"""
Methods for analyzing two-way contingency tables (i.e. frequency
tables for observations that are cross-classified with respect to two
categorical variables).
The main classes are:
* Table : implements methods that can be applied to any two-way
contingency table.
* SquareTable : implements methods that can be applied to a square
two-way contingency table.
* Table2x2 : implements methods that can be applied to a 2x2
contingency table.
* StratifiedTable : implements methods that can be applied to a
collection of contingency tables.
Also contains functions for conducting Mcnemar's test and Cochran's q
test.
Note that the inference procedures may depend on how the data were
sampled. In general the observed units are independent and
identically distributed.
"""
from __future__ import division
from statsmodels.tools.decorators import cache_readonly, resettable_cache
import numpy as np
from scipy import stats
import pandas as pd
from statsmodels import iolib
from statsmodels.tools.sm_exceptions import SingularMatrixWarning
def _make_df_square(table):
"""
Reindex a pandas DataFrame so that it becomes square, meaning that
the row and column indices contain the same values, in the same
order. The row and column index are extended to achieve this.
"""
if not isinstance(table, pd.DataFrame):
return table
# If the table is not square, make it square
if table.shape[0] != table.shape[1]:
ix = list(set(table.index) | set(table.columns))
table = table.reindex(ix, axis=0)
table = table.reindex(ix, axis=1)
# Ensures that the rows and columns are in the same order.
table = table.reindex(table.columns)
return table
class _Bunch(object):
def __repr__(self):
return "<bunch object containing statsmodels results>"
[docs]class Table(object):
"""
Analyses that can be performed on a two-way contingency table.
Parameters
----------
table : array-like
A contingency table.
shift_zeros : boolean
If True and any cell count is zero, add 0.5 to all values
in the table.
Attributes
----------
table_orig : array-like
The original table is cached as `table_orig`.
marginal_probabilities : tuple of two ndarrays
The estimated row and column marginal distributions.
independence_probabilities : ndarray
Estimated cell probabilities under row/column independence.
fittedvalues : ndarray
Fitted values under independence.
resid_pearson : ndarray
The Pearson residuals under row/column independence.
standardized_resids : ndarray
Residuals for the independent row/column model with approximate
unit variance.
chi2_contribs : ndarray
The contribution of each cell to the chi^2 statistic.
local_logodds_ratios : ndarray
The local log odds ratios are calculated for each 2x2 subtable
formed from adjacent rows and columns.
local_oddsratios : ndarray
The local odds ratios are calculated from each 2x2 subtable
formed from adjacent rows and columns.
cumulative_log_oddsratios : ndarray
The cumulative log odds ratio at a given pair of thresholds is
calculated by reducing the table to a 2x2 table based on
dichotomizing the rows and columns at the given thresholds.
The table of cumulative log odds ratios presents all possible
cumulative log odds ratios that can be formed from a given
table.
cumulative_oddsratios : ndarray
The cumulative odds ratios are calculated by reducing the
table to a 2x2 table based on cutting the rows and columns at
a given point. The table of cumulative odds ratios presents
all possible cumulative odds ratios that can be formed from a
given table.
See also
--------
statsmodels.graphics.mosaicplot.mosaic
scipy.stats.chi2_contingency
Notes
-----
The inference procedures used here are all based on a sampling
model in which the units are independent and identically
distributed, with each unit being classified with respect to two
categorical variables.
References
----------
Definitions of residuals:
https://onlinecourses.science.psu.edu/stat504/node/86
"""
def __init__(self, table, shift_zeros=True):
self.table_orig = table
self.table = np.asarray(table, dtype=np.float64)
if shift_zeros and (self.table.min() == 0):
self.table = self.table + 0.5
@classmethod
[docs] def from_data(cls, data, shift_zeros=True):
"""
Construct a Table object from data.
Parameters
----------
data : array-like
The raw data, from which a contingency table is constructed
using the first two columns.
shift_zeros : boolean
If True and any cell count is zero, add 0.5 to all values
in the table.
Returns
-------
A Table instance.
"""
if isinstance(data, pd.DataFrame):
table = pd.crosstab(data.iloc[:, 0], data.iloc[:, 1])
else:
table = pd.crosstab(data[:, 0], data[:, 1])
return cls(table, shift_zeros)
[docs] def test_nominal_association(self):
"""
Assess independence for nominal factors.
Assessment of independence between rows and columns using
chi^2 testing. The rows and columns are treated as nominal
(unordered) categorical variables.
