Weibull Analysis: Results (Grouped Data) - Estimate Tab

Select the Estimate tab of the Weibull Analysis: Results (Grouped Data) dialog box to access the options described here.

Parameter values/estimates. Use the options in this group box to change and/or display the parameters for the Weibull distribution. By default, the values in the Shape parameter and Scale parameter boxes will show the weighted least squares parameter estimates for the current data for the two-parameter Weibull distribution unless the parameter estimation procedure failed to converge (see the Introductory Overview).

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.

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 option button is selected in the Estimate parameters (weighted least squares) group box on the Estimate tab.

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 option button is selected in the Estimate parameters (weighted least squares) group box on the Estimate tab.

Summary: Parameters

Click this button to produce a spreadsheet with the current parameter values. If the current parameter values are the weighted least squares parameter estimates (as estimated by selecting the ML shape & scale parameters option button - see below), the spreadsheet will also contain the standard errors and confidence intervals for the parameter estimates. Note that the confidence intervals are adjusted to reflect the fact that the valid parameter space is bounded (i.e., the shape parameter must be greater than 0). See Nelson (1990; Section 5.7) or Dodson (1994) for computational details.

CL (Confidence limit)

The percentiles for the confidence limits for the parameter estimates (see above) depend on the value entered into the CL (Confidence limit) box. By default, Statistica will compute 95% confidence intervals.

Estimate parameters (weighted least squares)

This group box contains two options:

ML shape & scale params

Select this option button for Statistica to compute the parameters for the two-parameter Weibull distribution (via weighted least squares methods; see also the Introductory Overview for details). The estimates will be computed based on the currently specified Offset (threshold/location) parameter (the location parameter will be subtracted from the interval boundary values prior to fitting the Weibull distribution) and will automatically be transferred into the Shape parameter and Scale parameter boxes when you click the Recompute button (see below). When the ML shape & scale params option button is selected, the analysis spreadsheets will also contain the standard errors and confidence intervals for the parameter estimates.

The weighted least squares algorithm is based on a method proposed by Gehan and Siddiqui (1973; see also Lee, 1992), and the same one used in the Survival Analysis module (for fitting distributions to life tables; see also Survival Analysis - Notes and Technical Information). However, note that the parameterization of the Weibull distribution in the Survival Analysis (life table) module is different than the one used in this module; refer to the description of the Weibull distribution in the Glossary for details. Refer also to the Introductory Overview section for details about the two- and three-parameter Weibull distribution and the issues involved in maximum likelihood estimation.

Recompute

Click this button 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.

User-defined parameters

Select this option button in order adjust the values in the Shape parameter and Scale parameter boxes. If the values are changed in the Shape parameter or Scale parameter boxes (see above), the resulting  spreadsheets will contain only the current parameter values (not the standard errors and confidence intervals).

Goodness of fit

Use the options in this group box to 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.

Quantile-Quantile (Q-Q)

Click this button to produce a standard quantile-quantile (Q-Q) plot for the current parameter values/estimates (see above) and for the lower interval boundaries. This plot is based on the life table estimates of the observed cumulative distribution. 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 this 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 for additional details). 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; see also Weibull CDF, reliability, and hazard functions).

Time-to-failure vs. cum. hazard.

hazard. Click this 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 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; see also Weibull CDF, reliability, and hazard functions).

Log(t) vs. log cumul. hazard.

hazard. Click this 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 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; see also Weibull CDF, reliability, and hazard functions).

Histogram

Click this button to produce a stacked histogram of the number of failures and censored observations at the observed lower interval boundaries, 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 this check box to show only the observed failure times on the histogram (see above).