Contingency tables¶
Statsmodels supports a variety of approaches for analyzing contingency tables, including methods for assessing independence, symmetry, homogeneity, and methods for working with collections of tables from a stratified population.
The methods described here are mainly for two-way tables. Multi-way
tables can be analyzed using log-linear models. Statsmodels does not
currently have a dedicated API for loglinear modeling, but Poisson
regression in statsmodels.genmod.GLM can be used for this
purpose.
A contingency table is a multi-way table that describes a data set in which each observation belongs to one category for each of several variables. For example, if there are two variables, one with r levels and one with c levels, then we have a r \times c contingency table. The table can be described in terms of the number of observations that fall into a given cell of the table, e.g. T_{ij} is the number of observations that have level i for the first variable and level j for the second variable. Note that each variable must have a finite number of levels (or categories), which can be either ordered or unordered. In different contexts, the variables defining the axes of a contingency table may be called categorical variables or factor variables. They may be either nominal (if their levels are unordered) or ordinal (if their levels are ordered).
The underlying population for a contingency table is described by a distribution table P_{i, j}. The elements of P are probabilities, and the sum of all elements in P is 1. Methods for analyzing contingency tables use the data in T to learn about properties of P.
The statsmodels.stats.Table is the most basic class for
working with contingency tables. We can create a Table object
directly from any rectangular array-like object containing the
contingency table cell counts:
In [1]: import numpy as np
In [2]: import pandas as pd
In [3]: import statsmodels.api as sm
In [4]: df = sm.datasets.get_rdataset("Arthritis", "vcd", cache=True).data
---------------------------------------------------------------------------
gaierror Traceback (most recent call last)
/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1253 try:
-> 1254 h.request(req.get_method(), req.selector, req.data, headers)
1255 except OSError as err: # timeout error
/usr/lib/python3.5/http/client.py in request(self, method, url, body, headers)
1106 """Send a complete request to the server."""
-> 1107 self._send_request(method, url, body, headers)
1108
/usr/lib/python3.5/http/client.py in _send_request(self, method, url, body, headers)
1151 body = _encode(body, 'body')
-> 1152 self.endheaders(body)
1153
/usr/lib/python3.5/http/client.py in endheaders(self, message_body)
1102 raise CannotSendHeader()
-> 1103 self._send_output(message_body)
1104
/usr/lib/python3.5/http/client.py in _send_output(self, message_body)
933
--> 934 self.send(msg)
935 if message_body is not None:
/usr/lib/python3.5/http/client.py in send(self, data)
876 if self.auto_open:
--> 877 self.connect()
878 else:
/usr/lib/python3.5/http/client.py in connect(self)
1252
-> 1253 super().connect()
1254
/usr/lib/python3.5/http/client.py in connect(self)
848 self.sock = self._create_connection(
--> 849 (self.host,self.port), self.timeout, self.source_address)
850 self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)
/usr/lib/python3.5/socket.py in create_connection(address, timeout, source_address)
693 err = None
--> 694 for res in getaddrinfo(host, port, 0, SOCK_STREAM):
695 af, socktype, proto, canonname, sa = res
/usr/lib/python3.5/socket.py in getaddrinfo(host, port, family, type, proto, flags)
732 addrlist = []
--> 733 for res in _socket.getaddrinfo(host, port, family, type, proto, flags):
734 af, socktype, proto, canonname, sa = res
gaierror: [Errno -3] Temporary failure in name resolution
During handling of the above exception, another exception occurred:
URLError Traceback (most recent call last)
<ipython-input-4-82c338cb1ccf> in <module>()
----> 1 df = sm.datasets.get_rdataset("Arthritis", "vcd", cache=True).data
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in get_rdataset(dataname, package, cache)
288 "master/doc/"+package+"/rst/")
289 cache = _get_cache(cache)
--> 290 data, from_cache = _get_data(data_base_url, dataname, cache)
291 data = read_csv(data, index_col=0)
292 data = _maybe_reset_index(data)
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in _get_data(base_url, dataname, cache, extension)
219 url = base_url + (dataname + ".%s") % extension
220 try:
--> 221 data, from_cache = _urlopen_cached(url, cache)
222 except HTTPError as err:
223 if '404' in str(err):
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in _urlopen_cached(url, cache)
210 # not using the cache or didn't find it in cache
211 if not from_cache:
--> 212 data = urlopen(url).read()
213 if cache is not None: # then put it in the cache
214 _cache_it(data, cache_path)
/usr/lib/python3.5/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
161 else:
162 opener = _opener
