EXPERIMENTAL DESIGNS AND ANALYSES TO MEASURE A …

EXPERIMENTAL DESIGNS AND ANALYSES TO MEASURE

PROCESS VARIATION CAPABILITIES

by

P. RAJU, B.E.

A THESIS

IN

INDUSTRIAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

INDUSTRIAL ENGINEERING

Approved

Accepted

December, 1986

//O. I^X^ ACKNOWLEDGEMENTS

I am deeply indebted to Professor William J. Kolarik

for his direction of this thesis and to other members of my

committee. Professors Milton L. Smith and Brian K. Lambert

for their assistance and helpful criticism during my

research. In addition, I would like to acknowledge the FSI

Corporation of Chaska, Minnesota and Mr. Robert Blackwood of

the FSI Advanced Technology Center in Lubbock, Texas for

allowing me to use experimental data from a product develop

ment effort in this research.

11

CONTENTS

ACKNOWLEDGEMENTS i i

LIST OF FIGURES iv

LIST OF TABLES vi

I. INTRODUCTION 1

Background 1 Research Questions 3

Objectives 3

II. LITERATURE REVIEW 5

Process Capability 5 Experimental Design 14 Capability Indices 26 Sample Size 29

III. PROCEDURE 34 Model Development 34 Organization of the Software Package 51 General Specification of the Software 53

IV. VALIDATION AND RESULTS 56

Validation 56

Results 57

V. CONCLUSIONS AND CONTRIBUTIONS 65

Conclusions 65 Contributions 67

BIBILIOGRAPHY 69

APPENDIX A. PROGRAM LISTINGS 72

APPENDIX B. USER'S MANUAL 102

111

LIST OF FIGURES

1. Capability Plots 7

2. Variation - Common and Special Causes 10

3. Sources of Variability 13

4 (a). The Analysis Formula - Model I 39

(b). The Analysis Formula - Model II 39

5. The Analysis Formula - Model III 42

6. The Analysis Formula - Model IV 46

7. Flow Chart of the Software 52

8. Title of the Software 104

9. Brief Description of the Software 106

10. Main Menu 106

11. Description of Model III 107

12. Field Layout Questions 107

13. Field Layout for Model III 108

14. Set of Interactive Questions 108

15. Order of Randomization 110

16. Data Entry 110

17. Data Display H I

18. Outlier Identification 113

19. Saving the Data 113

20. Model of the Experiment 114

21. ANOVA Table 114

22. Statistical Tests (F-Tests) 115

iv

23. Process Capabilities

24. Reading Data from an External File

25. Structure of the External File

26. Blank Work Sheet

27. Data for the Example Problem

116

116

118

119

120

v

LIST OF TABLES

1. Results of Model I 58

2. Results of Model II 59

3. Results of Model III 60

4. Results of Model IV 61

VI

CHAPTER I

INTRODUCTION

Background

There is a growing interest in the study of process

variation capability problems associated with manufacturing

equipment and processes. Research has been carried out to

measure the process variation capabilities for many types of

manufacturing equipment and processes. Papers have been

published giving the details of various capability analysis

procedures. However, most documented procedures concentrate

on the process as a whole and do not allow for a study of

the various components of variation.

Variation is always present in the measured quality

characteristics of manufactured products. There are three

primary sources of variation in any process [11] :

(1) the operator

(2) the material and

(3) the process equipment.

The variation from the first two can be reduced to a

minimum by using a properly trained operator and high qual

ity homogeneous material. A measure of the remaining varia

tion can be viewed as the capability of the process. This

measure is especially useful in assigning production jobs to

specific machines. Process capability or natural tolerance

describes the best effort of a process in the sense that

assignable causes of process variation have been minimized.

Today, there exists, among manufacturing people, an

interest in reducing variability in processes and in the

concept of linking experimental design with the process

capability study. Since in most experimental work, the cost

of experimental runs is considerably more than the cost of

statistical analysis, the use of designed experiments is a

profitable practice. This is particularly true in today's

factory atmosphere, because of the great variety of

processes being performed.

Manufacturing equipment and processes undergo several

stages of development prior to actual production. The stages

of prove out testing at the machine supplier, initial test

ing at the manufacturing facility, and pre-production test

ing all seek to determine whether machinery can produce, on

an on-going basis, production units that meet the required

engineering specifications.

In this study, the objective is focused on how to

efficiently measure and analyze the components of process

variation capabilities and to develop experimental designs

for obtaining the above. Finally, an experimental design

package, using a computer program for analyzing the experi-'

ment, has been designed and developed.

Research Questions

Basically, there are three major questions related to

this study :

(1) For a given manufacturing process, how should one

determine the repeatability or capability ?

(2) How can the process variation be measured and

broken down into its components efficiently ?

(3) What effect does sample size have on the above ?

These three questions focus on the issues of process

variation components, experimental design and sample sizing.

Objectives

The purpose of this study is to :

(1) Determine a method to efficiently measure (with a

minimum amount of experimentation) and analyze the

components of process variation capabilities,

(2) Develop experimental designs for obtaining the

above, and

(3) Provide a microcomputer package of standard

procedures for designing the experiments and a

program to analyze the experimental results, with

interpretation aids.

CHAPTER II

LITERATURE REVIEW

A number of papers have been published which deal with

the process capability and variability in a manufactured

product. But,.they are mostly written in the context of

control charting. Another approach is to use experimental

designs to measure and analyze the components of process

variation capabilities.

Process Capability

In the literature, the term "process" refers to any

combination of conditions which work together to produce a

given result. The term "process capability" refers to the

"normal behavior of a process when operating in a state of

statistical control; the predictable series of effects pro

duced by a process when allowed to operate without inter

ference from outside causes."[22]

In manufacturing terminology, process capability refers

to the inherent ability of the process to turn out similar

parts; the natural behavior of the process that can be

maintained in statistical control for a sustained period of

time under a given set of conditions after unnatural distur

bances are eliminated [22]. Process capability may be

expressed in terms of either attributes, such as percent

defective, or in terms of variables such as a measurement.

Elwood G. Kirkpatrik [11], defines process capability

(or natural tolerance) as the "best effort of the process in

the sense that assignable causes of variation have been

eliminated or at least minimized." Assignable causes are

sources of variation which can be recognized and corrected

readily; process operator, product material, process setup,

process adjustment, etc..

A good first impression for a given operation can be

obtained by plotting individual measurements [11] as indi

cated in Figure 1. Plot(a) indicates adequate capability,

(b) very good capability, and (c) good capability but a

misdirected process setup. Plot (d) shows poor capability,

and plot(e) adequate capability but a rapid tool wear trend.

A.V. Feigenbaum [5], states that with given combina

tions of materials, speeds, feeds, temperatures, flow, cool

ants, and so on, almost all processing operations have an

inherent variation. This process capability is largely

independent of specification tolerances for parts to be

manufactured by the process.

(a)

Maximum scecified limit

Nominal

Minimum specified limit

Maximum specified limit

(b) • > • • Nominal


- y ^ — \ - X ' ' — — — — Maximum specified limit

(c) Nominal


'.d)

— — — - ^ ———— . . . Maximum specified limit

Nominal


(9)

Maximum scecnied limit

Nominal

Minimum specifiec! limit

Time

Figure I : Capability Plots [Li

7

8

Feigenbaum's definition of process capability is

"quality performance capability of the process with given

factors and under normal, in control conditions." Two signi

ficant elements in this concept of process capability are :

(1) Process factors and

(2) Process conditions.

Variation is always present in the measured quality of

manufactured products [1]. This variation is composed of two

components. It is produced by "chance causes" and "assigna

ble causes." Variation due to the former is inevitable. But

variation due to the latter can usually be detected and

corrected by appropriate methods.

No two products or quality characteristics are exactly

alike, because processes contain many sources of variabili

ty. The differences among products may be large, or they may

be almost unmeasurably small, but they are always present

[6]. The time period and conditions under which measurements

are made will also affect the amount of measured variation

that will be present.

From the classical standpoint of minimum requirements,

the issue of variation is often simplified: parts within

specification tolerances are acceptable, parts beyond speci

fication tolerances are not acceptable. However, to manage

any process and reduce variation, the variation must be

traced back to its source (s). The first step is to make the

distinction between common and special causes of variation.

Common cause refers to the many sources of variation

within a process that is in statistical control. They behave

like a constant system of chance causes. While individual

measured values are all different, as a group they tend to

form a pattern that can be described as a distribution as

shown in Figure 2. This distribution can be characterized

by :

(1) Location (typical value)

(2) Spread (amount by which the smaller values differ

from larger ones), and

(3) Shape (the pattern of variation - whether it is

symmetrical, peaked, e t c . ) .

In Figure 2, (a) shows that pieces can vary from each

other, but they form a pattern that, if stable, is called a

distribution, as shown in (b). The distributions can differ

in location, spread, shape or any combination of these, as

plotted in (c).

Special cause refers to any factors causing variation

that cannot be adequately explained by any single

10

m e s s VARY PfK3M EACH OTHCR:

(a) Sin » « • ^ — » SIM - Hit

BUT THCY TORM A PATTERN THAT. \P STABLE. IS CAU.EO A 0ISTRI8UTION:

(b) f • I

sai

oisTnauTiONS CAM O#ÊR m

(c)

on ANY COMSMATION 0# THCSC.

SHAPE

/**/< / /

/ / //

s \

N \

N-, s;a

IP ONLY COMMON CAUSES OP VARI* ATKM AAC PRESCNT. THf OUTPUT OP A PnOCSSS PORMS A OISTRISU* nON THAT IS STABLE OVER TIME AND IS PRCOICTABLE:

(d)

IP SPCCML CAUSES OP VARiAnON ARE PRESENT. THE PROCESS OUTPUT IS NOT STABLE OVER TIME AND tS NOT PREDICTABLE:

(e)

Figure 2 : Variation - Common and Special Causes [6]

11

distribution of the process output, as would be the case if

the process were in statistical control. Unless all the

special causes of variation are identified and corrected,

they will continue to affect the process output in unpre

dictable ways. The common and special causes of variation

are also shown in Figure 2.

From Figure 2, it is clear that if only common causes

of variation are present, the output of a process forms a

distribution that is stable over time and is predictable, as

in the case of (d). Whereas, on the other hand, referring

to (e), if special causes of variation are present, the

process output is not stable over time and is not predict

able.

As much of the discussion of process capability concen

trates on the analysis of sources of variability, it is

worthwhile, therefore, to consider the possible sources of

variation in manufactured product. A diagram of the break

down of sources of variability in manufactured product is

discussed by Grant and Leavenworth [7] and shown in Figure

3, described below :

(a) The long term variation in product,for convenience

termed the product spread, may be measured from a

histogram made up from inspection data taken over a

12

(a) y\

Lot-to-iot variation

rs. (b) Stream-to-stream variation

(c) y\ Time-lo-time variation

z\ id) -^

Piece positional variation

Error of measurement

(e)

Product spread in form of frequency histogram

Wittiin lot variation

Within stream variation

^ ^

Within time variation

^ ^

Piece-to-piece vorialion

InherenI process variation

Equipment error Human error

Figure 3 : Sources of Variability [7]

13

substantial period of time. If the process is

shifting, there will be some difference between the

process average, and possibly the standard devia

tion, from lot to lot. One of the objectives is to

eliminate or markedly reduce this lot to lot

variability.

(b) The distribution of product flowing from several

streams is formed by a weighted average of the

distributions of each individual producing unit.

The variability, termed stream to stream variabi

lity, of this weighted average will frequently be

much greater than the variabilities inherent in the

individual streams.

(c) Another factor contributing to product spread is

the time to time variation, as shown in this

figure.

(d) In many cases, physical inspection measurements may

be taken at many different points on a given unit,

which may not lead to consistent results. Such

differences are referred to as piece positional

variability. This within piece variability neces

sitate changes in tooling, material or machinery.

14

(e) There are many examples in industry where the

inherent error of measurement constitutes a signi

ficant portion of the apparent product spread. The

remaining source of variability is the piece to

piece variability of a single production entity,

the inherent process capability.

>

In this context, it is worthwhile to mention the Pareto

principle, as applied to quality problems. This principle is

useful in any effort to reduce costs. It states [2] that

"quality losses are always maldistributed in such a way that

a vital few quality characteristics always contribute a high

percentage of the quality losses." The identification of

this maldistribution is of great help to managers because it

helps to identify the few projects that can be undertaken to

reduce the bulk of the quality losses.

Experimental Design

Experiments are carried out by investigators in all

fields of study either to discover something about a parti

cular process or to compare the effect of several conditions

on some phenomena. If an experiment is to be performed most

efficiently, then a scientific approach to planning the

experiment should be considered. Statistical design of

experiments refer [14] to the process of planning the

15

experiment, so that appropriate data will be collected,

which may be analyzed by statistical methods resulting in

valid and objective conclusions.

Many of the early applications of experimental design

methodology were in the agricultural and biological

sciences. Applied to industrial problems, it has been found

that experiments can be designed so that a relatively large

amount of information can be obtained from relatively few

experimental runs, through the use of statistical methods.

The objective of many research projects is an experi

mental investigation of the effects of a number of variables

upon some response of interest. The design of the experiment

is crucial because it determines the quantity of information

in the experiment relevant to the various unknown parameters

in the process.

The crucial step in any capability study is the tracing

and identification of causes. If the causes remain deeply

hidden, then use of a designed experiment to break up the

variation into component parts is a suitable technique [21].

Before reviewing the experimental designs used in this

section, the following definitions of some of the terms used

in this chapter are provided [20]:

16

Experimental Unit - eu - the largest collection of experi

mental material to which a single independent application of

treatment is made at random, i.e., animal, group of animals,

leaf, piece of metal, batch of chemicals, etc.

Treatment - what is done to the experimental material.

Sampling Unit - su - the unit of material on which the

treatment effect is measured, a fraction or part of an eu.

Experimental Error - usually a measure of the variation

which exists among observations on eu's treated alike.

Sampling Error - usually a measure of the variation which

exists among observations on su's treated alike.

Randomization - the ordering of the experiment which helps

assure that valid or unbiased estimates of error and treat

ment means are obtained by "averaging out" effects of un

controlled variables which are present. Randomization is

the key to design of experiments.

The two basic principles of experimental design are

replication and randomization [14]. Replication refers to a

repetition of the basic experiment. Randomization refers to

both the allocation of the experimental material and the

order in which the individual runs or trials of the

experiment are to be performed. These are randomly deter

mined, depending on the design utilized.

17

In order to use the statistical approach to designing

and analyzing an experiment, it is necessary that everyone

involved in the experiment should have a clear idea, in

advance, of exactly what is to be studied, how the data is

to be collected, and at least a qualitative understanding of

how this data is to be analyzed. Montgomery [14], gives a

brief outline of the recommended procedure for the above as.

follows :

(1

(2

(3

(4

(5

(6

(7

Recognition of and statement of the problem.

Choice of factors and levels.

Selection of a response variable.

Choice of experimental design.

Performing the experiment.

Data analysis.

Conclusions and recommendations.

The different types of experiments that can be per

formed and also the related experimental design process can

be discussed now. Basically, there are two types of experi

ments [8] :

(1) Single factor and

(2) Two or more factor experiments, where the factors

can be

18

(a) crossed or

(b) nested .

Whenever only one factor is varied, whether the levels

be quantitative or qualitative, fixed or random, the experi

ment is referred to as a single factor experiment. A nested

factorial experiment is an experiment in which some factors

are crossed with others and some factors are nested within

others.

Two factors say A and B are said to be crossed if

every level of A occurs with every level of B. Factor B is

said to be nested in factor A if every level of B occurs in

one and only one level of A and levels of B within A are not

identical from A level to A level.

If the order of experimentation applied to the several

levels of the factor is completely random, the design is

called a completly randomized design. Experimental material

for this type of experiment should be nearly homogeneous.

The randomized block design is an experimental design

in which the treatments are randomized within a block and

several blocks are run. When each treatment appears once and

only once in each block, the design is referred to as a

complete randomized block. If not all treatment combination

19

can be included in one block, then it is termed an incom

plete block design.

A randomized complete block design is typically used

when the experimental units can be meaningfully grouped. The

number of experimental units in a group is equal to the

number of treatments or some multiple of it. Such a group is

called a block or replicate.

A block is a collection of experimental units put

together so that the anticipated variability among units in

the same block is less than that of different blocks [19].

The objective of blocking is to compensate before hand for

the suspected fact that certain experimental units, if

treated alike, will behave differently or to remove a

recognized source of variation from experimental error by

virtue of a restriction on randomization. Variability among

blocks does not affect differences among treatment means

since each treatment appears in every block.

The randomized complete block design is a balanced

design, where each treatment appears once in each block and

each block contains all treatments. This design is used more

frequently than any other design. When the experimental

units have been assigned to blocks, they are numbered in

some convenient manner. Treatments are also numbered and

20

then randomly assigned to the units within any block. A new

randomization is carried out for each blodk.

The randomized complete block design has many advantages

over other designs. It is usually possible to group experi

mental units into blocks, so that more precision is obtained

than with the completely randomized design. There is no

restriction on the number of treatments or blocks. If, as a

result of a mishap, the data from a complete block or for

certain treatments are unusable, these data may be omitted

without complicating the analysis. If the experimental error

is hetrogeneous, unbiased components applicable to testing

specific comparisons can be obtained [8].

The chief disadvantage of the randomized complete block

design is that when the variation among experimental units

within a block is large, a large error term results. This

may occur when the number of treatments is large.

In many experimental situations, several observations

are made within the experimental unit; the unit to which the

treatment is applied. Such observations are made on subsara-

ples or sampling units. When more than one observation is

made on at least one experimental unit, it is necessary to

distinguish between sampling error and experimental error.

These are the two sources of variation which contribute to

21

the variance applicable to comparisons among treatment

means.

Sampling error measures the failure of the observations

made in any experimental unit to be precisely alike. Experi

mental error is often expected to be larger than sampling

error. In other words, variation among experimental units is

often expected to be larger than variation among subsamples

of the same unit.

For a proper evaluation of the experimental data, the

model must be specifically stated and specified rules of

randomization must be followed. Two common models are the

fixed effects model and the random effects model. A fixed

effect model is appropriate when two or more blocks (or

treatments) are deliberately (non-randomly) selected for

testing. All blocks (or treatments) about which inferences

are to be drawn should be selected. Inferences are made

about selected blocks (or treatments) only. A random effect

model is appropriate when two or more blocks (or treatments)

are selected (from a population of blocks or treatments) for

testing. Inferences are made about the populations of blocks

(or treatments).

