Testing Goodness-Of-Fit
It is often necessary to verify whether the distribution of a variable fits a certain theoretical distribution. Goodness-of-fit tests can be used to perform this verification. Goodness of fit tests require all sample values. They can't be performed using only the summary statistics.
The Chi-Square Test for Goodness-of-Fit
The chi-square goodness-of-fit test compares observed cell frequencies from a sample with the cell frequencies expected from the proposed underlying distribution. The test is based on the assumption that the variable is categorical in nature. When the variable is continuous, the chi-square test cannot be used directly. It is possible to group the data into cells and use the categorized data in the test. The outcome of the test depends on how the continuous data is grouped, so it may not be as reliable.
Two other assumptions are made, namely that the sample is randomly selected from the population, and that the expected frequency of each cell is large enough. A common lower limit is 5. If either of these assumptions is violated, the reliability of the chi-square test may be compromised.
The test statistic is calculated from the difference between the expected and actual cell frequencies. The distribution of the statistic is approximated by the chi square distribution. The approximation is better the higher the expected cell frequencies.
The null hypothesis is that the observed cell frequencies are equal to the expected frequencies. The alternative hypothesis is that at least one cell frequency is different from its expected value.
This test should not be confused with the chi-square test for the variance of a distribution.
The chi-square goodness-of-fit test is implemented by the OneSampleChiSquareTest class.
Example 1 - Fitting a Discrete Distribution
In a gambling game, the payout is directly proportional to the number of sixes that are thrown. A very successful customer has the following results:
# sixes | # throws |
|---|---|
0 | 52 |
1 | 35 |
2 | 11 |
3 | 2 |
The casino management suspects that the customer may be using weighted dice. The significance level for this test is 0.01.
The number of sixes thrown follow a binomial distribution with p = 1/6. The expected values can be calculated easily using the GetExpectedHistogram method of the BinomialDistribution. We then compare the results to the actual:
var sixesDistribution = new BinomialDistribution(3, 1 / 6.0);
var expected = sixesDistribution.GetExpectedHistogram(100);
var actual = Vector.Create<double>(51, 35, 12, 2);
var chiSquare = new ChiSquareGoodnessOfFitTest(actual, expected);
chiSquare.SignificanceLevel = 0.01;
Console.WriteLine("Test statistic: {0:F4}", chiSquare.Statistic);
Console.WriteLine("P-value: {0:F4}", chiSquare.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
chiSquare.Reject() ? "yes" : "no");The value of the chi-square statistic is 9.6013 giving a p-value of 0.0223. As a result, the hypothesis that the dice are weighted is rejected at the 0.01 level.
The One Sample Kolmogorov-Smirnov Test
The one sample Kolmogorov-Smirnov test (KS test) is a one sample test that is used to test the hypothesis that a given sample was taken from a proposed continuous distribution. The test statistic is based on a comparison of the empirical distribution of the sample to the proposed distribution.
One of the advantages of the KS test is that it can be applied to any continuous distribution. On the other hand, it can't be applied to discrete distributions, and is more sensitive near the center of the distribution than at the tails.
The biggest drawback is that the distribution must be completely specified. If one or more of the distribution's parameters is estimated, the distribution of the test statistic is different from the Kolmogorov-Smirnov distribution.
The null hypothesis is always that the population underlying the sample has the proposed distribution. The alternative hypothesis is that the population does not have the proposed distribution.
There is also a two sample Kolmogorov-Smirnov test, which is used to test whether two samples were taken from the same, unknown distribution.
The one sample Kolmogorov-Smirnov test is implemented by the OneSampleKolmogorovSmirnovTest class. It has three constructors. The first constructor takes no arguments. All test parameters must be specified through properties of the test object. The second constructor takes two arguments. The first is a Vector<T> object that specifies the sample. The second is a Func<T, TResult> delegate, which specifies the cumulative distribution function of the distribution being tested. The third constructor also takes two arguments. The first argument is once again vector. The second argument must be of a type derived from ContinuousDistribution.
