Sequential Experimental Designs For Sensitivity Experiments

NIST GLM ConferenceApril 18-20, 2002

Joseph G. VoelkelCenter for Quality and Applied StatisticsCollege of EngineeringRochester Institute of Technology

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Sensitivity Experiments ASTM method D 1709–91 Impact resistance of plastic film by free-falling

dart method

Film

D art Pass

Fail

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Objectives

Engineer Specify a probability of

failure – 0.50, 0.10, … Find dart weight x= such

that Prob(F; )=

Statistician Find a strategy for

selecting weights {xi} so that is estimated as precisely as possible

0.1

X

p

Darts are dropped one at a time. Weight of ith dart may depend on results obtained up to date

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Data Collection PossibilitiesNon-sequential Specify n and all the {xi} before any Pass-Fail data {Yi} are

obtained. Find dose of drug at which 5% of mice develop tumors

Group-sequential Example: two-stage. Specify n1 and the {x1i}. Obtain data {Y1i} Use this info to specify n2 and the {x2i}. Obtain data {Y2i}

Same mice example, but with more time.

(Fully) Sequential Use all prior knowledge: x1Y1 x2Y2 x3Y3 x4Y4

Dart-weight example. One machine, one run at a time.

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Model and Objectives

Objective: Example Estimate weight at

which 10% of the samples fail

0.1

X

p

1 ( )

1 (1 exp( / ) )

x xP Y p F x

x

x

ln( /(1 ))

logit(p )x xx p p

logit( )

ln(0.10 / 0.90)

logit(0.10)

= 2.2

ˆvar( ) So, try to set the {xi} to

minimize

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Our Interest

(Fully) Sequential experiments Estimating a corresponding to a given , e.g. 0.10. The real problem. = 0.50? = 0.001?

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A Quick Tour of Some Past Work

Up-Down Method. Dixon and Mood (1948)

Only for =0.50 Robbins-Munro (1951)

wanted {xi} to converge to .

Like Up-Down, but with decreasing increments

far from 0.50 convergence is too slow

151050

7

6

5

Run

Set

ting

151050

7

6

5

Run

Set

ting

Pass

Fail

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A Quick Tour of Some Past Work

Wu’s (1985) Sequential-Solving Method Similar in spirit to the R-M procedure Collect some initial data to get estimates of and Choose the next setting, xn1 , to solve

( )F xn n 1 . So xn n 1

Better than R-M, much better than Up-and-Down Performance depends somewhat heavily on initial runs Asymptotically optimal, in a certain sense

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Some Non-Sequential Bayesian Results

Tsutakawa (1980) How to create design for estimation of for a given . Certain priors on and Some approximations Assumed constant number of runs made at equally

spaced settings. Chaloner and Larntz (1989)

Includes how to create design for estimation of for a given

Some reasonable approximations used Not restricted to constant number of runs or equally

spaced settings.

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Examples of Optimal Designs for estimating when =0.50.

(Based on Chaloner and Larntz)

~ , , .Unif 1 1 14a f ~ , , .Unif 10 10 14a f

100-10x

-10 0 10x

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Examples of Optimal Designs for estimating when =0.05.

(Based on Chaloner and Larntz)

~ , , .Unif 1 1 14a f ~ , , .Unif 10 10 14a f

100-10x

100-10x

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This Talk. Bayesian Sequential Design

A way to specify priors Measures of what we are learning about , ,

and —AII and Information Specifying the next setting, with some insights Some examples and comparisons Rethinking the priors

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Specifying Priors Consider the related tolerance-distribution problem The r.v. Xi represents the (unobservable) speed at which

the ith sample of film would have failed. Say from a location-scale family (e.g., logistic, normal, …)

1

1 (1 exp( / ) )

xP Y P X x

x

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Specifying Priors

Two-parameter distribution Could specify priors on

(, ) (, ) (, )

For simplicity, want to assume independence so only need to specify marginals of each parameter

, , not ,s X s X z s

0.50

(, )

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Specifying Priors

Instead of (,) … Consider =0.10

example Consider

distance

from to = 2.2

1050

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

x

c5/.7

logit(0.10)= 2.2

Easier for engineer to understand

(, )

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Specifying Priors

Ask engineer for Best guess and 95% range for

5.0 ± 3.0 Best guess and 95% range for – distance

6.6 / 2.0 Translate –=2.2 into terms: 3.0 / 2.0 Translate into normal, independent, priors on

and ln() We used a discrete set of 1515=225 values as

prior distribution of (,) (, )

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Specifying Priors

More natural for engineer to think about priors on and . We let engineer do this as follows.

We created 27 combinations of prior distributions: best guess—10 uncertainty (95% limits)— ± 2, 4, 6. best guess— 1, 3, 5 uncertainty (95% limits)— / 2, 4, 6.

We graphed these in terms of (,)

(, )

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Example of Prior Distributions of =10± 4 /

10 4

= 1 = 3 = 5 2

15105 100-10 100-10 100-10 4

15105 100-10 100-10 100-10 6

15105 100-10 100-10 100-10

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Finding the next setting xn+1 to runAssume we have collected data 1,..., nY Y based on

settings 1,..., nx x (This includes the case of not having any data, for which n=0.)

B a s e d o n t h i s , u p d a t e t h e o r i g i n a l p r i o r s o n a n d ( a n d h e n c e ) t o f o r m 1D i s t , | , . . . , D i s t ,n nY Y ,

o u r c u r r e n t p r i o r o n a n d . We would like to select a new setting 1nx so that if we run

there and get the result xY, we want Varn xY , the (new)

posterior variance of , to be as small as possible. However, we don’t know xY, so can’t find Varn xY . So find 1nx to minimize result on average, or EVarnnxY, where expectation is wrt xY. The optimal setting.

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AII Measure

Before we make the ( )n1th run, current prior variance of

, denoted by 1Var | ,..., =Varn nY Y , is determined. 1Varn (total) information about after n runs. E q u iva le n t w a y to fin d o p tim a l x is th e x th a t m a x im ize s

1 1

A IIV arE (V ar ) nn n x

xY

AII = anticipated increase in information. Plotting AIIxaf versus x gives a good indication for how sensitive our results are to the choice of the next x

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Simple Example

Objective: find the corresponding to =0.10

Prob

8 0.25

9 0.50

10 0.25

Prob

1 0.25

2 0.50

3 0.25

Pr8 1 5.8 .0625

8 2 3.6 .1250

8 3 1.4 .0625

9 1 6.8 .1250

9 2 4.6 .2500

9 3 2.4 .1250

10 1 7.8 .0625

10 2 5.6 .1250

10 3 3.4 .0625

0E 5.8 0.0625 ... 3.4 0.0625 4.6

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Simple Example

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Simple Example

Finding the AII for various x settings

20100

0.02

0.01

0.00

x

AII

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How AII “Thinks”

20100

0.02

0.01

0.00

x

AII

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First Simulation (,)=(8,1.82). Makes =4.0

403020100

6

5

4

3

2

Run

Set

ting,

E(

P assFailSettingX

Setting increment = 1

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Example with a More Diffuse Prior

=10 ± 4, =5 / 6

Simulation againdone with =8,=4

100-10

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Example with a More Diffuse Prior

6050403020100

7

6

5

4

3

2

1

0

Run

Set

ting,

E(

P assFailSettingX

403020100

6

5

4

3

2

Run

Set

ting,

E(

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Behavior of AII after 0, 2, 10, 20, 60 runs

20100

0.010

0.005

0.000

Setting

AII

0

2

10

20

60

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Information on , , = 2.2

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0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Run

Info

rmat

ion

on

A serious problem—all the information on was obtained through The simulation trusted the relative tight prior on … Another problem: more

objective methods of estimation, such as MLE, will likely not work wellAre there other ways to specify priors that might be better? Two methods…

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Equal-Contribution Priors For =2.2, restrict original prior so that

Var0()=Varo(2.2) Results of another simulation

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Run

Info

rmat

ion

on

6050403020100

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

Run

Info

rmat

ion

on

Problem: fails for case =: Var0()=Varo(0)?

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Relative Priors Consider the tolerance-distribution problem The r.v. Xi represents the (unobservable) speed at which

the ith sample of film would have failed. Say from a location-scale family (e.g., logistic, normal, …)

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Relative Priors

We observe only the ( x, Yx ) ’s If we could observe the X ’s, the problem would

be a simple one-sample problem of finding the 100 percentile of a distribution.

Assume the distribution of the X ’s has a finite fourth moment.

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Relative Priors

Using delta-method to find Var(s) and m-1m

22sd sd

4X m s

m

So, to a good approximation

1sd sds X k

2 2 4 22Var Var

1X m s

m m

After m runs, observing X1, X2, …, Xm, we have

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Relative Priors

So, with k1 and k2 known,

1 2ˆ ˆsd = sdk k

So, in this sense it is defensible to specify only the prior precision with which is know, and base the prior precision of upon it.

2ˆ ˆˆ ˆ= , , corr , =0 X k s

Now assume tolerance distribution is symmetric and its shape is know, e.g. logistic. Then

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Logistic Example

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1.0

0.5

0.0

Run

Info

rmat

ion

on

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Summary

AII as a useful measure of value of making next run at x. Combination of shift in posterior mean & probability that a failure will occur at x

Informal comparison to non-Bayesian methodsBayesian x-strategy is more subtle

Danger of simply using any prior, and recommended way to set priors