Patsy: Contrast Coding Systems for categorical variables¶
Note
This document is based on this excellent resource from UCLA.
A categorical variable of K categories, or levels, usually enters a regression as a sequence of K-1 dummy variables. This amounts to a linear hypothesis on the level means. That is, each test statistic for these variables amounts to testing whether the mean for that level is statistically significantly different from the mean of the base category. This dummy coding is called Treatment coding in R parlance, and we will follow this convention. There are, however, different coding methods that amount to different sets of linear hypotheses.
In fact, the dummy coding is not technically a contrast coding. This is because the dummy variables add to one and are not functionally independent of the model’s intercept. On the other hand, a set of contrasts for a categorical variable with k levels is a set of k-1 functionally independent linear combinations of the factor level means that are also independent of the sum of the dummy variables. The dummy coding is not wrong per se. It captures all of the coefficients, but it complicates matters when the model assumes independence of the coefficients such as in ANOVA. Linear regression models do not assume independence of the coefficients and thus dummy coding is often the only coding that is taught in this context.
To have a look at the contrast matrices in Patsy, we will use data from UCLA ATS. First let’s load the data.
Example Data¶
In [1]: import pandas
In [2]: url = 'https://stats.idre.ucla.edu/stat/data/hsb2.csv'
In [3]: hsb2 = pandas.read_csv(url)
---------------------------------------------------------------------------
ConnectionRefusedError Traceback (most recent call last)
/usr/lib/python3.9/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1345 try:
-> 1346 h.request(req.get_method(), req.selector, req.data, headers,
1347 encode_chunked=req.has_header('Transfer-encoding'))
/usr/lib/python3.9/http/client.py in request(self, method, url, body, headers, encode_chunked)
1278 """Send a complete request to the server."""
-> 1279 self._send_request(method, url, body, headers, encode_chunked)
1280
/usr/lib/python3.9/http/client.py in _send_request(self, method, url, body, headers, encode_chunked)
1324 body = _encode(body, 'body')
-> 1325 self.endheaders(body, encode_chunked=encode_chunked)
1326
/usr/lib/python3.9/http/client.py in endheaders(self, message_body, encode_chunked)
1273 raise CannotSendHeader()
-> 1274 self._send_output(message_body, encode_chunked=encode_chunked)
1275
/usr/lib/python3.9/http/client.py in _send_output(self, message_body, encode_chunked)
1033 del self._buffer[:]
-> 1034 self.send(msg)
1035
/usr/lib/python3.9/http/client.py in send(self, data)
973 if self.auto_open:
--> 974 self.connect()
975 else:
/usr/lib/python3.9/http/client.py in connect(self)
1440
-> 1441 super().connect()
1442
/usr/lib/python3.9/http/client.py in connect(self)
944 """Connect to the host and port specified in __init__."""
--> 945 self.sock = self._create_connection(
946 (self.host,self.port), self.timeout, self.source_address)
/usr/lib/python3.9/socket.py in create_connection(address, timeout, source_address)
843 try:
--> 844 raise err
845 finally:
/usr/lib/python3.9/socket.py in create_connection(address, timeout, source_address)
831 sock.bind(source_address)
--> 832 sock.connect(sa)
833 # Break explicitly a reference cycle
ConnectionRefusedError: [Errno 111] Connection refused
During handling of the above exception, another exception occurred:
URLError Traceback (most recent call last)
<ipython-input-3-4735750ff492> in <module>
----> 1 hsb2 = pandas.read_csv(url)
/usr/lib/python3/dist-packages/pandas/io/parsers.py in read_csv(filepath_or_buffer, sep, delimiter, header, names, index_col, usecols, squeeze, prefix, mangle_dupe_cols, dtype, engine, converters, true_values, false_values, skipinitialspace, skiprows, skipfooter, nrows, na_values, keep_default_na, na_filter, verbose, skip_blank_lines, parse_dates, infer_datetime_format, keep_date_col, date_parser, dayfirst, cache_dates, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, doublequote, escapechar, comment, encoding, dialect, error_bad_lines, warn_bad_lines, delim_whitespace, low_memory, memory_map, float_precision)
686 )
687
--> 688 return _read(filepath_or_buffer, kwds)
689
690
/usr/lib/python3/dist-packages/pandas/io/parsers.py in _read(filepath_or_buffer, kwds)
434 # though mypy handling of conditional imports is difficult.
