The accuracy of reliability estimation depends on the models that relate the failure data under severe conditions, or high stress, to that at normal operating conditions, or design stress. Elsayed (1996) classifies these models into three groups: statistics-
based models, physics-statistics models, and physics-experimental models.
Furthermore, he classifies the statistics models into two sub-categories: parametric and nonparametric models. We limit the models in this chapter to the statistics models as they are more general while the physics-statistics and physics-experi- mental models are usually developed for particular applications such as fatigue testing, creep testing and electromigration models.
7.3.1 Statistics-based Models: Parametric
The failure times at each stress level are used to determine the most appropriate failure time distribution along with its parameters. We refer to these models as AFT (accelerated failure time). Parametric models assume that the failure times at different stress levels are related to each other by a common failure time distribu- tion with different parameters. Usually, the shape parameter of the failure time distribution remains unchanged for all stress levels, but the scale parameter may present a multiplicative relationship with the stress levels. For practical purposes, we assume that the time scale transformation (also referred to as acceleration factor,AF >1) is constant, which implies that we have a true linear acceleration.
Thus the relationships between the accelerated and normal conditions are sum- marized as follows (Tobias and Trindade 1986; Elsayed 1996). Let the subscripts o and s refer to the operating conditions and stress conditions, respectively.
The relationship between the time to failure at operating conditions and stress conditions is
o F S.
t =A t× (7.1)
The cumulative distribution functions are related as
( ) .
F
o s t
F t F A
⎛ ⎞
= ⎜ ⎟
⎝ ⎠ (7.2)
The probability density functions are related as
( ) 1 .
F
o s
F
f t f t
A A
⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ (7.3)
The failure rates are given by
( ) 1 .
F F
o s t
h t h
A A
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= (7.4)
The acceleration factor is obtained by determining the median lives of units tested at two different accelerated stresses and extrapolating to the median life at normal operating stress. It can also be estimated by replacing the medians with some quartiles.
The accuracy of the reliability estimates suffers when small samples are tested at the stress conditions since the determination of proper failure time distribution that describes these failures becomes difficult. More importantly, the assumption of having the same failure time distributions at different stress levels is difficult to justify especially when small numbers of failures are observed. In these cases, it is more appropriate to use nonparametric models as described next.
7.3.2 Statistics-based Models: Nonparametric
Nonparametric models relax the requirement of the common failure time distribution, i.e., no common failure time distribution is required among all stress levels. Several nonparametric models have been developed and validated in recent years. We describe these models below.
7.3.2.1 Proportional Hazards Model
Cox’s Proportional Hazards (PH) model (Cox 1972, 1975) is the most popular non- parametric model. It has become the standard nonparametric regression model for accelerated life testing in the past few years. The PH model is distribution-free requiring only the ratio of hazard rates between two stress levels to be constant with time.
The proportional hazards model has the following form:
( ; )t 0( )exp( )t
λ z =λ βz (7.5)
The base line hazard function λ0( )t is an arbitrary function; it is modified multiplicatively by the covariates (i.e. applied stresses).
Elsayed and Zhang (2006) assume λ0( )t to be linear with time:λ0( )t =γ0+γ1t. Substituting λ0( )t into the PH model, we obtain: λ( ; ) (t z = γ0+γ1t)exp( )βz , where z=( , ,z z1 2…zp)T is a column vector of the covariates (or applied stresses).
For ALT, the column vector represents the stresses used in the test and/or their interactions. β =( , ,β β1 2…βp) is a row vector of the unknown coefficients corre- sponding to the covariates z . These coefficients can be estimated using a partial likelihood estimation procedure.
This model usually produces “good” reliability estimation with failure data for which the proportional hazards assumption holds and even when it does not exactly hold.
7.3.2.2 Extended Linear Hazards Regression Model
The PH and AFT models have different assumptions. The only model that satisfies both assumptions is the Weibull regression model (Kalbfleisch and Prentice 2002).
For generalization, the Extended Hazard Regression (EHR) model (Ciampi and
Etezadi-Amoli 1985; Etezadi-Amoli and Ciampi 1987; Shyur et al. 1999) is pro- posed to combine the PH and AFT models into one form:
0 '
( ; )t (e t)exp( ' )
λ z =λ zβ zα (7.6)
The unknowns of this model are the regression coefficients α β, and the unspecified baseline hazard function λ0( )t . The model reflects that the covariate z has both the time scale changing effect and hazard multiplicative effect. It becomes the PH model when β =0 and the AFT model when α β= .