Example
In this example, we take samples of a lognormal distribution, and test whether it could come from a similar looking Weibull distribution.
var weibull = new WeibullDistribution(2, 1);
var logNormal = new LognormalDistribution(0, 1);
var logNormalSample = logNormal.Sample(25);
var ksTest = new OneSampleKolmogorovSmirnovTest(logNormalSample, weibull);
Console.WriteLine("Test statistic: {0:F4}", ksTest.Statistic);
Console.WriteLine("P-value: {0:F4}", ksTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
ksTest.Reject() ? "yes" : "no");First we create a Weibull and a lognormal distribution. We then get 25 random samples from the lognormal distribution using its Sample(Int32) method.
Because we use random samples, the results of the test are different on each run. The trend is that the p-value is anywhere from 0.03 to 0.3. We can conclude from this that it is not possible to distinguish a lognormal distribution from a Weibull distribution using only 25 sample points.
The Two Sample Kolmogorov-Smirnov Test
The two sample Kolmogorov-Smirnov test is used to test the hypothesis that two samples come from a population with the same, unknown distribution.
The null hypothesis is always that the two samples come from the same underlying distribution. The alternative hypothesis is always that the two samples come from different distributions.
The two sample Kolmogorov-Smirnov test is implemented by the TwoSampleKolmogorovSmirnovTest class. It has two constructors. The first constructor takes no arguments. The second constructor takes two arguments. Both are vectors that specify the two samples that are being compared.
Example
We investigate whether we can distinguish a sample taken from a lognormal distribution from a sample taken from a similar looking Weibull distribution. We use the lognormal samples we created in the previous section.
var weibullSample = weibull.Sample(25);
var ksTest2 = new TwoSampleKolmogorovSmirnovTest(logNormalSample, weibullSample);
Console.WriteLine("Test statistic: {0:F4}", ksTest2.Statistic);
Console.WriteLine("P-value: {0:F4}", ksTest2.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
ksTest2.Reject() ? "yes" : "no");The Anderson-Darling Test for Normality
The Anderson-Darling test is a one sample test of normality. It is a variation of the Kolmogorov-Smirnov test that assigns more weight to the tails of the distribution. Unlike the Kolmogorov-Smirnov test, the distribution of the test statistic is dependent on the distribution. The parameters of the distribution are estimated from the sample.
The null hypothesis is always that the population underlying the sample follows a normal distribution. The alternative hypothesis is always that the underlying population does not follow a normal distribution.
The Anderson-Darling test is implemented by the AndersonDarlingTest class. It has three constructors. The first constructor has no arguments. The second constructor has one argument: a vector that specifies the sample to be tested. The third constructor takes three arguments. The first is once again a vector that specifies the sample. The second and third arguments are the mean and standard deviation of the normal distribution being tested. If no values are provided, the values are estimated from the sample.
Example
We investigate the strength of polished airplane windows. We want to verify that the measured strengths follow a normal distribution. We have a total of 31 samples.
var strength = Vector.Create(
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);
var adTest = new AndersonDarlingTest(strength, 30.81, 7.38);
Console.WriteLine("Test statistic: {0:F4}", adTest.Statistic);
Console.WriteLine("P-value: {0:F4}", adTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
adTest.Reject() ? "yes" : "no");The value of the Anderson-Darling statistic is 0.5128, corresponding to a p-value of 0.1795. We conclude that the window strengths do follow a normal distribution.
The Shapiro-Wilk Test for Normality
The Shapiro-Wilk test is a one sample test of normality. The parameters of the distribution are estimated from the sample. The Shapiro-Wilk test is generally considered more reliable than the Anderson-Darling or Kolmogorov-Smirnov test. It is valid for sample sizes between 3 and 5000.
The null hypothesis is always that the population underlying the sample follows a normal distribution. The alternative hypothesis is always that the underlying population does not follow a normal distribution.
The Shapiro-Wilk test is implemented by the ShapiroWilkTest class. It has two constructors. The first constructor has no arguments. The second constructor has one argument: a vector that specifies the sample to be tested.
Example
As above, we investigate the strength of polished airplane windows. We want to verify that the measured strengths follow a normal distribution. We have a total of 31 samples.
var swTest = new ShapiroWilkTest(strength);
Console.WriteLine("Test statistic: {0:F4}", swTest.Statistic);
Console.WriteLine("P-value: {0:F4}", swTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
swTest.Reject() ? "yes" : "no");The value of the Shapiro-Wilk statistic is 0.9511, corresponding to a p-value of 0.1675. We conclude that the window strengths do follow a normal distribution.