math602: applied statistics

LECTURE 8(Winter'99)1

MATH602: APPLIED STATISTICS

Dr. Srinivas R. ChakravarthyDepartment of Science and Mathematics

KETTERING UNIVERSITYFlint, MI 48504-4898

Lecture 8 Winter 1999

LECTURE 8(Winter'99) 2

LECTURE 8

DESIGN OF EXPERIMENTS

-Earlier we talked about the quality of a product and

how statistics is used to continuously to improve the

quality of a product. We saw a number of statistical

methods to analyze the data and make interpretations.

-One of the important tools of statistics that has been

widely used in evaluating the quality of a product,


identifying the sources that affect the quality, setting up

the values of the parameters that will optimize the

response variable, is the Design of Experiments.

-Designing an experiment is like designing a product.

-The purpose should be clearly defined to begin with.

-The experiment should be set up to answer a specific

question or a set of questions.


WHAT IS A DESIGNED EXPERIMENT?

- Enables us to observe the behavior of a particular

aspect of reality.

-Experimental design is an organized approach to the

collection of information.

-In most practical problems, many variables influence

the outcome of an experiment.

-Usually these interact in very complex ways.


-A good design allows for estimation and interpretation

of these interactions.

-An experimenter chooses certain factors and in a

controlled environment varies these factors so as to

observe the effects.

-No statistical tool can come to rescue data obtained

from designs conducted haphazardly.


OBJECTIVES

- Maximize the amount of information

- Identify factors that (a) affect the average response; (b) affect

the variability; (c) do not contribute significantly.

- Identify the mathematical model relating the response to the

factors

- Identify Aoptimum@ settings for the factors

- CONFIRM the settings


STARTING POINT OF DOE: Consider the

following scenario: A process engineer in the

manufacture of reinforced pet moldings using injection-

molding process asks the following question: We are

manufacturing two different parts using two-cavity

injection molds. One part, the shaft, is molded in a 55%

glass fiber reinforced PET polyester, while the other

part, the tube is produced from a 45% fiber reinforced


PET. Both parts are end gated and we also know where

the areas of failure during a physical testing for these

two parts. We want to find the optimum molding

process. That is, what should be the levels of the

factors: melt temperature, mold pressure, hold time,

injection speed, and hold pressure that will optimize the

strength of the reinforced pet moldings?

-Almost all DOE’s in practice start with such a


statement.

MAJOR STEPS IN DOE: Design of experiment

(DOE) is an iterative decision-making process. Like

any area of applied science, the steps involved in DOE

can be grouped into three stages: analysis, synthesis,

and evaluation. These phases are characterized as:

Analysis: (a) Recognition of the problem; (b)


formulating the experimental problem; (c) analysis of

the experiment.

Synthesis: (a) Designing the experimental model; (b)

designing the analytical model.

Evaluation: (a) Conducting the experiment; (b)

Deriving solution(s) from the model; (c) Make

appropriate conclusions and recommendations.


Basic concepts in Design of Experiments: Factor,

level, treatment, effect, response, test run, interaction,

blocking, confounding, experimental unit, replication,

randomization, and covariate. Some of these were seen

in our lecture on ANOVA.

Block: A factor that has influence on the variability of

the response variable.


Randomization: This refers to assigning the

experimental units randomly to treatments.

Replication: This refers to the repetition of an

experiment. This should be practiced in all

experimental work in order to increase the precision.


Block: A group of homogeneous experimental units.

Confounding: When one or more effects that cannot

be unambiguously be attributed to a single factor or

interaction.

Covariate: An uncontrollable variable that influences

the response but is unaffected by any other

experimental factors. Covariates are not additional

responses and hence their values are not affected by the


factors in the experiment.

Test run: Single combination of factor levels that

yields an observation on the response.

SELECTION OF RESPONSE (or dependent)

VARIABLES AND FACTORS: Usually there will be

only one response variable and the objective of the


experiment will indicate the response variable. The

response variable can be qualitative or quantitative.

