Generalized Estimating Equations

Generalized Estimating Equations estimate generalized linear models for panel, cluster or repeated measures data when the observations are possibly correlated withing a cluster but uncorrelated across clusters. It supports estimation of the same one-parameter exponential families as Generalized Linear models (GLM).

See Module Reference for commands and arguments.

Examples

The following illustrates a Poisson regression with exchangeable correlation within clusters using data on epilepsy seizures.

In [1]: import statsmodels.api as sm

In [2]: import statsmodels.formula.api as smf

In [3]: data = sm.datasets.get_rdataset('epil', package='MASS', 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-3-ba4a6e64d7bb> in <module>()
----> 1 data = sm.datasets.get_rdataset('epil', package='MASS', 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 [4]: fam = sm.families.Poisson()

In [5]: ind = sm.cov_struct.Exchangeable()

In [6]: mod = smf.gee("y ~ age + trt + base", "subject", data,
   ...:               cov_struct=ind, family=fam)
   ...: 
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-6-ff1703408948> in <module>()
      1 mod = smf.gee("y ~ age + trt + base", "subject", data,
----> 2               cov_struct=ind, family=fam)

/build/statsmodels-0.8.0/.pybuild/pythonX.Y_3.5/build/statsmodels/genmod/generalized_estimating_equations.py in from_formula(cls, formula, groups, data, subset, time, offset, exposure, *args, **kwargs)
    668 
    669         if type(groups) == str:
--> 670             groups = data[groups]
    671 
    672         if type(time) == str:

KeyError: 'subject'

In [7]: res = mod.fit()
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-deef2687e692> in <module>()
----> 1 res = mod.fit()

NameError: name 'mod' is not defined

In [8]: print(res.summary())
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-8-a8dc848a1f25> in <module>()
----> 1 print(res.summary())

NameError: name 'res' is not defined

Several notebook examples of the use of GEE can be found on the Wiki: Wiki notebooks for GEE

References

  • KY Liang and S Zeger. “Longitudinal data analysis using generalized linear models”. Biometrika (1986) 73 (1): 13-22.
  • S Zeger and KY Liang. “Longitudinal Data Analysis for Discrete and Continuous Outcomes”. Biometrics Vol. 42, No. 1 (Mar., 1986), pp. 121-130
  • A Rotnitzky and NP Jewell (1990). “Hypothesis testing of regression parameters in semiparametric generalized linear models for cluster correlated data”, Biometrika, 77, 485-497.
  • Xu Guo and Wei Pan (2002). “Small sample performance of the score test in GEE”. http://www.sph.umn.edu/faculty1/wp-content/uploads/2012/11/rr2002-013.pdf
  • LA Mancl LA, TA DeRouen (2001). A covariance estimator for GEE with improved small-sample properties. Biometrics. 2001 Mar;57(1):126-34.

Module Reference

Model Class

GEE(endog, exog, groups[, time, family, …]) Estimation of marginal regression models using Generalized Estimating Equations (GEE).

Results Classes

GEEResults(model, params, cov_params, scale) This class summarizes the fit of a marginal regression model using GEE.
GEEMargins(results, args[, kwargs]) Estimated marginal effects for a regression model fit with GEE.

Dependence Structures

The dependence structures currently implemented are

CovStruct([cov_nearest_method]) A base class for correlation and covariance structures of grouped data.
Autoregressive([dist_func]) A first-order autoregressive working dependence structure.
Exchangeable() An exchangeable working dependence structure.
GlobalOddsRatio(endog_type) Estimate the global odds ratio for a GEE with ordinal or nominal data.
Independence([cov_nearest_method]) An independence working dependence structure.
Nested([cov_nearest_method]) A nested working dependence structure.

Families

The distribution families are the same as for GLM, currently implemented are

Family(link, variance) The parent class for one-parameter exponential families.
Binomial([link]) Binomial exponential family distribution.
Gamma([link]) Gamma exponential family distribution.
Gaussian([link]) Gaussian exponential family distribution.
InverseGaussian([link]) InverseGaussian exponential family.
NegativeBinomial([link, alpha]) Negative Binomial exponential family.
Poisson([link]) Poisson exponential family.