Returns
-------
A bunch containing the following attributes:
statistic : float
The chi^2 test statistic.
df : integer
The degrees of freedom of the reference distribution
pvalue : float
The p-value for the test.
"""
statistic = np.asarray(self.chi2_contribs).sum()
df = np.prod(np.asarray(self.table.shape) - 1)
pvalue = 1 - stats.chi2.cdf(statistic, df)
b = _Bunch()
b.statistic = statistic
b.df = df
b.pvalue = pvalue
return b
[docs] def test_ordinal_association(self, row_scores=None, col_scores=None):
"""
Assess independence between two ordinal variables.
This is the 'linear by linear' association test, which uses
weights or scores to target the test to have more power
against ordered alternatives.
Parameters
----------
row_scores : array-like
An array of numeric row scores
col_scores : array-like
An array of numeric column scores
Returns
-------
A bunch with the following attributes:
statistic : float
The test statistic.
null_mean : float
The expected value of the test statistic under the null
hypothesis.
null_sd : float
The standard deviation of the test statistic under the
null hypothesis.
zscore : float
The Z-score for the test statistic.
pvalue : float
The p-value for the test.
Notes
-----
The scores define the trend to which the test is most sensitive.
Using the default row and column scores gives the
Cochran-Armitage trend test.
"""
if row_scores is None:
row_scores = np.arange(self.table.shape[0])
if col_scores is None:
col_scores = np.arange(self.table.shape[1])
if len(row_scores) != self.table.shape[0]:
raise ValueError("The length of `row_scores` must match the first dimension of `table`.")
if len(col_scores) != self.table.shape[1]:
raise ValueError("The length of `col_scores` must match the second dimension of `table`.")
# The test statistic
statistic = np.dot(row_scores, np.dot(self.table, col_scores))
# Some needed quantities
n_obs = self.table.sum()
rtot = self.table.sum(1)
um = np.dot(row_scores, rtot)
u2m = np.dot(row_scores**2, rtot)
ctot = self.table.sum(0)
vn = np.dot(col_scores, ctot)
v2n = np.dot(col_scores**2, ctot)
# The null mean and variance of the test statistic
e_stat = um * vn / n_obs
v_stat = (u2m - um**2 / n_obs) * (v2n - vn**2 / n_obs) / (n_obs - 1)
sd_stat = np.sqrt(v_stat)
zscore = (statistic - e_stat) / sd_stat
pvalue = 2 * stats.norm.cdf(-np.abs(zscore))
b = _Bunch()
b.statistic = statistic
b.null_mean = e_stat
b.null_sd = sd_stat
b.zscore = zscore
b.pvalue = pvalue
return b
@cache_readonly
[docs] def marginal_probabilities(self):
# docstring for cached attributes in init above
n = self.table.sum()
row = self.table.sum(1) / n
col = self.table.sum(0) / n
if isinstance(self.table_orig, pd.DataFrame):
row = pd.Series(row, self.table_orig.index)
col = pd.Series(col, self.table_orig.columns)
return row, col
@cache_readonly
[docs] def independence_probabilities(self):
# docstring for cached attributes in init above
row, col = self.marginal_probabilities
itab = np.outer(row, col)
if isinstance(self.table_orig, pd.DataFrame):
itab = pd.DataFrame(itab, self.table_orig.index,
self.table_orig.columns)
return itab
@cache_readonly
[docs] def fittedvalues(self):
# docstring for cached attributes in init above
probs = self.independence_probabilities
fit = self.table.sum() * probs
return fit
@cache_readonly
[docs] def resid_pearson(self):
# docstring for cached attributes in init above
fit = self.fittedvalues
resids = (self.table - fit) / np.sqrt(fit)
return resids
@cache_readonly
[docs] def standardized_resids(self):
# docstring for cached attributes in init above
row, col = self.marginal_probabilities
sresids = self.resid_pearson / np.sqrt(np.outer(1 - row, 1 - col))
return sresids
@cache_readonly
[docs] def chi2_contribs(self):
# docstring for cached attributes in init above
return self.resid_pearson**2
@cache_readonly
[docs] def local_log_oddsratios(self):
# docstring for cached attributes in init above
ta = self.table.copy()
a = ta[0:-1, 0:-1]
b = ta[0:-1, 1:]
c = ta[1:, 0:-1]
d = ta[1:, 1:]
tab = np.log(a) + np.log(d) - np.log(b) - np.log(c)
rslt = np.empty(self.table.shape, np.float64)
rslt *= np.nan
rslt[0:-1, 0:-1] = tab
if isinstance(self.table_orig, pd.DataFrame):
rslt = pd.DataFrame(rslt, index=self.table_orig.index,
columns=self.table_orig.columns)
return rslt
@cache_readonly
[docs] def local_oddsratios(self):
# docstring for cached attributes in init above
return np.exp(self.local_log_oddsratios)
@cache_readonly
[docs] def cumulative_log_oddsratios(self):
# docstring for cached attributes in init above
ta = self.table.cumsum(0).cumsum(1)
a = ta[0:-1, 0:-1]
b = ta[0:-1, -1:] - a
c = ta[-1:, 0:-1] - a
d = ta[-1, -1] - (a + b + c)
tab = np.log(a) + np.log(d) - np.log(b) - np.log(c)
rslt = np.empty(self.table.shape, np.float64)
rslt *= np.nan
rslt[0:-1, 0:-1] = tab
if isinstance(self.table_orig, pd.DataFrame):
rslt = pd.DataFrame(rslt, index=self.table_orig.index,
columns=self.table_orig.columns)
return rslt
@cache_readonly
[docs] def cumulative_oddsratios(self):
# docstring for cached attributes in init above
return np.exp(self.cumulative_log_oddsratios)
[docs]class SquareTable(Table):
"""
Methods for analyzing a square contingency table.
Parameters
----------
table : array-like
A square contingency table, or DataFrame that is converted
to a square form.
shift_zeros : boolean
If True and any cell count is zero, add 0.5 to all values
in the table.
These methods should only be used when the rows and columns of the
table have the same categories. If `table` is provided as a
Pandas DataFrame, the row and column indices will be extended to
create a square table. Otherwise the table should be provided in
a square form, with the (implicit) row and column categories
appearing in the same order.
"""
def __init__(self, table, shift_zeros=True):
table = _make_df_square(table) # Non-pandas passes through
k1, k2 = table.shape
if k1 != k2:
raise ValueError('table must be square')
super(SquareTable, self).__init__(table, shift_zeros)
[docs] def symmetry(self, method="bowker"):
"""
Test for symmetry of a joint distribution.
This procedure tests the null hypothesis that the joint
distribution is symmetric around the main diagonal, that is
.. math::
p_{i, j} = p_{j, i} for all i, j
Returns
-------
A bunch with attributes:
statistic : float
chisquare test statistic
p-value : float
p-value of the test statistic based on chisquare distribution
df : int
degrees of freedom of the chisquare distribution
Notes
-----
The implementation is based on the SAS documentation. R includes
it in `mcnemar.test` if the table is not 2 by 2. However a more
direct generalization of the McNemar test to larger tables is
provided by the homogeneity test (TableSymmetry.homogeneity).
The p-value is based on the chi-square distribution which requires
that the sample size is not very small to be a good approximation
of the true distribution. For 2x2 contingency tables the exact
distribution can be obtained with `mcnemar`
See Also
--------
mcnemar
homogeneity
"""
if method.lower() != "bowker":
raise ValueError("method for symmetry testing must be 'bowker'")
k = self.table.shape[0]
upp_idx = np.triu_indices(k, 1)
tril = self.table.T[upp_idx] # lower triangle in column order
triu = self.table[upp_idx] # upper triangle in row order
statistic = ((tril - triu)**2 / (tril + triu + 1e-20)).sum()
df = k * (k-1) / 2.
pvalue = stats.chi2.sf(statistic, df)
b = _Bunch()
b.statistic = statistic
b.pvalue = pvalue
b.df = df
return b
[docs] def homogeneity(self, method="stuart_maxwell"):
"""
Compare row and column marginal distributions.
Parameters
----------
method : string
Either 'stuart_maxwell' or 'bhapkar', leading to two different
estimates of the covariance matrix for the estimated
difference between the row margins and the column margins.
Returns a bunch with attributes:
statistic : float
The chi^2 test statistic
pvalue : float
The p-value of the test statistic
df : integer
The degrees of freedom of the reference distribution
Notes
-----
For a 2x2 table this is equivalent to McNemar's test. More
generally the procedure tests the null hypothesis that the
marginal distribution of the row factor is equal to the
marginal distribution of the column factor. For this to be
meaningful, the two factors must have the same sample space
(i.e. the same categories).
