In the Devalt.jmp data, units are stressed by heating, in order to make them fail soon enough to obtain enough failures to fit the distribution.
Note: The data in Devalt.jmp comes from Meeker & Escobar (1998, p. 493).
1.
Open the Devalt.jmp sample data table, located in the Reliability folder.
2.
Select Analyze > Fit Y by X.
3.
Select Hours and click Y, Response.
4.
Select Temp and click X, Factor.
5.
Bivariate Plot of Hours by Log Temp
6.
Select Analyze > Reliability and Survival > Survival.
7.
Select Hours and click Y, Time to Event.
8.
Select Censor and click Censor.
9.
Select Temp and click Grouping.
10.
Select Weight and click Freq.
11.
12.
From the red triangle menu, select LogNormal Plot and LogNormal Fit.
Lognormal Plot
13.
Select Analyze > Reliability and Survival > Fit Parametric Survival.
14.
Select Hours and click Time to Event.
15.
Select x and click Add.
16.
Select Censor and click Censor.
17.
Select Weight and click Freq.
18.
Change the Distrib type to Lognormal.
19.
Click Run.
Devalt Parametric Output
20.
From the red triangle menu, select Estimate Survival Probability.
21.
Estimating Survival Probabilities
22.
Survival Probabilities
The ICdevice02.jmp data shows failures that were found to have happened between inspection intervals. The model uses two y-variables, containing the upper and lower bounds on the failure times. Right-censored times are shown with missing upper bounds.
Note: The data in ICdevice02.jmp comes from Meeker & Escobar (1998, p. 640).
1.
Open the ICdevice02.jmp sample data table, located in the Reliability folder.
2.
Select Analyze > Reliability and Survival > Fit Parametric Survival.
3.
Select HoursL and HoursU and click Time to Event.
4.
Select Count and click Freq.
5.
Select x and click Add.
6.
Click Run.
ICDevice Output
Note the following about the Tobit2.jmp data table:
Age and Liquidity are independent variables.
The table also includes the model and tobit loss function. The model in residual form is durable-(b0+b1*age+b2*liquidity). To see the formula associated with Tobit Loss, right-click on the column and select Formula.
1.
Open the Tobit2.jmp data table, located in the Reliability folder.
2.
Select Analyze > Modeling > Nonlinear.
3.
Select Model and click X, Predictor Formula.
4.
Select Tobit Loss and click Loss.
5.
6.
7.
Click Confidence Limits.
Solution Report
1.
Open the Tobit2.jmp sample data table, located in the Reliability folder.
2.
Select Analyze > Reliability and Survival > Fit Parametric Survival.
3.
Select YLow and YHigh and click Time to Event.
4.
Select age and liquidity and click Add.
5.
Change the Distrib type to Lognormal.
6.
Click Run.
Tobit Model Results
1.
Open the VA Lung Cancer.jmp sample data table.
2.
Select Analyze >Modeling >  Nonlinear.
3.
Select Model and click X, Predictor Formula.
4.
5.
Initial Parameter Values in the Nonlinear Fit Control Panel
6.
Click Save Estimates.
The Weibull column contains the Weibull formula, explained in Weibull Loss Function.
7.
Select Analyze >Modeling >  Nonlinear again.
8.
Select Model and click X, Predictor Formula.
9.
Select Weibull loss and click Loss.
10.
The Nonlinear Fit Control Panel on the left in Nonlinear Model with Custom Loss Function appears. There is now the additional parameter called sigma in the loss function. Because it is in the denominator of a fraction, a starting value of 1 is reasonable for sigma. When using any loss function other than the default, the Loss is Neg LogLikelihood box on the Control Panel is checked by default.
11.
Nonlinear Model with Custom Loss Function
12.
(Optional) Click Confidence Limits to show lower and upper 95% confidence limits for the parameters in the Solution table.
Solution Report
The Loss Function Templates folder has templates with formulas for exponential, extreme value, loglogistic, lognormal, normal, and one-and two-parameter Weibull loss functions. To use these loss functions, copy your time and censor values into the Time and censor columns of the loss function template. To run the model, select Nonlinear and assign the loss column as the Loss variable. Because both the response model and the censor status are included in the loss function and there are no other effects, you do not need a prediction column (model variable).
The Fan.jmp data table can be used to illustrate the Exponential, Weibull, and Extreme value loss functions discussed in Nelson (1982). The data are from a study of 70 diesel fans that accumulated a total of 344,440 hours in service. The fans were placed in service at different times. The response is failure time of the fans or run time, if censored.
Tip: To view the formulas for the loss functions, in the Fan.jmp data table, right-click the Exponential, Weibull, and Extreme value columns and select Formula.
1.
Open the Fan.jmp data table, located in the Reliability folder.
2.
Select Analyze > Modeling > Nonlinear.
3.
Select Exponential and click Loss.
4.
5.
Make sure that the Loss is Neg LogLikelihood check box is selected.
6.
7.
Click Confidence Limits.
8.
Repeat these steps, selecting Weibull and Extreme value instead of Exponential.
Nonlinear Fit Results
The Locomotive.jmp data can be used to illustrate a lognormal loss. The lognormal distribution is useful when the range of the data is several powers of e.
Tip: To view the formula for the loss function, in the Locomotive.jmp data table, right-click on the logNormal column and select Formula.
The lognormal loss function can be very sensitive to starting values for its parameters. Because the lognormal distribution is similar to the normal distribution, you can create a new variable that is the natural log of Time and use Distribution to find the mean and standard deviation of this column. Then, use those values as starting values for the Nonlinear platform. In this example, the mean of the natural log of Time is 4.72 and the standard deviation is 0.35.
1.
Open the Locomotive.jmp sample data table, located in the Reliability folder.
2.
Select Analyze > Modeling > Nonlinear.
3.
Select logNormal and click Loss.
4.
5.
6.
Click Confidence Limits.
Solution Report
The maximum likelihood estimates of the lognormal parameters are 5.11692 for Mu and 0.7055 for Sigma (in natural logs). The corresponding estimate of the median of the lognormal distribution is the antilog of 5.11692 (e5.11692), which is approximately 167. This represents the typical life for a locomotive engine.