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8. Assessing Product Reliability
8.4. Reliability Data Analysis
8.4.1. How do you estimate life distribution parameters from censored data?

Graphical estimation

The line on a probability plot uniquely identifies distributional parameters Once you have calculated plotting positions from your failure data, and have generated the probability plot for your chosen model, parameter estimation follows easily. But along with the mechanics of graphical estimation, be aware of both the advantages and the disadvantages of graphical estimation methods.
Most probability plots have simple procedures to calculate underlying distribution parameter estimates Graphical Estimation Mechanics:

If you draw a line through points on a probability plot, there are usually simple rules to find estimates of the slope (or shape parameter) and the scale parameter. On lognormal probability plot with time on the \(x\)-axis and cumulative percent on the \(y\)-axis, draw horizontal lines from the 34th and the 50th percentiles across to the fitted line, and drop vertical lines to the time axis from these intersection points. The time corresponding to the 50th percentile is the \(T_{50}\) estimate. Divide \(T_{50}\) by the time corresponding to the 34th percentile (this is called \(T_{34}\)). The natural logarithm of that ratio is the estimate of sigma, or the slope of the line (\(\sigma = \mbox{ln } (T_{50} / T_{34})\)).

For a Weibull probability plot draw a horizontal line from the \(y\)-axis to the fitted line at the 62.3 percentile point. That estimation line intersects the line through the points at a time that is the estimate of the characteristic life parameter \(\alpha\). In order to estimate the slope of the fitted line (or the shape parameter \(\gamma\)), choose any two points on the fitted line and divide the change in the \(y\) variable by the change in \(x\) variable.

Using a computer generated line fitting routine removes subjectivity and can lead directly to computer parameter estimates based on the plotting positions To remove the subjectivity of drawing a line through the points, a least-squares (regression) fit can be performed using the equations described in the section on probability plotting. An example of this for the Weibull was also shown in that section. Another example of a Weibull plot for the same data appears later in this section.

Finally, if you have exact times and complete samples (no censoring), many software packages have built-in Probability Plotting functions. Examples were shown in the sections describing various life distribution models.

Do probability plots even if you use some other method for the final estimates Advantages of Graphical Methods of Estimation:
  • Graphical methods are quick and easy to use and make visual sense.
  • Calculations can be done with little or no special software needed.
  • Visual test of model (i.e., how well the points line up) is an additional benefit.
Disadvantages of Graphical Methods of Estimation
Perhaps the worst drawback of graphical estimation is you cannot get legitimate confidence intervals for the estimates The statistical properties of graphical estimates (i.e., how precise are they on average) are not good:
  • they are biased,
  • even with large samples, they are not minimum variance (i.e., most precise) estimates,
  • graphical methods do not give confidence intervals for the parameters (intervals generated by a regression program for this kind of data are incorrect), and
  • formal statistical tests about model fit or parameter values cannot be performed with graphical methods.
As we will see in the next section, Maximum Likelihood Estimates overcome all these disadvantages - at least for reliability data sets with a reasonably large number of failures - at a cost of losing all the advantages listed above for graphical estimation.
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