QUT Digital Repository: http://eprints.qut.edu.au/

Gorjian, Nima and Ma, Lin and Mittinty, Murthy and Yarlagadda, Prasad and Sun, Yong (2009) A review on reliability models with covariates. In: Proceedings of the 4th World Congress on Engineering Asset Management, 28-30 September 2009, Marriott Athens Ledra Hotel, Athens.

© Copyright 2009 Springer

Proceedings of the 4th

World Congress on Engineering Asset Management

Athens, Greece

28 - 30 September 2009

A REVIEW ON RELIABILITY MODELS WITH COVARIATES

Nima Gorjian a, b

, Lin Ma a, b

, Murthy Mittinty c, Prasad Yarlagadda

b, Yong Sun

a, b

a Cooperative Research Centre for Integrated Engineering Asset Management (CIEAM), Brisbane, Australia

b School of Engineering Systems, Queensland University of Technology (QUT), Brisbane, Australia

c School of Mathematical Sciences, Queensland University of Technology (QUT), Brisbane, Australia

Modern Engineering Asset Management (EAM) requires the accurate assessment of current and the prediction of

future asset health condition. Suitable mathematical models that are capable of predicting Time-to-Failure (TTF)

and the probability of failure in future time are essential. In traditional reliability models, the lifetime of assets is

estimated using failure time data. However, in most real-life situations and industry applications, the lifetime of

assets is influenced by different risk factors, which are called covariates. The fundamental notion in reliability

theory is the failure time of a system and its covariates. These covariates change stochastically and may

influence and/or indicate the failure time. Research shows that many statistical models have been developed to

estimate the hazard of assets or individuals with covariates. An extensive amount of literature on hazard models

with covariates (also termed covariate models), including theory and practical applications, has emerged. This

paper is a state-of-the-art review of the existing literature on these covariate models in both the reliability and

biomedical fields. One of the major purposes of this expository paper is to synthesise these models from both

industrial reliability and biomedical fields and then contextually group them into non-parametric and semi-

parametric models. Comments on their merits and limitations are also presented. Another main purpose of this

paper is to comprehensively review and summarise the current research on the development of the covariate

models so as to facilitate the application of more covariate modelling techniques into prognostics and asset

health management.

Key Words: Covariate model, Hazard, Reliability analysis, Survival analysis, Asset health, Life prediction

1 INTRODUCTION

In recent years, the emphasis on prognostics and asset life prediction has increased in the area of Engineering Asset

Management (EAM) due to longer-term planning and budgeting requirements. One essential scientific research problem in

EAM is the development of mathematical models that are capable of predicting Time-To-Failure (TTF) and the probability of

failures in future time. In most real-life situations and industry applications, the hazard (failure rate) of assets is influenced

and/or indicated by different risk factors, which are often termed as covariates. Probabilistic modelling of assets lifetime using

covariates (i.e. diagnostic factors and operating environment factors) is one of the indispensible scientific research problems

for prognostics and asset life prediction.

Until now, a number of statistical models have been developed to estimate the hazard of an asset/individual with covariates

in both the reliability and biomedical fields. Most of these models are developed based on the Proportional Hazard Model

(PHM) theory which was proposed by Cox in 1972 [1]. The basic theory of these covariate models is to build the baseline

hazard function using historical failure data and the covariate function using covariate data. There are few review papers on the

covariate models which have been reported in the literature. Kumar and Klefsjo [2] reviews the existing literature on the

proportional hazard model. Kumar and Westberg [3] provides a review of some reliability models for analysing the effect of

operating conditions on equipment lifetime. Ma [4] discusses new research directions for Condition Monitoring (CM) and

reviews some prognostic models in EAM.

Almost all existing covariate models have been applied in the biomedical field. However, some of them have been applied

in the reliability area. This expository paper is a collective review of the existing literature on covariate models in both the

reliability and biomedical fields. In this paper, each individual covariate model has been contextually grouped into non-

parametric and semi-parametric models. Moreover, comments on their merits and limitations are discussed. Applications of

each model in both biomedical and reliability field are also presented. The purpose of this study is to facilitate the application

of more covariate modelling techniques into prognostics and asset life prediction.

The reminder of this paper is organised as follows. Section 2 classifies these covariate models into two groups and then

explains them in greater detail. In this section, the merits, limitations, and applications of each model are discussed. Section 3

provides the conclusions of this paper.

