Dr. Gary Blau, Sean HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION III SINGLE...

Monday, Aug 13, 2007Dr. Gary Blau, Sean Han

Statistical Design of Experiments

SECTION III

SINGLE FACTOR EXPERIMENTS


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


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


WHEN TO USE DOE

Want to develop new knowledge

Already know some things about process

Have resources tomake several runs

NOT

NOT

NOT

Only want to “makeproduct” or demonstrate

New, unfamiliar process

Can afford many more runs


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


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)


PHASE OF A PROJECT

The Design

– Number of Experiments

– Sequential experimentation

Hypothesis Consequence Consequence

Experimentation Experimentation

– Randomization/Blocking/Repeating

– Mathematical model to describe experiment


PHASE OF A PROJECT

The Analysis

– Data collection and processing

– Computation of test statistics

– Interpretation of results


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


DEFINITION OF TERMS

• Factor - controllable variable

• Level - value of the factor

• Treatment - distinct collection of factor levels


CONSIDERATIONS IN PLANNING THE EXPERIMENTS

– Factor Levels

– Replicates

– Randomization

– Blocking (Restriction on Randomization)


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)


FACTOR LEVELS FOR QUANTITATIVE FACTORS For quantitative factors space the levels

reasonably far apart so both detection and estimation of effects are possible (but not too far apart)

Level of Factor

Res

pons

e

(-) (+)

Eff

ect

Response

Experimental Error


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


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.

2/2Zn = ( ) ,

D


RANDOMIZATION

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


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


RANDOMIZATION

What should be randomized?

• Assignment of experimental Units to Treatments

• Order of running Experiments

• Order of evaluating Experimental Results


A SINGLE FACTOR AT THREE LEVELS EXAMPLE

Randomly apply treatments A, B and C to nine objects

Results:Treatment

A B C1 2 55 4 43 6 6

Average 3 4 5

Is there a difference between the treatments?


PLOT OF EXAMPLE DATA IN JMP


ANALYSIS OF EXAMPLE DATA IN JMP

There are no significant differences between the factors.


SAME EXAMPLE WITH DIFFERENT RESULTS

A B C 2.8 4.2 5.2 3.2 3.8 4.8

3.0 4.0 5.0Average 3.0 4.0 5.0

Now is there a difference between the treatments?


PLOT OF EXAMPLE DATA IN JMP


ANALYSIS OF EXAMPLE WITH NEW DATA IN JMP

There are significant differences between the factors.


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


% 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


RESULTS OF % TABLET DISSOLUTION EXAMPLE

• Results of completely randomized design for four incipient types:

1 2 3 456 64 45 4255 61 46 3962 50 45 4559 55 39 4360 56 43 4158.4 57.2 43.6 42.0 Overall = 50.3

• Is there a difference between the excipients?


JMP PLOT OF % DISSOLUTION DATA


JMP ANALYSIS OF % DISSOLUTION DATA

There are significant differences between the factors.


JMP ANALYSIS OF % DISSOLUTION DATA (CONTINUED)


BLOCKING

Blocking

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


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


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?


JMP PLOT OF % API


JMP ANALYSIS OF CRD

There are no significant differences between the factors.


API SHELF LIFE STUDY WITH RANDOMIZED BLOCK DESIGN

This time the production lots of the tablets were identifiedand 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 16.4Lot #2 13.9 13.7 13.1Lot #3 15.5 15.3 15.0Lot #4 13.8 13.2 13.1AVE 14.95 14.65 14.4

Now is there a difference between coatings?


JMP PLOT OF RANDOMIZED BLOCK DESIGN


JMP ANALYSIS OF RANDOMIZED BLOCK DESIGN

Both factors are significant.


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


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?


DIFFERENT DESIGNS FOR TIRE WEAR EXAMPLE

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

Good – Reproducibility Bad – Cars confused with brands



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

Good – Randomized Bad – Not each brand on each car



(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


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):


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=25RF B=16 C=26 D=30 A=37LR C=17 D=40 A=27 B=24RR D=25 A=32 B=20 C=30


JMP ANALYSIS OF TIRE WEAR DATA

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


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.


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


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.


GRAPHICAL ILLUSTRATION OF DICHOTOMOUS SEARCH


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.


GRAPHICAL ILLUSTRATION OF GOLDEN SECTION

(A) Locate the first two experiments as shown:

(B) The optimum (maximum) lies in the interval [a, d] by unimodality. Discard [d, b].

(C) Locate third experiment symmetrically such that new d is old c, new c is calculated by b-c = d-a:


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 HanMonday, Aug 13, 2007 Statistical Design of Experiments SECTION III SINGLE...

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