Step Stress Accelerated Life Testing: Classical Methods Mikhail Nikulin IMB UMR 5251,University Victor Segalen, Bordeaux, France
[email protected]
Abstract. We consider here the accelerated models describing dependence of the lifetime distribution on the step stresses. Such models are often used in reliability and survival analysis to study the reliability of highly reliable complex technical and bio-technical systems in dynamic environments. Keywords Accelerated trials, AFT model, Arrhenius Model, Cox model, Models with finite supports, GPH model, Log-linear model, Power rule model, Resource, Step-stress, Weibull family 1. Introduction in Accelerated Life testing The failure time data are used for product reliability estimation. Failures of highly reliable units are rare and other information should be used than traditional censored failure time data. One way of obtaining this complementary reliability information is to apply the methods of the so-called accelerated life-testing (ALT) for improvements in reliability life testing, to obtain wished reliability results for the products more quickly then under normal operating conditions, given in terms of step-stresses. Step-stress ALT gives an interesting approach to study the possibility to take into account the cumulative effect of the applied stresses on aging, fatigue and degradation of testing items, systems, etc... The idea to use the step-stresses in ALT was proposed at first by Miner [17], Pieruschka [25], Sedyakin [26], Singpurwalla [30]. Later this idea was actively developed in publications of many statisticians, see, for example, [18],[28], [14], [27],[21-23],[32],[29], [31],[19],[112],[24], etc..., where one can find a list of references on applications and estimation in ALT. In ALT any accelerated life model is defined on a set E = {x(·)} of all possible or admissible stresses for considered problem, where any stress (explanatory variable) x(·) ∈ E is a deterministic (or random) time function: x(·) = (x1 (·), ..., xm (·))T : [0, ∞) → Rm , xi (·) is univariate explanatory variable. If x(·) is constant in time, x(·) ≡ x = const, we shall write x instead of x(·) in all formulas. Let E1 denote a set of constant in time stresses, E1 ⊂ E. Using a set E1 one may construct a 1
set E2 , E2 ⊂ E of step-stresses of the form
½
x1 , x2 ,
x(u) =
0 ≤ u < t1 , u ≥ t1 ,
for any t1 > 0, where x1 , x2 ∈ E1 . By the same way for any k, k = 2, 3, ... one may consider a class Ek ,
Ek ⊂ E, of step-stresses of the form x1 , 0 ≤ u < t1 , x2 , t1 ≤ u < t2 , x(u) = ··· ··· xk , tk−1 ≤ u < tk ≤ ∞,
(1)
where x1 , · · · , xk are constant in time stresses from E1 . It is clear that E1 ⊂ E2 ⊂ ... ⊂ Ek ⊂ E, where E is a set of all admissible stresses. The mostly used stresses in ALT are step-stresses from just defined classes E2 , E3 or E4 , and so on. The units are placed on test at an initial low constant stress x1 and if they do not fail in a predetermined time t1 , the stress is increased. If they do not fail in a predetermined time t2 > t1 , the constant stress x2 is increased once more, and so on. Let Tx(·) be the failure time under x(·) ∈ E. Then the cumulative distribution function, the hazard rate function and the cumulative hazard function given x(·) ∈ E are : © ª Fx(·) (t) = P Tx(·) ≤ t | x(s) : 0 ≤ s ≤ t ,
λx(·) (t) =
0 Fx(·) (t)
1 − Fx(·) (t)
Z ,
Λx(·) (t) =
t
0
λx(·) (s)ds,
from where one can see their dependence on the life-history up to time t. Denote by λ0 (t) a baseline rate function. It corresponds to the failure rate of given items when hypothetically all of them are testing in the same ”standard”, ” normal”, or ”usual” conditions, given by a stress x0 (·), x0 (·) ∈ E. The two most known general models of ALT with time varying stresses are: the Accelerated Failure Time (AFT) model and the Proportional Hazards (PH) or Cox model . The AFT model on E: Z Fx(·) (t) = F0 (
t
r[x(τ )]dτ ),
x(·) ∈ E,
(2)
0
where r(·) is a positive function on E. In practice often a baseline cumulative distribution function F0 is taken from some class of simple parametric distributions, such as Weibull, lognormal, log-logistic, etc., and in this case we obtain the parametric AFT model. We have an interesting family when F0 belongs to the Power Generalized Weibull (PGW) family of distributions (see [6]). In terms of the cumulative distribution functions the PGW family is given by the next formula: ( · µ ¶ν ¸ γ1 ) t F (t, σ, ν, γ) = 1 − exp 1 − 1 + , σ 2
t > 0, γ > 0, ν > 0, σ > 0.