2 SURVIVAL / RELIABILITY MODELS WITH COVARIATES

Survival / reliability analysis (also called failure time analysis) is a specific field of statistics that studies failure time and its

probability on a group or groups of assets/individuals. Failure time is a defined point event, often called failures, occurring

after a length of time. Some examples of failure times include the lifetimes of machine components in reliability field, the

survival times of patients in a clinical trial, and durations of economic recessions in economics. Survival analysis was

advanced at the UC Berkeley in 1960s to present a better analysis method for Life Table data [5]. The development of

statistical procedures and models for survival analysis exploded in the 1970s. In the 1980s and early 1990s, survival models

with covariates have been widely applied in both the reliability and biomedical research. In general, the survival / reliability

models with covariates can be classified into two groups as: non-parametric and semi-parametric models, which are discussed

in the following sub-sections.

2.1 Non-Parametric Models

In non-parametric models, the form of degradation paths or distribution of degradation measure is unspecified [6]. When

the failure time data involve complex distributions that are largely unknown, or when the number of observations is small, it is

difficult to accurately fit a known failure time distribution. In other words, non-parametric models are used to avoid making

unrealistic assumptions that would be difficult to test [7]. Such models to be reviewed include:

Proportional hazard model

Stratified proportional hazard model

Two-step regression model

Additive hazard model

Mixed (additive-multiplicative) model

Accelerated failure time model

Extended hazard regression model

Proportional intensity model

Proportional odds model

Proportional covariate model

2.1.1 Proportional Hazard Model

The Proportional Hazard Model (PHM), which is a multivariate regression analysis, was first proposed by Cox [1] in 1972.

This model estimates the effects of different covariates influencing TTF of a system. This model has been employed for

different applications in lifetime data analysis. Due to its generality and flexibility, PHM was quickly and widely adopted in

the biomedical, reliability, and economics from 1970s to early 1990s. Almost all covariate models are based on PHM theory.

Cox’s PHM for static explanatory variables is expressed as [1, 8]:

𝑕 𝑡; 𝒛 = 𝑕0 𝑡 𝜓(𝜸𝒛) (1)

Where, 𝑕0(𝑡) is the unspecified baseline hazard function which is dependent on time only and without influence of

covariates. The positive functional term,𝜓(𝜸𝒛), is dependent on the effects of different factors, which have multiplicative

effect (rather than additive) on the baseline hazard function.

The proportionality assumption in PHM is that 𝑕(𝑡; 𝑧𝑋) 𝑕 𝑡; 𝑧𝑌 = 𝑕0 𝑡 exp(𝑧𝑋𝛾) 𝑕0 𝑡 exp(𝑧𝑌𝛾) = exp 𝛾(𝑧𝑋 − 𝑧𝑌) .

The hazard at different 𝑧 values are in constant proportion for all 𝑡 > 0, hence the name for PHM [9-11].

Cox’s PHM for dynamic explanatory variables is [8, 12]:

𝑕 𝑡; 𝒛(𝑡) = 𝑕0 𝑡 𝜓(𝜸𝒛(𝑡)) (2)

Three parameterizations of 𝜓 may be considered as log linear, linear, and logistic forms [8]. The log linear form has

become the most popular for good reasons. Covariates are represented by a row vector consisting of the covariates 𝒛 and 𝜸 is a

column vector consisting of the regression coefficients. The covariate 𝑧 is associated with the system, and 𝛾 is the unknown

parameter of the model, defining the effects of the covariates. Cox [1, 13] propose the conditional likelihood, which later is so

called partial likelihood, to estimate regression coefficients (regression coefficients) (𝛾). Kalbfleisch and Prentice [12, 14]

propose the marginal likelihood which is identical to partial likelihood. Their likelihood can deal with tied data

(assets/individuals which failed simultaneously), censored data and uncensored (observed) data. Regression coefficients are

estimated by maximising the partial likelihood.

A number of graphical techniques, goodness-of-fit techniques, and confidence intervals techniques can be employed to

examine the appropriateness and fit of the PHM [15-17]. Since 1972, a number of applications of PHM in the reliability [10,

11, 15, 18-43] and biomedical fields have been developed.

Key merits

PHM is an influential technique which can be used to investigate the effects of various explanatory variables on hazard

of assets/individuals This approach is essentially distribution free, thus it does not have to assume a specific form for the baseline hazard

function Regression coefficients are estimated using partial likelihood without the need of specifying the baseline hazard

function This model is available for both static and dynamic explanatory variables

Explanatory variables have a multiplicative effect (rather than additive effect) on the baseline hazard function, thus it is

a more realistic and reasonable assumption

This model handles truncated, non-truncated data, and tied values

Many goodness-of-fit tests and graphical methods are available for this model [17]

Key limitations

PHM is a vulnerable approach when covariates are deleted or the precision of covariate measurements is changed.