"""
if self.table.shape[0] < 1:
raise ValueError('table is empty')
elif self.table.shape[0] == 1:
b = _Bunch()
b.statistic = 0
b.pvalue = 1
b.df = 0
return b
method = method.lower()
if method not in ["bhapkar", "stuart_maxwell"]:
raise ValueError("method '%s' for homogeneity not known" % method)
n_obs = self.table.sum()
pr = self.table.astype(np.float64) / n_obs
# Compute margins, eliminate last row/column so there is no
# degeneracy
row = pr.sum(1)[0:-1]
col = pr.sum(0)[0:-1]
pr = pr[0:-1, 0:-1]
# The estimated difference between row and column margins.
d = col - row
# The degrees of freedom of the chi^2 reference distribution.
df = pr.shape[0]
if method == "bhapkar":
vmat = -(pr + pr.T) - np.outer(d, d)
dv = col + row - 2*np.diag(pr) - d**2
np.fill_diagonal(vmat, dv)
elif method == "stuart_maxwell":
vmat = -(pr + pr.T)
dv = row + col - 2*np.diag(pr)
np.fill_diagonal(vmat, dv)
try:
statistic = n_obs * np.dot(d, np.linalg.solve(vmat, d))
except np.linalg.LinAlgError:
import warnings
warnings.warn("Unable to invert covariance matrix",
SingularMatrixWarning)
b = _Bunch()
b.statistic = np.nan
b.pvalue = np.nan
b.df = df
return b
pvalue = 1 - stats.chi2.cdf(statistic, df)
b = _Bunch()
b.statistic = statistic
b.pvalue = pvalue
b.df = df
return b
[docs] def summary(self, alpha=0.05, float_format="%.3f"):
"""
Produce a summary of the analysis.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the interval.
float_format : string
Used to format numeric values in the table.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
fmt = float_format
headers = ["Statistic", "P-value", "DF"]
stubs = ["Symmetry", "Homogeneity"]
sy = self.symmetry()
hm = self.homogeneity()
data = [[fmt % sy.statistic, fmt % sy.pvalue, '%d' % sy.df],
[fmt % hm.statistic, fmt % hm.pvalue, '%d' % hm.df]]
tab = iolib.SimpleTable(data, headers, stubs, data_aligns="r",
table_dec_above='')
return tab
[docs]class Table2x2(SquareTable):
"""
Analyses that can be performed on a 2x2 contingency table.
Parameters
----------
table : array-like
A 2x2 contingency table
shift_zeros : boolean
If true, 0.5 is added to all cells of the table if any cell is
equal to zero.
Attributes
----------
log_oddsratio : float
The log odds ratio of the table.
log_oddsratio_se : float
The asymptotic standard error of the estimated log odds ratio.
oddsratio : float
The odds ratio of the table.
riskratio : float
The ratio between the risk in the first row and the risk in
the second row. Column 0 is interpreted as containing the
number of occurences of the event of interest.
log_riskratio : float
The estimated log risk ratio for the table.
log_riskratio_se : float
The standard error of the estimated log risk ratio for the
table.
Notes
-----
The inference procedures used here are all based on a sampling
model in which the units are independent and identically
distributed, with each unit being classified with respect to two
categorical variables.
Note that for the risk ratio, the analysis is not symmetric with
respect to the rows and columns of the contingency table. The two
rows define population subgroups, column 0 is the number of
'events', and column 1 is the number of 'non-events'.
"""
def __init__(self, table, shift_zeros=True):
if (table.ndim != 2) or (table.shape[0] != 2) or (table.shape[1] != 2):
raise ValueError("Table2x2 takes a 2x2 table as input.")
super(Table2x2, self).__init__(table, shift_zeros)
@classmethod
[docs] def from_data(cls, data, shift_zeros=True):
"""
Construct a Table object from data.
Parameters
----------
data : array-like
The raw data, the first column defines the rows and the
second column defines the columns.
shift_zeros : boolean
If True, and if there are any zeros in the contingency
table, add 0.5 to all four cells of the table.
"""
if isinstance(data, pd.DataFrame):
table = pd.crosstab(data.iloc[:, 0], data.iloc[:, 1])
else:
table = pd.crosstab(data[:, 0], data[:, 1])
return cls(table, shift_zeros)
@cache_readonly
[docs] def log_oddsratio(self):
# docstring for cached attributes in init above
f = self.table.flatten()
return np.dot(np.log(f), np.r_[1, -1, -1, 1])
@cache_readonly
[docs] def oddsratio(self):
# docstring for cached attributes in init above
return self.table[0, 0] * self.table[1, 1] / (self.table[0, 1] * self.table[1, 0])
@cache_readonly
[docs] def log_oddsratio_se(self):
# docstring for cached attributes in init above
return np.sqrt(np.sum(1 / self.table))
[docs] def oddsratio_pvalue(self, null=1):
"""
P-value for a hypothesis test about the odds ratio.
