Reliability HotWire: eMagazine for the Reliability Professional
Reliability HotWire

Issue 53, July 2005

Reliability Basics

Using Categorical Variables in Accelerated Life Testing and Life-Stress Analysis

While ALTA Pro allows the use of numerical (continuous or discrete) variables, it also allows you to incorporate factors that represent categories. Examples include a product used in four different geographical locations, a product produced in three different manufacturing sites or a product that uses material supplied by two different suppliers. These types of variables are called categorical or qualitative variables.

It is important to note that categorical variables should not be represented (coded) the same way discrete variables are incorporated into a life-stress model. For example, Supplier A, Supplier B, Supplier C should not be coded as 1, 2, 3. Such coding would suggest some sort of ordering (suggesting that one supplier has a higher value than another) among the different suppliers and would lead to misleading results.

Indicator variables (Xi) are used to represent categorical variables. Indicator variables only take a value of 1 or 0. There are multiple appropriate schemes for coding a categorical factor using indicators. A categorical variable that has k categories is represented with k-1 indicator variables. The most common way to use indicator variables is as follows: For each category i within a categorical factor, Xi = 1 and Xj = 0, for all j i and j k-1.

Note also that it is preferable to use actual numeric measures that describe a variable and that categorical variables should be dedicated to variables that cannot be quantified easily. For example, if temperature is the main distinguishable factor describing location, then instead of considering location as a categorical variable and using coding to describe Location A, Location B, Location C and Location D, it would be better to describe the locations with their respective temperatures. Using numerical variables allows extrapolations and interpolations of the life estimate, whereas using categorical factors only allows predictions for the specific studied categories.

Example

A manufacturing site has three different lots. The manufacturer wants to determine the effect of the different lots on the life of the product. An accelerated test was conducted by selecting a number of these units from Lot 1, Lot 2 and Lot 3 and testing them to failure under elevated temperature levels. These three lots can be represented with the use of indicator variables as follows.

       Define two indicator variables, X1 and X2.
       For the units from Lot 1,
X1 = 1 and X2 = 0.
       For the units from Lot 2,
X1 = 0 and X2 = 1.
       For the units from Lot 3,
X1 = 0 and X2 = 0.

An accelerated test was performed with these units and temperature was the accelerated stress. In this case, the General Log-Linear model can be used to analyze the data. (Note that the General Log-Linear model is available only in ALTA Pro, not in ALTA Standard.) From the equation of the General Log-Linear model, we obtain the following relationship:

where:

  • α0 , α1, α2 , α3  are model parameters.
  • X1 and X2 are the indicator variables, as defined above.
  • X3 =1/T  where T is the temperature.
  • X is a vector of n stresses.

For this example, three columns are needed: one column for the Temperature variable and two columns for the Lots factor. In ALTA, the number of columns can be defined when creating the project as follows.

The data are shown next (the header names on the two Other columns were changed to X1 and X2).

After selecting the General Log-Linear model as the choice of life-stress relationship, choose Select Stress Column(s) from the Data menu. In the window that appears, select the stress columns to be used in the calculations as follows.

Next, choose Stress Relations from the Data menu. The stress relationships are defined as follows.

Using Weibull as the life distribution, the parameter values are β = 13.1469, α0  = -1.0066, α1 = 2394.9334, α2  = 0.3646 and α3  = 0.5316.

To calculate metrics that need settings for the different variables, the same scheme presented above for coding the categorical variable can be used. For example, to calculate the Mean Life for Temperature = 300K and Lot = Lot 1 (X1 = 1 and X2 = 0), the variables values could be set in the QCP as follows to calculate the mean life.

The expected mean life in this case is 1483.01 hr.

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