
GOODNESS OF FITName:
Detailed descriptions of each of these methods is given below in the Notes section. As a general comment, goodness of fit methods are typically based on comparing the cumulative distribution of the data with a theoretical distribution or comparing the quantiles of the data with the a theoretical percent point function. Previous versions of Dataplot supported separate commands (ANDERSON DARLING TEST, KOLMOGOROV SMIRNOV GOODNESS OF FIT TEST, and CHISQUARE GOODNESS OF FIT TEST). These separate commands have been replaced with the unified GOODNESS OF FIT command and are no longer available). Some comments on this command.
<SUBSET/EXCEPT/FOR qualification> where <dist> is one Dataplot's supported distributions; <method> is one of ANDERSON DARLING, KOLMOGOROV SMIRNOV, CHISQUARE, or PPCC; <y> is the response variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional. Enter HELP PROBABILITY DISTRIBUTIONS for a list of supported distributions and the name of any required parameters.
<SUBSET/EXCEPT/FOR qualification> where <dist> is one Dataplot's supported distributions; <method> is one of ANDERSON DARLING, KOLMOGOROV SMIRNOV, CHISQUARE, or PPCC; <y1> ... <yk> is a list of 1 to 30 response variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax will generate the goodness of fit statistic for each variable in the list. Note that the syntax
is supported. This is equivalent to
<SUBSET/EXCEPT/FOR qualification> where <dist> is one Dataplot's supported distributions; <method> is one of ANDERSON DARLING, KOLMOGOROV SMIRNOV, CHISQUARE, or PPCC; <y> is the response variable; <x1> ... <xk> is a list of 1 to 6 groupid variables; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
This syntax peforms a crosstabulation of Note that the syntax
<SUBSET/EXCEPT/FOR qualification> where <dist> is one Dataplot's supported distributions; <y> is a variable of precomputed frequencies; <x> is a variable containing the midpoints of the bins; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax is used for the case where you have binned data with equal size bins. Currently, only the chisquare goodness of fit method is supported for grouped data (although this may change in future releases).
<SUBSET/EXCEPT/FOR qualification> where <dist> is one Dataplot's supported distributions; <y> is a variable of precomputed frequencies; <xlow> is a variable containing the lower limits of the bins; <xhigh> is a variable containing the upper limits of the bins; and where the <SUBSET/EXCEPT/FOR qualification> is optional. This syntax is used for the case where you have binned data with unequal size bins. Currently, only the chisquare goodness of fit method is supported for grouped data (although this may change in future releases).
LET KSLOC = 5 LET KSSCALE = 10 WEIBULL ANDERSON DARLING GOODNESS OF FIT Y WEIBULL KOLMOGOROV SMIRNOV GOODNESS OF FIT Y WEIBULL PPCC GOODNESS OF FIT Y WEIBULL CHISQUARE GOODNESS OF FIT Y
where n_{i} is the number of points less than Y_{i} This is a step function that increases by 1/N at the value of each data point. The KolmogorovSmirnov goodness of fit test statistic is defined as
where F is the theoretical cumulative distribution of the distribution being tested. We can graph a plot of the empirical distribution function with a cumulative distribution function for a given distribution. The KS test is based on the maximum distance between these two curves. An example of this plot for a sample of 100 normal random numbers is given here.
An attractive feature of this test is that the distribution of the KS test statistic itself does not depend on the underlying cumulative distribution function being tested. Another advantage is that it is an exact test (the chisquare goodness of fit depends on an adequate sample size for the approximations to be valid). Despite these advantages, the KS test has several important limitations:
To specify whether published tables or simulation will be used to generate the critical values, enter the command (if the specified distribution does not support published tables, simulation will automatically be used).
Currently, Dataplot supports critical values from published tables for the following distributions:
Dynamic simulation of critical values for other distributions is available when there is a builtin maximum likelihood estimation procedure available (see the Note section below for the SET DISTRIBUTIONAL FIT TYPE command for a complete list of supported distributions). Note that the uniform (0,1) case can be used for fully specified distributions (i.e., the shape, location, and scale parameters are not estimated from the data). Simply apply the appropriate CDF function to the data (this transforms it to a (0,1) interval) and apply the uniform (0,1) test to the transformed data. The AndersonDarling test statistic is
where
where F is the cumulative distribution function of interest.
