Competing Cause Report

Tip: If you find that the report window is too long, select Tabbed Report from the red triangle menu.

The Competing Cause report appears if you have assigned a Failure Cause column in the launch window. Use this report to analyze the competing causes to determine which causes are influential. For an example of this report, see Omit Competing Causes. For technical details, see Competing Cause Details.

The Competing Cause report contains the following content and options:

•	Cause Combination

•	Competing Cause: Statistics

•	Individual Causes

•	Competing Cause Report Options

Competing Cause Workflow

Follow these steps to facilitate your use of the Competing Cause report:

1.	For convenience, select the Tabbed Report option from the Competing Cause red triangle menu.

2.	Select the Individual Causes tab. For each failure cause, use the options in its individual Life Distribution outline to select a distributional fit. See Individual Causes.

3.	Select the Cause Combination tab. For each failure cause, specify the desired distribution in each Distribution list. See Cause Combination.

4.	Click Update Model.

5.	Select the Statistics tab to explore and save results for the model. See Competing Cause: Statistics.

Note: Customizations made to the competing cause model report in the Statistics outline might be lost if you change the model and click Update Model again.

Competing Cause Model

In a competing cause situation, the aggregated failure function can be written as follows:

where Fi(x) is the cumulative failure distribution for the ith cause and k is the total number of causes. The function Fi(x) is cause-specific. It reflects the probability of failure due to cause i alone and does not account for other causes of failure.

An alternative formulation is given as follows:

where each

is a monotone increasing function with values in the interval [0, 1]. The function

is called a subdistribution. This form of the aggregated failure distribution is used to predict proportions of failures that are associated with individual causes, accounting for failure due to other causes.

Cause Combination

The Cause Combination report lets you fit and compare different failure time distributions (Fi(x)) for the various causes. Different distributions are available based on selections that you have made in the launch window. Negative and zero-failure times are not allowed.

The default plot is based on a Linear scale and shows the following:

•	Nonparametric estimates (Kaplan-Meier-Turnbull) for the uncensored data values and their confidence intervals. The confidence intervals are represented by horizontal blue lines.

•

Fits of cumulative failure distributions (Fi(x)) to each of the causes. The initial distribution is the one that you selected in the launch window. If you selected Individual Best, the best distribution is computed for each group and these fits are shown. (This selection can be time-intensive.) If you selected Manual Pick, the initial Distribution is Nonparametric for all groups and nonparametric fits are shown. A legend appears to the right of the plot.

•	The Aggregated cumulative failure distribution, F(x), represented by a black line. This function is computed based on the selected cause distribution.

As you interact with the report, statistics for the aggregated model are re-evaluated.

The Cause Combination report contains these elements:

Scale

Select the probability scale for the plot’s vertical axis. If a distribution fits well, then the points should follow a straight line when plotted on that scale. Change the Scale illustrates how changing the scale affects the distribution fit.

Omit

Check a box to remove the fit for the corresponding cause. Use this when a particular cause has been fixed. The Aggregated model updates to reflect the removal of the failures due to that cause. Omit Competing Causes illustrates the effect of omitting causes.

Cause

Lists the causes in the Cause column.

Distribution

Lists the available distributions for each cause. To change the distribution for a specific failure cause, select the distribution from the Distribution list. Click Update Model to show the new distribution fit on the plot. The Cause Summary report is also updated.

Count

Gives the number of observations with the given failure cause.

Update Model

Shows the selected distributions in the plot; updates the Cause Summary report with the selected models; adds the selected model as the most recent one in the Individual Causes report.

Competing Cause: Statistics

The Statistics report for Competing Cause contains the following reports:

•	Cause Summary Report

•

Profilers

Cause Summary Report

The Cause Summary report shows information for the fit defined by the current selections under Distribution in the Cause Summary report. The report shows the number of failures for each cause and the parameter estimates for the distribution fit to each failure cause. When you change distribution fits in the Cause Combination report and click Update Model, the Cause Summary report is updated.

The following information is provided:

•	The Cause column shows either labels of causes or the censor code.

•	The Counts column lists the number of failures for each cause.

•	A numerical entry in the Counts column indicates that the cause has enough failure events to consider. A cause with fewer than two events is considered right censored. The column also identifies missing causes.

•	The Distribution column specifies the selected distribution for each cause.

•	Depending on the selected distributions, various columns display the parameters of the distributions:

‒	The location column specifies location parameters for various distributions.

‒	The scale column specifies scale parameters for various distributions. The Weibull α and Weibull β columns show Weibull estimates of alpha and beta.

‒	Other columns show parameters of other selected distributions.

‒	A Convergence column appears if there are convergence issues.

