Weibull Analysis: Results (Raw Data) - Reliability & Distribution Function Tab

Use the Variable << and >> buttons (or use the adjacent menu) to select a current variable for which to compute results. These buttons are only available when multiple variables with failure times are selected on the Weibull & Reliability/Failure Time Analysis - Raw Data tab.

R(t). Click the Time-to-fail. vs. R(t) button to display a plot of the reliability function (complement of the cumulative distribution function; see the Introductory Overview section). If the Max. likelihood option button is selected (see below), then the values for the reliability function will be computed from the current parameter values/estimates; if those parameters are the maximum likelihood parameters (i.e., they are the default parameters), then the plot will contain the estimates computed from those maximum likelihood parameter estimates, along with the confidence interval for the reliability function. If the Nonparametric option button is selected (see below), then the plot will show the nonparametric (rank-based) estimates of the reliability function, along with the nonparametric confidence interval.

Note: Computation of confidence limits for maximum likelihood parameter estimates. The reliability function is bounded in the 0-1 interval, and the approximate normal confidence interval is computed adjusting for this restriction in range. Refer to Nelson (1990, Section 5.7) for computational details. Also, Dodson (1994) cautions against the interpretation of confidence intervals computed from maximum likelihood estimates when the shape parameter is less than 2; in that case the variance estimates computed for maximum likelihood estimates lack accuracy.

Note: Computation of nonparametric estimates of reliability and confidence intervals. When the Nonparametric option button is selected in the Conf. intervals box (see below), then the estimate of the reliability for each observed failure time is computed based on the respective ranks. For data without censoring or single censored, the method for computing the estimate of the cumulative distribution function (and reliability function) can be selected in the Failure orders for no/single censoring group box (see below); for multiple censoring data, a weighted average ordered failure is computed as described in Dodson (1994). Note that in addition to the nonparametric confidence intervals (e.g., 95% confidence interval), these plots will also show the center (50th percentile) of the confidence interval.

Click the Probability plot button to display a Weibull probability plot. This plot will show on the left y-axis the quantity log(log(1/(1-F(t)))), where F(t) denotes the cumulative Weibull distribution function; the x-axis will show the quantity log(failure time minus location parameter). If confidence intervals are included in the plot (see below), then the percentile value for the confidence interval will be taken from the Conf. limit field (see Current parameter values/estimates).

When the Max. likelihood option button is set (see below), the values for the cumulative distribution function will be computed from the

c

urrent parameter values/estimates

; if those parameters are the maximum likelihood parameters (i.e., they are the default parameters), then the plot will contain the estimates computed from those maximum likelihood parameter estimates, along with the confidence interval. Note that the cumulative distribution function F(t) is the complement of the reliability function (i.e., R(t)=1-F(t)), and the same caveats and comments regarding the confidence limits for maximum likelihood estimate of the reliability function apply (see option Time-to-fail vs. R(t), above).

When the Nonparametric option button is selected (see below), the plot will be based on the nonparametric (rank-based) estimates of the cumulative distribution function, along with the nonparametric confidence intervals and the center (50th percentile) of the interval; refer to the description of the Time-to-fail vs. R(t) option for comments and references regarding the computation of these values (note that R(t)=1-F(t)). This plot can be used to derive parameter estimates for the two-parameter Weibull distribution. For this purpose, the plot includes the linear fit line (by default in the same color as the points) for the plotted observed failure times (failure orders), and the parameter estimates derived from the plot will be shown in the graph title. Specifically, the shape parameter is equal to the slope of the linear fit-line, and the scale parameter can be estimated as exp(-intercept/slope). See also Hahn and Shapiro (1967) for a detailed description of distribution fitting techniques based on probability plotting.

Reliability values and conf. intervals. Click the Reliability values and conf. intervals button to display a spreadsheet with the reliability and cumulative distribution function values (and confidence intervals based on the maximum likelihood parameter values, or nonparametric confidence intervals) for each observed failure time. These values are computed in the same manner as for the plots (i.e., they depend on the setting of the Max likelihood or Nonparametric option buttons; see below).

The Conf. intervals group box contains two options: Max. likelihood and Nonparametric.

If the Max. likelihood option button is selected, you can plot (or display) the reliability function for the current parameters (see also Weibull CDF, reliability, and hazard functions)

If the Nonparametric option button is selected, then the values displayed in the spreadsheet and plots are computed from rank-based estimates of the cumulative distribution function and the standard error of that function (see Failure order for single or uncensored data). Note that the percentile for the confidence interval is taken from the CL box in the Parameter values/estimates group box (see Weibull Analysis: Results (Raw Data) - Quick tab or Advanced tab).