If γ = 1 we have the Weibull family of distributions. If γ = 1 and ν = 1, we have the exponential family of distributions. The PGW family of distributions has very nice probability properties. All moments of this distribution are finite. Depending on the parameter values the hazard rate can be T S constant, monotone (increasing or decreasing), unimodal or -shaped, and bathtube or -shaped. Another interesting family, the Exponentiated Weibull Family of distributions, was proposed in [20]. The PH model on E: λx(·) (t) = r[x(t)]λ0 (t),
x(·) ∈ E.
(3)
The function r in the AFT model (as in the PH model) can be parametrized in the following way : r[x(τ )] = eβϕ(x(τ )) , where ϕ is some known function of the stress. For example, if ϕ(x) = x, ln x, 1/x, we have generalizations of the log-linear, power rule, Arrhenius models, respectively, see [19,21]. The AFT and PH models are rather restrictive. In the case of AFT model the stress changes locally only the scale. Suppose that the PH model holds on E, x0 is a design stress, x1 is an accelerated stress with respect to the stress x0 , x0 , x1 ∈ E1 , so that Fx0 (t) ≤ Fx1 (t) for all t ≥ 0, and let
½ x(t) =
x1 , 0 ≤ t < t1 , x0 , t ≥ t1 ,
(4)
is a step-stress. Then in the case of the PH model λx(·) (t) = λx0 (t) for all t > t1 . So if one group of items is tested under the design stress x0 and the second group is tested at first under an accelerated stress x1 until the moment t1 and after this moment is tested under the design stress x0 , so that both groups are observed under the same design stress x0 after the moment t1 , the failure rate after the moment t1 is the same for both groups. If items are aging, it is not natural to apply the PH model. 2. AFT and PH models for constant and step-stresses In terms of cumulative distribution function the PH model (3) is written: Z Fx(·) (t) = 1 − exp{−
t
r[x(τ )]dΛ0 (τ )},
x(·) ∈ E.
(5)
0
In the particular case of constant in time stress xj and the step-stress (1) the AFT model (2) implies Fxj (t) = F0 (rj t),
3
and for t ∈ [ti−1 , ti ), (i = 1, 2, ..., k) i−1 X Fx(·) (t) = F0 rj ∆tj + ri (t − ti−1 ) . j=1
In terms of Fxj , Fx(·) (t) = Fxi (si−1 + t − ti−1 ),
(6)
where si can be determined recurrently from the equations Fxi+1 (si ) = Fxi (si−1 + ti − ti−1 ).
(7)
For the step-stress (4) and k = 2 the PH model (5) implies ½ λx(·) (τ ) =
λx1 (τ ), 0 ≤ τ ≤ t, , ρλx1 (τ ), τ > t.
where ρ = r(x1 )/r(x0 ). In this particular case of step-stresses the Cox model is sometimes called the Tampered Failure Rate (TFR) model, see [10]. If k ≥ 2, the formula (5) implies Fxj (t) = 1 − e−rj Λ0 (t) and for t ∈ [ti−1 , ti ) Fx(·) (t) = 1 − exp{−
i−1 X
rj (Λ0 (tj ) − Λ0 (tj−1 )) − ri (Λ0 (t) − Λ0 (ti−1 ))}.
(8)
j=1
In the case when the distribution under the design stress x0 is Weibull : Fx0 (t) = 1 − e−t
δ
/α0
,
Λ0 (t) =
tδ , r0 α 0
we have Fxi (t) = 1 − e−t and for t ∈ [ti−1 , ti )
δ
/θi
,
i−1 δ X tj − tδj−1 tδ − tδi−1 . Fx(·) (t) = 1 − exp − − θj θi j=1
This particular case of the PH model was considered in [13]. 3. Models with finite supports Following [2] we consider generalisation of both the AFT and PH models: GPH model. For this purpose we formulate these models in other terms. For any x(·) the random variable R = Λx(·) (Tx(·) ) 4
has the standard exponential distribution with the survival function SR (t) = e−t , t ≥ 0, and is called the exponential resource, the number Λx(·) (t) being the exponentional resource used until the moment t under the stress x(·). So the failure rate λx(·) (t) is the rate of exponential resource usage. Under the PH model we have Λ0x(·) (t) = r[x(t)]λ0 (t), i.e. the rate of resource usage is proportional to some baseline rate and the constant of proportionality is a function of a stress. Let consider now the Generalized Proportional Hazards Model (GPH) model, proposed in [3]. GPH model holds on E if λx(·) (t) = r{x(t)}q{Λx(·) (t)}λ0 (t),
x(·) ∈ E.