--> 163 return opener.open(url, data, timeout)
164
165 def install_opener(opener):
/usr/lib/python3.5/urllib/request.py in open(self, fullurl, data, timeout)
464 req = meth(req)
465
--> 466 response = self._open(req, data)
467
468 # post-process response
/usr/lib/python3.5/urllib/request.py in _open(self, req, data)
482 protocol = req.type
483 result = self._call_chain(self.handle_open, protocol, protocol +
--> 484 '_open', req)
485 if result:
486 return result
/usr/lib/python3.5/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
442 for handler in handlers:
443 func = getattr(handler, meth_name)
--> 444 result = func(*args)
445 if result is not None:
446 return result
/usr/lib/python3.5/urllib/request.py in https_open(self, req)
1295 def https_open(self, req):
1296 return self.do_open(http.client.HTTPSConnection, req,
-> 1297 context=self._context, check_hostname=self._check_hostname)
1298
1299 https_request = AbstractHTTPHandler.do_request_
/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1254 h.request(req.get_method(), req.selector, req.data, headers)
1255 except OSError as err: # timeout error
-> 1256 raise URLError(err)
1257 r = h.getresponse()
1258 except:
URLError: <urlopen error [Errno -3] Temporary failure in name resolution>
In [5]: tab = pd.crosstab(df['Treatment'], df['Improved'])
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-5-0b25032f7dc0> in <module>()
----> 1 tab = pd.crosstab(df['Treatment'], df['Improved'])
NameError: name 'df' is not defined
In [6]: tab = tab.loc[:, ["None", "Some", "Marked"]]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-6-91139cd86e09> in <module>()
----> 1 tab = tab.loc[:, ["None", "Some", "Marked"]]
NameError: name 'tab' is not defined
In [7]: table = sm.stats.Table(tab)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-7-486230d614bc> in <module>()
----> 1 table = sm.stats.Table(tab)
NameError: name 'tab' is not defined
Alternatively, we can pass the raw data and let the Table class construct the array of cell counts for us:
In [8]: table = sm.stats.Table.from_data(df[["Treatment", "Improved"]])
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-8-b9d5bf0bff71> in <module>()
----> 1 table = sm.stats.Table.from_data(df[["Treatment", "Improved"]])
NameError: name 'df' is not defined
Independence¶
Independence is the property that the row and column factors occur independently. Association is the lack of independence. If the joint distribution is independent, it can be written as the outer product of the row and column marginal distributions:
P_{ij} = sum_k P_{ij} cdot sum_k P_{kj} forall i, j
We can obtain the best-fitting independent distribution for our observed data, and then view residuals which identify particular cells that most strongly violate independence:
In [9]: print(table.table_orig)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-9-a5d915d9c8d9> in <module>()
----> 1 print(table.table_orig)
NameError: name 'table' is not defined
In [10]: print(table.fittedvalues)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-10-65678b12aad0> in <module>()
----> 1 print(table.fittedvalues)
NameError: name 'table' is not defined
In [11]: print(table.resid_pearson)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-11-aba8d1b98407> in <module>()
----> 1 print(table.resid_pearson)
NameError: name 'table' is not defined
In this example, compared to a sample from a population in which the rows and columns are independent, we have too many observations in the placebo/no improvement and treatment/marked improvement cells, and too few observations in the placebo/marked improvement and treated/no improvement cells. This reflects the apparent benefits of the treatment.
If the rows and columns of a table are unordered (i.e. are nominal factors), then the most common approach for formally assessing independence is using Pearson’s \chi^2 statistic. It’s often useful to look at the cell-wise contributions to the \chi^2 statistic to see where the evidence for dependence is coming from.
In [12]: rslt = table.test_nominal_association()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-12-44f3c4aa3774> in <module>()
----> 1 rslt = table.test_nominal_association()
NameError: name 'table' is not defined
In [13]: print(rslt.pvalue)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-13-185fc35d15cc> in <module>()
----> 1 print(rslt.pvalue)
NameError: name 'rslt' is not defined
In [14]: print(table.chi2_contribs)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-14-96fadb1722ee> in <module>()
----> 1 print(table.chi2_contribs)
NameError: name 'table' is not defined
For tables with ordered row and column factors, we can us the linear by linear association test to obtain more power against alternative hypotheses that respect the ordering. The test statistic for the linear by linear association test is
sum_k r_i c_j T_{ij}
where r_i and c_j are row and column scores. Often these scores are set to the sequences 0, 1, …. This gives the ‘Cochran-Armitage trend test’.
