Source code for statsmodels.stats.rates
'''
Test for ratio of Poisson intensities in two independent samples
Author: Josef Perktold
License: BSD-3
'''
import numpy as np
import warnings
from scipy import stats, optimize
from statsmodels.stats.base import HolderTuple
from statsmodels.stats.weightstats import _zstat_generic2
from statsmodels.stats._inference_tools import _mover_confint
# shorthand
norm = stats.norm
method_names_poisson_1samp = {
"test": [
"wald",
"score",
"exact-c",
"midp-c",
"waldccv",
"sqrt-a",
"sqrt-v",
"sqrt",
],
"confint": [
"wald",
"score",
"exact-c",
"midp-c",
"jeff",
"waldccv",
"sqrt-a",
"sqrt-v",
"sqrt",
"sqrt-cent",
"sqrt-centcc",
]
}
[docs]
def test_poisson(count, nobs, value, method=None, alternative="two-sided",
dispersion=1):
"""Test for one sample poisson mean or rate
Parameters
----------
count : array_like
Observed count, number of events.
nobs : arrat_like
Currently this is total exposure time of the count variable.
This will likely change.
value : float, array_like
This is the value of poisson rate under the null hypothesis.
method : str
Method to use for confidence interval.
This is required, there is currently no default method.
See Notes for available methods.
alternative : {'two-sided', 'smaller', 'larger'}
alternative hypothesis, which can be two-sided or either one of the
one-sided tests.
dispersion : float
Dispersion scale coefficient for Poisson QMLE. Default is that the
data follows a Poisson distribution. Dispersion different from 1
correspond to excess-dispersion in Poisson quasi-likelihood (GLM).
Dispersion coeffficient different from one is currently only used in
wald and score method.
Returns
-------
HolderTuple instance with test statistic, pvalue and other attributes.
Notes
-----
The implementatio of the hypothesis test is mainly based on the references
for the confidence interval, see confint_poisson.
Available methods are:
- "score" : based on score test, uses variance under null value
- "wald" : based on wald test, uses variance base on estimated rate.
- "waldccv" : based on wald test with 0.5 count added to variance
computation. This does not use continuity correction for the center of
the confidence interval.
- "exact-c" central confidence interval based on gamma distribution
- "midp-c" : based on midp correction of central exact confidence interval.
this uses numerical inversion of the test function. not vectorized.
- "sqrt" : based on square root transformed counts
- "sqrt-a" based on Anscombe square root transformation of counts + 3/8.
See Also
--------
confint_poisson
"""
n = nobs # short hand
rate = count / n
if method is None:
msg = "method needs to be specified, currently no default method"
raise ValueError(msg)
if dispersion != 1:
if method not in ["wald", "waldcc", "score"]:
msg = "excess dispersion only supported in wald and score methods"
raise ValueError(msg)
dist = "normal"
if method == "wald":
std = np.sqrt(dispersion * rate / n)
statistic = (rate - value) / std
elif method == "waldccv":
# WCC in Barker 2002
# add 0.5 event, not 0.5 event rate as in waldcc
# std = np.sqrt((rate + 0.5 / n) / n)
# statistic = (rate + 0.5 / n - value) / std
std = np.sqrt(dispersion * (rate + 0.5 / n) / n)
statistic = (rate - value) / std
elif method == "score":
std = np.sqrt(dispersion * value / n)
statistic = (rate - value) / std
pvalue = stats.norm.sf(statistic)
elif method.startswith("exact-c") or method.startswith("midp-c"):
pv1 = stats.poisson.cdf(count, n * value)
pv2 = stats.poisson.sf(count - 1, n * value)
if method.startswith("midp-c"):
pv1 = pv1 - 0.5 * stats.poisson.pmf(count, n * value)
pv2 = pv2 - 0.5 * stats.poisson.pmf(count, n * value)
if alternative == "two-sided":
pvalue = 2 * np.minimum(pv1, pv2)
elif alternative == "larger":
pvalue = pv2
elif alternative == "smaller":
pvalue = pv1
else:
msg = 'alternative should be "two-sided", "larger" or "smaller"'
raise ValueError(msg)
statistic = np.nan
dist = "Poisson"
elif method == "sqrt":
std = 0.5
statistic = (np.sqrt(count) - np.sqrt(n * value)) / std
elif method == "sqrt-a":
# anscombe, based on Swift 2009 (with transformation to rate)
std = 0.5
statistic = (np.sqrt(count + 3 / 8) - np.sqrt(n * value + 3 / 8)) / std
elif method == "sqrt-v":
# vandenbroucke, based on Swift 2009 (with transformation to rate)
std = 0.5
crit = stats.norm.isf(0.025)
statistic = (np.sqrt(count + (crit**2 + 2) / 12) -
# np.sqrt(n * value + (crit**2 + 2) / 12)) / std
np.sqrt(n * value)) / std
else:
raise ValueError("unknown method %s" % method)
if dist == 'normal':
statistic, pvalue = _zstat_generic2(statistic, 1, alternative)
res = HolderTuple(
statistic=statistic,
pvalue=np.clip(pvalue, 0, 1),
distribution=dist,
method=method,
alternative=alternative,
rate=rate,
nobs=n
)
return res
[docs]
def confint_poisson(count, exposure, method=None, alpha=0.05):
"""Confidence interval for a Poisson mean or rate
The function is vectorized for all methods except "midp-c", which uses
an iterative method to invert the hypothesis test function.
All current methods are central, that is the probability of each tail is
smaller or equal to alpha / 2. The one-sided interval limits can be
obtained by doubling alpha.
Parameters
----------
count : array_like
Observed count, number of events.
exposure : arrat_like
Currently this is total exposure time of the count variable.
This will likely change.
method : str
Method to use for confidence interval
This is required, there is currently no default method
alpha : float in (0, 1)
Significance level, nominal coverage of the confidence interval is
1 - alpha.
Returns
-------
tuple (low, upp) : confidence limits.
Notes
-----
Methods are mainly based on Barker (2002) [1]_ and Swift (2009) [3]_.
Available methods are:
- "exact-c" central confidence interval based on gamma distribution
- "score" : based on score test, uses variance under null value
- "wald" : based on wald test, uses variance base on estimated rate.
- "waldccv" : based on wald test with 0.5 count added to variance
computation. This does not use continuity correction for the center of
the confidence interval.
- "midp-c" : based on midp correction of central exact confidence interval.
this uses numerical inversion of the test function. not vectorized.
- "jeffreys" : based on Jeffreys' prior. computed using gamma distribution
- "sqrt" : based on square root transformed counts
- "sqrt-a" based on Anscombe square root transformation of counts + 3/8.
- "sqrt-centcc" will likely be dropped. anscombe with continuity corrected
center.
(Similar to R survival cipoisson, but without the 3/8 right shift of
the confidence interval).
sqrt-cent is the same as sqrt-a, using a different computation, will be
deleted.
sqrt-v is a corrected square root method attributed to vandenbrouke, which
might also be deleted.
Todo:
- missing dispersion,
- maybe split nobs and exposure (? needed in NB). Exposure could be used
to standardize rate.
- modified wald, switch method if count=0.
See Also
--------
test_poisson
References
----------
.. [1] Barker, Lawrence. 2002. “A Comparison of Nine Confidence Intervals
for a Poisson Parameter When the Expected Number of Events Is ≤ 5.”
The American Statistician 56 (2): 85–89.
https://doi.org/10.1198/000313002317572736.
.. [2] Patil, VV, and HV Kulkarni. 2012. “Comparison of Confidence
Intervals for the Poisson Mean: Some New Aspects.”
REVSTAT–Statistical Journal 10(2): 211–27.
.. [3] Swift, Michael Bruce. 2009. “Comparison of Confidence Intervals for
a Poisson Mean – Further Considerations.” Communications in Statistics -
Theory and Methods 38 (5): 748–59.
https://doi.org/10.1080/03610920802255856.