The Min/max for plots group box contains two options: Minimum and Maximum. Use these options to enter the values for the scaling of the axes denoting the failure times in the Time-to-fail. vs. R(t) and Probability plot (see above).

Enter the minimum value for the scaling of the axes denoting the failure times in the Time-to-fail. vs. R(t) and Probability plot (see above).

Enter the maximum value for the scaling of the axes denoting the failure times in the Time-to-fail. vs. R(t) and Probability plot (see above).

Use the options in this group box to select the method to be used for computing the nonparametric estimate of the cumulative distribution function. Available methods (option buttons) are: Median rank method, Mean rank method, and White's plotting position. See Weibull & Reliability Analysis - Failure order for single or uncensored data for further details on these options.

Click this button to produce a plot of the cumulative distribution function (see also Weibull CDF, reliability, and hazard functions). The method of computation of the cumulative distribution function depends on the setting of the Max. likelihood or Nonparametric option buttons in the Conf. intervals group box (see above). If the Max. likelihood option button is selected, then the values for the cumulative distribution function will be computed from the Current parameter values/estimates (see Quick tab); if those parameters are the maximum likelihood parameters (i.e., they are the default parameters), then the plot will contain the estimates computed from those maximum likelihood parameter estimates, along with the confidence interval for the cumulative distribution function. If the Nonparametric option button is selected, then the plot will show the nonparametric (rank-based) estimates of the cumulative distribution function, along with the nonparametric confidence interval (see also Failure order for single or uncensored data and Modified failure order for multiple-censored data for additional computational details).

Note: Computation of confidence limits for maximum likelihood parameter estimates. The cumulative distribution function is bounded in the 0-1 interval, and the approximate normal confidence interval is computed adjusting for this restriction in range. Refer to Nelson (1990, Section 5.7) for computational details. Also, Dodson (1994) cautions against the interpretation of confidence intervals computed from maximum likelihood estimates when the shape parameter is less than 2; in that case the variance estimates computed for maximum likelihood estimates lack accuracy.

Note: Computation of nonparametric estimates of reliability and confidence intervals. When the Nonparametric option button is selected in the Conf. intervals group box, then the estimate of the cumulative distribution function is computed based on the ranked failure times. For data without censoring or single censoring, the method for computing the estimate of the cumulative distribution function can be selected in the Failure orders for no/single censoring group box; for multiple censored data, a weighted average ordered failure is computed as described in Dodson (1994). Note that in addition to the nonparametric confidence intervals (e.g., 95% confidence interval), these plots will also show the center (50th percentile) of the confidence interval.

Click this button to produce a spreadsheet containing the percentiles for the cumulative Weibull distribution, given the Current parameter values/estimates (see Quick tab or Advanced tab); if those parameters are the maximum likelihood parameters (i.e., they are the default parameters), then the spreadsheet will contain the estimates computed from those maximum likelihood parameter estimates, along with the confidence interval for the percentiles.

Click this button to produce a plot of the Kaplan-Meier product limit estimator of the reliability (survivorship) function. The Kaplan-Meier estimator S(t) for failure time t is computed as:

S(t)=Õ_j=^t₁[(n-j) / (n-j+1)]^δ (j)

In this equation, n is the total number of cases, and Õ denotes the multiplication (geometric sum) across all cases less than or equal to t; δ(j) is a constant that is either 1 if the j'th case is uncensored (complete), and 0 if it is censored.

In addition to the standard Kaplan-Meier estimate, the confidence intervals for the estimates are also computed and shown in the plot. The confidence limits are computed based on the standard errors of the Kaplan-Meier estimates (e.g., see Lawless, 1982), and using the formulas for 0-1 range restricted estimators as described in Nelson (1990). Note that the percentile value for the confidence interval will be taken from the CL (Confidence limit) box on the Advanced tab.

Click the Descriptive statistics button to produce a spreadsheet with the standard descriptive statistics (mean, standard deviation, minimum, maximum, N) for the observed failure times, censored times, and all cases combined. Note that you can also use the Basic Statistics/Tables option on the Statistics menu to compute additional detailed descriptive statistics (e.g., quantiles), broken down for censored and complete observations, or for all observations combined.