This option fits a Crow-AMSAA model (MIL-HDBK-189 1981). A Crow-AMSAA model is a nonhomogeneous Poisson process with failure intensity as a function of time t given by ρ(t) = λβtβ-1. Here, λ is a scale parameter and β is a growth parameter. This function is also called a Weibull intensity, and the process itself is also called a power law process (Rigdon and Basu 2000; Meeker and Escobar 1998). Note that the Recurrence platform fits the Power Nonhomogeneous Poisson Process, which is equivalent to the Crow-AMSAA model, though it uses a different parameterization. See Fit Model in Recurrence Analysis for details.
The intensity function is a concept applied to repairable systems. Its value at time t is the limiting value of the probability of a failure in a small interval around t, divided by the length of this interval; the limit is taken as the interval length goes to zero. You can think of the intensity function as measuring the likelihood of the system failing at a given time. If β < 1, the system is improving over time. If β > 1, the system is deteriorating over time. If β = 1, the rate of occurrence of failures is constant.
When the Crow AMSAA option is selected, the Cumulative Events plot updates to show the cumulative events curve estimated by the model. For each time point, the shaded band around this curve defines a 95% confidence interval for the true cumulative number of events at that time. The Model List report also updates. Figure 9.8 shows the Observed Data report for the data in TurbineEngineDesign1.jmp.
A Crow-AMSAA report opens within the Models report. If Time to Event Format is used, the Crow AMSAA report shows an MTBF plot with both axes scaled logarithmically. See MTBF Plot.
The mean time between failures (MTBF) plot is displayed by default (Figure 9.9). For each time point, the shaded band around the MTBF plot defines a 95% confidence interval for the true MTBF at time t. If Time to Event Format is used, the plot is shown with both axes logarithmically scaled. With this scaling, the MTBF plot is linear. If Dates Format is used, the plot is not logarithmically scaled.
Figure 9.9 MTBF Plot
To see why the MTBF plot is linear when logarithmic scaling is used, consider the following. The mean time between failures is the reciprocal of the intensity function. For the Weibull intensity function, the MTBF is 1/(λβtβ-1), where t represents the time since testing initiation. It follows that the logarithm of the MTBF is a linear function of log(t), with slope 1 ‑ β. The estimated MTBF is defined by replacing the parameters λ and β by their estimates. So the log of the estimated MTBF is a linear function of log(t).
Maximum likelihood estimates for lambda (λ), beta (β), and the Reliability Growth Slope (1 ‑ β), appear in the Estimates report below the plot. (See Figure 9.9.) Standard errors and 95% confidence intervals for λ, β, and 1 ‑ β are given. For details about calculations, see Parameter Estimates for Crow-AMSAA Models.
This plot shows the estimated intensity function (Figure 9.10). The Weibull intensity function is given by ρ(t) = λβtβ-1, so it follows that log(Intensity) is a linear function of log(t). If Time to Event Format is used, both axes are scaled logarithmically.
Figure 9.10 Intensity Plot
This plot shows the estimated cumulative number of events (Figure 9.11). The observed cumulative numbers of events are also displayed on this plot. If Time to Event Format is used, both axes are scaled logarithmically.
Figure 9.11 Cumulative Events Plot
For the Crow-AMSAA model, the cumulative number of events at time t is given by λtβ. It follows that the logarithm of the cumulative number of events is a linear function of log(t). So, the plot of the estimated Cumulative Events is linear when plotted against logarithmically scaled axes.
Three profilers are displayed, showing estimated MTBF, Failure Intensity, and Cumulative Events (Figure 9.12). These profilers do not use logarithmic scaling. By dragging the red vertical dashed line in any profiler, you can explore model estimates at various time points; the value of the selected time point is shown in red beneath the plot. Also, you can set the time axis to a specific value by pressing the Ctrl key while you click in the plot. A blue vertical dashed line denotes the time point of the last observed failure.
The profilers also display 95% confidence bands for the estimated quantities. For the specified time setting, the estimated quantity (in red) and 95% confidence limits (in black) are shown to the left of the profiler. For further details, see Profilers.
Note that you can link these profilers by selecting Factor Settings > Link Profilers from any of the profiler red triangle menus. For further details about the use and interpretation of profilers, see Factor Profiling in the Fitting Linear Models book. Also see Profiler in the Profilers book.
Figure 9.12 Profilers
A confidence interval for the MTBF at the point when testing concludes is often of interest. For uncensored failure time data, this report gives an estimate of the Achieved MTBF and a 95% confidence interval for the Achieved MTBF. You can specify a 100*(1-α)% confidence interval by entering a value for Alpha. The report is shown in Figure 9.13. For censored data, only the estimated MTBF at test termination is reported.
Figure 9.13 Achieved MTBF Report
There are infinitely many possible failure-time sequences from an NHPP; the observed data represent only one of these. Suppose that the test is failure terminated at the nth failure. The confidence interval computed in the Achieved MTBF report takes into account the fact that the n failure times are random. If the test is time terminated, then the number of failures as well as their failure times are random. Because of this, the confidence interval for the Achieved MTBF differs from the confidence interval provided by the MTBF Profiler at the last observed failure time. Details can be found in Crow (1982) and Lee and Lee (1978).
The Goodness of Fit report tests the null hypothesis that the data follow an NHPP with Weibull intensity. Depending on whether one or two time columns are entered, either a Cramér-von Mises (see Cramér-von Mises Test for Data with Uncensored Failure Times) or a chi-squared test (see Chi-Squared Goodness of Fit Test for Interval-Censored Failure Times) is performed.
The entry below the p-Value heading indicates how unlikely it is for the test statistic to be as large as what is observed if the data come from a Weibull NHPP model. The platform computes p-values up to 0.25. If the test statistic is smaller than the value that corresponds to a p-value of 0.25, the report indicates that its p-value is >=0.25. Details about this test can be found in Crow (1975).
Figure 9.14 shows the goodness-of-fit test for the fit of a Crow-AMSAA model to the data in TurbineEngineDesign1.jmp. The computed test statistic corresponds to a p-value that is less than 0.01. We conclude that the Crow-AMSAA model does not provide an adequate fit to the data.
Figure 9.14 Goodness of Fit Report - Cramér-von Mises Test