"""
n = exposure # short hand
rate = count / exposure
alpha = alpha / 2 # two-sided
if method is None:
msg = "method needs to be specified, currently no default method"
raise ValueError(msg)
if method == "wald":
whalf = stats.norm.isf(alpha) * np.sqrt(rate / n)
ci = (rate - whalf, rate + whalf)
elif method == "waldccv":
# based on WCC in Barker 2002
# add 0.5 event, not 0.5 event rate as in BARKER waldcc
whalf = stats.norm.isf(alpha) * np.sqrt((rate + 0.5 / n) / n)
ci = (rate - whalf, rate + whalf)
elif method == "score":
crit = stats.norm.isf(alpha)
center = count + crit**2 / 2
whalf = crit * np.sqrt((count + crit**2 / 4))
ci = ((center - whalf) / n, (center + whalf) / n)
elif method == "midp-c":
# note local alpha above is for one tail
ci = _invert_test_confint(count, n, alpha=2 * alpha, method="midp-c",
method_start="exact-c")
elif method == "sqrt":
# drop, wrong n
crit = stats.norm.isf(alpha)
center = rate + crit**2 / (4 * n)
whalf = crit * np.sqrt(rate / n)
ci = (center - whalf, center + whalf)
elif method == "sqrt-cent":
crit = stats.norm.isf(alpha)
center = count + crit**2 / 4
whalf = crit * np.sqrt((count + 3 / 8))
ci = ((center - whalf) / n, (center + whalf) / n)
elif method == "sqrt-centcc":
# drop with cc, does not match cipoisson in R survival
crit = stats.norm.isf(alpha)
# avoid sqrt of negative value if count=0
center_low = np.sqrt(np.maximum(count + 3 / 8 - 0.5, 0))
center_upp = np.sqrt(count + 3 / 8 + 0.5)
whalf = crit / 2
# above is for ci of count
ci = (((np.maximum(center_low - whalf, 0))**2 - 3 / 8) / n,
((center_upp + whalf)**2 - 3 / 8) / n)
# crit = stats.norm.isf(alpha)
# center = count
# whalf = crit * np.sqrt((count + 3 / 8 + 0.5))
# ci = ((center - whalf - 0.5) / n, (center + whalf + 0.5) / n)
elif method == "sqrt-a":
# anscombe, based on Swift 2009 (with transformation to rate)
crit = stats.norm.isf(alpha)
center = np.sqrt(count + 3 / 8)
whalf = crit / 2
# above is for ci of count
ci = (((np.maximum(center - whalf, 0))**2 - 3 / 8) / n,
((center + whalf)**2 - 3 / 8) / n)
elif method == "sqrt-v":
# vandenbroucke, based on Swift 2009 (with transformation to rate)
crit = stats.norm.isf(alpha)
center = np.sqrt(count + (crit**2 + 2) / 12)
whalf = crit / 2
# above is for ci of count
ci = (np.maximum(center - whalf, 0))**2 / n, (center + whalf)**2 / n
elif method in ["gamma", "exact-c"]:
# garwood exact, gamma
low = stats.gamma.ppf(alpha, count) / exposure
upp = stats.gamma.isf(alpha, count+1) / exposure
if np.isnan(low).any():
# case with count = 0
if np.size(low) == 1:
low = 0.0
else:
low[np.isnan(low)] = 0.0
ci = (low, upp)
elif method.startswith("jeff"):
# jeffreys, gamma
countc = count + 0.5
ci = (stats.gamma.ppf(alpha, countc) / exposure,
stats.gamma.isf(alpha, countc) / exposure)
else:
raise ValueError("unknown method %s" % method)
ci = (np.maximum(ci[0], 0), ci[1])
return ci
[docs]
def tolerance_int_poisson(count, exposure, prob=0.95, exposure_new=1.,
method=None, alpha=0.05,
alternative="two-sided"):
"""tolerance interval for a poisson observation
Parameters
----------
count : array_like
Observed count, number of events.
exposure : arrat_like
Currently this is total exposure time of the count variable.
prob : float in (0, 1)
Probability of poisson interval, often called "content".
With known parameters, each tail would have at most probability
``1 - prob / 2`` in the two-sided interval.
exposure_new : float
Exposure of the new or predicted observation.
method : str
Method to used for confidence interval of the estimate of the
poisson rate, used in `confint_poisson`.
This is required, there is currently no default method.
alpha : float in (0, 1)
Significance level for the confidence interval of the estimate of the
Poisson rate. Nominal coverage of the confidence interval is
1 - alpha.
alternative : {"two-sider", "larger", "smaller")
The tolerance interval can be two-sided or one-sided.
Alternative "larger" provides the upper bound of the confidence
interval, larger counts are outside the interval.
Returns
-------
tuple (low, upp) of limits of tolerance interval.
The tolerance interval is a closed interval, that is both ``low`` and
``upp`` are in the interval.
Notes
-----
verified against R package tolerance `poistol.int`
See Also
--------
confint_poisson
confint_quantile_poisson
References
----------
.. [1] Hahn, Gerald J., and William Q. Meeker. 1991. Statistical Intervals:
A Guide for Practitioners. 1st ed. Wiley Series in Probability and
Statistics. Wiley. https://doi.org/10.1002/9780470316771.
.. [2] Hahn, Gerald J., and Ramesh Chandra. 1981. “Tolerance Intervals for
Poisson and Binomial Variables.” Journal of Quality Technology 13 (2):
100–110. https://doi.org/10.1080/00224065.1981.11980998.
"""
prob_tail = 1 - prob
alpha_ = alpha
if alternative != "two-sided":
# confint_poisson does not have one-sided alternatives
alpha_ = alpha * 2
low, upp = confint_poisson(count, exposure, method=method, alpha=alpha_)
if exposure_new != 1:
low *= exposure_new
upp *= exposure_new
if alternative == "two-sided":
low_pred = stats.poisson.ppf(prob_tail / 2, low)
upp_pred = stats.poisson.ppf(1 - prob_tail / 2, upp)
elif alternative == "larger":
low_pred = 0
upp_pred = stats.poisson.ppf(1 - prob_tail, upp)
elif alternative == "smaller":
low_pred = stats.poisson.ppf(prob_tail, low)
upp_pred = np.inf
# clip -1 of ppf(0)
low_pred = np.maximum(low_pred, 0)
return low_pred, upp_pred
[docs]
def confint_quantile_poisson(count, exposure, prob, exposure_new=1.,
method=None, alpha=0.05,
alternative="two-sided"):
"""confidence interval for quantile of poisson random variable
Parameters
----------
count : array_like
Observed count, number of events.
exposure : arrat_like
Currently this is total exposure time of the count variable.
prob : float in (0, 1)
Probability for the quantile, e.g. 0.95 to get the upper 95% quantile.
With known mean mu, the quantile would be poisson.ppf(prob, mu).
exposure_new : float
Exposure of the new or predicted observation.
method : str
Method to used for confidence interval of the estimate of the
poisson rate, used in `confint_poisson`.
This is required, there is currently no default method.
alpha : float in (0, 1)
Significance level for the confidence interval of the estimate of the
Poisson rate. Nominal coverage of the confidence interval is
1 - alpha.
alternative : {"two-sider", "larger", "smaller")
The tolerance interval can be two-sided or one-sided.
Alternative "larger" provides the upper bound of the confidence
interval, larger counts are outside the interval.
Returns
-------
tuple (low, upp) of limits of tolerance interval.
The confidence interval is a closed interval, that is both ``low`` and
``upp`` are in the interval.
See Also
--------
confint_poisson
tolerance_int_poisson
References
----------
Hahn, Gerald J, and William Q Meeker. 2010. Statistical Intervals: A Guide
for Practitioners.
