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DOE is more efficient that a standardapproach of changing one variable at a timein order to observe the variables impact on agiven response.

DOE generates information on the effectvarious factors have on a response variable

and in some cases may be able to determineoptimal settings for those factors.


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DOE encourages brainstorming activitiesassociated with discussing key factors thatmay affect a given response and allows the

experimenter to identify the key factors forfuture studies.

DOE is readily supported by numerousstatistical software packages available on themarket.


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BASIC STEPS IN DOE

Four elements associated with DOE:

1. The design of the experiment,

2. The collection of the data,

3. The statistical analysis of the data, and

4. The conclusions reached and

recommendations made as a result of theexperiment.


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TERMINOLOGY

Replication repetition of a basic experimentwithout changing any factor settings, allowsthe experimenter to estimate theexperimental error (noise) in the system usedto determine whether observed differences inthe data are real or just noise, allows theexperimenter to obtain more statistical power

(ability to identify small effects)


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TERMINOLOGY

.Randomization a statistical tool used tominimize potential uncontrollable biases inthe experiment by randomly assigningmaterial, people, order that experimental

trials are conducted, or any other factor notunder the control of the experimenter.Results in averaging out the effects of the

extraneous factors that may be present inorder to minimize the risk of these factorsaffecting the experimental results.


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TERMINOLOGY

Blocking technique used to increase theprecision of an experiment by breaking theexperiment into homogeneous segments(blocks) in order to control any potential

block to block variability (multiple lots of rawmaterial, several shifts, several machines,several inspectors). Any effects on theexperimental results as a result of the

blocking factor will be identified andminimized.


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FACTORIAL (2k) DESIGNS

Experiments involving several factors ( k = #of factors) where it is necessary to study thejoint effect of these factors on a specific

response. Each of the factors are set at two levels (alow level and a high level) which may bequalitative (machine A/machine B, fan

on/fan off) or quantitative (temperature800/temperature 900, line speed 4000 perhour/line speed 5000 per hour).


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Factors are assumed to be fixed (fixed effectsmodel)

Designs are completely randomized(experimental trials are run in a random

order, etc.) The usual normality assumptions are

satisfied.


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Particularly useful in the early stages ofexperimental work when you are likely tohave many factors being investigated and youwant to minimize the number of treatment

combinations (sample size) but, at the sametime, study all k factors in a completefactorial arrangement (the experimentcollects data at all possible combinations offactor levels).


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As k gets large, the sample size will increaseexponentially. If experiment is replicated, the# runs again increases.

k # of runs

2 4

3 8

4 16

5 32

6 64

7 128

8 256

9 512

10 1024

6


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FACTORIAL (2k) DESIGNS (k = 2)

Two factors set at two levels (normally

referred to as low and high) would result inthe following design where each level of factorA is paired with each level of factor B.

RUN Factor A Factor B RESPONSE RUN Factor A Factor B RESPONSE

1 low low y1 1 -1 -1 y1

2 high low y2 2 +1 -1 y2

3 low high y3 3 -1 +1 y3

4 high high y4 4 +1 +1 y4

Generalized Settin s Ortho onal Settin s


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Estimating main effects associated with

changing the level of each factor from low tohigh. This is the estimated effect on theresponse variable associated with changing

factor A or B from their low to high values.

2

)(

2

)( 3142 yyyyEffectAFactor

2

)(

2

)( 2143 yyyyEffectBFactor

8


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FACTORIAL (2k

) DESIGNS (k = 2): GRAPHICALOUTPUT

Neither factor A nor Factor B have an effect onthe response variable.


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FACTORIAL (2k) DESIGNS (k = 2):

GRAPHICAL OUTPUT

Factor A has an effect on the responsevariable, but Factor B does not.

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FACTORIAL (2k) DESIGNS (k = 2):

GRAPHICAL OUTPUT

Factor A and Factor B have an effect on theresponse variable.

21k


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21FACTORIAL (2k) DESIGNS (k = 2):

GRAPHICAL OUTPUT

Factor B has an effect on the response variable, butonly if factor A is set at the High level. This iscalled interactionand it basically means that theeffect one factor has on a response is dependent onthe level you set other factors at. Interactions can be

major problems in a DOE if you fail to account for theinteraction when designing your experiment.

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EXAMPLE:


A microbiologist is interested in the effect of twodifferent culture mediums [medium 1 (low) and

medium 2 (high)] and two different times [10hours (low) and 20 hours (high)] on the growthrate of a particular CFU [Bugs].

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EXAMPLE:


Since two factors are of interest, k =2, and wewould need the following four runs resulting

inRUN Medium Time Growth Rate

1 low low 17

2 high low 153 low high 38

4 high high 39

Generalized Settin s

24A


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24EXAMPLE:


Estimates for the medium and timeeffects are

Medium effect = [(15+39)/2] [(17 + 38)/2]= -0.5

Time effect = [(38+39)/2] [(17 + 15)/2] =

22.5

25EXAMPLE


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25EXAMPLE:


26EXAMPLE


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26EXAMPLE:


A statistical analysis using the appropriatestatistical model would result in the followinginformation. Factor A (medium) and Factor B(time)

Type III Sums of Squares

------------------------------------------------------------------------------------

Source Sum of Squares Df Mean Square F-Ratio P-Value

------------------------------------------------------------------------------------

FACTOR A 0.25 1 0.25 0.11 0.7952

FACTOR B 506.25 1 506.25 225.00 0.0424

Residual 2.25 1 2.25

------------------------------------------------------------------------------------

Total (corrected) 508.75 3

All F-ratios are based on the residual mean square error.

27EXAMPLE


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27EXAMPLE:

CONCLUSIONS

In statistical language, one would concludethat factor A (medium) is not statisticallysignificant at a 5% level of significance sincethe p-value is greater than 5% (0.05), but

factor B (time) is statistically significant at a 5% level of significance since this p-value isless than 5%.

28EXAMPLE


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28EXAMPLE:

CONCLUSIONS

In layman terms, this means that we have noevidence that would allow us to conclude thatthe medium used has an effect on the growthrate, although it may well have an effect (our

conclusion was incorrect).

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EXAMPLE:CONCLUSIONS

Additionally, we have evidence that would allowus to conclude that time does have an effect on

the growth rate, although it may well not have aneffect (our conclusion was incorrect).

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EXAMPLE:

CONCLUSIONS

In general we control the likelihood of reachingthese incorrect conclusions by the selection of

the level of significance for the test and theamount of data collected (sample size).


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Interactions for 2k Designs (k = 3)

Interactions between various factors can beestimated for different designs above bymultiplying the appropriate columnstogether and then subtracting the average

response for the lows from the averageresponse for the highs.

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Interactions for 2k Designs (k = 3)

a b c ab ac bc abc

-1 -1 -1 1 1 1 -1

+1 -1 -1 -1 -1 1 1

-1 +1 -1 -1 1 -1 1

+1 +1 -1 1 -1 -1 -1

-1 -1 +! 1 -1 -1 1

+1 -1 +1 -1 1 -1 -1-1 +1 +1 -1 -1 1 -1

+1 +1 +1 1 1 1 1


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2kDESIGNS (k > 2)

For example, if there are no significantinteractions present, you can estimate a responseby the following formula. (for quantitative factors

only)

Y = (average of all responses) + )](*)2

[( LfactorLEVECTfactorEFFE

= BAY BA *)2

(*)2

(

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