-The selection of factors is a critical one and involves

a detailed plan. At first all possible factors, irrespective

whether they are practical to be measured or not,

should be included in the experiment.

-A common approach is to use a cause-and-effect

diagram (refer to Lecture 1 notes for details on this)


listing all the factors.

-To better understand these concepts, let us look at

some illustrative examples.

ILLUSTRATIVE EXAMPLE 1: A new brand of

printing paper is being considered by a leading

photographic company. The study will be focusing on


the effects of various factors on the development time.

So, the response variable for this is the development

time. The experiment will consists of the following

steps: (i) a test negative will be placed on the glass top

of a contact printer; (ii) a sample of printing paper will

be placed on top of the negative; (iii) the light on the

contact printer will be turned on for a specific amount

of time; and (iv) the printing paper will be placed on a


developing tray until an image appears. The following

factors are considered to play a role: (1) exposure time;

(2) density of test negative; (3) temperature of the

laboratory where the developing is done; (4) intensity

of exposing light; (5) types of developer; (6) amount of

developer; (7) grade of printing paper; (8) condition of

printing paper; (9) voltage fluctuations during the

experiment; (10) humidity; (11) number of times the


developer will be used; (12) size of printing paper; and

(13) operator. After careful study, the company decided

to use three factors: exposure time, type of developer,

and grade of printing paper in the experiment and the

remaining factors are either controlled or made as

experimental error.


ILLUSTRATIVE EXAMPLE 2: An experiment is

conducted to study the flow of suspended particles in

two types of coolants used in industrial equipment. The

coolants are to be forced through a slit aperture in the

middle of a fixed length pipe. This experiment is

conducted with three different flow rates and four

different angles of inclination at which the pipe will be

kept. The study will focus on the buildup of the


particles in the coolant on the edge of the aperture.

For this experiment, the response variable is the flow

rate of suspended particles; the factors are: coolant, a

qualitative factor at three levels (1 and 2); pipe angle,

a quantitative factor at four levels (15, 25, 45, 60

degrees from horizontal position); and flow rate, a

quantitative factor at three levels (60, 90 and 120

ft/sec). All 24 combinations of the factor levels are to


be included in the experiment. All the test runs are to

be conducted on a particular so as to eliminate day-to-

day variation in the response variable. The sequence of

tests was determined randomly to minimize any bias in

the experimentation. Since the temperature will vary

from early morning to late evening (during the time of

the test runs), this may affect the test results and so the

temperature is taken to be a covariate.


ILLUSTRATIVE EXAMPLE 3: The engineering

application of a particular printed circuit board requires

that the variability of the thickness of solder coating on

these boards be as small as possible. In order to

determine what factors may cause the variability in the


thickness an experiment was proposed. The response

variable is solder coating thickness and the following

factors were identified as key ones: (1) tool type-

measuring instrument in measuring the solder

thickness; (2) inspectors-persons performing the

inspection of the boards; (3) position of measurment-

past experience has shown that error measurement

tends to be larger at position close to the component


part; and (4) boards.


The design of experiments refers to the structure of the

experiment with reference to

- the set of treatments included

- the set of experimental units

- the rules by which the treatments are assigned to the

units

- the measurements taken

-For example if a teacher wishes to compare the


relative merits of four teaching aids: text book only,

text book and class notes, text book and lab manual,

text book, lab manual and class notes.

-Treatments: four teaching aids

-Experimental units: participating students (or classes)

Rules: Once the treatments and the experimental units

are selected the rules are required for assigning the

treatments to the experimental units.


RANDOMIZATION: (Sir R. A. Fisher) assigning the

units randomly to treatments. This tends to eliminate

the influence of external factors (or noise factors) not

under the direct control of the experimenter; avoid any

selection bias. Also the variation from these noise

factors can bias the estimated effects. Hence in order to


minimize this source of bias, randomization technique

should be adopted in all experimental work.

REPLICATION: repetition of an experiment.

For example if we have 3 treatments and 6 units, the

assignment of 3 units at random to the 3 treatments

constitute one replication and the assignment of the

remaining 3 units to the 3 treatments constitute another


replication of the experiment.

