# T-test p-value precision hampered by machine epsilon

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#### Details

• Bug
• Status: Closed
• Minor
• Resolution: Fixed
• 1.2
• None
• None

#### Description

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.

#### People

Brent Worden
Peter Wyngaard
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#### Dates

Created:
Updated:
Resolved: