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General Log-Linear Model for Accelerated Life Data Analysis

Accelerated tests are becoming increasingly popular in today's industry due to the need to quickly obtain life data. However, the rate of increase of this popularity has been very slow because of the limited options that are currently available for performing adequate life data analyses on data obtained from accelerated tests. Most of the tools available today for accelerated life data analyses are limited to performing analyses on data obtained from tests of one or two stress variables. 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 other engineering variable is sought. The theory for such cases has been developed for some time now. The means of its application, however, has been a subject of wishful thinking.

This article presents a general model that incorporates most of the widely used life-stress relationships, such as Arrhenius and inverse power. The model can be used for single or multiple accelerated stresses. For example, when performing the analysis for data obtained in an accelerated test with multiple stress types, the engineer can now choose which types of stresses follow an inverse power relationship and which follow an exponential relationship. In addition, indicator and categorical variables, such as material and vendors, can also be included in the analysis. Isolating the effects of these factors becomes essential in estimating their influence. These factors must be identified and quantified by numeric variables known as covariates, diagnostic or explanatory variables. The new version of the ALTA software, ALTA PRO, includes this flexible new model, the general log-linear relationship. ALTA PRO will be used in analyzing a sample data set.

General Log-Linear Relation
The formulation of the general log-linear model begins with the assumption of a log-linear relation for the characteristic life, tp, as follows:

The coefficients of this equation are the parameters of the life-stress relationship. The variables x1, x2,…, xm are the covariates. These covariates can represent indicator variables, categorical variables, stress, or some transformation of stress. This equation can be rewritten in a more compact form as:

Or, in terms of the characteristic life:

The characteristic life, tp, can represent any percentile. The percentile is chosen according to the assumed underlying life distribution. For the Weibull, exponential and lognormal distributions, the characteristic lives are (eta), mean and median life respectively.

The advantage of representing the characteristic life with the log-linear relationship is that relationships such as the Arrhenius and inverse power models can be assumed for the covariates by performing a simple transformation. The appropriate transformations for some widely used life-stress relationships are given in Table 1.

Example
To illustrate the use of the general log-linear model in accelerated life data analysis, consider the data summarized in Table 2 and Table 3. This data set represents a typical three stress type accelerated test.

The data from Table 3 will be analyzed assuming a Weibull distribution, an Arrhenius life-stress relationship for temperature, and an inverse power life-stress relationship for voltage. No transformation is performed on the operation type. The operation type variable is treated as an indicator variable, taking the discrete values of 0 and 1 to represent on/off and continuous operation, respectively.

The general log-linear equation (in terms of the characteristic life) then becomes:

The resulting relationship after performing these transformations is:

Therefore, the parameter B of the Arrhenius relationship is equal to the log-linear coefficient , and the parameter n of the inverse power relationship is equal to . Therefore can also be written as:

The activation energy of the Arrhenius relationship can be calculated by multiplying B with Boltzmann's constant. The best fit values for the parameters in this case are:

= 3.74830;

= -6.02201; = 5776.94047; = -1.43404; = 0.62424

Once the parameters are estimated, we can perform further analysis on the data. Figures 1, 2, 3 and 4 display probability and life vs. stress plots generated in ALTA PRO.

First, we generate a Weibull probability plot of the data, as shown in Figure 1. We can obtain several types of information about the model as well as the data from a probability plot. For example, we can examine the choice of an underlying distribution and the assumption of a common slope (shape parameter). In this example, the linearity of the data supports the use of the Weibull distribution. In addition, the data appear parallel on this plot, therefore reinforcing the assumption of a common beta. Further statistical analysis can and should be performed for these purposes as well.

 Figure 1: Weibull probability plot for all covariates

The life vs. stress plot is a very common plot for the analysis of accelerated data. Life vs. stress plots can be very useful in assessing the effect of each stress on a product's failure. In this case, since the life is a function of three stresses, we can create three different plots, displayed in Figures 2, 3 and 4. Such plots are created by holding two of the stresses constant at the desired use level and varying the remaining one. The use stress levels for this example are 328K for temperature and 10V for voltage. For the operation type, the engineers must make a decision as to whether they implement on/off or continuous operation. In these plots, "on" (1) is considered to be the use level for operation type.

Figure 2 displays the effect of temperature on life. The voltage and operation type are held constant at the use stress levels. Figure 3 displays the effect of voltage on life. The effects of the two different operation types on life can be observed in Figure 4. It can be seen that the on/off cycling has a greater effect on the life of the product in terms of accelerating failure than the continuous operation. In other words, a higher reliability can be achieved by running the product continuously.

 Figure 2: Effects of temperature on life

 Figure 3: Effects of voltage on life

 Figure 4: Effect of operation type on life

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