The rate of resource usage at the moment t depends not only on a stress applied at this moment but also on the resource used until t. The particular cases of the GPH model are the PH model (q(u) ≡ 1) and the AFT model (λ0 (t) ≡ λ0 = const). Consider the case when the rate of exponential resource usage with respect to the baseline rate is proportional to a monotone function of the used resource. One of the most natural parametrizations of the function q in this case would be q(u) = eγu ,
γ ∈ R1 .
So we consider the so-called Generalized Linear Proportional Hazards (GLPH) model on E : λx(·) (t) = eβx(t)+γΛx(·) (t) λ0 (t),
x(·) ∈ E.
(9)
This model implies, that if γ ≤ 0, the support of distribution is [0, ∞), and if γ > 0, we have the model with finite support [0, spx(·) ), where spx(·) verifies the equation Z
spx(·)
eβx(u) dΛ0 (u) = 1.
0
In the particular cases of the step-stress (1) we have the formula (8) when γ = 0. In the case γ 6= 0 and x(·) ∈ Ek , we have for all t ∈ [ti−1 , ti ) Fx(·) (t) = 1 − {1 − γ
i−1 X
eβxj (Λ0 (tj ) − Λ0 (tj−1 )) − γeβxi (Λ0 (t) − Λ0 (ti−1 ))}1/γ ,
j=1
5
(10)
and for the constant stress xj Fxj (t) = 1 − {1 − γeβxj Λ0 (t)}1/γ . If the distribution of time-to-failure under the design stress x0 is Weibull, i.e. Fx0 (t) = 1 − e−t
δ
/α0
,
λ0 (t) =
´ γ 1 −βx0 ³ e 1 − e− α0 +δ , γ
then under the GLPH model (9) and the step-stress (1) Fx(·) (t) = 1 − {1 +
i−1 X
δ
δ
eβ(xj −x0 ) [e−γtj /α0 − e−γtj−1 /α0 ]+
j=1
eβ(xi −x0 ) [e−γt
δ
/α0
δ
− e−γti−1 /α0 ]}1/γ .
(11)
Note that if the rate of resource usage depends on used resource ( i.e. γ 6= 0), the distribution of the time-to-failure under constant in time stresses xj 6= x0 is not Weibull : Fxj (t) = 1 − {1 − eβ(xj −x0 ) [1 − e−γt
δ
/α0
]}1/γ .
Under the GLPH model distributions of times-to-failure under constant in time stresses are from the same interesting family of distributions if we take one of the following two families of distributions K1 and K2 . The family K1 : Fx0 (t) = 1 − (1 + α0 tδ )1/γ ,
t ≥ 0,
(α0 > 0, δ > 0, γ < 0).
(12)
If 0 < δ ≤ 1, the failure rate λx0 (t) is decreasing. If δ > 1, then the failure rate is increasing until ´1/δ ³ T and decreasing after this moment, i.e. has the -shape. the moment t = δ−1 α0 The family K2 : Fx0 (t) = 1 − (1 + α0 tδ )1/γ ,
µ ¶1/δ 1 t ∈ [0, − ), α0
(α0 < 0, δ > 0, γ > 0).
³ ´1/δ If 0 < δ < 1, the hazard rate is decreasing in the interval [0, δ−1 ) and increasing in the interval α0 ³ ´1/δ ³ ´1/δ S , − α10 ). The hazard rate has -shape. So this family is very appealing. If δ > 1, the ( δ−1 α0 ´1/δ ³ hazard rate is increasing on (0, − α10 ). Suppose that the time-to-failure distribution is from one of the above mentioned families. If Fx0 (t) ∈ Ki (i=1,2) and the GLPH model holds, then Λ0 (t) = −
α0 −βx0 δ e t γ
Fxj (t) = 1 − (1 + αj tδ )1/γ ,
and 6
where αj = α0 eβ(xj −x0 ) . If Fx0 ∈ K2 , the support of the distribution is finite : [0,
³
1 −αj
´ δ1
).
If x(·) is a step stress of the form (1) then Fx(·) (t) = 1 − {1 + α0 ϕ(t, β, δ)}1/γ ,
0 < t < spx(·) ,
(13)
where ( P ∗ k −1 ϕ(t, β, δ) = where ∗
k = {k :
e
k−1 X
e
β(xj −x0 ) δ (tj j=1 e β(x1 −x0 ) δ
β(xj −x0 )
t ,
(tδj
−
tδj−1 )
j=1
− tδj−1 ) + eβ(xk∗ −x0 ) (tδ − tδk∗ −1 ), k ∗ > 1 , k ∗ = 1. 1 ≤− , α0
k X
eβ(xj −x0 ) (tδj − tδj−1 ) > −
j=1
1 } α0
If α0 > 0, γ < 0, then spx(·) = +∞. If α0 < 0, γ > 0, the right limit of the support (0, spx(·) ) verifies the equation ϕ(spx(·) , β, δ) = −1/α0 , i.e.