"""
alpha_ = alpha
if alternative != "two-sided":
# confint_poisson does not have one-sided alternatives
alpha_ = alpha * 2
low, upp = confint_poisson(count, exposure, method=method, alpha=alpha_)
if exposure_new != 1:
low *= exposure_new
upp *= exposure_new
if alternative == "two-sided":
low_pred = stats.poisson.ppf(prob, low)
upp_pred = stats.poisson.ppf(prob, upp)
elif alternative == "larger":
low_pred = 0
upp_pred = stats.poisson.ppf(prob, upp)
elif alternative == "smaller":
low_pred = stats.poisson.ppf(prob, low)
upp_pred = np.inf
# clip -1 of ppf(0)
low_pred = np.maximum(low_pred, 0)
return low_pred, upp_pred
def _invert_test_confint(count, nobs, alpha=0.05, method="midp-c",
method_start="exact-c"):
"""invert hypothesis test to get confidence interval
"""
def func(r):
v = (test_poisson(count, nobs, value=r, method=method)[1] -
alpha)**2
return v
ci = confint_poisson(count, nobs, method=method_start)
low = optimize.fmin(func, ci[0], xtol=1e-8, disp=False)
upp = optimize.fmin(func, ci[1], xtol=1e-8, disp=False)
assert np.size(low) == 1
return low[0], upp[0]
def _invert_test_confint_2indep(
count1, exposure1, count2, exposure2,
alpha=0.05,
method="score",
compare="diff",
method_start="wald"
):
"""invert hypothesis test to get confidence interval for 2indep
"""
def func(r):
v = (test_poisson_2indep(
count1, exposure1, count2, exposure2,
value=r, method=method, compare=compare
)[1] - alpha)**2
return v
ci = confint_poisson_2indep(count1, exposure1, count2, exposure2,
method=method_start, compare=compare)
low = optimize.fmin(func, ci[0], xtol=1e-8, disp=False)
upp = optimize.fmin(func, ci[1], xtol=1e-8, disp=False)
assert np.size(low) == 1
return low[0], upp[0]
method_names_poisson_2indep = {
"test": {
"ratio": [
"wald",
"score",
"score-log",
"wald-log",
"exact-cond",
"cond-midp",
"sqrt",
"etest-score",
"etest-wald"
],
"diff": [
"wald",
"score",
"waldccv",
"etest-score",
"etest-wald"
]
},
"confint": {
"ratio": [
"waldcc",
"score",
"score-log",
"wald-log",
"sqrtcc",
"mover",
],
"diff": [
"wald",
"score",
"waldccv",
"mover"
]
}
}
[docs]
def test_poisson_2indep(count1, exposure1, count2, exposure2, value=None,
ratio_null=None,
method=None, compare='ratio',
alternative='two-sided', etest_kwds=None):
'''Test for comparing two sample Poisson intensity rates.
Rates are defined as expected count divided by exposure.
The Null and alternative hypothesis for the rates, rate1 and rate2, of two
independent Poisson samples are
for compare = 'diff'
- H0: rate1 - rate2 - value = 0
- H1: rate1 - rate2 - value != 0 if alternative = 'two-sided'
- H1: rate1 - rate2 - value > 0 if alternative = 'larger'
- H1: rate1 - rate2 - value < 0 if alternative = 'smaller'
for compare = 'ratio'
- H0: rate1 / rate2 - value = 0
- H1: rate1 / rate2 - value != 0 if alternative = 'two-sided'
- H1: rate1 / rate2 - value > 0 if alternative = 'larger'
- H1: rate1 / rate2 - value < 0 if alternative = 'smaller'
Parameters
----------
count1 : int
Number of events in first sample, treatment group.
exposure1 : float
Total exposure (time * subjects) in first sample.
count2 : int
Number of events in second sample, control group.
exposure2 : float
Total exposure (time * subjects) in second sample.
ratio_null: float
Ratio of the two Poisson rates under the Null hypothesis. Default is 1.
Deprecated, use ``value`` instead.
.. deprecated:: 0.14.0
Use ``value`` instead.
value : float
Value of the ratio or difference of 2 independent rates under the null
hypothesis. Default is equal rates, i.e. 1 for ratio and 0 for diff.
.. versionadded:: 0.14.0
Replacement for ``ratio_null``.
method : string
Method for the test statistic and the p-value. Defaults to `'score'`.
see Notes.
ratio:
- 'wald': method W1A, wald test, variance based on observed rates
- 'score': method W2A, score test, variance based on estimate under
the Null hypothesis
- 'wald-log': W3A, uses log-ratio, variance based on observed rates
- 'score-log' W4A, uses log-ratio, variance based on estimate under
the Null hypothesis
- 'sqrt': W5A, based on variance stabilizing square root transformation
- 'exact-cond': exact conditional test based on binomial distribution
This uses ``binom_test`` which is minlike in the two-sided case.
- 'cond-midp': midpoint-pvalue of exact conditional test
- 'etest' or 'etest-score: etest with score test statistic
- 'etest-wald': etest with wald test statistic
diff:
- 'wald',
- 'waldccv'
- 'score'
- 'etest-score' or 'etest: etest with score test statistic
- 'etest-wald': etest with wald test statistic
compare : {'diff', 'ratio'}
Default is "ratio".
If compare is `ratio`, then the hypothesis test is for the
rate ratio defined by ratio = rate1 / rate2.
If compare is `diff`, then the hypothesis test is for
diff = rate1 - rate2.
alternative : {"two-sided" (default), "larger", smaller}
The alternative hypothesis, H1, has to be one of the following
- 'two-sided': H1: ratio, or diff, of rates is not equal to value
- 'larger' : H1: ratio, or diff, of rates is larger than value
- 'smaller' : H1: ratio, or diff, of rates is smaller than value
etest_kwds: dictionary
Additional optional parameters to be passed to the etest_poisson_2indep
function, namely y_grid.
Returns
-------
results : instance of HolderTuple class
The two main attributes are test statistic `statistic` and p-value
`pvalue`.
See Also
--------
tost_poisson_2indep
etest_poisson_2indep
Notes
-----
The hypothesis tests for compare="ratio" are based on Gu et al 2018.
The e-tests are also based on ...
- 'wald': method W1A, wald test, variance based on separate estimates
- 'score': method W2A, score test, variance based on estimate under Null
- 'wald-log': W3A, wald test for log transformed ratio
- 'score-log' W4A, score test for log transformed ratio
- 'sqrt': W5A, based on variance stabilizing square root transformation
- 'exact-cond': exact conditional test based on binomial distribution
- 'cond-midp': midpoint-pvalue of exact conditional test
- 'etest': etest with score test statistic
- 'etest-wald': etest with wald test statistic
The hypothesis test for compare="diff" are mainly based on Ng et al 2007
and ...
- wald
- score
- etest-score
- etest-wald
Note the etests use the constraint maximum likelihood estimate (cmle) as
parameters for the underlying Poisson probabilities. The constraint cmle
parameters are the same as in the score test.
The E-test in Krishnamoorty and Thomson uses a moment estimator instead of
the score estimator.
References
----------
.. [1] Gu, Ng, Tang, Schucany 2008: Testing the Ratio of Two Poisson Rates,
Biometrical Journal 50 (2008) 2, 2008
.. [2] Ng, H. K. T., K. Gu, and M. L. Tang. 2007. “A Comparative Study of
Tests for the Difference of Two Poisson Means.”
Computational Statistics & Data Analysis 51 (6): 3085–99.
https://doi.org/10.1016/j.csda.2006.02.004.
