{ Analytica Model Anderson_Darling_tes, encoding="UTF-8" }
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Model Anderson_Darling_tes
Title: Anderson Darling Test Example
Description: The Anderson-Darling test is used to detect whether a set of univariate continuous samples comes from a known distribution. The Kolmogorov-Smirnov Tests are used for the same thing, but differ in the choice of metric used to measure the distance between distributions. The Anderson-Darling test is more powerful (i.e., more likely to detect a difference in distribution when one exists) in many cases.~
~
Here, instead of assuming any particular distribution, I show how to compute the p-value for any distribution null hypothesis in Analytica. The trade-off is that this is indeed distribution-free, but requires a Monte Carlo simulation to compute every p-value.~
~
See <a href="https://docs.analytica.com/index.php/Anderson-Darling_test">Anderson-Darling test</a> for more details on this example model.
Author: Dale Rice~
Lumina Decision Systems
Date: Mon, Mar 27, 2023 2:58 PM
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FileInfo: 0,Model Anderson_Darling_tes,2,2,0,0,W:\TestModels\Anderson-Darling test.ana

Variable X
Title: X
Description: Our example data.
Definition: Random(Logistic(10,2,over:I))
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Index I
Title: I
Description: This indexes our data points.
Definition: 1..15
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Function F(x)
Title: F
Description: The analytic CDF function for the null hypothesis distribution.
Definition: DetermTable(Hypothesis)(CumNormal(x, 10, 2 ), CumLogistic(x, 10, 2 ) )
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Variable A2
Title: A^2
Description: This is the Anderson-Darling statistic, which is usually written an A^2.  It is a measure of distance between a sample of points and a known distribution.
Definition: AD_dist(x,I,F)
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Chance Xr
Title: Xr
Description: Simulated points from the hypothesized distribution, used for computing the p-Value in the Anderson-Darling hypothesis test.
Definition: DetermTable(Hypothesis)(Normal(10,2,over:I),Logistic(10,2,over:I))
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Function AD_dist( x : [I] ; I : Index ; F : Function(x) atom )
Title: AD dist
Description: Returns the A^2 statistic that measures the distance of the distribution of points in «x» from a reference distribution whose CDF is given by F(x).
Definition: Local xs := Sort(x,I);~
Local n := sum(x<>Null, I);~
Local S := Sum( (2*@I-1)/n * (ln(F(xs)) + ln(1-F(Reverse(xs,I)))), I );~
-n - S
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Variable Sim_AD_stat_under_nu
Title: Sim AD stat under null hyp
Description: The Anderson-Darling statistic for each of the Monte Carlo simulation runs.
Definition: AD_dist(Xr, I, F)
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Variable p_value
Title: p value
Description: The probability of seeing an Anderson-Darling distance larger than our measured AD-distance if our null-hypothesis is true, i.e., if the data really is generated from the hypothesized distribution.
Definition: Probability( Sim_AD_stat_under_nu > A2 )
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Decision Hypothesis
Title: Hypothesis
Definition: Choice(Self,2)
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Close Anderson_Darling_tes