Weibull Analysis: Results (Raw Data) - Advanced Tab

Select the Advanced tab of the Weibull Analysis: Results (Raw Data) dialog box to access the options described here. See the Introductory Overview for further discussion of these options.

Variable

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.

Parameter values/estimates

The default parameter values displayed in the Offset (threshold/location), Shape parameter, and Scale parameter boxes in the Parameter values/estimates group box are the maximum likelihood estimates. The values in these boxes will affect all results, spreadsheets, and graphs.

Note: if the current parameter values are the maximum likelihood estimates (the default values have not been changed in the Shape parameter, or Scale parameter boxes), then the resulting spreadsheets will contain the standard errors and confidence intervals for the parameter estimates. If the values have been changed in the Shape parameter or Scale parameter boxes, then the spreadsheet will contain only the current parameter values. See also Weibull and Reliability/Failure Time Analysis - Two- and three-parameter Weibull distribution.

Offset (threshold/location)

Enter a value in this box (or use the accompanying microscrolls) to change the offset (threshold/location) parameter for the Weibull distribution. This box is only available if the ML shape and scale parameters option button or the User-defined parameters options button is selected.

Shape parameter

Enter a value in this box (or use the accompanying microscrolls) to change the shape parameter for the Weibull distribution. This box is only available if the User-defined parameters options button is selected.

Scale parameter

Enter a value in this box (or use the accompanying microscrolls) to change the scale parameter for the Weibull distribution. This box is only available if the User-defined parameters options button is selected.

See the Introductory Overview for additional details concerning the two- and three-parameter Weibull distribution, and the issues involved in maximum likelihood estimation.

Summary: Parameters

Click this button to display a spreadsheet containing the current parameter values. If the current parameter values are the maximum likelihood estimates (the default values have not been changed in the Shape parameter or Scale parameter boxes), then the spreadsheet will also contain the standard errors and confidence intervals for the parameter estimates. Note that the confidence intervals are based on the information matrix from the maximum likelihood estimation procedure, and they are only approximations (based on the normal distribution). In addition, in the case of the shape and scale parameters, they are adjusted to reflect the fact that the valid parameter space is bounded (i.e., shape and scale must be greater than 0). See Nelson (1990; Section 5.7) or Dodson (1994) 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.

If the values have been changed in the Shape parameter or Scale parameter boxes, then the spreadsheet will contain only the current parameter values.

CL

Enter a value into the CL (Confidence Limit) box to reflect the percentiles for the confidence limits for the parameter estimates (see above). By default, Statistica will compute 95% confidence intervals. Note that the (percentile) value entered in the Confidence limit box will also be used to construct the confidence intervals for the various results spreadsheets and graphs involving the reliability and cumulative distribution functions (see Weibull Analysis: Results (Raw Data) - Reliability & distribution function tab).

ML shape & scale params

Select this option button to compute maximum likelihood parameters estimates for the two-parameter Weibull distributions, i.e., to estimate the shape and scale parameters, contingent on the current (user-defined) location parameter (the two-parameter Weibull distribution will be fit to the data after subtracting the location parameter).

Recompute

Click the Recompute button in order for Statistica to "read" the current value of the Offset (threshold/location) parameter, and then compute maximum likelihood parameter estimates for the Shape and Scale parameters based on the respective Offset (threshold/location) parameter. Note that this button is only available when you select the ML shape & scale params option button.

ML location, shape, scale parameters

Select the ML location, shape, scale parameters option button to compute maximum likelihood parameters estimates for the three-parameter Weibull distributions, i.e., to estimate the location, scale, and shape parameters.

The estimation of the location parameter for the three-parameter Weibull distribution poses a number of special problems, which are detailed in Lawless (1982). Specifically, when the shape parameter is less than 1, then a maximum likelihood solution does not exist for the parameters. In other instances, the likelihood function may contain more than one maximum (i.e., multiple local maxima). In the latter case, Lawless recommends using the smallest failure time (or a value that is a little bit less) as the estimate of the location parameter. In general, Statistica follows these recommendations. However, the results dialog provides interactive options for "experimenting" with different parameters (e.g., you may set the location parameter to a particular value, and then compute the maximum likelihood parameter estimates, contingent on this user-defined value of the location parameter, see above); alternative options (e.g., based on probability plotting) for estimating parameters are also provided.

User-defined parameters

Select the User-defined parameters option button if you want to adjust the values in the Offset (threshold/location), Shape parameter, and Scale parameter boxes.

Refer to the Introductory Overview section for additional details concerning the estimation procedures implemented in Statistica. In general, a Newton-Raphson iterative method is used to maximize the likelihood of the data. Detailed discussions of maximum likelihood estimation methods for fitting the Weibull distribution to censored data can be found in Dodson (1994) or Lawless (1982); a detailed description of an algorithm similar to that used in Statistica for estimating the maximum likelihood parameters for the two-parameter distribution is presented in Keats and Lawrence (1997).

Maximum number of iterations

Enter a value in the Maximum number of iterations box to specify the maximum number of iterations for the iterative parameter estimation. Normally, the default value (50) is more than sufficient to achieve convergence.

Convergence criterion

Enter a value in the Convergence criterion box to specify a convergence criterion for the iterative parameter estimation procedure. The specific interpretation of this parameter is different for the two-parameter versus the three-parameter estimation procedure. Either way, it is roughly equal to the maximum (proportional) change (from iteration to iteration) in the parameter values that would stop the iterations. The default value is appropriate in most cases, and should be changed only when the iterative estimation procedure fails to converge (for different values of the location parameter), in which case the Weibull distribution may not provide an appropriate fit to the data (use the nonparametric probability plots to observe the "general closeness" of the data to the Weibull distribution).