Elsayed et al. (2006) propose a new model called Extended Linear Hazard Regression (ELHR) model. The ELHR model (e.g., with one covariate) assumes those coefficients to be changing linearly with time:
( )
0 1
( )
0 0 1
( ; )t z (teβ βt z)exp ( t z)
λ =λ + α +α (7.7)
The model considers the proportional hazards effect, time scale changing effect as well as time-varying coefficients effect. It encompasses all previously developed models as special cases. It may provide a refined model fit to failure time data and a better representation regarding complex failure processes.
Since the covariate coefficients and the unspecified baseline hazard cannot be expressed separately, the partial likelihood method is not suitable for estimating the unknown parameters. Elsayed et al. (2006) propose the maximum likelihood method which requires the baseline hazard function to be specified in a parametric form. In the EHR model, the baseline hazards function has two specific forms; one is a quadratic function and the other is a quadratic spline. In the proposed ELHR model, we assume the baseline hazard function λ0( )t to be a quadratic function:
0( )t 0 1t 2t2
λ =γ +γ +γ (7.8)
Substituting λ0( )t into the ELHR model yields
0 1 0 1 2 0 1
0 1 2
( ; ) t z eαz αzt teθ z θzt t eωz ωzt
λ = γ + + γ + + γ + (7.9)
where
0 0 0
θ =α +β , θ1=α1+β1, ω0=α0+2β0, ω α1= 1+2β1 The cumulative hazard rate function is obtained as
( ) ( )
( ) ( )
0 1 0 1 0 1
0 1 0 0 1 0 1 0
0 1 0 1 0 1
0 1 2 2
0 0 0 0
0 0 1 1 1
2 2
1 1 1 1 1
22 2 2
2 3
1 1 1
( ; ) ( ; )
2 2
t t z zu t z zu t z zu
z zt z z zt z zt z
z zt z zt z zt
t z u z du e du ue du u e du
e e te e e
z z z z z
t t
e e e
z z z
α α θ θ ω ω
α α α θ θ θ θ θ
ω ω ω ω ω ω
λ γ γ γ
γ γ γ γ γ
α α θ θ θ
γ γ γ
ω ω ω
+ + +
+ + +
+ + +
Λ = = + +
= − + − +
+ − + −
∫ ∫ ∫ ∫
( )123 0
2 e z
z γ ω
ω
The reliability function, R t z( ; ) and the probability density functions f t z( ; ) are obtained as
( ; ) exp( ( ; )) ( ; ) ( ; )exp( ( ; ))
R t z t z
f t z λ t z t z
= −Λ
= −Λ
Although the ELHR model is developed based on the distribution-free concept, a close investigation of the model reveals its capability of capturing the features of commonly used failure time distributions. The main limitation of this model is that
“good” estimates of the many parameters of the model require a large number of test units.
7.3.2.3 Proportional Mean Residual Life Model
Oakes and Dasu (1990) originally propose the concept of the Proportional Mean Residual Life (PMRL) by analogy with PH model. Two survivor distributions
( )
F t and F t0( ) are said to have PMRL if ( ) 0( )
e x =θe x (7.10)
where e x( ) is the mean residual life at time x.
We extend the model to a more general framework with a covariate vector Z (applied stress)
) ( ) exp(
)
|
(t z z e0 t
e = βT (7.11)
We refer to this model as the proportional mean residual life regression model which is used to model accelerated life testing. Clearly e x0( ) serves as the MRL corresponding to a baseline reliability function R t0( )and is called the baseline mean residual function; e t z( ) is the conditional mean residual life function of T t− given T t> and Z z= . Where zT =( , ; , )z z1 2 zp is the vector of covariates,
1 2
( , ; , )
T p
β = β β β is the vector of coefficients associated with the covariates, and p is the number of covariates. Typically, we can experimentally obtain {( , ),t z ii i =1,2, , } n the set of failure time and the vectors of covariates for each unit (Zhao and Elsayed, 2005). The main assumption of this model is the pro- portionality of mean residual lives with applied stresses. In other words, the mean
residual life of a unit subjected to high stress is proportional to the mean residual life of a unit subjected to low stress.