-Replication should be practiced in all DOE work.

-Also replication is used to assess the error mean square

as well as to increase the precision.


SOME COMMON PROBLEMS IN DOE

(a) experimental variation hides true factor effects;

(b) uncontrolled factors compromise experimental

conclusions;

(c) one-factor-at-a-time designs will not give a true

picture of many-factor experiments.


Experimental variation hides true factor effects

CASE 1:

CASE 2:


Uncontrolled factors compromise experimental conclusions

Suppose that methods (such as weight-loss, groove-

depth, etc) of determining wear and tear of tires are

under study. In order to find a relationship between

various methods (for calibration purposes), one has to

be aware of large variation associated with the

"uncontrolled" factors such as road conditions,

vehicles, drivers, and weather.


One-factor-at-a-time

While it looks that this approach requires very minimum

number of experimental runs, the following example will

illustrate how one can be way off from the optimum.

Suppose that two factors: A (temperature) and B (time) are

under study to look at the effect of yield in a chemical

experiment.


ILLUSTRATIVE EXAMPLE

(ONE-FACTOR-AT-A-TIME)


COMMONLY USED DESIGNS

- Completely Randomized Designs (CRD)

- Randomized Block Designs (RBD)

- Latin Square Designs (LSD)

- 2n Factorial Designs.

- Fractional Factorial Designs (including Taguchi=s

orthogonal designs)


Completely Randomized Design (CRD)

-This is the basic design.

-All other randomized designs stem from it by imposing

restrictions upon the allocation of the treatments to the units.

- The units are assigned to treatments at random.

-Thus every unit chosen for the study has an equal chance of

being assigned to any treatment.

-This is useful when the units are homogeneous.

-Most useful in laboratory techniques.


Advantages and Disadvantages of a CRD

(1) it is felxible

(2) its MSE has a larger degrees of freedom

(3) it allows for missing observations

(4) it has fewer assumptions

-Heterogeneous; # of treatments is large.


ANALYSIS OF A CRD

-The analysis of single-factor studies that we discussed

in ANOVA is applicable and there is no need to repeat

the analysis here.


Randomized Block Design (RBD)

-When experimental units are heterogeneous to reduce

experimental error variability we need to sort the units

into homogeneous groups called blocks.

-The treatments are then randomly assigned within

blocks.

-That is, randomization is restricted.


-This procedure is called BLOCKING.

-Since the development of RBD in 1925 this design has

become very popular among all designs.

-As an example of this design, suppose that a company

is considering buying one of 5 word processors for use

in its offices. In order to study the average time for its

employees to learn the word processors, if all have the

same ability we could use a CRD. However this will be


the case. We can sort the employees into blocks of 5

and assign randomly the 5 word processors for

learning. If we had used a CRD any effect that should

have been attributed to blocks would end up in the error

term. By blocking we remove a source of variation

from the error term.


Advantages and Disadvantages of a RBD

(1) provides precise results with proper blocking

(2) No need to have equal sample sizes

(3) the analysis is simple

(4) one can bring in more variability among the

experimental units, which usually is the case in practice.

(1) missing observations; (2) DF are not as large as with a

CRD; (3) Need more assumptions.


ANALYSIS OF A RBD

The analysis of multi-factor studies that we discussed in

ANOVA is applicable and there is no need to repeat the

analysis here.


Latin Square Design (LSD)

-RBD is one of many block designs. In it one source of

variation is blocked.

-If there are two sources of variations that need to be

blocked we need to use different design called the Latin

Square design.

-The treatments are grouped into two blocks, once in

rows and once in columns.


-Through this, the error variance is reduced.

-This design has a wide variety of applications in industrial,

field, laboratory, greenhouse, educational, marketing,

medicine, and sociology.

-According to Fisher, if experimentation were only

concerned with the comparison of four to eight treatments

or varieties, LSD would therefore be not merely the

principal but almost the universal design employed.


