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Probability Plotting

Probability plots are useful tools for the display and analysis of lifetime data. Refer to Abernethy (1996) for examples using probability plots in the analysis of reliability data. Probability plots use a special scale so that a cumulative distribution function (CDF) plots as a straight line. Thus, if lifetime data are a sample from a distribution, the CDF estimated from the data plots approximately as a straight line on a probability plot for the distribution.

You can use the RELIABILITY procedure to construct probability plots for data that are complete, right censored, or interval censored (in readout form) for each of the probability distributions in Table 30.37.

A random variable Y belongs to a location-scale family of distributions if its CDF F is of the form

Pr\{Y \leq y\} = F(y) = G(\frac{y-\mu}{\sigma})

where \mu is the location parameter, and \sigma is the scale parameter. Here, G is a CDF that cannot depend on any unknown parameters, and G is the CDF of Y if \mu=0 and \sigma=1. For example, if Y is a normal random variable with mean \mu and standard deviation \sigma,

G(u) = \Phi(u) = \int_{-\infty}^u\frac{1}{\sqrt{2\pi}}\exp(-\frac{u^2}2) \, du
F(y) = \Phi(\frac{y-\mu}{\sigma})

Of the distributions in Table 30.37, the normal, extreme value, and logistic distributions are location-scale models. As shown in Table 30.38, if T has a lognormal, Weibull, or log-logistic distribution, then log(T) has a distribution that is a location-scale model. Probability plots are constructed for lognormal, Weibull, and log-logistic distributions by using log(T) instead of T in the plots.

Let y_{(1)} \le y_{(2)} \le  ...  \le y_{(n)} be ordered observations of a random sample with distribution function F(y). A probability plot is a plot of the points y(i) against mi=G-1(ai), where a_{i}=\hat{F}(y_{i})is an estimate of the CDF F(y_{(i)})=G(\frac{y_{(i)}-\mu}{\sigma}).The points ai are called plotting positions. The axis on which the points mis are plotted is usually labeled with a probability scale (the scale of ai).

If F is one of the location-scale distributions, then y is the lifetime; otherwise, the log of the lifetime is used to transform the distribution to a location-scale model.

If the data actually have the stated distribution, then \hat{F} \approx F,

m_{i}=G^{-1}(\hat{F}(y_{i})\approx G^{-1}(G(\frac{y_{(i)}-\mu}{\sigma}))=
and points (y(i), mi) should fall approximately on a straight line.

There are several ways to compute plotting positions from failure data. These are discussed in the next two sections.

Complete and Right-Censored Data

The censoring times must be taken into account when you compute plotting positions for right-censored data. The RELIABILITY procedure provides several methods for computing plotting positions. These are specified with the PPOS= option in the ANALYZE, PROBPLOT, and RELATIONPLOT statements. All of the methods give similar results, as illustrated in the following sections, "Expected Ranks, Kaplan-Meier, and Modified Kaplan-Meier Methods" and "Median Ranks."

Expected Ranks, Kaplan-Meier, and Modified Kaplan-Meier Methods

Let y_{(1)} \le y_{(2)} \le  ...  \le y_{(n)} be ordered observations of a random sample including failure times and censor times. Order the data in increasing order. Label all the data with reverse ranks ri, with r1 = n, ... , rn = 1. For the failure corresponding to reverse rank ri, compute the reliability
Ri = [[(ri)/(ri+1)]]Ri-1
with R0=1. The expected rank plotting position is computed as ai=1-Ri . The option PPOS=EXPRANK specifies the expected rank plotting position.

For the Kaplan-Meier method,

Ri = [[(ri-1)/(ri)]]Ri-1
The Kaplan-Meier plotting position is then computed as a'i=1-Ri. The option PPOS=KM specifies the Kaplan-Meier plotting position.

For the modified Kaplan-Meier method, use

R'i = [(Ri + Ri-1)/2]
where Ri is computed from the Kaplan-Meier formula with R0=1. The plotting position is then computed as a''i=1-R'i. The option PPOS=MKM specifies the modified Kaplan-Meier plotting position. If the PPOS option is not specified, the modified Kaplan-Meier plotting position is used as the default method.