'''
# shortcut names
y1, n1, y2, n2 = map(np.asarray, [count1, exposure1, count2, exposure2])
d = n2 / n1
rate1, rate2 = y1 / n1, y2 / n2
rates_cmle = None
if compare == 'ratio':
if method is None:
# default method
method = 'score'
if ratio_null is not None:
warnings.warn("'ratio_null' is deprecated, use 'value' keyword",
FutureWarning)
value = ratio_null
if ratio_null is None and value is None:
# default value
value = ratio_null = 1
else:
# for results holder instance, it still contains ratio_null
ratio_null = value
r = value
r_d = r / d # r1 * n1 / (r2 * n2)
if method in ['score']:
stat = (y1 - y2 * r_d) / np.sqrt((y1 + y2) * r_d)
dist = 'normal'
elif method in ['wald']:
stat = (y1 - y2 * r_d) / np.sqrt(y1 + y2 * r_d**2)
dist = 'normal'
elif method in ['score-log']:
stat = (np.log(y1 / y2) - np.log(r_d))
stat /= np.sqrt((2 + 1 / r_d + r_d) / (y1 + y2))
dist = 'normal'
elif method in ['wald-log']:
stat = (np.log(y1 / y2) - np.log(r_d)) / np.sqrt(1 / y1 + 1 / y2)
dist = 'normal'
elif method in ['sqrt']:
stat = 2 * (np.sqrt(y1 + 3 / 8.) - np.sqrt((y2 + 3 / 8.) * r_d))
stat /= np.sqrt(1 + r_d)
dist = 'normal'
elif method in ['exact-cond', 'cond-midp']:
from statsmodels.stats import proportion
bp = r_d / (1 + r_d)
y_total = y1 + y2
stat = np.nan
# TODO: why y2 in here and not y1, check definition of H1 "larger"
pvalue = proportion.binom_test(y1, y_total, prop=bp,
alternative=alternative)
if method in ['cond-midp']:
# not inplace in case we still want binom pvalue
pvalue = pvalue - 0.5 * stats.binom.pmf(y1, y_total, bp)
dist = 'binomial'
elif method.startswith('etest'):
if method.endswith('wald'):
method_etest = 'wald'
else:
method_etest = 'score'
if etest_kwds is None:
etest_kwds = {}
stat, pvalue = etest_poisson_2indep(
count1, exposure1, count2, exposure2, value=value,
method=method_etest, alternative=alternative, **etest_kwds)
dist = 'poisson'
else:
raise ValueError(f'method "{method}" not recognized')
elif compare == "diff":
if value is None:
value = 0
if method in ['wald']:
stat = (rate1 - rate2 - value) / np.sqrt(rate1 / n1 + rate2 / n2)
dist = 'normal'
"waldccv"
elif method in ['waldccv']:
stat = (rate1 - rate2 - value)
stat /= np.sqrt((count1 + 0.5) / n1**2 + (count2 + 0.5) / n2**2)
dist = 'normal'
elif method in ['score']:
# estimate rates with constraint MLE
count_pooled = y1 + y2
rate_pooled = count_pooled / (n1 + n2)
dt = rate_pooled - value
r2_cmle = 0.5 * (dt + np.sqrt(dt**2 + 4 * value * y2 / (n1 + n2)))
r1_cmle = r2_cmle + value
stat = ((rate1 - rate2 - value) /
np.sqrt(r1_cmle / n1 + r2_cmle / n2))
rates_cmle = (r1_cmle, r2_cmle)
dist = 'normal'
elif method.startswith('etest'):
if method.endswith('wald'):
method_etest = 'wald'
else:
method_etest = 'score'
if method == "etest":
method = method + "-score"
if etest_kwds is None:
etest_kwds = {}
stat, pvalue = etest_poisson_2indep(
count1, exposure1, count2, exposure2, value=value,
method=method_etest, compare="diff",
alternative=alternative, **etest_kwds)
dist = 'poisson'
else:
raise ValueError(f'method "{method}" not recognized')
else:
raise NotImplementedError('"compare" needs to be ratio or diff')
if dist == 'normal':
stat, pvalue = _zstat_generic2(stat, 1, alternative)
rates = (rate1, rate2)
ratio = rate1 / rate2
diff = rate1 - rate2
res = HolderTuple(statistic=stat,
pvalue=pvalue,
distribution=dist,
compare=compare,
method=method,
alternative=alternative,
rates=rates,
ratio=ratio,
diff=diff,
value=value,
rates_cmle=rates_cmle,
ratio_null=ratio_null,
)
return res
def _score_diff(y1, n1, y2, n2, value=0, return_cmle=False):
"""score test and cmle for difference of 2 independent poisson rates
"""
count_pooled = y1 + y2
rate1, rate2 = y1 / n1, y2 / n2
rate_pooled = count_pooled / (n1 + n2)
dt = rate_pooled - value
r2_cmle = 0.5 * (dt + np.sqrt(dt**2 + 4 * value * y2 / (n1 + n2)))
r1_cmle = r2_cmle + value
eps = 1e-20 # avoid zero division in stat_func
v = r1_cmle / n1 + r2_cmle / n2
stat = (rate1 - rate2 - value) / np.sqrt(v + eps)
if return_cmle:
return stat, r1_cmle, r2_cmle
else:
return stat
[docs]
def etest_poisson_2indep(count1, exposure1, count2, exposure2, ratio_null=None,
value=None, method='score', compare="ratio",
alternative='two-sided', ygrid=None,
y_grid=None):
"""
E-test for ratio of two sample Poisson rates.
Rates are defined as expected count divided by exposure. The Null and
alternative hypothesis for the rates, rate1 and rate2, of two independent
Poisson samples are:
for compare = 'diff'
- H0: rate1 - rate2 - value = 0
- H1: rate1 - rate2 - value != 0 if alternative = 'two-sided'
- H1: rate1 - rate2 - value > 0 if alternative = 'larger'
- H1: rate1 - rate2 - value < 0 if alternative = 'smaller'
for compare = 'ratio'
- H0: rate1 / rate2 - value = 0
- H1: rate1 / rate2 - value != 0 if alternative = 'two-sided'
- H1: rate1 / rate2 - value > 0 if alternative = 'larger'
- H1: rate1 / rate2 - value < 0 if alternative = 'smaller'
Parameters
----------
count1 : int
Number of events in first sample
exposure1 : float
Total exposure (time * subjects) in first sample
count2 : int
Number of events in first sample
exposure2 : float
Total exposure (time * subjects) in first sample
ratio_null: float
Ratio of the two Poisson rates under the Null hypothesis. Default is 1.
Deprecated, use ``value`` instead.
.. deprecated:: 0.14.0
Use ``value`` instead.
value : float
Value of the ratio or diff of 2 independent rates under the null
hypothesis. Default is equal rates, i.e. 1 for ratio and 0 for diff.
.. versionadded:: 0.14.0
Replacement for ``ratio_null``.
method : {"score", "wald"}
Method for the test statistic that defines the rejection region.
alternative : string
The alternative hypothesis, H1, has to be one of the following
- 'two-sided': H1: ratio of rates is not equal to ratio_null (default)
- 'larger' : H1: ratio of rates is larger than ratio_null
- 'smaller' : H1: ratio of rates is smaller than ratio_null
y_grid : None or 1-D ndarray
Grid values for counts of the Poisson distribution used for computing
the pvalue. By default truncation is based on an upper tail Poisson
quantiles.
ygrid : None or 1-D ndarray
Same as y_grid. Deprecated. If both y_grid and ygrid are provided,
ygrid will be ignored.
.. deprecated:: 0.14.0
Use ``y_grid`` instead.
Returns
-------
stat_sample : float
test statistic for the sample
pvalue : float
References
----------
Gu, Ng, Tang, Schucany 2008: Testing the Ratio of Two Poisson Rates,
Biometrical Journal 50 (2008) 2, 2008
Ng, H. K. T., K. Gu, and M. L. Tang. 2007. “A Comparative Study of Tests
for the Difference of Two Poisson Means.” Computational Statistics & Data
Analysis 51 (6): 3085–99. https://doi.org/10.1016/j.csda.2006.02.004.
"""
y1, n1, y2, n2 = map(np.asarray, [count1, exposure1, count2, exposure2])
d = n2 / n1
eps = 1e-20 # avoid zero division in stat_func
if compare == "ratio":
if ratio_null is None and value is None:
# default value
value = 1
elif ratio_null is not None:
warnings.warn("'ratio_null' is deprecated, use 'value' keyword",
FutureWarning)
value = ratio_null
r = value # rate1 / rate2
r_d = r / d
rate2_cmle = (y1 + y2) / n2 / (1 + r_d)
rate1_cmle = rate2_cmle * r
if method in ['score']:
def stat_func(x1, x2):
return (x1 - x2 * r_d) / np.sqrt((x1 + x2) * r_d + eps)
# TODO: do I need these? return_results ?
# rate2_cmle = (y1 + y2) / n2 / (1 + r_d)
# rate1_cmle = rate2_cmle * r
# rate1 = rate1_cmle
# rate2 = rate2_cmle
elif method in ['wald']:
def stat_func(x1, x2):
return (x1 - x2 * r_d) / np.sqrt(x1 + x2 * r_d**2 + eps)
# rate2_mle = y2 / n2
# rate1_mle = y1 / n1
# rate1 = rate1_mle
# rate2 = rate2_mle
else:
raise ValueError('method not recognized')
elif compare == "diff":
if value is None:
value = 0
tmp = _score_diff(y1, n1, y2, n2, value=value, return_cmle=True)
_, rate1_cmle, rate2_cmle = tmp
if method in ['score']:
def stat_func(x1, x2):
return _score_diff(x1, n1, x2, n2, value=value)
elif method in ['wald']:
def stat_func(x1, x2):
rate1, rate2 = x1 / n1, x2 / n2
stat = (rate1 - rate2 - value)
stat /= np.sqrt(rate1 / n1 + rate2 / n2 + eps)
return stat
else:
raise ValueError('method not recognized')
# The sampling distribution needs to be based on the null hypotheis
# use constrained MLE from 'score' calculation
rate1 = rate1_cmle
rate2 = rate2_cmle
mean1 = n1 * rate1
mean2 = n2 * rate2
stat_sample = stat_func(y1, y2)
if ygrid is not None:
warnings.warn("ygrid is deprecated, use y_grid", FutureWarning)
