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

8.4.1.1.

Graphical estimation

Every line on probability paper uniquely identifies distribution parameters

Once you have calculated plotting positions from your failure data, and put the points on the appropriate graph paper 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 papers have simple procedures that go from a line to the underlying distribution parameter estimates

Graphical Estimation Mechanics:

If you draw a line through the points, and the paper is commercially designed probability paper, there are usually simple rules to find estimates of the slope (or shape parameter) and the scale parameter. On lognormal paper with time on the x-axis and cum percent on the y-axis, draw horizontal lines from the 34th and the 50th percentiles across to the line, and drop vertical lines to the time axis from these intersection points. The time corresponding to the 50th percentile is the T₅₀ estimate. Divide T₅₀ by the time corresponding to the 34th percentile (this is called T₃₄). The natural logarithm of that ratio is the estimate of sigma, or the slope of the line ( = ln (T₅₀ / T₃₄). 

On commercial Weibull probability paper there is often a horizontal line through 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. In order to estimate the line slope (or the shape parameter ), some papers have a special point on them called an estimation point. You drop a line from the estimation point perpendicular to the fitted line and look at where it passes through a special estimation scale. The estimate of  is read off the estimation scale where the line crosses it.

Other papers may have variations on the methods described above.

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 how special papers work. An example of this for the Weibull, using the Dataplot FIT program, was also shown in that section. A SAS JMP™ example of a Weibull plot for the same data is shown later in this section.

Finally, if you have exact times and complete samples (no censoring), Dataplot has built-in Probability Plotting functions and built-in Weibull paper - examples were shown in the sections on the 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 the average) are not good

• they are biased
• even with large samples, they do not become 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)
• 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.