For complete samples, ai=i/(n+1) for the expected rank method, a'i=i/n for the Kaplan-Meier method, and a''i=(i-.5)/n for the modified Kaplan-Meier method. If the largest observation is a failure for the Kaplan-Meier estimator, then Fn=1 and the point is not plotted. These three methods are shown for the field winding data in Table 30.40 and Table 30.41.

Table 30.40: Expected Rank Plotting Position Calculations
Ordered Reverse ri/(ri+1) ×Ri-1 =Ri ai=1-Ri
Observation Rank        
+ Censored Times

Table 30.41: Kaplan-Meier and Modified Kaplan-Meier Plotting Position Calculations
Ordered Reverse (ri-1)/ri ×Ri-1 =Ri a'i=1-Ri a''i
Observation Rank          
+ Censored Times 

Median Ranks

Refer to Abernethy (1996) for a discussion of the methods described in this section. Let y_{(1)} \le y_{(2)} \le  ...  \le y_{(n)} be ordered observations of a random sample including failure times and censor times. A failure order number ji is assigned to the ith failure: j_{i}=j_{i-1}+\Delta,where j0=0. The increment \Delta is initially 1 and is modified when a censoring time is encountered in the ordered sample. The new increment is computed as
\Delta = \frac{(n+1) - { previous failure order number }}
 {1 + { number of items beyond previous censored item }}
The plotting position is computed for the ith failure time as
ai = [(ji - .3)/(n + .4)]
For complete samples, the failure order number ji is equal to i, the order of the failure in the sample. In this case, the preceding equation for ai is an approximation to the median plotting position computed as the median of the ith-order statistic from the uniform distribution on (0, 1). In the censored case, ji is not necessarily an integer, but the preceding equation still provides an approximation to the median plotting position. The PPOS=MEDRANK option specifies the median rank plotting position.

For complete data, an alternative method of computing the median rank plotting position for failure i is to compute the exact median of the distribution of the ith order statistic of a sample of size n from the uniform distribution on (0,1). If the data are right censored, the adjusted rank ji, as defined in the preceding paragraph, is used in place of i in the computation of the median rank. The PPOS=MEDRANK1 option specifies this type of plotting position.

Nelson (1982, p.148) provides the following example of multiply right-censored failure data for field windings in electrical generators. Table 30.42 shows the data, the intermediate calculations, and the plotting positions calculated by exact (a'i) and approximate (ai) median ranks.

Table 30.42: Median Rank Plotting Position Calculations
Ordered Increment Failure Order    
Observation \Delta Number ji ai a'i
+ Censored Times

Interval-Censored Data

Readout Data

The RELIABILITY procedure can create probability plots for interval-censored data when all units share common interval endpoints. This type of data is called readout data in the RELIABILITY procedure. Estimates of the cumulative distribution function are computed at times corresponding to the interval endpoints. Right censoring can also be accommodated if the censor times correspond to interval endpoints. See "Weibull Analysis of Interval Data with Common Inspection Schedule" for an example of a Weibull plot and analysis for interval data. Table 30.43 illustrates the computational scheme used to compute the CDF estimates. The data are failure data for microprocessors (Nelson 1990, p.147). In Table 30.43, ti are the interval upper endpoints, in hours, fi is the number of units failing in interval i, and ni is the number of unfailed units at the beginning of interval i.

Note that there is right censoring as well as interval censoring in these data. For example, two units fail in the interval (24, 48) hours, and there are 1414 unfailed units at the beginning of the interval, 24 hours. At the beginning of the next interval, (48, 168) hours, there are 573 unfailed units. The number of unfailed units that are removed from the test at 48 hours is 1414 - 2 - 573 = 839 units. These are right-censored units.

The reliability at the end of interval i is computed recursively as

Ri = (1-(fi/ni))Ri-1

with R0=1. The plotting position is ai=1-Ri.