Example of a LSD

-Testing 3 electronic components (A-C) for its

fascimile telephone to determine whether or not the

average transmission speed of a page was

approximately the same for the 3 components.

-Three different kinds of pages: text only, picture only

and text and picture


-Three kinds of transmitting were used.

-The design that blocks using two variables (page type

and device type) and tests for differences among the

same number of treatments (electronic components) is

a 3x3 LSD.

-The rows of the square stand for the page type and the

columns stand for the device type. The Latin square is:


DEVICEPAGE

1 2 3

Text B A C

Picture A C B

Text & Picture C B A


-Thus, a LSD has r treatments; two blocking variables, each

having r levels and each row and each column in the design

square contains all treatments.

Advantages and Disadvantages of a LSD:

(1) reduces the variability of experimental error.

(2) effects studied from a small-scale experiment.

# of levels for each blocking variable must equal the

number of treatments and the assumptions of the model are

restrictive.


STATISTICAL MODEL

.εγβτµ ijkkjiijk + + + + = y

yijk = obs. of the j-th row and k-th column for the i-th

treatment effect.

ANOVA:

STOT = SSTR + SSROW + SSCOL + SSE

These are calculated as follows:


2...

1 1 1

2 YNYSSTr

i

r

j

r

kijk −= ∑∑∑

= = =

2...

1

2.. YN

r

TSS

r

i

iTr −= ∑

=

2...

1

2.. YN

r

TSS

r

j

jRow −= ∑

=

2...

1

2.. YNr

TSS

r

k

kCol −= ∑

=


EXAMPLE 1: Suppose that four cars and four drivers

are used in the study of possible differences between

four gasoline additives in reducing the oxides of

nitrogen in the auto emissions. Because of the

possibility of systematic differences among the cars and

among the drivers, it was decided to use LSD. Table 1

gives the (coded) results for the reduction in the oxides

of nitrogen.


CarDriver

1 2 3 4

I A

21

B

26

D

20

C

25

II D

23

C

26

A

20

B

27

III B

15

D

13

C

16

A

16

IV C

17

A

15

B

20

D

20


FACTORIAL DESIGNS• In many experiments the success or failure may depend

more on the selection of treatments for comparisons to bemade than on the design itself.

• Thus an experimenter should be careful not only in theselection of the design but also on the treatments.

• Factorial experiments are very commonly used inmanufacturing and engineering experiments.

• Here several factors at two or more levels, are controlledto carefully measure the effects.

• The effects in a factorial experiment are composed of maineffects and interactions.


• All the units are used to evaluate main effects andinteractions.

• The main effect of a factor is composed of a set ofsingle-degree of freedom contrasts among the totalnumber of levels of that factor.

• The interaction of two factors is the failure of the levels ofone factor to retain the same order and magnitude ofperformance throughout all levels of the second factor.

• Full factorial designs are used to assess all possiblecombinations of the factor levels under study.

• For example, a full factorial experiment consisting of 8two-level factors require 256 trials.


• This will have information not only on the main effects ofall 8 factors, but also on the interaction of the factorsincluding whether all 8 factors work in conjunction toaffect the response variable.

• Usually in practice, one will be interested in the maineffects and the interaction of 2 factors at a time, as thehigher order interaction will be negligibly small.


Advantages and Disadvantages of a Factorial Design:The advantages of a factorial design are:(1) all experimental units are utilized in evaluating the main

effects and interactions(2) the effects are evaluated over a wide range of conditions

with the minimum resources(3) a factorial set of treatments is optimum for estimating the

main effects and interactionsThe main disadvantage is the large number of combinationsneeded to study the effects.However, an alternative approach to use fractional factorialdesigns. We will see this later.


• For a general factorial design an experimenter selects afixed number of levels for each of the factors that areunder study and then runs experiments with all possiblecombinations.

• For example, if there are 3 factors with levels 3, 4, and 6,then the experiment requires a minimum of 3x4x6 = 72runs. This design is called a 3 x 4 x 6 factorial design.

• A factorial design with k factors, all of which have only 2levels is a 2k factorial design.