In experimental design problems where the fixed

effects model is appropriate, two or more treatments are

22

selected for testing. These are not randomly drawn from a

population of possible treatments but are selected, perhaps

as those which show the most promise or those most readily

available. All treatments about which inferences are to be

drawn are included in the experiment.

For the fixed model with negligible interaction, both

block and treatment effects can be tested with the residual

or error mean square. However when there are fixed inter

action effects of treatments, it is not possible to make

valid F - tests.

For the random model, both treatments and blocks are

drawn at random from populations of treatment and block

effects. Inferences are drawn about the population of treat

ments and blocks rather than the particular treatments and

blocks. In most cases the blocking variable is not of major

concern other than to reduce the error. Therefore, it is

usually not interpreted beyond the general ANOVA F-test.

Normally, the expected mean squares (EMS) obtained from the

ANOVA table are appropriate for the hypothesis testing for

all the above models [8].

An experiment in which all levels of each factor in the

experiment are combined with all levels of every other

23

factor is known as a factorial arrangement of treatment

experiment. If only a fraction of a complete factorial is

run, then it is called a fractional replication.

Another experimental design, the Latin square, is one

in which each level of a factor is combined only once with

each level of two blocking variables. An incomplete Latin

square is called a Youden square. An extension of the Latin

square is called a Graeco-Latin square.

In some multifactor designs involving randomized

blocks, one may be unable to completely randomize the order

of runs within the block. This often results in a generali

zation of the randomized block design called a split plot

design. The split plot design, a commonly used design, has

its main effect confounded with blocks due to the practical

necessites of the order of experimentation.

The analysis of variance (ANOVA), probably the most

useful technique in the field of statistical inference, is

essentially an arithmetic process for partitioning a total

sum of squares into components associated with recognized

sources of variation [20]. It has been used to advantage all

fields of research where data are measured quantitatively.

The name analysis of variance is derived from a parti

tioning of total variability into it's component parts. For

24

example, in a completely randomized design with only one

factor, the total corrected sum of squares is used as a

measure of overall variability. The sum of squares due to

treatments is an estimate of the common variance between

treatments, and sum of squares due to error is an estimate

of the common variance within each of the treatments.

The analysis of variance, under a true null hypothesis,

provides two independent estimates of the experimental error

2

(a ); one based on the inherent variability within treat

ments and one based on the variability between treatments.

If there are no differences in treatment means (a true null

hypothesis), these two estimates are very similar and if

they are not, the observed difference must be caused by

differences in treatment means [14].

In the analysis of variance where tests of significance

are made, the basic assumptions are [19] :

1. Treatment and environmental effects are additive,

2. Experimental errors are random, independently and

normally distributed about zero mean, with a common

variance.

3. The variances of the experimental errors are equal

throughout the experiment [8].

25

Various techniques for estimating the variability ulti

mately use the measure of the standard deviation [16]. These

techniques may include a direct calculation of the standard

deviation from all the data, the use of normal probability

paper, etc. But, when they are used in the traditional

approach, none of them assumes that the process is in

statistical control.

In general, moderate departures from normality are of

little concern in the fixed effects analysis of variance

[14]. The random effects model is more severly impacted by

nonnormality. A very common defect that shows up on normal

probability plots is one residual that is very much larger

than any of the others. Such a residual is often called an

outlier. The presence of one or more outliers can seriously

distort the analysis of variance; so when a potential out

lier is located, careful investigation is called for. The

outlier may be a result of the process or it may be a result

of an assignable cause. In the first case, a serious depar

ture may cast doubt on the validity of the analysis. In the

second case, the experimental combination may need to be re

run, to reflect the actual process.

26

Capability Indices

Lawrence P. Sullivan [21] recommends that one focus on

methods to reduce the variability of process output, not

just meeting specifications. He suggests a new definition of

"manufacturing quality" as "product uniformity around the

target rather than conformance to specification limits."

In the same paper, Sullivan mentions a Japanese statis

tical consultant, Genichi Tanguchi, who showed that even

though some shipped parts from Sony (a manufacturing firm),

were out of specification in Japan, the San Diego (USA)

plant showed a greater loss per unit than the loss regis

tered by the Japanese plant. He showed that the more uniform

distribution around the target was ultimately less expen

sive.

Capability indices can be used to efficiently summarize

information in a succint manner [10]. The indices Cp, CPU,

CPL, k and Cpk form a group of complementary measures that

comprise a convenient unitless system. These measures

collectively determine whether a process has sufficiently

low variability to meet the process specification.

A typical baseline is the assessment of whether the

natural tolerance (6<r) of a process is within specification

limits. An alternate formulation is to evaluate the

27

capability index Cp. The process potential index (Cp) is the

ratio of the allowable process spread to the actual process

spread. A Cp of 1.0 indicates that a process is judged to be

capable. A minimum value of Cp = 1.33 is generally used for

an ongoing process. This ensures a very low rejection rate.

If the value of Cp exceeds 1.33 or decreases below 1.0, then

the process is judged not to be capable, and the process

will produce a relatively high proportion of nonconforming

items, resulting in a higher rejection rate.

The Cp index is used to measure the process potential

for two sided specification limits. The Cp index measures

potential process variability, since only the process spread

is related to the specification limits. The location of the

process mean is not considered. It would be possible to have

any percentage of parts outside the specification limits

with a high Cp by merely locating the process mean suffi

ciently close to a specification limit. Thus, only potential

performance of a process is quantified by Cp and can only be

attained for sure when the process mean is equal to the

midpoint of the specification limits. In general, Cp is the

ratio between the allowable process spread to the actual

process spread.

28

The CPU and CPL are a measure of process performance

relative to the upper specification limit and the lower

specification limit, respectively, i.e.,

CPU = allowable upper spread / actual upper spread

CPL = allowable lower spread / actual lower spread.

The Cpk index is related to the Cp index, but utilizes

the process mean and can be considered a measure of process

performance for two sided specification limits, i.e., Cpk =

Min{CPL, CPU}. Thus, the Cpk index relates the scaled

distance between the process mean and the closest speci

fication limit.

And lastly, the k index is a measure of the deviation

of the process mean from the midpoint of the specification

limits. The k = actual deviation / (allowable spread / 2).

Generally k has a value between 0 and 1. In the single

limit case, a CPU or CPL =1.0 implies half as many noncon

forming items as when Cp = 1.0 in the two limit case. A

typical benchmark of Cpk is 1.33, which will make noncon

forming units unlikely in many situations.

Perhaps the greatest value of these indices is that

their use encourages efforts to prevent the production of

nonconforming products. They provide a method to monitor

29

continuous improvement on a broad scale. Also, these

measures enable effective communication of process potential

and performance information in a language that can be easily

understood.

Sample Size

The question of how large a sample to take from a

population for making a test is one often asked of the

statistician. Unfortunately, there is no correct sample size

that can be determined without additional information. The

size of the sample required for a given experiment is in

fluenced by the values selected for alpha and beta risks

and by the value or values of the population variance. The

correct sample size "assures" the experimenter that risks of

error will be equal to or less than alpha and beta when the

experiment is completed.[4]

When comparing properties of interest betweeen two

populations, the experimenter will be confronted with two

alternative possibilities, which in statistical terms can be

stated as :

(1) Null hypothesis (HQ) - no statistically significant

difference exists between the properties of in

terest in the two populations (i.e., Hg : -: "-:

for population means).

30

(2) Alternate hypothesis (H,) - a statistically sin^nifi a

cant difference does exist between the properties

of interest in the two populations (i.e. H : M^ Û2 a

for population means).

The experimenter runs the risk of making either of two

statistical errors in any experiment :

(1) Alpha error (a) - the experimenter accepts the

alternative hypothesis as being true when the null

hypothesis is actually true.

(2) Beta error (g) - the experimenter accepts the null

hypothesis as being true when the alternate hypo

thesis is actually true.

In choosing a sample size to detect a particular <differ

ence, one must admit the possibility of either a Type I or

Type II error and choose the sample size accordingly [20].

Calculations of the number of replicates required depends on 2

(1) An estimate of o,

(2) The size of the difference to be detected,

(3) The assurance with which it is desired to detect the

difference (1 - 6 ),

(4) The level of significance to be used in the actual

experiment (Type I error), (5) Whether a one or two - tailed test is required.

31

Several approaches have been made to the general prob

lem of size of an experiment [20]. OC curves often play an

important role in the choice of sample size in experimental

design problems [14]. The OC curve defines "the probable

outcome of an experiment for every possible value of u "

[4].

The difference betweeny- and the potential value ofu

at which the 3 risk applies is termed delta and is denoted

by the Greek letter delta (5). The 5 reflects the engineer

ing requirements of the experiment [4]. The MO is a fixed

number, to which the experimenter makes a comparison with

the population, e.g., a value in a quality control specifi

cation [4] .

Under assumptions of normality and a single population,

the formulas for calculating sample size are siiiiple and easy

to use, and, more importantly, they are the primary criteria

for the validity of experiments. If, for a given experiment, 2

the values for a , 8 , and 5 are carefully chosen, and if, a

is accurately known, then the value for N (sample size) can

be computed [4]. With even one sample less than N, the

experiment would be invalid, since either a or 6 would then

be greater than the a or 6 that was specified as absolutely

necessary. On the other hand, if even one sample more than N

32

is tested, the experimenter would waste time and money,

since either a or 8 would then be smaller than the " or g

that was specified as necessary. The correct sample size

assures that the risks of error will be equal to or less

than a and S when the experiment is completed.

An experiment would be suspect if an experimenter pro

perly calculated an N of say 20 samples, but then decided

that this sample size was excessive for the time or money

available and proceeded to test a sample of size 10 instead

of the 20 required. Nevertheless, it is always necessary to

be realistic, and sometimes it will be necessary to chose an

N based on time and resources available, rather than on a

calculation from the proper formula. In this type of situa

tion, it is recommended that the efficiency of the experi

ment be evaluated before it is started. If the a error is

critical, the way to determine the efficiency of the experi

ment is to compute the value of ci that would be obtained

by using the correct 8 and 5 an^ the proposed sample

size. On the other hand, if the 8 error is critical, the

efficiency of a proposed experiment is determined by

computing the value of 8 with the proposed sample size and

correct a and S • Also, a limited pilot study be performed

before actually performing the full experiment. This may

33

help provide a general idea of how large a sample size would

be necessary.

Generally, the experimenter tends to increase the sam

ple size to make sure that the results will be conclusive.

In such cases, the sample size may be chosen, based on past

experience.

In more complex cases, i.e. in multifactor designed

experiments, usually a pilot study of the experiment will be

helpful for developing the full experiment. In such cases,

it seems to be intuitively obvious that the larger the

sample size, the more accurate and precise the result.

In general, determining the sample size in these type of

experiments becomes very difficult and complex, if not im

possible, from a practical standpoint. In these cases, past

experience with the process, or other similar processes, and

pilot study results are considered along with budget,

resources and timeliness. The result many times is a design

and sample size based to a large degree on professional

judgement.

CHAPTER III

PROCEDURE

Model Development

For the development of this research a randomized

complete block design was used as the basis and four models

were adapted and developed to analyze the process variation

capabilities :

Model I Single factor (random effect) experiment

with subsampling.

Model II : Single factor (fixed effect) experiment

Model III

Model IV

with subsampling.

Two factor experiment with subsampling and

Three factor experiment with subsampling.

The above four models were chosen for the development

of this research, as they represent simple cases which are

frequently encountered in actual practice. The need and

the general explanation for all four models is discussed

below.

Model I

The need for this model arises in equipment development

work when it is important, to assess effects of reoccuring

conditions, such as on-off cycles on equipment capability.

A need arises here to separate the possible reoccurring

34

35

"disruption" variation from the between and within experi-

mental unit (eu) variation. The response to this need is

Model I.

In this model, a repetition of the experiment will

bring in a new set of treatments, but from same population,

of treatments. By using this model one can draw infer

ences about the population of treatments. The number of

factors involved in this model is one. The blocks in the

experiment provide a measure of run to run variability, such

as that due to reoccuring daily cycles. Each run will con

tain N different experimental units. EU's within each block

are to be selected from a homogeneous batch or lot of

experimental units.

The statistical model and notation are:

ijk 1 D(I) k(i3)

i indicates run, i = 1, 2, ..., r

j indicates factor 1 within run, j = 1, 2, ..., t

k indicates locations within run and factor 1, k = 1,

A f . . . , n

Y. ., = A response measurement. ijk

A symbolic summary of the working and definition for

mulas for the sums of squares, degrees of freedom, mean

36

squares and expected mean squares in analysis of variance

of data from the randomized complete block design. Model I,

is given in Figure 4(a).

The "F" column in the ANOVA (Figure 4a) represents a

sequence of statistical tests that can be performed :

1. MSB / MSW :

This term tests for a s-signifleant (statistically

significant) between eu variability. If the test is not

statistically significant, then one concludes that between

eu variability is relatively small and it cannot be

detected in the presence of the within eu variability (at a

specified s-significance level).

2. MSR / MSB :

This term tests for a s-significant between run varia

bility. If the test is not s-significant, then one con

cludes that the between run variability is relatively small

and that the between run variability cannot be detected in

the presence of the between eu and within eu variability (at

a specified s-significant level).

Model II

This is an enhancement of Model I. Whenever one wants

to analyse the process or an experiment, in the presence of

an independent factor such as temperature, then this model

3 7.

yields the inferences about that particular treatment, along

with the run to run variability. The experiment should be

performed at different levels of the selected factor. How

ever, care should be taken to structure the factor levels in

a meaningful fashion. For example the levels should be far

enough apart to potentially yield differences, but not so

far as to yield an experiment which leaves serious doubts as

to factor level responses between those selected.

In this experiment, a repetition of the experiment

would bring the same set of treatments into the new experi

ment. By using this experiment, one can draw inferences

about the particular treatments. The number of factors in

volved in this model is one. The blocks in the experiment

provide a measure of run to run variability. The factor

levels, for the single factor considered, should appear

once and only once in each run. Factor levels should be run

in a random order during each run, and each run should be

re-randomized.


Y . ., =» U + R . + T . + B, ,. ., + W . .^ i lm 1 3 id:) md^l)


j indicates factor 1, j = 1, 2, ..., t

1 indicates eu's within run and factor 1, 1 = 1

38

m indicates locations within run, eu and factor 1, m =

X , £,f . . . , n

iilm ~ ^ Response measurement.




of data from the randomized complete block design is

given in Figure 4(b).

The "F" column in the ANOVA (Figure 4b) represents a


1. MSB / MSW :

This term tests for a statistically significant between

eu variability. If the test is not statistically signifi

cant, then one concludes that the between eu variability is

relatively small and cannot be detected in the presence of

the within eu variability (at a specified s-significance

level).

2. MST / MSB :

This term tests for a factor 1 effect. If no signifi

cant interaction is present and this test is s-significant,

it indicates that the response of all levels of factor 1

are not the same.

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3. MSR / MSB :

This term tests>for a s-significant between run varia

bility. If the test is not s-significant, then one con

cludes that the between run variability is relatively small

and cannot be detected in the presence of the between eu and

within eu variability (at a specified s-significant level).

Model III

This is an enhancement of Model II. Whenever one wants

to analyse the process or an experiment in the presence of

two factors, such as temperature and humidity, then this

model provides inferences about the particular treatments,

along with the run to run variability. The experiment should

be performed at different levels of the selected factors.

In this experiment the number of factors involved are

two. The factor combinations are run in a random (pairwise)

order during each run. The point in doing this is to avoid

the introduction of any systematic biases in the machine

settings or performance of the experiment. The blocks in the

experiment provide a measure of run to run variability. All

treatment combinations should appear once and only once in

each run. Also, the combinations should be run in a random

order during each run, and each run should be re-randomized.

41


^ i j k l m = y + R i + T j + C k + T C j k + B i ( i j k ) + Wni(ijik)

i indicates run, i = 1, 2, . . . , r


k indicates factor 2, k = 1, 2, ..., c

1 indicataes EU's within run, factor 1 & factor 2,1 = 1

m indicates locations within run,eu,factor 1 & factor 2

m = 1, 2, ..., n

^ ij]tLm - A Response measurement

A symbolic summary of the working and definition form

ulas for the sums of squares, degrees of freedom, mean


of data from the randomized complete block design is

given in Figure 5.

The "F" column in the ANOVA (Figure 5) represent a


1. MSB / MSW :

This term tests for a s-significant (statistically

significant) between eu variability. If the test is not

statistically significant, then one concludes that the bet

ween eu variability is relatively small and it cannot be

detected in the presence of the within eu variability (at a

specified s-significance level).

42

1 ^

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f p« r*

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U H OO CO CO CO

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CO

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43

2. MSTC / MSB :

This term tests for a factor 1 * factor 2 interaction

effect. If the test is s-significant, it indicates that the

factor 1 or factor 2 effect is not the same at each level of

factor 2 or factor 1, respectively. Therefore, the factor 1

and 2 effects should not be interpreted independent of one

another.

3. MSC / MSB :




are not the same.

4. MST / MSB :



it indicates that the response of all levels of factor 1 are

not the same.

5. MSR / MSB :

This term tests for a s-significant between run varia

bility. If the test is not s-significant then one con

cludes, that between run variability is relatively small

and that the between run variability cannot be detected in

44

the presence of the between eu and within eu variability (at

a specified s-significant level).

Model IV

This is an enhancement of Model III. Whenever, one

wants to analyze a process or an experiment in the presence

of three factors, such as temperature, humidity and

vibration, then this model provides inferences about the

particular treatments, along with the run to run varia

bility. The experiment should be performed at different

levels of the selected factors.

In this experiment, the number of factors involved are

three. The factor combinations are run in a random order

during each run. The point in doing this is to avoid any

systematic biases in the machine settings or performance of

the experiment. The blocks in the experiment provide a

measure of run to run variability. All treatment combina

tions should appear once and only once in each run. Also,

the treatment combinations should be run in a random order

during each run, and each run should be re-randomized.


Y = u + R + T - » - C + D + T C + T D + i jkpl in i j 1< P D^ DP

CD j ^ + T C D j ] ^ + 3 i ( i j k p ) + Wn^(ijkpi)

45



k indicates factor 2, k = 1, 2, ..., c

p indicates factor 3, p = 1, 2, ..., d

1 indicates eu's within run, factor 1, factor 2 and

factor 3, 1 = 1

m indicates locations witnin run, eu, factor i, factor

Y

2 and factor 3, m = 1, 2, ..., n

i- kplm " - Response measurement




of data from this design is given in Figure 6.

