Publication date: 08/13/2020

Life Distribution: Statistics

The Statistics report includes the following sub-reports:

Model Comparisons

Summary of Data

Nonparametric Estimate

Parametric Estimate - <Distribution Name> (one report appears for each distribution that you select in the Compare Distributions report)

Model Comparisons

The Model Comparisons report provides the AICc, -2Loglikelihood, and BIC statistics for each fitted distribution. Smaller values of each of these statistics indicate a better fit. For more information about these statistics, see Likelihood, AICc, and BIC in Fitting Linear Models.

Initially, the rows are sorted by AICc. To change the statistic used to sort the report, click the Life Distribution red triangle and select Comparison Criterion. See Life Distribution Report Options for more information about this option.

Summary of Data

The Summary of Data report shows the total number of units observed, the number of uncensored units, and the numbers of right-censored, left-censored, and interval-censored units.

Nonparametric Estimate

The Nonparametric Estimate report shows nonparametric estimates for each observation. For right-censored data specified as a single Time to Event column, the report gives the following:

Midpoint Estimate

Midpoint-adjusted Kaplan-Meier estimates.

Lower 95%, Upper 95%

Pointwise 95% confidence intervals. You can change the confidence level by selecting Change Confidence Level from the report options.

Simultaneous Lower 95% (Nair), Simultaneous Upper 95% (Nair)

Simultaneous 95% confidence intervals. You can change the confidence level by selecting Change Confidence Level from the report options. See Nair (1984) and Meeker and Escobar (1998).

Kaplan-Meier Estimate

Standard Kaplan-Meier estimates.

If failure times are represented by two Time to Event columns, the report gives Turnbull estimates (in a column called Estimate), pointwise confidence intervals, and simultaneous confidence intervals (Nair).

See Nonparametric Fit for more information about nonparametric estimates.

Parametric Estimate - <Distribution Name>

A report called Parametric Estimate - <Distribution Name> appears for each distribution that is fit. The report gives the distribution’s parameter estimates, their standard errors, and confidence intervals. The criteria that appear in the Model Comparisons report are shown under Criterion.

Note: Whenever an estimate of the mean is provided, its confidence interval is computed as a Wald interval even if you select Likelihood as the Confidence Interval Method in the launch window. In this case, the notation Mean (Wald CI) appears in the Parameter column to indicate that the confidence interval for the mean is a Wald interval.

For more information about how the distributions are parametrized, see Parametric Distributions.

The Parametric Estimate report contains the following reports:

Covariance Matrix

Profilers

Additional reports can be added by selecting report options from the Parametric Estimate red triangle menu. These include the Fix Parameter, Bayesian Estimates, Custom Estimation (Estimate Probability, Estimate Quantile), and Mean Remaining Life reports. See Parametric Estimate Options.

Covariance Matrix

For each distribution, the Covariance Matrix report shows the covariance matrix for the estimates.

Profilers

Four types of profilers appear for each distribution:

The Distribution Profiler shows cumulative failure probability as a function of time.

The Quantile Profiler shows failure time as a function of cumulative probability.

The Hazard Profiler shows the hazard rate as a function of time.

The Density Profiler shows the density function for the distribution.

The profilers contain the following red triangle menu options:

Confidence Intervals

The Distribution, Quantile, and Hazard profilers show Wald-based confidence curves for the plotted functions. This option shows or hides the confidence curves.

Reset Factor Grid

Displays a window for each factor enabling you to enter a specific value for the factor’s current setting, to lock that setting, and to control aspects of the grid. See Reset Factor Grid in Profilers.

Factor Settings

Provides a menu that consists of several options. See Factor Settings in Profilers.

Note: The confidence intervals provided in the profilers are based on the Wald method even if the Likelihood Confidence Interval Method is selected in the launch window. This is done to reduce computation time.

Parametric Estimate Options

The Parametric Estimate red triangle menu has the following options:

Save Probability Estimates

Saves the estimated failure probabilities and confidence intervals to the data table.

Save Quantile Estimates

Saves the estimated quantiles and confidence intervals to the data table.

Save Hazard Estimates

Saves the estimated hazard values and confidence intervals to the data table.

Show Likelihood Contour

Shows or hides a contour plot of the log-likelihood function. If you have selected the Weibull distribution, a second contour plot appears for the alpha-beta parameterization. This option is available only for distributions with two parameters.

Show Likelihood Profiler

Shows or hides a profiler of the log-likelihood function. This option is not available for the threshold (TH) distributions.

