Using JMP's Nonlinear Fit Platform to do Weibull Regression with Voltage and Temperature Stresses

by Barry Eggleston

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The following tutorial includes:

Building the negative loglikelihood function for logweibull (Gumbel) distributions.
How one can use JMP's nonlinear fit platform to find maximum likelihood estimates of the model parameters.
Fit a regression model with voltage and temperature stresses and right censored data.

In section 19.3.3 of Statistical Methods for Reliability Data by William Q. Meeker and Luis A. Escobar, a Tantalum
Electrolytic Capacitor data set is analyzed. Two models are considered. One with voltage and temperature effects, another
which in addition includes their interaction. The data contains the times to failure or removal. The times are in hours, the
temperature is measured in the centigrade scale. The following demonstration shows how one can use JMP's nonlinear platform to estimate the parameters of the no-interaction model. Click here for the description of the no-interaction model given in tutorial "Weibull Regression with voltage and temperature stresses". We can also do the interaction model by adding the appropriate crossterm, but we don't do this for the sake of simplicity. It is assumed that you have gone through the tutorial "Weibull Regression with voltage and temperature stresses."

The voltage levels are 35.0, 40.6, 46.5, 51.5, 57.0, and 62.5 Volts.

The temperature levels are 5, 45, and 85 degrees C.

To begin using JMP's nonlinear fit platform to estimate the mle's for the model, open the Tantalum Electrolytic Capacitors data set. Or open up the JMP data table created and saved in the "Weibull Regression with voltage and temperature stresses" tutorial.

On page 512 of Meeker and Escobar the distribution of the data is considered. They decide on using weibull errors, so in the following analysis we will assume the errors are weibull. Their decision was based on a weibull probability plot for the individual combinations of voltage and temperature. One can use JMP to perform this non-parametric approach to distribution identification. See web tuturial "Accelerated Life Model: Distribution Identification" by Ramon Leon for details. The ability of this method to identify distributions improves as the number of failures occurring at each voltage/temperature combination increases. Also see tutorial "Weibull Regression with voltage and temperature stresses" from a weibull plot of the data used in this tutorial.

In the tutorial "Weibull Regression with voltage and temperature stresses", JMP's Survival fit platform was used to estimate the model parameters of the no-interaction model.

In order to use JMP's nonlinear platform for mle estimation of a Weibull regression model with voltage and temperature stresses, one needs to create a column that contains summands of the negative loglikelihood for the Gumbel distribution. I called the column "GumbelLoss". The GumbelLoss column will contain the following formula:

Notice how mu reflects the assumed no-interaction model.

Create the GumbelLoss column, and enter the above formula. See Creating the GumbelLoss Column for a tutorial on creating the above formulas.

For all initial values use estimates calculated from the Survival fit platform. See tutorial, "Weibull Regression with voltage and temperature stresses".

Once the loss function has been built, click on "Evaluate", and exist the calculator window by clicking on the "X" button in the upper right hand corner of the calculator window.
Note: It might be a good idea to save the file at this time. A lot of work was put into building the GumbelLoss column, be safe and save the JMP data table.

The data table should look like (only part of the data table is shown):

Now click on "Analyze", then click on "Nonlinear Fit"

The Nonlinear fit window will appear:

Enter the GumbelLoss column as the Loss function by clicking on the GumbelLoss column, then clicking on the "Loss" button.
Enter the Frequency column as a frequency by clicking on "Frequency" then clicking on the "Freq" button
Click on OK

The nonlinear fitting window will appear.

Click on the box next to "Loss is negative loglikelihood"
Click on "Reset" and "Go".

A "Confid Limits" button will appear. Right below it, a message that the routine has converged on a solution is written. Click on the "Confid Limits" button and likelihood ratio confidence limits will be calculated.

After the likelihood confidence limits are calculated, the results are as follows:

Right under "SSE" is the opposite value of the loglikelihood function for logdata at the mle's. So the maximum of the loglikelihood function for logdata is about -246.773.

In the solution report we have the mle's, and their approximate standard error. Go to the regression output window in tutorial one "Weibull Regression with voltage and temperature stresses" for comparison of results. The results are the same. The confidence limits are not based on the standard error, but are likelihood ratio intervals. The above confidence limits are not the same as those in Table 19.6 of Meeker and Escobar, because the confidence intervals in Table 19.6 are based on standard errors. These likelihood ratio confidence intervals are the extreme points of a subspace within the likelihood space. All combinations of parameter estimates on the surface of this subspace produce a likelihood value of -248.694. This value is located right below the maximum likelihood value at the mle's.

The "Correlation of Estimates" is the reason for the extra work of this tutorial. Using the ApproxStdErr and these correlations, we can calculate delta method confidence intervals for quantiles at ambient stress.

Click here and go to the "Calculating Delta Method Confidence Intervals for quantiles at ambient stress" tutorial.