Survival Models for Step-Stress Experiments With Lagged Effects

Abstract

In this chapter, we consider models for experiments in which the stress levels are altered at intermediate stages during the exposure. These experiments, referred to as step-stress tests, belong to the class of accelerated models that are extensively used in reliability and life-testing applications. Models for step-stress tests have largely relied on the cumulative exposure model (CEM) discussed by Nelson. Unfortunately, the assumptions of the model are fairly restrictive and quite unreasonable for applications in survival analysis. In particular, under the CEM the hazard function has discontinuities at the points at which the stress levels are changed.

We introduce a new step-stress model where the hazard function is continuous. We consider a simple experiment with only two stress levels. The hazard function is assumed to be constant at the two stress levels and linear in the intermediate period. This model allows for a lag period before the effects of the change in stress are observed. Using this formulation in terms of the hazard function, we obtain the maximum likelihood estimators of the unknown parameters. A simple least squares-type procedure is also proposed that yields closed-form solutions for the underlying parameters. A Monte Carlo simulation study is performed to study the behavior of the estimators obtained by the two methods for different choices of sample sizes and parameter values. We analyze a real data set and show that the model provides an excellent fit.

Appendix

1.1 The likelihood equations

Here we show that the MLE of b can be obtained as the solution of (23.7). Note that the log-likelihood function can be written from (23.6) as

$$L(a, b) = \ln l(a, b) = - a T - b K + n_1 \ln (a + b \tau_1) + \sum_{i \in I_2} \ln ( a + b t_i) + n_3 \ln ( a + b \tau_2).$$

Therefore, $\hat{a}$ and $\hat{b}$ can be obtained by solving the equations

$$\frac{\partial L}{\partial a} = 0 \quad\textrm{and}\quad\frac{\partial L}{\partial b} = 0.$$

((23.15))

From (23.15), we obtain

$$a \times \frac{\partial L}{\partial a} + b \frac{\partial L}{\partial b} = 0.$$

((23.16))

After simplification, we obtain

$$a = \frac{n - b K}{T}.$$

((23.17))

Substituting this expression of a from (23.17) in $\frac{\partial L}{\partial a} = 0$, we obtain the equation in (23.7).

1.2 Fisher information matrix

In this subsection, we obtain the explicit expressions for elements of the observed Fisher information matrix. We can use the asymptotic normality property of the MLEs to construct approximate confidence intervals of a and b for large n.

Let $O(a,b) = O_{ij}(a,b), i, j = 1,2$, denote the observed Fisher information matrix of a and b. Then

$$\begin{aligned}O_{11} & = \frac{n_1}{(\hat{a} + \hat{b} \tau_1)^2} + \sum_{i \in I_2} \frac{1}{ (\hat{a} + \hat{b} t_i)^{2}} + \frac{n_3}{(\hat{a} +\hat{b}\tau_2)^2}\\O_{22} & = \frac{n_1 \tau_1^2}{(\hat{a} + \hat{b} \tau_1)^2} +\sum_{i \in I_2} \frac{t_i^2}{ (\hat{a} + \hat{b} t_i)^2} +\frac{n_3 \tau_2^2}{(\hat{a} +\hat{b} \tau_2)^2}\\O_{12} & = \frac{n_1 \tau_1}{(\hat{a} + \hat{b} \tau_1)^2} +\sum_{i \in I_2} \frac{t_i}{ (\hat{a} + \hat{b} t_i)^2} + \frac{n_3\tau_2}{(\hat{a} +\hat{b} \tau_2)^2}.\\\end{aligned}$$

The approximate variances of $\hat{a}$ and $\hat{b}$ can then be obtained through the observed information matrix as

$$V_a = \frac{O_{22}}{O_{11} O_{22} - O_{12}^2}\textrm{and} V_b = \frac{O_{11}}{O_{11} O_{22} - O_{12}^2}.$$

((23.18))

Using the observed asymptotic variances of $\hat{a}$ and $\hat{b}$, approximate confidence intervals can be easily constructed.