spx(·)
1/δ ∗ kX −1 1 , = tδk∗ −1 − e−β(xk∗ −x0 ) + eβ(xj −x0 ) (tδj − tδj−1 ) α0 j=1
and the failure rate δ tδ−1 λx(·) (t) = −α0 eβ(xk∗ −x0 ) , γ (1 + α0 ϕ(t, β, δ))
0 < t < spx(·) ,
has the same shape as in the case of constant in time stresses (decreasing or S K1 , increasing or -shape for the family K2 ).
T
-shape for the family
4. Remarks on estimation Suppose that all units are tested under stresses from Ek , i.e. they are initially placed on test at x1 and run until t1 when the stress is changed to x2 > x1 . At x2 , testing continues until t2 when stress is changed to x3 > x2 , and so on, until xk . Under xk , testing continues until all remaining units fail or until tf whichever comes first. Denote by D the number of failures during the experiment, T(1) ≤ ... ≤ T(D) the observed moments of failures. In the case of a finite support the following reparametrization can be introduced: σ = spx(·) , β, γ, δ. Then α0 = −1/ϕ(σ, β, δ). The likelihood function for the family K1 is given by the formula
µ ¶D PD δ β (xk(T(i) ) −x0 ) i=1 L(α0 , β, γ, δ) = −α0 e × γ D Y
1
δ−1 T(i) {1 + α0 ϕ(T(i) ; β, δ)} γ −1 {1 + α0 ϕ(tf ; β, δ)}
i=1
7
n−D γ
and for the family K2 by the next one µ ¶D PD µ ¶D δ 1 β (xk(T(i) ) −x0 ) i=1 L(σ, β, γ, δ) = e × γ ϕ(σ, β, δ) µ ¶ 1 −1 µ ¶ n−D γ ϕ(T(i) , β, δ)) γ ϕ(tf , β, δ)) δ−1 T(i) 1− 1− 1{T(D) < σ}. ϕ(σ, β, δ) ϕ(σ, β, δ) i=1 n Y
The estimator for the cumulative distribution function Fx0 (t) is ˆ 1 Fˆx0 (t) = 1 − (1 + α ˆ 0 tδ ) γˆ .
In the case of the class K1 approximate confidence intervals for Fx0 (t) and other reliability characteristics under the design stress x0 are obtained by standard ML (maximum-likelihood) methods using the estimated Fisher information matrix. In the case of the class K2 when the support is finite, rigorous asymptotic theory is to be established. If the time-to-failure under the design stress x0 has the Weibull (or another) distribution, then estimation under the GLPH model can be done similarly taking the formula (11) (or (10) with respective Λ0 ) for Fx(·) instead of (13) in all formulas. References . 1. Bagdonaviˇcius V, Nikulin M. Semiparametric models in accelerated life testing. Queen’s papers in pure and applied mathematics, 1995 98: Queen’s University, Kingston, Ontario, Canada. 2. Bagdonaviˇcius V, Gerville-R´eache L, Nikulin M. On parametric inference for step-stresses models. IEEE Transactions on Reliability, 2002 1: 27-31. 3. Bagdonaviˇcius V, Nikulin M. Generalized proportional hazards model based on modified partial likelihood, Lifetime Data Analysis, 1999 5: 329-350. 4. Bagdonaviˇcius V. Testing the hypothesis of the additive accumulation of damage. Prob. Theory and Its Applications, 1978 23: 403-408. 5. Bagdonaviˇcius V, Gerville-R´eache L, Nikoulina V and Nikulin M. Exp´eriences acc´el´er´ees: Analyse statistique du mod`ele standard de vie acc´el´er´ee. Revue de Statistique Appliqu´ee, 2000 3: 5-38. 6. Bagdonaviˇcius V, Nikulin M. Accelerated life models. Chapman&Hall/CRC: Boca Raton, 2002. 7. Bagdonaviˇcius V, Cheminade O, Nikulin, M. Statistical planning and inference in accelerated life testing using the CHSS model. Journal of Statistical Planning and Inference, 2004 126: 535-551. 8. Bagdonaviˇcius V, Nikulin M. On semiparametric estimation of reliabilty from accelerated life data, in: Statistical and Probabilistic Models in Reliability. Ionescu D, Limnios N (eds). Birkhauser: Boston, 1998; 75-89. 8
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Goettingen: Vandenhoeck and
