A Bayesian Semiparametric Analysis

of the Reliability an d Maintenance

of Machine Tools

Jason R. W. Merrick

Department of Statistical Sciences and

Operations Research

Virginia Commonwealth University

Richmond, VA 23284

( jrmerric@vcu.edu)

Re’ k Soyer

Department of Management Science

The George Washington University

Washington, DC 20052

( soyer@gwu.edu)

Thomas A. Mazzuchi

Department of Engineering Management & Systems Engineering

The George Washington University, Washington, DC 20052

( mazzuchi@seas.gwu.edu)

A Bayesian semiparametric proportional hazards model is presented to describe the failure behavior of

machine tools. The semiparametric setup is introduced using a mixture of Dirichlet processes prior. A

Bayesian analysis is performed on real machine tool failure data using the semiparametric setup, and

development of optimal replacement strategies are discussed. The results of the semiparametric analysis

and the replacement policies are compared with those under a parametric model.

KEY WORDS: Bayesian; Markov chain Monte Carlo; Mixtures of Dirichlet processes prior; Propor-

tional hazards model; Semiparametric inference.

1. INTRODUCTION

The useful life of a mac hine tool is the duration of t ime that

the tool maintains an acceptable quality of performance. As an

alternative to replacing tools on observing unacceptable qual-

ity of performance, using planned replacements can reduce

costs associated with in-service failures of m achine tools,

thereby resulting in increased productivity due t o decreased

downtime and scrapping of material and decreased inventory

costs due to improved planning. Key to the determination

of an optimal replacement strategy is the development of

an adequate statist ical model for tool life. The development

of such a model is complicated because o f the fact that the

conditions under which a tool operates (called the operational

environment) vary even for tools on the same shop oor.

In addition, commercial engineering materials require vari-

ability within speci ed ranges of chemistry and mechanical

properties for their economical production, corresponding to

variability in the properties that degrade the tool during its

operational li fe. Variabilities can also be expected in machine

tool dynamics and materials-handling performance of the

processing systems. Thus, each machine tool may exhibit

inherent variabilities in its useful life.

In the early literature, the lack of a universally acceptable

physical theory of tool failure led to the use of an empirical

model describing the relationsh ip between tool life and oper-

ational variables, including cutting speed, feed rate, and depth

of cut. Taraman (1974) performed an experiment designed

to estimate the parameters of this empirical model. Balakr-

ishnan and DeVries (1985) extended this analysis to allow

sequential updating of parameter estimates and inclusion of

prior information in the estimation procedure. Mazzuchi and

Soyer (1989) noted that the empirical model proposed by Tara-

man (1974) accounted for the effect of the machine operating

environment bu t failed to account for aging (or wear out)

characteristics of the tool. To account for both aging and the

characteristics of the machine operating environment, a pro-

portional hazards model (PHM) was proposed to assess tool

life. In specifying the PHM, a Weibull model was assumed for

the baseline failure rate to incorporate aging of the tool, and

the effect of machining environment, as speci e d by Taraman

(1974), was used to modulate the baseline failure rate.

Because of t he variation in individual characteristics of

machine tools and the possible omission of relevant variables

(describing the operational environme nt) from the model,

a fully parametric PHM for tool life may not be adequate.

This may re sult in suboptimal replacement strategies. Thus

a more general model that relaxes some of these parametric

assumptions may be more desirable for modeli ng machine

tool life. As pointed out by Gelfand (1999), in a situation

where the parametric assumptions may be too restrictive, a

semiparametric mo del can be developed by a nonparametric

speci cation of some portions of the model.

