def _DegreesOfFreedom(stats1, stats2):
"""Calculates degrees of freedom using the Welch-Satterthwaite formula.
Degrees of freedom is a measure of sample size. For other types of tests,
degrees of freedom is sometimes N - 1, where N is the sample size. However,
for the Welch's t-test, the degrees of freedom is approximated with the
"Welch-Satterthwaite equation".
The degrees of freedom returned from this function should be at least 1.0
because the first row in the t-table is for degrees of freedom of 1.0.
Args:
stats1: An SampleStats named tuple for the first sample.
stats2: An SampleStats named tuple for the second sample.
Returns:
An estimate of degrees of freedom. Guaranteed to be at least 1.0.
Raises:
RuntimeError: Invalid input.
"""
# When there's no variance in either sample, return 1.
if stats1.var == 0 and stats2.var == 0:
return 1.0
if stats1.size < 2:
raise RuntimeError('Sample 1 size < 2. Actual size: %s' % stats1.size)
if stats2.size < 2:
raise RuntimeError('Sample 2 size < 2. Actual size: %s' % stats2.size)
df = math_utils.Divide(
(stats1.var / stats1.size + stats2.var / stats2.size)**2,
math_utils.Divide(stats1.var**2,
(stats1.size**2) * (stats1.size - 1)) +
math_utils.Divide(stats2.var**2, (stats2.size**2) * (stats2.size - 1)))
return max(1.0, df)
def _TValue(stats1, stats2):
"""Calculates a t-statistic value using the formula for Welch's t-test.
The t value can be thought of as a signal-to-noise ratio; a higher t-value
tells you that the groups are more different.
Args:
stats1: An SampleStats named tuple for the first sample.
stats2: An SampleStats named tuple for the second sample.
Returns:
A t value, which may be negative or positive.
"""
# If variance of both segments is zero, then a very high t-value should
# be returned because any difference between the two samples could be
# considered a very clear difference. Also, in the equation, as the
# variance approaches zero, the quotient approaches infinity.
if stats1.var == 0 and stats2.var == 0:
return float('inf')
return math_utils.Divide(
stats1.mean - stats2.mean,
math.sqrt(stats1.var / stats1.size + stats2.var / stats2.size))
def testDivide_UsesFloatArithmetic(self):
self.assertEqual(1.5, math_utils.Divide(3, 2))
def testDivide_ByZero_ReturnsZero(self):
self.assertTrue(math.isnan(math_utils.Divide(1, 0)))
def _Normalize(values):
"""Makes a series with the same shape but with variance = 1, mean = 0."""
mean = math_utils.Mean(values)
zeroed = [x - mean for x in values]
stddev = math_utils.StandardDeviation(zeroed)
return [math_utils.Divide(x, stddev) for x in zeroed]