Table 30.43: Interval-Censored Plotting Position Calculations
Interval Interval fi/ni R'i= Ri= ai=1-Ri
i Endpoint ti   1-(fi/ni) R'iRi-1  

The variance v(ai) of the cumulative probability estimate ai=1-Ri is computed using the exact variance method of Nelson (1990 pp. 150-151).

If no right censoring has occurred before ti, then ai is a binomial probability, and exact binomial confidence limits for ai are computed. See Binomial Distribution for a description of this method.

If right censoring has occurred before ti, then two-sided approximate 100\gamma\% confidence limits for ai are computed as

a_{L} & = & a_{i} - K_{\gamma}\sqrt{v(a_{i}}) \ 
 a_{U} & = & a_{i} + K_{\gamma}\sqrt{v(a_{i}})

where K_{\gamma} is the (1+\gamma)/2 x 100\%percentile of the standard normal distribution.

The estimates ai, confidence limits aL and aU, and standard errors \sqrt{v(a_{i}}) are tabulated in the ANALYZE, PROBPLOT, and RELATIONPLOT statements for readout data. The PCONFPLT option requests that the confidence limits be displayed on probability plots.

Arbitrarily-Censored Data

The RELIABILITY procedure can create probability plots for data that consists of combinations of exact, left-censored, right-censored, and interval-censored lifetimes. Unlike the method in the previous section, failure intervals need not share common endpoints. The RELIABILITY procedure uses an iterative algorithm developed by Turnbull (1976) to compute a nonparametric maximum likelihood estimate of the cumulative distribution function for the data. Since the technique is maximum likelihood, standard errors of the cumulative probability estimates are computed from the inverse of the associated Fisher information matrix. A technique developed by Gentleman and Geyer (1994) is used to check for convergence to the maximum likelihood estimate. Also see Meeker and Escobar (1998, chap. 3) for more information.

Although this method applies to more general situations, where the intervals may be overlapping, the example of the previous section will be used to illustrate the method. Table 30.44 contains the microprocessor data of the previous section, arranged in intervals. A missing (.) lower endpoint indicates left censoring, and a missing upper endpoint indicates right censoring. These can be thought of as semi-infinite intervals with lower (upper) endpoint of negative (positive) infinity for left (right) censoring.

Table 30.44: Interval-Censored Data
Lower Upper Number
Endpoint Endpoint Failed

The following SAS program will compute the Turnbull estimate and create a lognormal probability plot.

   data micro;                                
      input t1 t2 f ;                          
      . 6 6                                    
      6 12 2                                   
      12 24 0                                  
      24 48 2                                  
      24 .  1                                  
      48 168 1                                 
      48 .   839                               
      168 500 1                                
      168 .   150                              
      500 1000 2                               
      500 .    149                             
      1000 2000 1                              
      1000 . 147                               
      2000 . 122                               
   proc reliability data=micro;               
      distribution lognormal;                  
      freq f;                                  
      pplot ( t1 t2 ) / itprintem              
                        maxitem = (1000,25)
                        cframe = ligr;
      inset / cfill = ywh;                 

The nonparametric maximum likelihood estimate of the CDF can only increase on certain intervals, and must remain constant between the intervals. The Turnbull algorithm first computes the intervals on which the nonparametric maximum likelihood estimate of the CDF can increase. The algorithm then iteratively estimates the probability associated with each interval. The ITPRINTEM option along with the PRINTPROBS option instructs the procedure to print the intervals on which probability increases can occur and the iterative history of the estimates of the interval probabilities. The PPOUT option requests tabular output of the estimated CDF, standard errors, and confidence limits for each cumulative probability.