In this article we present a semiparametric analysis for

machine t ool life that treats the baseline fai lure rate function

in a nonparametric manner while treating the effect of the

covariates parametrically, as implied by Taraman (1974),

using the PHM approach. A mixture of Dirichlet processes

© 2003 American Statistical Association and

the American Society for Quality

TECHNOMETRICS, FEBRUARY 2003, VOL. 45, NO. 1

DOI 10.1198/004017002188618707

58

BAYESIAN SEMIPARAMETRIC ANALYSIS OF MACHINE TOOL RELIABILITY AND MAINTENANCE 59

(MDP) prior, proposed by Antoniak (1974), is assumed for

the part of the model that de nes the underlying fa ilure

distribution. Whereas the foregoing set-up leads t o a complex

analysis, inference is developed using the framework of

Escobar and West (1995) and the ef cient sampling algorithms

of MacEachern (1994). The MDP approach reduces the

restriction of the parametric baseline failure rate and allows

assessment of differences between the failure characteristics

of tools that cannot be described by the covariates in the

model of Taraman (1974). An additional bene t of the MDP

approach for the PHM is the ability to compare the parameters

of the baseline failure rate corresponding to each individual

machine tool, thus addressing the question of whether all

necessary covariates are included in th e model.

The development of optimal replacement strategies using a

nonparametric form for the failure model has been considered

from a samplin g theory perspective by Frees and Ruppert

(1985) and Aras and Whitaker (1991). Neither group con-

sidered the effect of covariate information, however. Neither

a nonparametric nor semiparametric approach nor the use of

covariates has been considered from a Bayesian point of v iew

for developing optimal replacement strategies. The B ayesian

decision theoretic framework for replacement policies for

machine tools presented h ere represents an ext ension of

the work of Mazzuchi and Soyer (1995, 1996) to include

semiparametric models and covariate effects.

While building on existing analytical techniques proposed

in the literature for semiparametri c models, the analysis herein

has four basic aims:

1. To develop a model describing the failure behavior of

machine tools accounting for the operational environment

effect as well as aging, thereby c apturing a wi de class of tool

failure behavior

2. To illustrate the use of this mo del as a diagnostic tool for

assessing model adequacy in describing the differences among

the machine tools

3. To illustrate the use of this model in predicting the useful

life of a machine tool op erating in a particular operational

environment

4. To illustrate the use of this model in developing optimal

replacement strategies for single machine tools and groups of

tools.

The PHM for machine tool life is discussed in Section 2,

along with the relaxation of parametric assumpt ions. In

Section 3 the semiparametric model is used to analyze the

machine tool failure data given by Taraman (1974). The

posterior distribution of the parameters of the model are

compared to a parametric model, and the predictive ability

of the two approaches is compared using both posterior

predictive densities and the decision information criterion

from Spiegelhalter, Best, Carlin, and van der Linde (2002).

The model is used to develop Bayesian semiparametric

replacement strategies in Section 4. Conclusions are presented

in Section 5.

2. A BAYESIAN SEMIPARAMETRIC

PROPORTIONAL HAZARDS MODEL

This section discusses the use of a PHM to analyze machine

tool failures under varying operating conditions. Unlike the

parametric analysis of Mazzuchi and Soyer (1989), we

develop a semiparametric inference using an MDP approach,

with inference furnished using the ef cient algorithms of

MacEachern (1994). This development is, therefore, a nontriv-

ial combination of previously published techniques that allows

a full analysis in this particular application. This section

outlines the necessary methodology for this combination of

techniques.

2.1 A Proportional Hazards Model

for Machine Tool Failure

The PHM was proposed by Cox (1972) to incorporate

covariate information in a survival or time to failure model.

The model has been widely applied in survival and reliabilit y

analysis. The PHM is de ned using the concept of the failure

rate. Let T

i

be the life length of ma chine tool i. Assuming that

T

i

is continuous, the failure rate function of the distribution

of T

i

is de ned as

‹

i

4t5 D

f

i

4t5

R

i

4t5

1

where f

i

4t5 is the probability density function of T

i

and

R

i

4t5 D P4T

i

¶ t5 D exp8ƒå

i

4t59

is the reliability of machine tool i at time t, with å

i

4t5 D

R

t

0

‹

i

4s5 ds the cumulative failure rate.