• 2k factorial designs play an important role for a number ofreasons.


• First of all they require only few runs per factor studied.These can be used to see whether there are any majortrends and to determine the direction for further study.

• When a more thorough study is needed they could besupplemented to form composite designs. Also where weneed to determine whether there are any interactionspresent among the factors we could quickly use thisdesign.

• When the factors to be studied are large, because of thesize of the design we could either have only onereplication for the experiment or obtain most of theinformation by looking only at a fraction of the design.


- Effect of a factor: Change in response produced by a

change in the level of that factor averaged over the levels

of the other factor(s).

- Magnitude and direction of factor effects are to be

examined to see which are likely to be important.


INTERACTION

- Exists if the difference in response between the levels of

one factor is not the same at all levels of the other

factor(s).

- Calculated as the average difference between the effect of

A at high level of B and the effect of A at the low level of

B.


In 2k design:

• All factor effects will have 1 d.f

• If there are n replicates, SSE will have (n-1)2k d.f.

• Replicates are very important in testing for lack of fit [

Recall this from Regression Analysis]

• If n=1, we have estimate for error [Why?]

• Use higher order interactions to get an estimate.

- Plot the estimates on a normal probability paper. All

effects that are insignificant will fall on a line.


22 FACTORIAL DESIGNS

• Two factors, say, A and B, at two levels.

• A full factorial design will consist of 4 runs.

• We can estimate two main effects and one two-factor

effect from these 4 trials.

• The total sum of squares will be split as: SSA, SSB and

SSAB.


The Regression Model: It is easy to see how the effect

estimates in a 2k factorial design into a regression model, which

can be used to predict the response for the factor space. For

example, a 22 design to estimate linear effects is transformed to

a multiple linear regression model as:

Y = b0 + b1 X1 + b2 X2 + e,

where X1 and X2 are coded variables.

2/)(

2/)(

LowHigh

LowHigh

AA

AAAX

−+−

=


The fitted model is:

22110ˆˆˆˆ XXY βββ ++=

which in our case reduces to

21 22ˆ XXYY BA ∆

+∆

+=

[Why?]


n

b

n

aabYY

AAA 2

)1(

2

+−

+=−=∆ −+

n

a

n

babYY

BBB 2

)1(

2

+−

+=−=∆ −+

n

ba

n

abYY

ABABAB 22

)1( +−

+=−=∆ −+


Note: The numerator terms in the above average effects are

refered as CONTRASTS. A contrast is a linear combination

of the parameters:

iicL µ∑=

.0such that =∑ ic

In general, SS (due to a contrast) = (contrast)2/ n 2k.


EXAMPLE 2: A study was conducted to determine the effectof mixing time and the temperature on the yellowness index ofa paint to be used on commercial products. Let y = yellownessindex, x1 = mixing temperature, 50oF and 70oF; x1 = mixingtime, 0.5 and 2 hours. The study yielded the following data.

Obs. No. A B Index

1 50 0.5 6.0

2 70 0.5 8.03 50 2.0 7.5

4 70 2.0 8.5


23 FACTORIAL DESIGNS

-Three factors, say, A, B and C, at two levels.

-A full factorial design will consist of 8 runs.

-We can estimate three main effects, three two-factor

effects and one three-factor effect from these 8 trials.

-The total sum of squares will be split similar to 22

design.


EXAMPLE 3: Suppose that a chemist is interested in

the yield of a chemical process that has three factors:

temperature (A), concentration (B) and catalyst (C).

The first two variables are quantitative while the

variable catalyst is qualitative. The data for these

variables and the yield are given below. Note that there

are 2 replications.


Yield(in gms)

Obs.No.

A B C

1 2 TOTAL

1 160 20 I 60 58

2 180 20 I 72 68

3 160 40 I 54 64

4 180 40 I 68 62

5 160 20 II 52 45

6 180 20 II 83 76

7 160 40 II 45 53


8 180 40 II 80 76

USE OF MINITAB IN 2k DESIGNS

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