For the chisquare goodness of fit, the data is divided into k bins and the test statistic is defined as
where O_{i} is the observed frequency for bin i and E_{i} is the expected frequency for bin i. The expected frequency is calculated by
where F is the cumulative distribution function for the distribution being tested, Y_{u} is the upper limit for class i, and Y_{l} is the lower limit for class i. This test is sensitive to the choice of bins. There is no optimal choice for the bin width (since the optimal bin width depends on the distribution). Most reasonable choices should produce similar, but not identical, results. This test is most frequently used when the data are received in prebinned form (for raw data, the AndersonDarling test is more powerful). However, you can use the chisquare test for raw data (you typically will want to have a reasonably large data set before doing this). For raw data, you can specify the binning with the commands CLASS WIDTH, CLASS LOWER, and CLASS UPPER. The default class width is 0.3 times the sample standard deviation. To specify other default algorithms, enter HELP HISTOGRAM CLASS WIDTH. For the chisquare approximation to be valid, the expected frequency should be at least 5. The chisquare approximation may not be valid for small samples, and if some of the counts are less than five, you may need to combine some bins in the tails. The test statistic follows, approximately, a chisquare distribution with (k  c) degrees of freedom where k is the number of nonempty cells and c = the number of parameters (including location and scale parameters and shape parameters) for the distribution + 1. For example, for a 3parameter Weibull distribution, c = 4. The primary advantage of the chi square goodnes of fit test is that it is quite general. It can be applied for any distribution, either discrete or continuous, for which the cumulative distribution function can be computed. Dataplot supports the chisquare goodness of fit test for all distributions for which it supports a CDF function. There are several disadvantages:
For more information on the PPCC method, enter HELP PPCC PLOT. Also see the NIST/SEMATACH eHandbook of Statistical Methods:
For several distributions, you can choose an alternative estimation method using the command
where <value> can be one of the following (since this applies to the AndersonDarling or KolmogorovSmirnov methods, only continuous distributions are listed).
Note that the above list gives the distributions for which dynamic critical values can be obtained by simulation when the parameters are assumed unknown for the AndersonDarling and KolmogorovSmirnov methods. If a particular distribution only supports a single method (e.g., several currently only support Lmoment estimates), that method will always be used. If you specify a method that is not supported for a given distribution, the default method (usually maximum likelihood) will be used. Also note that a given estimation method for a particular distribution may fail for certain data sets. Since a large number of simulated data sets are generated, this may be an issue for some distributions. The output will return the number of times a failure in the estimation procedure was detected in the simulations.
GOF is a synonym for GOODNESS OF FIT
Stephens, M. A. (1976), "Asymptotic Results for GoodnessofFit Statistics with Unknown Parameters," Annals of Statistics, Vol. 4, pp. 357369. Stephens, M. A. (1977), "Goodness of Fit for the Extreme Value Distribution," Biometrika, Vol. 64, pp. 583588. Stephens, M. A. (1977), "Goodness of Fit with Special Reference to Tests for Exponentiality," Technical Report No. 262, Department of Statistics, Stanford University, Stanford, CA. Stephens, M. A. (1979), "Tests of Fit for the Logistic Distribution Based on the Empirical Distribution Function," Biometrika, Vol. 66, pp. 591595. "MILHDBK17 Volume 1: Guidelines for Characterization of Structural Materials", Depeartment of Defense, chapter 8. The URL for MILHDBK17 is http://mil17.udel.edu/. V. Choulakian and M. A. Stephens (2001), "GoodnessofFit Tests for the Generalized Pareto Distribution", Technometrics, Vol. 43, No. 4, pp. 478484. James J. Filliben (1975), "The Probability Plot Correlation Coefficient Test for Normality," Technometrics, Vol. 17, No. 1. Chakravart, Laha, and Roy (1967), "Handbook of Methods of Applied Statistics, Volume I," John Wiley, pp. 392394. Snedecor and Cochran (1989), "Statistical Methods", Eight Edition, Iowa State, 1989, pp. 7679.
