STATISTICA Basic Program: Lillief.stb

Products Services Solutions Support Downloads Customers Partners Company

Home I Electronic Statistics Textbook I Contact Us I Request Quote I Search

STATISTICA BASIC Program Lillief.stb

{ This program will generate a userdefined number of samples of fixed size from the standard normal distribution; then the Kolmogorov-Smirnov D statistic is computed for each sample, based on the means and standard deviations estimated from the samples. Finally, the program will compute the 80'th, 85'th, 90'th, 95'th, and 99'th percentile of the distribution of D over the respective number of samples.These percentile values can be used as critical values of D (for alpha = 1-.80=.20, .15, .10, .05, .01) for the Kolmogorov-Smirnov test for normality, when the mean and standard deviation are estimated from the data these are also called the Lilliefors critical value; see Lilliefors, 1967).

This program can easily be modified to derive critical values for the Kolmogorov-Smirnov D statistic for other distributions. Note that you can generate random numbers following any other distribution by using:

x:=vdistrib(rnd(1),p1,p2,...);

where vdistrib stands for the respective inverse distribution function
[e.g., vWeibull(alpha, x, b, theta)], and rnd(1) stands for the random
number function that will produce values for a unit rectangular variate
(see Evans, Hastings, & Peacock, 1993).

References:

Evans, M., Hastings, N., & Peacock, B. (1993).Statistical distributions. New York: Wiley.
Lilliefors, H. W. (1967) "On the Kolmogorov-Smirnov test for normality with mean and variance unknown." ASA Journal, June, 399-402.

NOTE: For large numbers of replications, and for large samples, this program may require some time to complete the compuations.

Program written, modified, or edited at StatSoft, Inc.}

randomaccess;
NoDataFileVariableNames;


	Sub DisplayHistogramOfD(Dvalues,samplesize,nreplications)
{make a histogram of the D values}
	begin
	 dim dvalues(nreplications);
	 VectorSort (dvalues, SORT_ASCENDING);
	 xmax:=dvalues(nreplications);
	 xmin:=0;
	 xstep:=.05;
	 ncats:=0;
	 ncats:=xmax/xstep;
	 if ncats<8 then xstep:=.01;
	 ncats:=xmax/xstep+2;
	 redim xvals(ncats),yvals(ncats);
	 MatrixSetToZero (xvals);
	 MatrixSetToZero (yvals);
	 k:=1;
	 for i:=1 to ncats do begin
	  x:=i*xstep;
	  xvals(i):=x;
nextcase:
	  if dvalues(k)<=x then begin
	   yvals(i):=yvals(i)+1;
	   k:=k+1;
	   if k>nreplications then goto makegraph;
	   goto nextcase;
	  end;
	 end;
makegraph:;
	 Graph:=NewGraph (BARPLOT,
	 'Distribution of K-S D|Samplesize='+str(samplesize,4,0)+
       ', Replications='+str(nreplications,6,0),
	 'Frequencies',
	 'D<=x',ncats, xvals, yvals);
	 GraphSetPlot2DLayout (Graph, 1, BARPLOT,
	 ?DataLabels, ?BarStyle,
	 xstep, ?DevLevel, ?IsRightAxis);
	end;

	Sub DisplayResults(dvalues,samplesize,nreplications)
	begin
	 dim dvalues(nreplications);
	 redim cutoffs(1,5);
	 ValPercentile (dvalues, 1, nreplications, 80, cutoffs(1,1));
	 ValPercentile (dvalues, 1, nreplications, 85, cutoffs(1,2));
	 ValPercentile (dvalues, 1, nreplications, 90, cutoffs(1,3));
	 ValPercentile (dvalues, 1, nreplications, 95, cutoffs(1,4));
	 ValPercentile (dvalues, 1, nreplications, 99, cutoffs(1,5));
	 NewScrollsheet (?NoOfRows, ?NoOfColumns, cutoffs,
	  'Table of Critical Values of K-S D',
	  'N='+Str (samplesize, 3, 0),
	  '  .20   |  .15   |  .10   |  .05   |  .01   ');
	end;

	Function GetParameters(byref samplesize,byref nreplications)
	begin;
	 redim params(2);
	 params(1):=samplesize;
	 params(2):=nreplications;
	 GetParameters:=DisplayNumericInputBox (
	  'Enter the sample size and number of replications:',
	  'Sample size:|Replications:',
	  params);
	 samplesize:=params(1);
	 nreplications:=params(2);
	end;

	Sub GenerateNormalSample(d, n);
	begin
	 dim d(n);
	 for i:=1 to n do d(i):=normal(1);
	end;

	Function ComputeKSDStatistic(d, n);
	begin
	 dim d(n);
	 ValMean (d, 1, n, ex);
	 ValStDeviation (d, 1, n, xt);
	 VectorSort(d, SORT_ASCENDING);
	 f0:=0.0;
	 fn:=0;ff:=0;en:=0;dt:=0;x8:=0;
       f0:=0;
       en:=n;
	 ComputeKSDStatistic:=0.0;
	 for j:=1 to n do begin
     	  fn:=j/en;
	  ff:=INormal (d(j), ex, xt);
	  dt:=MAX(abs(f0-ff),abs(fn-ff));
	  if dt > ComputeKSDStatistic then
	   ComputeKSDStatistic:=dt;
	  f0:=fn;
       end;
	end;

{ Start of main program }
{ Get sample size and number of replications}
	samplesize:=5;
	nreplications:=1000;
      if not GetParameters(samplesize,nreplications)
	 then goto thatsit;
	redim d(samplesize);
	redim dvalues(nreplications);
	for i:=1 to nreplications do begin
{ generate the data}
	  GenerateNormalSample(d,samplesize);
{compute d statistic}
	  dvalues(i):=ComputeKSDStatistic(d,samplesize);
	end;
{prepare results}
	DisplayResults(dvalues,samplesize,nreplications);
{make histogram of dvalues}
	DisplayHistogramOfD(Dvalues,samplesize,nreplications);

thatsit:;
	stop;

Back to List of Programs

[StatSoft]
2300 East 14th Street, Tulsa, OK 74104
Phone: (918) 749-1119; Fax: (918) 749-2217
e-mail: info@statsoft.com