def equalSampleSize(self, seq1, seq2, totalSeq1, totalSeq2, alpha, beta): oneMinusAlpha = 1.0 - alpha oneMinusBeta = 1.0 - beta p1 = float(seq1) / totalSeq1 p2 = float(seq2) / totalSeq2 q1 = 1.0 - p1 q2 = 1.0 - p2 d = p1 - p2 if d == 0: return 1 return (zScore(oneMinusAlpha) * math.sqrt((p1 + p2)*(q1 + q2)/2) + zScore(oneMinusBeta)*math.sqrt((p1*q1) + (p2*q2)))**2 / (d**2)
def power(self, seq1, seq2, totalSeq1, totalSeq2, alpha):
# The chi-square test is equivalent to the difference between proportions
# test as illustrated by Rivals et al., 2007. Here we use the standard
# asymptotic power formulation for a difference between proportions test.
oneMinusAlpha = 1.0 - alpha
p1 = float(seq1) / totalSeq1
p2 = float(seq2) / totalSeq2
d = p1 - p2
stdDev = math.sqrt((p1 * (1 - p1)) / totalSeq1 +
(p2 * (1 - p2)) / totalSeq2)
if stdDev != 0:
p = float(totalSeq1 * p1 + totalSeq2 * p2) / (totalSeq1 +
totalSeq2)
q = 1 - p
pooledStdDev = math.sqrt((p * q) / totalSeq1 + (p * q) / totalSeq2)
zScore = zScore(oneMinusAlpha)
zLower = (-zScore * pooledStdDev - d) / stdDev
zUpper = (zScore * pooledStdDev - d) / stdDev
return standardNormalCDF(zLower) + (1.0 -
standardNormalCDF(zUpper))
else:
return 1.0
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage):
'''
Calculate ratio of proportions (relative risk) confidence interval.
'''
note = ''
if seq1 == 0 or seq2 == 0:
pseudocount = self.preferences['Pseudocount']
seq1 += pseudocount
seq2 += pseudocount
totalSeq1 += 2 * pseudocount
totalSeq2 += 2 * pseudocount
note = 'degenerate case: CI calculation used pseudocount'
effectSize = (float(seq1) / totalSeq1) / (float(seq2) / totalSeq2)
logEffectSize = math.log(effectSize)
logSE = math.sqrt(1.0 / seq1 - 1.0 / totalSeq1 + 1.0 / seq2 -
1.0 / totalSeq2)
z = zScore(coverage)
logLowerCI = logEffectSize - z * logSE
logUpperCI = logEffectSize + z * logSE
lowerCI = math.exp(logLowerCI)
upperCI = math.exp(logUpperCI)
return lowerCI, upperCI, effectSize, note
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage): ''' Calculate confidence interval using Newcombe-Wilson method. Results are report as percent difference. ''' note = '' if totalSeq1 == 0: totalSeq1 = self.preferences['Pseudocount'] note = 'degenerate case: CI calculation used pseudocount' if totalSeq2 == 0: totalSeq2 = self.preferences['Pseudocount'] note = 'degenerate case: CI calculation used pseudocount' z = zScore(coverage) roots1 = self.NewcombeWilsonFindRoots(seq1, totalSeq1, z) roots2 = self.NewcombeWilsonFindRoots(seq2, totalSeq2, z) diff = float(seq1)/totalSeq1 - float(seq2)/totalSeq2 lowerCI = z*math.sqrt(roots1[0]*(1-roots1[0])/totalSeq1 + roots2[1]*(1-roots2[1])/totalSeq2) upperCI = z*math.sqrt(roots1[1]*(1-roots1[1])/totalSeq1 + roots2[0]*(1-roots2[0])/totalSeq2) return (diff-lowerCI)*100, (diff+upperCI)*100, diff*100, note
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage):
'''
Calculate confidence interval using asymptotic method with a continuity correction.
Results are report as percent difference.
'''
note = ''
if totalSeq1 == 0:
totalSeq1 = self.preferences['Pseudocount']
note = 'degenerate case: CI calculation used pseudocount'
if totalSeq2 == 0:
totalSeq2 = self.preferences['Pseudocount']
note = 'degenerate case: CI calculation used pseudocount'
R1 = float(seq1) / totalSeq1
R2 = float(seq2) / totalSeq2
diff = R1 - R2
stdErr = math.sqrt(
(R1 * (1 - R1)) / totalSeq1 +
(R2 *
(1 - R2)) / totalSeq2) + (1.0 / totalSeq1 + 1.0 / totalSeq2) / 2
offset = zScore(coverage) * stdErr
return (diff - offset) * 100, (diff + offset) * 100, diff * 100, note
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage):
'''
Calculate confidence interval using Newcombe-Wilson method.
Results are report as percent difference.
'''
note = ''
if totalSeq1 == 0:
totalSeq1 = self.preferences['Pseudocount']
note = 'degenerate case: CI calculation used pseudocount'
if totalSeq2 == 0:
totalSeq2 = self.preferences['Pseudocount']
note = 'degenerate case: CI calculation used pseudocount'
z = zScore(coverage)
roots1 = self.NewcombeWilsonFindRoots(seq1, totalSeq1, z)
roots2 = self.NewcombeWilsonFindRoots(seq2, totalSeq2, z)
diff = float(seq1) / totalSeq1 - float(seq2) / totalSeq2
lowerCI = z * math.sqrt(roots1[0] *
(1 - roots1[0]) / totalSeq1 + roots2[1] *
(1 - roots2[1]) / totalSeq2)
upperCI = z * math.sqrt(roots1[1] *
(1 - roots1[1]) / totalSeq1 + roots2[0] *
(1 - roots2[0]) / totalSeq2)
return (diff - lowerCI) * 100, (diff + upperCI) * 100, diff * 100, note
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage): ''' Calculate ratio of proportions (relative risk) confidence interval. ''' note = '' if seq1 == 0 or seq2 == 0: pseudocount = self.preferences['Pseudocount'] seq1 += pseudocount seq2 += pseudocount totalSeq1 += 2*pseudocount totalSeq2 += 2*pseudocount note = 'degenerate case: CI calculation used pseudocount' effectSize = (float(seq1) / totalSeq1) / (float(seq2) / totalSeq2) logEffectSize = math.log(effectSize) logSE = math.sqrt(1.0/seq1 - 1.0/totalSeq1 + 1.0/seq2 - 1.0/totalSeq2) z = zScore(coverage) logLowerCI = logEffectSize - z*logSE logUpperCI = logEffectSize + z*logSE lowerCI = math.exp(logLowerCI) upperCI = math.exp(logUpperCI) return lowerCI, upperCI, effectSize, note
def testNormalDist(self): """Verify computation of normal distribution methods""" from stamp.metagenomics.stats.distributions.NormalDist import standardNormalCDF, zScore self.assertAlmostEqual(standardNormalCDF(-2), 0.022750131948179209) self.assertAlmostEqual(standardNormalCDF(-1), 0.15865525393145705) self.assertAlmostEqual(standardNormalCDF(0), 0.5) self.assertAlmostEqual(standardNormalCDF(1), 0.84134474606854293) self.assertAlmostEqual(standardNormalCDF(2), 0.97724986805182079) self.assertAlmostEqual(standardNormalCDF(-1e-6), 1.0 - standardNormalCDF(1e-6)) self.assertAlmostEqual(standardNormalCDF(-1e-12), 1.0 - standardNormalCDF(1e-12)) self.assertAlmostEqual(zScore(0.90), 1.6448536269514722) self.assertAlmostEqual(zScore(0.95), 1.959963984540054) self.assertAlmostEqual(zScore(0.98), 2.3263478740408408) self.assertAlmostEqual(zScore(0.99), 2.5758293035489004) self.assertAlmostEqual(zScore(0.80), 1.2815515655446004)
def equalSampleSize(self, seq1, seq2, totalSeq1, totalSeq2, alpha, beta):
# The chi-square test is equivalent to the difference between proportions
# test as illustrated by Rivals et al., 2007. Here we use the standard
# equal sample size formulation for a difference between proportions test.
oneMinusAlpha = 1.0 - alpha
oneMinusBeta = 1.0 - beta
p1 = float(seq1) / totalSeq1
p2 = float(seq2) / totalSeq2
q1 = 1.0 - p1
q2 = 1.0 - p2
d = p1 - p2
if d == 0:
return 1
return (zScore(oneMinusAlpha) * math.sqrt((p1 + p2)*(q1 + q2)/2) + zScore(oneMinusBeta)*math.sqrt((p1*q1) + (p2*q2)))**2 / (d**2)
def equalSampleSize(self, seq1, seq2, totalSeq1, totalSeq2, alpha, beta):
# The chi-square test is equivalent to the difference between proportions
# test as illustrated by Rivals et al., 2007. Here we use the standard
# equal sample size formulation for a difference between proportions test.
oneMinusAlpha = 1.0 - alpha
oneMinusBeta = 1.0 - beta
p1 = float(seq1) / totalSeq1
p2 = float(seq2) / totalSeq2
q1 = 1.0 - p1
q2 = 1.0 - p2
d = p1 - p2
if d == 0:
return 1
return (zScore(oneMinusAlpha) * math.sqrt((p1 + p2) * (q1 + q2) / 2) +
zScore(oneMinusBeta) * math.sqrt((p1 * q1) +
(p2 * q2)))**2 / (d**2)
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage):
'''
Calculate odds ratio confidence interval.
'''
a, b, c, d, note = self.tableValues(seq1, seq2, totalSeq1, totalSeq2)
effectSize = (float(a) * d) / (float(b) * c)
logEffectSize = math.log(effectSize)
logSE = math.sqrt(1.0 / a + 1.0 / b + 1.0 / c + 1.0 / d)
z = zScore(coverage)
logLowerCI = logEffectSize - z * logSE
logUpperCI = logEffectSize + z * logSE
lowerCI = math.exp(logLowerCI)
upperCI = math.exp(logUpperCI)
return lowerCI, upperCI, effectSize, note
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage): ''' Calculate odds ratio confidence interval. ''' a, b, c, d, note = self.tableValues(seq1, seq2, totalSeq1, totalSeq2) effectSize = (float(a) * d) / (float(b) * c) logEffectSize = math.log(effectSize) logSE = math.sqrt(1.0/a + 1.0/b + 1.0/c + 1.0/d) z = zScore(coverage) logLowerCI = logEffectSize - z*logSE logUpperCI = logEffectSize + z*logSE lowerCI = math.exp(logLowerCI) upperCI = math.exp(logUpperCI) return lowerCI, upperCI, effectSize, note
def power(self, seq1, seq2, totalSeq1, totalSeq2, alpha): oneMinusAlpha = 1.0 - alpha p1 = float(seq1) / totalSeq1 p2 = float(seq2) / totalSeq2 d = p1 - p2 stdDev = math.sqrt( (p1 * (1-p1)) / totalSeq1 + (p2 * (1 - p2)) / totalSeq2 ) if stdDev != 0: p = float(totalSeq1*p1 + totalSeq2*p2) / (totalSeq1 + totalSeq2) q = 1-p pooledStdDev = math.sqrt( (p*q) / totalSeq1 + (p*q) / totalSeq2 ) zScore = zScore(oneMinusAlpha) zLower = ( -zScore * pooledStdDev - d ) / stdDev zUpper= ( zScore * pooledStdDev - d ) / stdDev return standardNormalCDF(zLower) + (1.0 - standardNormalCDF(zUpper)) else: return 1.0
def run(self, seq1, seq2, totalSeq1, totalSeq2, coverage): ''' Calculate confidence interval using standard asymptotic method. Results are report as percent difference. ''' note = '' if totalSeq1 == 0: totalSeq1 = self.preferences['Pseudocount'] note = 'degenerate case: CI calculation used pseudocount' if totalSeq2 == 0: totalSeq2 = self.preferences['Pseudocount'] note = 'degenerate case: CI calculation used pseudocount' R1 = float(seq1) / totalSeq1 R2 = float(seq2) / totalSeq2 diff = R1 - R2 stdErr = math.sqrt((R1*(1-R1)) / totalSeq1 + (R2*(1-R2)) / totalSeq2) offset = zScore(coverage) * stdErr return (diff - offset) * 100, (diff + offset) * 100, diff * 100, note
def power(self, seq1, seq2, totalSeq1, totalSeq2, alpha):