. Step 1: Read the data . . Following data from Jeffery Fong of the NIST . Applied and Computational Mathematics Division. . This is strength data in ksi units. . read y 18.830 20.800 21.657 23.030 23.230 24.050 24.321 25.500 25.520 25.800 26.690 26.770 26.780 27.050 27.670 29.900 31.110 33.200 33.730 33.760 33.890 34.760 35.750 35.910 36.980 37.080 37.090 39.580 44.045 45.290 45.381 end of data . . Step 2: Apply goodness of fit tests for Weibull distribution . based on ML estimates . set write decimals 5 3parameter weibull mle y let ksloc = locml let ksscale = scaleml let gamma = shapeml . . AndersonDarling . set anderson darling critical values table weibull anderson darling goodness of fit y set anderson darling critical values simulation weibull anderson darling goodness of fit y . . Step 3: Apply goodness of fit tests for normal distribution . normal mle y let ksloc = xmean let ksscale = xsd . set anderson darling critical values table normal anderson darling goodness of fit y set anderson darling critical values simulation normal anderson darling goodness of fit y set kolmogorov smirnov critical values simulation normal kolmogorov smirnov goodness of fit yThe following output is generated. ********************************* ** 3parameter weibull mle y ** ********************************* ThreeParameter Weibull (Minimum) Parameter Estimation: Full Sample Case Summary Statistics: Number of Observations: 31 Sample Mean: 30.81141 Sample Standard Deviation: 7.25338 Sample Skewness: 0.39880 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Zanakis Percentile Method: Estimate of Location: 18.65836 Estimate of Scale: 15.10163 Estimate of Shape: 1.86735 Value of LogLikelihood Function: 104.60286 AIC: 215.20572 AICC: 216.09461 BIC: 219.50768 WycoffBainEnglehardt Percentile Method Estimate of Location: 16.64362 Estimate of Scale: 16.41275 Estimate of Shape: 1.92760 Value of LogLikelihood Function: 103.63967 AIC: 213.27934 AICC: 214.16823 BIC: 217.58131 Modified Moments: Estimate of Location: 15.60378 Estimate of Scale: 17.17121 Estimate of Shape (Gamma): 2.21477 Standard Error of Location: 0.71154 Standard Error of Scale: 0.52547 Standard Error of Shape: 0.09924 Value of LogLikelihood Function: 103.56460 AIC: 213.12921 AICC: 214.01810 BIC: 217.43118 Maximum Likelihood: Estimate of Location: 17.64420 Estimate of Scale: 14.83507 Estimate of Shape (Gamma): 1.91358 Value of LogLikelihood Function: 103.26267 AIC: 212.52535 AICC: 213.41423 BIC: 216.82731 ************************* ** let ksloc = locml ** ************************* THE COMPUTED VALUE OF THE CONSTANT KSLOC = 0.1764420E+02 ***************************** ** let ksscale = scaleml ** ***************************** THE COMPUTED VALUE OF THE CONSTANT KSSCALE = 0.1483507E+02 *************************** ** let gamma = shapeml ** *************************** THE COMPUTED VALUE OF THE CONSTANT GAMMA = 0.1913580E+01 *********************************** ** . AndersonDarling ** *********************************** ************************************************** ** set anderson darling critical values table ** ************************************************** THE FORTRAN COMMON CHARACTER VARIABLE ANDEDARL HAS JUST BEEN SET TO TABL ************************************************** ** weibull anderson darling goodness of fit y ** ************************************************** AndersonDarling Goodness of Fit Test (Critical Values from Published Tables) Response Variable: Y H0: The distribution fits the data Ha: The distribution does not fit the data Distribution: WEIBULL Location Parameter: 17.64420 Scale Parameter: 14.83507 Shape Parameter 1: 1.91358 Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Sample Mean: 30.81141 Sample SD: 7.25338 AndersonDarling Test Statistic Value: 0.33805 Adjusted Test Statistic Value: 0.35019 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 0.637 Accept H0 5% 95% 0.757 Accept H0 2.5% 97.5% 0.877 Accept H0 1% 99% 1.038 Accept H0 ******************************************************* ** set anderson darling critical values simulation ** ******************************************************* THE FORTRAN COMMON CHARACTER VARIABLE ANDEDARL HAS JUST BEEN SET TO SIMU ************************************************** ** weibull anderson darling goodness of fit y ** ************************************************** AndersonDarling Goodness of Fit Test (Fully Specified Model) Response Variable: Y H0: The distribution fits the data Ha: The distribution does not fit the data Distribution: WEIBULL Location Parameter: 17.64420 Scale Parameter: 14.83507 Shape Parameter 1: 1.91358 Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Sample Mean: 30.81141 Sample SD: 7.25338 AndersonDarling Test Statistic Value: 0.33805 Number of Monte Carlo Simulations: 10000.00000 CDF Value: 0.09370 PValue 0.90630 Percent Points of the Reference Distribution  Percent Point Value  0.0 = 0.000 50.0 = 0.772 