In [15]: rslt = table.test_ordinal_association()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-15-17dbaedf54cd> in <module>()
----> 1 rslt = table.test_ordinal_association()
NameError: name 'table' is not defined
In [16]: print(rslt.pvalue)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-16-185fc35d15cc> in <module>()
----> 1 print(rslt.pvalue)
NameError: name 'rslt' is not defined
We can assess the association in a r\times x table by constructing a series of 2\times 2 tables and calculating their odds ratios. There are two ways to do this. The local odds ratios construct 2\times 2 tables from adjacent row and column categories.
In [17]: print(table.local_oddsratios)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-17-13e04c251e53> in <module>()
----> 1 print(table.local_oddsratios)
NameError: name 'table' is not defined
In [18]: taloc = sm.stats.Table2x2(np.asarray([[7, 29], [21, 13]]))
In [19]: print(taloc.oddsratio)
0.149425287356
In [20]: taloc = sm.stats.Table2x2(np.asarray([[29, 7], [13, 7]]))
In [21]: print(taloc.oddsratio)
2.23076923077
The cumulative odds ratios construct 2\times 2 tables by dichotomizing the row and column factors at each possible point.
In [22]: print(table.cumulative_oddsratios)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-22-18bf33b6e78f> in <module>()
----> 1 print(table.cumulative_oddsratios)
NameError: name 'table' is not defined
In [23]: tab1 = np.asarray([[7, 29 + 7], [21, 13 + 7]])
In [24]: tacum = sm.stats.Table2x2(tab1)
In [25]: print(tacum.oddsratio)
0.185185185185
In [26]: tab1 = np.asarray([[7 + 29, 7], [21 + 13, 7]])
In [27]: tacum = sm.stats.Table2x2(tab1)
In [28]: print(tacum.oddsratio)
1.05882352941
A mosaic plot is a graphical approach to informally assessing dependence in two-way tables.
from statsmodels.graphics.mosaicplot import mosaic
mosaic(data)
Symmetry and homogeneity¶
Symmetry is the property that P_{i, j} = P_{j, i} for every i and j. Homogeneity is the property that the marginal distribution of the row factor and the column factor are identical, meaning that
sum_j P_{ij} = sum_j P_{ji} forall i
Note that for these properties to be applicable the table P (and T) must be square, and the row and column categories must be identical and must occur in the same order.
To illustrate, we load a data set, create a contingency table, and
calculate the row and column margins. The Table class
contains methods for analyzing r \times c contingency tables.
The data set loaded below contains assessments of visual acuity in
people’s left and right eyes. We first load the data and create a
contingency table.
In [29]: df = sm.datasets.get_rdataset("VisualAcuity", "vcd").data
---------------------------------------------------------------------------
gaierror Traceback (most recent call last)
/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1253 try:
-> 1254 h.request(req.get_method(), req.selector, req.data, headers)
1255 except OSError as err: # timeout error
/usr/lib/python3.5/http/client.py in request(self, method, url, body, headers)
1106 """Send a complete request to the server."""
-> 1107 self._send_request(method, url, body, headers)
1108
/usr/lib/python3.5/http/client.py in _send_request(self, method, url, body, headers)
1151 body = _encode(body, 'body')
-> 1152 self.endheaders(body)
1153
/usr/lib/python3.5/http/client.py in endheaders(self, message_body)
1102 raise CannotSendHeader()
-> 1103 self._send_output(message_body)
1104
/usr/lib/python3.5/http/client.py in _send_output(self, message_body)
933
--> 934 self.send(msg)
935 if message_body is not None:
/usr/lib/python3.5/http/client.py in send(self, data)
876 if self.auto_open:
--> 877 self.connect()
878 else:
/usr/lib/python3.5/http/client.py in connect(self)
1252
-> 1253 super().connect()
1254
/usr/lib/python3.5/http/client.py in connect(self)
848 self.sock = self._create_connection(
--> 849 (self.host,self.port), self.timeout, self.source_address)
850 self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)
/usr/lib/python3.5/socket.py in create_connection(address, timeout, source_address)
693 err = None
--> 694 for res in getaddrinfo(host, port, 0, SOCK_STREAM):
695 af, socktype, proto, canonname, sa = res
/usr/lib/python3.5/socket.py in getaddrinfo(host, port, family, type, proto, flags)
732 addrlist = []
--> 733 for res in _socket.getaddrinfo(host, port, family, type, proto, flags):
734 af, socktype, proto, canonname, sa = res
gaierror: [Errno -3] Temporary failure in name resolution