Let Z

i

be a vector of p measured covariates de scribing the

operational environments of machine tool i. The cova riates

available for t he machine tool analysis are constant with

respect to time and are known before any failure data are

observed. Cox (1972) propo sed that the distribution of T

i

could be made dependent on Z

i

via the failure rate by

assuming that the failure rate of the ith item is a product of

a common base failure rate function and a function of the

covariates, explicitly

‹

i

4t3 Z

i

5 D ‹

0

4t5e

‚

T

Z

i

1

where ‚ is a vector of p regression parameters and ‹

0

4t5 is

a baseline failure rate function. The parameters of the model

are suppressed in the notation ‹

i

4t3 Z

i

5.

Often a parametric form is assumed for the baseline

failure rate. This is equivalent to choosing a common family

of distributions for the life lengths of the machine tools.

Mazzuchi and Soyer (1989) performed an analysis of the

Weibull parametric model for the machine tool fai lure data.

Their analysis used integral approximation techniques to nd

the marg inal posterior distributions of the model parameters.

An alternative analysis of the parametric model was developed

by Dellaportas and Smith (1993) using Markov chain Monte

Carlo techniques, although an analysis of the machine tool

failure data using these methods has not yet been published.

Approaches for modeling the baseline cumulative fail-

ure rate function include the gamma process propo sed

by Kalb eisch (1978) an d criticized by Hjort (1990), the

extended gamma process p resented by Laud, Damien, and

Smith (1996), and the beta process presented by Hjort (1990),

with a computational model developed by Laud et al. (1998).

(For a full review of other semiparametric approaches to

inference on regression models, see Gelfand 1999.) In the

TECHNOMETRICS, FEBRUARY 2003, VOL. 45, NO. 1

60 JASON R. W. MERRICK, REFIK SOYER, AND THOMAS A. MAZZUCHI

next section we present an MDP prior, as de ned by Antoniak

(1974), for the baseline failure rate of the PHM to an alyze the

machine tool problem. This prior distribution allows a large

family of continuous failure time distributions, thus relaxing

the full parametric assumption. Our setup is similar to the

semiparametric accelerated failure time model proposed by

Kuo and Mallick (1997), but applied to the PHM.

2.2 A Mixture of Dirichlet Processes Prior

for the Proportional Hazards Model

Under the MDP approach, the baseline failure rate is

assumed to be some continuous function ‹

0

4t3 ˆ

i

5, where ˆ

i

is

the vector of unknown parameters speci c to the ith machine

tool. Uncertainty about the ˆ

i

’s is described by specifying

a prior distribu tion G. If the form of G is known but the

hyperparameters are unknown, then this class of problems is

referred to as hierarchical Bayes problems in the se nse of

Lindley and Smith (1972). If the form of G is unknown, then

uncertainty about G must be modeled. One way to model

this uncertainty is to follow the development of MacEachern

(1994) and West, Muller, and Escobar (1994) and describe

uncertainty about G by a Dirichlet process prior denoted by

G ¹ DP4G

0

1 M51

where G

0

is the baseline prior and M is the strength of belief

parameter. (See Ferguson 1973 for a discussion of Dirichlet

process priors.)

By specifying a form for ‹

0

4t

i

3 ˆ

i

5 conditional on ˆ

i

, we

specify a conditional parametric model for T

i

whose density

f 4t

i

—ˆ

i

1 ‚1 Z

i

5 is given by

‹

0

4t

i

3 ˆ

i

5e

‚

T

Z

i

exp8ƒå

0

4t

i

3 ˆ

i

5e

‚

T

Z

i

91 (1)

where å

0

4t

i

3 ˆ

i

5 D

R

t

i

0

‹

0

4s3 ˆ

i

5 ds. Speci cation of the semi-

parametric PHM is completed by specifying a parametric prior

for the covariate effects ‚, denoted by 4‚5, which is inde-

pendent of the ˆ

i

’s. The nonparametric nature of the model

arises because the distribution of T

i

, unconditional on ˆ

i

, is an

unknown mixture of f 4t

i

—ˆ

i

1 ‚1 Z

i

5 given by

f 4t

i

—G1 ‚1 Z

i

5 D

Z

f 4t

i

—ˆ

i

1 ‚1 Z

i

5 dG4ˆ

i

51

where the distribution of T

i

results from mixing with respect

to G. These models were termed Dirichlet process mixed

models by M ukhopadhyay and Gelfand (1997), because G is

assumed a priori to be a Dirichlet process.

