Weibull & Reliability Analysis - Failure Order for Single or Uncensored Data

The Failure orders for no/single censoring group box is located on the Weibull Analysis: Results (Raw Data) - Reliability & Distribution Function tab. 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.

The Failure orders for no/single censoring group box is only available if the current data set contains no censored observations, or if single-censoring only is found in the data file (see also the Introductory Overview for a discussion of different types of censoring; Statistica will check when reading the data file whether all censoring occurs at the same time, and at or after the last observed failure). In that case there are several ways in which the cumulative distribution function can be estimated from the ranks of the failure times (failure orders); these ranks are used in the Time to fail. vs. R(t) and Probability plot, and the Reliability values and conf. intervals computations if the Nonparametric option button is selected in the Conf. intervals group box, and in all other plots that involve the estimation of the cumulative Weibull distribution function from ranks (i.e., the Quantile-Quantile (Q-Q) plot; the R-square vs. location parameter plot, which is based on the Q-Q plot; and the Cumulative distribution function plot).

Select the method to be used for computing the nonparametric estimate of the cumulative distribution function:

Median rank method.

  F(t) = (j-0.3)/(n+0.4)

Mean rank method.

  F(t) = j/(n+1)

White's plotting position.

  F(t) = (j-3/8)/(n+1/4)

where j denotes the failure order (rank; for multiple-censored data a weighted average ordered failure is computed; see Dodson, p. 21, for details), and n is the total number of observations. In practice, the difference in the estimates for the cumulative distribution function resulting from these different methods are often very small.

Confidence interval

The confidence interval for the cumulative distribution function estimated from ranks can be estimated by the following expressions:

LCL_α = [j/(n-j+1)] / [F1-_α, 2*(n-j+1), 2*j + j/(n-j+1)]

UCL_α = {[ j/(n-j+1)]*F_α, 2*j, 2*(n-j+1)} / {1+[j/(n-j+1)] * F_α, 2*j, 2*(n-j+1) / (n-j+1)}

where:

LCL_α is the lower confidence limit (with α<.5)

UCL_α is the upper confidence limit (with α>.5)

j is the failure order

n is the total number of data points

F_{α, v1 ,v2} is the respective F distribution value for p=α, with degrees of freedom v1 and v2