Statistical Experimental Design

A Primer

by

H. B. Oblad (Bruce)

Getting Answers Easier - Overview

The Old Method

The Better Method

Simple Statistics for the Lab

Let’s Try It Out!

The Old Method

Experiments one variable at a time in sequence. Effect of Temperature on Yield

Yie

ld,

wt

%

Temperature, °C

Pressure = 1000 psi

Time = 20 min

Next Set of Experiments

• Effect of PressureY

ield

, w

t %

Pressure, psi

Temperature = 300 °C

Time = 20 min

More Experiments

• Effect of Time on YieldY

ield

, w

t %

Time, min

Temperature = 300 °C

Pressure = 1000 psi

What Have We Learned?• 13 Experiments in 3 Factors

Temp

Pre

ssur

e

Time

• What combinations of conditions have we covered? What’s still unknown?

• Do we know anything about the repeatability of our lab technique?

• Are the responses straight or curved?• Can we build a meaningful model that

leads to a mechanism?• Could we have done less work and gotten

more information?• Minor information about effects of factors.• Know nothing about interactions.

Temp

Pre

ssur

e

Time

A Smarter Way

2-Level Factorial Design

• 8 Tests (XY X = levels, Y= factors)• Now know what happens over a large

experimental volume.• Now know the effects of factors at two

surfaces. Effect of factors tested 4 x each• Some information about interactions

between factors.• Repeatability is still unknown.• Curvature?

Temp

Pre

ssur

e

Time

An Even Smarter Way

2-Level Factorial Design w/ Center Points

• 11 Tests (3 cntr pts), 13 Tests (5 cntr pts)

• Now know the effects of factors at two surfaces and within the volume.

• More information about interactions between factors.

• Repeatability is now estimated or known.

• Curvature can be estimated.

• Predictive model is easy to create.

Box-Behnken Design 3 Factor, 3 Level

Spherical, so extrapolation is less risky. 15 tests (3 cp), 17 tests (5 cp)A fractional factorial design

Simple Statistics

• Bell Curve = Normal Dist. = Gaussian Dist.• Total population or very large sample• Errors in lab methodology are assumed random

and normally distributed except for time. Must randomize order to bury effect of time into error.

• Repeated tests may be pooled to estimate std. dev. and variance.

Bell Curve = Normal Dist.

68% of area is <>+/-1 std. dev.

94% of area us <>+/- 2 std. dev.

99% of area is <>+/- 3 std. dev.

Means Testing• If the means and standard deviations of

the measurements are equal, the things being measured are of the same population. Opposite is true also (null hyp.) Use Student’s t-test.

Means Testing

• If the means are the same, the things are of the same population. Use Welch’s t-test

Analysis of Variance(ANOVA)

• Variance (standard deviation2) of means of several sample groups is determined by F-test. Probability criterion is used for pass/fail or probability of F being equal is given.

Factors, Responses and Interactions

• Numeric Factors are variable inputs to a process e.g. feed rate, temperature, pressure, component concentration, knobs, levers

• Categorical Factors are discrete inputs e.g. catalyst type, feed material, operator

• Responses are effects of changes in factors e.g. Reaction rate increases w/ temp.

• Factors that affect each other are said to interact e.g. drinking, driving, vs drunken driving

Rubber Band Experiment

• What affects the distance traveled?

• Factors? How many?

• Numeric or categorical?

• Which design to use?

• Can we make a predictive model?

• Any interaction of factors?

• Can we understand the problem better?