y_grid = y_grid if y_grid is not None else ygrid
# The following uses a fixed truncation for evaluating the probabilities
# It will currently only work for small counts, so that sf at truncation
# point is small
# We can make it depend on the amount of truncated sf.
# Some numerical optimization or checks for large means need to be added.
if y_grid is None:
threshold = stats.poisson.isf(1e-13, max(mean1, mean2))
threshold = max(threshold, 100) # keep at least 100
y_grid = np.arange(threshold + 1)
else:
y_grid = np.asarray(y_grid)
if y_grid.ndim != 1:
raise ValueError("y_grid needs to be None or 1-dimensional array")
pdf1 = stats.poisson.pmf(y_grid, mean1)
pdf2 = stats.poisson.pmf(y_grid, mean2)
stat_space = stat_func(y_grid[:, None], y_grid[None, :]) # broadcasting
eps = 1e-15 # correction for strict inequality check
if alternative in ['two-sided', '2-sided', '2s']:
mask = np.abs(stat_space) >= (np.abs(stat_sample) - eps)
elif alternative in ['larger', 'l']:
mask = stat_space >= (stat_sample - eps)
elif alternative in ['smaller', 's']:
mask = stat_space <= (stat_sample + eps)
else:
raise ValueError('invalid alternative')
pvalue = ((pdf1[:, None] * pdf2[None, :])[mask]).sum()
return stat_sample, pvalue
[docs]
def tost_poisson_2indep(count1, exposure1, count2, exposure2, low, upp,
method='score', compare='ratio'):
'''Equivalence test based on two one-sided `test_proportions_2indep`
This assumes that we have two independent poisson samples.
The Null and alternative hypothesis for equivalence testing are
for compare = 'ratio'
- H0: rate1 / rate2 <= low or upp <= rate1 / rate2
- H1: low < rate1 / rate2 < upp
for compare = 'diff'
- H0: rate1 - rate2 <= low or upp <= rate1 - rate2
- H1: low < rate - rate < upp
Parameters
----------
count1 : int
Number of events in first sample
exposure1 : float
Total exposure (time * subjects) in first sample
count2 : int
Number of events in second sample
exposure2 : float
Total exposure (time * subjects) in second sample
low, upp :
equivalence margin for the ratio or difference of Poisson rates
method: string
TOST uses ``test_poisson_2indep`` and has the same methods.
ratio:
- 'wald': method W1A, wald test, variance based on observed rates
- 'score': method W2A, score test, variance based on estimate under
the Null hypothesis
- 'wald-log': W3A, uses log-ratio, variance based on observed rates
- 'score-log' W4A, uses log-ratio, variance based on estimate under
the Null hypothesis
- 'sqrt': W5A, based on variance stabilizing square root transformation
- 'exact-cond': exact conditional test based on binomial distribution
This uses ``binom_test`` which is minlike in the two-sided case.
- 'cond-midp': midpoint-pvalue of exact conditional test
- 'etest' or 'etest-score: etest with score test statistic
- 'etest-wald': etest with wald test statistic
diff:
- 'wald',
- 'waldccv'
- 'score'
- 'etest-score' or 'etest: etest with score test statistic
- 'etest-wald': etest with wald test statistic
Returns
-------
results : instance of HolderTuple class
The two main attributes are test statistic `statistic` and p-value
`pvalue`.
References
----------
Gu, Ng, Tang, Schucany 2008: Testing the Ratio of Two Poisson Rates,
Biometrical Journal 50 (2008) 2, 2008
See Also
--------
test_poisson_2indep
confint_poisson_2indep
'''
tt1 = test_poisson_2indep(count1, exposure1, count2, exposure2,
value=low, method=method,
compare=compare,
alternative='larger')
tt2 = test_poisson_2indep(count1, exposure1, count2, exposure2,
value=upp, method=method,
compare=compare,
alternative='smaller')
# idx_max = 1 if t1.pvalue < t2.pvalue else 0
idx_max = np.asarray(tt1.pvalue < tt2.pvalue, int)
statistic = np.choose(idx_max, [tt1.statistic, tt2.statistic])
pvalue = np.choose(idx_max, [tt1.pvalue, tt2.pvalue])
res = HolderTuple(statistic=statistic,
pvalue=pvalue,
method=method,
compare=compare,
equiv_limits=(low, upp),
results_larger=tt1,
results_smaller=tt2,
title="Equivalence test for 2 independent Poisson rates"
)
return res
[docs]
def nonequivalence_poisson_2indep(count1, exposure1, count2, exposure2,
low, upp, method='score', compare="ratio"):
"""Test for non-equivalence, minimum effect for poisson.
This reverses null and alternative hypothesis compared to equivalence
testing. The null hypothesis is that the effect, ratio (or diff), is in
an interval that specifies a range of irrelevant or unimportant
differences between the two samples.
The Null and alternative hypothesis comparing the ratio of rates are
for compare = 'ratio':
- H0: low < rate1 / rate2 < upp
- H1: rate1 / rate2 <= low or upp <= rate1 / rate2
for compare = 'diff':
- H0: rate1 - rate2 <= low or upp <= rate1 - rate2
- H1: low < rate - rate < upp
Notes
-----
This is implemented as two one-sided tests at the minimum effect boundaries
(low, upp) with (nominal) size alpha / 2 each.
The size of the test is the sum of the two one-tailed tests, which
corresponds to an equal-tailed two-sided test.
If low and upp are equal, then the result is the same as the standard
two-sided test.
The p-value is computed as `2 * min(pvalue_low, pvalue_upp)` in analogy to
two-sided equal-tail tests.
In large samples the nominal size of the test will be below alpha.
References
----------
.. [1] Hodges, J. L., Jr., and E. L. Lehmann. 1954. Testing the Approximate
Validity of Statistical Hypotheses. Journal of the Royal Statistical
Society, Series B (Methodological) 16: 261–68.
.. [2] Kim, Jae H., and Andrew P. Robinson. 2019. “Interval-Based
Hypothesis Testing and Its Applications to Economics and Finance.”
Econometrics 7 (2): 21. https://doi.org/10.3390/econometrics7020021.
"""
tt1 = test_poisson_2indep(count1, exposure1, count2, exposure2,
value=low, method=method, compare=compare,
alternative='smaller')
tt2 = test_poisson_2indep(count1, exposure1, count2, exposure2,
value=upp, method=method, compare=compare,
alternative='larger')
# idx_min = 0 if tt1.pvalue < tt2.pvalue else 1
idx_min = np.asarray(tt1.pvalue < tt2.pvalue, int)
pvalue = 2 * np.minimum(tt1.pvalue, tt2.pvalue)
statistic = np.choose(idx_min, [tt1.statistic, tt2.statistic])
res = HolderTuple(statistic=statistic,
pvalue=pvalue,
method=method,
results_larger=tt1,
results_smaller=tt2,
title="Equivalence test for 2 independent Poisson rates"
)
return res
[docs]
def confint_poisson_2indep(count1, exposure1, count2, exposure2,
method='score', compare='ratio', alpha=0.05,
method_mover="score",
):
"""Confidence interval for ratio or difference of 2 indep poisson rates.
Parameters
----------
count1 : int
Number of events in first sample.
exposure1 : float
Total exposure (time * subjects) in first sample.
count2 : int
Number of events in second sample.
exposure2 : float
Total exposure (time * subjects) in second sample.
method : string
Method for the test statistic and the p-value. Defaults to `'score'`.
see Notes.
ratio:
- 'wald': NOT YET, method W1A, wald test, variance based on observed
rates
- 'waldcc' :
- 'score': method W2A, score test, variance based on estimate under
the Null hypothesis
- 'wald-log': W3A, uses log-ratio, variance based on observed rates
- 'score-log' W4A, uses log-ratio, variance based on estimate under
the Null hypothesis
- 'sqrt': W5A, based on variance stabilizing square root transformation
- 'sqrtcc' :
- 'exact-cond': NOT YET, exact conditional test based on binomial
distribution
This uses ``binom_test`` which is minlike in the two-sided case.
- 'cond-midp': NOT YET, midpoint-pvalue of exact conditional test
- 'mover' :
diff:
- 'wald',
- 'waldccv'
- 'score'
- 'mover'
compare : {'diff', 'ratio'}
Default is "ratio".
If compare is `diff`, then the hypothesis test is for
diff = rate1 - rate2.
