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The RELIABILITY Procedure 
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 locationscale family of distributions if its CDF F is of the form
where is the location parameter, and is the scale parameter. Here, G is a CDF that cannot depend on any unknown parameters, and G is the CDF of Y if and . For example, if Y is a normal random variable with mean and standard deviation ,
Of the distributions in Table 30.37, the normal, extreme value, and logistic distributions are locationscale models. As shown in Table 30.38, if T has a lognormal, Weibull, or loglogistic distribution, then log(T) has a distribution that is a locationscale model. Probability plots are constructed for lognormal, Weibull, and loglogistic distributions by using log(T) instead of T in the plots.
Let be ordered observations of a random sample with distribution function F(y). A probability plot is a plot of the points y_{(i)} against m_{i}=G^{1}(a_{i}), where is an estimate of the CDF .The points a_{i} are called plotting positions. The axis on which the points m_{i}s are plotted is usually labeled with a probability scale (the scale of a_{i}).
If F is one of the locationscale distributions, then y is the lifetime; otherwise, the log of the lifetime is used to transform the distribution to a locationscale model.
If the data actually have the stated distribution, then ,
There are several ways to compute plotting positions from failure data. These are discussed in the next two sections.
For the KaplanMeier method,
For the modified KaplanMeier method, use
For complete samples, a_{i}=i/(n+1) for the expected rank method, a'_{i}=i/n for the KaplanMeier method, and a''_{i}=(i.5)/n for the modified KaplanMeier method. If the largest observation is a failure for the KaplanMeier estimator, then F_{n}=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  r_{i}/(r_{i}+1)  ×R_{i1}  =R_{i}  a_{i}=1R_{i} 
Observation  Rank  
31.7  16  16/17  1.0000  0.9411  0.0588 
39.2  15  15/16  0.9411  0.8824  0.1176 
57.5  14  14/15  0.8824  0.8235  0.1765 
65.0+  13  
65.8  12  12/13  0.8235  0.7602  0.2398 
70.0  11  11/12  0.7602  0.6968  0.3032 
75.0+  10  
75.0+  9  
87.5+  8  
88.3+  7  
94.2+  6  
101.7+  5  
105.8  4  4/5  0.6968  0.5575  0.4425 
109.2+  3  
110.0  2  2/3  0.5575  0.3716  0.6284 
130.0+  1  
+ Censored Times 
Ordered  Reverse  (r_{i}1)/r_{i}  ×R_{i1}  =R_{i}  a'_{i}=1R_{i}  a''_{i} 
Observation  Rank  
31.7  16  15/16  1.0000  0.9375  0.0625  0.0313 
39.2  15  14/15  0.9375  0.8750  0.1250  0.0938 
57.5  14  13/14  0.8750  0.8125  0.1875  0.1563 
65.0+  13  
65.8  12  11/12  0.8125  0.7448  0.2552  0.2214 
70.0  11  10/11  0.7448  0.6771  0.3229  0.2891 
75.0+  10  
75.0+  9  
87.5+  8  
88.3+  7  
94.2+  6  
101.7+  5  
105.8  4  3/4  0.6771  0.5078  0.4922  0.4076 
109.2+  3  
110.0  2  1/2  0.5078  0.2539  0.7461  0.6192 
130.0+  1  
+ Censored Times 
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 j_{i}, 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 rightcensored 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 (a_{i}) median ranks.
Table 30.42: Median Rank Plotting Position CalculationsOrdered  Increment  Failure Order  
Observation  Number j_{i}  a_{i}  a'_{i}  
31.7  1.0000  1.0000  0.04268  0.04240 
39.2  1.0000  2.0000  0.1037  0.1027 
57.5  1.0000  3.0000  0.1646  0.1637 
65.0+  1.0769  
65.8  1.0769  4.0769  0.2303  0.2294 
70.0  1.0769  5.1538  0.2960  0.2953 
75.0+  1.1846  
75.0+  1.3162  
87.5+  1.4808  
88.3+  1.6923  
94.2+  1.9744  
101.7+  2.3692  
105.8  2.3692  7.5231  0.4404  0.4402 
109.2+  3.1590  
110.0  3.1590  10.6821  0.6331  0.6335 
130.0+  6.3179  
+ Censored Times 
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 rightcensored units.
The reliability at the end of interval i is computed recursively as
Interval  Interval  f_{i}/n_{i}  R^{'}_{i}=  R_{i}=  a_{i}=1R_{i} 
i  Endpoint t_{i}  1(f_{i}/ni)  R^{'}_{i}R_{i1}  
1  6  6/1423  0.99578  0.99578  .00421 
2  12  2/1417  0.99859  0.99438  .00562 
3  24  0/1415  1.00000  0.99438  .00562 
4  48  2/1414  0.99859  0.99297  .00703 
5  168  1/573  0.99825  0.99124  .00876 
6  500  1/422  0.99763  0.98889  .01111 
7  1000  2/272  0.99265  0.98162  .01838 
8  2000  1/123  0.99187  0.97364  .02636 
The variance v(a_{i}) of the cumulative probability estimate a_{i}=1R_{i} is computed using the exact variance method of Nelson (1990 pp. 150151).
If no right censoring has occurred before t_{i}, then a_{i} is a binomial probability, and exact binomial confidence limits for a_{i} are computed. See Binomial Distribution for a description of this method.
If right censoring has occurred before t_{i}, then
twosided approximate confidence limits for a_{i}
are computed as
The estimates a_{i}, confidence limits a_{L} and a_{U}, and standard errors 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.
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 semiinfinite intervals with lower (upper) endpoint of negative (positive) infinity for left (right) censoring.
Table 30.44: IntervalCensored DataLower  Upper  Number 
Endpoint  Endpoint  Failed 
.  6  6 
6  12  2 
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 
The following SAS program will compute the Turnbull estimate and create a lognormal probability plot.
data micro; input t1 t2 f ; datalines; . 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 printprobs maxitem = (1000,25) nofit pconfplt ppout cframe = ligr; inset / cfill = ywh; run;
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 loglikelihood 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 expectationmaximization (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.
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.

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.

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