The Cause Summary red triangle menu options enable you to save the probability, quantile, hazard, and density estimates for the aggregated failure distribution to the data table.

Profilers

The Distribution, Quantile, Hazard, and Density profilers help you visualize the aggregated failure distribution. The Distribution, Quantile, and Hazard profilers show 95% level confidence bands. For further details, see Profilers.

Note: If a nonparametric distribution is specified for a cause, the Hazard and Density profilers are not provided. Also, confidence limits are not provided in the Distribution and Quantile profilers.

Individual Causes

The Individual Causes report contains Life Distribution - Failure Cause: <Distribution Name> subreports for each of the individual causes. Each Life Distribution - Failure Cause: <Distribution Name> report shows plots and distributional fit statistics for the individual failure cause indicated in the report title.

Whenever you click Update Model in the Cause Combination report, any new cause distribution that you select is added to the Life Distribution - Failure Cause: <Distribution Name> report for that cause. In the Life Distribution - Failure Cause <Distribution Name> report, the following occur:

•	The distribution is selected in the Compare Distributions report.

•	The distribution is added to the Model Comparisons report.

•	A Parametric report is added for that distribution.

Each Life Distribution - Failure Cause <Distribution Name> report is a Life Distribution: Statistics report as described in Life Distribution: Statistics. However, all confidence intervals are Wald intervals. These intervals are not affected by the selection of Likelihood as the Confidence Interval Method in the launch window.

Available Distributions for Competing Cause Compare Distributions Reports

If you specify a Failure Cause in the launch window, you can specify which groupings of distributions and models that you want to allow in the resulting Compare Distributions reports for Individual Causes. You can select ZI (Zero-Inflated), TH (Threshold), and DS (Defective Subpopulation) distributions. You can also select fixed parameter and Bayesian models.

Note: If you have disallowed any distributions in Preferences, these do not appear. Also, rules that govern which distributions appear for Life Distribution apply. See Available Parametric Distributions.

Competing Cause Report Options

The red triangle menu next to Competing Cause contains the following options:

Tabbed Report

Shows graphs and data on individual tabs rather than in the default outline style.

Tabbed Report for Individual Causes

Shows the Life Distribution - Failure Cause: <Distribution Name> reports in tabs, rather than as a stack of Life Distribution reports.

Show Points

Shows or hides data points in the Cause Combination plot. The Life Distribution platform uses the midpoint estimates of the step function to construct probability plots. When you deselect Show Points, the midpoint estimates are replaced by Kaplan-Meier estimates.

Show Subdistributions

Shows the profiler for each individual cause subdistribution

. See Individual Subdistribution Profiler for Cause.

Show Remaining Life Distribution

Shows the remaining life distribution through the Distribution Profiler. This is conditional upon the unit surviving through a given time.

Mean Remaining Life

Estimates the mean remaining life of a unit, given a survival time. In the Mean Remaining Life Calculator, enter a time value and press Enter to see the estimate and confidence limits. Click the plus sign to enter additional times. Click the minus sign to remove the most recent entry.

To obtain a confidence interval, select the Configuration option from the Mean Remaining Life Calculator. Check Use bootstrap to construct confidence intervals. Enter appropriate values, keeping in mind that computation can be time-intensive. For details, see Mean Remaining Life Calculator.

Export Bootstrap Results

Appears only when a Bayesian model is selected and when Update Model has been applied.

Bootstrap Sample Size

When Bayesian Estimates or Weibayes results are used for any cause, the confidence limits for aggregated functions that appear in the Distribution Profiler must be simulated using parametric bootstrap. Use this option to specify the number of samples to be used in the bootstrap. See Specify a Bayesian Model for a Cause.

Script

Contains options that are available to all platforms. For more information, see Using JMP.

Individual Subdistribution Profiler for Cause

For a given cause, the Individual Subdistribution Profiler for Cause shows the estimated probability of failure,

, from that cause at time t. The estimated probability takes into account failures from competing causes. See Competing Cause Model.

To show a profiler of the subdistribution for each cause, select Show Subdistributions from the Competing Cause red triangle menu. The Individual Subdistribution Profiler for Cause report appears beneath the other profilers. It consists of a profiler and a calculator.

Note: When you select Show Subdistributions, the Cause Combination plot is updated to show the subdistribution functions for all causes.

Select a Cause from the list to the right of the profiler to see its profiler. Options that apply to the profiler are provided in the Individual Subdistribution Profiler for Cause red triangle menu. See Profilers.

Use the Calculator panel to find values of the subdistribution functions for all causes at one or more Time to Event values. Enter a value for the Time to Event variable. When you press Enter (or click outside the text box), values for each of the causes are updated. To add a Time to Event value, click the plus sign. To remove the most recent value, click the minus sign.