# The chi-square test is equivalent to the difference between proportions
# test as illustrated by Rivals et al., 2007. Here we use the standard
# asymptotic power formulation for a difference between proportions test.
oneMinusAlpha = 1.0 - alpha
p1 = float(seq1) / totalSeq1
p2 = float(seq2) / totalSeq2
d = p1 - p2
stdDev = math.sqrt( (p1 * (1-p1)) / totalSeq1 + (p2 * (1 - p2)) / totalSeq2 )
if stdDev != 0:
p = float(totalSeq1*p1 + totalSeq2*p2) / (totalSeq1 + totalSeq2)
q = 1-p
pooledStdDev = math.sqrt( (p*q) / totalSeq1 + (p*q) / totalSeq2 )
zScore = zScore(oneMinusAlpha)
zLower = ( -zScore * pooledStdDev - d ) / stdDev
zUpper= ( zScore * pooledStdDev - d ) / stdDev
return standardNormalCDF(zLower) + (1.0 - standardNormalCDF(zUpper))
else:
return 1.0
To facilitate calling this function on several different binomial random variables this is taken as a parameter so it only needs to be calculated once. ''' totalSeqs = max(totalSeqs, 1.0) z = zScore zSqrd = z*z p = float(posSeqs) / totalSeqs q = 1.0 - p term1 = p + zSqrd / (2*totalSeqs) offset = z * math.sqrt(p*q / totalSeqs + zSqrd / (4*totalSeqs*totalSeqs)) denom = 1 + zSqrd / totalSeqs lowerCI = (term1 - offset) / denom upperCI = (term1 + offset) / denom # Good correction, but computationally expensive #if posSeqs >= 1 and posSeqs <=3: # use one-sided Poisson approximation when probability ~= 0 (see Brown et al., 2001) # lowerCI = 0.5*chi2.isf(coverage, 2*posSeqs) / totalSeqs return lowerCI, upperCI, p if __name__ == "__main__": wilsonCI = WilsonCI() lowerCI, upperCI, p = wilsonCI.run(10,100, 0.95, zScore(0.95)) print lowerCI, upperCI, p
'''
totalSeqs = max(totalSeqs, 1.0)
z = zScore
zSqrd = z * z
p = float(posSeqs) / totalSeqs
q = 1.0 - p
term1 = p + zSqrd / (2 * totalSeqs)
offset = z * math.sqrt(p * q / totalSeqs + zSqrd /
(4 * totalSeqs * totalSeqs))
denom = 1 + zSqrd / totalSeqs
lowerCI = (term1 - offset) / denom
upperCI = (term1 + offset) / denom
# Good correction, but computationally expensive
#if posSeqs >= 1 and posSeqs <=3:
# use one-sided Poisson approximation when probability ~= 0 (see Brown et al., 2001)
# lowerCI = 0.5*chi2.isf(coverage, 2*posSeqs) / totalSeqs
return lowerCI, upperCI, p
if __name__ == "__main__":
wilsonCI = WilsonCI()
lowerCI, upperCI, p = wilsonCI.run(10, 100, 0.95, zScore(0.95))
print lowerCI, upperCI, p