The semiparametric PHM using a an MDP approach can be

summarized as

4T

i

—ˆ

i

1 ‚1 Z

i

5 ¹ f 4t

i

—ˆ

i

1 ‚1 Z

i

51

4ˆ

i

—G5 ¹ G1

4G5 ¹ DP4G

0

1 M51

4‚5 ¹ 4‚50 (2)

It is also assumed a priori that ‚ and ä D 4ˆ

1

1 : : : 1 ˆ

n

5 are

independent of each other.

In addition to its exibility and ability to capture individual

characteristics of the machine tools, the proposed semipara-

metric PHM also provides an assessment of the completeness

of the set of covariates included in the analysis. In the clas-

sical literature, a residual analysis is performed to determine

whether differences in the fail ure characteristics among the

machine tools remain after th e effect of the covariates has

been removed. In the proposed model, differences between

the individual m achine tools can be assessed by differences

between the distributions of the ˆ

i

’s.

2.3 Posterior Inference and Prediction

Given fail ure and covariate data D D 8T

1

D t

1

1 : : : 1 T

n

D

t

n

1 Z

1

1 : : : 1 Z

n

9 on n machine tools, the likelihood fu nction of

G and ‚ given the data D is obtained as

L4G1 ‚—D5 D

n

Y

i

D1

Z

f 4t

i

—ˆ

i

1 ‚1 Z

i

5 dG4ˆ

i

50

Given an arbitrary prior on ‚, say 4‚5, which is independent

of ä and G, following Kuo and Mallick (1997), the posterior

distribution of G given ‚ and D can be obtained as a mixture

of Dirichlet processes,

4G—‚1D5 ¹

Z

DP

³

MG

0

C

n

X

j

D1

„

ˆ

j

´

dç4ä—‚1 D51 (3)

where „

ˆ

j

denotes a point mass distribution concentrated at ˆ

j

and dç4ä—‚1 D5 is proportional to

n

Y

i

D1

f 4t

i

—ˆ

i

1 ‚1 Z

i

5

µ

MG

0

C

i

ƒ1

X

j

D1

„

ˆ

j

¶

dˆ

i

0

It is dif cult to sample from the distribution 4G—‚1 D5 given in

(3), because G is effect ively an in nite-d imensional parameter

(see, e.g., Kuo 1986).

Instead of attempting to perform inference on the mixing

distribution G directly, one can perform simple inference using

the Markov chain Monte Carlo methods in algorithm 1 of

Escobar and West (1995) to obtain a sample from the posterior

distribution of ä and ‚ given the data D. For our problem,

the attractive feature of this approach is that computation of

4ä1 ‚—D 5 based on the Gibbs sampler can be achieved with-

out sampling from the posterior distribution of 4G—‚1 D5, thus

reducing the problem to n dimensions.