The "F" column in the ANOVA (Figure 6) represent a


1. MSB / MSW :

This term tests for a statistically si jni f i.: in t o-twein

eu variability. If the test is not statistically signifi

cant, then one concludes that the between ea v. r u b i 1 11 / i .s

relatively small and it cannot be detected in the ocisence

of the within eu variability (at a specified s-s i-j n i !. i : i nc,

level).

46

^" in o s

» % s o

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a _ 2 S • • •

> > > ^ \f 7 \? ^ \? • •

^4

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a CO

2 m

CO

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a CO

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73

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47

2. MSTCD / MSB :

This term tests for a factor 1 * factor 2 * factor 3

interaction effect. If the test is s-significant, it indi

cates that any one of the factor effects is not the same at

each level of the other two factors, respectively. There

fore, the factor 1, 2, and 3 effects should not be inter

preted independent of one another.

3. MSTC / MSB :





and 2 effects should not be interpreted independent of one

another.

4. MSTD / MSB :





and 3 effects should not be interpreted independent.

5. MSCD / MSB :



48



and factor 3 effects should not be interpreted independent

of one another.

6. MSD / MSB :




are not the same.

7. MSC / MSB :




not the same.

8. MST / MSB :




not the same.

9. MSR / MSB :

This term tests for a s-significant between run

variability. If the test is not s-significant, then one

49

concludes, that the between run variability is relatively

small and that the between run variability cannot be

detected in the presence of the between eu and within eu

variability (at a specified s-significant level).

The capability estimates for all models are developed

from the expected mean squares which are estimated by

the mean square column in the analysis of variance table.

Provided the process selected for the analysis is generally

in control, functioning in statistically consistent fashion,

reasonably accurate process capability estimates can be

made. If the process is operating in an erratic fashion,

poor process capabilities will be obtained. This condition

would result in a conservative estimate, in terms of stating

a capability interval width larger than would be encountered

in a condition of stastical control.

The calculations below show how one would obtain 4

sigma capabilities :

4 Sigma within eu capability = 4 ^ w

where ^ w = V MSW

4 sigma between eu capability =4 ^ b

where ^ b = \/ (MSB - MSW) / n A

4 Sigma between runs capability = 4 ^ r

50

where ^ r = \/ (MSR - MSB) / A

(A = n for Model I,

A = t * n for Model II,

A = t * c * n for Model III,

A = t * c * d * n for Model IV )

4 sigma total capability = 4 ' total

A A A

where a total = V/ a w + o b + a r

Problems with outliers may be expected to occur. The

outlier identification in this research applies to the sub-

samples of the experiment. The outliers are to be identified

in the course of the experiment due to the reasons listed in

Chapter II. The user should first identify the assignable

cause (s), before rerunning the treatment combination con

taining the outliers.

There are several formal statistical procedures for

detecting outliers. This research used a simple procedure of

measuring the distance from the outlier to the mean in terms

of thQ standard deviation [4]. For calculating the outliers,

the following formula is used :

Sigma = VZ:(Y - Y ) / (N - 1)

Low = Y - (a / 2) * Sigma

51

High = Y + (a / 2) * Sigma

If Y£ < Low (or) > High, then it is considered an

outlier.

Where a = 1, 2, 3 or 4, depending on one's desires as

to outlier identification policy. An a = 2 or a = 3 value

seems to be the most reasonable, when associated with the

normal distribution.

Organization of the Software Package

In order to utilize the capability experiment models

previously described for a realistic experiment, the ana

lysis must be computer coded. The experimental designs and

analysis package developed to carry out the objectives

stated in Chapter I are organized in the fashion outlined in

the flowchart in Figure 7. Appendix I gives the listings of

the program that form the package, and Appendix II contains

a manual for using the software package.

The software consists of a number of sections that may

easily be separated. This modular design allows for poten

tial future modifications such as adding other experimental

designs. Modules can also be removed and utilized in other

programs that use the function implemented by a module.

52

TITLE DISPLAY

BRIEF DESCRIPTION OF SOFTWARE!

FIELD LAYOUT DISPLAY

SELECTION OF DESIGN

INPUT PARAMETERS

RANDOMIZATION DISPLAY

INTERACTIVE INPUT FILE OPTION EXTERNAL FILE OPTION

OUTLIER IDENTIFICATION

DISPLAY DATA

MAKE CHANGES, IF ANY

CALCULATE AND DISPLAY RESULTS

PRINT OUT RESULTS

Figure 7: Flow Chart of the Software

53

The usefulness and versatility of the package have been

enhanced by allowing the user various menu choices such as

generated or external data sets, one factor, two factor, or

three factor experiments, display and change any data (op

tional), identification of the outliers in the subsamples

(optional), and the ANOVA table (with the statistical tests

and the observed significance levels) and capability

estimates. In addition to above, a brief description and a

field layout for all the four models is provided. Also, the

randomization display i.e., the ordering of the experiment

is another feature of the software.

General Specification of the Software

The package has been written in IBM BASICA to run on

IBM Personal Computers and computers compatible with the IBM

Personal Computer such as Zenith, Compaq, etc., with a

computer memory of 256K or more. The package is split into

two modules, namely, the title display module and the cal

culation module. Both are compiled and linked togther to

form a batch file. This procedure helps in faster cal

culations of the results, thereby increasing the efficiency

of the software.

This software can run up to 8 replications (blocks) and

samples containing up to 5 subsamples (observations). On the

54

other hand, subsamples can be increased to 10 by reducing

the replications to 5. It can handle up to 3 factors,

each factor having a maximum of 5 levels. It can accommodate

an unequal number of factor levels. For example, consider

there are 2 factors. Each factor can have different levels,

say, factor one has 3 levels and factor two has 5 levels.

For any type of input parameters, the software can display

the ordering of the experiment i.e., a randomization

pattern.

After a brief description of the software, including

the field layout and the input of the parameters, the

software creates an interactive input file for the data, or

an external data set can be selected. This external file can

be created using any text editor, such as the EDLIN utility,

which is a part of the operating system software of the IBM

PC or by using some other text editor which develops an

ASCII input file. The feature of saving the input parameters

and the data into an external file is incorporated in this

package.

The entire data set is displayed in a readable form.

Changes can be made, if necessary, in the data set. Using

the outlier option, identification of subsample outliers at

1 sigma, 2 sigma, 3 sigma or 4 sigma levels, for a set of

55 .

subsamples, can be performed and, if necessary, the outlier

data can be changed, pending a rerun of the treatment combi

nations. Once everything is satisfactory, the program runs

and performs calculations to obtain the results. It displays

the model of the experiment chosen for the analysis, along

with the notations. The ANOVA table follows, which in

cludes the statistical tests (F - Tests) and the analysis,

along with the observed significance level. Finally, the

expected mean squares and the process capabilities are

printed. The level of sigma for the capabilities can be

specified by the user. Appropriate messages are displayed

on the screen in order to make the user aware of all the

possible options.

CHAPTER IV

VALIDATION AND RESULTS

Validation

Establishing the validity of a computer software

package is important in that a user can be assured that the

results generated are correct. The validity of the developed

package was established in a variety of ways. The package

was checked for accuracy by running various example problems

with known results, and other problems were checked with

solutions obtained by hand calculations. Results from the

package are discussed in the next section of this chapter.

First, data taken from worked examples by Hicks [8]

were used as external data set inputs to the package.

Results were obtained in cases of single factor, two factor

and three factor randomized complete block design experi

ments with subsampling, that corresponded very closely to

the results given by Hicks. In the cases of response data

with and without subsampling, the package gave the same

results as a manual solution of the problem with the same

set of data, using methods given in the aforementioned book.

Also, the ANOVA table and the observed significance

levels calculated from the package for one of the designs

were checked with the corresponding values generated by the

56

57

SAS package. Both results were very close to one another,

with a negligible difference between them.

Results

In all, four problems were run through the developed

package, one for each model. The results obtained from

the software are shown on the following pages in Tables 1,

2, 3, and 4. For all the problems chosen, the ANOVA tables,

along with the statistical tests and observed significance

level, obtained are shown. Then finally, the process

capability estimates are shown. The problem summaries are

shown below.

Problem 1 Summary :

Number of factors = 1

Number of factor levels = 5

Number of runs = 3

Number of subsamples = 5

Software Hand SAS

MSR 2.6757E+07 2.6753E+07 2.6758E+07

MSB 9.4338E+06 9.4336E+06 9.4337E+06

MSW 8.3872E+04 8.3874E+04 8.3872E+04

F(l) 2.5060E+00 2.5051E+00 2.5055E+00

F(2) 6.7490E+03 6.7494E+03 6.7491E+03

58

Table 1:, Results of Model I

THS ANOVA TABLS

SOUIICJ OP SS MS

TOTAU 74 1 .6<80B«08 »UK 2 5 . 3 5 1 4 B * 0 7 2 . 6 7 5 7 B * 0 7 2.SOfiEÔO BST1S8N 8U* • 12 1 .1321B»0a 9. 4338E4>06 6. 749B>03 WITMIM SU* • 60 8. 3872B*Q4 1.3979E<»03

SIGMA WITHIN BU • 37. 38809

SI6NA BBTHBEN BU • 1 3 7 3 . 4 9

SIGMA BBTVBBN RUN • 8 3 2 . 4 2 9 4

4 SIGMA CAPABILITIES ARE :

WITHIN BU CAPABILITT - 1 4 9 . 5 5 2 2

BBTWeSN BU CAPABILITT • 5 4 9 3 . 9 6

aBTWCSN RUN CAPABILITT • 3 3 2 9 . 7 1 9

TOTAL CAPABILITT • 6 4 2 5 . 9 6 2

59

Table 2: Results of Model II

TH8 ANOTA TABLS

SOUaCB OP ss MS

TOTAL 79 \.6937t*Qa RON 3 6 . 1 4 1 9 B * 0 7 2 . 0 4 7 3 E * 0 7 1 . 3 3 4 E * 0 0 Pr««aur« 3 7. 4Q72B4>06 2. 4691 E^06 2. 21 2E-01 BBTUBSN BU* • 9 1 .0046Bâ8 1 .1163E^07 a. 4S0Ei>Q3 VITHIN BU* 8 64 8. 4S44B«-04 1.3210E«-03

SIGMA WITHIN BU • 3 6 . 3 4 5 5 6

SIGMA BBT'WBBN BU • 1 4 9 4 . 0 7 7

SIGMA BBTUBBN RUN • 682 . 2861


WITHIN BU CAPABILITT • 1 4 5 . 0 8 2 2

B8TWESN BU CAPABILITT • 5 9 7 6 . 3 0 8

BBTHESN RUN CAPABILITT • 2 7 2 9 . 1 4 4

TOTAL CAPABILITT • 6 5 7 1 . 5 7 7

60

Table 3: Results of Model III

THI ANOVA TABLB

SOURCE

TOTAL HON P««4 Sp««tf P««4 *S9««4 BtTVtBM BU*• RUN 'Pcctf RUN 'Spc^d RUN •Pc^d WITHIN BU* s

0

22

•Sp««d 1 180

SS

6.7374E»08 4.6968B«06 5.1613E«08 1.3268E«08 1.8247B»07 1.7292E«06 1.3647B«06 2.36298^05 1.2416E«05 2.57S4B*05

MS

1.1742B*06 2.SaQ6B>08 6.6342E>07 4. S618E4'06 5. 3912B«>04

1 .43QaE^03

2 . 1 7 8 E * 0 1 4 . 7 8 7 E * 0 3 1 .231E*03 8. 462E^01 3 . 7 6 a E * 0 1

SIGMA WITHIN BU

SIGMA BITWBBN BU

SIGMA BBTWBBN RUN

3 7 . 8 2 S 3 3

1 0 2 . 4 5 1 2

1 5 7 . 7 8 3


WITHIN BU CAPABILITT - 1 5 1 . 3 0 1 3

BETWEEN BU CAPABILITT " 409. 8048

BETWEEN RUN CAPABILITT • 6 3 1 . 1 3 1 9

TOTAL CAPABILITT • 767 . 5673

Table 4: Results of Model IV

61

SOURCE

TOTAL RUN Tamp TiflM HuMxdl ty T«iap T«nip Ti iM Tamp

•Tlflw • H u m i d i t y « H u « i d i t y 'Timrn « H u i a i d i t y

BETWEEN BU* s RUN RUN RUN RUN RUN RUN RUN

•T«Mp • T i l M • H u m i d i t y *T«mp •T im« *T«mp ' H u m i d i t y •T im« " H u m i d i t y *T<mp *Tim« "Humid

WITHIN BU* a

OP

539 3 2 2 2 4 4 4 8

78 6 6 6

12 12 12 24

4 3 2

SS MS

1 2 0 4 8 * 0 9 2 0 7 1 E * 0 7 7586E'»06 3S42E>06 4019E«>07 11 14E*'05 5 5 3 0 E * 0 5 3698E-»-05 84728«>06 1 7 7 7 B » 0 7 6 6 3 1 E « 0 6 4200E-I-05 1 5 5 2 E ^ 0 6 439SE-»>Q6 1 1 3 0 E « 0 6 1 5 1 6 B ^ 0 6 1 5 1 2 E * 0 7 0 1 7 7 B * 0 9

1 . 4. 1 . 7. 1 . 1 . 1 . 2. 5.

2.

0 6 9 0 E * 0 7 3 7 9 3 E * 0 6 1 7 7 1 E * 0 6 0093E^Q6 2 7 7 8 E * 0 5 63a2E>Q5 S924E^0S 3090E«'05 3 5 6 0 E ^ 0 5

3559E«-Q6

1 . 3 . 2. 1 . 2. 3 . * • 4. 2.

996E*01 176E*00 198£>0Q 309E*01 3 a 6 E - 0 1 0 5 9 E - 0 1 973E-Q1 3 1 1 E - 0 1 : 7 3 E - 0 1

SIGMA WITHIN EU

SIGMA BETWEEN EU

SIGMA BETWEEN RUN

1 5 3 4 . 8 8 9

4 1 . 7 4 2

2 7 4 . 2 6 4 5


WITHIN EU CAPABILITT • 6 1 3 9 . 5 5 5

BETW88N EU CAPABILITY • 1 6 6 . 9 6 8

BETWEEN RUN CAPABILITT • 1 0 9 7 . 0 5 8

TOTAL CAPABILITT • S 2 3 6 . 3

62

Problem 2 Summary:


Name of the factor = Pressure

Number of factor levels = 4

Number of runs = 4


Software Hand SAS

MSR 2.0473E+07 2.0473E+07 2.0473E+07

MST 2.4691E+06 2.4690E+06 2.4691E+06

MSB 1.1163E+07 1.1164E+07 1.1163E+07

MSW 1.3210E+03 1.3210E+03 1.3210E+03

Problem 3 Summary:


Name of the factors = Feed & Speed

Number of levels for Feed = 3

Number of levels for Speed = 3

Number of runs = 5


Software Hand SAS

MSR 1.1742E+06 1.1742E+06 1.1742E+06

MST 2.5806E+08 2.5806E+08 2.5806E+08

MSC 6.6342E+07 6.6342E+07 6.6342E+07

MSTC 4.5618E+06 4.5618E+06 4.5618E+06

63

MSB 5.3912E+04 5.3912E+04 5.3912E+04

MSW 1.4308E+03 1.4307E+03 1.4308E+03

Problem 4 Summary:


Name of the factors = Temp, Time, & Humidity

Number of factor levels for each factor = 3

Number of runs = 4


Software SAS

MSR 1.0690E+07 1.0690E+07

MST 4.3793E+06 4.3793E+06

MSC 1.1771E+06 1.1771E+06

MSD 7.0093E+06 7.0093E+06

MSTC 1.2778E+05 1.2778E+05

MSTD 1.6382E+05 1.6382E+05

MSCD 1.5924E+05 1.5924E+05

MSTCD 2.3090E+05 2.3090E+05

MSB 5.3560E+05 5.3560E+05

MSW 2.3559E+06 2.3559E+06

Results Summary:

From this, it can be seen that there is only a negli

gible difference between the values calculated from the

package and the corresponding values generated by the SAS

64

package and by hand calculations. Also, every possible op

tions of the package was run and thoroughly checked.

CHAPTER V

CONCLUSIONS AND CONTRIBUTIONS

Conclusions

From this research, it can be concluded that relatively

simple designed experiments are applicable to process capa

bility studies, if proper consideration is given to their

development. From the given set of process capability experi

ments previously discussed and an extensive equipment

development data set from the FSI Corporation, the following

conclusions can be drawn :

(1) By using the ANOVA models dicussed, one can

effficiently measure and quantify the components

of process variation capabilities.

(2) This research has applied and integrated recogniz

able experimental designs to capability studies.

(3) The techniques developed are compatible with the

development process and testing of equipment as well

as the process of industrial experimentation to

assess production capabilities in a manufacturing

organization.

(4) This research can benefit the practitioner in

assessing the effect of the factors involved in the

experiment as well as quantifying the components of

process variation.

65

66

(5) The use of organized and recognized randomization

techniques will help assure that valid or unbiased

estimates of error are obtained by "averaging out"

effects of uncontrolled variables.

(6) The software developed for this research is

compatible with microcomputers, hence the data can

be entered into the package as the experiment is

conducted. This can allow for "on the spot" reruns

for combinations which are "outliers" by virtue of

assignable cause.

The meaningful calculation or the estimation of the

sample size is extremely complicated, if not impossible, in

complicated experimental designs. One other interesting

conclusion is that the usefulness and versatility of the

package have been enhanced by allowing the user various menu

choices, printing appropriate messages and a high degree of

interaction.

The execution time, for the calculations to be per

formed and printed out, varies from about 3 seconds for a

"typical" single factor model to about 8 seconds for a

"typical" three factor model. This short execution period is

acheived because of the compilation of the program. Due to

the compilation of the software, it uses only 256K memory

67

and the software can function on its own without any aid

from other software.

Contributions

This study yielded a general purpose process capability

estimating package, which consists of both research and

computer development, that can be used for one of two pur

poses :

(1) To aid in linking experimental design with

capability studies,

(2) To provide general guidance to practitioners

involved in the analysis of capability studies

through the development of an interactive analysis

tool.

Through the careful documentation in the program, to

aid in interpretation, randomization, analysis, etc., the

usage of such techniques should be advanced. Usually a

limiting factor, to the acceptance of such methods by prac

titioners, is their lack of understanding of such techniques

with respect to analysis and interpretation.