Therefore, if one pertinent covariate is omitted, even if it is independent of the other covariates in the model, averaging

on the omitted covariate gives a new model which leads to biased estimates of regression coefficients (𝛾) [10]

The estimated value of the regression coefficient is biased in the case of a small sample size

Mixing different types of covariates in one model may cause some problems

The main assumption of this model is that an asset/individual life is assumed to be terminated at the first failure time.

In other words, this model depends only on the time 𝑡 elapsed between the starting event (e.g. diagnosis) and the

terminal event (e.g. fail) and not on the chronological time 𝑡

The influence of a covariate in PHM is assumed to be time-independent [44]

Due to proportionality assumption, a common baseline hazard for all assets/individuals has been assumed in a case in

which the assets/individuals should be stratified according to baseline [12, 17]

Proportionality assumption imposes a severe limitation which that survival curves for assets/individuals with different

covariates can never cross [45-49]

2.1.2 Stratified Proportional Hazard Model

In the Stratified Proportional Hazard Model (SPHM), it is assumed that the population can be divided into 𝑞 strata (or

levels), based on the discrete values of a single covariate or combination of discrete values of a set of covariates [3]. For

example, an asset operates at three different temperature levels; say low, medium, and high. In SPHM, it is assumed that the

hazard is proportional within the same stratum (or level) but not necessary across different strata. The hazard of a system in the

𝑗𝑡𝑕 stratum can accordingly be expressed as [12]:

𝑕𝑗 𝑡; 𝒛 = 𝑕0𝑗 𝑡 exp(𝜸𝒛) (3)

A similar likelihood method to Cox’s PHM is used to estimate regression coefficients (𝜸). The model is applied in

biomedical by Kay [50]. This model applied by Kumar [31] in the reliability field.

Key merits

The arbitrary baseline hazard function is allowed to be different for each stratum whereas the regression coefficients

are the same across strata

SPHM is one of the simplest and most useful extensions of the PHM for its application in different situations

Similar to PHM, this approach is essentially distribution free, thus it does not have to assume a specific form for the

baseline hazard function Regression coefficients are estimated using partial likelihood without the need of specifying the baseline hazard

function Explanatory variables have a multiplicative effect (rather than additive effect) on the baseline hazard function, thus it is

a more realistic and reasonable assumption

Similar to PHM, many goodness-of-fit tests and graphical methods are available for PSHM

Key limitations

Due to multicollinearity, estimated values of regression coefficients (𝛾) are sensitive to omission, misclassification and

time dependence of explanatory variables [10]

The estimated value of the regression coefficient is biased in the case of a small sample size

The main assumption of this model is that an asset/individual life is assumed to be terminated at the first failure time.

In other words, this model depends only on the time 𝑡 elapsed between the starting event (e.g. diagnosis) and the

terminal event (e.g. fail) and not on the chronological time 𝑡

2.1.3 Two-Step Regression Model

Anderson and Senthilselvan [51] extends Cox’s PHM to allow (approximately) for changing covariate effects in time and

presents a method of estimation. This model is applied and tested in the biomedical field. This extension is called two-step

regression model. Anderson and Senthilselvan [51] also introduces a method for smooth estimates of the hazard. In general, the

prediction effect of a covariate measured at a particular point of time (the beginning of a study) becomes progressively less

important as time goes by. If there is a good deal of information about a particular process, it may be possible to model this

directly and express 𝛽𝑗 as a specific function of time, such as 𝛽𝑗 𝑡 = 𝛽𝑗∗𝑒−𝛾𝑗 𝑡 , 𝑗 = 1, … , 𝑝. Two-step regression model allows

a very simple form of time-dependent for 𝛽𝑗 . It assumes that [51]:

𝛽𝑗 𝑡 = 𝜶𝒋 for 𝑡 ≤ 𝐵 and 𝛽𝑗 𝑡 = 𝜸𝒋 for 𝑡 > 𝐵 (4)

For some 𝐵 , which is not assumed known a priori. In most situations this step-function must be regarded as a first

approximation to the true form of 𝛽, as a function of time. The regression coefficients 𝜶𝑇 = (𝛼1, … , 𝛼𝑝) and 𝜸𝑇 = (𝛾1, … , 𝛾𝑝)

refer to short-term and long-term dependence of the hazard of 𝑧 [51]. It is noticeable that approximation implies roughly equal

rates of change for the 𝛽𝑗 (𝑡). The hazard of two-step regression model is [51, 52].

𝑕 𝑡; 𝒛 = 𝑕0(𝑡) exp 𝜶𝑇𝒛 , 𝑡 ≤ 𝐵

𝑕1(𝑡) exp 𝜸𝑇𝒛 , 𝑡 > 𝐵

(5)

Anderson and Senthilselvan [51] proposes the conditional log-likelihood of Peto in Cox’s discussion paper [1] to estimate

regression coefficients. By maximising this log-likelihood regression coefficients will be estimated.