Following the derivations of Escobar and West (1995), it

can be shown that the full conditional for each ˆ

i

is

4ˆ

i

—ˆ

1

1 : : : 1 ˆ

i

ƒ1

1 ˆ

i

C1

1 : : : 1 ˆ

n

1 ‚1 D5

¹ q

i1

0

G

b

4ˆ

i

—t

i

1 ‚1 Z

i

5 C

X

j

6D

i

q

i1 j

„

ˆ

j

4ˆ

i

51

where „

ˆ

j

4ˆ

i

5 equals 1 if ˆ

i

D ˆ

j

and 0 otherwise. The term

G

b

4ˆ

i

—t

i

1 ‚1 Z

i

5 is the baseline posterior distribu tion

dG

b

4ˆ

i

—t

i

1 ‚1 Z

i

5 / f 4t

i

—ˆ

i

1 ‚1 Z

i

5 dG

0

4ˆ

i

50

The terms q

i1 j

, for j 6D i represent the positive probability that

some of the ˆ

i

’s will take the same values due to the discrete-

ness of G (as a result of the Dirichlet process prior). These

are given by

q

i1

0

/ M

Z

f 4t

i

—ˆ

i

1 ‚1 Z

i

5 dG

0

4ˆ

i

5

and

q

i1 j

/ f 4t

i

—ˆ

j

1 ‚1 Z

i

51

TECHNOMETRICS, FEBRUARY 2003, VOL. 45, NO. 1

BAYESIAN SEMIPARAMETRIC ANALYSIS OF MACHINE TOOL RELIABILITY AND MAINTENANCE 61

where f 4t

i

—ˆ

j

1 ‚1 Z

i

5 is the density of T

i

when ˆ

i

D ˆ

j

and

q

i1

0

C

X

j

6D

i

q

i1 j

D 10

An algorithm proposed by MacEachern (1994) exploits this

fact to increase the ef ciency of the samp ling processs by

updating the ˆ

i

’s in groups or cluste rs. Using this algorithm,

we may draw a sample from the c onditional distribution of

ä given ‚ and the data. The other distribution needed to

implement the Gibbs sampler for this model is the condi-

tional distribution of ‚ given ä and the data. Samples from

this distribution can be obtained using the methods discussed

by Dellaportas and Smith (1993) for the parametric model,

because given ä, a conditionally parametric model is speci ed

by (1).

In implementing t he Gibbs sampler, given a current ä and

‚, the general steps are as follows:

1. Generate a new ä conditional on ‚ using the ef cient

MDP algorithm of MacEachern (1994).

2. Generate a new ‚ conditional on ä using the methods

discussed by Dellaportas and Smith (1993).

An attractive feature of the MDP setup and the proposed

algorithm is that posterior predictive densities and reliability

functions can be easily evaluated once the posterior sample

from ä and ‚ is available. For example, in predicting T

n

C1

,

the posterior predictive density f 4t

n

C1

—D1Z

n

C1

5 is

Z

‚1 ä

f 4t

n

C1

—ä1 ‚1 Z

n

C1

5 4ä1 ‚—D 5 dä d‚1

where f 4t

n

C1

—ä1 ‚1 Z

n

C1

5 is

Z

ˆ

n

C1

f 4t

n

C1

—ˆ

n

C1

1 ‚1 Z

n

C1

5 4ˆ

n

C1

—ä5 dˆ

n

C1

and 4ˆ

n

C1

—ä5 is

M

M C n

G

0

4ˆ

n

C1

5 C

1

M C n

n

X

i

D1

„

ˆ

i

4ˆ

n

C1

50 (4)

Thus the posterior predictive density f 4t

n

C1

—D1 Z

n

C1

5 can be

written as

Z

‚1 ä1 ˆ

n

C1

f 4t

n

C1

—ˆ

n

C1

1 ‚1 Z

n

C1

5 4ˆ

n

C1

—ä5

4ä1 ‚—D5 dˆ

n

C1

dä d‚1 (5)

and using the posterior sample from 4ä1 ‚—D5, denoted

by 4ˆ

1

1 l

1 : : : ˆ

n1 l

1 ‚

l

5 for l D 11 : : : 1 S, and draws from (4),

denoted by ˆ

n

C1

1 l

, f 4t

n

C1

—D1 Z

n

C1

5 can be approximated as

1

S

S

X

l

D1

f 4t

n

C1

—ˆ

n

C1

1 l

1 ‚

l

1 Z

n

C1

50 (6)

