Analysis of Failure Data with Multiple Types of Censoring

The following demonstration includes:

• modeling lognormal failure data with left, right, and interval censoring using JMP's non-linear platform.
• The Arrhenius Relationship for temperature acceleration.
• Lognormal error assumption.

To get a copy of the data you can right click on the hyperlink below, and then click on "Save Link As..." to get a local copy of the data set.  If you open the data set by clicking on the hyperlink, you can go to "File|Save as" to save a local copy.
Click here to open a text version of the Integrated circuit device data set.

Once you import the data set into JMP, the JMP table should look like: (we have changed the column label to F in the frequency column)


Explaination of above table:
Use the Time1 column for left censored data, and the lower value of an interval censored observation.
Use the Time2 column for right censored data, and the upper value of an interval censored observation

For example:

• Row 1 has one recorded value in column Time2, none in columns Time1 or Exact, so it is a right censored observation(s).  This row records that 50 integrated circuit devices operating at 150 degrees C were right censored at 1536 hrs.
• Row 2 has one recorded value in column Time2, none in columns Time1 or Exact, so it is also a right censored observation(s).  This row records that 50 integrated circuit devices operating at 175 degrees C were right censored at 1536 hrs as well.
• Row 4 has two recorded values, 384 in column Time1, and 788 in column Time2, none is column Exact, so it is an interval censored observation(s).  This row records that one integrated circuit device operating at 250 degrees C failed between 384 and 788 hrs.
• Row 6 has only one recorded value which is in column Exact, so it is an exact failure time.
• Row 11 has one recorded value in column Time1, none in columns Time2 or Exact, so it is a left censored observation(s).  This row records that 2 integrated circuit devices operating at 300 degrees C failed before 192 hrs.

The missing data points identified by the ? mark will not create a problem.  The CensorCode column identifies the censoring type for the computer.  0 is an exact failure time.  1 is a right censored observation.  2 is a left censored observation.  3 is an interval censored observation.

The CensorCode column is used to direct the program to use data from a specific column for a specific row.  If the CensorCode is 0, then for that observation only data from the "Exact" column is used in the analysis.  If the CensorCode is 3, then for that observation only data from columns "Time1" and "Time2" are used.

To begin you will need to create a new column I called "Loss", and enter the following formula for the column.  See tutorial "Creating the GumbelLoss Column" from the tutorial "Using JMP's nonlinear fit platform to do weibull regression with voltage and temperature stresses" for an example of how to create new columns, and build formulas within them.  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.

Given our assumption of lognormal errors, we will model the log of the data.  Therefore, the formulas below contain the negative log of the normal pdf, survival, and cdf functions.  The "normDist(z)" function is located in the probability list.  See picture below the formulas for location of the "normDist(z)" function within the JMP calculator.  For the initial parameter values use 1 for both sigma and b0 (estimate of beta0).  Use 0 for b1 (estimate of beta1).  (If these initial values do not converge, one can obtain initial estimates by least squares.)

 



 

Once the loss function is created, the non-linear fit platform can be used to estimate the mle's of the model.

Click on the non-linear fit in the analysis window.
Enter the "Loss" column as the "loss", and enter the "Frequency" column as the "Freq".
Click on OK.


The following report will appear.

• Click on "Loss is -LogLikelihood",
• Click on Reset,
• Click on Go.

Once the algorthym has converged in the gradent, click on the "Confid Limits" button.  The Confidence limits will be calculated as below:





We have convergance on all parameter estimates and likelihood ratio confidence intervals; however, notice the large correlation between the b0 and b1 estimates. This is most likely due to failures occuring only at 250 degrees C, and 300 degrees C. We will calculate a confidence interval for b10 (10th quantile) at 150 degrees C, but the confidence interval will be suspect.

Next delta method confidence intervals for quantiles at ambient stress could be calculated using the above information as described in Calculating Delta Method Confidence Intervals for quantiles at ambient stress.  Calculate an approx. 95% confidence interval for the 10th quantile, b10, at 150 degrees C.  Remember, one first creates a confidence interval for ln(b10), (log of the 10th quantile), then exponentiates.  This prevents the confidence interval from having a negative lower end point.  Estimates of ln(b10), and the standard error of ln(b10) are needed.  For the calculation of the standard error, an estimate of the ln(b10) variance is calculated.

So the appropriate formulas are:

and


The -1.282 in the above formulas is the appropriate a_q value for the 10th quantile of a lognormal distribution.  See the table of appropriate a_q values in the Calculating Delta Method Confidence Intervals for quantiles at ambient stress tutorial.
The estimated ln(b10) is: 11.85562773
The estimated variance-covariance matrix is:
 

	sigma	b0	b1
sigma	0.0038199124	-0.0369661556	0.0018269169
b0	-0.0369661556	2.596206513	-0.1239393115
b1	0.0018269169	-0.1239393115	0.0059285426

The above table was calculated from the standard errors and the correlation matrix obtained from the JMP nonlinear platform output.  See Calculating Delta Method Confidence Intervals for quantiles at ambient stress tutorial for review of calculating the variance-covariance matrix from the standard errors and the correlation matrix.

Using the formula and table above:
The estimated std error for ln(b10) is the square root of 0.2304943848, or 0.4800983074
The delta method 95% confidence interval for ln(b10) at 150 degrees C is (11.85562773 +/- 1.96*0.4800983074) or (10.914635, 12.796620)

The estimated b10 is 140874.93 hours (16 yeas), and by exponentiating:  the approx.. 95% confidence interval of b10 is (54975.07 hrs, 360995.39 hrs), or (6.3 yrs, 41.2 yrs) .

However, what if one could use JMP to calculate a likelihood based confidence interval by reparameterizing the model in terms of b10 at 150 degrees.  Click here to see how to use JMP to calculate likelihood ratio confidence intervals.