Key merits

Literature shows this model fits the data much better than Cox’s PHM [51, 52]

This model is more appropriate to deal with time-dependent covariates rather than Cox’s PHM

This model handles heavy censoring circumstance better than Cox’s PHM [51, 52]

Approximation of the model replaces a time-dependent coefficient by the step function, which is representing short-

term and long-term effects

This model can be extended to three-step regression function

Key limitations

This model has difficulty for estimating regression parameters due to large values of breakpoint 𝐵 [51]

This model has a common breakpoint for all covariates [51, 52]

2.1.4 Additive Hazard Model

The Additive Hazard Model (AHM) can be described as [53]:

𝑕 𝑡; 𝒛 = 𝑕0 𝑡 + 𝜓(𝒛𝜸) (6)

Where, 𝑕0(𝑡) is the unspecified baseline hazard function which is dependent on time only and without influence of

covariates. The positive or negative functional term,𝜓(𝜸𝒛), is dependent on the effects of different factors, which have additive

effect (rather than multiplicative) on the baseline hazard function. This model provides the means for modelling a circumstance

when the hazard is not zero at time zero. Maximum likelihood procedures can be used to estimate this model’s parameters

[54].

Pijnenburg [53] tests this model in modelling the reliability of the air conditioning system of aircrafts. Newby [55] asserts

that AHM applications are restricted by an identifiability problem. Theoretical limitations leads to identification problems

while estimating parameters of the model [56]. Due to that this model is not identifiable, the observation of explanatory

variables does not add anything to the knowledge obtained from the event data [55].

Key merits

AHM is intuitively an attractive model when a system after repair is better than it was just before the repair, but not as

good as new

This model offers the hazard that is not zero at time zero

Key limitations

An additive assumption often leads to estimated hazard less than zero, as a result it is not a realistic and reasonable

assumption

This model cannot handle tied values (which have likelihood zero under the continuous random variable assumption)

The model cannot handle failure times equal to zero

This model can only be used in a phenomenological way to measure the magnitude of the jump 𝜓 in the hazard, and

models for the 𝜓𝑗 which makes use of explanatory variables are unlikely to produce satisfactory estimators of the

parameters [56]

2.1.5 Mixed (Additive-Multiplicative) Model

To enhance modelling capability about covariates, the mixed model considers the hazard of an asset/ individual, which

contains both a multiplicative and an additive component [57, 58]. The additive-multiplicative hazard model specifies that the

hazard for the counting process associates with a multidimensional covariate process 𝑍 = (𝑊𝑇 , 𝑋𝑇)𝑇 . Therefore, a general

additive-multiplicative hazard model takes the below form [59]:

𝑕(𝑡 𝑍) = 𝑔 𝛽0𝑇𝑊(𝑡) + 𝑕0(𝑡) 𝑓 𝛾0

𝑇𝑋(𝑡) (7)

Where, 𝜃0 = (𝛽0𝑇 , 𝛾0

𝑇)𝑇 is a vector of unknown regression coefficients, 𝑔 and 𝑓 are known link functions and 𝑕0 is an

unspecified baseline hazard function under 𝑔 = 0 and = 1 . Lin and Ying [59] develops a class of simple estimating functions

for 𝜃0, which contains the partial likelihood score function in the special case of proportional hazard model. The mixed model

is applied in the biomedical field [57].

Key merits

This model sometimes gives a better fit to the data rather than PHM [59]

This model allows covariates to have both the additive and multiplicative effects

Key limitation

Extremely limited testing

2.1.6 Accelerated Failure Time Model

The Accelerated Failure Time Model (AFTM) is one of the most common approaches used in obtaining reliability and

failure rate estimates of devices and components in a much shorter time [47, 60-62]. AFTM is to assume that the log

lifetime 𝑌 = log(𝑡), given applied stress vector 𝒛, has a distribution with a location parameter 𝜇(𝒛) and a constant scale

parameter 𝜎. AFTM can be expressed as [47, 62]:

𝑌 = log 𝑡 = 𝜇(𝒛) + 𝜎𝜖 (8)

Where, 𝜎 > 0 and 𝜖 is a random variable whose distribution does not depend on 𝒛. The hazard of AFTM can be written as

[8, 12]:

𝑕 𝑡; 𝒛 = 𝑕0 𝑡 ∙ 𝜓(𝜸𝒛) ∙ 𝜓(𝜸𝒛) (9)

Where, 𝑕0(𝑡) denotes the baseline hazard function, and 𝜸 is a vector of regression coefficients. The effects of a covariate is

to accelerate or decelerate the failure time relative to the baseline hazard function according to 𝜓 𝜸𝒛 > 1 or 𝜓 𝜸𝒛 < 1 [60].