435 # See https://github.com/python/mypy/issues/1297
--> 436 fp_or_buf, _, compression, should_close = get_filepath_or_buffer(
437 filepath_or_buffer, encoding, compression
438 )
/usr/lib/python3/dist-packages/pandas/io/common.py in get_filepath_or_buffer(filepath_or_buffer, encoding, compression, mode, storage_options)
181 if isinstance(filepath_or_buffer, str) and is_url(filepath_or_buffer):
182 # TODO: fsspec can also handle HTTP via requests, but leaving this unchanged
--> 183 req = urlopen(filepath_or_buffer)
184 content_encoding = req.headers.get("Content-Encoding", None)
185 if content_encoding == "gzip":
/usr/lib/python3/dist-packages/pandas/io/common.py in urlopen(*args, **kwargs)
135 import urllib.request
136
--> 137 return urllib.request.urlopen(*args, **kwargs)
138
139
/usr/lib/python3.9/urllib/request.py in urlopen(url, data, timeout, cafile, capath, cadefault, context)
212 else:
213 opener = _opener
--> 214 return opener.open(url, data, timeout)
215
216 def install_opener(opener):
/usr/lib/python3.9/urllib/request.py in open(self, fullurl, data, timeout)
515
516 sys.audit('urllib.Request', req.full_url, req.data, req.headers, req.get_method())
--> 517 response = self._open(req, data)
518
519 # post-process response
/usr/lib/python3.9/urllib/request.py in _open(self, req, data)
532
533 protocol = req.type
--> 534 result = self._call_chain(self.handle_open, protocol, protocol +
535 '_open', req)
536 if result:
/usr/lib/python3.9/urllib/request.py in _call_chain(self, chain, kind, meth_name, *args)
492 for handler in handlers:
493 func = getattr(handler, meth_name)
--> 494 result = func(*args)
495 if result is not None:
496 return result
/usr/lib/python3.9/urllib/request.py in https_open(self, req)
1387
1388 def https_open(self, req):
-> 1389 return self.do_open(http.client.HTTPSConnection, req,
1390 context=self._context, check_hostname=self._check_hostname)
1391
/usr/lib/python3.9/urllib/request.py in do_open(self, http_class, req, **http_conn_args)
1347 encode_chunked=req.has_header('Transfer-encoding'))
1348 except OSError as err: # timeout error
-> 1349 raise URLError(err)
1350 r = h.getresponse()
1351 except:
URLError: <urlopen error [Errno 111] Connection refused>
It will be instructive to look at the mean of the dependent variable, write, for each level of race ((1 = Hispanic, 2 = Asian, 3 = African American and 4 = Caucasian)).
In [4]: hsb2.groupby('race')['write'].mean()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-4-a0004b337d47> in <module>
----> 1 hsb2.groupby('race')['write'].mean()
NameError: name 'hsb2' is not defined
Treatment (Dummy) Coding¶
Dummy coding is likely the most well known coding scheme. It compares each level of the categorical variable to a base reference level. The base reference level is the value of the intercept. It is the default contrast in Patsy for unordered categorical factors. The Treatment contrast matrix for race would be
In [5]: from patsy.contrasts import Treatment
In [6]: levels = [1,2,3,4]
In [7]: contrast = Treatment(reference=0).code_without_intercept(levels)
In [8]: print(contrast.matrix)
[[0. 0. 0.]
[1. 0. 0.]
[0. 1. 0.]
[0. 0. 1.]]
Here we used reference=0, which implies that the first level, Hispanic, is the reference category against which the other level effects are measured. As mentioned above, the columns do not sum to zero and are thus not independent of the intercept. To be explicit, let’s look at how this would encode the race variable.
In [9]: contrast.matrix[hsb2.race-1, :][:20]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-9-eae0b0d66a00> in <module>
----> 1 contrast.matrix[hsb2.race-1, :][:20]
NameError: name 'hsb2' is not defined
This is a bit of a trick, as the race category conveniently maps to zero-based indices. If it does not, this conversion happens under the hood, so this will not work in general but nonetheless is a useful exercise to fix ideas. The below illustrates the output using the three contrasts above
In [10]: from statsmodels.formula.api import ols
In [11]: mod = ols("write ~ C(race, Treatment)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-11-3bdf176f3042> in <module>
----> 1 mod = ols("write ~ C(race, Treatment)", data=hsb2)
NameError: name 'hsb2' is not defined
In [12]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-12-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [13]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-13-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
We explicitly gave the contrast for race; however, since Treatment is the default, we could have omitted this.
Simple Coding¶
Like Treatment Coding, Simple Coding compares each level to a fixed reference level. However, with simple coding, the intercept is the grand mean of all the levels of the factors. See User-Defined Coding for how to implement the Simple contrast.
