Patsy: Contrast Coding Systems for categorical variables¶
Note
This document is based heavily 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 isn’t 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 = 'http://www.ats.ucla.edu/stat/data/hsb2.csv'
In [3]: hsb2 = pandas.read_table(url, delimiter=",")
---------------------------------------------------------------------------
URLError Traceback (most recent call last)
<ipython-input-3-626fbea21bbc> in <module>()
----> 1 hsb2 = pandas.read_table(url, delimiter=",")
/usr/lib/python2.7/dist-packages/pandas/io/parsers.pyc in parser_f(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, iterator, chunksize, compression, thousands, decimal, lineterminator, quotechar, quoting, escapechar, comment, encoding, dialect, tupleize_cols, error_bad_lines, warn_bad_lines, skip_footer, doublequote, delim_whitespace, as_recarray, compact_ints, use_unsigned, low_memory, buffer_lines, memory_map, float_precision)
539 skip_blank_lines=skip_blank_lines)
540
--> 541 return _read(filepath_or_buffer, kwds)
542
543 parser_f.__name__ = name
/usr/lib/python2.7/dist-packages/pandas/io/parsers.pyc in _read(filepath_or_buffer, kwds)
291 filepath_or_buffer, _, compression = get_filepath_or_buffer(
292 filepath_or_buffer, encoding,
--> 293 compression=kwds.get('compression', None))
294 kwds['compression'] = (inferred_compression if compression == 'infer'
295 else compression)
/usr/lib/python2.7/dist-packages/pandas/io/common.pyc in get_filepath_or_buffer(filepath_or_buffer, encoding, compression)
306
307 if _is_url(filepath_or_buffer):
--> 308 req = _urlopen(str(filepath_or_buffer))
309 if compression == 'infer':
310 content_encoding = req.headers.get('Content-Encoding', None)
/usr/lib/python2.7/urllib2.pyc in urlopen(url, data, timeout, cafile, capath, cadefault, context)
152 else:
153 opener = _opener
--> 154 return opener.open(url, data, timeout)
155
156 def install_opener(opener):
/usr/lib/python2.7/urllib2.pyc in open(self, fullurl, data, timeout)
427 req = meth(req)
428
--> 429 response = self._open(req, data)
430
431 # post-process response
/usr/lib/python2.7/urllib2.pyc in _open(self, req, data)
445 protocol = req.get_type()
446 result = self._call_chain(self.handle_open, protocol, protocol +
--> 447 '_open', req)
448 if result:
449 return result
/usr/lib/python2.7/urllib2.pyc in _call_chain(self, chain, kind, meth_name, *args)
405 func = getattr(handler, meth_name)
406
--> 407 result = func(*args)
408 if result is not None:
409 return result
/usr/lib/python2.7/urllib2.pyc in http_open(self, req)
1226
1227 def http_open(self, req):
-> 1228 return self.do_open(httplib.HTTPConnection, req)
1229
1230 http_request = AbstractHTTPHandler.do_request_
/usr/lib/python2.7/urllib2.pyc in do_open(self, http_class, req, **http_conn_args)
1196 except socket.error, err: # XXX what error?
1197 h.close()
-> 1198 raise URLError(err)
1199 else:
1200 try:
URLError: <urlopen error [Errno -3] Temporary failure in name resolution>
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)).
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 [4]: from patsy.contrasts import Treatment
In [5]: levels = [1,2,3,4]
In [6]: contrast = Treatment(reference=0).code_without_intercept(levels)
In [7]: 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 [8]: contrast.matrix[hsb2.race-1, :][:20]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-8-7f2c2108d810> 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 won’t work in general but nonetheless is a useful exercise to fix ideas. The below illustrates the output using the three contrasts above
In [9]: from statsmodels.formula.api import ols
In [10]: mod = ols("write ~ C(race, Treatment)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-10-726467a33d67> in <module>()
----> 1 mod = ols("write ~ C(race, Treatment)", data=hsb2)
NameError: name 'hsb2' is not defined
In [11]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-11-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [12]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-12-8b05a733a78f> 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 [13]: contrast = Simple().code_without_intercept(levels)
In [14]: 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 [15]: mod = ols("write ~ C(race, Simple)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-15-093844a49cfd> in <module>()
----> 1 mod = ols("write ~ C(race, Simple)", data=hsb2)
NameError: name 'hsb2' is not defined
In [16]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-16-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [17]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-17-8b05a733a78f> 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 [18]: from patsy.contrasts import Sum
In [19]: contrast = Sum().code_without_intercept(levels)
In [20]: print contrast.matrix
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]
[-1. -1. -1.]]
