Multivariable Relationships: General Log-Linear and Proportional Hazards

Introduction

So far in this reference the life-stress relationships presented have been either single stress relationships or two stress relationships. In most practical applications, however, life is a function of more than one or two variables (stress types). In addition, there are many applications where the life of a product as a function of stress and of some engineering variable other than stress is sought. In this chapter, the general log-linear relationship and the proportional hazards model are presented for the analysis of such cases where more than two accelerated stresses (or variables) need to be considered.

Introduction

So far in this reference the life-stress relationships presented have been either single stress relationships or two stress relationships. In most practical applications, however, life is a function of more than one or two variables (stress types). In addition, there are many applications where the life of a product as a function of stress and of some engineering variable other than stress is sought. In this chapter, the general log-linear relationship and the proportional hazards model are presented for the analysis of such cases where more than two accelerated stresses (or variables) need to be considered.

Indicator Variables

Another advantage of the models presented in this chapter is that they allow for simultaneous analysis of continuous and categorical variables. Categorical variables are variables that take on discrete values such as the lot designation for products from different manufacturing lots. In this example, lot is a categorical variable, and it can be expressed in terms of indicator variables. Indicator variables only take a value of 1 or 0. For example, consider a sample of test units. A number of these units were obtained from Lot 1, others from Lot 2, and the rest from Lot 3. These three lots can be represented with the use of indicator variables, as follows:

• Define two indicator variables, [math]\displaystyle{ {{X}_{1}} }[/math] and [math]\displaystyle{ {{X}_{2}}. }[/math]

• For the units from Lot 1, [math]\displaystyle{ {{X}_{1}}=1, }[/math] and [math]\displaystyle{ {{X}_{2}}=0. }[/math]

• For the units from Lot 2, [math]\displaystyle{ {{X}_{1}}=0, }[/math] and [math]\displaystyle{ {{X}_{2}}=1. }[/math]

• For the units from Lot 3, [math]\displaystyle{ {{X}_{1}}=0, }[/math] and [math]\displaystyle{ {{X}_{2}}=0. }[/math]

Assume that an accelerated test was performed with these units, and temperature was the accelerated stress. In this case, the GLL relationship can be used to analyze the data. From the GLL relationship we get:

[math]\displaystyle{ L(\underline{X})={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}{{X}_{1}}+{{\alpha }_{2}}{{X}_{2}}+{{\alpha }_{3}}{{X}_{3}}}} }[/math]

where:
• [math]\displaystyle{ {{X}_{1}} }[/math] and [math]\displaystyle{ {{X}_{2}} }[/math] are the indicator variables, as defined above.

• [math]\displaystyle{ {{X}_{3}}=\tfrac{1}{T}, }[/math] where [math]\displaystyle{ T }[/math] is the temperature.

The data can now be entered in ALTA and, with the assumption of an underlying life distribution and using MLE, the parameters of this model can be obtained.

Indicator Variables

Another advantage of the models presented in this chapter is that they allow for simultaneous analysis of continuous and categorical variables. Categorical variables are variables that take on discrete values such as the lot designation for products from different manufacturing lots. In this example, lot is a categorical variable, and it can be expressed in terms of indicator variables. Indicator variables only take a value of 1 or 0. For example, consider a sample of test units. A number of these units were obtained from Lot 1, others from Lot 2, and the rest from Lot 3. These three lots can be represented with the use of indicator variables, as follows:

• Define two indicator variables, [math]\displaystyle{ {{X}_{1}} }[/math] and [math]\displaystyle{ {{X}_{2}}. }[/math]

• For the units from Lot 1, [math]\displaystyle{ {{X}_{1}}=1, }[/math] and [math]\displaystyle{ {{X}_{2}}=0. }[/math]

• For the units from Lot 2, [math]\displaystyle{ {{X}_{1}}=0, }[/math] and [math]\displaystyle{ {{X}_{2}}=1. }[/math]

• For the units from Lot 3, [math]\displaystyle{ {{X}_{1}}=0, }[/math] and [math]\displaystyle{ {{X}_{2}}=0. }[/math]

Assume that an accelerated test was performed with these units, and temperature was the accelerated stress. In this case, the GLL relationship can be used to analyze the data. From the GLL relationship we get:

[math]\displaystyle{ L(\underline{X})={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}{{X}_{1}}+{{\alpha }_{2}}{{X}_{2}}+{{\alpha }_{3}}{{X}_{3}}}} }[/math]

where:
• [math]\displaystyle{ {{X}_{1}} }[/math] and [math]\displaystyle{ {{X}_{2}} }[/math] are the indicator variables, as defined above.

• [math]\displaystyle{ {{X}_{3}}=\tfrac{1}{T}, }[/math] where [math]\displaystyle{ T }[/math] is the temperature.

The data can now be entered in ALTA and, with the assumption of an underlying life distribution and using MLE, the parameters of this model can be obtained.