8.
Assessing Product Reliability
8.1. Introduction 8.1.6. What are the basic lifetime distribution models used for nonrepairable populations?


The proportional hazards model is often used in survival analysis (medical testing) studies. It is not used much with engineering data 
The proportional hazards model,
proposed by Cox (1972), has been used primarily in medical testing analysis,
to model the effect of secondary variables on survival. It is more like
an acceleration model than a specific life distribution model, and its
strength lies in its ability to model and test many inferences about survival
without making any specific assumptions about the form of the life distribution
model.
This section will give only a brief description of the proportional hazards model, since it has limited engineering applications. Proportional Hazards Model Assumption Let \(z = \{x, \, y, \, \ldots\}\) be a vector of one or more explanatory variables believed to affect lifetime. These variables may be continuous (like temperature in engineering studies, or dosage level of a particular drug in medical studies) or they may be indicator variables with the value 1 if a given factor or condition is present, and 0 otherwise. Let the hazard rate for a nominal (or baseline) set \(z_0 = \{x_0, \, y_0, \, \ldots\}\) of these variables be given by \(h_0(t)\), with \(h_0(t)\) denoting a legitimate hazard function (failure rate) for some unspecified life distribution model. 

The proportional hazard model assumes changing a stress variable (or explanatory variable) has the effect of multiplying the hazard rate by a constant. 
The proportional hazards model assumes we can
write the changed hazard function for a new value of \(z\)
$$ h_z(t) = g(z) h_0(t) \,\, .$$
In other words, changing \(z\), the explanatory variable vector, results in a new hazard function that is proportional to the nominal hazard function, and the proportionality constant is a function of \(z,\, g(z)\) independent of the time variable \(t\). A common and useful form for \(f(z)\) is the Log Linear Model which has the equation: \(g(x) = e^{\alpha x}\) for one variable, \(g(x,y) = e^{ax + by}\) for two variables, etc. Properties and Applications of the Proportional Hazards Model