Because the results u nder the MDP setup may be sensitive

to the choice of M , we use a further extension by incorporat-

ing M into the Gibbs sampler. Our approach follows that of

Escobar and West (1995), assuming a priori that M follows

an arbitrary prior 4M5. In their development, Escobar and

West de ned K as the number of unique values of ˆ

1

1 : : : ˆ

n

,

also referred t o as the number of cliques by MacEachern

(1998). Conditioned on K, th e full conditional distribution of

M is independent of all other parameters with density propor-

tional to

M

K

ƒ1

4M C n5B4M C 11 n5 4M 51

where B4¢ 5 is the standard beta function. Escobar and West

(1995) offered a simple two-step process for sampling from

this distribution if 4M 5 is assumed to be a gamma distribu-

tion. The approach involves using a data-augmentation step at

each iteration of the Gibbs sampler. Thus K is a lso recorded

in the Gibbs sample, because the distribution of the number

of cliques is of interest in the reliability analysis in Section 3.

We note that at each iteration of t he Gibbs sampler, once M

is drawn, the rest of the quantities are sampled as discussed

in steps 1 and 2.

3. ANALYSIS OF MACHINE TOOL

FAILURE DATA USING PARAMETRIC

AND SEMIPARAMETRIC INFERENCE

The data used in this analysis, given in Table 1, were rst

presented by Taraman (1974). They consist of the failure times

of 24 machine tools and their corresponding cutting speed,

feed rate, and depth of cut. Each experimental run used a

workpiece material of SAE 1018 cold-rolled steel, 4 inches

in diameter and 2 feet long. The 24 machine tools used for

the cutting were tungsten carbide disposable inserts mounted

in a tool h older. A 7.5-horsepower engine lathe equipped with

a three-jaw universal chuck and a live center mounted in the

tailstock was used to perform the cutting operation. The cut-

ting operations were performed without using cutting ui ds.

In this section we use both parametric and semiparametric

inference methods to assess the effect of the covariates on the

useful life of these machine tools. The parametric analysis mir-

rors the development of Mazzuchi and Soyer (1989) by speci-

fying ‹

i

4t3 z

i

5 D ‹

0

4t5 exp8‚

1

ln Z

i1

1

C ‚

2

ln Z

i1

2

C ‚

3

ln Z

i1

3

9,

Table 1. The Machine Tool Failure Data

Machine Speed Depth of cut Tool life

tool (fpm) Feed (ipr) (inches) (min.)

1 340 .00630 .02100 70.0

2 570 .00630 .02100 29.0

3 340 .01410 .02100 60.0

4 570 .01416 .02100 28.0

5 340 .00630 .02100 64.0

6 570 .00630 .04000 32.0

7 340 .01416 .04000 44.0

8 570 .01416 .04000 24.0

9 440 .00905 .02900 35.0

10 440 .00905 .02900 31.0

11 440 .00905 .02900 38.0

12 440 .00905 .02900 35.0

13 305 .00905 .02900 52.0

14 635 .00905 .02900 23.0

15 440 .00472 .02900 40.0

16 440 .01732 .02900 28.0

17 440 .00905 .01350 46.0

18 440 .00905 .04550 33.0

19 305 .00905 .02900 46.0

20 635 .00905 .02900 27.0

21 440 .00472 .02900 37.0

22 440 .01732 .02900 34.0

23 440 .00905 .01350 41.0

24 440 .00905 .04550 28.0

TECHNOMETRICS, FEBRUARY 2003, VOL. 45, NO. 1

62 JASON R. W. MERRICK, REFIK SOYER, AND THOMAS A. MAZZUCHI

where Z

i1

1

is cutting speed, Z

i1

2

is feed rate, Z

i1

3

is depth of

cut, and ‹

0

4t5 D ƒt

ƒ

ƒ1

. For computational ef cien cy, we use

the techniqu es developed by Dellaportas and Smith (1993)

for inference in the parametric model. The semiparametric

inference follows the methods developed in Section 2. The two

approaches are compared in several ways, using the posteri or

distributions of the model parameters, posterior distributions

of the predicted reliabilities, and posterior predictive densit ies,

as discussed by Gelfand (1996), and the deviance information

criteria (DIC), as described by Spiegelhalter et al. (2002).

