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/python3.6/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1317 h.request(req.get_method(), req.selector, req.data, headers,
-> 1318 encode_chunked=req.has_header('Transfer-encoding'))
1319 except OSError as err: # timeout error
/usr/lib/python3.6/http/client.py in request(self, method, url, body, headers, encode_chunked)
1238 """Send a complete request to the server."""
-> 1239 self._send_request(method, url, body, headers, encode_chunked)
1240
/usr/lib/python3.6/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
1284 body = _encode(body, 'body')
-> 1285 self.endheaders(body, encode_chunked=encode_chunked)
1286
/usr/lib/python3.6/http/client.py in endheaders(self, message_body, encode_chunked)
1233 raise CannotSendHeader()
-> 1234 self._send_output(message_body, encode_chunked=encode_chunked)
1235
/usr/lib/python3.6/http/client.py in _send_output(self, message_body, encode_chunked)
1025 del self._buffer[:]
-> 1026 self.send(msg)
1027
/usr/lib/python3.6/http/client.py in send(self, data)
963 if self.auto_open:
--> 964 self.connect()
965 else:
/usr/lib/python3.6/http/client.py in connect(self)
1391
-> 1392 super().connect()
1393
/usr/lib/python3.6/http/client.py in connect(self)
935 self.sock = self._create_connection(
--> 936 (self.host,self.port), self.timeout, self.source_address)
937 self.sock.setsockopt(socket.IPPROTO_TCP, socket.TCP_NODELAY, 1)
/usr/lib/python3.6/socket.py in create_connection(address, timeout, source_address)
703 err = None
--> 704 for res in getaddrinfo(host, port, 0, SOCK_STREAM):
705 af, socktype, proto, canonname, sa = res
/usr/lib/python3.6/socket.py in getaddrinfo(host, port, family, type, proto, flags)
744 addrlist = []
--> 745 for res in _socket.getaddrinfo(host, port, family, type, proto, flags):
746 af, socktype, proto, canonname, sa = res
gaierror: [Errno -2] Name or service not known
During handling of the above exception, another exception occurred:
URLError Traceback (most recent call last)
<ipython-input-4-30b371d59b8a> in <module>()
----> 1 df = sm.datasets.get_rdataset("Arthritis", "vcd", cache=True).data
/build/statsmodels-pH_Txj/statsmodels-0.8.0/.pybuild/pythonX.Y_3.6/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-pH_Txj/statsmodels-0.8.0/.pybuild/pythonX.Y_3.6/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-pH_Txj/statsmodels-0.8.0/.pybuild/pythonX.Y_3.6/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/python3.6/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
221 else:
222 opener = _opener
--> 223 return opener.open(url, data, timeout)
224
225 def install_opener(opener):
/usr/lib/python3.6/urllib/request.py in open(self, fullurl, data, timeout)
524 req = meth(req)
525
--> 526 response = self._open(req, data)
527
528 # post-process response
/usr/lib/python3.6/urllib/request.py in _open(self, req, data)
542 protocol = req.type
543 result = self._call_chain(self.handle_open, protocol, protocol +
--> 544 '_open', req)
545 if result:
546 return result
/usr/lib/python3.6/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
502 for handler in handlers:
503 func = getattr(handler, meth_name)
--> 504 result = func(*args)
505 if result is not None:
506 return result
/usr/lib/python3.6/urllib/request.py in https_open(self, req)
1359 def https_open(self, req):
1360 return self.do_open(http.client.HTTPSConnection, req,
-> 1361 context=self._context, check_hostname=self._check_hostname)
1362
1363 https_request = AbstractHTTPHandler.do_request_
/usr/lib/python3.6/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1318 encode_chunked=req.has_header('Transfer-encoding'))
1319 except OSError as err: # timeout error
-> 1320 raise URLError(err)
1321 r = h.getresponse()
1322 except:
URLError: <urlopen error [Errno -2] Name or service not known>
In [5]: tab = pd.crosstab(df['Treatment'], df['Improved'])