
DISTRIBUTIONAL LIKELIHOOD RATIO TESTName:
The likelihood ratio test given here was proposed by Dumonceaux, Antle, and Haas. The basic algorithm is as follows:
Currently, Dataplot only supports this test for uncensored and ungrouped data from continuous distributions. Also, Dataplot only supports this command for distributions for which it supports maximum likelihood estimation. Dumonceaux, Antle, and Haas proposed some simplified tests for a few specific cases. Dataplot supports the following specific cases:
It is also important to note that it matters which distribution is specified for the null hypothesis and which is specified for the alternative hypothesis. The power of the test is estimated by running 5,000 simulations from the alternative hypothesis distribution (as with the critical values, location and scale parameters are set to 0 and 1, respectively, and the shape parameter is obtained from the maximum likelihood fit). When the power is relatively low, the distribution specified in the null hypothesis may be favored. For example, suppose you are testing a Weibull and a lognormal. It is quite possible that if the Weibull is given as the null hypothesis distribution that the null hypothesis will not be rejected and likewise if the lognormal is given as the null hypothesis it will not be rejected either.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
TEST <y1> ... <yk> <SUBSET/EXCEPT/FOR qualification> where <y1> ... <yk> is a list of up to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will generate the test for each of the listed response variables. Although the word MULTIPLE is optional, it can be useful to distinguish this from the REPLICATED case. Note that the syntax
RATIO TEST Y1 TO Y4 is supported. This is equivalent to
RATIO TEST Y1 Y2 Y3 Y4
<y> <x1> ... <xk> <SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <x1> ... <xk> is a list of one to six groupid variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax peforms a crosstabulation of <x1> ... <xk> and performs the test for each unique combination of crosstabulated values. For example, if X1 has 3 levels and X2 has 2 levels, there will be a total of 6 likelihood ratio tests performed. The word REPLICATED is required to distinguish the replication case from the multiple case (if there are multiple variables and neither MULTIPLE or REPLICATED is specified, Dataplot assumes MULTIPLE). Note that the syntax
RATIO TEST Y X1 TO X4 is supported. This is equivalent to
RATIO TEST Y X1 X2 X3 X4
TEST Y1 NORMAL AND EXPONENTIAL MULTIPLE DISTRIBUTIONAL LIKELIHOOD ... RATIO TEST Y1 TO Y5 NORMAL AND EXPONENTIAL REPLICATED DISTRIBUTIONAL ... LIKELIHOOD RATIO TEST Y X NORMAL AND EXPONENTIAL DISTRIBUTIONAL LIKELIHOOD RATIO ... TEST Y1 SUBSET Y1 > 0
The word TEST is optional.
If the MULTIPLE or REPLICATED option is used, these values will be written to the file "dpst1f.dat" instead.
DISTRIBUTIONAL LIKELIHOOD RATIO REPLICATED is a synonym for REPLICATED DISTRIBUTIONAL LIKELIHOOD RATIO
Dumonceaux and Antle (1973), "Discrimination Between the LogNormal and Weibull Distributions", Technometrics, Vol. 15, No. 4, pp. 923926.
. Step 1: Create the data for the example on page 25 of the . Dumonceaux, Antle, and Hass Technometrics paper . serial read y 35.15 44.62 40.85 45.32 36.08 38.97 32.48 34.36 38.05 26.84 33.68 42.90 33.57 36.64 33.82 42.26 37.88 38.57 32.05 41.50 end of data . . Step 2: Perform Test . set write decimals 4 normal and exponential distributional likelihood ratio test y normal and double exponential distributional likelihood ratio test y Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  NORMAL Ha: Data are from distribution  EXPONENTIAL Summary Statistics: Total Number of Observations: 20 Sample Mean: 37.2795 Sample Standard Deviation: 4.7235 Sample Minimum: 26.8400 Sample Maximum: 45.3200 H0 Distribution: Estimate of Location Parameter: 37.2795 Estimate of Scale Parameter: 4.7235 Ha Distribution: Estimate of Location Parameter: 26.8400 Estimate of Scale Parameter: 10.4395 Test: Test Statistic Value: 0.4525 CDF of Test Statistic: 0.1818 PValue: 0.8182 Number of Simulations for CV: 10000 Number of Simulations for Power: 4999 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 0.538 75.0 = 0.612 80.0 = 0.631 90.0 = 0.683 95.0 = 0.736 99.0 = 0.824 99.9 = 0.936 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.98 0.683 Accept H0 5% 95% 0.95 0.736 Accept H0 1% 99% 0.85 0.824 Accept H0 Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  NORMAL Ha: Data are from distribution  DOUBLE EXPONENTIAL Summary Statistics: Total Number of Observations: 20 Sample Mean: 37.2795 Sample Standard Deviation: 4.7235 Sample Minimum: 26.8400 Sample Maximum: 45.3200 H0 Distribution: Estimate of Location Parameter: 37.2795 Estimate of Scale Parameter: 4.7235 Ha Distribution: Estimate of Location Parameter: 37.2600 Estimate of Scale Parameter: 3.8125 Test: Test Statistic Value: 1.2390 CDF of Test Statistic: 0.2723 PValue: 0.7277 Number of Simulations for CV: 10000 Number