During handling of the above exception, another exception occurred:
URLError Traceback (most recent call last)
<ipython-input-29-06c2c6b80004> in <module>()
----> 1 df = sm.datasets.get_rdataset("VisualAcuity", "vcd").data
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in get_rdataset(dataname, package, cache)
288 "master/doc/"+package+"/rst/")
289 cache = _get_cache(cache)
--> 290 data, from_cache = _get_data(data_base_url, dataname, cache)
291 data = read_csv(data, index_col=0)
292 data = _maybe_reset_index(data)
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in _get_data(base_url, dataname, cache, extension)
219 url = base_url + (dataname + ".%s") % extension
220 try:
--> 221 data, from_cache = _urlopen_cached(url, cache)
222 except HTTPError as err:
223 if '404' in str(err):
/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/datasets/utils.py in _urlopen_cached(url, cache)
210 # not using the cache or didn't find it in cache
211 if not from_cache:
--> 212 data = urlopen(url).read()
213 if cache is not None: # then put it in the cache
214 _cache_it(data, cache_path)
/usr/lib/python3.5/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
161 else:
162 opener = _opener
--> 163 return opener.open(url, data, timeout)
164
165 def install_opener(opener):
/usr/lib/python3.5/urllib/request.py in open(self, fullurl, data, timeout)
464 req = meth(req)
465
--> 466 response = self._open(req, data)
467
468 # post-process response
/usr/lib/python3.5/urllib/request.py in _open(self, req, data)
482 protocol = req.type
483 result = self._call_chain(self.handle_open, protocol, protocol +
--> 484 '_open', req)
485 if result:
486 return result
/usr/lib/python3.5/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
442 for handler in handlers:
443 func = getattr(handler, meth_name)
--> 444 result = func(*args)
445 if result is not None:
446 return result
/usr/lib/python3.5/urllib/request.py in https_open(self, req)
1295 def https_open(self, req):
1296 return self.do_open(http.client.HTTPSConnection, req,
-> 1297 context=self._context, check_hostname=self._check_hostname)
1298
1299 https_request = AbstractHTTPHandler.do_request_
/usr/lib/python3.5/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1254 h.request(req.get_method(), req.selector, req.data, headers)
1255 except OSError as err: # timeout error
-> 1256 raise URLError(err)
1257 r = h.getresponse()
1258 except:
URLError: <urlopen error [Errno -3] Temporary failure in name resolution>
In [30]: df = df.loc[df.gender == "female", :]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-30-d8d829c2caf6> in <module>()
----> 1 df = df.loc[df.gender == "female", :]
NameError: name 'df' is not defined
In [31]: tab = df.set_index(['left', 'right'])
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-31-a0870d2ccda3> in <module>()
----> 1 tab = df.set_index(['left', 'right'])
NameError: name 'df' is not defined
In [32]: del tab["gender"]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-32-bbac1d0bf131> in <module>()
----> 1 del tab["gender"]
NameError: name 'tab' is not defined
In [33]: tab = tab.unstack()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-33-01759d3a056d> in <module>()
----> 1 tab = tab.unstack()
NameError: name 'tab' is not defined
In [34]: tab.columns = tab.columns.get_level_values(1)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-34-bd31950d0ac5> in <module>()
----> 1 tab.columns = tab.columns.get_level_values(1)
NameError: name 'tab' is not defined
In [35]: print(tab)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-35-27689641d70c> in <module>()
----> 1 print(tab)
NameError: name 'tab' is not defined
Next we create a SquareTable object from the contingency
table.
In [36]: sqtab = sm.stats.SquareTable(tab)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-36-2c9b0224c8f0> in <module>()
----> 1 sqtab = sm.stats.SquareTable(tab)
NameError: name 'tab' is not defined
In [37]: row, col = sqtab.marginal_probabilities
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-37-9306dee26be2> in <module>()
----> 1 row, col = sqtab.marginal_probabilities
NameError: name 'sqtab' is not defined
In [38]: print(row)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-38-d25e8148caa1> in <module>()
----> 1 print(row)
NameError: name 'row' is not defined
In [39]: print(col)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-39-289245b3b2b4> in <module>()
----> 1 print(col)
NameError: name 'col' is not defined
The summary method prints results for the symmetry and homogeneity
testing procedures.