If compare is `ratio`, then the hypothesis test is for the
rate ratio defined by ratio = rate1 / rate2.
alternative : string
The alternative hypothesis, H1, has to be one of the following
- 'two-sided': H1: ratio of rates is not equal to ratio_null (default)
- 'larger' : H1: ratio of rates is larger than ratio_null
- 'smaller' : H1: ratio of rates is smaller than ratio_null
alpha : float in (0, 1)
Significance level, nominal coverage of the confidence interval is
1 - alpha.
Returns
-------
tuple (low, upp) : confidence limits.
"""
# shortcut names
y1, n1, y2, n2 = map(np.asarray, [count1, exposure1, count2, exposure2])
rate1, rate2 = y1 / n1, y2 / n2
alpha = alpha / 2 # two-sided only
if compare == "ratio":
if method == "score":
low, upp = _invert_test_confint_2indep(
count1, exposure1, count2, exposure2,
alpha=alpha * 2, # check how alpha is defined
method="score",
compare="ratio",
method_start="waldcc"
)
ci = (low, upp)
elif method == "wald-log":
crit = stats.norm.isf(alpha)
c = 0
center = (count1 + c) / (count2 + c) * n2 / n1
std = np.sqrt(1 / (count1 + c) + 1 / (count2 + c))
ci = (center * np.exp(- crit * std), center * np.exp(crit * std))
elif method == "score-log":
low, upp = _invert_test_confint_2indep(
count1, exposure1, count2, exposure2,
alpha=alpha * 2, # check how alpha is defined
method="score-log",
compare="ratio",
method_start="waldcc"
)
ci = (low, upp)
elif method == "waldcc":
crit = stats.norm.isf(alpha)
center = (count1 + 0.5) / (count2 + 0.5) * n2 / n1
std = np.sqrt(1 / (count1 + 0.5) + 1 / (count2 + 0.5))
ci = (center * np.exp(- crit * std), center * np.exp(crit * std))
elif method == "sqrtcc":
# coded based on Price, Bonett 2000 equ (2.4)
crit = stats.norm.isf(alpha)
center = np.sqrt((count1 + 0.5) * (count2 + 0.5))
std = 0.5 * np.sqrt(count1 + 0.5 + count2 + 0.5 - 0.25 * crit)
denom = (count2 + 0.5 - 0.25 * crit**2)
low_sqrt = (center - crit * std) / denom
upp_sqrt = (center + crit * std) / denom
ci = (low_sqrt**2, upp_sqrt**2)
elif method == "mover":
method_p = method_mover
ci1 = confint_poisson(y1, n1, method=method_p, alpha=2*alpha)
ci2 = confint_poisson(y2, n2, method=method_p, alpha=2*alpha)
ci = _mover_confint(rate1, rate2, ci1, ci2, contrast="ratio")
else:
raise ValueError(f'method "{method}" not recognized')
ci = (np.maximum(ci[0], 0), ci[1])
elif compare == "diff":
if method in ['wald']:
crit = stats.norm.isf(alpha)
center = rate1 - rate2
half = crit * np.sqrt(rate1 / n1 + rate2 / n2)
ci = center - half, center + half
elif method in ['waldccv']:
crit = stats.norm.isf(alpha)
center = rate1 - rate2
std = np.sqrt((count1 + 0.5) / n1**2 + (count2 + 0.5) / n2**2)
half = crit * std
ci = center - half, center + half
elif method == "score":
low, upp = _invert_test_confint_2indep(
count1, exposure1, count2, exposure2,
alpha=alpha * 2, # check how alpha is defined
method="score",
compare="diff",
method_start="waldccv"
)
ci = (low, upp)
elif method == "mover":
method_p = method_mover
ci1 = confint_poisson(y1, n1, method=method_p, alpha=2*alpha)
ci2 = confint_poisson(y2, n2, method=method_p, alpha=2*alpha)
ci = _mover_confint(rate1, rate2, ci1, ci2, contrast="diff")
else:
raise ValueError(f'method "{method}" not recognized')
else:
raise NotImplementedError('"compare" needs to be ratio or diff')
return ci
[docs]
def power_poisson_ratio_2indep(
rate1, rate2, nobs1,
nobs_ratio=1,
exposure=1,
value=0,
alpha=0.05,
dispersion=1,
alternative="smaller",
method_var="alt",
return_results=True,
):
"""Power of test of ratio of 2 independent poisson rates.
This is based on Zhu and Zhu and Lakkis. It does not directly correspond
to `test_poisson_2indep`.
Parameters
----------
rate1 : float
Poisson rate for the first sample, treatment group, under the
alternative hypothesis.
rate2 : float
Poisson rate for the second sample, reference group, under the
alternative hypothesis.
nobs1 : float or int
Number of observations in sample 1.
nobs_ratio : float
Sample size ratio, nobs2 = nobs_ratio * nobs1.
exposure : float
Exposure for each observation. Total exposure is nobs1 * exposure
and nobs2 * exposure.
alpha : float in interval (0,1)
Significance level, e.g. 0.05, is the probability of a type I
error, that is wrong rejections if the Null Hypothesis is true.
value : float
Rate ratio, rate1 / rate2, under the null hypothesis.
dispersion : float
Dispersion coefficient for quasi-Poisson. Dispersion different from
one can capture over or under dispersion relative to Poisson
distribution.
method_var : {"score", "alt"}
The variance of the test statistic for the null hypothesis given the
rates under the alternative can be either equal to the rates under the
alternative ``method_var="alt"``, or estimated under the constrained
of the null hypothesis, ``method_var="score"``.
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is False, then only the power is returned.
If return_results is True, then a results instance with the
information in attributes is returned.
power : float
Power of the test, e.g. 0.8, is one minus the probability of a
type II error. Power is the probability that the test correctly
rejects the Null Hypothesis if the Alternative Hypothesis is true.
Other attributes in results instance include :
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
References
----------
.. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two
Poisson or Negative Binomial Rates in Noninferiority or Equivalence
Trials.” Statistics in Biopharmaceutical Research, March.
https://doi.org/10.1080/19466315.2016.1225594
.. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for
Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3):
376–87. https://doi.org/10.1002/sim.5947.
.. [3] PASS documentation
"""
# TODO: avoid possible circular import, check if needed
from statsmodels.stats.power import normal_power_het
rate1, rate2, nobs1 = map(np.asarray, [rate1, rate2, nobs1])
nobs2 = nobs_ratio * nobs1
v1 = dispersion / exposure * (1 / rate1 + 1 / (nobs_ratio * rate2))
if method_var == "alt":
v0 = v1
elif method_var == "score":
# nobs_ratio = 1 / nobs_ratio
v0 = dispersion / exposure * (1 + value / nobs_ratio)**2
v0 /= value / nobs_ratio * (rate1 + (nobs_ratio * rate2))
else:
raise NotImplementedError(f"method_var {method_var} not recognized")
std_null = np.sqrt(v0)
std_alt = np.sqrt(v1)
es = np.log(rate1 / rate2) - np.log(value)
pow_ = normal_power_het(es, nobs1, alpha, std_null=std_null,
std_alternative=std_alt,
alternative=alternative)
p_pooled = None # TODO: replace or remove
if return_results:
res = HolderTuple(
power=pow_,
p_pooled=p_pooled,
std_null=std_null,
std_alt=std_alt,
nobs1=nobs1,
nobs2=nobs2,
nobs_ratio=nobs_ratio,
alpha=alpha,
tuple_=("power",), # override default
)
return res
return pow_
[docs]
def power_equivalence_poisson_2indep(rate1, rate2, nobs1,
low, upp, nobs_ratio=1,
exposure=1, alpha=0.05, dispersion=1,
method_var="alt",
return_results=False):
"""Power of equivalence test of ratio of 2 independent poisson rates.
Parameters
----------
rate1 : float
Poisson rate for the first sample, treatment group, under the
alternative hypothesis.
rate2 : float
Poisson rate for the second sample, reference group, under the
alternative hypothesis.
nobs1 : float or int
Number of observations in sample 1.
low : float
Lower equivalence margin for the rate ratio, rate1 / rate2.
upp : float
Upper equivalence margin for the rate ratio, rate1 / rate2.
nobs_ratio : float
Sample size ratio, nobs2 = nobs_ratio * nobs1.
exposure : float
Exposure for each observation. Total exposure is nobs1 * exposure
and nobs2 * exposure.
alpha : float in interval (0,1)
Significance level, e.g. 0.05, is the probability of a type I
error, that is wrong rejections if the Null Hypothesis is true.
value : float
Difference between rates 1 and 2 under the null hypothesis.
method_var : {"score", "alt"}
The variance of the test statistic for the null hypothesis given the
rates uder the alternative, can be either equal to the rates under the
alternative ``method_var="alt"``, or estimated under the constrained
of the null hypothesis, ``method_var="score"``.