Maximum likelihood estimation can be used to estimate the parameters of the AFTM.

Key merit

AFTM is used to obtain reliability and failure rate estimates of assets and components in a much shorter time

Key limitation

AFTM is a time consuming process as well as is costly to set up this test

2.1.7 Extended Hazard Regression Model

The Extended Hazard Regression Model (EHRM) that includes PHM and AFTM is developed at 1985 [45, 46]. This model

assumes that a covariate vector changes the baseline hazard function, 𝑕0(𝑡), according to [45, 46]:

𝑕 𝑡; 𝒛 = 𝑔1 𝜶𝒛 𝑕0(𝑔2 𝜷𝒛 𝑡) (10)

Where, 𝑔1(𝑥) and 𝑔2(𝑥) are positive functions equal to 1 at zero. 𝑕0(𝑡) denotes the baseline hazard function, and 𝜶 and 𝜷

are vectors of regression coefficients. For simplicity, it is assumed that 𝑔1 𝑥 = 𝑔2 𝑥 = exp(𝑥). The general case describes a

situation in which 𝒛 affects survival by changing both the time scale by the factor 𝑔2 𝜷𝒛 , and the scale in which the hazard is

measured by the factor 𝑔1 𝜶𝒛 𝑔2 𝜷𝒛 . This model reduces to the PHM for 𝜷 = 0 and to AFTM for 𝜶 = 𝜷. This model

describes a situation in which 𝒛 influences survival by changing both the time scale by the factor 𝑔2(𝜷𝒛), and the scale in

which the hazard is measured by the factor 𝑔1(𝜶𝒛) 𝑔2(𝜷𝒛) .Maximum likelihood function based on polynomial spline

approximation is developed to estimate all of the parameters of the model. This model is applied in both the biomedical [45,

46] and reliability fields [47].

Key merits

The appropriate applications of this model are AFTM dependent on the failure data analysis and types of failure

mechanisms

EHRM is a general model for hazard which includes both PHM and AFTM

Key limitations

Maximum likelihood estimation has a restricted assumption due to choosing a quadratic splines in order to maintain

the number of parameters small

Maximum likelihood estimation for this model is based on approximation of 𝑕0(∙) by splines; however, a spline

function cannot guarantee that any point in 𝑕0 ∙ is always positive

2.1.8 Proportional Intensity Model

The Proportional Intensity Model (PIM) was first introduced by Cox [1]. PIM is similar to PHM, but with the underlying

failure mechanism following a stochastic point process rather than a probabilistic distribution [63]. PIM is used to model the

intensity process of failures and repairs of a repairable system which incorporates explanatory variables [54, 64]. Volk et al.

[65] introduces PIM for both non-repairable and repairable systems utilising historic failure data and corresponding diagnostic

measurements. PIM assumes that a system enters stratum 1 at 𝑡 = 0 and that it enters stratum 𝑖 immediately following the

𝑗 − 1 𝑡𝑕 failure, 𝑗 = 1,2, … , 𝑚 [3]. The classes are based on two time scales, namely the global time 𝑡 and the time from the

immediately preceding failure, 𝑡 − 𝑡𝑛(𝑡), respectively [66, 67]:

𝑕 𝑡 𝑁 𝑡 , 𝒛(𝑡) = 𝑕0𝑗 𝑡 exp 𝒛(𝑡)𝜸𝑗

𝑕 𝑡 𝑁 𝑡 , 𝒛(𝑡) = 𝑕0𝑗 𝑡 − 𝑡𝑛(𝑡) exp 𝒛(𝑡)𝜸𝑗

(11)

Where, 𝑁(𝑡) represents a random variable for the number of failure in (0, 𝑡], and 𝒛(𝑡) denotes the covariate process up to

time 𝑡. 𝑕 𝑡 𝑁 𝑡 , 𝒛(𝑡) and 𝑕0𝑗 𝑡 are the intensity function and the baseline intensity function, respectively, and 𝜸𝑗 is the

regression coefficient for the 𝑗𝑡𝑕 stratum. The baseline intensity function can have three different forms: constant intensity,

log-linear intensity, and power-law intensity [68]. Maximum likelihood estimation is used to estimate regression coefficients.

This model is applied in the reliability field [31, 34, 64, 66].

