frequency column)
For an explaination of the above data table, see tutorial "Analysis of failure data with multiple types of censoring."
For this tutorial, it is assumed one has completed the tutorial "Analysis of failure data with multiple types of censoring".
To complete this tutorial, open up the Integrated circuit device JMP
file created in the tuturial "Analysis
of failure data with multiple types of censoring", or get a copy of
the data by right clicking on the hyperlink below, and then click on "Save
Link As..." 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.
The JMP table should look like the table below (we have changed the
column label to F in the
frequency column)
For an explaination of the above data table, see tutorial "Analysis
of failure data with multiple types of censoring."
For a review of the model applied to the data in this tutorial, see "Analysis of failure data with multiple types of censoring."
In this tutorial we will calculate the likelihood ratio confidence interval
for the 10th quantile at a temperature of 150 degrees C.
Note: b10 is the 10th percentile at temperature equal to 150.
b0 and b1 are coefficients in the model for mu.
The formula for b10( 10th quantile) is:
Substituting the assumed model for muhat we have the following formula
for b10 at 150 degrees C:
Note: The 150 added to 273 in the denominator makes this a formula
for b10 at 150 degrees C. If we had 75 instead of 150, then the formula
would be for b10 at 75 degrees C.
Solving this formula for b0 (estimate of beta0) we have:
In the JMP formulation we can make b10 a parameter, choose a specific value of temperature (say 150 degrees C) and then make b0 a variable. This will allow us to estimate b10 at 150 degrees C, and calculate a 95% likelihood ratio confidence interval.
To begin you will need to create a new column which I call "b10Loss", and enter the following formula for the column. See tutorial "Creating the GumbelLoss Column" from "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.
For the initial parameter values use those calculated in "Analysis of failure data with multiple types of censoring." Sigma = 0.5898. b1 (estimate of beta1) = 0.85128. b10 =140874.93. b0 (beta0) now becomes a variable, not a parameter. Remember to enclose the denominator in parentheses before entering "Temperature + 273".
The formula is as follows:
Once the formula has been entered, click on "Evaluate" and exit the calculator window. This formula will allow us to estimate b10 at 150 degrees C, and calculate a 95% likelihood ratio confidence interval.
Go to the non-linear fit platform and enter the following information:
Click on OK
When the nonlinear window appears:
The following output results:
- Click on "loss is -loglikelihood"
- Click on Reset.
- Click on Go.
Click on the "Confid Limits" button, the Solution window will contain
likelihood ratio confidence intervals for sigma, b1, and b10 (10th percentile
at 150 degrees C.)
The approx. 95% likelihood ratio confidence interval for b10 at 150 degrees C is 57577.36 hrs to 393312.85 hrs, or 6.6 yrs to 44.9 yrs. With an estimated b10 life of 140874.93 hrs, or 16.1 yrs.
Compare this cofidence interval with the approx. 95% confidence interval of b10 obtained in tutorial "Analysis of failure data with multiple types of censoring."
To calculate a confidence interval for another value of temperature, go back to the b10 loss formula and change to 150 in the b0 variable statement to the desired level of temperature.
Return to index.