In [14]: contrast = Simple().code_without_intercept(levels)
In [15]: print(contrast.matrix)
[[-0.25 -0.25 -0.25]
[ 0.75 -0.25 -0.25]
[-0.25 0.75 -0.25]
[-0.25 -0.25 0.75]]
In [16]: mod = ols("write ~ C(race, Simple)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-16-6ce0487a5b61> in <module>
----> 1 mod = ols("write ~ C(race, Simple)", data=hsb2)
NameError: name 'hsb2' is not defined
In [17]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-17-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [18]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-18-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
Sum (Deviation) Coding¶
Sum coding compares the mean of the dependent variable for a given level to the overall mean of the dependent variable over all the levels. That is, it uses contrasts between each of the first k-1 levels and level k In this example, level 1 is compared to all the others, level 2 to all the others, and level 3 to all the others.
In [19]: from patsy.contrasts import Sum
In [20]: contrast = Sum().code_without_intercept(levels)
In [21]: print(contrast.matrix)
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]
[-1. -1. -1.]]
In [22]: mod = ols("write ~ C(race, Sum)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-22-fcaa6b96ccfd> in <module>
----> 1 mod = ols("write ~ C(race, Sum)", data=hsb2)
NameError: name 'hsb2' is not defined
In [23]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-23-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [24]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-24-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
This corresponds to a parameterization that forces all the coefficients to sum to zero. Notice that the intercept here is the grand mean where the grand mean is the mean of means of the dependent variable by each level.
In [25]: hsb2.groupby('race')['write'].mean().mean()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-25-b21947de62cb> in <module>
----> 1 hsb2.groupby('race')['write'].mean().mean()
NameError: name 'hsb2' is not defined
Backward Difference Coding¶
In backward difference coding, the mean of the dependent variable for a level is compared with the mean of the dependent variable for the prior level. This type of coding may be useful for a nominal or an ordinal variable.
In [26]: from patsy.contrasts import Diff
In [27]: contrast = Diff().code_without_intercept(levels)
In [28]: print(contrast.matrix)
[[-0.75 -0.5 -0.25]
[ 0.25 -0.5 -0.25]
[ 0.25 0.5 -0.25]
[ 0.25 0.5 0.75]]
In [29]: mod = ols("write ~ C(race, Diff)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-29-e28888f23177> in <module>
----> 1 mod = ols("write ~ C(race, Diff)", data=hsb2)
NameError: name 'hsb2' is not defined
In [30]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-30-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [31]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-31-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
For example, here the coefficient on level 1 is the mean of write at level 2 compared with the mean at level 1. Ie.,
In [32]: res.params["C(race, Diff)[D.1]"]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-32-f27d60ed9ed4> in <module>
----> 1 res.params["C(race, Diff)[D.1]"]
NameError: name 'res' is not defined
In [33]: hsb2.groupby('race').mean()["write"][2] - \
....: hsb2.groupby('race').mean()["write"][1]
....:
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-33-c6e56faf0508> in <module>
----> 1 hsb2.groupby('race').mean()["write"][2] - \
2 hsb2.groupby('race').mean()["write"][1]
NameError: name 'hsb2' is not defined
Helmert Coding¶
Our version of Helmert coding is sometimes referred to as Reverse Helmert Coding. The mean of the dependent variable for a level is compared to the mean of the dependent variable over all previous levels. Hence, the name ‘reverse’ being sometimes applied to differentiate from forward Helmert coding. This comparison does not make much sense for a nominal variable such as race, but we would use the Helmert contrast like so:
In [34]: from patsy.contrasts import Helmert
In [35]: contrast = Helmert().code_without_intercept(levels)
In [36]: print(contrast.matrix)
[[-1. -1. -1.]
[ 1. -1. -1.]
[ 0. 2. -1.]
[ 0. 0. 3.]]
In [37]: mod = ols("write ~ C(race, Helmert)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-37-c991b20b1c77> in <module>
----> 1 mod = ols("write ~ C(race, Helmert)", data=hsb2)
NameError: name 'hsb2' is not defined
In [38]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-38-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [39]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-39-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
To illustrate, the comparison on level 4 is the mean of the dependent variable at the previous three levels taken from the mean at level 4
In [40]: grouped = hsb2.groupby('race')
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-40-717e908ca802> in <module>
----> 1 grouped = hsb2.groupby('race')
NameError: name 'hsb2' is not defined
In [41]: grouped.mean()["write"][4] - grouped.mean()["write"][:3].mean()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-41-a29d75116d4c> in <module>
----> 1 grouped.mean()["write"][4] - grouped.mean()["write"][:3].mean()
NameError: name 'grouped' is not defined
As you can see, these are only equal up to a constant. Other versions of the Helmert contrast give the actual difference in means. Regardless, the hypothesis tests are the same.
