### 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
2323.7  26kv    7.74    30kv
17.05   30kv    20.46   30kv
21.02   30kv    22.66   30kv
43.4    30kv    47.3    30kv
139.07  30kv    144.12  30kv
175.88  30kv    194.90  30kv
0.19     34kv    .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    .09     38kv
0.39     38kv    .47     38kv
0.73     38kv    .74     38kv
1.13    38kv    1.40    38kv
2.38    38kv
;
```

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 16.5 for the variable `Time` at each level of the group-variable `Voltage`:

```proc reliability data=fluid;
distribution Weibull;
pplot time=voltage  / overlay
noconf;
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.

A summary table that contains information for all groups is displayed. In addition, information identical to that shown in Figure 16.3 is tabulated for each level of voltage. The summary table for all groups and the tables for the 26kV group are shown in Figure 16.6 and Figure 16.7.

Figure 16.6: Partial Listing of the Tabular Output for the Insulating Fluid Data

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

 Algorithm converged for group 26kv.

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

Figure 16.7: Partial Listing of the Tabular Output for the Insulating Fluid Data

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

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
Standard Deviation 3279.0212 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