Methods and formulas for probability plot in Distribution Overview Plot (Arbitrary Censoring)

Probability plot

A probability plot includes the following:
  • Plot points, which are the estimated percentiles for corresponding probabilities of an ordered data set.
  • Fitted line, which is the expected percentile from the distribution based on maximum likelihood parameter estimates.
  • Confidence intervals, which are the confidence intervals for the percentiles.

Because the plot points do not depend on any distribution, they would be the same (before being transformed) for any probability plot made. However, the fitted line differs depending on the parametric distribution chosen. So you can use the probability plot to assess whether a particular distribution fits your data. In general, the closer the points fall to the fitted line, the better the fit.

Plot points

The plot points of the probability plot represent the likelihood of a product failing before time, t. For right-censored or uncensored data, Minitab calculates the plot points using the following methods:
  • Median rank method (default)
  • Modified Kaplan-Meier method
  • Herd-Johnson method
  • Kaplan-Meier method

If the data contain tied failure times (identical failure times), either all points (default), the average (median), or the maximum of the tied points is plotted. If the tie involves failures and suspensions, the failures are considered to occur before the suspensions.

Each of these methods generates nonparametric estimates of F(t), the cumulative distribution function for the random variable T, which is time to failure.

For a sample of n observations, let x(1), x(2),...,x(n) be the order statistics, or the data ordered from smallest to largest. Then i is the rank of the I th ordered observation x(I). The formula for each method is as follows:

Median rank (Benard's method)

Formula for uncensored data

Formula for censored data

Modified Kaplan-Meier

Formula for uncensored data

Formula for censored data

Herd-Johnson estimate

Formula for uncensored data

Formula for censored data

Kaplan-Meier product limit estimate

Note

If the largest observation is uncensored, the Kaplan-Meier method results in p = 1 for the largest uncensored observation. In this case, the Kaplan-Meier estimate for the largest observation results in a number that cannot be used in the plot. This problem is corrected by recalculating the largest p as 90% of the distance between the prior p and 1.

Note

For arbitrarily-censored data, Minitab estimates the cumulative probabilities using the Turnbull method1.

Formula for uncensored data

Formula for censored data

Notation

TermDescription
irank of the data point, with ties given consecutive ranks
nnumber of observations in the data
δj 0 if the j th observation is censored, or 1 if the j th observation is uncensored
ARi
AR0equals 0
p'i

Fitted line

The following table shows how the x and y coordinates for the fitted line are constructed. Note the following:
  • Minitab transforms the x-axis to a log scale when you are using the Weibull, 3-parameter Weibull, exponential, lognormal, or loglogistic distribution.
  • Minitab transforms the y-axis to a percent scale by default. If you change the y-scale type to probability, Minitab transforms the y-axis to a probability scale.

Distribution x coordinate y coordinate
Smallest extreme value failure time ln(–ln(1 – p))
Weibull ln(failure time) ln(–ln(1 – p))
3-parameter Weibull ln(failure time – threshold) ln(–ln(1 – p))
Exponential ln(failure time) ln(–ln(1 – p))
2-parameter exponential ln(failure time – threshold) ln(–ln(1 – p))
Normal failure time Φ –1 (p)
Lognormal ln(failure time) Φ –1 (p)
3-parameter lognormal ln(failure time – threshold) Φ –1 (p)
Logistic failure time
Loglogistic ln(failure time)
3-parameter loglogistic ln(failure time – threshold)

Notation

TermDescription
Φ –1 inverse cdf for the standard normal distribution
ln (x)natural log of x
1 B.W. Turnbull (1976). "The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data", Journal of the Royal Statistical Society, 38, 290-295.
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