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is False, then only the power is returned.
If return_results is True, then a results instance with the
information in attributes is returned.
power : float
Power of the test, e.g. 0.8, is one minus the probability of a
type II error. Power is the probability that the test correctly
rejects the Null Hypothesis if the Alternative Hypothesis is true.
Other attributes in results instance include :
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
References
----------
.. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two
Poisson or Negative Binomial Rates in Noninferiority or Equivalence
Trials.” Statistics in Biopharmaceutical Research, March.
https://doi.org/10.1080/19466315.2016.1225594
.. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for
Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3):
376–87. https://doi.org/10.1002/sim.5947.
.. [3] PASS documentation
"""
rate1, rate2, nobs1 = map(np.asarray, [rate1, rate2, nobs1])
nobs2 = nobs_ratio * nobs1
v1 = dispersion / exposure * (1 / rate1 + 1 / (nobs_ratio * rate2))
if method_var == "alt":
v0_low = v0_upp = v1
elif method_var == "score":
v0_low = dispersion / exposure * (1 + low * nobs_ratio)**2
v0_low /= low * nobs_ratio * (rate1 + (nobs_ratio * rate2))
v0_upp = dispersion / exposure * (1 + upp * nobs_ratio)**2
v0_upp /= upp * nobs_ratio * (rate1 + (nobs_ratio * rate2))
else:
raise NotImplementedError(f"method_var {method_var} not recognized")
es_low = np.log(rate1 / rate2) - np.log(low)
es_upp = np.log(rate1 / rate2) - np.log(upp)
std_null_low = np.sqrt(v0_low)
std_null_upp = np.sqrt(v0_upp)
std_alternative = np.sqrt(v1)
pow_ = _power_equivalence_het(es_low, es_upp, nobs2, alpha=alpha,
std_null_low=std_null_low,
std_null_upp=std_null_upp,
std_alternative=std_alternative)
if return_results:
res = HolderTuple(
power=pow_[0],
power_margins=pow[1:],
std_null_low=std_null_low,
std_null_upp=std_null_upp,
std_alt=std_alternative,
nobs1=nobs1,
nobs2=nobs2,
nobs_ratio=nobs_ratio,
alpha=alpha,
tuple_=("power",), # override default
)
return res
else:
return pow_[0]
def _power_equivalence_het_v0(es_low, es_upp, nobs, alpha=0.05,
std_null_low=None,
std_null_upp=None,
std_alternative=None):
"""power for equivalence test
"""
s0_low = std_null_low
s0_upp = std_null_upp
s1 = std_alternative
crit = norm.isf(alpha)
pow_ = (
norm.cdf((np.sqrt(nobs) * es_low - crit * s0_low) / s1) +
norm.cdf((np.sqrt(nobs) * es_upp - crit * s0_upp) / s1) - 1
)
return pow_
def _power_equivalence_het(es_low, es_upp, nobs, alpha=0.05,
std_null_low=None,
std_null_upp=None,
std_alternative=None):
"""power for equivalence test
"""
s0_low = std_null_low
s0_upp = std_null_upp
s1 = std_alternative
crit = norm.isf(alpha)
# Note: rejection region is an interval [low, upp]
# Here we compute the complement of the two tail probabilities
p1 = norm.sf((np.sqrt(nobs) * es_low - crit * s0_low) / s1)
p2 = norm.cdf((np.sqrt(nobs) * es_upp + crit * s0_upp) / s1)
pow_ = 1 - (p1 + p2)
return pow_, p1, p2
def _std_2poisson_power(
rate1, rate2, nobs_ratio=1, alpha=0.05,
exposure=1,
dispersion=1,
value=0,
method_var="score",
):
rates_pooled = (rate1 + rate2 * nobs_ratio) / (1 + nobs_ratio)
# v1 = dispersion / exposure * (1 / rate2 + 1 / (nobs_ratio * rate1))
if method_var == "alt":
v0 = v1 = rate1 + rate2 / nobs_ratio
else:
# uaw n1 = 1 as normalization
_, r1_cmle, r2_cmle = _score_diff(
rate1, 1, rate2 * nobs_ratio, nobs_ratio, value=value,
return_cmle=True)
v1 = rate1 + rate2 / nobs_ratio
v0 = r1_cmle + r2_cmle / nobs_ratio
return rates_pooled, np.sqrt(v0), np.sqrt(v1)
[docs]
def power_poisson_diff_2indep(rate1, rate2, nobs1, nobs_ratio=1, alpha=0.05,
value=0,
method_var="score",
alternative='two-sided',
return_results=True):
"""Power of ztest for the difference between two independent poisson rates.
Parameters
----------
rate1 : float
Poisson rate for the first sample, treatment group, under the
alternative hypothesis.
rate2 : float
Poisson rate for the second sample, reference group, under the
alternative hypothesis.
nobs1 : float or int
Number of observations in sample 1.
nobs_ratio : float
Sample size ratio, nobs2 = nobs_ratio * nobs1.
alpha : float in interval (0,1)
Significance level, e.g. 0.05, is the probability of a type I
error, that is wrong rejections if the Null Hypothesis is true.
value : float
Difference between rates 1 and 2 under the null hypothesis.
method_var : {"score", "alt"}
The variance of the test statistic for the null hypothesis given the
rates uder the alternative, can be either equal to the rates under the
alternative ``method_var="alt"``, or estimated under the constrained
of the null hypothesis, ``method_var="score"``.
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is False, then only the power is returned.
If return_results is True, then a results instance with the
information in attributes is returned.
power : float
Power of the test, e.g. 0.8, is one minus the probability of a
type II error. Power is the probability that the test correctly
rejects the Null Hypothesis if the Alternative Hypothesis is true.
Other attributes in results instance include :
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
References
----------
.. [1] Stucke, Kathrin, and Meinhard Kieser. 2013. “Sample Size
Calculations for Noninferiority Trials with Poisson Distributed Count
Data.” Biometrical Journal 55 (2): 203–16.
https://doi.org/10.1002/bimj.201200142.
.. [2] PASS manual chapter 436
"""
# TODO: avoid possible circular import, check if needed
from statsmodels.stats.power import normal_power_het
rate1, rate2, nobs1 = map(np.asarray, [rate1, rate2, nobs1])
diff = rate1 - rate2
_, std_null, std_alt = _std_2poisson_power(
rate1,
rate2,
nobs_ratio=nobs_ratio,
alpha=alpha,
value=value,
method_var=method_var,
)
pow_ = normal_power_het(diff - value, nobs1, alpha, std_null=std_null,
std_alternative=std_alt,
alternative=alternative)
if return_results:
res = HolderTuple(
power=pow_,
rates_alt=(rate2 + diff, rate2),
std_null=std_null,
std_alt=std_alt,
nobs1=nobs1,
nobs2=nobs_ratio * nobs1,
nobs_ratio=nobs_ratio,
alpha=alpha,
tuple_=("power",), # override default
)
return res
else:
return pow_
def _var_cmle_negbin(rate1, rate2, nobs_ratio, exposure=1, value=1,
dispersion=0):
"""
variance based on constrained cmle, for score test version
for ratio comparison of two negative binomial samples
value = rate1 / rate2 under the null
"""
# definitions in Zhu
# nobs_ratio = n1 / n0
# value = ratio = r1 / r0
rate0 = rate2 # control
nobs_ratio = 1 / nobs_ratio
a = - dispersion * exposure * value * (1 + nobs_ratio)
b = (dispersion * exposure * (rate0 * value + nobs_ratio * rate1) -
(1 + nobs_ratio * value))
c = rate0 + nobs_ratio * rate1
if dispersion == 0:
r0 = -c / b
else:
r0 = (-b - np.sqrt(b**2 - 4 * a * c)) / (2 * a)
r1 = r0 * value
v = (1 / exposure / r0 * (1 + 1 / value / nobs_ratio) +
(1 + nobs_ratio) / nobs_ratio * dispersion)
r2 = r0
return v * nobs_ratio, r1, r2
[docs]
def power_negbin_ratio_2indep(
rate1, rate2, nobs1,
nobs_ratio=1,
exposure=1,
value=1,
alpha=0.05,
dispersion=0.01,
alternative="two-sided",
method_var="alt",
return_results=True):
"""
Power of test of ratio of 2 independent negative binomial rates.