Parameters
----------
null : float
The null value of the odds ratio.
"""
return self.log_oddsratio_pvalue(np.log(null))
[docs] def log_oddsratio_pvalue(self, null=0):
"""
P-value for a hypothesis test about the log odds ratio.
Parameters
----------
null : float
The null value of the log odds ratio.
"""
zscore = (self.log_oddsratio - null) / self.log_oddsratio_se
pvalue = 2 * stats.norm.cdf(-np.abs(zscore))
return pvalue
[docs] def log_oddsratio_confint(self, alpha=0.05, method="normal"):
"""
A confidence level for the log odds ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
confidence interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
f = -stats.norm.ppf(alpha / 2)
lor = self.log_oddsratio
se = self.log_oddsratio_se
lcb = lor - f * se
ucb = lor + f * se
return lcb, ucb
[docs] def oddsratio_confint(self, alpha=0.05, method="normal"):
"""
A confidence interval for the odds ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
confidence interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
lcb, ucb = self.log_oddsratio_confint(alpha, method=method)
return np.exp(lcb), np.exp(ucb)
@cache_readonly
[docs] def riskratio(self):
# docstring for cached attributes in init above
p = self.table[:, 0] / self.table.sum(1)
return p[0] / p[1]
@cache_readonly
[docs] def log_riskratio(self):
# docstring for cached attributes in init above
return np.log(self.riskratio)
@cache_readonly
[docs] def log_riskratio_se(self):
# docstring for cached attributes in init above
n = self.table.sum(1)
p = self.table[:, 0] / n
va = np.sum((1 - p) / (n*p))
return np.sqrt(va)
[docs] def riskratio_pvalue(self, null=1):
"""
p-value for a hypothesis test about the risk ratio.
Parameters
----------
null : float
The null value of the risk ratio.
"""
return self.log_riskratio_pvalue(np.log(null))
[docs] def log_riskratio_pvalue(self, null=0):
"""
p-value for a hypothesis test about the log risk ratio.
Parameters
----------
null : float
The null value of the log risk ratio.
"""
zscore = (self.log_riskratio - null) / self.log_riskratio_se
pvalue = 2 * stats.norm.cdf(-np.abs(zscore))
return pvalue
[docs] def log_riskratio_confint(self, alpha=0.05, method="normal"):
"""
A confidence interval for the log risk ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
confidence interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
f = -stats.norm.ppf(alpha / 2)
lrr = self.log_riskratio
se = self.log_riskratio_se
lcb = lrr - f * se
ucb = lrr + f * se
return lcb, ucb
[docs] def riskratio_confint(self, alpha=0.05, method="normal"):
"""
A confidence interval for the risk ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
confidence interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
lcb, ucb = self.log_riskratio_confint(alpha, method=method)
return np.exp(lcb), np.exp(ucb)
[docs] def summary(self, alpha=0.05, float_format="%.3f", method="normal"):
"""
Summarizes results for a 2x2 table analysis.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the confidence
intervals.
float_format : string
Used to format the numeric values in the table.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
def fmt(x):
if type(x) is str:
return x
return float_format % x
headers = ["Estimate", "SE", "LCB", "UCB", "p-value"]
stubs = ["Odds ratio", "Log odds ratio", "Risk ratio", "Log risk ratio"]
lcb1, ucb1 = self.oddsratio_confint(alpha, method)
lcb2, ucb2 = self.log_oddsratio_confint(alpha, method)
lcb3, ucb3 = self.riskratio_confint(alpha, method)
lcb4, ucb4 = self.log_riskratio_confint(alpha, method)
data = [[fmt(x) for x in [self.oddsratio, "", lcb1, ucb1, self.oddsratio_pvalue()]],
[fmt(x) for x in [self.log_oddsratio, self.log_oddsratio_se, lcb2, ucb2,
self.oddsratio_pvalue()]],
[fmt(x) for x in [self.riskratio, "", lcb2, ucb2, self.riskratio_pvalue()]],
[fmt(x) for x in [self.log_riskratio, self.log_riskratio_se, lcb4, ucb4,
self.riskratio_pvalue()]]]
tab = iolib.SimpleTable(data, headers, stubs, data_aligns="r",
table_dec_above='')
return tab
[docs]class StratifiedTable(object):
"""
Analyses for a collection of 2x2 contingency tables.
Such a collection may arise by stratifying a single 2x2 table with
respect to another factor. This class implements the
'Cochran-Mantel-Haenszel' and 'Breslow-Day' procedures for
analyzing collections of 2x2 contingency tables.