of Simulations for Power: 5000 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 1.282 75.0 = 1.336 80.0 = 1.351 90.0 = 1.391 95.0 = 1.432 99.0 = 1.510 99.9 = 1.634 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.50 1.391 Accept H0 5% 95% 0.37 1.432 Accept H0 1% 99% 0.19 1.510 Accept H0Program 2: . Step 1: Create the data for the example on page 22 of the . Dumonceaux, Antle, and Hass Technometrics paper . let y = data 47 38 29 92 41 44 47 62 59 44 47 41 . . Step 2: Perform Test . set write decimals 4 normal and cauchy distributional likelihood ratio test y Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  NORMAL Ha: Data are from distribution  CAUCHY Summary Statistics: Total Number of Observations: 12 Sample Mean: 49.2500 Sample Standard Deviation: 16.0348 Sample Minimum: 29.0000 Sample Maximum: 92.0000 H0 Distribution: Estimate of Location Parameter: 49.2500 Estimate of Scale Parameter: 16.0348 Ha Distribution: Estimate of Location Parameter: 44.4556 Estimate of Scale Parameter: 4.3886 Test: Test Statistic Value: 1.2468 CDF of Test Statistic: 0.9945 PValue: 0.0055 Number of Simulations for CV: 10000 Number of Simulations for Power: 5000 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 0.828 75.0 = 0.893 80.0 = 0.911 90.0 = 0.971 95.0 = 1.033 99.0 = 1.180 99.9 = 1.451 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.81 0.971 Accept H0 5% 95% 0.74 1.033 Accept H0 1% 99% 0.62 1.180 Accept H0Program 3: . Step 1: Create the data for the example on page 926 of the . Dumonceaux and Antle Technometrics paper . serial read y 0.654 0.613 0.315 0.449 0.297 0.402 0.379 0.423 0.379 0.3235 0.269 0.740 0.418 0.412 0.494 0.416 0.338 0.392 0.484 0.265 end of data . . Step 2: Perform Test . set write decimals 4 normal and gumbel distributional likelihood ratio test y Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  NORMAL Ha: Data are from distribution  GUMBEL Summary Statistics: Total Number of Observations: 20 Sample Mean: 0.4231 Sample Standard Deviation: 0.1253 Sample Minimum: 0.2650 Sample Maximum: 0.7400 H0 Distribution: Estimate of Location Parameter: 0.4231 Estimate of Scale Parameter: 0.1253 Ha Distribution: Estimate of Location Parameter: 0.3841 Estimate of Scale Parameter: 0.1434 Test: Test Statistic Value: 0.9869 CDF of Test Statistic: 0.8517 PValue: 0.1483 Number of Simulations for CV: 10000 Number of Simulations for Power: 4999 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 0.944 75.0 = 0.975 80.0 = 0.981 90.0 = 0.994 95.0 = 1.002 99.0 = 1.016 99.9 = 1.026 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.29 0.994 Accept H0 5% 95% 0.17 1.002 Accept H0 1% 99% 0.04 1.016 Accept H0Program 4: . Step 1: Create the data for the example on page 925 of the . Dumonceaux and Antle Technometrics paper . serial read y 17.88 28.92 33.00 41.52 42.12 45.60 48.48 51.84 51.96 54.12 55.56 67.80 68.64 68.64 68.88 84.12 93.12 98.64 105.12 105.84 127.92 128.04 173.40 end of data . . Step 2: Perform Test . set write decimals 4 lognormal and weibull distributional likelihood ratio test y weibull and lognormal distributional likelihood ratio test y Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  LOGNORMAL Ha: Data are from distribution  WEIBULL Summary Statistics: Total Number of Observations: 23 Sample Mean: 72.2243 Sample Standard Deviation: 37.4887 Sample Minimum: 17.8800 Sample Maximum: 173.4000 H0 Distribution: Estimate of Scale Parameter: 63.4628 Estimate of Shape Parameter 1: 0.5334 Ha Distribution: Estimate of Scale Parameter: 81.8783 Estimate of Shape Parameter 1: 2.1021 Test: Test Statistic Value: 0.9763 CDF of Test Statistic: 0.6753 PValue: 0.3247 Number of Simulations for CV: 10000 Number of Simulations for Power: 4999 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 0.945 75.0 = 0.990 80.0 = 1.002 90.0 = 1.033 95.0 = 1.062 99.0 = 1.117 99.9 = 1.193 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.65 1.033 Accept H0 5% 95% 0.51 1.062 Accept H0 1% 99% 0.29 1.117 Accept H0 Distributional Likelihood Ratio Test Response Variable: Y H0: Data are from distribution  WEIBULL Ha: Data are from distribution  LOGNORMAL Summary Statistics: Total Number of Observations: 23 Sample Mean: 72.2243 Sample Standard Deviation: 37.4887 Sample Minimum: 17.8800 Sample Maximum: 173.4000 H0 Distribution: Estimate of Scale Parameter: 81.8783 Estimate of Shape Parameter 1: 2.1021 Ha Distribution: Estimate of Scale Parameter: 63.4628 Estimate of Shape Parameter 1: 0.5334 Test: Test Statistic Value: 1.0243 CDF of Test Statistic: 0.8752 PValue: 0.1248 Number of Simulations for CV: 10000 Number of Simulations for Power: 5000 Percent Points of the Reference Distribution  Percent Point Value  50.0 = 0.940 75.0 = 0.990 80.0 = 1.003 90.0 = 1.033 95.0 = 1.062 99.0 = 1.118 99.9 = 1.183 Conclusions (Upper 1Tailed Test)  Power Critical Alpha CDF (1Beta) Value Conclusion  10% 90% 0.64 1.033 Accept H0 5% 95% 0.49 1.062 Accept H0 1% 99% 0.22 1.118 Accept H0  
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Date created: 01/31/2015 