Parameters
----------
rate1 : float
Poisson rate for the first sample, treatment group, under the
alternative hypothesis.
rate2 : float
Poisson rate for the second sample, reference group, under the
alternative hypothesis.
nobs1 : float or int
Number of observations in sample 1.
low : float
Lower equivalence margin for the rate ratio, rate1 / rate2.
upp : float
Upper equivalence margin for the rate ratio, rate1 / rate2.
nobs_ratio : float
Sample size ratio, nobs2 = nobs_ratio * nobs1.
exposure : float
Exposure for each observation. Total exposure is nobs1 * exposure
and nobs2 * exposure.
value : float
Rate ratio, rate1 / rate2, under the null hypothesis.
alpha : float in interval (0,1)
Significance level, e.g. 0.05, is the probability of a type I
error, that is wrong rejections if the Null Hypothesis is true.
dispersion : float >= 0.
Dispersion parameter for Negative Binomial distribution.
The Poisson limiting case corresponds to ``dispersion=0``.
method_var : {"score", "alt"}
The variance of the test statistic for the null hypothesis given the
rates under the alternative, can be either equal to the rates under the
alternative ``method_var="alt"``, or estimated under the constrained
of the null hypothesis, ``method_var="score"``, or based on a moment
constrained estimate, ``method_var="ftotal"``. see references.
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is False, then only the power is returned.
If return_results is True, then a results instance with the
information in attributes is returned.
power : float
Power of the test, e.g. 0.8, is one minus the probability of a
type II error. Power is the probability that the test correctly
rejects the Null Hypothesis if the Alternative Hypothesis is true.
Other attributes in results instance include :
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
References
----------
.. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two
Poisson or Negative Binomial Rates in Noninferiority or Equivalence
Trials.” Statistics in Biopharmaceutical Research, March.
https://doi.org/10.1080/19466315.2016.1225594
.. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for
Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3):
376–87. https://doi.org/10.1002/sim.5947.
.. [3] PASS documentation
"""
# TODO: avoid possible circular import, check if needed
from statsmodels.stats.power import normal_power_het
rate1, rate2, nobs1 = map(np.asarray, [rate1, rate2, nobs1])
nobs2 = nobs_ratio * nobs1
v1 = ((1 / rate1 + 1 / (nobs_ratio * rate2)) / exposure +
(1 + nobs_ratio) / nobs_ratio * dispersion)
if method_var == "alt":
v0 = v1
elif method_var == "ftotal":
v0 = (1 + value * nobs_ratio)**2 / (
exposure * nobs_ratio * value * (rate1 + nobs_ratio * rate2))
v0 += (1 + nobs_ratio) / nobs_ratio * dispersion
elif method_var == "score":
v0 = _var_cmle_negbin(rate1, rate2, nobs_ratio,
exposure=exposure, value=value,
dispersion=dispersion)[0]
else:
raise NotImplementedError(f"method_var {method_var} not recognized")
std_null = np.sqrt(v0)
std_alt = np.sqrt(v1)
es = np.log(rate1 / rate2) - np.log(value)
pow_ = normal_power_het(es, nobs1, alpha, std_null=std_null,
std_alternative=std_alt,
alternative=alternative)
if return_results:
res = HolderTuple(
power=pow_,
std_null=std_null,
std_alt=std_alt,
nobs1=nobs1,
nobs2=nobs2,
nobs_ratio=nobs_ratio,
alpha=alpha,
tuple_=("power",), # override default
)
return res
return pow_
[docs]
def power_equivalence_neginb_2indep(rate1, rate2, nobs1,
low, upp, nobs_ratio=1,
exposure=1, alpha=0.05, dispersion=0,
method_var="alt",
return_results=False):
"""
Power of equivalence test of ratio of 2 indep. negative binomial rates.
Parameters
----------
rate1 : float
Poisson rate for the first sample, treatment group, under the
alternative hypothesis.
rate2 : float
Poisson rate for the second sample, reference group, under the
alternative hypothesis.
nobs1 : float or int
Number of observations in sample 1.
low : float
Lower equivalence margin for the rate ratio, rate1 / rate2.
upp : float
Upper equivalence margin for the rate ratio, rate1 / rate2.
nobs_ratio : float
Sample size ratio, nobs2 = nobs_ratio * nobs1.
alpha : float in interval (0,1)
Significance level, e.g. 0.05, is the probability of a type I
error, that is wrong rejections if the Null Hypothesis is true.
dispersion : float >= 0.
Dispersion parameter for Negative Binomial distribution.
The Poisson limiting case corresponds to ``dispersion=0``.
method_var : {"score", "alt"}
The variance of the test statistic for the null hypothesis given the
rates under the alternative, can be either equal to the rates under the
alternative ``method_var="alt"``, or estimated under the constrained
of the null hypothesis, ``method_var="score"``, or based on a moment
constrained estimate, ``method_var="ftotal"``. see references.
alternative : string, 'two-sided' (default), 'larger', 'smaller'
Alternative hypothesis whether the power is calculated for a
two-sided (default) or one sided test. The one-sided test can be
either 'larger', 'smaller'.
return_results : bool
If true, then a results instance with extra information is returned,
otherwise only the computed power is returned.
Returns
-------
results : results instance or float
If return_results is False, then only the power is returned.
If return_results is True, then a results instance with the
information in attributes is returned.
power : float
Power of the test, e.g. 0.8, is one minus the probability of a
type II error. Power is the probability that the test correctly
rejects the Null Hypothesis if the Alternative Hypothesis is true.
Other attributes in results instance include :
std_null
standard error of difference under the null hypothesis (without
sqrt(nobs1))
std_alt
standard error of difference under the alternative hypothesis
(without sqrt(nobs1))
References
----------
.. [1] Zhu, Haiyuan. 2017. “Sample Size Calculation for Comparing Two
Poisson or Negative Binomial Rates in Noninferiority or Equivalence
Trials.” Statistics in Biopharmaceutical Research, March.
https://doi.org/10.1080/19466315.2016.1225594
.. [2] Zhu, Haiyuan, and Hassan Lakkis. 2014. “Sample Size Calculation for
Comparing Two Negative Binomial Rates.” Statistics in Medicine 33 (3):
376–87. https://doi.org/10.1002/sim.5947.
.. [3] PASS documentation
"""
rate1, rate2, nobs1 = map(np.asarray, [rate1, rate2, nobs1])
nobs2 = nobs_ratio * nobs1
v1 = ((1 / rate2 + 1 / (nobs_ratio * rate1)) / exposure +
(1 + nobs_ratio) / nobs_ratio * dispersion)
if method_var == "alt":
v0_low = v0_upp = v1
elif method_var == "ftotal":
v0_low = (1 + low * nobs_ratio)**2 / (
exposure * nobs_ratio * low * (rate1 + nobs_ratio * rate2))
v0_low += (1 + nobs_ratio) / nobs_ratio * dispersion
v0_upp = (1 + upp * nobs_ratio)**2 / (
exposure * nobs_ratio * upp * (rate1 + nobs_ratio * rate2))
v0_upp += (1 + nobs_ratio) / nobs_ratio * dispersion
elif method_var == "score":
v0_low = _var_cmle_negbin(rate1, rate2, nobs_ratio,
exposure=exposure, value=low,
dispersion=dispersion)[0]
v0_upp = _var_cmle_negbin(rate1, rate2, nobs_ratio,
exposure=exposure, value=upp,
dispersion=dispersion)[0]
else:
raise NotImplementedError(f"method_var {method_var} not recognized")
es_low = np.log(rate1 / rate2) - np.log(low)
es_upp = np.log(rate1 / rate2) - np.log(upp)
std_null_low = np.sqrt(v0_low)
std_null_upp = np.sqrt(v0_upp)
std_alternative = np.sqrt(v1)
pow_ = _power_equivalence_het(es_low, es_upp, nobs1, alpha=alpha,
std_null_low=std_null_low,
std_null_upp=std_null_upp,
std_alternative=std_alternative)
if return_results:
res = HolderTuple(
power=pow_[0],
power_margins=pow[1:],
std_null_low=std_null_low,
std_null_upp=std_null_upp,
std_alt=std_alternative,
nobs1=nobs1,
nobs2=nobs2,
nobs_ratio=nobs_ratio,
alpha=alpha,
tuple_=("power",), # override default
)
return res
else:
return pow_[0]
Last update:
Dec 14, 2023