Goodness of fit

The options available in the Goodness of fit group box produce various tests and graphs to evaluate the closeness of the fit of the Weibull distribution for the current parameter values/estimates (see above) to the observed failure time data.

Goodness of fit tests

Click the Goodness of fit tests button to display tests of goodness of fit for the current parameter values/estimates (see above). A number of different tests have been proposed for evaluating the quality of the fit of the Weibull distribution to the observed data. These tests are discussed and compared in detail in Lawless (1982).

Hollander-Proschan

This test compares the theoretical reliability function to the Kaplan-Meier estimate. The actual computations for this test are relatively complex, and you may refer to Dodson (1994, Chapter 4) for a detailed description of the computational formulas. The Hollander-Proschan test is applicable to complete, single-censored, and multiple-censored data sets; however, Dodson (1994) cautions that the test may sometimes indicate a poor fit when the data are heavily single-censored. The Hollander-Proschan C statistic can be tested against the normal distribution (z), and STATISTICA will report both the test statistic and the respective (two-sided) p-value.

Mann-Scheuer-Fertig

This test, proposed by Mann, Scheuer, and Fertig (1973), is described in detail in, for example, Dodson (1994) or Lawless (1982). The null hypothesis for this test is that the population follows the Weibull distribution with the estimated (or user-defined) parameters. Nelson (1982) reports this test to have reasonably good power, and this test can be applied to Type II censored data (see Introductory Overview). For computational details refer to Dodson (1994) or Lawless (1982); the critical values for the test statistic have been computed based on Monte Carlo studies, and have been tabulated for N (sample sizes) between 3 and 25; for N greater than 25, this test is not computed.

Anderson-Darling

The Anderson-Darling procedure is a general test to compare the fit of an observed cumulative distribution function to an expected cumulative distribution function. However, this test is only applicable to complete data sets (without censored observations); otherwise the Anderson-Darling test will not be computed. The critical values for the Anderson-Darling statistic have been tabulated (see, for example, Dodson, 1994, Table 0.5) for sample sizes between 10 and 40; this test is not computed for N less than 10 and greater than 40.

Location parameters

Click the R-square vs. location parameter (see below) button to display a plot of the R-square values (shown on the y-axis) as a function of the location parameter in the range From: and To:, as specified in these respective edit fields.

R² vs. location parameter. Click the R² vs. location parameter button to display a plot of the R-square values (shown on the y-axis) as a function of the location parameter. The R-square value is computed for the correlation between x- and y-axis values in the nonparametric (rank-based) probability plot (see Weibull Analysis: Results (Raw Data) - Reliability tab). In general, the closer the points in a probability plot fall onto a straight line, the better is the fit of the respective distribution to the data. Thus, this plot provides a nonparametric method for estimating the location parameter for the three-parameter Weibull distribution. Note that the location parameter value for which the highest R-square value was obtained, is listed in the title of the plot. The location parameter estimated via this option will not be identical to the maximum likelihood estimate; that is because these are two very different methods of estimation. For additional details, see also the Introductory Overview section, or Dodson (1994).

Quantile-Quantile (Q-Q)

Click the Quantile-Quantile (Q-Q) button to produce a standard quantile-quantile (Q-Q) plot for the current parameter values/estimates (see above). The closer the points in this plot follow a straight line, the better is the fit of the Weibull distribution to the observed failure times.

Hazard function

Click the Hazard function button to produce a spreadsheet of the hazard and cumulative hazard functions (the hazard function describes the probability of failure during a very small time increment, assuming that no failures have occurred prior to that time; see also the Introductory Overview section and Weibull CDF, reliability, and hazard functions for additional details). If the Max. likelihood option button is set in the Conf. intervals box (see Weibull Analysis: Results (Raw Data) - Reliability & distribution function tab), then the hazard function will be computed from the estimate of the probability density and cumulative distribution functions for the current parameter values/estimates (see Dodson, 1994, Evans, Hastings, and Peacock, 1992; or Hahn and Shapiro, 1967). If the Nonparametric option button is set, then the hazard and cumulative hazard functions are estimated from the ranked failure times (failure orders; see also Failure order for single or uncensored data and Modified failure order for multiple-censored data for additional details).

Time-to-failure vs. cum. hazard.

hazard. Click the Time-to-failure vs. cum. hazard button to produce a plot of the cumulative hazard function (shown on the y-axis) versus the failure times (shown on the x-axis). The method of computation of the cumulative hazard function depends on the setting of the Max. likelihood or Nonparametric option buttons in the Conf. intervals box (see Weibull Analysis: Results (Raw Data) - Reliability & distribution function tab), as described in the context of the hazard function spreadsheet described above.

Log(t) vs. log cumul. hazard.

hazard. Click the Log(t) vs. log cumul. hazard button to produce a plot of the log of the cumulative hazard function (shown on the y-axis) versus the log of the failure time minus the current location parameter. The method of computation of the cumulative hazard function depends on the setting of the Max. likelihood or Nonparametric option buttons in the Conf. intervals box (see Weibull Analysis: Results (Raw Data) - Reliability & distribution function tab), as described in the context of the hazard function spreadsheet described above. Note that, when the log of the nonparametric estimate for the cumulative hazard function is plotted, the fitted regression line can be used to derive parameter estimates for the two-parameter Weibull distribution (shape parameter = slope, scale parameter = exp(-intercept/slope)). Refer to the Introductory Overview section for details.

Histogram

Click the Histogram button to produce a stacked histogram of the observed failures times and the censoring times, with the Weibull distribution for the current parameter values/estimates (see above) superimposed. This plot is useful for evaluating the overall fit of the current distribution to the observed data.

Obs. failures only.

Select the Obs. failures only check box to show only the observed failure times on the histogram (see above).