T-test p-value precision hampered by machine epsilon
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Key: MATH-201
URL: https://issues.apache.org/jira/browse/MATH-201
Project: Commons Math
Issue Type: Bug
Affects Versions: 1.2
Reporter: Peter Wyngaard
Priority: Minor
The smallest p-value returned by TTestImpl.tTest() is the machine epsilon,
which is 2.220446E-16 with IEEE754 64-bit double precision floats.
We found this bug porting some analysis software from R to java, and noticed
that the p-values did not match up. We believe we've identified why this is
happening in commons-math-1.2, and a possible solution.
Please be gentle, as I am not a statistics expert!
The following method in org.apache.commons.math.stat.inference.TTestImpl
currently implements the following method to calculate the p-value for a
2-sided, 2-sample t-test:
protected double tTest(double m1, double m2, double v1, double v2, double n1,
double n2)
and it returns:
1.0 - distribution.cumulativeProbability(-t, t);
at line 1034 in version 1.2.
double cumulativeProbability(double x0, double x1) is implemented by
org.apache.commons.math.distribution.AbstractDisstribution, and returns:
return cumulativeProbability(x1) - cumulativeProbability(x0);
So in essence, the p-value returned by TTestImpl.tTest() is:
1.0 - (cumulativeProbability(t) - cumulativeProbabily(-t))
For large-ish t-statistics, cumulativeProbabilty(-t) can get quite small, and
cumulativeProbabilty(t) can get very close to 1.0. When
cumulativeProbability(-t) is less than the machine epsilon, we get p-values
equal to zero because:
1.0 - 1.0 + 0.0 = 0.0
An alternative calculation for the p-value of a 2-sided, 2-sample t-test is:
p = 2.0 * cumulativeProbability(-t)
This calculation does not suffer from the machine epsilon problem, and we are
now getting p-values much smaller than the 2.2E-16 limit we were seeing
previously.
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