Key merits

PHM assumes a system is renewed at failure, while PIM does not necessary make this assumption

This model is suitable for optimising maintenance and repair policy in a cost-effective manner

This model is simple, and directly related to the two most common models (PHM and AFTM) already in use, and

effective in discriminating between them by means of 𝑥2 statistics based on a few degrees of freedom only

Key limitation

If covariates are deleted from the model or measured with different level of precision, the proportionality is in general

destroyed

2.1.9 Proportional Odds Model

In 1980, McCullagh [69] generalises the idea of constant odds ratio to more than two samples by means of a regression

model which is termed as the Proportional Odds Model (POM). The theoretical basis for the model assumes that prognostics

factors have a multiplicative effect on the odds against survival beyond any given time [70]. POM can be expressed as [12, 69,

71]:

𝐹(𝑡; 𝒛)

1 − 𝐹(𝑡; 𝒛)=

𝐹0 𝑡

1 − 𝐹0 𝑡 𝜓(𝒛𝜸)

(12)

Where, 𝐹(𝑡; 𝒛) is the cumulative distribution function of the occurrence of events in group 𝑖 and 𝐹0(𝑡) is an underlying

unknown cumulative distribution function. The term 𝐹𝑖(𝑡) 1 − 𝐹𝑖(𝑡) is called proportional odds ratio. The full maximum

likelihood function of this model is derived by Bennett and McCullagh [69, 71]. POM is developed [69] and applied [70] in the

biomedical field.

Key merits

Hazard for separate groups of asset/individual converges with time

Due to hazard converges with time, it is useful model when the effects of covariates diminish or disappear with time

increases

Key limitation

It is necessary to use some time transformation of the failure times to estimate the parameters of the model

2.1.10 Proportional Covariate Model

The Proportional Covariate Model (PCM) is developed by Sun et al. [72] in 2006. PCM assumes that covariate of a system,

or a function of those covariates, are proportional to the hazard of the system. Sun et al. [72] and Sun and Ma [73] claim that

PCM is developed due to some shortcomings of PHM. The generic form of PCM is expressed as [72]:

𝑍𝑟 𝑡 = 𝐶 𝑡 𝑕(𝑡) (13)

Where, 𝑍𝑟(𝑡) is the covariate function which is usually time dependent. The variable 𝐶(𝑡) is the baseline covariate function

which is also usually time dependent. The function 𝑕(𝑡) is the hazard of a system. Maximum likelihood estimation is used to

estimate parameters. Due to the novelty of PCM, this model is only tested by laboratory data in the reliability field.

Key merits

Baseline covariate function 𝐶(𝑡) can take into account both historical failure data and historical condition monitoring

data

Baseline covariate function can be updated according to newly observed failure data and covariates

PCM is used to update the hazard of a system

Key limitations

This model does not consider operating environment data. Thus, this model is not sensitive to severe environments (e.g.

high ambient temperature)

Unlike the authors’ claims as an advantage of the model, PCM requires historical failure data to establish the covariate

baseline

PCM has not been applied widely in literature as it is a relatively new model

2.2 Semi-Parametric Models

In parametric models, the form of degradation paths or distribution of degradation measure is specified and/or partially

specified [6, 74]. These models can be classified as:

Weibull proportional hazard model

Logistic regression model

Log-Logistic regression model

Aalen’s regression model

2.2.1 Weibull Proportional Hazard Model

The Weibull Proportional Hazard Model (WPHM) is a special case of PHM when the Weibull distribution is assumed for

the failure times. Thus, in this model the baseline hazard function is Weibull distribution [33, 75, 76]. This model, as employed

by Jardine [21, 26], introduces a new concept in reliability of utilising diagnostic factors (condition monitoring data) as

explanatory variables. This model is expressed as [21, 77]:

𝑕 𝑡; 𝒛(𝑡) =𝛽

𝜂 𝑡

𝜂

𝛽−1

𝜓(𝜸𝒛(𝑡))

(14)

Jardine et al. [21] derives a new likelihood function. By maximising this likelihood function, all parameters (i.e. regression

coefficients and parameters of Weibull distribution in the baseline hazard function) are estimated. Moreover, EXAKT software

is developed based on WPHM by Jardine et al. [78, 79] in University of Toronto.

Key merits

WPHM is an influential technique which can be used to investigate the effects of various explanatory variables on the

life length of assets/individuals

Explanatory variables have a multiplicative effect (rather than additive effect) on the baseline hazard function, thus it is

a more realistic and reasonable assumption

The model handles truncated or non-truncated data

Key limitations

Due to multicollinearity, estimated values of regression coefficients (𝛾) are sensitive to omission, misclassification and

time dependence of explanatory variables [10]

The estimated value of the regression coefficient is biased in the case of a small sample size

Mixing different types of covariates in one model may cause some problems

The main assumption of this model is that an asset/individual life is assumed to be terminated at the first failure time,

in other words this model depends only on the time 𝑡 elapsed between the starting event (e.g. diagnosis) and the

terminal event (e.g. death) and not on the chronological time 𝑡

Unlike PHM, this model does assume a specified form for the baseline hazard function

