(and how it can save a company time and money)
2004 by John Zorich ( johnzorich@yahoo.com )
To understand the theoretical and historical basis of reliability plotting for reliability statistics, the following quotations and annotations are useful. Each quote is followed by its author's name; the full references are provide in the "References" table below. Note that "Reliability Plotting" is referred to by various names in reliability literature, such as "Probability Plotting", "Rectification", "Linear Regression Analysis", as well as "Reliability Plotting" and others.
"Reliability engineering discipline is quite young, being born out of the unreliability of war material during World War II and the Korean War [the same time frame during which was developed most other statistical quality tools, such as SPC (statistical process control), "K" factors and other types of reliability statistics, Gage R&R, power curve Analysis, statistical sampling plans, and so on]. The basic underlying premise of the reliability theory is that products of the same design and manufacture are essentially homogeneous and that their failures follow a definite pattern. Once this premise is accepted, the next task is to define that pattern. This pattern is known as the distribution and lends itself to statistical resolution. Much of the early efforts in the development of the reliability discipline were along statistical lines. Attempts were made at curve fitting of empirical data. There is no such thing as a perfect fit so the search is for a representative distribution. The statistical techniques used in reliability discipline are varied and depend on the specific application. ....There are normal (Gaussian) distributions, binomial distributions, Wiebull distributions, and other distributions ad infinitum. Many of these distributions are subsets of each other and, in most cases are used to fit a unique situation. ....These statistical relationships are approximate models that attempt to match empirical data. The word approximate must be emphasized; it must be remembered that none of these models precisely fit any specific case." Pyzdek, pp. 389-390.
Such curves represent a model of the reliability "life" of the product, a "life distribution" so to speak. "How do we pick a life distribution model from a practical point of view? The approach we recommend is as follows: Use a life distribution model primarily because it works; that is, fits the data well and leads to reasonable projection when extrapolating beyond the range of data. Look for a new model when the one previously used no longer 'works'. " Tobias, p. 91.
The purpose in identifying a life distribution model for a component is to predict the component's reliability under conditions that are impractical to test. "If we have 100 units on test, the smallest percentile we can actually observe is the 1 percent point. We need a parametric model to project to the earlier time when 0.001 percent of the population might fail, or to estimate a proportion of failures for a time much smaller than the time of the first observed test failure" or beyond the last observed test failure. Tobias, p. 126.
"In many cases...it is possible by simple transformations of the variables to represent the... curve as a linear relation between the transformed variables....If the points...when transformed...are grouped about a straight line in such a manner that [they are] normally distributed...then the theory of linear regression may be applied to the transformed observations. The transformation functions are chosen on the basis of a graphical analysis of the observations..." Hald, pp. 558-559.
"Many physical situations produce data that are normally distributed; others, data that follow some other known distribution; and still others, data that can be transformed to normal data." Crow, p. 88. "After the transformation is made...standard methods applicable to normally distributed variables can be used to determine reliability-confidence relationships." Amstadter, p. 90.
In contrast to our understanding of curved lines, our understanding of confidence limits for extrapolated straight lines is very well developed. Almost every advanced statistical textbook, and some quality control textbooks (e.g., Juran), provide the basic equations for establishing confidence limits on the mean of the data at a point on the X-axis --- that is, at the stress level for which a reliability statement is sought. Such an equation provides the equivalent of the standard error of the mean at the extrapolated point.
The process of Reliability Plotting starts by trying to find a transformation that yields a straight line. "Probability paper was developed initially to allow data analysts to plot data, obtain estimates, and assess fit of a particular model by comparing with a straight line....However, because there are many different combinations of possible probability and data axes that might be needed, it is useful to have a computer implementation of probability plotting methods...." Meeker, p.123.
"From a purely statistical point of view, the regression curve provides a description of the interrelation between the two variables within the limited range of the observations....It is therefore absolutely necessary that extrapolations [to values outside this range] be based upon professional knowledge concerning the data...." Hald, p. 559. These convenient methods do require judgment (e.g., how "straight must the line be?) because the sample is never a perfect fit...." Juran, p. 23.44.
After determining the upper-one-sided confidence limit on the extrapolated value (using the equation mentioned above), the %Failure transformation is reversed, the result of which, when subtracted from 1.000, yields the Reliability at the extrapolated stress level, with a confidence equal to whatever was used to determine the "t-table" value in the calculation of the one-sided confidence limit.
CONCLUSION: Reliability plotting is a time-honored, standard statistical technique that can save time and money by providing reliability data that is adequate to the task but is based on smaller sample sizes or is collected in a shorter time-frame than would otherwise be possible.
Literature REFERENCES
Amstadter, B. L., Reliability Mathematics: Fundamentals, Practices, Procedures,1971 by McGraw-Hill
Crow, E. L., Statistics Manual With Examples From Ordnance Development,1960 by Dover Publications, NY.
Hald, A., Statistical Theory with Engineering Applications, 1952 by John Wiley & Sons.
Juran, J. M., and Gryna, F. M., editors, Juran's Quality Control Handbook, 4th edition, 1988 by McGraw-Hill Inc.
Meeker, W. Q., and Escobar, L. A., Statistical Methods for Reliability Data, 1998 by John Wiley & Sons Inc.
Pyzdek, T., and Berger, R. W., editors, Quality Engineering Handbook, 1992 by Marcel Dekker Inc.
Tobias, P. A., and Trinidade, D. C., Applied Reliability, 2nd edition, 1995 by Chapman and Hall