The following tutorial includes:

• How one can use JMP's Survival fit platform to fit a regression model with voltage and temperature stresses and right censored data.

In section 19.3.3 of Meeker and Escobar the distribution of the data is considered.  They decide to use weibull errors, so in the following analysis we will assume the errors are weibull.
The no-interaction model is:

Here G is a Gumbel random variable with location parameter 0 and scale parameter.  Hence Log(T) is Gumbel with location parameter  and scale parameter .  It follows that T has a weibull distribution with scale parameter  and shape parameter .  In Statistical Methods for Reliability Data, the Gumbel distribution is identified as the Smallest Extreme Value Distribution.

 In the formula for  , equals the activation energy, in electron volts, eV, and 11605 is the inverse of Boltzmann's constant in electron volts per degrees C.
Using, in the formula for is a result of the Arrhenius relationship.
 

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 C.

To get a copy of the data  right click on the hyperlink below, and then click on "Save Link As..." to get a local copy of the data set.  If one opens the data set by clicking on the hyperlink, go to "File|Save" as to save a local copy.

Click here to open a text version of the Tantalum Electrolytic Capacitors data set.

The JMP table should look as follows (this picture shows only part of the data set):

A look at the combinations of voltage and temperature will show a lack of balance in the data.  Censor code 0 means a failure.  Censor code 1 means a removal time.  The frequency column gives the number of capacitors that failed or were removed at a given time.  The next tutorial uses the same data set, so save the JMP data table before you quit this tutorial.

The decision to use weibull errors 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.

Below is a weibull plot generated as described above:

Each line represents a temperature voltage combination.
The steep blue line for 85C and 46.5 volts, labeled as 3, is a lot different than the others; however, this combination only had 2 out of 50 failures.  As mentioned by Meeker/Escobar the slope of this line could be due to random variation.
 

To use JMP to estimate the coefficents of the model, one needs to create columns that contain transformations of voltage and temperature.

One can also see JMP's User's Guide p76-78 for assistance on column creation, and chapter 5 of the same manual for assistance on formula creation using the JMP calculator.

When done the data table will look as follows: (we have changed the column label to F in the frequency column)

When two new columns have been create, enter the following formulas in the appropriate column:

Now JMP's Survival fit platform can be used to estimate the model parameters.

Click on Analyze, then click on Survival.  The Survival Time modeling window will appear.

Click on the "Parametric Model.." option button.

The Survival Model window will appear.

The above screen shot was obtained by doing the following:

• Click on FailureTime in the column list, then click on the Time button.
• Click on Censor in the column list, then click on the Censor button.
• Click on Frequency in the column list, then click on the Freq button.
• Click on LogVoltage, then click on the Add button.
• Click on TempTrans, then click on the Add button.

Click on "Run Model" button, and a model fit window will appear,

The whole model test report within this window, is the result of a likelihood ratio test between the following models:

Reduced model, Ho: 
mu and sigma are independent of voltage and temperature.
 

Full model, H1:
where 
mu is a function of voltage, and temperature. sigma is independent of voltage and temperature.
 
 

Discription of the likelihood ratio test produced in the whole model test output:

2*[L(f) - L(r)] is approximately distributed as chi-square with degrees of freedom equal to the number of restrictions.
2[-246.773495-(-280.55842)]=67.56985.

This is the Chi-square value in the Whole-Model test output is calculated.
The p-value of something less than 0.0001 is evidence that the full model represents the data much better than the restricted model.
 

Click on the check box at the lower left-hand corner of the model fit window, and select "Likelihood Ratio Tests".  Click on the same check box again and select "Confidence Intervals".

Resize the window so all of the report is visible, and the results should look like:


 

• The parameter Estimates section contains approx. 95% confidence intervals for coefficient estimates.
• The estimated coefficient for logVoltage is -20.09, ie the estimate of .
• The approx. 95%confidence interval for the coefficient of logVoltage is (-30.44, -12.27), therefore we have evidence that Voltage affects the failure times of Tantalum Electrolytic Capacitors.
• The estimated coefficient for TempTrans is 0.33, ie the estimate of .  The confidence interval for TempTrans is (-0.06, 0.71), so there is little statistical evidence that temperature affects the failure times of Tantalum Electrolytic Capacitors.  Yet, the coefficient for TempTrans is the activation energy and theory says it is non-negative, so the negative endpoint for the confidence interval is an approximation error.  See Meeker and Escobar page 514 for a discussion.
• On page 515 of Meeker and Escobar Table 19.6 contains values for the parameter estimates, standard errors, and approx. 95% confidence intervals.  The confidence limits given by  JMP output above are not the same as the one's given in Table 19.6.  The confidence limits given by JMP are likelihood ratio limits.  The confidence intervals given in Table 19.6 are based on the standard errors.
• Delta is the estimated value for the gumbel parameter sigma.

To fit the interaction model do the following:

1. Click on Analyze, then click on Survival.  The Survival Time modeling window will appear.
2. Click on the "Parametric Model.." option button.  The Survival Model window will appear.
3. Enter the appropriate information, including the crossterm.  The Survival Model window should look like:

Drawback of this approach to regression modeling