Fix Parameter

Opens a report where you can specify the value of parameters. Enter your fixed parameter values, select the appropriate check box, and then click Update. JMP re-estimates the other parameters, covariances, and profilers based on the new parameters, and shows them in the Fix Parameter report. A distribution profiler of the unconstrained model is shown below the distribution profiler for the fixed parameter model. For an example in a competing cause situation, see Specify a Fixed Parameter Model as a Distribution for a Cause.

For the Weibull distribution, the Fix Parameter option lets you select the Weibayes method. For an example, see Weibayes Estimates. The Weibayes option is not available for interval-censored data.

Bayesian Estimates

Performs Bayesian estimation of parameters for certain distributions based on three methods of specifying prior distributions (Location and Scale Priors, Quantile and Parameter Priors, and Failure Probability Priors). See Bayesian Estimation - <Distribution Name>. This option is available only for the following distributions: Lognormal, Weibull, Loglogistic, Fréchet, Normal, SEV, Logistic, LEV.

Custom Estimation

Provides calculators that enable you to predict failure probabilities, survival probabilities, and quantiles for specific time and failure probability values. Each calculated quantity includes confidence intervals, which can be two-sided or one-sided (in either direction). Two reports appear: Estimate Probability and Estimate Quantile. See Custom Estimation.

Mean Remaining Life

Provides a calculator that enables you to estimate the mean remaining life of a unit. In the Mean Remaining Life Calculator, enter a Time and press Enter to see the estimate. Click the plus sign to enter additional times. This calculator is available for the following distributions: Lognormal, Weibull, Loglogistic, Fréchet, Normal, SEV, Logistic, LEV, and Exponential.

Bayesian Estimation - <Distribution Name>

For certain distributions, you can fit Bayesian models. This is done using a Markov Chain Monte Carlo (MCMC) algorithm. More specifically, the platform attempts a basic rejection sampler. If the rejection sampler produces valid results, these results are reported. If the rejection sampler cannot produce valid results, the platform uses a random walk Metropolis-Hastings algorithm and adds a note to the top of the Bayesian Estimation report. See Robert and Casella (2004).

From the Parametric Estimate - <Distribution Name> report outline, select Bayesian Estimates. This opens an outline called Bayesian Estimation - <Distribution Name>. The initial report is a control panel where you can specify the parameters for the priors and control aspects of the simulation.

The workflow is as follows:

Select a prior specification method from the Bayesian Estimation red triangle menu and set values for the parameters of the priors. See Bayesian Estimation Red Triangle Options.

Specify the simulation options. See Bayesian Estimates - Result <N>.

Select Fit Model to fit a model. See Bayesian Estimates - Result <N>.

Bayesian Estimation Red Triangle Options

You can choose from the following prior specification methods in the Bayesian Estimation red triangle menu:

Location and Scale Priors

Enables you to specify hyperparameters for prior distributions on generic parameters (location and scale parameters). Select the Prior Distribution red triangle menu to select a distribution for each parameter. You can enter new values for the hyperparameters of the priors. The initial values that are provided are estimates consistent with the MLEs. See Prior Distributions for Bayesian Estimation.

Quantile and Parameter Priors

Enables you to specify prior information about a quantile and the scale parameter (or Weibull β if the parametric fit is Weibull). The quantile is defined by the value next to Probability. The default Probability value is 0.10, but you can specify a value that corresponds to the quantile of interest. Specify information about the prior information in terms of Lower and Upper 99% limits on the range of each prior distribution. See Meeker and Escobar (1998). The initial values that are provided are estimates consistent with the MLEs. See Prior Distributions for Bayesian Estimation.

Failure Probability Priors

Enables you to specify prior information about failure probabilities at two distinct time points. You can specify the two time points. The prior distribution for each time point is Beta. You can specify the prior distributions using either of two synchronized approaches:

1. Specify failure probability by estimates and error percentages. The prior information for each Beta prior distribution can be specified using a probability estimate and an estimate error. See Kaminskiy and Krivtsov (2005).

2. Specify failure probability estimate ranges. You can specify the 99% range for the two Beta distributions in the following ways:

For each failure time, enter an initial value for the Lower and Upper 99% Limits.

Click the vertical line segments in the graph and drag them to your two time points. Adjust the vertical spread of each marker to specify the 99% limits.

Simulation Options

For any of the prior specification methods that you select in the Bayesian Estimation red triangle menu, the following options appear at the bottom of the panel:

Number of Monte Carlo Iterations

Controls the sample size that will be drawn from the posterior distribution after a burn-in procedure.

Random Seed

Sets the initial state of the simulation. By default, it is the clock time. The number should be a positive integer greater than 1. If you specify 1, the current clock time is used.

