Reliability and Survival Methods • Survival Analysis • Survival Platform Options • Exponential, Weibull, and Lognormal Plots and Fits
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Exponential, Weibull, and Lognormal Plots and Fits
For each of the three distributions that JMP supports, there is a plot command and a fit command. Use the plot command to see whether the event markers seem to follow a straight line. The markers tend to follow a straight line when the distributional fit is suitable for the data. Then, use the fit commands to estimate the parameters.
Exponential, Weibull, and Lognormal Plots and Reports
The following table shows what to plot to make a straight line fit for that distribution:
Straight Line Fits for Distribution 
Distribution Plot
X Axis
Y Axis
Interpretation
Exponential
time
-log(S)
slope is 1/theta
Weibull
log(time)
log(-log(S))
slope is beta
Lognormal
log(time)
Probit(1-S)
slope is 1/sigma
Note: S = product-limit estimate of the survival distribution.
The exponential distribution is the simplest, with only one parameter, called theta. It is a constant-hazard distribution, with no memory of how long it has survived to affect how likely an event is. The parameter theta is the expected lifetime.
The Weibull distribution is the most popular for event-time data. There are many ways in which different authors parameterize this distribution (as shown in Various Weibull Parameters in Terms of JMP’s alpha and beta). JMP reports two parameterizations, labeled the lambda-delta extreme value parameterization and the Weibull alpha-beta parameterization. The alpha-beta parameterization is used in the reliability literature. See Nelson (1990). Alpha is interpreted as the quantile at which 63.2% of the units fail. Beta is interpreted as follows: if beta>1, the hazard rate increases with time; if beta<1, the hazard rate decreases with time; and if beta=1, the hazard rate is constant, meaning it is the exponential distribution.
Various Weibull Parameters in Terms of JMP’s alpha and beta 
JMP Weibull
alpha
beta
Wayne Nelson
alpha=alpha
beta=beta
Meeker and Escobar
eta=alpha
beta=beta
Tobias and Trindade
c = alpha
m = beta
Kececioglu
eta=alpha
beta=beta
Hosmer and Lemeshow
exp(X beta)=alpha
lambda=beta
Blishke and Murthy
beta=alpha
alpha=beta
Kalbfleisch and Prentice
lambda = 1/alpha
p = beta
JMP Extreme Value
lambda=log(alpha)
delta=1/beta
Meeker and Escobar s.e.v.
mu=log(alpha)
sigma=1/beta
The lognormal distribution is also very popular. This is the distribution where if you take the log of the values, the distribution is normal. If you want to fit data to a normal distribution, you can take the exp() of it and analyze it as lognormal. See Additional Examples of Fitting Parametric Survival in Fit Parametric Survival.
Additional Options
To see additional options for the exponential, Weibull, and lognormal fits, hold down the Shift key, click the red triangle of the Product-Limit Survival Fit menu, and click on the desired fit.
Use these options to:
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Set the confidence level for the limits.
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Set the constrained value for theta (in the case of an exponential fit), sigma (in the case of a lognormal fit) or beta (in the case of a Weibull fit). See WeiBayes Analysis.
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Obtain a Confidence Contour Plot for the Weibull and lognormal fits (when there are no constrained values). See Confidence Contour Plot.
Confidence Contour Plot
WeiBayes Analysis
JMP can constrain the values of the Theta (Exponential), Beta (Weibull), and Sigma (LogNormal) parameters when fitting these distributions. This feature is needed in WeiBayes situations, for example:
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Where there are few or no failures
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There are existing historical values for beta
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There is still a need to estimate alpha
For more details about WeiBayes situations, see Abernethy (1996).
With no failures, the standard technique is to add a failure at the end and the estimates would reflect a type of lower bound on what the alpha value would be, rather than a real estimate. The WeiBayes feature allows for a true estimation.

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Help created on 9/19/2017