Parameters
----------
tables : list or ndarray
Either a list containing several 2x2 contingency tables, or
a 2x2xk ndarray in which each slice along the third axis is a
2x2 contingency table.
Attributes
----------
logodds_pooled : float
An estimate of the pooled log odds ratio. This is the
Mantel-Haenszel estimate of an odds ratio that is common to
all the tables.
log_oddsratio_se : float
The estimated standard error of the pooled log odds ratio,
following Robins, Breslow and Greenland (Biometrics
42:311-323).
oddsratio_pooled : float
An estimate of the pooled odds ratio. This is the
Mantel-Haenszel estimate of an odds ratio that is common to
all tables.
risk_pooled : float
An estimate of the pooled risk ratio. This is an estimate of
a risk ratio that is common to all the tables.
Notes
-----
This results are based on a sampling model in which the units are
independent both within and between strata.
"""
def __init__(self, tables, shift_zeros=False):
if isinstance(tables, np.ndarray):
sp = tables.shape
if (len(sp) != 3) or (sp[0] != 2) or (sp[1] != 2):
raise ValueError("If an ndarray, argument must be 2x2xn")
table = tables
else:
# Create a data cube
table = np.dstack(tables).astype(np.float64)
if shift_zeros:
zx = (table == 0).sum(0).sum(0)
ix = np.flatnonzero(zx > 0)
if len(ix) > 0:
table = table.copy()
table[:, :, ix] += 0.5
self.table = table
self._cache = resettable_cache()
# Quantities to precompute. Table entries are [[a, b], [c,
# d]], 'ad' is 'a * d', 'apb' is 'a + b', 'dma' is 'd - a',
# etc.
self._apb = table[0, 0, :] + table[0, 1, :]
self._apc = table[0, 0, :] + table[1, 0, :]
self._bpd = table[0, 1, :] + table[1, 1, :]
self._cpd = table[1, 0, :] + table[1, 1, :]
self._ad = table[0, 0, :] * table[1, 1, :]
self._bc = table[0, 1, :] * table[1, 0, :]
self._apd = table[0, 0, :] + table[1, 1, :]
self._dma = table[1, 1, :] - table[0, 0, :]
self._n = table.sum(0).sum(0)
@classmethod
[docs] def from_data(cls, var1, var2, strata, data):
"""
Construct a StratifiedTable object from data.
Parameters
----------
var1 : int or string
The column index or name of `data` containing the variable
defining the rows of the contingency table. The variable
must have only two distinct values.
var2 : int or string
The column index or name of `data` containing the variable
defining the columns of the contingency table. The variable
must have only two distinct values.
strata : int or string
The column index of name of `data` containing the variable
defining the strata.
data : array-like
The raw data. A cross-table for analysis is constructed
from the first two columns.
Returns
-------
A StratifiedTable instance.
"""
if not isinstance(data, pd.DataFrame):
data1 = pd.DataFrame(index=data.index, column=[var1, var2, strata])
data1.loc[:, var1] = data[:, var1]
data1.loc[:, var2] = data[:, var2]
data1.loc[:, strata] = data[:, strata]
else:
data1 = data[[var1, var2, strata]]
gb = data1.groupby(strata).groups
tables = []
for g in gb:
ii = gb[g]
tab = pd.crosstab(data1.loc[ii, var1], data1.loc[ii, var2])
tables.append(tab)
return cls(tables)
[docs] def test_null_odds(self, correction=False):
"""
Test that all tables have odds ratio equal to 1.
This is the 'Mantel-Haenszel' test.
Parameters
----------
correction : boolean
If True, use the continuity correction when calculating the
test statistic.
Returns
-------
A bunch containing the chi^2 test statistic and p-value.
