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8. Assessing Product Reliability
8.4. Reliability Data Analysis
8.4.1. How do you estimate life distribution parameters from censored data?

8.4.1.3.

A Weibull maximum likelihood estimation example

Reliability analysis with a popular statistical software package SAS JMPTM Example

SAS JMP software has excellent survival analysis (i.e., reliability analysis) capabilities, incorporating both graphical plotting and maximum likelihood estimation and covering the exponential, Weibull, lognormal and extreme value distribution models. 

Use of JMP (Release 3) for plotting Weibull censored data and estimating parameters will be illustrated using data from a previous example

Steps in a Weibull analysis using JMP software Weibull Data Example

Failure times were 55, 187, 216, 240, 244, 335, 361, 373, 375, and 386 hours, and 10 unfailed units were removed from test at 500 hours. The steps in creating a JMP worksheet and analyzing the data are as follows:

1. Set up three columns, one for the failure and censoring times ("Time"), another to indicate whether the time is a failure or a censoring time ("Cens") and the third column to show how many units failed or were censored at that time ("Freq"). Fill in the 11 times above, using "0" in Cens to indicate a failure and "1" in Cens to indicate a censoring time. The spreadsheet will look as follows:

JMP worksheet showing data for Weibull example

You can obtain a copy of this JMP worksheet by clicking here mleex.jmp . If your browser is configured to bring up JMP automatically, you can try out the example as you read about it.

2. Click on Analyze, choose "Survival" and then choose "Kaplan - Meier Method". Note: Some software packages (and other releases of JMP) might use the name "Product Limit Method" or "Product Limit Survival Estimates" instead of the equivalent name "Kaplan-Meier". 

3. In the box that appears, select the columns from mleex that correspond to "Time", "Censor" and "Freq", put them in the corresponding slots on the right (see below) and click "OK". 

JMP worksheet showing Kaplan-Meier menu

4. Click "OK" and the analysis results appear. You may have to use the "check mark" tab on the lower left to select Weibull Plot (other choices are Lognormal and Exponential). You may also have to open the tab next to the words "Weibull Plot" and select "Weibull Estimates". The results are shown below. 

JMP worksheet showing analysis of Weibull data, shows Weibull plot, parameter estimates for extreme value and Weibull MLE

Note: JMP uses the parameter alpha for the Weibull characteristic life (as does Dataplot), and the parameterbeta for the shape (Dataplot uses gamma). The Extreme Value distribution parameter estimates are for the distribution of "ln time to fail" and have the relationship 

lambda = LN(alpha), delta = 1/beta;   Dataplot uses mu instead of lambda and beta instead of delta

5. There is an alternate way to obtain some of the same results, which can also be used to fit models when there are additional "effects" such as temperature differences or vintage or plant of manufacturing differences. Instead of clicking "Kaplan - Meier Method" in step 2, chose "Parametric Model" after selecting "Survival" from the "Analysis" choices. The screen below appears. Repeat step 3 and make sure "Weibull" appears as the "Get Model" choice. In this example there are no other effects to "Add" (the acceleration model example later on will illustrate how to add a temperature effect). Click "Run Model" to obtain the results below. This time, you need to use the check symbol tab to obtain confidence limits. Only the Extreme Value distribution parameter estimates are displayed. 

JMP worksheet showing extreme value model fit

Limitations and a warning about the Likelihood calculation in JMP Notes

1. The built in reliability analysis routine that iscurrently part of JMP only handles exact time of failure data with possible right censoring. However, the use of templates (provided later in the Handbook) for either Weibull or lognormal data extends JMP analysis capabilities to handle readout (interval) data and any type of censoring or truncation. This will be described in the acceleration model example later on. 

2. The "Model Fit" screen for the Weibull model gives a value for -Loglikelihood for the Weibull fit. This should be the negative of the maximized likelihood function. However, JMP leaves out a term consisting of the sum of all the natural logarithms of the times of failures in the data set. This does not affect the calculation of MLE's or confidence bounds but can be confusing when comparing results between different software packages. In the example above, the sum of the ln times is ln 55 + ln 187 + . . . + ln 386 = 55.099 and the correct maximum log likelihood is - (20.023 + 55.099) = - 75.122. 

3. The omission of the sum of the ln times of failures in the likelihood also occurs when fitting lognormal and exponential models. 

4. Different releases of JMP may, of course, operate somewhat differently. The analysis shown here used release 3.2.2.

Conclusions

MLE analysis is an accurate and easy way to estimate life distribution parameters, provided that a good software analysis package is available. The package should also calculate confidence bounds and loglikelihood values. JMP has this capability, as do several other commercial statistical analysis packages.

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