Graphical Method

Graphical analysis is the simplest method for obtaining results in both life data and accelerated life testing analyses. Although they have limitations (presented in Comments on the Graphical Method section) in general graphical methods are easily implemented and easy to interpret.

The graphical method for estimating the parameters of accelerated life data involves generating two types of plots. First, the life data at each individual stress level are plotted on a probability paper appropriate to the assumed life distribution (i.e. Weibull, exponential, or lognormal). The parameters of the distribution at each stress level are then estimated from the plot. Once these parameters have been estimated at each stress level, the second plot is created on a paper that linearizes the assumed life-stress relationship (i.e. Arrhenius, inverse power law, etc.). The parameters of the life-stress relationship are then estimated from the second plot. The life distribution and life-stress relationship are then combined to provide a single model that describes the accelerated life data. Figure 1 illustrates these two plots.

Fig. 1: Illustration of the graphical method of parameter estimation.

With this general understanding of the graphical parameter estimation method, we will continue with a more specific discussion of each step.

Life Distribution Parameters at Each Stress Level

The first step in the graphical analysis of accelerated data is to calculate the parameters of the assumed life distribution at each stress level. Because life data are collected at each test stress level in accelerated life tests, the assumed life distribution is fitted to data at each individual stress level. The parameters of the distribution at each stress level are then estimated using the probability plotting method described next.

Life Distribution Probability Plotting

The easiest parameter estimation method (to use by hand) for complex distributions, such as the Weibull distribution, is the method of probability plotting. Probability plotting involves a physical plot of the data on specially constructed probability plotting paper. This method is easily implemented by hand as long as one can obtain the appropriate probability plotting paper.

Probability plotting looks at the cdf (cumulative density function) of the distribution and attempts to linearize it by employing a specially constructed paper. For example, in the case of the 2-parameter Weibull distribution, the cdf and unreliability Q(T) can be shown to be:

(The Life Distributions chapter of this reference presents derivations of this equation.) This function can then be linearized (i.e. put into the common form of y = a + bx) as follows:

(20)

Then setting:

and:

the equation can be rewritten as:

which is now a linear equation with a slope of β and an intercept of β ln(η) .

The next task is to construct a paper with the appropriate x- and y- axes. The x-axis is easy since it is simply logarithmic. The y-axis, however, must represent:

where Q(T) is the unreliability. Such papers have been created by different vendors and are called Weibull probability plotting papers.

To illustrate, consider the following probability plot on a Weibull Probability Paper (created using Weibull++).

For free downloads of probability plotting papers from ReliaSoft, visit http://www.weibull.com/GPaper/index.htm.

This paper is constructed based on the y and x transformation mentioned previously where the y-axis represents unreliability and the x-axis represents time. Both of these values must be known for each point (or time-to-failure) we want to plot.

Then, given the y and x value for each point, the points can easily be placed on the plot. Once the points are placed on the plot, the best possible straight line is drawn through these points. Once the line is drawn, the slope of the line can be obtained (most probability papers include a slope indicator to facilitate this) and thus the parameter β, which is the value of the slope, can be obtained.

To determine the scale parameter, η (also called the characteristic life by some authors), a little more work is required. Note that from before:

so at T = η :

Thus if we entered the y axis at Q(T) = 63.2%, the corresponding value of T will be equal to η. Using this simple, but rather time-consuming methodology, then, the parameters of the Weibull distribution can be determined. For data obtained from accelerated tests, this procedure is repeated for each stress level.

Determining the X and Y Position of the Plot Points

The points plotted on the probability plot represent our data, or more specifically in life data analysis, times-to-failure data. So if we tested four units that failed at 10, 20, 30 and 40 hours at a given stress level, we would use these times as our x values or time values. Determining the appropriate y plotting position, or the unreliability, is a little more complex. To determine the y plotting positions, we must first determine a value called the median rank for each failure.

Median Ranks

Median ranks are used to obtain an estimate of the unreliability, Q(Tj) for each failure. It represents the value that the true probability of failure, Q(Tj), should have at the jth failure out of a sample of N units, at a 50% confidence level. This is an estimate of the value based on the binomial distribution. The rank can be found for any percentage point, P, greater than zero and less than one, by solving the cumulative binomial distribution for Z (rank for the jth failure) [34]:

where N is the sample size and j the order number.

The median rank is obtained by solving the following equation for:

(21)

For example if N = 4 and we have four failures at that particular stress level, we would solve the median rank equation, Eqn. (21), four times; once for each failure with j = 1, 2, 3 and 4, for the value of Z. This result can then be used as the unreliability for each failure, or the y plotting position. Solution of equation (21) requires numerical methods.

A more straightforward and easier method of estimating median ranks is to apply two transformations to Eqn. (21), first to the beta distribution and then to the F distribution, resulting in [34]:

F0.50;m;n denotes the F distribution at the 0.50 point, with m and n degrees of freedom, for the jth failure out of N units.

A quick and less accurate approximation of the median ranks is also given by [34]:

(22)

This approximation of the median ranks is also known as Benard's approximation.

Some Shortfalls of Manual Probability Plotting

Besides the most obvious shortfall of probability plotting, the amount of effort required, manual probability plotting is not always consistent in the results. Two people plotting a straight line through a set of points will not always draw this line the same way and they will therefore come up with slightly different results. In addition, when dealing with accelerated test data a probability plot must be constructed for each stress level. This implies that sufficient failures must be observed at each stress level, which is not always possible.

Probability Plotting with Censored Data

Probability plotting can also be performed with censored data. The methodology involved is rather laborious. ReliaSoft [34] presents this methodology.

This section includes the following topics:

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