Proportionality assumption imposes a severe limitation which that survival curves for assets/individuals with different

covariates can never cross [45-49]

2.2.2 Logistic Regression Model

The Logistic regression model is a special case of POM. Logistic regression model is usually adopted to relate the

probability of an event to a set of covariates [80]. This concept can be used in degradation analysis. If current degradation

features are 𝒛(𝑡), Liao [80] defines the odds ratio between the reliability function 𝑅(𝑡 𝒛(𝑡)) and the cumulative distribution

function as:

𝑅(𝑡 𝒛(𝑡))

1 − 𝑅(𝑡 𝒛(𝑡)) = exp(𝛼 + 𝜸𝒛 𝑡 )

(15)

Where 𝛼 > 0 and 𝜸 are the model parameters to be estimated. Therefore, the reliability function can be expressed as:

𝑅(𝑡 𝒛(𝑡)) =exp(𝛼 + 𝜸𝒛 𝑡 )

1 + exp(𝛼 + 𝜸𝒛 𝑡 )

(16)

Liao [80] asserts that maximum likelihood function for the model parameters can be obtained by maximising the log-

likelihood function using the Nelder-Mead’s algorithm. This model is applied in reliability analysis [80].

Key merit

Based on its likelihood function, it needs less computation effort to estimate parameters rather than PHM [80]

Key limitations

Unlike POM, this model assumes a specified distribution

To estimate parameters and evaluate the reliability function, this model takes into account only the current covariates,

whereas PHM the entire history ones [80]

2.2.3 Log-Logistic Regression Model

The Log-Logistic regression model is a special case of POM when a Log-Logistic distribution is assumed for the failure

times [3]. The Log-Logistic regression model is described in which the hazard for separate samples converges with time.

Therefore, this provides a linear model for the log odds on survival by any chosen time. This model is developed to overcome

some shortcomings of Weibull distribution in the modelling of failure time data.

The distribution used frequently in the modelling of survival and failure time data is the Weibull distribution. However, its

application is limited by the fact that its hazard, while may be increasing or decreasing, must be monotonic, whatever the

values of its parameters. Bennett [71] claims that Weibull distribution may be inappropriate where the course of the failure

(e.g. disease in individuals) is such that mortality reaches a peak after some finite period, and then slowly declines. The hazard

of Log-Logistic regression model is [71]:

𝑕 𝑡; 𝒛 =𝛿

𝑡 1 + 𝑡−𝛿 exp(−𝒛𝜸)

(17)

Which has its maximum value at 𝑡 = 1 − 𝛿 exp( −𝒛𝜸) 1 𝛿 . Where, 𝛿 is a measure of precision and 𝜸 is a measure of

location. The hazard is assumed to be increasing first and then decreasing with a change at the time. Parameters of this model

are estimated by maximising the likelihood function [71]. This model is tested and applied in biomedical [71]. The ratio of the

hazard for a covariate 𝒛 taking two values 𝑧1 and 𝑧2, which converges to unity as 𝑡 increases, is given by [71]:

𝑕(𝑡; 𝑧1)

𝑕(𝑡; 𝑧2)=

1 + 𝑡−𝛿 exp(−𝛾1𝑧2)

1 + 𝑡−𝛿 exp(−𝛾2𝑧1)

(18)

Key merits

It is more suitable to apply in the analysis of survival data rather than Log-Normal distribution

It is suitable model where hazard reaches a peak after some finite period, and the slowly declines

It has mathematical tractability when dealing with the censored observations

The hazard for different samples is not proportional through time as in the Weibull model, but that their ratio trends to

unity. Thus, this property is desirable when the initial effects of covariates (e.g. treatment) trend to diminish with time,

and the survival probabilities of different groups of asset/ individual become more similar

Key limitation

Unlike POM, this model assumes a specified distribution

2.2.4 Aalen’s Regression Model

The Aalen’s regression model is introduced by Aalen in 1980 [81]. This model is based on Aalen’s multiplicative intensity

model for counting processes [82] to assess additive time-dependent covariate effects in possibly right-censored survival data.

At first glance, this model seems to be non-parametric due to the absence of specified distribution. In this model linearity

represents a kind of distributional assumption; however, no finite-dimensional parameter is introduced in the model [82].