In [42]: k = 4
In [43]: 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-43-8956699e0a60> in <module>
----> 1 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
NameError: name 'grouped' is not defined
In [44]: k = 3
In [45]: 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-45-8956699e0a60> in <module>
----> 1 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
NameError: name 'grouped' is not defined
Orthogonal Polynomial Coding¶
The coefficients taken on by polynomial coding for k=4 levels are the linear, quadratic, and cubic trends in the categorical variable. The categorical variable here is assumed to be represented by an underlying, equally spaced numeric variable. Therefore, this type of encoding is used only for ordered categorical variables with equal spacing. In general, the polynomial contrast produces polynomials of order k-1. Since race is not an ordered factor variable let’s use read as an example. First we need to create an ordered categorical from read.
In [46]: _, bins = np.histogram(hsb2.read, 3)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-46-55fa0c232e59> in <module>
----> 1 _, bins = np.histogram(hsb2.read, 3)
NameError: name 'hsb2' is not defined
In [47]: try: # requires numpy master
....: readcat = np.digitize(hsb2.read, bins, True)
....: except:
....: readcat = np.digitize(hsb2.read, bins)
....:
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-47-4311ab0dca44> in <module>
1 try: # requires numpy master
----> 2 readcat = np.digitize(hsb2.read, bins, True)
3 except:
NameError: name 'hsb2' is not defined
During handling of the above exception, another exception occurred:
NameError Traceback (most recent call last)
<ipython-input-47-4311ab0dca44> in <module>
2 readcat = np.digitize(hsb2.read, bins, True)
3 except:
----> 4 readcat = np.digitize(hsb2.read, bins)
NameError: name 'hsb2' is not defined
In [48]: hsb2['readcat'] = readcat
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-48-e199b94a103d> in <module>
----> 1 hsb2['readcat'] = readcat
NameError: name 'readcat' is not defined
In [49]: hsb2.groupby('readcat').mean()['write']
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-49-97d196c364b0> in <module>
----> 1 hsb2.groupby('readcat').mean()['write']
NameError: name 'hsb2' is not defined
In [50]: from patsy.contrasts import Poly
In [51]: levels = hsb2.readcat.unique().tolist()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-51-4d972b7909b9> in <module>
----> 1 levels = hsb2.readcat.unique().tolist()
NameError: name 'hsb2' is not defined
In [52]: contrast = Poly().code_without_intercept(levels)
In [53]: print(contrast.matrix)
[[-0.6708 0.5 -0.2236]
[-0.2236 -0.5 0.6708]
[ 0.2236 -0.5 -0.6708]
[ 0.6708 0.5 0.2236]]
In [54]: mod = ols("write ~ C(readcat, Poly)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-54-e9324312786f> in <module>
----> 1 mod = ols("write ~ C(readcat, Poly)", data=hsb2)
NameError: name 'hsb2' is not defined
In [55]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-55-fa3ccf53f431> in <module>
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [56]: print(res.summary())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-56-ba064a039ab1> in <module>
----> 1 print(res.summary())
NameError: name 'res' is not defined
As you can see, readcat has a significant linear effect on the dependent variable write but not a significant quadratic or cubic effect.
User-Defined Coding¶
If you want to use your own coding, you must do so by writing a coding class that contains a code_with_intercept and a code_without_intercept method that return a patsy.contrast.ContrastMatrix instance.
In [57]: from patsy.contrasts import ContrastMatrix
....:
....: def _name_levels(prefix, levels):
....: return ["[%s%s]" % (prefix, level) for level in levels]
....:
In [58]: class Simple(object):
....: def _simple_contrast(self, levels):
....: nlevels = len(levels)
....: contr = -1./nlevels * np.ones((nlevels, nlevels-1))
....: contr[1:][np.diag_indices(nlevels-1)] = (nlevels-1.)/nlevels
....: return contr
....:
....: def code_with_intercept(self, levels):
....: contrast = np.column_stack((np.ones(len(levels)),
....: self._simple_contrast(levels)))
....: return ContrastMatrix(contrast, _name_levels("Simp.", levels))
....:
....: def code_without_intercept(self, levels):
....: contrast = self._simple_contrast(levels)
....: return ContrastMatrix(contrast, _name_levels("Simp.", levels[:-1]))
....:
File "<tokenize>", line 13
def code_without_intercept(self, levels):
^
IndentationError: unindent does not match any outer indentation level
In [60]: mod = ols("write ~ C(race, Simple)", data=hsb2)
....: res = mod.fit()
....: print(res.summary())
....:
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-60-b06f36e8be6e> in <module>
----> 1 mod = ols("write ~ C(race, Simple)", data=hsb2)
2 res = mod.fit()
3 print(res.summary())
NameError: name 'hsb2' is not defined