75.0 = 1.248 90.0 = 1.964 95.0 = 2.579 97.5 = 3.230 99.0 = 4.115 99.5 = 4.814 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 1.964 Accept H0 5% 95% 2.579 Accept H0 2.5% 97.5% 3.230 Accept H0 1% 99% 4.115 Accept H0 *Critical Values Based on 10000 Monte Carlo Simulations ******************** ** normal mle y ** ******************** Normal Parameter Estimation Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Maximum Likelihood: Estimate of Location (Mean): 30.81141 Standard Error of Location: 1.30274 Estimate of Scale (SD): 7.25338 Standard Error of Scale: 0.93640 Loglikelihood: 0.1049126E+03 AIC: 0.2138252E+03 AICc: 0.2142538E+03 BIC: 0.2166932E+03 Confidence Interval for Location Parameter (Normal Approximation)  Confidence Lower Upper Coefficient Limit Limit  50.00 29.92196 31.70087 75.00 29.28321 32.33962 90.00 28.60032 33.02251 95.00 28.15085 33.47198 99.00 27.22887 34.39396 99.90 26.06166 35.56117  Confidence Interval for Scale Parameter (Normal Approximation)  Confidence Lower Upper Coefficient Limit Limit  50.00 6.73462 8.03002 75.00 6.35897 8.58825 90.00 6.00479 9.23849 95.00 5.79626 9.69540 99.00 5.42284 10.69967 99.90 5.03893 12.08652  ************************* ** let ksloc = xmean ** ************************* THE COMPUTED VALUE OF THE CONSTANT KSLOC = 0.3081142E+02 ************************* ** let ksscale = xsd ** ************************* THE COMPUTED VALUE OF THE CONSTANT KSSCALE = 0.7253381E+01 ************************************************** ** set anderson darling critical values table ** ************************************************** THE FORTRAN COMMON CHARACTER VARIABLE ANDEDARL HAS JUST BEEN SET TO TABL ************************************************* ** normal anderson darling goodness of fit y ** ************************************************* AndersonDarling Goodness of Fit Test (Critical Values from Published Tables) Response Variable: Y H0: The distribution fits the data Ha: The distribution does not fit the data Distribution: NORMAL Location Parameter: 30.81141 Scale Parameter: 7.25338 Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Sample Mean: 30.81141 Sample SD: 7.25338 AndersonDarling Test Statistic Value: 0.53219 Adjusted Test Statistic Value: 0.58701 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 0.616 Accept H0 5% 95% 0.735 Accept H0 2.5% 97.5% 0.861 Accept H0 1% 99% 1.020 Accept H0 ******************************************************* ** set anderson darling critical values simulation ** ******************************************************* THE FORTRAN COMMON CHARACTER VARIABLE ANDEDARL HAS JUST BEEN SET TO SIMU ************************************************* ** normal anderson darling goodness of fit y ** ************************************************* AndersonDarling Goodness of Fit Test (Fully Specified Model) Response Variable: Y H0: The distribution fits the data Ha: The distribution does not fit the data Distribution: NORMAL Location Parameter: 30.81141 Scale Parameter: 7.25338 Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Sample Mean: 30.81141 Sample SD: 7.25338 AndersonDarling Test Statistic Value: 0.53219 Number of Monte Carlo Simulations: 10000.00000 CDF Value: 0.29750 PValue 0.70250 Percent Points of the Reference Distribution  Percent Point Value  0.0 = 0.000 50.0 = 0.764 75.0 = 1.231 90.0 = 1.919 95.0 = 2.478 97.5 = 3.115 99.0 = 3.942 99.5 = 4.535 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 1.919 Accept H0 5% 95% 2.478 Accept H0 2.5% 97.5% 3.115 Accept H0 1% 99% 3.942 Accept H0 *Critical Values Based on 10000 Monte Carlo Simulations ********************************************************* ** set kolmogorov smirnov critical values simulation ** ********************************************************* THE FORTRAN COMMON CHARACTER VARIABLE KOLMSMIR HAS JUST BEEN SET TO SIMU *************************************************** ** normal kolmogorov smirnov goodness of fit y ** *************************************************** KolmogorovSmirnov Goodness of Fit Test Response Variable: Y H0: The distribution fits the data Ha: The distribution does not fit the data Distribution: NORMAL Location Parameter: 30.81141 Scale Parameter: 7.25338 Summary Statistics: Number of Observations: 31 Sample Minimum: 18.82999 Sample Maximum: 45.38100 Sample Mean: 30.81141 Sample SD: 7.25338 KolmogorovSmirnov Test Statistic Value: 0.15139 Number of Monte Carlo Simulations: 10000.00000 CDF Value: 0.57660 PValue 0.42340 (Fully Specified Model) Percent Points of the Reference Distribution  Percent Point Value  0.0 = 0.000 50.0 = 0.143 75.0 = 0.176 90.0 = 0.213 95.0 = 0.236 97.5 = 0.256 99.0 = 0.284 99.5 = 0.305 Conclusions (Upper 1Tailed Test)  Alpha CDF Critical Value Conclusion  10% 90% 0.213 Accept H0 5% 95% 0.236 Accept H0 1% 99% 0.284 Accept H0 *Critical Values Based on 10000 Monte Carlo Simulations  
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Date created: 09/22/2011 