Compare Distributions in Life Distribution
In the Life Distribution platform, the Compare Distribution report enables you to fit and compare different failure time distributions. The report contains two lists.
Distribution
Select a distribution for the response. Different distributions appear based on characteristics of the data. For more information about which distributions are available, see Available Parametric Distributions.
Scale
Select a scale for the probability axis. The probability scale corresponds to the distribution listed to the left of the Scale button. Using this scale, the fitted model is represented by a line. Suppose that you fit a given distribution and then scale the axis using that distribution. If the points generally fall along a line, this indicates that the distribution provides a reasonable fit.
The Life Distribution platform uses the midpoint estimates of the nonparametric Kaplan-Meier step function to construct probability plots. The default plot shows the midpoint estimates of the step function for the uncensored data values and their confidence intervals. The confidence intervals are indicated by horizontal blue lines.
Tip: To customize the plot of the nonparametric estimates, select File > Preferences > Platforms > Life Distribution and select one or more of the following preferences: Show Shaded Pointwise Intervals, Show Shaded Simultaneous Intervals, or Show Staircase Style Function.
By default, there is a panel at the top of the plot that displays times of right-censored observations.
Tip: To hide the panel that contains markers for right-censored observations, select File > Preferences > Platforms > Life Distribution and uncheck Show Markers for Right Censored Observations.
For each distribution that you select, the Compare Distributions report is updated to show the following:
• the estimated cumulative distribution curve, which appears on the probability plot
• a shaded region that indicates confidence intervals for the cumulative distribution
• a Distribution Profiler that shows the cumulative probability of failure for a given period of time
Figure 3.9 shows an example of the Compare Distributions report. The Logistic (yellow) and Exponential (magenta) distributions are shown. The plot is scaled using the Exponential distribution.
Figure 3.9 Compare Distributions Report and Distribution Profiler
Available Parametric Distributions
This section addresses the distributions available in the Compare Distributions report.
Note: Distributions for the Competing Cause report are covered in Available Distributions for Competing Cause Compare Distributions Reports.
The available distributions are listed and described in detail in Parametric Distributions. There are four major groupings of parametric distributions:
• Basic Failure-Time Distributions
• Defective Subpopulation Distributions
Tip: To restrict which distributions are available by default, select File > Preferences > Platforms > Life Distribution and uncheck the distributions that you do not want to appear. The distributions listed include the Threshold, Defective Subpopulation, Zero-Inflated, LogGenGamma, and GenGamma distributions. By default, all of the distributions are checked and available.
The rules that determine which distributions appear in the Compare Distributions panel depend on the particular implementation. As a general guide, distributions are available if they are not disabled in Preferences and if they are appropriate in the given situation.
Basic Failure-Time Distributions
The basic failure-time distributions are available whenever all failure times are positive. They include the following:
• Lognormal
• Weibull
• Loglogistic
• Fréchet
• Normal
• SEV
• Logistic
• LEV
• Exponential
• LogGenGamma
• GenGamma
Note: When there are negative or zero failure times, only the Normal, SEV, Logistic, LEV, and LogGenGemma are available.
Threshold Distributions
The threshold (TH) distributions are always available. Threshold distributions are log-location-scale distributions with threshold parameters. The threshold parameter shifts the distribution away from 0. These distributions assume that all units survive until the threshold value. Threshold distributions are useful for fitting moderate to heavily shifted distributions. The threshold distributions are the following:
• TH Lognormal
• TH Weibull
• TH Loglogistic
• TH Fréchet
Defective Subpopulation Distributions
The defective subpopulation (DS) distributions are available when all failure times are positive. These distributions are useful when only a fraction of the population has a particular defect leading to failure. Use the DS distribution options to model failures that occur on only a subpopulation. The DS distributions are the following:
• DS Lognormal
• DS Weibull
• DS Loglogistic
• DS Fréchet
Zero-Inflated Distributions
When the time-to-event data contain zero as the minimum value in the Life Distribution platform, the following zero-inflated distributions are available:
• Zero-Inflated Lognormal (ZI Lognormal)
• Zero-Inflated Weibull (ZI Weibull)
• Zero-Inflated Loglogistic (ZI Loglogistic)
• Zero-Inflated Fréchet (ZI Fréchet)
Zero-inflated distributions are used when some proportion of units fails at time zero. When the data contain more zeros than expected by a standard model, the number of zeros is inflated.
Zero-Failure Data
In the case of zero-failure data, none of the above distributions are available by default. To obtain Bayesian fits for those distributions where the Bayesian Estimate option is available, select File > Preferences > Platforms > Life Distribution and uncheck Weibayes Only for Zero Failure Data. See Weibayes Only for Zero Failure Data.
Parametric Distributions That Allow Bayesian Estimation
Bayesian estimation is available for the following parametric distributions:
• Lognormal
• Weibull
• Loglogistic
• Fréchet
• Normal
• SEV
• Logistic
• LEV
A list of distributions that are available as priors for hyperparameters of these distributions is given in Prior Distributions for Bayesian Estimation.