
MCCOOL WEIBULL LOCATION TESTName:
McCool described a test for this purpose that can be applied to either censored or uncensored data. The derivation of this test is given in the references listed below. The test statistic is \[ W = \frac{\hat{\gamma}_{L}} {\hat{\gamma}_{A}} \] where $$ \hat{\gamma}_{A} $$ is the maximum likelihood estimate of the shape parameter from a 2parameter Weibull distribution based on the full data set and $$ \hat{\gamma}_{L} $$ is the maximum likelihood estimate of the shape parameter based on the first r_{1} uncensored observations. McCool performed power studies to determine the optimal value of r_{1}. He recommends
The critical values for this test are determined via simulation. The simulation uses 10,000 samples from a 2parameter Weibull distribution with a shape parameter of 1 and a scale parameter of 10 (this corresponds to a tenth percentile of 1.0). Currently, Dataplot limits this test to values of n between 10 and 100. Also, the number of uncensored observations must be greater than r_{1}. Although the test and simulation can be computed for n > 100, optimal values of r_{1} have not been published.
<SUBSET/EXCEPT/FOR qualification> where <y> is a response variable containing failure times; <x> is a censoring variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Dataplot uses the combined Fibonacci/multiplicative congruential generator. If you do not set the seed for the generator explicitly, then a default value of 23709 will be used. If you choose to use a different value for the seed, you may see slight differences in the critical values. These slight differences in the critical values should be small enough for practical applications of the test.
If you have uncensored data, you can create the censoring variable as follows
LET X = 1 FOR I = 1 1 N MCCOOL WEIBULL LOCATION TEST Y X
LET A = MCCOOL WEIBULL LOCATION TEST CDF Y X LET A = MCCOOL WEIBULL LOCATION TEST PVALUE Y X LET A = MCCOOL WEIBULL LOCATION TEST CV50 Y X LET A = MCCOOL WEIBULL LOCATION TEST CV90 Y X LET A = MCCOOL WEIBULL LOCATION TEST CV95 Y X Enter HELP STATISTICS for a list of commands that can be used with Dataplot supported statistics. See the Program section below for an example.
This defines the r_{1} value to use for the test. Be aware that the optimal value (in the sense of maximizing the power of the test) of r_{1} is not known in this case.
John McCool (2012), "Using the Weibull Distribution: Reliability, Modeling, and Inference," Wiley, pp. 301307. Horst Rinne (2009), "The Weibull Distribution: A Handbook," CRC Press, pp. 640642.
. Data set from McCool's paper . let y = data 90.4 94.2 97.8 101.8 104.6 113.0 118.0 154.9 181.3 186.2 let n = size y let tag = 1 for i = 1 1 n . let statval = mccool weibull location test y tag let statcdf = mccool weibull location test cdf y tag let pvalue = mccool weibull location test pvalue y tag let cv50 = mccool weibull location test cv50 y tag let cv90 = mccool weibull location test cv90 y tag let cv95 = mccool weibull location test cv95 y tag . set write decimals 4 print statval statcdf pvalue cv50 cv90 cv95 . mccool weibull location test y tagThe following output is generated PARAMETERS AND CONSTANTS STATVAL  4.4566 STATCDF  0.9909 PVALUE  0.0091 CV50  1.1547 CV90  2.1077 CV95  2.6370 McCool Weibull Location Test Response Variable: Y Censoring Variable: TAG H0: mu = 0 Ha: mu > 0 Summary Statistics: Total Number of Observations: 10 Number of Uncensored Observations: 10 Value of R1: 5 Value of Shape Parameter for All Data: 3.7785 Value of Shape Parameter for R1 Data: 16.8391 Test Statistic Value: 4.4565 CDF of Test Statistic: 0.9920 PValue for Test Statistic: 0.0080 Conclusions (Upper 1Tailed Test)  Null Null Significance Test Critical Hypothesis Hypothesis Level Statistic Region (>=) Conclusion  mu = 0 50.0% 4.4565 1.1457 REJECT mu = 0 75.0% 4.4565 1.5476 REJECT mu = 0 90.0% 4.4565 2.1408 REJECT mu = 0 95.0% 4.4565 2.6455 REJECT mu = 0 97.5% 4.4565 3.1822 REJECT mu = 0 99.0% 4.4565 4.1601 REJECT mu = 0 99.9% 4.4565 7.6829 ACCEPT  
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Date created: 04/15/2013 