Show Prior Scatter Plot

Select this option to draw random samples from the prior distributions and to plot results on a scatter plot. After you select Fit Model, the scatter plot appears in an outline entitled Prior Scatter Plot in the Bayesian Estimates - Results <N> report.

Overlay Likelihood Contour

Overlays likelihood-based contours on scatter plots in the Bayesian Estimates Results report.

Fit Model

Estimates the posterior lifetime distribution based on prior distributions that JMP fits using the values that you specified. Adds a report entitled Bayesian Estimates - Results <N>, where N is an integer that consecutively numbers the Bayesian Results reports.

Bayesian Estimates - Result <N>

Once you have specified priors using one of the red triangle menu options, select Fit Model. A Bayesian Estimates - Result <N> report is provided for each selection of priors. This report contains these headings:

Priors

Documents the specifications that you entered in the Bayesian Estimation report to fit the Bayesian model. The Priors report also specifies the random seed.

Posterior Estimates

Shows five marginal statistics and one joint statistic describing the posterior distribution of the generic parameters (location and scale parameters). The marginal statistics are the median, 0.025 quantile (Lower Bound), 0.975 quantile (Upper Bound), mean, and standard deviation computed from the Monte Carlo samples. The parameter values listed beneath Joint HPD are the values where the joint posterior density is maximized. When the posterior estimates are generated using the Quantile and Parameter Priors specification, this table also includes the posterior estimate of the quantile and Slope β (for the Weibull distribution).

To compute statistics for other derived variables based on the posterior estimates of the generic parameters, click the Export Monte Carlo Samples link.

Prior Scatter Plot

Appears when you select Show Prior Scatter Plot before clicking Fit Model. Shows prior scatter plots of parameters or equivalent quantities associated with the prior specification method for the distribution.

Posterior Scatter Plot

Shows posterior scatter plots of parameters or equivalent quantities associated with the prior specification method for the distribution.

Profilers

Shows two profilers based on samples from the posterior distribution.

The values shown in the Distribution Profiler, at a given time t, are calculated as follows:

For each set of sampled parameter values from the posterior distribution, the value of the cumulative distribution function at time t is calculated.

The predicted value is the median of these calculated values.

The upper and lower confidence limits are the 0.025 and 0.975 quantiles of these calculated values.

The plot and confidence limits shown in the Quantile Profiler are obtained in a similar fashion. For a given Probability value p, the quantiles corresponding to p are calculated from the distributions associated with the posterior parameter values.

Weibayes Only for Zero Failure Data

In a zero-failure situation, no units fail. All observations are right censored. If you have zero-failure data, it is possible to conduct either Bayesian estimation or Weibayes inference. See Weibayes Report.

Note: By default, zero-failure data is analyzed using the Weibayes method. If you want to conduct a broader Bayesian analysis on zero-failure data, select File > Preferences > Platforms > Life Distribution and uncheck Weibayes Only for Zero Failure Data.

Custom Estimation

The Custom Estimation option produces two reports: Estimate Probability and Estimate Quantile. The Estimate Probability report contains a calculator that enables you to predict failure and survival probabilities for specific time values. The Estimate Quantile report contains a calculator that enables you to predict quantiles for specific failure probability values. Both Wald-based and likelihood-based confidence intervals appear for each estimated quantity. The confidence level for these intervals is determined by the Change Confidence Level option in the Life Distribution red triangle menu.

Estimate Probability

In the Estimate Probability calculator, enter a value for Time. Press Enter to see the estimates of failure probabilities, survival probabilities, and corresponding confidence intervals. To calculate multiple probability estimates, click the plus sign, enter another Time value in the box, and press Enter. Click the minus sign to remove the last entry.

The Estimate Probability calculator contains an option, Side, that enables you to change the form of the intervals. Select one of the following suboptions:

Two Sided

Provides two-sided confidence intervals for failure probability and survival probability.

Upper Failure Probability

Provides one-sided confidence intervals that contain upper limits for the failure probability and lower limits for the survival probability.

Lower Failure Probability

Provides one-sided confidence intervals that contain lower limits for the failure probability and upper limits for the survival probability.

Estimate Quantile

In the Estimate Quantile report, enter a value for Failure Probability. Press Enter to see the quantile estimates and corresponding confidence intervals. To calculate multiple quantile estimates, click the plus sign, enter another Failure Probability value in the box, and press Enter. Click the minus sign to remove the last entry.

The Estimate Quantile calculator contains an option, Side, that enables you to change the form of the intervals. Select one of the following suboptions:

Two Sided

Provides two-sided confidence intervals for quantiles.

Lower

Provides one-sided confidence intervals that contain lower limits for quantiles.

Upper

Provides one-sided confidence intervals that contain upper limits for quantiles.

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