"""
statistic = np.sum(self.table[0, 0, :] - self._apb * self._apc / self._n)
statistic = np.abs(statistic)
if correction:
statistic -= 0.5
statistic = statistic**2
denom = self._apb * self._apc * self._bpd * self._cpd
denom /= (self._n**2 * (self._n - 1))
denom = np.sum(denom)
statistic /= denom
# df is always 1
pvalue = 1 - stats.chi2.cdf(statistic, 1)
b = _Bunch()
b.statistic = statistic
b.pvalue = pvalue
return b
@cache_readonly
[docs] def oddsratio_pooled(self):
# doc for cached attributes in init above
odds_ratio = np.sum(self._ad / self._n) / np.sum(self._bc / self._n)
return odds_ratio
@cache_readonly
[docs] def logodds_pooled(self):
# doc for cached attributes in init above
return np.log(self.oddsratio_pooled)
@cache_readonly
[docs] def risk_pooled(self):
# doc for cached attributes in init above
acd = self.table[0, 0, :] * self._cpd
cab = self.table[1, 0, :] * self._apb
rr = np.sum(acd / self._n) / np.sum(cab / self._n)
return rr
@cache_readonly
[docs] def logodds_pooled_se(self):
# doc for cached attributes in init above
adns = np.sum(self._ad / self._n)
bcns = np.sum(self._bc / self._n)
lor_va = np.sum(self._apd * self._ad / self._n**2) / adns**2
mid = self._apd * self._bc / self._n**2
mid += (1 - self._apd / self._n) * self._ad / self._n
mid = np.sum(mid)
mid /= (adns * bcns)
lor_va += mid
lor_va += np.sum((1 - self._apd / self._n) * self._bc / self._n) / bcns**2
lor_va /= 2
lor_se = np.sqrt(lor_va)
return lor_se
[docs] def logodds_pooled_confint(self, alpha=0.05, method="normal"):
"""
A confidence interval for the pooled log odds ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
Returns
-------
lcb : float
The lower confidence limit.
ucb : float
The upper confidence limit.
"""
lor = np.log(self.oddsratio_pooled)
lor_se = self.logodds_pooled_se
f = -stats.norm.ppf(alpha / 2)
lcb = lor - f * lor_se
ucb = lor + f * lor_se
return lcb, ucb
[docs] def oddsratio_pooled_confint(self, alpha=0.05, method="normal"):
"""
A confidence interval for the pooled odds ratio.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
interval.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
Returns
-------
lcb : float
The lower confidence limit.
ucb : float
The upper confidence limit.
"""
lcb, ucb = self.logodds_pooled_confint(alpha, method=method)
lcb = np.exp(lcb)
ucb = np.exp(ucb)
return lcb, ucb
[docs] def test_equal_odds(self, adjust=False):
"""
Test that all odds ratios are identical.
This is the 'Breslow-Day' testing procedure.
Parameters
----------
adjust : boolean
Use the 'Tarone' adjustment to achieve the chi^2
asymptotic distribution.
Returns
-------
A bunch containing the following attributes:
statistic : float
The chi^2 test statistic.
p-value : float
The p-value for the test.
"""
table = self.table
r = self.oddsratio_pooled
a = 1 - r
b = r * (self._apb + self._apc) + self._dma
c = -r * self._apb * self._apc
# Expected value of first cell
e11 = (-b + np.sqrt(b**2 - 4*a*c)) / (2*a)
# Variance of the first cell
v11 = 1 / e11 + 1 / (self._apc - e11) + 1 / (self._apb - e11) + 1 / (self._dma + e11)
v11 = 1 / v11
statistic = np.sum((table[0, 0, :] - e11)**2 / v11)
if adjust:
adj = table[0, 0, :].sum() - e11.sum()
adj = adj**2
adj /= np.sum(v11)
statistic -= adj
pvalue = 1 - stats.chi2.cdf(statistic, table.shape[2] - 1)
b = _Bunch()
b.statistic = statistic
b.pvalue = pvalue
return b
[docs] def summary(self, alpha=0.05, float_format="%.3f", method="normal"):
"""
A summary of all the main results.
Parameters
----------
alpha : float
`1 - alpha` is the nominal coverage probability of the
confidence intervals.
float_format : string
Used for formatting numeric values in the summary.
method : string
The method for producing the confidence interval. Currently
must be 'normal' which uses the normal approximation.
"""
def fmt(x):
if type(x) is str:
return x
return float_format % x
co_lcb, co_ucb = self.oddsratio_pooled_confint(alpha=alpha, method=method)
clo_lcb, clo_ucb = self.logodds_pooled_confint(alpha=alpha, method=method)
headers = ["Estimate", "LCB", "UCB"]
stubs = ["Pooled odds", "Pooled log odds", "Pooled risk ratio", ""]
data = [[fmt(x) for x in [self.oddsratio_pooled, co_lcb, co_ucb]],
[fmt(x) for x in [self.logodds_pooled, clo_lcb, clo_ucb]],
[fmt(x) for x in [self.risk_pooled, "", ""]],
['', '', '']]
tab1 = iolib.SimpleTable(data, headers, stubs, data_aligns="r",
table_dec_above='')
headers = ["Statistic", "P-value", ""]
stubs = ["Test of OR=1", "Test constant OR"]
rslt1 = self.test_null_odds()
rslt2 = self.test_equal_odds()
data = [[fmt(x) for x in [rslt1.statistic, rslt1.pvalue, ""]],
[fmt(x) for x in [rslt2.statistic, rslt2.pvalue, ""]]]
tab2 = iolib.SimpleTable(data, headers, stubs, data_aligns="r")
tab1.extend(tab2)
headers = ["", "", ""]
stubs = ["Number of tables", "Min n", "Max n", "Avg n", "Total n"]
ss = self.table.sum(0).sum(0)
data = [["%d" % self.table.shape[2], '', ''],
["%d" % min(ss), '', ''],
["%d" % max(ss), '', ''],
["%.0f" % np.mean(ss), '', ''],
["%d" % sum(ss), '', '', '']]
tab3 = iolib.SimpleTable(data, headers, stubs, data_aligns="r")
tab1.extend(tab3)
return tab1
[docs]def mcnemar(table, exact=True, correction=True):
"""
McNemar test of homogeneity.