Therefore, in this study, it is classified as a semi-parametric model. Aalen develops this model in order to improve some

restrictions of Cox’s PHM. In his model, 𝑕𝑖(𝑡) denotes the intensity of the event happening at time 𝑡 for the 𝑖𝑡𝑕

asset/individual, (𝑕𝑖 𝑡 dt is the probability that the event occurs in some small time interval between 𝑡 and 𝑡 + 𝑑𝑡 given that it

has not happened before). 𝑛 is the number of assets/individuals and 𝑟 is the number of covariates in the analysis. Aalen [82]

considers the following linear model for the vector,𝒉(𝑡; 𝒛), of intensities 𝑕𝑖 𝑡 , 𝑖 = 1,2, … , 𝑛:

𝒉 𝑡; 𝒛 = 𝒀 𝑡 𝜸(𝑡) 0 ≤ 𝑡 ≤ 1 (19)

The 𝑛 × (𝑟 + 1) matrix 𝒀(𝑡) is constructed as follows: if the 𝑖𝑡𝑕 asset/individual is a member of the risk set at time 𝑡 then

the 𝑖𝑡𝑕 row of 𝒀(𝑡) is the vector 𝒛𝑖 𝑡 = (1, 𝑧1𝑖 𝑡 , 𝑧2

𝑖 𝑡 , … , 𝑧𝑟𝑖 𝑡 )′, where 𝑧𝑗

𝑖 𝑡 , 𝑗 = 1,2, … , 𝑟, are time-dependent covariate

values. If the 𝑖𝑡𝑕 asset/individual is not in the risk set at time 𝑡, thus the corresponding row of 𝒀(𝑡) contains only zeros. The

first element of the vector 𝜸 𝑡 , 𝛾0(𝑡), is interpreted as a baseline parameter function, while the remaining elements, 𝛾𝑖 𝑡 , 𝑖 =1,2, … , 𝑟, are called regression coefficients, which measure the influence of the respective covariates. Aalen’s regression model

explicitly allows for contributions of the covariates that change over time, since the regression functions may vary arbitrary

with time [52]. Consequently, if the effect of covariates is zero, the slope will also be zero. If the covariate has a constant

influence over time, the plot will be approximately a straight line. If the slope is positive (or negative), it shows that the effect

of covariates is to increase (or decrease) the hazard. If the plot is a curve with an increasing (or decreasing) slope this indicates

an increase (or decrease) in the magnitude of influence of covariates [3]. Aalen’s regression model has been applied in

biomedical [52, 82, 83] and reliability [44] fields. The merits and limitations of the model are illustrated in the following table.

Key merits

The main advantage of this model is its linearity

It is less vulnerable approach than Cox’s PHM to problem of inconsistency when covariates are deleted or the

precision of covariate measurements is changed [82]

If a covariate which is independent of the other covariates is removed from the model, then the new model is still linear

with unchanged regression coefficients for the remaining covariates, only the baseline parameter function is affected

This model shows the influence of time-dependent covariates better than Cox’s PHM [82]

Key limitations

If a covariate which is not independent of the other covariates is removed from the model, then it is necessary to

assume a multivariate normal distribution for the covariates in order to the new model stay linear, thus normal

assumption cannot be taken literally, since that might give positive probability to negative intensities

The obvious weakness of this model compared to Cox’s PHM is the expression for 𝑕𝑖(𝑡) does not restrict naturally to

non-negative values

The consequence of the lack of restriction to non-negative values is that the estimated survival function may not be

monotonically decreasing throughout, but can have occasional lapses where it increases slightly

Suitable methods of parameter estimation and its goodness-of-fit require to be investigated

3 CONCLUSIONS

The hazard model with covariates is one of the most common statistical models in reliability and survival analysis. This

expository paper reviews the existing literature on hazard models with covariates (termed covariate models) in both the

industrial and biomedical fields. This paper synthesises these models from both industrial reliability and biomedical fields and

then contextually groups them into: non-parametric and semi-parametric models. The merits and limitations of these models

have been discussed so as to establish suitable potential applications, especially for the information of fellow researchers.

This review paper demonstrates that all of these covariate models have been developed and then tested in the biomedical

field to estimate the survival time of patients or individuals. Only a few of these models appear in reliability literature and have

been applied in industrial cases. Most covariate models have not yet been effectively applied in reliability analysis. One reason

for this situation may be the lack of awareness of these covariate models by reliability engineers. Moreover, due to the

prominence of some models (e.g. PHM and WPHM), attempts to develop and apply alternative covariate models in the

reliability field have been somehow stifled. It may be beneficial to introduce and evaluate more covariate models to reliability

analysis for industrial cases.

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Acknowledgement

The authors gratefully acknowledge the financial support provided by both the Cooperative Research Centre for Integrated

Engineering Asset Management (CIEAM), established and supported under the Australian Government’s Cooperative

Research Centres Programme, and the School of Engineering Systems of Queensland University of Technology (QUT).