3.1 Comparison of the Posterior Distributions

of the Model Parameters

For comparison of the semiparametric i nference method

to the parametric method proposed by Mazzuchi and

Soyer (1989), the conditional baseline failure rate in (2)

is assumed to be a Weibull density with scale parameter

i

and shape parameter ƒ. Thus, under the notation of

Section 2, ˆ

i

D 4

i

1 ƒ5, with n D 24. Note that each item

is assumed to wear at the same rate; thus ƒ is common

to each item, whereas the scale parameter

i

is allowed

to vary from one machine tool to the next. The condi-

tional parametric density in (2) is then f 4t

i

—

i

1 ƒ1 ‚1 Z

ü

i

5 D

i

ƒt

ƒ

ƒ1

i

exp8‚

T

Z

ü

i

9 exp8ƒ

i

t

ƒ

i

exp8‚

T

Z

ü

i

99, where Z

ü

i

D

4ln Z

i1

1

1 ln Z

i1

2

1 ln Z

i1

3

5

T

. For the prior best guess of the

mixing distribution of the

i

’s, G

0

in (2), a gamma distribution

is chosen.

The pri or assumptions of Mazzuchi and Soyer (1989) had

low varia nces and seemingly speci c values fo r the means of

each paramete r. However, no motivation was given for these

prior assumptions. Thus our prior distributions are noninfor-

mative with large varia nces assumed on each pa rameter. A pri-

ori, ƒ, ‚

1

, ‚

2

, and ‚

3

are independent of each other and

1

1 : : : 1

n

. A normal prior, with mean 0 and variance 20,

was assumed for each of the covariate effect p arameters, ‚

1

,

‚

2

, and ‚

3

, where the covariate values had been scaled so that

they were of the same order of magnitude. This re ects our

lack of knowledge of whether the c ovariates would increase

or decrease failure time. The prior distribution of the shape

parameter, ƒ, was assumed to be a normal distribution trun-

cated at 0 with mean 1 and variance 10 re ec ting no strong

prior belief concerning the failure rate behavior (whether it is

increasing or decreasing). The best-guess prior distribution G

0

for the scale parameters was assumed to be a gamma distri-

bution with mean 1 and variance 10. A priori, M is assumed

to follow an un informative gamma distribution with mean 24

and standard deviation 100.

A single-chain Gibbs sampler was run to obtain 2,500 sam-

ples with a warmup of 5,000 and a lag of 25 between succes-

sive samples. Boxplots of th e marginal posterior distributions

of the natural log of the scale parameters for the tools in the

data obtained under the semiparametric model are shown in

Figure 1. A second Gibbs sampler was run for the parametric

model using the methods of Dellaportas and Smith (1993).

The same pri or distributions were assumed, except that the

prior distribution of the scale parameter was assumed to be the

best-guess prior, G

0

, in the semiparametric model. The box-

plot of the distribution obtained under the parametric model

is also shown in Figure 1, denoted by a P on the x-axis.

Figure 1. Boxplots of the Marginal Posterior Distributions of the Log

Scale Parameters of the Machine Tools Under the Parametric and Semi-

parametric Models.

Figure 1 shows marked differences between the posterior

distributions of the scale parameters. The posterior distribu-

tion of the scale parameter under the parametric model has a

smaller variance and obviously cannot represent the variability

among the tools demonstrated by the individual scale param-

eters under the M DP setup. In the semiparametric model, the

individual scale parameters express the differences between

the machine tools that are not ex plained by the covariates.

We can examine such differences among the machine tools

through the marginal posterior distribution of K, the num-

ber of groups of

i

’s in the MDP setup, shown in Figure 2.

Figure 2 indicates t hat there is little support for 1 group, as i n

Figure 2. Marginal Posterior Distribution of the Number of Groups

of

i

’s in the Semiparametric Model.

TECHNOMETRICS, FEBRUARY 2003, VOL. 45, NO. 1