Figure 30.25 shows every 25th iteration and the last iteration for the Turnbull estimate of the CDF for the microprocessor data. The initial estimate assigns equal probabilities to each interval. You can specify different initial values with the PROBLIST= option. The algorithm converges in 130 iterations for this data. Convergence is determined if the change in the log-likelihood between two successive iterations less than delta, where the default value of delta is 10-8. You can specify a different value for delta with the TOLLIKE= option. This algorithm is an example of an expectation-maximization (EM) algorithm. EM algorithms are known to converge slowly, but the computations within each iteration for the Turnbull algorithm are moderate. Iterations will be terminated if the algorithm does not converge after a fixed number of iterations. The default maximum number of iterations is 1000. Some data may require more iterations for convergence. You can spoecify the maximum allowed number of iterations with the MAXITEM= option in the PROBPLOT, ANALYZE, or RPLOT statements.


Iteration History for the Turnbull Estimate of the CDF
Iteration Loglikelihood (., 6) (6, 12) (24, 48) (48, 168) (168, 500) (500, 1000) (1000, 2000) (2000, .)
0 -1133.4051 0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
25 -104.16622 0.00421644 0.00140548 0.00140648 0.00173338 0.00237846 0.00846094 0.04565407 0.93474475
50 -101.15151 0.00421644 0.00140548 0.00140648 0.00173293 0.00234891 0.00727679 0.01174486 0.96986811
75 -101.06641 0.00421644 0.00140548 0.00140648 0.00173293 0.00234891 0.00727127 0.00835638 0.9732621
100 -101.06534 0.00421644 0.00140548 0.00140648 0.00173293 0.00234891 0.00727125 0.00801814 0.97360037
125 -101.06533 0.00421644 0.00140548 0.00140648 0.00173293 0.00234891 0.00727125 0.00798438 0.97363413
130 -101.06533 0.00421644 0.00140548 0.00140648 0.00173293 0.00234891 0.00727125 0.007983 0.97363551
Figure 30.25: Iteration History for Turnbull Estimate

If an interval probability is smaller than a tolerance (10-6 by default) after convergence, the probability is set to zero, the interval probabilities are renormalized so that they add to one, and iterations are restarted. Usually the algorithm converges in just a few more iterations. You can change the default value of the tolerance with the TOLPROB= option. You can specify the NOPOLISH option to avoid setting small probabilities to zero and restarting the algorithm.

If you specify the ITPRINTEM option, the table in Figure 30.26 summarizing the Turnbull estimate of the interval probabilities is printed. The columns labeled 'Reduced Gradient' and 'Lagrange Multiplier' are used in checking final convergence to the maximum likelihood estimate. The Lagrange multipliers must all be greater than or equal to zero, or the solution is not maximum likelihood. Refer to Gentleman and Geyer (1994) for more details of the convergence checking.


Lower Lifetime Upper Lifetime Probability Reduced Gradient Lagrange Multiplier
. 6 0.0042 0 0
6 12 0.0014 0 0
24 48 0.0014 0 0
48 168 0.0017 0 0
168 500 0.0023 0 0
500 1000 0.0073 -7.219342E-9 0
1000 2000 0.0080 -0.037063236 0
2000 . 0.9736 0.0003038877 0
Figure 30.26: Final Probability Estimates for Turnbull Algorithm

Figure 30.27 shows the final estimate of the CDF, along with standard errors and confidence limits. Figure 30.28 shows the CDF and pointwise confidence limits plotted on a lognormal probability plot.


Cumulative Probability Estimates
Lower Lifetime Upper Lifetime Cumulative
95% Confidence Limits Standard Error
Lower Upper
6 6 0.0042 0.0019 0.0094 0.0017
12 24 0.0056 0.0028 0.0112 0.0020
48 48 0.0070 0.0038 0.0130 0.0022
168 168 0.0088 0.0047 0.0164 0.0028
500 500 0.0111 0.0058 0.0211 0.0037
1000 1000 0.0184 0.0094 0.0357 0.0063
2000 2000 0.0264 0.0124 0.0553 0.0101
Figure 30.27: Final CDF Estimates for Turnbull Algorithm

turnbull.gif (5496 bytes)

Figure 30.28: Lognormal Probability Plot for the Microprocessor Data

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.