As a result of this research, equipment manufacturers

will be able to more readily assess their equipment's capa

bility and analyze its associated components. This can be

68

done in both the equipment development phase as well as in

the implementation of the equipment into a production envi

ronment. It will also aid the producer (equipment user) to

assess the potential of his equipment in order to help plan

work assignments to the equipment. Compared with more

general analysis packages, like SAS [17], this software has

some unique features; randomization of the experiment, out

lier identification, interpretation aids, and the calculation

of process capabilities.

BIBILIOGRAPHY

(1) Allan, Douglas H.W., Statistical Quality Control,

Reinhold Publishing Corporation, New York, 1959.

(2) Case, Kenneth E., and Jones, Lynn L., Profit Through

Quality: Quality Assurance Programs For Manufacturers,

American Institute Of Industrial Engineers Inc., Norcross,

Georgia, 1978.

(3) Clifford, Paul C , "A Process Capability Study Using

Control Charts", Journal of Quality Technology, Vol. 3, No.

3, pp 107-111, July 1971.

(4) Diamond, William J., Practical Experiment Designs for

Engineers and Scientists, Lifetime Learning Publications,

Belmont, California, 1981.

(5) Feigenbaum, A.V., Total Quality Control, Third Edition,

McGraw Hill, New York, 1983.

(6) Ford Motor Company Operations Support Staffs

(Statistical Methods Office), Continuing Process Control and

Process Capability Improvement, Dearborn, Michigan, Ford

Motor Co., 1984.

(7) Grant, Eugene L., and Leavenworth, Richard S.,

Statistical Quality Control, McGraw Hill Book Co., New York,

1980.

(8) Hicks, Charles R., Fundamental Concepts in the Design of

Experiments, CBS College Publishing, New York, 1982.

69

70

(9) Juran, J.M., Quality Control Handbook, McGraw Hill, New

York, 1962.

(10) Kane, Victor E., "Process Capability Indicies", Journal

of Quality Technology, Vol. 18, No. 1, January 1986.

(11) Kirkpatrick, Elwood G., Quality Control for Managers

and Engineers, John Wiley and Sons Inc., New York, 1970.

(12) Mendenhall, William, Intoduction to Linear Models and

The Design and Analysis of Experiments, Wadsworth Publishing

Co. Inc., Belmont, California, 1968.

(13) Mentch, C.C., "Manufacturing Process Quality

Optimization Studies", Journal of Quality Technology, Vol.

12, No. 3, pp 119-122, July 1980.

(14) Montgomery, D.C., Design and Analysis of Experiments,

John Wiley and Sons Inc., New York, 1976.

(15) Ott, Ellis R., Process Quality Control, McGraw Hill

Inc., New York, 1975.

(16) Pitt, Hy, "A Modern Strategy for Process Improvement",

Quality Progress, pp 22-28, May 1985.

(17) SAS Institute Inc., SAS User' s Guide j_ Statistics,

Version 5 Edition, Gary, NC:SAS Institute Inc., 1985.

(18) Sedar, L.A., and Cowan, D., Span Plan Method of Process

Capability Analysis, American Society for Quality Control,

Milwaukee, Wisconsin, 1956.

(19) Shewart, W.A., Statistical Method From the Viewpoint of

71

(19) Shewart, W.A., Statistical Method From the Viewpoint of

Quality Control, Edited by W. Edwards Dennis, United States

Department of Agriculture, 1939.

(20) Steel, Robert G.D., and Torrie, James H., Principles

and Procedures of Statistics, McGraw Hill Book Co. Inc., New

York, 1960.

(21) Sullivan, L.P., "Reducing Variability : A New Approach

to Quality", Quality Progress, pp 15-21, July 1984.

(22) Western Electric Co. Inc., Statistical Quality Control

Handbook, Mack Printing Co., Easton, Pennsylvania, 1958.

APPENDIX A

PROGRAM LISTINGS

72

73

MAIN PROGRAM

(PROCAP)

ECHO OFF

THTIT.EXE

THPROG.EXE

TITLE DISPLAY MODULE

(THTIT.EXE)

PROGRAM FOR THE ANALYSIS OF PROCESS VARIATION CAPABILITIES

USING EXPERIMENTAL DESIGNS

BY

P. KAJU

10 »»»«»»*»»*«*»*«»*»*»»»»*««»»«»»»»»#»»»»»*»»»«»»»»**»«»«•»*»«»»»«»»»«*»*»»*»»*

20

30

40

50

60

70

80

90

100 '

120 '

130 KEY OFF

140 REM&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

150 REMSi &&&&

160 REMi TITLE DISPLAY ROUTINE 4i4&

170 REM& &&&&

180 REM&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

190 CLS

200 COLOR 4:LUCATE 1,1,0 'COLOR 4 FOR THE I.B.M. COMPUTER

210 PRINT TA8(22) CHR$(20I);

220 FOR 1=1 TO 34:PRINT CHR$(205);:NEXT I

230 PRINT CHR$(187)

240 PRINT TAB(22) CHR$(186),TAB(57) CHR$(186)

250 PRINT TAB(22) CHR$(186),TAfcl(57) CHRJ(186)

260 PRINT TAB(22) CHRJ(186),TAB(57) CHR$(186)

270 PRINT TAB(22) CHR$(186);" ";CHR$(201);;FOR 1=1 TO 8:PRINT CHR$(205);

:NEXT l:PRINT CHRi(187);:PRINT" '^tPRINT CHR$(201);:FOR 1=1 TO 8

:PRINT CHR$(205);:NEXT I:PRINT CHR$(187);" ";CHR$(186)

280 PRINT TAB(22) CHR$(200);CHRJ(205);CHR$(205);CHR$(205);CHR$(188);

TA8(35) CHRJ(186);TAB(44) CHR$(186);TAB(53) CHR$(200);CHRJ(205);CHK$(205);

CHR$(205);CHR$(188);

290 FOR 1=1 TO 13:PRINT TAB(35) CHR$(186);TAB(44) CHKJ(18b):NEXT I

300 PRINT TAB(32) CHRJ(201);CHR$(205);CHR$(205);CHR$(188);" ";

CHR$(200);CHR$(205);CHR$(205);CHR$(187)

74

:PRINT CHR$(188);

310 PRINT TAB(32) CHR$(186);TAB(47) CHR$(186)

320 PRINT TAB(32) CHR$(186);TAB(47) CHR$(186)

330 PRINT TAB(32) CHR$(200);:FOR 1=1 TO 14:PRINT CHR$(205);:NEXT

340 LXATE 8,1

350 PRINT TAB(28) CHR$(201);

360 FOR 1=1 TO 22:PR1NT CHR$(205);:NEXT I 370 PRINT CHR$(187)

380 PRINT TAB(28) CHR$(186),TAB(51) CHR$(186)

390 PRINT TAB(28) CHR$(186),TA8(51) CHR$(186)

400 PRINT TAB(2b) CHRJ(186);" ";CHR$(201); :FOR 1 = 1 TO 4:PRINT cm$(2Q5); :NEXT l:PRINT CHR$(187);:PRINT" ";:PRINT C^«$(201);:FUR 1=1 TO 4

:PRINT CHR$(205);:NEXT l:PRINT CHR$(187);" ";CHRJ(1d6)

410 PRINT TAB(28) CHR$(200) ;CrtR$(205) ;CHR$(205) ;CHR$(205) ;Cmj( 188) ;TAB(35)

CHR$(186);" ";CHR$(186);TAB(42) CHR$(186);" ••;CHRJ(186);TAB(47) CHRJ(200);

CHR$(205) ;CHR$(205) ;CHR$(205) ;CW4( 188)

420 FOR 1=1 TO 8:L0CATE ,37:PRINT CHR$(186);TAB(42) CHRJ(186):NEXT I

430 LOCATE ,35:PRINT Ch«$(201);CHR$(205);ChR$(188);" ";CHR$(200);CHR$(205);CHR$(187)

440 LOCATE ,35:PRINT (JHRS(200);:FOR 1=1 TO 8;PRINT CHR$(205);:NEXT l:PKINT CHRi(18b); 450 FOR 1=1 TO lOOOiNEXT I

460 COLOR 15,0:LOCATE 3,23:PRINT "EXPERIMENTAL DESIGNS AND ANALYSIS"

470 LOCATE 10,35:PR I NT "BY P. RAJU"

480 LOCATE 14,25:PR I NT "DEPARTMENT OF INDUSTRIAL ENGINEERING"

490 LOCATE 17,31:PR I NT "TEXAS TECH UNIVERSITY"

500 COLOR 17:LOCATE 24,24:PR I NT" <PRESS ANY KEY FOR CUNT INUAT10N>";

510 AA$=INKEY$: IF AAJ ="" THEN 510

520 LOCATE 5:COLOR 7

530 CLS

540 '

550 • BRIEF DESCRIPTION OF THE SOFTWARE 560 »

570 '

580 PRINT :PRINT :PRINT iPRINT

590 REM

600 REM

610 PRINT " "

620 PRINT "

630 PRINT "

640 PRINT "

650 PRINT "

660 PRINT "

670 PRINT "

680 PRINT "

690 PRINT "

700 PRINT "

710 PRINT "

720 PRINT "

730 PRINT "

740 PRINT "

EXPERIMENTAL DESIGNS AND ANALYSIS

OF PROCESS VARIATION CAPABILITIES

RANDOMIZED COMPLETE BLXK DESIGN WITH SUBSAMPLING IS THE

EXPERIMENT USED IN THIS SOFTWARE TO ANALYZE THE PROCESS

VARIATION CAPABILITIES.

!"

!"

!"

!" Ml

1"

!"

Ml

Ml

RANDOMIZED COMPLETE BLXK DESIGN IS AN EXPERIMENTAL DESIGN !"

IN WHICH THE TREATMENT COMBINATIONS ARE RANDOMIZED WITHIN

A BLOCK AND SEVERAL BLOCKS ARE RUN. THIS IS A BALANCED

DESIGN - EACH TREATMENT APPEARS ONCE IN EACH BLOCK AND

M l

M l

I II

75

1230 PRINT " REPRESENT DAYS IN THIS EXPERIMENT. EACH RUN WILL CONTAIN N DIFFERENT "

1240 PRINT " EXPERIMENTAL UNITS (EU). EU'S WITHIN EACH BLOCK ARE TO BE RANDOMLY SELE-"

1250 PRINT " CTED FROM A HOMOGENEOUS BATCH OR LOT OF EU'S."

1260 GOTO 1930

1270 PRINT " SINGLE FACTOR (FIXED EFFECT) RANDOMIZED COMPLETE BLOCK"

1280 PRINT " EXPERIMENT WITH SUBSAMPLING"

1290 PRINT " M

1300 PRINT

1310 PRINT " IN THIS EXPERIMENT A REPETITION OF THE EXPERIMENT WOULD BRING THE SAME "

1320 PRINT " SET OF TREATMENTS INTO THE NEW EXPERIMENT. BY USING THIS EXPERIMENT ONE"

1330 PRINT " CAN DRAW INFERENCES ABOUT THE PARTICULAR TREATMENTS."

1340 PRINT " THE NUMBER OF FACTORS INVOLVED IN THIS MODEL IS ONE."

1350 PRINT

1360 PRINT " FOR EXAMPLE, IN A SINGLE FACTOR (FIXED EFFECT) CASE, THE FACTOR A WITH "

1370 PRINT " THREE LEVELS CAN BE CONSIDERED, IN ANY CHEMICAL PROCESS. FACTOR LEVELS OF"

1380 PRINT " X UNITS, Y UNITS & Z UNITS CAN BE SELECTED FOR THE EXPERIMENT. IT IS"

1390 PRINT " BEST TO CHOOSE EQUALLY SPACED LEVELS OF FACTOR A. IN THIS EXPERIMENT"

1400 PRINT " ONLY ONE LEVEL OF FACTOR 2 SHOULD BE SELECTED FOR THE ENTIRE EXPERIMENT."

1410 PRINT " THE RUNS WILL REPRESENT BLOCKS IN THE EXPERIMENT AND PROVIDE A MEASURE"

1420 PRINT " OF RUN TO RUN VARIABILITY. RUNS MAY REPRESENT DAYS IN THE EXPERIMENT."

1430 PRINT " ALL A'S SHOULD APPEAR ONCE AND ONLY IN EACH RUN. THE A'S SHOULD BE RUN"

1440 PRINT " IN A RANDOM ORDER DURING EACH RUN. EACH RUN SHOULD BE RERANDOMIZED."

1450 PRINT " ALL EXPERIMENTAL UNITS USED IN THIS EXPERIMENT ARE TO BE SELECTED FRW "

1460 PRINT " A HOMOGENEOUS BATCH OR LOT OF EXPERIMENTAL UNITS."

1470 GOTO 1930

1480 PRINT " TWO FACTOR RANDOMIZED COMPLETE BLOCK WITH SUBSAMPLING"

1490 PRINT " "

1500 PRINT

1510 PRINT " IN THIS EXPERIMENT THE NUMBER OF FACTORS INVOLVED ARE TWO. THE FACTOR"

1520 PRINT " COMBINATIONS ARE RUN IN A RANDOM ORDER DURING EACH RUN. THE POINT IN "

1530 PRINT " DOING THIS IS TO AVOID THE INTRODUCTION OF ANY SYSTEMATIC BIASES IN THE"

1540 PRINT " MACHINE SETTINGS OR PERFORMANCE OF THE EXPERIMENT."

1550 PRINT

1560 PRINT " FOR EXAMPLE, IN A TWO FACToR CASE, THE FACTORS A & B WITH THKEE LEVELb "

1570 PRINT " EACH CAN BE CONSIDERED. FACTOR LEVELS (FOR FACTOR A) OF X UNITS, Y UNITS"

1580 PRINT " & Z UNITS AND FACTOR LEVELS (FOR FACTOR ti) OF U UNITS, V UNITb i w UNITb "

1590 PRINT " CAN BE CHOSEN FOR THE EXPERIMENT. THIS LEADS TO THE TREATMENT (A IB 1 , A IbZ. .'•

1600 PRINT " ..A3B3) COMBINATIONS FOR THE EXPERIMENT. CARE MUST bE TAKEN To AbbURL"

1610 PRINT " THAT THE DIFFERENCE BETWEEN THE EXTREMES (AlBl & AiB3) IS NOT SO GREAT"

1620 PRINT " OR SO LITTLE SO AS TO PRODUCE AN EXPERIMENT wITH LITTLE USEFUL INFuKM-"

1630 PRINT " AT I ON. THE RUNS WILL REPRESENT BLOCKS IN THE EXPERIMENT AND PKOVlut A"

1640 PRINT " MEASURE OF RUN TO RUN VARIABILITY.RUNS MAY KEPKEStNT JAYS IN THL LXPt-"

1650 PRINT " RIMENT. ALL FACTOR A * FACTOR B (TREATMENT) COMBINATIONS SHOULD AHPhAK"

1660 PRINT " ONCE AND ONLY ONCE IN EACH RUN. A * ti COMBINATIONS SHOULD tiE KUN IN A"

1670 PRINT " RANDOM ORDER DURING EACH RUN. EACH RUN SHOULD BE RERANDOMIZED. ALL "

1680 PRINT " EXPERIMENTAL UNITS WITHIN A BLOCK SHOULD Bt SELECTED FROM A HOMOGENEOUS "

1690 PRINT " BATCH OR LOT OF EXPERIMENTAL UNITS."

17U0 GOTO 1930

75

750 PRINT " ! EACH BLOCK CONTAINS ALL TREATMENTS. !"

760 PRINT " ! !„

770 PRINT " ! Ml

780 PRINT :PRINT :PRINT

790 FOR 1=1 TO 1000:NEXT I

bOO COLOR 17:LOCATE 24,24:PR I NT" <PRESS ANY KEY FOR CONTINUATION>";

810 AA$=INKEY$: IF AAJ ="" THEN 810

820 LOCATE 5:COLOR 7

830 CLS

840 '

850 ' SELECTION OF MODELS TO BE USED FOR THE ANALYSIS

850 ' __

870 '

880 PRINT "THE FOLLOWING FOUR MODELS ARE USED FOR THE ANALYSIS" 890 PRINT : PRINT

900 PRINT "A) MODEL I : SINGLE FACTOR (RANDOM EFFECT)"

910 PRINT "8) MODEL II : SINGLE FACTOR (FIXED EFFECT)"

920 PRINT " O MODEL III : TWO FACTOR EXPERIMENT"

930 PRINT "D) MODEL IV : THREE FACTOR EXPERIMENT"

940 PRINT : PRINT

950 '

960 ' BRIEF DESCRIPTION OF THE MODELS (OPTIONAL) 970 I

980 '

990 INPUT "00 YOU WANT A BR IEF DESCRIPTION OF THE ABOVE MODELS(Y/N) ",TX$

1000 PRINT : PRINT

1010 IF TXJ = "N" OR TX$ = "n" THEN 2010

1020 INPUT "SELECT THE MODEL (A,B,C,D) ",TX$

1030 CLS

1040 PRINT : PRINT : PRINT : PRINT

1050 IF TXJ = "A" OR TX$ = "a" THEN 1090

1060 IF TX$ = "B" OR TX$ = "b" THEN 1270

1070 IF TX$ = "C" OR TX$ = "c" THEN 1480

1080 IF TXJ = "D" OR TXJ = "d" THEN 1710 1090 PRINT " SINGLE FACTOR (RANDOM EFFECT) RANDOMIZED COMPLETE BLOCK"

1100 PRINT " EXPERIMENT WITH bUBSAMPLING"

1110 PRINT " "

1120 PRINT

1130 PRINT " IN THIS EXPERIMENT A REPETITION OF THE EXPERIMENT WOULD BR INb IN A NEW btT

1140 PRINT " OF TREATMENTS BUT FROM THE SAME POPULATION OF TREATMENTS. bY USING THIb "

1150 PRINT " EXPERIMENT ONE CAN DRAW INFERENCES AbOUT THE POPULATION oF TKEATMLNTO."

1160 PRINT " THE NUMBER OF FACTORS INVOLVED IN THIS MODEL IS ONE."

1170 PR I NT

1180 PRINT " FOR EXAMPLE, CONSIDER AN EXPERIMENTAL UNIT IN A MANUFACTURING PRoCEbb. "

1190 PRINT " IN THIS CASE THE NUMBER OF EXPERIMENTAL UNITS PER RUN WILL BE THE FACTOR "

1 00 PRINT " LEVELS. ALSO, ONE FACTOR 1 » FACTOR 2 COMBINATION IS SELECTED FOR THE "

1210 PRINT " ENTIRE EXPERIMENT. THE RUNS WILL REPRESENT BLOCKS IN THE EXPERIMENT AMU"

1220 PRINT " PROVIDES A MEASURE OF A RUN TO RUN OR BETWEEN RUN VARIABILITY. KUNb MAY"

1710

1720

1730

1740

1750

1760

1770

1780

1790

1800

1810

1820

1830

1840

1850

1860

1870

1880

1890

1900

1910

1920

1930

1940

1950

1960

1970

1980

1990

2000

2010

2020

2030

2040

2050

2060

2070

2080

2090

2100

2110

2120

213U

214U

2150

2160

2170

2180

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

PRINT

THREE FACTOR RANDOMIZED COMPLETE BLOCK WITH SUBSAMPLING"

' IN THIS EXPERIMENT THE NUMBER OF FACTORS INVOLVED ARE THREE. THE FACTOR"

• COMBINATIONS ARE RUN IN A RANDOM ORDER DURING EACH RUN. THE POINT IN "

' DOING THIS IS TO AVOID THE INTRODUCTION oF ANY SYSTEMATIC BIASES IN THE"

' MACHINE SETTINGS OR PERFORMANCE OF THE EXPERIMENT."

• FOR EXAMPLE, IN A THREE FACTOR CASE, THE FACTORS A, B, & C WITH THREE "

' LEVELS EACH CAN BE CONSIDERED."

• FOR FACTOR A, FACTOR LEVELS OF X, Y, & Z UNITS, FOR FACTOR B FACTOR"

• LEVELS OF U, V & W UNITS AND FOR C, FACTOR LEVELS OF L, M, i N UNITS CAN"

• BE CHOSEN FOR THE EXPERIMENT. THIS LEADS TO THE TREATMENT (A iBlCl,A IB1C2"

' ...A3B3C3) COMBINATIONS FOR THE EXPERIMENT. CARE MUST BE TAKEN TO ASSURE"

' THAT THE DIFFERENCE BETWEEN THE EXTREMES (AIBICI & A3B3C3) IS NOT SO GRE-"

• AT OR SO LITTLE SO AS TO PRODUCE AN EXPERIMENT WITH LITTLE USEFUL INFUR-"

• MAT I ON. THE RUNS WILL REPRESENT BLOCKS IN THE EXPERIMENT AND PROVIDE A"

• MEASURE OF RUN TO RUN VARIABILITY.RUNS MAY REPRESENT DAYS IN THE EXPE-"

' RIMENT. ALL A * B » C (TREATMENT) COMBINATIONS SHOULD APPEAR ONCE AND"

' ONLY ONCE IN EACH RUN. A * B » C COMBINATIONS SHOULD BE RUN IN A RANDOM"

' ORDER DURING EACH RUN. EACH RUN SHOULD BE RERANDOMIZED. ALL EU'S WITHIN"

' A BLOCK SHOULD BE SELECTED FROM A HOMOGENEOUS BATCH OR LOT OF EU'S." PRINT

INPUT "DO YOU WANT THE BRIEF UESCRIPTION OF ANY OTHER MODELS ?(Y/N) ",TXJ

IF TXJ = "Y" OR TXJ = "y" THEN 1020

PRINT : PRINT I

' FIELD LAYOUT DISPLAY FOR THE ABOVE MODELS (OPTIONAL)

INPUT "DO YOU WANT THE FIELD LAYOUT ? (Y/N) ",TXJ

PRINT : PRINT

IF TXJ = "N" OR TXJ = "n" THEN 2810

INPUT "SELECT THE DESIGN (A,B,C,D) ",TXJ

CLS

PRINT : PRINT

IF TXJ = "A" OR TXJ = "a" THEN 2110

IF TXJ = "B" OR TXJ = "b" THEN 2140

IF TXJ = "C" OR TXJ = "c" THEN 2330

IF TXJ = "D" OR TXJ = "d" THEN 2560

PRINT " FIELD LAYOUT FOR MODEL I"

PRINT " "

GOTO 21bO

PRINT " FIELD LAYOUT FOR MODEL II"

PRINT " "

PRINT PRINT "

PRINT " 1 1 1

II

II "

78

2190 PRINT

2200 PRINT

2210 PRINT

2220 PRINT

2230 PRINT

2240 PRINT

2250 PRINT

2260 PRINT

2270 PRINT

2280 PRINT

2290 PRINT

2300 PRINT

2310 PRINT

2320 GOTO :

2330 PRINT

2340 PRINT

2350 PRINT

2360 PRINT

2370 PRINT

2380 PRINT

2390 PRINT

2400 PRINT

2410 PRINT

2420 PRINT

2430 PRINT

2440 PRINT

2450 PRINT

2460 PRINT

2470 PRINT

2480 PRINT

2490 PRINT

2500 PRINT

2510 PRINT

2520 PRINT

2530 PRINT

2540 PRINT

II

II

II

II

II

11

II

11

II

II

II

II

II

2780 II

II

II

II

II

II

If

II

11

II

II

II

II

II

II

II

II

" A 4

" TO

2550 GOTO 2780

2560 PRINT

2570 PRINT

2580 PRINT

2590 PRINT

2600 PRINT

2610 PRINT

2620 PRINT

2630 PRINT

2640 PRINT

2650 PRINT

2660 PRINT

II

II

II

II

II

II

II

II

II

II

BLOCK 1

BLOCK 2

11

II

II

II

BLOCK 5

BLOCK 1

BLOCK 2

f1

• •

II

• II

•

BLOCK 5

B REPRESENT

FACTOR NAMES

BLOCK 1

BLOCK 2

II

1LEVEL11LEVEL21 1LEVEL51 "

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 "

FIELD LAYOUT FOR MODEL III"

1 1 1

1 Al B11 Al B21

1 1 1

1 1 1

1 Al 811 Al B21

1 1 1

1 A3 B31

1 A3 B31

II

1 " 1 1 1 1

1 Al B11 Al B21 1 A3 B31 "

1 1 1 1 1 "

THE NAMES OF THE 2 FACTORS. THE NUMERIC NUMBER ADJACENT"

REPRESENTS THE LEVEL OF THE CORRESPONDING FACTOR"

FIELD LAYOUT FOR MODEL IV" - _ _ — _ _ — _ — — — _ _ . — — — — 1 1

A1B1C11A1B1C2

1

1

A1B1C11A1B1C2

,1A3BX3

1

"l

, 1 A 3 B X J

1

II

It

11

II

II

II

79

2670 PRINT " .11 2680 PRINT " .1. 2690 PRINT " .11 2700 PRINT " .1 2710 PRINT " 1 ^ ^ i 'l 11

2720 PRINT " BLOCK 5 1A1B1C11A1B1C21 1A3B3C31 " 2730 PRINT " 1 1 1 1 1 " 2740 PRINT 2750 PRINT 2760 PRINT " A, B & C REPRESENT THE NAMES OF THE 3 FACTORS. THE NUMERIC NUMBER" 2770 PRINT " ADJACENT TO FACTOR NAMES REPRESENTS THE LEVEL OF THE CORRESPONDING FACTOR" 2780 PRINT 2790 INPUT "DO YOU WANT THE FIELD LAYOUT OF ANY OTHER MODELS ? (Y/N) ",TXJ 2800 IF TXJ = "Y" OR TXJ = "y" THEN 2040 2810 END

80

CALCULATION MODULE

(THPROG.EXE)

10 CLS

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

340

350

360 DIM A(5,5,5,5,5)

370 DIM F(10)

380 DIM P(10)

390 DIM NldO)

400 DIM N2(10)

410 DIM 0UTLY(5)

420 DIM TXJ(3)

430 '

VARIABLE TABLE

NUM : NUMBER OF FACTORS INVOLVED

T : NU^BER OF FACTOR LEVELS FOR FACTOR 1

C : NUMBER OF FACTOR LEVELS FOR FACTOR 2

D : NUMBER OF FACTOR LEVELS FOR FACTOR 3

TXJ : NAME OF THE FACTORS

R : NUMBER OF RUNS (REPLICATIONS)

N : NUMBER OF SUBSAMPLES

SS : SUM OF SQUARES

OF : DEGREES OF FREEDOM

MS : MEAN SQUARE

F : STATISTICAL TEST (F-TEST)

EMS : EXPECTED MEAN SQUARE

PC : PROCESS CAPABILITY

MATRIX DEFINITION

81

440 » INITIALIZATION 450 I

460 '

470 LINENOS = 12

480 NCOUNT = 100

490 T = 1

500 C = 1

510 D = 1

520 '

530 ' INPUT THE PARAMETERS 540 I

550 '

560 INPUT "DO YOU WANT TO ENTER THE DATA FROM THE KEYBOARD ? (Y/N) ",TX1J

570 IF TXIJ = "Y" OR TXIJ = "y" THEN 610

580 INPUT "NAME OF THE DATA FILE (DRIVE:FILENAME) ",FILEJ

590 OPEN FILEJ FOR INPUT AS #1

600 GOTO 1370

610 INPUT "SELECT THE DESIGN FOR THE ANALYSIS (A,B,C,D) ",SEL3S

620 IF SEL3J = "A" OR SEL3J = "a" OR SEL3J = "B" OR SEL3J = "b" THEN NUM = 1

630 IF SEL3J = "C" OR SEL3J = "c" THEN NUM = 2

640 IF SEL3$ = "D" OR SEL3J = "d" THEN NUM = 3

650 PRINT " THE NUMBER OF FACTORS INVOLVED FOR THE SELECTED MODEL = "NUM

660 FOR B = 1 TO NUM

670 PRINT "THE NAME OF THE FACTOR";B;

680 INPUT TXJ(B)

690 PRINT "THE NUMBER OF FACTOR LEVELS FOR FACTOR ";TXJ(B);

700 INPUT " ",OUTLY(B)

710 NEXT a

720 T = OUTLY(l)

730 IF 0UTLY(2) < 2 THEN 750

740 C = 0UTLY(2)

750 IF 0UTLY(3) < 2 THEN 770

760 D = 0UTLY(3) 770 INPUT "THE NUMBER OF RUNS ",R

780 INPUT "DO YOU WANT THE RANDOMIZATION ? (Y/N) ",TEMPJ

790 ' 800 ' DISPLAY THE RANDOMIZATION OF THE EXPERIMENT (OPTIONAL)

810 »

820 '

830 IF TEMPJ = "N" OR TEMPJ = "n" THEN 1310

840 CLS

850 RNl = 1

860 FOR J = 1 TO T

870 FOR K = 1 TO C

880 FOR P = 1 TO 0

890 A(1,J,K,P,1) = RNl

900 RNl = RNl + 1

910 NEXT P

82

920 NEXT K

930 NEXT J

940 FOR I = 1 TO R

950 PRINT " FOR RUN"I"THE RANDOMIZATION IS AS SHOWN BELOW :"

960 FQH COUNT = 1 TO NCOUNT

970 JJl = INT(RND » T) + 1

980 KKl = INT(RND « C) + 1

990 PPl = INT(RND * 0) + 1

1000 JJ2 = INT(RNO * T) + 1

1010 KK2 = INTCRNO * C) + 1

1020 PP2 = INT(RN0 * 0) + 1

1030 TEMP = A(1,JJ1,KK1,PP1,1)

1040 A(1,JJ1,KK1,PP1,1) = A(1,JJ2,KK2,PP2,1)

1050 A(1,JJ2,KK2,PP2,1) = TEMP

1060 NEXT COUNT

1070 FOR P = 1 TO D

1080 IF D = 1 THEN 1110

1090 PRINT "LEVEL OF FACTOR "TXJ(3)" = "P

1100 PRINT

1110 PRINT ,

1120 FOR J = 1 TO T

1130 PRINT TXJ(1);J,

1140 NEXT J

1150 PRINT

1160 FOR K = 1 TO C

1170 IF C = 1 THEN 1190

1180 PRINT TX$(2);K,

1190 IF C > 1 THEN 1210

1200 PRINT ,

1210 FOR J = 1 TO T

1220 PRINT " "A(1,J,K,P,1),

1230 NEXT J

1240 PRINT

1250 NEXT K

1260 PRINT

1270 NEXT P

1280 PRINT " PRESS ANY KEY TO PROCEED"

1290 AAJ = INKEYJ : IF AAJ = "" THEN 1290

1300 NEXT I

1310 INPUT "THE NUMBER OF SUBSAMPLES ",N


1330 •

1340 ' READING THE DATA FROM AN EXTERNAL FILE

1350 •

1360 '

1370 INPUT#1, SEL3J

1380 IF SEL3J = "A" OR SEL3J = "a" OR SEL3J = "B" OR SEL3J = "b" THEN NUM = I 1390 IF SEL3J = "C" OR SEL3J = "c" THEN NUM = 2

83

1400 IF SEL3J = »D" OR SEL3J = "d" THEN NUM = 3


1420 INPUT#1, TXJ(8)

1430 INPUT#1, OUTLY(B)

1440 NEXT 8

1450 T = OUTLY(l)

1460 IF 0UTLY(2) < 2 THEN 1480

1470 C = 0UTLY(2)

1480 IF 0UTLY(3) < 2 THEN 1500

1490 D = 0UTLY(3)

1500 INPUT#1, R

1510 INPUT#1, N

1520 FOR I = 1 TO R

1530 FOR J = 1 TO T

1540 FOR K = 1 TO C

1550 FOR P = 1 TO D

1560 FOR M = 1 TO N

1570 IF TXIJ = "N" OR TXIJ = "n" THEN 1770

1580 PRINT "FOR RUN "liPRINT " FOR FACTOR "TXJ(1)",LEVEL "J

1590 IF C = 1 THEN 1610

1600 PRINT " FOR FACTOR "TXJ(2)",LEVEL "K

1610 IF D = 1 THEN 1630

1620 PRINT " FOR FACTOR "TXJ(3)",LEVEL "P

1630 PRINT " FOR SUBSAMPLE "M

1640 '

1650 ' ENTERING THE DATA INTERACTIVELY

1660 '

1670 '

1680 INPUT " ENTER THE DATA ",DJ

1690 IF DJ = "" THEN 1720

1700 A(I,J,K,P,M) = VAL(DJ)

1710 GOTO 1740

1720 PRINT "NO DATA ENTERED. THE EXPERIMENT DOES NOT ACCEPT MISSING VALUES"

1730 GOTO 1680

1740 INPUT " HAS CORRECT DATA BEEN ENTERED ? (Y/N) ",TXJ

1750 IF TXJ = "N" OR TXJ = "n" THEN 1680

1760 GOTO 1780

1770 INPUT#1, A(I,J,K,P,M)

1780 NEXT M

1790 NEXT P

1800 NEXT K

1310 NEXT J

1820 NEXT I

1830 CLS

1840 PRINT "THE DATA HAS BEEN RECORDED"


1860 CLOSE i^] 1870 '

84

1880 • DISPLAY THE ENTERED DATA (OPTIONAL) 1890 '

1900 •

1910 PRINT "00 YOU WANT IT TO BE DISPLAYED ?"

1920 INPUT "ENTER Y/N ",TXJ

1930 IF TXJ = "N" OR TXJ = "n" GOTO 2350

1940 LINECT = 0

1950 PRINT "RUN";: PRINT " LEVEL OF";

1960 IF C = 1 THEN 1980

1970 PRINT " LEVEL OF";

1980 IF D = 1 THEN 2000

1990 PRINT " LEVEL OF";

2000 PRINT " SUBSAMPLE DATA"

2010 PRINT " FACTOR "TXJ(1);

2020 IF C = 1 THEN 2040


2040- IF 0 = 1 THEN 2060


2060 PRINT " NUMBER "

2070 PRINT

2080 FOR I = 1 TO R

2090 FOR J = 1 TO T

2100 FOR K = 1 TO C

2110 FOR P = 1 TO D

2120 FOR M = 1 TO N

2130 PRINT " "I;" "J;

2140 IF C = 1 THEN 2160

2150 PRINT " "K;

2160 IF D = 1 THEN 2180

2170 PRINT " "P;

2180 PRINT " "M;

2190 PRINT " "A(I,J,K,P,M)

2200 LINECT = LINECT + 1

2210 IF LINECT < LINENOS THEN 2250



2240 LINECT = 0

2250 NEXT M

2260 PRINT

2270 NEXT P

2280 NEXT K

2290 NEXT J

2300 NEXT I

2310 '

2320 ' MAKE CHANGES, IF ANY, IN THE DATA (OPTIONAL)

2330 '

2340 '

2350 INPUT "ANY DATA TO BE CHANGED ? (Y/N) ",TXJ

85

2360 IF TXJ = "N» OR TXJ = "n" THEN 2550

2370 INPUT "RUN NUI4iER ",1

2380 PRINT "LEVEL OF FACTOR "TXJ(l);

2390 INPUT J

2400 K = 1

2410 P = 1

2420 IF C = 1 THEN 2450

2430 PRINT "LEVEL OF FACTOR "TXJ(2);

2440 INPUT K

2450 IF D = 1 THEN 2480

2460 PRINT "LEVEL OF FACTOR "TXJ(3);

2470 INPUT P

2480 INPUT "SUBSAMPLE ",M

2490 IF I >= 1 AND J >= 1 AND K >= 1 Arc M >= 1 THEN 2520

2500 PRINT "WRONG VALUES ENTERED"

2510 GOTO 2370

2520 INPUT "ENTER DATA ",A(I,J,K,P,M)

2530 PRINT "NEW DATA ENTERED"

2540 GOTO 2350

2550 INPUT "WANT TO REDISPLAY THE DATA ? (Y/N) ",TXJ

2560 IF TXJ = "Y" OR TXJ = "y" THEN 1940

2570 CLS

2580 '

2590 ' OUTLIER IDENTIFICATION (OPTIONAL)

2600 '

2610 '

2620 INPUT "00 YOU WANT THE OUTLIER TO BE IDENTIFIED ? (Y/N) ",TXJ


2640 PRINT

2650 INPUT "AT WHAT LEVELd SIGMA,2 SIGMA,3 SIGMA,4 SIGMA)? (1,2,3,4) ",MAGIC

2660 FOR I = 1 TO R

2670 FOR J = 1 TO T

2680 FOR K = 1 TO C

2690 FOR P = 1 TO D

2700 MEAN = 0

2710 SIGMA = 0

2720 FOR M= 1 TO N

2730 MEAN = MEAN + A(I,J,K,P,M)

2740 NEXT M

2750 MEAN = MEAN / N

2760 '

2770 ' CALCULATION OF OUTLIERS

2780 •

2790 •

2800 FOR M = 1 TO N

2810 SIGMA = SIGMA + (MEAN - A(I,J,K,P,M)) * (MEAN - A(I,J,K,P,M))

2820 NEXT M

2830 SIGMA = SQR(SIGMA / (N - 1))

86

2840 FUG = 0

2850 LOW = MEAN - MAGIC / 2 * SIGMA

2860 HIGH = MEAN + MAGIC / 2 » SIGMA

2870 FOR M = 1 TO N

2880 F(M) = 0

2890 IF A(I,J,K,P,M) >= LOW AND A(I,J,K,P,M) <= HIGH THEN 2920

2900 F(M) = 1

2910 FLAG = 1

2920 NEXT M

2930 '

2940 ' POINTING OUT THE OUTLIER 2950 »

2960 •

2970 IF FLAG = 0 THEN 3220

2980 PRINT "IN RUN "l:PRINT " FACTOR "TXJ(1);",LEVEL "J

2990 IF C = 1 THEN 3010

3000 PRINT " FACTOR "TXJ(2);",LEVEL "K

3010 IF 0 = 1 THEN 3040

3020 PRINT

3030 PRINT " FACTOR "TXJ(3);",LEVEL "P

3040 PRINT " SUBSAMPLE DATA"

3050 FOR M = 1 TO N

3060 IF F(M) = 1 THEN 3090

3070 PRINT " ";A(I,J,K,P,M)

3080 GOTO 3110

3090 PRINT " ";A(I,J,K,P,M);

3100 PRINT "^ IS AN OUTLIER"

3110 NEXT M

3120 F(3R M = 1 TO N

3130 '

3140 ' MAKE ANY CHANGES IN THE OUTLIER (OPTIONAL) 3150 1

3160 '

3170 PRINT " DO YOU WANT TO CHANGE DATA "M

3180 INPUT " ENTER Y/N ",TXJ


3200 INPUT " ENTER DATA ",A(I,J,K,P,M)

3210 NEXT M

3220 NEXT P

3230 NEXT K

3240 NEXT J

3250 NEXT I

3260 PRINT : PRINT

3270 PRINT " NO MORE OUTLIERS"

3280 PRINT : PRINT

3290 '

3300 ' SAVING THE INPUT DATA

3310 •

87

3320 •

3330 INPUT " DO YOU WANT TO SAVE THE DATA ? (Y/N) ",TXJ


3350 INPUT " ENTER THE NAME OF THE FILE TO SAVE THE DATA (DRIVE:FILENAME) ",FILEJ 3360 OPEN FILEJ FOR OUTPUT AS #2 3370 •

3380 • PRINTING THE ENTERED DATA TO THE OUTPUT FILE 3390 '

3400 '

3410 PRINT#2, SEL3J


3430 PRINT#2, TXJ(B)

3440 PRINT#2, 0UTLY(8)

3450 NEXT B

3460 PRINT#2, R

3470 PRINT#2, N

3480 FOR I = 1 TO R

3490 FOR J = 1 TO T

3500 FOR K = 1 TO C

3510 FOR P = 1 TO D

3520 FOR M = 1 TO N

3530 PRINT#2, A(I,J,K,P,M)

3540 NEXT M

3550 NEXT P

3560 NEXT K

3570 NEXT J

3580 NEXT I

3590 CLOSE #2

3600 PRINT " CALCULATION TO OBTAIN ANOVA TABLE IS GOING ON"

3610 '

3620 • CALCULATIONS TO OBTAIN ANOVA TABLE

3630 •

3640 '

3650 FOR I = 1 TO R

3660 SSR1 = 0

3670 SSBl = 0

3680 FOR J = 1 TO T

3690 SSRT1 = 0

3700 FOR K = 1 TO C

3710 SSRCT1 = 0

3720 FOR P = 1 TO D

3730 SSRTCD1 = 0

3740 FOR M = 1 TO N

3750 CF = CF + A(l,J,K,P,M)

3760 SQI = SQI + A(I,J,K,P,M) » A(I,J,K,P,M)

3770 SSRl = SSRI + A(I,J,K,P,M)

3780 SSRCT1 = SSRCTl + A(I,J,K,P,M)

3790 SSRTl = SSRTl + A(I,J,K,P,M)

88

3800 SSRTCDl = SSRTC01 + A(I,J,K,P,M)

3810 NEXT M

3820 SSRTC02 = SSRTCD2 + SSRTCDl • SSRTCDl

3830 NEXT P

3840 SSRCT2 = SSRCT2 + SSRCTl * SSRCTl

3850 NEXT K

3860 SSRT2 = SSRT2 + SSRTl * SSRTl

3870 SS81 = SSBl + SSRTl » SSRTl

3880 NEXT J

3890 SSR2 = SSR2 + SSRl * SSRI

3900 SS82 = SSB2 + SSBl / N - SSRl * SSRl / (T * N)

3910 NEXT I

3920 CF = CF * CF / (R * T » C » D * N)

3930 SSTOT = SQI - CF

3940 SSR = SSR2 / (T » C * D » N) - CF

3950 FOR J = 1 TO T

3960 SST1 = 0

3970 FOR I = 1 TO R

3980 FOR K = 1 TO C

3990 FOR P = 1 TO D

4000 FOR M = 1 TO N

4010 SST1 = SSTl + A(I,J,K,P,M)

4020 NEXT M

4030 NEXT P

4040 NEXT K

4050 NEXT I

4060 SST2 = SST2 + SSTl * SSTl

4070 NEXT J

4080 FOR P = 1 TO D

4090 SSD1 = 0

4100 FOR I = 1 TO R

4110 FOR J = 1 TO T

4120 FOR K = 1 TO C

4130 FOR M = 1 TO N

4140 SSDl = SSD1 + A(I,J,K,P,M)

4150 NEXT M

4160 NEXT K

4170 NEXT J

4180 NEXT I

4190 SSD2 = SSD2 + SSDl * SSDl

4200 NEXT P

4210 FOR J = 1 TO T

4220 FOR P = 1 TO D

4230 SSTD1 = 0

4240 FOR I = 1 TO R

4250 FOR K = 1 TO C

4260 FOR M = 1 TO N

4270 SSTDl = SSTDl + A(I,J,K,P,M)

89

4280 NEXT M

4290 NEXT K

4300 NEXT I

4310 SSTD2 = SST02 + SSTDl * SSTDl

4320 NEXT P

4330 NEXT J

4340 FOR K = 1 TO C

4350 FOR P = 1 TO 0

4360 SSCDl = 0

4370 FOR I = 1 TO R

4380 FOR J = 1 TO T

4390 FOR M = 1 TO N

4400 SSC01 = SSCDl + A(I,J,K,P,M)

4410 NEXT M

4420 NEXT J

4430 NEXT I

4440 SSC02 = SSCD2 + SSCDl * SSCDl

4450 NEXT P

4460 NEXT K

4470 FOR J = 1 TO T

4480 FOR K = 1 TO C

4490 FOR P = 1 TO D

4500 SSTC01 = 0

4510 FOR I = 1 TO R

4520 FOR M = 1 TO N

4530 SSTCDl = SSTCD1 + A(I,J,K,P,M)

4540 NEXT M

4550 NEXT I

4560 SSTCD2 = SSTCD2 + SSTCDl * SSTCDl

4570 NEXT P

4580 NEXT K

4590 NEXT J

4600 FOR I = 1 TO R

4610 FOR P = 1 TO D

4620 SSRDl = 0

4630 FOR J = 1 TO T

4640 FOR K = 1 TO C

4650 FOR M = 1 TO N

4660 SSRDl = SSRDl + A(I,J,K,P,M)

4670 NEXT M

4680 NEXT K

4690 NEXT J

4700 SSR02 = SSRD2 + SSRDl * SSRDl

4710 NEXT P

4720 NEXT I

4730 FOR I = 1 TO R

4740 FOR J = 1 TO T

4750 FOR P = 1 TO 0

90

4760 SSRTOl = 0

4770 FOR K = 1 TO C

4780 FOR M = 1 TO N

4790 SSRTDl = SSRTDl + A(I,J,K,P,M)

4800 NEXT M

4810 NEXT K

4820 SSRTD2 = SSRTD2 + SSRTDl * SSRTDl

4830 NEXT P

4840 NEXT J

4850 NEXT I

4860 FOR I = 1 TO R

4870 FOR K = 1 TO C

4880 FOR P = 1 TO 0

4890 SSRCD1 = 0

4900 FOR J = 1 TO T

4910 FOR M = I TO N

4920 SSRCDl = SSRCDl + A(I,J,K,P,M)

4930 NEXT M

4940 NEXT J

4950 SSRC02 = SSRCD2 + SSRCDl * SSRCDl

4960 NEXT P

4970 NEXT K

4980 NEXT I

4990 FOR K = 1 TO C

5000 SSC1 = 0

5010 FOR I = 1 TO R

5020 FOR J = 1 TO T

5030 FOR P = 1 TO D

5040 FOR M = 1 TO N

5050 SSCl = SSC1 + A(I,J,K,P,M)

5060 NEXT M

5070 NEXT P

5080 NEXT J

5090 NEXT I

5100 SSC2 = SSC2 + SSCl * SSCl

5110 NEXT K

5120 FOR J = 1 TO T

5130 FOR K = 1 TO C

5140 SSTCl = 0

5150 FOR I = 1 TO R

5160 FOR P = 1 TO D

5170 FOR M = 1 TO N

5180 SSTCl = SSTCl + A(I,J,K,P,M)

5190 NEXT M

5200 NEXT P

5210 NEXT I

5220 SSTC2 = SSTC2 + SSTCl * SSTCl

5230 NEXT K

91

5240 NEXT J

5250 FOR I = I TO R

5260 FOR K = 1 TO C

5270 SSRCl = 0

5280 FOR J = 1 TO T

5290 FOR P = 1 TO D

5300 FOR M= 1 TO N

5310 SSRCl = SSRCl + A(I,J,K,P,M)

5320 NEXT M

5330 NEXT P

5340 NEXT J

5350 SSRC2 = SSRC2 + SSRCl * SSRCl

5360 NEXT K

5370 NEXT I

5380 •

5390 ' CALCULATE SUM OF SQUARES

5400 '

5410 '

5420 SST = SST2 / (R»C»0*N) - CF

5430 SSB = SSRT2 / (C«D*N) - CF - SSR - SST

5440 SSW = SSTOT - SSR - SST - SSB

5450 SSW2 = SSTOT - SSR - SSB2

5460 IF C = 1 THEN 5540

5470 SSC = SSC2 / (R*T»D»N) - CF

5480 SSTC = SSTC2 / (R*D»N) -CF - SST - SSC

5490 SSRT = SSRT2 / (C » D * N) - CF - SSR - SST

5500 SSRC = SSRC2 / (T»0»N) - CF - SSR - SSC

5510 SSRCT = SSRCT2 / (D»N) - CF - SSR - SST - SSC - SSTC - SSRT - SSRC

5520 SSB = SSB + SSRC + SSRCT

5530 SSW = SSTOT - SSR - SST - SSC - SSTC - SSB

5540 IF D = 1 THEN 5700

5550 SSD = SSD2 / (R*T*C*N) - CF

5560 SSTD = SSTD2 / (R»C*N) - CF - SST - SSD

5570 SSCD = SSCD2 / (R*T»N) - CF - SSC - SSD

5580 SSTCD = SSTCD2 / (R*N) - CF - SST - SSC - SSD - SSTC - SSTD - SSCD

5590 SSRD = SSRD2 / (T»C«N) - CF - SSR - SSD

5600 SSRTD = SSRTD2 / (C*N) - CF - SSR - SST - SSD - SSRT - SSRD - SSTD

5610 SSRCD = SSRCD2 / (T*N) - CF - SSR - SSC - SSD - SSRC - SSRD - SSCD

5620 TEMP = SSRD - SSTC - SSTD - SSCD - SSRCT - SSRTD - SSRCD - SSTCD

5630 SSRTCD = SSRTCD2 / N - CF - SSR - SST - SSC - SSD - SSRT - SSRC - TEMP

5640 SSB = SSB + SSRD + SSRTD + SSRCD + SSRTCD

5650 SSW = SSTOT - SSR - SST - SSC - SSD - SSTC - SSTD - SSCD - SSTCD - SSB

5660 '

5670 ' CALCULATE DEGREES OF FREEDOM

5680 '

5690 '

5700 DFTOT = ( R * T » C » D » N ) - 1

5710 DFR = R - 1

92

5720 OFT = T - 1

5730 DFC = C - 1

5740 OFD = D - 1

5750 OFTC = (T- 1) » (C - 1)

5760 DFTD = (T - 1) » (D - 1)

5770 DFCD = (C - 1) » (D - 1)

5780 DFTCD = (T -1) » (C - 1) * (D - 1)

5790 DFRT = (R - 1) » (T - 1)

5800 DFRC = (R - 1) » (C - 1)

5810 OFRO = (R - 1) * (D - 1)

5820 DFRCT = (R - 1) * (C - 1) * (T -1)

5830 DFRTD = (R - 1) » (T - 1) * (D - 1)

5840 OFRCD = (R - 1) » (C - 1) » (0 - 1)

5850 DFRTCD = (R - 1) * (T - 1) * (C - 1) * (D - 1)

5860 DFB = DFRT + DFRC + DFRD + DFRCT + DFRTD + DFRCD + DFRTCD

5870 DFB2 = R » (T - 1)

5880 DFW = R * T * C * D » ( N - 1 )

5890 '

5900 ' CALCULATE MEAN SQUARES

5910 •

5920 '

5930 MSR = SSR / DFR

5940 MST = SST / OFT

5950 IF C = 1 THEN 6040

5960 MSC = SSC / DFC

5970 IF D = 1 THEN 5990

5980 MSD = SSD / DFD

5990 MSTC = SSTC / DFTC

6000 IF D = 1 THEN 6040

6010 MSTD = SSTD / DFTD

6020 MSCD = SSCD / DFCD

6030 MSTCD = SSTCD / DFTCD

6040 MSB = SSB / DFB

6050 MSB2 = SSB2 / DFB2

6060 MSW = SSW / DFW

6070 MSW2 = SSW2 / DFW

6080 '

6090 ' CALCULATE F

6100 '

6110 '

6120 F(l) = MSR / MSB

6130 F(2) = MST / MSB

6140 IF C = 1 THEN 6230

6150 F(3) = MSC / MSB

6160 IF D = 1 THEN 6180

6170 F(4) = MSD / MSB

6180 F(5) = MSTC / MSB

6190 IF D = 1 THEN 6230

93

6200 F(6) = MSTD / MSB

6210 F(7) = MSCD / MSB

6220 F(8) = MSTCD / MSB

6230 F(9) = MSB / MSW

6240 F(IO) = MSB2 / MSW2

6250 '

6260 ' CALCULATE EXPECTED MEAN SQUARES

6270 '

6280 '

6290 IF SEL3J = "A" OR SEL3J = "a" THEN 6350

6300 IF MSW < 0 THEN EMSW = 0 ELSE EMSW = SQR(MSW)

6310 IF (MSB - MSW) < 0 THEN EMSB = 0 ELSE EMSB = S0R((MSB - MSW) / N)

6320 IF (MSR - MSB) < 0 THEN EMSR = 0 ELSE EMSR = SQR((MSR - MSB) / (T»C*D*N))

6330 EMSTOT = SQR{(EMSW * EMSW) + (EMSB » EMSB) + (EMSR » EMSR))

6340 GOTO 6390

6350 IF MSW2 < 0 THEN EMSW2 = 0 ELSE EMSW2 = SQR(MSW2)

6360 IF (MS82 - MSW2) < 0 THEN EMSB2 = 0 ELSE EMSB2 = SQR((MSB2 - MSW2)/N)

6370 IF (MSR - MSB2) < 0 THEN EMSR2 = 0 ELSE EMSR2 = SQR((MSR - MSB2)/(T * N))

6380 EMST0T2 = SQR((EMSW2 * EMSW2) + (EMSB2 * EMSB2) + (EMSR2 * EMSR2))

6390 CLS

6400 '

6410 ' DISPLAY THE MOOEL OF THE EXPERIMENT (OPTIONAL)

6420 '

6430 '

6440 INPUT "DO YOU WANT THE MOOEL OF THE EXPERIMENT TO BE DISPLAYED ?(Y/N) ",TXJ

6450 IF TXJ = "N" CR TXJ = "n" THEN 6850 6460 PRINT "THE MOOEL OF THE EXPERIMENT IS "

6470 PRINT : PRINT


6490 IF SEL3J = "B" OR SEL3J = "b" THEN 6590

6500 IF SEL3J = "C" OR SEL3J = "c" THEN 6660

6510 IF SEL3J = "0" OR SEL3J = "d" THEN 6740

6520 PRINT "Yijk = mu + Ri + Bj(i) + Wk(ij)"

6530 PRINT : PRINT

6540 PRINT "i INDICATES RUN"

6550 PRINT "j INDICATES "TXJ(l)" WITHIN RUN"

6560 PRINT "k INDICATES LOCATIONS WITHIN RUN AND "TXJ(l)

6570 PRINT "Yijk = A RESPONSE MEASUREMENT"

6580 GOTO 6820 6590 PRINT "Yijim = mu + Ri + Tj + BKij) + Wmd j I )" :PRINT : PRINT


6610 PRINT "j INDICATES "TXJ(1)

6620 PRINT "I INDICATES EU'S WITHIN RUN AND "TXJ(1)

6630 PRINT "m INDICATES LOCATIONS WITHIN RUN, EU AND "TXJ(1)

6640 PRINT "Yljim = A RESPONSE MEASUREMENT"

6650 GOTO 6820 6660 PRINT "Yijkim = mu + Ri + Tj + Ck + TCjk + Bl(ijk) + Wm(ijlk)"

6670 PRINT : PRINT

94

6680 PRINT "1 INDICATES RUN"

6690 PRINT "J INDICATES "TXJ(l)

6700 PRINT "k INDICATES "TXJ(2)

6710 PRINT "I INDICATES EU's WITHIN RUN, "TXJ(l)" AND "TXJ(2)

6720 PRINT "m INDICATES LXATIONS WITHIN RUN, EU, "TXJ(l)" AND "TXJ(2)

6730 PRINT "Yijkim = A RESPONSE MEASUREMENT" : GOTO 6820

6740 PRINT "Yljkplm = mu + RI + Tj + Ck + Dp + TCjk + TDjp + CDkp + TCDjkp + Bl(ijkp)


6760 PRINT "j INDICATES "TXJ(1)

6770 PRINT "k INDICATES "TXJ(2)

6780 PRINT "p INDICATES "TXJ(3)

6790 PRINT "I INDICATES EU's WITHIN RUN, "TXJ(l)", "TXJ(2)" AND "TXJ(3)

6800 PRINT "m INDICATES LOCATIONS WITHIN RUN, EU, "TXJ(1)", "TXJ(2)" AND "TXJ(3)

6810 PRINT "Yljkplm = A RESPONSE MEASUREMENT "

6820 PRINT : PRINT

6830 PRINT "PRESS ANY KEY TO PROCEED"


6850 CLS

6860 '

6870 ' DISPLAY THE ANOVA TABLE (OPTIONAL)

6880 '

6890 '

6900 INPUT "DO YOU WANT THE ANOVA TABLE TO BE DISPLAYED ? (Y/N) ",TXJ


6920 PRINT "THE ANOVA TABLE"

F" 6930 PRINT "

6940 PRINT "SOURCE

6950 PRINT "

6960 PRTJ = "###

6970 PRTIJ = "®

6980 PRT2J = " ®

6990 PRT3J = " ®

7000 PRT4J = " ®

##.####««<»

Dg^<S>

^<S <S^9

((j^(9 Qj^®

OF SS —

##.;«###<««» ##.##(C«ee® ® II

® II

• II

sg,® ® II

MS —

##.#.j(#®®®®"

'

7010 PRINT USING PRTIJ;"TOTAL";: PRINT " ";

7020 PRINT USING PRTJ;DFTOT,SSTOT

7030 PRINT USING PRTIJ;" RUN";: PRINT " ";

7040 PRINT USING PRTJ;DFR,SSR,MSR,F(1)


7060 PRINT " ";

7070 PRINT USING PRT1J;TXJ(1);

7080 PRINT USING PRTJ;DFT,SST,MST,F(2)

7090 IF C = 1 THEN 7260

7100 PRINT " ";


7120 PRINT USING PRTJ;DFC,SSC,MSC,F(3)

7130 IF D = 1 THEN 7170

7140 PRINT " ";


95

7160 PRINT USING PRTJ;DFD,SSD,MSD,F(4)

7170 PRINT USING PRT2J;TXJ(1),"»",TXJ(2);: PRINT " ";

7180 PRINT USING PRTJ;0FTC,SSTC,MSTC,F(5) 7190 IF 0 = 1 THEN 7260

7200 PRINT USING PRT2J;TXJ(1),"«",TXJ(3);: PRINT " ";

7210 PRINT USING PRTJ;DFTD,SSTD,MSTD,F(6)

7220 PRINT USING PRT2J;TXJ(2),"*",TXJ(3);: PRINT " ";

7230 PRINT USING PRTJ;DFCD,SSCD,MSCD,F(7)

7240 PRINT USING PRT3J;TXJ(1),"*",TXJ(2),"*",TXJ(3);: PRINT " ";

7250 PRINT USING PRTJ;DFTCD,SSTCD,MSTCD,F(8)

7260 PRINT USING PRTIJ;" BETWEEN EU's";: PRINT " ";

7270 IF SEL3J = "A" OR SEL3J = "a" THEN 7280 ELSE 7300

7280 PRINT USING PRTJ;0FB2,SSB2,MSB2,F(10)

7290 GOTO 7310

7300 PRINT USING PRTJ;0FB,SSB,MS8,F(9)

7310 IF C = 1 THEN 7550

7320 PRINT " ";

7330 PRINT USING PRT2J;"RUN","«",TXJ(1);

7340 PRINT USING PRTJ;DFRT,SSRT

7350 PRINT " ";

7360 PRINT USING PRT2J;"RUN","»",TXJ(2);

7370 PRINT USING PRTJ;DFRC,SSRC

7380 IF D = 1 THEN 7420

7390 PRINT " ";

7400 PRINT USING PRT2J;"RUN","*",TXJ(3);

7410 PRINT USING PRTJ;DFRD,SSRD

7420 PRINT " ";

7430 PRINT USING PRT3J;"RUN","*",TXJ(1),"*",TXJ(2);

7440 PRINT USING PRTJ;DFRCT,SSRCT

7450 IF D = 1 THEN 7550

7460 PRINT " ";


7480 PRINT USING PRTJ;DFRTD,SSRTD

7490 PRINT " ";


7510 PRINT USING PRTJ;DFRCD,SSRCD

7520 PRINT " "; 7530 PRINT USING PRT4J;"RUN","*",TXJ(1),"*",TXJ(2),"*",TXJ(3);: PRINT " ";

7540 PRINT USING PRTJ;DFRTCD,SSRTCD

7550 PRINT USING PRTIJ;" WITHIN EU's";: PRINT " ";


7570 PRINT USING PRTJ;DFW,SSW2,MSW2 : PRINT

7580 GOTO 7600

7590 PRINT USING PRTJ;DFW,SSW,MSW : PRINT

7600 PRINT "PRESS ANY KEY TO PROCEED"

7610 AAJ = INKEYJ: IF AAJ = "" THEN 7610

7620 CLS

7630 '

96

7640 ' CALCULATE THE OBSERVED SIGNIFICANCE LEVELS

7650 '

7660 '

7670 FOR U = 1 TO 8

7680 N2(U) = DFB

7690 NEXT U

7700 N2(9) = DFW

7710 N2(10) = DFW

7720 Nl(1) = DFR

7730 Nl(2) = OFT

7740 Nl(3) = DFC

7750 Nl(4) = DFD

7760 Nl(5) = OFTC

7770 NU6) = DFTD

7780 Nl(7) = DFCD

7790 Nl(8) = DFTCD

7800 Nl(9) = DFB

7810 Nl(lO) = DFB2

7820 FOR Q = 1 TO 10

7830 FF = F(Q)

7840 NNl = Nl(Q)

7850 NN2 = N2(Q)

7860 GOSUB 7900

7870 P(Q) = PP

7880 NEXT Q

7890 GOTO 8430 7900 IF FF = 0 OR NNl = 0 OR NN2 = 0 THEN 8370

7910 FFl = 2 * INT(NN1/2) - NNl + 2

7920 FF2 = 2 • INT(NN2/2) - NN2 + 2

7930 FF3 = FF » MNl / NN2

7940 FF4 = 1 / (1 + FF3)

7950 IF FFl <> 1 THEN 8050

7960 IF FF2 <> 1 THEN 8020

7970 PP = S(?R(FF3)

7980 FF7 = .318 7990 FF6 = FF7 * FF4 / PP

8000 PP = 2 » FF7 * ATN(PP)

8010 GOTO 8120

8020 PP = SQR(FF3 * FF4)

8030 FF6 = .5 * PP * FF4 / FF3

8040 GOTO 8120

8050 IF FF2 <> 1 THEN 8100

8060 PP = SQR(FF4)

8070 FF6 = .5 * FF4 » PP

8080 PP = 1 - PP

8090 GOTO 6120 8100 FF6 = FF4 » FF4

8110 PP = FF3 * FF4

97

8120 FF7 = 2 » FF3 / FF4

8130 IF FFl <> 1 THEN 8200

8140 FF5 = FF2 + 2

8150 IF FF5 > NN2 THEN 8230

8160 FF6 = (1 + FFl / (FF5 - 2)) * FF6 * FF4

8170 PP = PP + FF6 * FF7 / (FF5 - 1)

8180 FF5 = FF5 + 2

8190 GOTO 8150

8200 FF8 = FF4 « INT((NN2 - 1) / 2)

8210 FF6 = FF6 * FF8 » NN2 / FF2

3220 PP = PP * FF8 + FF3 » FF4 » (FF8 - 1) / (FF4 - 1)

3230 FF7 = FF3 / FF4

8240 FF4 = 2 / FF4

8250 FF2 = NN2 - 1

8260 FF9 = FFl -•• 2

8270 IF FF9 > NNl THEN 8330

8280 FF5 = FF9 + FF2

8290 FF6 » FF7 * FF6 * FF5 / (FF9 - 2)

8300 PP = PP - FF4 * FF6 / FF5

8310 FF9 = FF9 + 2

8320 GOTO 8270

8330 PP = 1 - PP

8340 IF PP < 1 THEN 8360

8350 PP = 1

8360 IF PP > 0 THEN 8380

8370 PP = 0

8380 RETURN

3390 '

8400 ' PERFORM THE STATISTICAL TESTS (F-TESTI

8410 '

3420 '

8430 PRINT " THE F - COLUMN REPRESENTS A SEQUENCE OF STATISTICAL TESTS THAT CAN"

8440 PRINT " BE PERFORMED :"

8450 PRINT


8470 PRINT " MSB / MSW : lF(10)i"

3480 EJ = "EU"

8490 GOSUB 9250

8500 Q = 10

8510 AJ = "variabiIity"

8520 GOSUB 9510

3530 GOSUB 9630

8540 GOTO 8620

3550 PRINT " MSB / MSW : (F(9)i"

8560 EJ = "EU"

3570 GOSUB 9250

8580 Q = 9

3590 AJ = "variabiIity"

98

3600 GOSUB 9510

8610 GOSUB 9630


8630 IF SEL3J = "B" OR SEL3J = "b" THEN 9100


3650 PRINT " MSTCD / MSB : IF(8)1"

8660 GOSUB 9440

8670 Q = 8

8680 AJ = "interaction"

3690 GOSUB 9510

3700 GOSUB 9630

3710 PRINT " MSTC / MSB : IF(5)1"

8720 EJ = TXJ(l)

3730 E U = TXJ(2)

8740 GOSUB 9310

3750 Q = 5


8770 GOSUB 9510

8780 GOSUB 9630


8800 PRINT " MSTD / MSB : IF(6)]"

8810 EJ = TXJ(l)

8820 E U = TXJ(3)

3830 GOSUB 9310

8840 Q = 6


3860 GOSUB 9510

3870 GOSUB 9630

8880 PRINT " MSCD / MSB : (F(7)i"

8390 EJ = TXJ(2)

8900 E U = TXJ(3)

8910 GOSUB 9310

8920 Q = 7


8940 GOSUB 9510

8950 GOSUB 9630

8960 PRINT " MSD / MSB : (F(4)l"

3970 EJ = TXJ(3)

3980 GOSUB 9390

8990 Q = 4

9000 AJ = "treatment"

9010 GOSUB 9510

9020 GOSUB 9630

9030 PRINT " MSC / MSB : IF(3)1"

9040 EJ = TXJ(2)

9050 GOSUB 9390

9060 Q = 3

9070 AJ = "treatment"

99

9080

9090

9100

9110

9120

9130

9140

9150

9160

9170

9180

9190

9200

9210

9220

9230

9240

9250

9260

9270

9280

9290

9300

9310

9320

9330

9340

9350

9360

9370

9380

9390

9400

9410

9420

9430

9440

9450

9460

9470

9480

9490

9500

9510

9520

9530

9540

9550

GOSUB 9510

GOSUB 9630

PRINT " MST / MSB

EJ = TXJ(l)

GOSUB 9390

Q = 2

AJ = "treatment"

GOSUB 9510

GOSUB 9630

PRINT » MSR / MSB

EJ = "RUN"

GOSUB 9250

Q = 1

AJ = "variabiIity"

GOSUB 9510

GOSUB 9630

GOTO 9720

PRINT

!F(2))"

IF(1)1"

PRINT "

PRINT "

PRINT "

PRINT "

RETURN

PRINT

PRINT "

PRINT "

PRINT "

PRINT "

PRINT "

PRINT

RETURN

PRINT

PRINT "

PRINT "

PRINT "

RETURN

PRINT

PRINT "

PRINT "

PRINT "

PRINT "

PRINT "

RETURN

PRINT

PRINT "

PRINT "

PRINT

PRINT "

Test for a statistically significant between "EJ" variability. If the"

test is not significant, then the between "EJ" variability is relatively"

small and cannot be detected in the presence of the within "EJ" varia-"

biIity (at a specified s-significance level)."

Test for a "EJ" * "ElJ" interaction effect. If the test is"

significant, it indicates that the "EJ" or "EU" effect is not"

the same at each level of "ElJ" or "EJ" respectively. Therefore,"

the "EJ" and "ElJ" effects should not be interpreted independent"

of one another."

Test for a "EJ" effect. If no s-significant interaction is "

present and this test is significant it indicates that the response"

of all levels of "EJ" are not the same."

Test for a "TXJ(1)", "TXJ(2)" and "TXJ(3)" interaction"

effect. If the test is s-significant, it indicates that the any one"

of the factor effect is not the same at each level of the other two"

factors respectively. Therefore, the "TXJ(I)", "TXJ(2)" and "TXJ(3)

effects should not be interpreted independent of one another."

The null hypothesis. Ho, states that there is no "AJ" effect"

and the alternate hypothesis states that the "AJ" effect is present."

In general. if"

100

9560 PRINT " OSL < 0.01 then this indicates very strong evidence against Ho"

9570 PRINT " 0.01 < OSL < 0.05 then this indicates strong evidence against Ho"

9580 PRINT " 0.05 < OSL < 0.1 then this indicates some evidence against Ho"

9590 PRINT " 0.1 < OSL then this indicates little or no evidence against Ho" 9600 PRINT

9610 PRINT " In this case the OSL is approximately = "P(Q)

9620 RETURN

9630 PRINT



9660 CLS

9670 RETURN

9680 '

9690 ' CALCULATE AND DISPLAY THE PROCESS CAPABILITIES

9700 '

9710 '

9720 INPUT "ENTER THE LEVEL OF SIGMA FOR THE PROCESS CAPABILITIES ",MAGIC


9740 PCW = MAGIC * EMSW

9750 PCS = MAGIC * EMSB

9760 PCR = MAGIC * EMSR

9770 PCTOT = MAGIC * EMSTOT

9780 GOTO 9830

9790 PCW2 = MAGIC * EMSW2

9800 PCB2 = MAGIC * EMS82

9810 PCR2 = MAGIC * EMSR2

9820 PCT0T2 = MAGIC » EMST0T2


9840 PRINT "SIGMA WITHIN EU = "EMSW : PRINT

9850 PRINT "SIGMA BETWEEN EU = "EMSB : PRINT

9860 PRINT "SIGMA BETWEEN RUN = "EMSR : PRINT : PRINT

9870 PRINT MAGIC;" SIGMA CAPABILITIES ARE :":PRINT

9880 PRINT "WITHIN EU CAPABILITY = "PCW : PRINT

9890 PRINT "BETWEEN EU CAPABILITY = "PCB : PRINT

9900 PRINT "BETWEEN RUN CAPABILITY = "PCR : PRINT

9910 PRINT "TOTAL CAPABILITY = "PCTOT

9920 IF PCW = 0 OR PCB = 0 OR PCR = 0 THEN 9930 ELSE 99o0

9930 PRINT

9940 PRINT " NOTE: A CAPABILITY ENTRY OF 0 RESULTS WHEN ITb VALUE(S)"

9950 PRINT " IS TOO SMALL TO DETECT."

9960 GOTO 10090

9970 PRINT "SIGMA WITHIN EU = "EMSW2 : PRINT

9980 PRINT "SIGMA BETWEEN EU = "EMS82 : PRINT

9990 PRINT "SIGMA BETWEEN RUN = "EMSR2 : PRINT : PRINT

10000 PRINT MAGIC;" bIGMA CAPABILITIES ARE :" : PRINT

10010 PRINT "WITHIN EU CAPABILITY = "PCW2 : PRINT

10020 PRINT "BETWEEN EU CAPABILITY = "PCti2 : PRINT 10030 PRINT "bETWEEN RUN CAPABILITY = "PCR2 : PRINT

101

10040 PRINT "TOTAL CAPABILITY = "PCT0T2

10050 IF PCW2 = 0 OR PC82 = 0 OR PCR2 = 0 THEN 10060 ELSE 10090 10060 PRINT

10070 PRINT " NOTE: A CAPABILITY ENTRY OF 0 RESULTS WHEN ITS VALUE(S)"

10080 PRINT " IS TOO SMALL TO DETECT."

10090 END

APPENDIX B

USER'S MANUAL

102

103

USER'S MANUAL

The interactive process capability package has been

designed to work on IBM personal computers and other person

al computers compatible with the IBM. The programs that

constitute the package have been written in BASICA. The two

programs i.e., the title display module and the calculations

module are compiled and subsequently a batch file, consist

ing of both of the compiled programs, has been created.

Booting the Computer

Put a PC DOS diskette in drive A, Disk drive A is the

drive on the left on most personal computers. It is the

upper one on computers with horizontal drives. Put the

diskette containing the package in drive B. Close the drive

by pulling the lever next to the disk slot over the slot

opening. Turn on the computer, if it is not on already. If

the computer is already on, keeping the CTRL and ALT >:ays

pressed, press the DEL key.

Input Information

When the "A >" sign appears on the display screen,

enter B: and a "B >" sign will appear. Then enter PROCAP.

The title of the software appears on the display screen, as

shown in Figure 8. By pressing any one of che keys co

104

EXPERiriENTAL DESIGNS AND ANALYSIS

by P . RAJU

DEPARTMENT OF INDUSTRIAL ENGINEERING

TEXAS TECH UNIVERSITY

C <PRESS ANY KEY FOR CONTINUATION)

Figure 8: Title of the Software

105

proceed, a brief description of the software is displayed,

.as shown in Figure 9. Press any key to obtain the menu shown

in Figure 10. After responding to the next question, enter A

or a for a display of the description and an example of

Model I, B or b for Model II, C or c for Model III and D or

d for Model IV. Figure 11 shows the description of Model III.

In response to the next question, shown in Figure 12,

enter Y or y to display the field layout for any one of the

models to be selected next. N or n must be entered if the

layout display is not required. This causes the title mo

dule to enter into the calculation module. Figure 13 illus

trates a field layout for Model III.

Figure 14 shows an example of responses to the next set

of questions. Enter Y or y if the data is to be entered from

the keyboard. Enter proper factor names (up to 8 characters)

and their respective factor levels with an integer value

between 2 and 5. Respond to the question about the number of

runs with an integer up to 5, for 256 K memory.

For a display of the randomization for the experiment,

enter Y or y in response to the next question. The ordering

display of the experiment, obtained from this option, is

shown in Figure 15. Respond to the question about the sub-

sample size with an integer up to 10, for 256 K memory.

106

EXPERinENTAL DESIGNS AND ANALYSIS

OF PROCESS VARIATION CAPABILITIES

RANOOniZED COMPLETE BLOCK DESIGN UITH SUBSAMPLING IS THE EXPERIMENT USED IN THIS SOFTWARE TO ANALYZE THE PROCESS VARIATION CAPABILITIES.

RANDOMIZED COMPLETE BLOCK DESIGN IS AN EXPERIMENTAL DESIGN IN WHICH THE TREATMENT COMBINATIONS ARE RANDOMIZED UITHIN A BLOCK AND SEVERAL BLOCKS ARE RUN. THIS IS A BALANCED DESIGN - EACH TREATMENT APPEARS ONCE IN EACH BLOCK AND EACH BLOCK CONTAINS ALL TREATMENTS.

<PRESS ANY KEY FOR CONTINUATION>

Figure 9: Brief Description of the Software

THE FOLLOUING FOUR MODELS ARE USED FOR THE ANALYSIS

A) MODEL I : SINGLE FACTOR (RANDOM EFFECT) B) MODEL II : SINGLE FACTOR (FIXED EFFECT) C) MODEL III : TUO FACTOR EXPERIHENT D) MODEL IV : THREE FACTOR EXPERIMENT

DO YOU UANT A BRIEF DESCRIPTION OF THE ABOVE nQDELS(Y/N) y

SELECT THE MODEL (A.S.C.O) C

Figure 13: Main Menu

107

TWO FACTOR RANDOMIZED COMPLETE BLOCK WITH SUBSAMPLING

IN THIS EXPERIMENT THE NUMBER OF FACTORS INVOLVED ARE TWO. THE FACTOR

COMBINATIONS ARE RUN IN A RANDOM ORDER DURING EACH RUN. THE POINT IN

DOING THIS IS TO AVOID THE INTRODUCTION OF ANY SYSTEMATIC BIASES IN THE

MACHINE SETTINGS OR PERFORMANCE OF THE EXPERIMENT.

FOR EXAMPLE, IN A TWO FACTOR CASE, THE FACTORS A & B WITH THREE LEVELS

EACH CAN BE CONSIDERED, FACTOR LEVELS (FOR FACTOR A) OF X UNITS, Y UNITS & Z

UNITS AND FACTOR LEVELS (FOR FACTOR B) OF U UNITS, V UNITS & W UNITS CAN

BE CHOSEN FOR THE EXPERIMENT. THIS LEADS TO THE TREATMENT (A1B1,A182..

..A3B3) COMBINATIONS FOR THE EXPERIMENT. CARE MUST BE TAKEN TO ASSURE

THAT THE DIFFERENCE BETWEEN THE EXTREMES (A1B1 i A3B3) IS NOT SO GREAT

OR SO LITTLE SO AS TO PRODUCE AN EXPERIMENT WITH LITTLE USEFUL INFORM

ATION. THE RUNS WILL REPRESENT BLOCKS IN THE EXPERIMENT AND PROVIDE A

MEASURE OF RUN TO RUN VARIABILITY.RUNS MAY REPRESENT DAYS IN THE EXPE

RIMENT. ALL FACTOR A » FACTOR B (TREATMENT) COMBINATIONS SHOULD APPEAR

ONCE AND ONLY ONCE IN EACH RUN. A • B COMBINATIONS SHOULD BE RUN IN A

RANDOM ORDER DURING EACH RUN. EACH RUN SHOULD BE RERANDOMIZED. ALL

EXPERIMENTAL UNITS WITHIN A BLOCK SHOULD BE SELECTED FROM A HOMOGENEOUS

BATCH OR LOT OF EXPERIMENTAL UNITS.

Figure 11: Description of Model III

00 YOU UANT THE BRIEF DESCRIPTION OF ANY OTHER MODELS ''(Y/N)

DO YOU UANT THE FIELD LAYOUT ? (Y/N) Y

SELECT THE DESIGN (A.S.C.O) C

Figure 12: Fielcf Layout Questions

108

FIELD LAYOUT FOR MOOEL III

BLOCK 1

BLOCK 2

BLOCK S

Al Bl

Al 81

Al 82:

Al 82!

A & 8 REPRESENT THE NAMES OF THE 2 FACTORS. THE NUMERIC NUMBER ADJACENT TO FACTOR NAMES REPRESENTS THE LEVEL OF THE CORRESPONDING FACTOR

OO YOU UANT THE FIELD LAYOUT OF ANY OTHER MODELS ? <Y/N)

Figure 13: Field Layout for Model III

00 YOU UANT TO ENTER THE DATA FROM THE KEYBOARD ? (Y/N) SELECT THE DESIGN FOR THE ANALYSIS (A.S.C.O) C THE NUMBER OF FACTORS INVOLVED FOR THE SELECTED MODEL THE NAME OF THE FACTOR I ? Tlm« THE NUMBER OF FACTOR LEVELS FOR FACTOR Tim* 3 THE NAME OF THE FACTOR 2 ? Pr«5SUPt THE NUMBER OF FACTOR LEVELS FOR FACTOR Pr«ssur« 3 THE NUMBER OF RUNS 3 DO YOU UANT THE RANDOMIZATION ? (Y/N) Y

Figure 14: Set of Interactive Questions

139

Enter the data (response measurement) as a response to

the next set of documented statements. If, the user presses

the carriage return key, accidently, without entering the

data, a proper message is displayed on the screen. All the

data are double checked when they are entered, by asking the

user a question "Has correct data been entered ?" after each

data entry. A message "The data has been recorded", at the

end of the data entry is printed out. Figure 16 illustrates

the above steps.

The next question asks whether data are to be displayed

or not. Enter Y or y to display and N or n if no display is

desired. A portion of the displayed data is shown in Figure

17. Respond with Y or y to the next question, to make

changes, if any, in the data. A message "New data entered"

is displayed and it is followed by a question for more

changes in data. Enter N or n if no changes are required.

After the changes, if a re-display of the data is

required, enter Y or y in response to the next question.

Outliers, for a set of subsamples, can be identifier by

entering Y or y in a response to the next question r n t ^ c i n

integer number between 1 to 4 as the sigma level for out

lier identification. The outlier (s), if any, is pointed OUL

by an arrow(s) and there is an option to chan-je any dat.-i

110

FOR RUN 1 THE RANDOMIZATION IS AS SHOUN BELOU Time 1 Time 2 Time 3

Pressure 1 4 5 9 Pressure 2 7 6 8 Pressure 3 1 3 2

PRESS ANY KEY TO PROCEED FOR RUN 2 THE RANDOMIZATION IS AS SHOUN BELOU

Time 1 Time 2 Time 3 Pressure 1 2 7 4 Pressure 2 5 9 1 Pressure 3 6 8 3

PRESS ANY KEY TO PROCEED FOR-RUN 3 THE RANDOMIZATION IS AS SHOUN BELOU

Time 1 Time 2 Time 3 Pressure 1 9 6 3 Pressure 2 1 7 8 Pressure 3 2 4 5

PRESS ANY KEY TO PROCEED THE NUMBER OF SUBSAMPLES 5

Figure 15: Order of Randomization

FOR RUN 1 FOR FACTOR Time,LEVEL 1 FOR FACTOR Pressure,LEVEL 1 FOR SUBSAMPLE ''1 ENTER THE DATA '42 HAS CORRECT DATA BEEN ENTERED ? <Y/N) y

FOR RUN 1 FOR FACTOR Time,LEVEL 1 FOR FACTOR Pressure.LEVEL 1 FOR SUBSAMPLE 2

NO |ATrENT lRED! \HE EXPERIMENT DOES NOT ACCEPT HISSING VALUES

HAS^COJRIC?' 'SATA1EEN ENTERED ? (Y/N) n

HAS^CoJRic?^SATA^BEEN ENTERED ? (Y/N) y FOR RUN 1 FOR FACTOR Time,LEVEL 1 FOR FACTOR Pressure,LEVEL 1 FOR SUBSAMPLE 3 ENTER THE DATA

Figure 16: Data Entry

I l l

THE DATA HAS SEEN RECORDED DC YOU WANT IT TO BE DISPLAYED ENTER Y/N y RUN LEVEL OF LEVEL

FACTOR Time FACTOR OF SUBSAMPLE Pressure NUMBER

DATA

I I

1 1 1 1 1

1 1 1 1 1

1 2 3 4 5

42 44 43 43 41

i I 1

I 1 1 1 1

2 2 2 2 2

1 2 3 4 5

35 35 34 34 35

1 1

1 1

3 3

PRESS ANY KEY TO PROCEED

1 2

46 46

3 3 3 3 3

3 3 3 3 3

3 3

3 3 3

3 3 3 3 3

3 3 3 3 3

3 3

PRESS ANY KEY 3 3 3

1 1 1 1 1

2 2 2 2 2

3 3

TO PROCEED 3 3 3

1 2 3 4 5

• 1

2 3 4 5

1 2

3 4 5

90 99 90 98 97

111 115 100 134 122

107 LJl

111 120 104

ANY DATA TO BE CHANGED ? (Y/N) n WANT TO REDISPLAY THE DATA ? (Y/N) n

Figure 17: Data Display

112

for that particular set, of subsamples, as illustrated in

Figure 18.

The external data file creation section of the package

is invoked by entering Y or y in response to the saving

option question in Figure 19. Once this selection is made,

the user is asked for the name of the file that the data are

to be saved on. The name of the file is of the form, DRIVE :

FILENAME. Using a drive name of A results in the file being

saved on a diskette in drive A.

A message is printed out saying that the "Calculation

to obtain the ANOVA table is going on" on the display

screen. The next set of questions ask whether the model of

the experiment and the ANOVA table are to be displayed.

Enter Y or y to display them, as shown in Figures 20 and 21.

The ANOVA table printed out is also shown and discussed in

Chapter IV. The statistical tests (F-tests) follow the ANOVA

table and appropriate tests for each model are displayed

along with the calculated observed significance levels. A

part of the displayed F-test section is shown in Figure 22.

Finally, the process capabilities are printed out (for

a specified level 4 sigma). This is illustrated in Figure

23. An appropriate message is displayed when any of the

capabilities are 0. Figure 24 shows how to read the data

113

DO YOU WANT THE OUTLIER TO BE IDENTIFIED ? (Y/N) y

AT WHAT LEVELd SIGMA,2 SIGMA,3 SIGMA,4 SIGMA)? (1,2,3,4) 3 IN RUN 2 FACTOR Time,LEVEL 2 FACTOR Pressure,LEVEL 2 SUBSAMPLE DATA

67 68 66 67 80 { IS AN OUTLIER

DO YOU. WANT TO CHANGE DATA 1 «

IN RUN 2 FACTOR Time,LEVEL 3 FACTOR Pressure,LEVEL 2 SUBSAMPLE DATA

57 57 5 { IS AN OUTLIER 54 52

DO YOU WANT TO CHANGE DATA 1 ENTER Y/N n DO YOU WANT TO CHANGE DATA 2 ENTER Y/N n DO YOU WANT TO CHANGE DATA 3 ENTER Y/N y ENTER DATA 55

DO YOU WANT TO CHANGE DATA 4 ENTER Y/N n DO YOU WANT TO CHANGE DATA 5 ENTER Y/N n

NO MORE OUTLIERS

Figure 18: Outlier Identification

DO YOU WANT TO SAVE THE DATA ? (Y/N) Y

ENTER THE NAME OF THE FILE TO SAVE THE DATA (DRIVE:FILENAME) B:DEMO

CALCULATION TO OBTAIN ANOVA TABLE IS GOING ON

Figure 19: Saving the Data

114

00 YOU UANT TH6 MOOEL OF THE EXPERIMENT TO BE DISPLAYED ?(Y/N) Y THE MOOEL OF THE EXPERIMENT IS

YlJklw « iwj • RI - Tj • Ck • TCjk * BKljk) • U«<ijlk)

1 INDICATES RUN J INOICATES Tifli« k INOICATES Pressure 1 INOICATES EU'S UITHIN RUN. Ti«e ANO Pressure • INOICATES LOCATIONS UITHIN RUN, EU. Tiaie ANO Pressure YljklM « A RESPONSE MEASUREMENT


Figure 20: Model of the Experiment

00 YOU »(ANT THE ANOVA TAdLE TO b£ DISPLAYED ? (Y/N) y THE ANOVA TABLE

SOURCE

TOTAL

RUN Time

Pressui

Time

re •Pressure

SETXEEN EU's

RUN RUN RUN «1TH1N

»Time

•Pressure

•Time

EU's

•Pressure

OF

134 2 2 2 4

16 4 4

8 10b

SS

7.!540E+04

1.2869E+04

2. 2076E-t.04

1.6132E+03

8.4008E+03

2.5349E+04

I.6449E+04

3.1744E+03

5.7263E+03

1.2316Ef03

MS

6.4344E+03

1.1038E+04

3.0659£•^02

2. 1002E+O3

1.5a43£+03

1.1404E+01

F

4.061E+00

6.9<37£- 00

5.U91E-01

1.326E+00

1.389t+02


Figure 21: ANOVA Table

115

BE^PIRFORSED"^ REPRESENTS A SEQUENCE OF STATISTICAL TESTS THAT CAN

MSB / MSW : [F(9) J

Test for a scacistically significant between EU variability. If tfte test is not significant, then the between EU variability is relatively small and cannot be detected in che presence of tne wicnm SU variability (at a specified s-significance level).

The null hypothesis, Ho, states that there is no variability effect and the alternate hypothesis states that the variability effect is present

In general, if OSL < 3.31 then this indicates very strong evidence againsr Ho

3.31 < OSL < 3.35 then this indicates strong evidence against Ho 3.35 < OSL < 3.1 then this indicates some evidence against Ho 3.1 < OSL then this indicates little or no evidence against Ho

In this case the OSL is approximately • Q


MSTC / MSB : [F(5)]

Test for a Time • Pressure interaction affect. If the test is significant, it indicates that the Time or Pressure effect is not the same at each level of Pressure or Time respectively. Therefore, the Time and Pressure effects should not be interpreted independent of one another.

The null hypothesis. Ho, states chat there is no interaction affect and t-he alternate hypothesis states that the interaction effect ;s present

In general. if OSL < 3.31 then this indicates very strong evidence against Ho

3.31 < OSL < 3.35 then this indicates strong evidence against Ho 3.35 < OSL < 3.1 then this indicates some evidence against Ho 3.1 < OSL then this indicates little or no evidence against Ho

In this case the OSL is approximately " .•i58

PRESS ANY .KEY TO PROCEED

Figure 22: Statistical Tests (F-Tests)

116

ENTER THE LEVEL OF SIGMA FOR THE PROCESS CAPABILITIES 4 SIGMA WITHIN EU » 3.376971

SIGMA BETWEEN EU » 17.73654

SIGMA BETWEEN RUN « 10.38171


WITHIN EU CAPABILITY » 13.50789

BETWEEN EU CAPABILITY » 70.94616

BETWEEN RUN CAPABILITY » 41.52683

TOTAL CAPABILITY » 83.30845

•NOTE:. A CAPABILITY ENTRY OF 0 RESULTS WHEN ITS VALUE (S) IS- TOO SMALL TO DETECT.

Figure 23: Process Capabilities

•DO YOU WANT TO ENTER THE DATA FROM THE KEYBOARD ? (Y/N) N NAME OF THE DATA FILE (DRIVE:FILENAME) 3:DEM0

Figure 24: Reading Data from an External File

117

from an external file, when N or n is entered as a response ,

to the question in Figure 14. The structure of the external

file is shown in Figure 25. The model selected (of the

experiment, A,B,C,D) is the first line, followed by the

factor names and their levels. The number of runs and the

number of subsamples constitute the next two lines, respec

tively. Then, the data is entered as shown.

Figure 26 shows a work sheet which will aid the experi

menter, while recording the data. Once this is completed,

the user can create an external file similar to that shown

in Figure 25. It must be remembered that the actual data

file is a single column file and that the extra columns in

the work sheet are provided only as an aid in developing the

data file. The data can then be displayed, after it is read

into the program, as in Figure 17.

The data for the example problem used in this section,

is shown in Figure 27. The same general procedure, as above,

is used for for all four models, pressing a key to continue

execution when appropriate.

Execution can be stopped by pressing Ctrl Break

("Break). If the execution is stopped, it can be restarted

by entering PROCAP.

118

Time 3

Pressure 3 3 5 42 44 43 43 41 35 35 34 34 35 46 46 46 46 46

58 56 59 53 54 90 99 90 98 97 111 115 100 134 122 107 101 111 108 104

Figure 25: Structure of the External File

Run Factor 1

Factor level of Factor 2 Factor 3

Subsample #

Response

li9

Figure 26: Blank Work Sheet

120

Run

1

1

1

1

1

1

1

1

Factor 1

1

1

1

2

2

2

3

3

Factor level of Factor 2

1

2

3

1

2

3

1

2

Factor 3 Subsample

#

1 2 3 k 5

1 2 3 4 5 1 2 3 4

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 2 ,3 4

5

1 2 3 4

Response

42 44 43 43 41

35 35 34 34 35

46 46 46 46 46

33 37 34 38 35 56 55 58 51 50 67 65 66 66 66

58 59 54

53 52

37 33 35 34 36

Figure 27: Data for the Example Problem

121

Run

1

2

2

2

2

2

2

2

Factor 1

3

1

1

1

2

2

2

3


3

1

2

3

1

2

3

1

Factor 3

•

Subsample #

1 2 3 4 5

1 2 3 4

1 2 3 4

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 2

3 4

Response

44 44 44 44 44

45 44 42

47 1*9

55 55 55 55 55

35 35 33 37 38

35 35 35 35 35 67 68 66 67 80

U 88 88 87

77 77 77

F i g u r e 27: (Cont inued)

122

Run

2

2

3

3

3

3

3

3


3

3

1

1

1

2

2

2

Factor 2

2

3

1

2

3

1

2

3

Factor 3 Subsample

1 •

2

3 4

1 2 3 4 5

1 2 3 4 5 1 2 3 4 5

1 2 3 4

1 2 3 4

1 2 3 4 5

1 2 3 4 5

Response

57 57 55 54 52 87 86 88 88 89

55 55 55 55 55 46 44 40 37 kk

33 33 33 33

48 48 46 45 48 74 72 77 71 76

58 56 59 53 54

Figure 27: (Continued)

L

123

Run

3

3

3

Factor 1

3

3

3


1

2

3

Factor 3 Subsample

#

1 2 3 4

1 2 3 4 5 1 2 3 4 5

Response

90 99 90 98

I l l 115 100 '

-134 122 107 101 111 108 104

F i g u r e 27: (Cont inued)

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requirements for a master's degree at Texas Tech University, I agree

that the Library and my major department shall make it freely avail

able for research purposes. Permission to copy this thesis for

scholarly purposes may be granted by the Director of the Library or

my major professor. It is understood that any copying or publication

of this thesis for financial gain shall not be allowed without my

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right infringement.

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