7.3.2.4 Proportional Odds Model
In many applications, however, it is often unreasonable to assume the effects of covariates on the hazard rates remain fixed over time. Brass (1971) observes that the ratio of the death rates, or hazard rates, of two populations under different stress levels (for example, one population for smokers and the other for non- smokers) is not constant with age, or time, but follows a more complicated course, in particular converging closer to unity for older people. So the PH model is not suitable for this case. Brass (1974) proposes a more realistic model: the pro- portional odds (PO) model. The proportional odds model has been successfully used in categorical data analysis (McCullagh 1980; Agresti and Lang 1993) and survival analysis (Hannerz 2001) in the medical fields. The PO model has a distinct different assumption on proportionality, and is complementary to the PH model. It has not been used in reliability analysis of accelerated life testing so far. Zhang and Elsayed (2005) extend this model for reliability estimates using ALT data.
We describe the PO model as follows. Let T>0 be a failure time associated with stress level z with cumulative distributionF t z( ; ), and that ratio ( ; )
1 ( ; ) F t z
F t z
− ,
or1 ( ; ) ( ; ) R t z R t z
− , be the odds on failure by timet. The PO model is then expressed as
0 0
( ) ( ; ) exp( )
1 ( ; ) 1 ( )
F t
F t z z
F t z = β F t
− − (7.12)
where F t0( )≡F t z( ; =0) is the baseline cumulative distribution function and β is unknown regression parameter. Let θ( ; )t z denote the odds function, then the above PO model is transformed to
( ; ) exp( ) ( )t z z 0 t
θ = β θ (7.13)
where θ0( )t ≡θ( ;t z=0) is the baseline odds function.
For two failure time samples with stress levels z1 and z2, the difference between the respective log odds functions is
1 2 1 2
log[ ( ; )] log[ ( ; )]θ t z − θ t z =β(z z− ),
which is independent of the baseline odds function θ0( )t and the time t. Hence, the odds functions are constantly proportional to each other. The baseline odds function could be any monotone increasing function of time t with the property of
0(0) 0
θ = . When θ0( )t =tϕ, PO model presented by Equation 7.13 becomes the
log-logistic accelerated failure time model (Bennett 1983), which is a special case of the general PO models.
In order to utilize the PO model in predicting reliability at normal operating conditions, it is important that both the baseline function and the covariate parameter,β, be estimated accurately. Since the baseline odds function of the general PO models could be any monotone increasing function, it is important to define a viable baseline odds function structure to approximate most, if not all, of the possible odds function. In order to find such a “universal” baseline odds function, we investigate the properties of odds function and its relation to the hazard rate function.
The odds function θ( )t is denoted by
( ) 1 ( ) 1
( ) 1
1 ( ) ( ) ( )
F t R t
t F t R t R t
θ = = − = −
− (7.14)
From the properties of reliability function and its relation to odds function shown in Equation 7.14, we could easily derive the following properties of odds function θ( )t :
1. θ(0) 0= , θ( )∞ = ∞
2. θ( )t is monotonically increasing function in time 3. ( ) 1 exp[ ( )] exp[ ( )] 1
exp[ ( )]
t t t
θ = − −Λt = −Λ −
−Λ , and Λ =( ) ln[ ( ) 1]t θ t + 4. ( ) ( )
( ) 1 t t
t λ θ
θ
= ′ +
Further investigation of such a “universal” odds function shows that it can be approximated by a polynomial function.
An appropriate ALT model is important since it explains the influences of the stresses on the expected life of a product based on its physical properties and the related statistical properties. On the other hand, a carefully designed test plan improves the accuracy and efficiency of the reliability estimation. The design of an accelerated life testing plan consists of the formulation of objective function, the determination of constraints and the definition of the decision variables such as stress levels, sample size, allocation of test units to each stress level, stress level changing time and test termination time, and others. Inappropriate values of the decision variable result in inaccurate reliability estimates and/or unnecessary test resources. Thus it is important to design test plans to minimize the objective function under specific time and cost constraints.