Parameters
----------
table : array-like
A square contingency table.
exact : bool
If exact is true, then the binomial distribution will be used.
If exact is false, then the chisquare distribution will be
used, which is the approximation to the distribution of the
test statistic for large sample sizes.
correction : bool
If true, then a continuity correction is used for the chisquare
distribution (if exact is false.)
Returns
-------
A bunch with attributes:
statistic : float or int, array
The test statistic is the chisquare statistic if exact is
false. If the exact binomial distribution is used, then this
contains the min(n1, n2), where n1, n2 are cases that are zero
in one sample but one in the other sample.
pvalue : float or array
p-value of the null hypothesis of equal marginal distributions.
Notes
-----
This is a special case of Cochran's Q test, and of the homogeneity
test. The results when the chisquare distribution is used are
identical, except for continuity correction.
"""
table = _make_df_square(table)
table = np.asarray(table, dtype=np.float64)
n1, n2 = table[0, 1], table[1, 0]
if exact:
statistic = np.minimum(n1, n2)
# binom is symmetric with p=0.5
pvalue = stats.binom.cdf(statistic, n1 + n2, 0.5) * 2
pvalue = np.minimum(pvalue, 1) # limit to 1 if n1==n2
else:
corr = int(correction) # convert bool to 0 or 1
statistic = (np.abs(n1 - n2) - corr)**2 / (1. * (n1 + n2))
df = 1
pvalue = stats.chi2.sf(statistic, df)
b = _Bunch()
b.statistic = statistic
b.pvalue = pvalue
return b
[docs]def cochrans_q(x, return_object=True):
"""
Cochran's Q test for identical binomial proportions.
Parameters
----------
x : array_like, 2d (N, k)
data with N cases and k variables
return_object : boolean
Return values as bunch instead of as individual values.
Returns
-------
Returns a bunch containing the following attributes, or the
individual values according to the value of `return_object`.
statistic : float
test statistic
pvalue : float
pvalue from the chisquare distribution
Notes
-----
Cochran's Q is a k-sample extension of the McNemar test. If there
are only two groups, then Cochran's Q test and the McNemar test
are equivalent.
The procedure tests that the probability of success is the same
for every group. The alternative hypothesis is that at least two
groups have a different probability of success.
In Wikipedia terminology, rows are blocks and columns are
treatments. The number of rows N, should be large for the
chisquare distribution to be a good approximation.
The Null hypothesis of the test is that all treatments have the
same effect.
References
----------
http://en.wikipedia.org/wiki/Cochran_test
SAS Manual for NPAR TESTS
"""
x = np.asarray(x, dtype=np.float64)
gruni = np.unique(x)
N, k = x.shape
count_row_success = (x == gruni[-1]).sum(1, float)
count_col_success = (x == gruni[-1]).sum(0, float)
count_row_ss = count_row_success.sum()
count_col_ss = count_col_success.sum()
assert count_row_ss == count_col_ss #just a calculation check
# From the SAS manual
q_stat = (k-1) * (k * np.sum(count_col_success**2) - count_col_ss**2) \
/ (k * count_row_ss - np.sum(count_row_success**2))
# Note: the denominator looks just like k times the variance of
# the columns
# Wikipedia uses a different, but equivalent expression
#q_stat = (k-1) * (k * np.sum(count_row_success**2) - count_row_ss**2) \
# / (k * count_col_ss - np.sum(count_col_success**2))
df = k - 1
pvalue = stats.chi2.sf(q_stat, df)
if return_object:
b = _Bunch()
b.statistic = q_stat
b.df = df
b.pvalue = pvalue
return b
return q_stat, pvalue, df
