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 The RELIABILITY Procedure

## Weibull Analysis Comparing Groups of Data

This example illustrates probability plotting and distribution fitting for data grouped by the levels of a special group-variable. The data are from an accelerated life test of an insulating fluid and are the times to electrical breakdown of the fluid under different high voltage levels. Each voltage level defines a subset of data for which a separate analysis and Weibull plot are produced. These data are the 26kV, 30kV, 34kV, and 38kV groups of the data provided by Nelson (1990, p. 129). The following statements create a SAS data set containing the lifetimes and voltages.
```   data fluid;
input time voltage\$ @@;
datalines;
5.79 26kV 1579.52 26kV
2323.70 26kV    7.74 30kV
17.05 30kV   20.46 30kV
21.02 30kV   22.66 30kV
43.40 30kV   47.30 30kV
139.07 30kV  144.12 30kV
175.88 30kV  194.90 30kV
0.19 34kV    0.78 34kV
0.96 34kV    1.31 34kV
2.78 34kV    3.16 34kV
4.15 34kV    4.67 34kV
4.85 34kV    6.50 34kV
7.35 34kV    8.01 34kV
8.27 34kV   12.06 34kV
31.75 34kV   32.52 34kV
33.91 34kV   36.71 34kV
72.89 34kV    0.09 38kV
0.39 38kV    0.47 38kV
0.73 38kV    0.74 38kV
1.13 38kV    1.40 38kV
2.38 38kV
;
run;
```

The variable TIME provides the time to breakdown in minutes, and the variable VOLTAGE provides the voltage level at which the test was conducted. These data are not censored.

The RELIABILITY procedure plots the data for the different voltage levels on the same Weibull probability plot, fits a separate distribution to the data at each voltage level, and superimposes distribution lines on the plot.

The following statements produce the probability plot shown in Figure 30.3 for the variable TIME at each level of the group-variable VOLTAGE.

```   symbol1 c=red v=plus;
symbol2 c=blue v=x;
symbol3 c=green v=square;
symbol4 c=yellow v=circle;
proc reliability data=fluid;
distribution weibull;
pplot time=voltage  / overlay
noconf
cframe = ligr;
run;
```
The input data set FLUID is specified by the DATA= option in the PROC RELIABILITY statement. The PROBPLOT statement option OVERLAY specifies that plots for the groups are to be overlaid rather than displayed separately. The option NOCONF specifies that no confidence bands are to be plotted, since these can interfere with one another on overlaid plots; confidence bands are displayed by default.

Figure 30.3: Weibull Probability Plot for the Insulating Fluid Data

A summary table that contains information for all groups is displayed. In addition, information identical to that shown in Figure 30.2 is tabulated for each level of voltage. The summary table for all groups and the tables for the 26kV group are shown in Figure 30.4.

 The RELIABILITY Procedure

 Model Information - All Groups Input Data Set WORK.FLUID Analysis Variable time Distribution Weibull Estimation Method Maximum Likelihood Confidence Coefficient 95% Observations Used 41

 The RELIABILITY Procedure

 Group Algorithm converged. 26kv

 Summary of Fit Group Observations Used 3 26kv Uncensored Values 3 26kv Maximum Loglikelihood -6.845551 26kv

 Weibull Parameter Estimates Parameter Estimate Standard Error Asymptotic Normal Group 95% Confidence Limits Lower Upper EV Location 6.8625 1.1040 4.6986 9.0264 26kv EV Scale 1.8342 0.9611 0.6568 5.1226 26kv Weibull Scale 955.7467 1055.1862 109.7941 8319.6794 26kv Weibull Shape 0.5452 0.2857 0.1952 1.5226 26kv

 Other Weibull Distribution Parameters Parameter Value Group Mean 1649.4882 26kv Mode 0.0000 26kv Median 487.9547 26kv

 Weibull Percentile Estimates Percent Estimate Standard Error Asymptotic Normal Group 95% Confidence Limits Lower Upper 0.1 0.00300636 0.02113841 3.11203E-9 2904.27046 26kv 0.2 0.01072998 0.06838144 4.03597E-8 2852.65767 26kv 0.5 0.0577713 0.31803193 1.19079E-6 2802.78862 26kv 1 0.20695478 1.00385021 0.00001538 2784.16263 26kv 2 0.74484901 3.12705686 0.00019885 2790.0941 26kv 5 4.1142692 13.7388263 0.00591379 2862.3304 26kv 10 15.406565 41.4763373 0.07873508 3014.69497 26kv 20 61.0231127 125.020566 1.10053199 3383.65475 26kv 30 144.246801 242.203982 5.36856883 3875.73303 26kv 40 278.770459 398.048692 16.9761581 4577.77125 26kv 50 487.954708 610.02855 42.0948552 5656.26835 26kv 60 814.147288 920.537706 88.770543 7466.84412 26kv 70 1343.42243 1433.97868 165.818889 10884.0666 26kv 80 2287.87124 2445.52431 281.5628 18590.3635 26kv 90 4412.96962 5148.34986 448.419608 43428.7452 26kv 95 7150.89745 9248.2654 566.892142 90202.9338 26kv 99 15735.8513 24666.0388 728.831025 339745.437 26kv 99.9 33104.172 62018.1074 841.826189 1301796.28 26kv
Figure 30.4: Partial Listing of the Tabular Output for the Insulating Fluid Data

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