In [40]: print(sqtab.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-40-0c58f2b319d6> in <module>()
----> 1 print(sqtab.summary())
NameError: name 'sqtab' is not defined
If we had the individual case records in a dataframe called data,
we could also perform the same analysis by passing the raw data using
the SquareTable.from_data class method.
sqtab = sm.stats.SquareTable.from_data(data[['left', 'right']])
print(sqtab.summary())
A single 2x2 table¶
Several methods for working with individual 2x2 tables are provided in
the sm.stats.Table2x2 class. The summary method displays
several measures of association between the rows and columns of the
table.
In [41]: table = np.asarray([[35, 21], [25, 58]])
In [42]: t22 = sm.stats.Table2x2(table)
In [43]: print(t22.summary())
Estimate SE LCB UCB p-value
-------------------------------------------------
Odds ratio 3.867 1.890 7.912 0.000
Log odds ratio 1.352 0.365 0.636 2.068 0.000
Risk ratio 2.075 0.636 2.068 0.000
Log risk ratio 0.730 0.197 0.345 1.115 0.000
-------------------------------------------------
Note that the risk ratio is not symmetric so different results will be obtained if the transposed table is analyzed.
In [44]: table = np.asarray([[35, 21], [25, 58]])
In [45]: t22 = sm.stats.Table2x2(table.T)
In [46]: print(t22.summary())
Estimate SE LCB UCB p-value
-------------------------------------------------
Odds ratio 3.867 1.890 7.912 0.000
Log odds ratio 1.352 0.365 0.636 2.068 0.000
Risk ratio 2.194 0.636 2.068 0.000
Log risk ratio 0.786 0.216 0.362 1.210 0.000
-------------------------------------------------
Stratified 2x2 tables¶
Stratification occurs when we have a collection of contingency tables
defined by the same row and column factors. In the example below, we
have a collection of 2x2 tables reflecting the joint distribution of
smoking and lung cancer in each of several regions of China. It is
possible that the tables all have a common odds ratio, even while the
marginal probabilities vary among the strata. The ‘Breslow-Day’
procedure tests whether the data are consistent with a common odds
ratio. It appears below as the Test of constant OR. The
Mantel-Haenszel procedure tests whether this common odds ratio is
equal to one. It appears below as the Test of OR=1. It is also
possible to estimate the common odds and risk ratios and obtain
confidence intervals for them. The summary method displays all of
these results. Individual results can be obtained from the class
methods and attributes.
In [47]: data = sm.datasets.china_smoking.load()
In [48]: mat = np.asarray(data.data)
In [49]: tables = [np.reshape(x, (2, 2)) for x in mat]
---------------------------------------------------------------------------
ValueError Traceback (most recent call last)
<ipython-input-49-023dddfffe37> in <module>()
----> 1 tables = [np.reshape(x, (2, 2)) for x in mat]
<ipython-input-49-023dddfffe37> in <listcomp>(.0)
----> 1 tables = [np.reshape(x, (2, 2)) for x in mat]
/usr/lib/python3/dist-packages/numpy/core/fromnumeric.py in reshape(a, newshape, order)
230 [5, 6]])
231 """
--> 232 return _wrapfunc(a, 'reshape', newshape, order=order)
233
234
/usr/lib/python3/dist-packages/numpy/core/fromnumeric.py in _wrapfunc(obj, method, *args, **kwds)
55 def _wrapfunc(obj, method, *args, **kwds):
56 try:
---> 57 return getattr(obj, method)(*args, **kwds)
58
59 # An AttributeError occurs if the object does not have
ValueError: cannot reshape array of size 1 into shape (2,2)
In [50]: st = sm.stats.StratifiedTable(tables)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-50-eead6f66e108> in <module>()
----> 1 st = sm.stats.StratifiedTable(tables)
NameError: name 'tables' is not defined
In [51]: print(st.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-51-640262627808> in <module>()
----> 1 print(st.summary())
NameError: name 'st' is not defined
Module Reference¶
Table(table[, shift_zeros]) |
Analyses that can be performed on a two-way contingency table. |
Table2x2(table[, shift_zeros]) |
Analyses that can be performed on a 2x2 contingency table. |
SquareTable(table[, shift_zeros]) |
Methods for analyzing a square contingency table. |
StratifiedTable(tables[, shift_zeros]) |
Analyses for a collection of 2x2 contingency tables. |
mcnemar(table[, exact, correction]) |
McNemar test of homogeneity. |
cochrans_q(x[, return_object]) |
Cochran’s Q test for identical binomial proportions. |