In [21]: mod = ols("write ~ C(race, Sum)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-21-74893cc350a7> in <module>()
----> 1 mod = ols("write ~ C(race, Sum)", data=hsb2)
NameError: name 'hsb2' is not defined
In [22]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-22-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [23]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-23-8b05a733a78f> in <module>()
----> 1 print res.summary()
NameError: name 'res' is not defined
This correspons 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 [24]: hsb2.groupby('race')['write'].mean().mean()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-24-05e782082dc3> 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 [25]: from patsy.contrasts import Diff
In [26]: contrast = Diff().code_without_intercept(levels)
In [27]: 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 [28]: mod = ols("write ~ C(race, Diff)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-28-06af35382249> in <module>()
----> 1 mod = ols("write ~ C(race, Diff)", data=hsb2)
NameError: name 'hsb2' is not defined
In [29]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-29-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [30]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-30-8b05a733a78f> 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 [31]: res.params["C(race, Diff)[D.1]"]
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-31-1ac50ef2ffcc> in <module>()
----> 1 res.params["C(race, Diff)[D.1]"]
NameError: name 'res' is not defined
In [32]: hsb2.groupby('race').mean()["write"][2] - \
....: hsb2.groupby('race').mean()["write"][1]
....:
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-32-86bf1877c8e1> in <module>()
----> 1 hsb2.groupby('race').mean()["write"][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 [33]: from patsy.contrasts import Helmert
In [34]: contrast = Helmert().code_without_intercept(levels)
In [35]: print contrast.matrix
[[-1. -1. -1.]
[ 1. -1. -1.]
[ 0. 2. -1.]
[ 0. 0. 3.]]
In [36]: mod = ols("write ~ C(race, Helmert)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-36-70f5b33a3695> in <module>()
----> 1 mod = ols("write ~ C(race, Helmert)", data=hsb2)
NameError: name 'hsb2' is not defined
In [37]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-37-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [38]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-38-8b05a733a78f> 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 [39]: grouped = hsb2.groupby('race')
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-39-7132042a695c> in <module>()
----> 1 grouped = hsb2.groupby('race')
NameError: name 'hsb2' is not defined
In [40]: grouped.mean()["write"][4] - grouped.mean()["write"][:3].mean()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-40-34ff38b41ae4> 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 [41]: k = 4
In [42]: 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-42-eba642883e4f> in <module>()
----> 1 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
NameError: name 'grouped' is not defined
In [43]: k = 3
In [44]: 1./k * (grouped.mean()["write"][k] - grouped.mean()["write"][:k-1].mean())
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-44-eba642883e4f> 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 [45]: _, bins = np.histogram(hsb2.read, 3)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-45-fe6483385cd5> in <module>()
----> 1 _, bins = np.histogram(hsb2.read, 3)
NameError: name 'hsb2' is not defined
In [46]: 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-46-9957f62bffd4> in <module>()
2 readcat = np.digitize(hsb2.read, bins, True)
3 except:
----> 4 readcat = np.digitize(hsb2.read, bins)
5
NameError: name 'hsb2' is not defined
In [47]: hsb2['readcat'] = readcat
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-47-314157e0b3ff> in <module>()
----> 1 hsb2['readcat'] = readcat
NameError: name 'readcat' is not defined
In [48]: hsb2.groupby('readcat').mean()['write']
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-48-10973d063fc2> in <module>()
----> 1 hsb2.groupby('readcat').mean()['write']
NameError: name 'hsb2' is not defined
In [49]: from patsy.contrasts import Poly
In [50]: levels = hsb2.readcat.unique().tolist()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-50-4696bbbc65e8> in <module>()
----> 1 levels = hsb2.readcat.unique().tolist()
NameError: name 'hsb2' is not defined
In [51]: contrast = Poly().code_without_intercept(levels)
In [52]: 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 [53]: mod = ols("write ~ C(readcat, