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## Statistical Design of Experiments

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INTRODUCTION TO DESIGN OF EXPERIMENTAL (DOE)

Definition

A systematic procedure for manipulating process variables to guide the search for the optimum of the process

- Based on a mathematical model used to approximate the process

Dr. Gary Blau, Sean Han

PURPOSE OF DOE

- Understand and quantify process fluctuations or variability
- Identify the most important variables affecting the output levels
- Maximize profitability and quality of product
- accomplish all of the above in the MINIMAL number of experiments and with LITTLE knowledge of the Process

Dr. Gary Blau, Sean Han

WHEN TO USE DOE

Only want to “make

product” or demonstrate

Want to develop new

knowledge

NOT

Already know some

things about process

NOT

New, unfamiliar process

Have resources to

make several runs

Can afford many more runs

NOT

Dr. Gary Blau, Sean Han

LIMITATIONS

- Yield “black box” polynomial models only

Answer “which” and “how” questions but not “why” questions

- Lack of physically significant of terms/parameters in the model equations
- Preclude extrapolation/scale-up

Dr. Gary Blau, Sean Han

PHASE OF A PROJECT

The Experiment

- Statement of the problem
- Choice of response
- Selection of factors that can be controlled and varied
- Feasible ranges and choice of levels of these factors (Prior Information)

Dr. Gary Blau, Sean Han

PHASE OF A PROJECT

The Design

- Number of Experiments
- Sequential experimentation

Hypothesis Consequence Consequence

ExperimentationExperimentation

- Randomization/Blocking/Repeating
- Mathematical model to describe experiment

Dr. Gary Blau, Sean Han

PHASE OF A PROJECT

The Analysis

- Data collection and processing
- Computation of test statistics
- Interpretation of results

Dr. Gary Blau, Sean Han

PURPOSE OF SINGLE FACTOR EXPERIMENTS

- Quantify relationship between a single factor and a single measured or response variable
- Compare the relative effectiveness of two or more treatments (levels of the factor).
- Estimate the level of the factor that optimizes the response variable

Dr. Gary Blau, Sean Han

DEFINITION OF TERMS

- Factor - controllable variable
- Level - value of the factor
- Treatment - distinct collection of factor levels

Dr. Gary Blau, Sean Han

CONSIDERATIONS IN PLANNING THE EXPERIMENTS

- Factor Levels
- Replicates
- Randomization
- Blocking (Restriction on Randomization)

Dr. Gary Blau, Sean Han

FACTOR LEVELS

The range of levels over which a factor is examined is determined by exploratory testing, subjective knowledge and/or the literature

- (this is your prior knowledge)

Dr. Gary Blau, Sean Han

Response

Effect

Response

(-)

(+)

Level of Factor

FACTOR LEVELS FOR QUANTITATIVE FACTORSFor quantitative factors space the levels reasonably far apart so both detection and estimation of effects are possible (but not too far apart)

Dr. Gary Blau, Sean Han

REPLICATION

- Repeat of an entire experiment or run (trial)
- Used to Define and understand sources of experimental error
- Used to test for significance of different levels

Dr. Gary Blau, Sean Han

NUMBER OF REPEATS / REPLICATES

The number of replicates, n, needed is a function of experimental error, σ2, the difference to be detected in the response, D, and degree of confidence you want in the result(α):

Where Zα/2 is the Z value at α/2.

Dr. Gary Blau, Sean Han

RANDOMIZATION

Is conducted in order to minimizes the effect of uncontrolled factors / nuisance variables

Dr. Gary Blau, Sean Han

RANDOMIZATION

- Possible nuisance variables:
- Warm-up time for tester
- Operator differences
- Time of day
- Raw material changes
- Batch/lot differences
- Equipment wear
- Systematic process change
- Catalyst deactivation

Dr. Gary Blau, Sean Han

RANDOMIZATION

What should be randomized?

- Assignment of experimental Units to Treatments
- Order of running Experiments
- Order of evaluating Experimental Results

Dr. Gary Blau, Sean Han

A SINGLE FACTOR AT THREE LEVELS EXAMPLE

Randomly apply treatments A, B and C to nine objects

Results:

Treatment

A B C

1 2 5

5 4 4

3 6 6

Average 3 4 5

Is there a difference between the treatments?

Dr. Gary Blau, Sean Han

PLOT OF EXAMPLE DATA IN JMP

Dr. Gary Blau, Sean Han

ANALYSIS OF EXAMPLE DATA IN JMP

There are no significant differences between the factors.

Dr. Gary Blau, Sean Han

SAME EXAMPLE WITH DIFFERENT RESULTS

A B C

2.8 4.2 5.2

3.2 3.8 4.8

3.0 4.0 5.0

Average 3.0 4.0 5.0

Now is there a difference between the treatments?

Dr. Gary Blau, Sean Han

PLOT OF EXAMPLE DATA IN JMP

Dr. Gary Blau, Sean Han

ANALYSIS OF EXAMPLE WITH NEW DATA IN JMP

There are significant differences between the factors.

Dr. Gary Blau, Sean Han

EXAMPLE COMPARISON

- Variability in the data affects the ability to discriminate levels.
- Without replication no estimation of experimental error is possible
- Intuition should guide analysis

Dr. Gary Blau, Sean Han

% TABLET DISSOLUTION EXAMPLE

- Problem: Determine the effects of four different excipients on tablet dissolution after 45 minutes.
- Background: Twenty tablets were selected at random. Four different treatments (excipients) were applied to the tablets. The % dissolution was measured and are shown in the data set below

Dr. Gary Blau, Sean Han

RESULTS OF % TABLET DISSOLUTION EXAMPLE

- Results of completely randomized design for four incipient types:

1 2 3 4

56 64 45 42

55 61 46 39

62 50 45 45

59 55 39 43

60 56 43 41

58.4 57.2 43.6 42.0 Overall= 50.3

- Is there a difference between the excipients?

Dr. Gary Blau, Sean Han

JMP PLOT OF % DISSOLUTION DATA

Dr. Gary Blau, Sean Han

JMP ANALYSIS OF % DISSOLUTION DATA

There are significant differences between the factors.

Dr. Gary Blau, Sean Han

JMP ANALYSIS OF % DISSOLUTION DATA (CONTINUED)

Dr. Gary Blau, Sean Han

BLOCKING

Blocking

- Sort treatments into “blocks” which are reasonably alike in order to reduce the impact of non-controlled sources of variability

Dr. Gary Blau, Sean Han

COMPLETELY RANDOMIZED AND RANDOMIZED BLOCK DESIGN

- Completely Randomized Design (CRD)
- Order in which treatments are selected and the runs are carried out is completely random
- Completely Randomized Block Design (CRBD)
- Treatments and runs are randomly assigned within the blocks
- Every treatment must occur in each block

Dr. Gary Blau, Sean Han

API SHELF LIFE STUDY (CRD)

A completely randomized design was conducted on

expensive tablets to determine the % API after one

year with three different coatings:

Residual API Concentration(%)

Coatings: A______ _ B C

16.6 16.4 16.4

13.9 13.7 13.1

15.5 15.3 15.0

13.8 13.2 13.1

AVG 14.95 14.65 14.4

Is there a difference between the coatings?

Dr. Gary Blau, Sean Han

JMP PLOT OF % API

Dr. Gary Blau, Sean Han

API SHELF LIFE STUDY WITH RANDOMIZED BLOCK DESIGN

This time the production lots of the tablets were identified

and tablets with different coatings were settled at random.

The data are the same as the previous example:

Residual API Concentration (%)

A B C

Lot #1 16.6 16.4 16.4

Lot #2 13.9 13.7 13.1

Lot #3 15.5 15.3 15.0

Lot #4 13.8 13.2 13.1

AVE 14.95 14.65 14.4

Now is there a difference between coatings?

Dr. Gary Blau, Sean Han

JMP PLOT OF RANDOMIZED BLOCK DESIGN

Dr. Gary Blau, Sean Han

EXAMPLE COMPARISON

- Blocking can be used to eliminate its effect on the comparison among treatments
- Without blocking no blocking effects is taken into consideration
- With blocking both factors became significant

Dr. Gary Blau, Sean Han

EXAMPLE OF A LATIN SQUARE DESIGN

- A fleet manager wants to select from among four brands of tires, A, B, C, and D, which will give the least amount of tire wear after 20,000 miles. The response variable is the difference in maximum tread thickness on a tire between the time it is mounted on the wheel of the car and after it has completed 20,000 miles. Four tires of each brand will be used to get an estimate of experiment error. In order to conduct this study, four cars, a BMW, a Ferrari, a Lamborghini, and a Mercedes Benz were selected to accommodate the 16 tires.

Problem:

- How do we assign the 16 tires to the four cars?

Dr. Gary Blau, Sean Han

DIFFERENT DESIGNS FOR TIRE WEAR EXAMPLE

(A) Randomly assign one brand to each car (randomized complete design):

Good – Reproducibility

Bad – Cars confused with brands

Dr. Gary Blau, Sean Han

DIFFERENT DESIGNS FOR TIRE WEAR EXAMPLE

(B) Randomly assign brands to cars (completely randomized design):

Good – Randomized

Bad – Not each brand on each car

Dr. Gary Blau, Sean Han

DIFFERENT DESIGNS FOR TIRE WEAR EXAMPLE

(C) Randomly assign brands to the different cars and make sure each brand is on each car:

Good – Each brand on each car

Bad – Have not balanced positions on cars

Dr. Gary Blau, Sean Han

ADDITIONAL RESTRICTION ON RANDOMIZATION

(D) Randomly assign brands to cars and positions so that each brand is on each car and at each position (Latin Square Design):

Dr. Gary Blau, Sean Han

TIRE WEAR DATA

Design (C) is called a latin square design with two restrictions on randomization. The results for this design are:

Cars

BMW Ferrari Lambo MB

Positions

LF A=23 B=22 C=19 D=25

RF B=16 C=26 D=30 A=37

LR C=17 D=40 A=27 B=24

RR D=25 A=32 B=20 C=30

Dr. Gary Blau, Sean Han

JMP ANALYSIS OF TIRE WEAR DATA

Therefore significant differences exist between brands and cars, but positions are not significant.

Dr. Gary Blau, Sean Han

DETERMINE THE OPTIMUM LEVEL

In the case of quantitative factors, (i.e. factors which have measurable levels) it is often important to determine the level which optimizes (i.e. maximizes/minimizes) the response variable.

By spacing the levels uniformly over the range the optimum can be observed. However this may require a large number of levels and only the optimal one may be of interest.

A more useful approach is to sequentially search out the optimal level.

Dr. Gary Blau, Sean Han

OPTIMUM SEEKING METHOD

- Unimodality Assumption:

Assume the response only has one optimum value and is monotonically decreasing (in the case of a maximum) or increasing (in the case of a minimum) from this value.

Two Methods:

- Dichotomous Search
- Golden Section Search

Dr. Gary Blau, Sean Han

DICHOTOMOUS SEARCH

The dichotomous search works by repeatedly locating two experiments symmetrically about the midpoint and sufficiently far apart to detect a difference. The interval which contains the optimum value is selected for further subdivision.

Dr. Gary Blau, Sean Han

GRAPHICAL ILLUSTRATION OF DICHOTOMOUS SEARCH

Dr. Gary Blau, Sean Han

GOLDEN SECTION SEARCH

Analogous to the dichotomous search strategy the golden section search finds the optimum (minimum or maximum) by successively narrowing the range of values inside which the optimum is known to exist. The technique derives its name from the fact that rather than halving the interval the experiments are symmetrically located by the “golden section” ratio.

Dr. Gary Blau, Sean Han

GRAPHICAL ILLUSTRATION OF GOLDEN SECTION

- Locate the first two experiments as shown:
- The optimum (maximum) lies in the interval [a, d] by unimodality. Discard [d, b].
- Locate third experiment symmetrically such that new d is old c, new c is calculated by b-c = d-a:

Dr. Gary Blau, Sean Han

SUMMARY

- Randomization, Blocking and Replication are key design considerations in testing for significance of levels in single factor experiments.
- Optimum seeking methods are useful tools to reduce the number of experiments required to find the best conditions.

Dr. Gary Blau, Sean Han

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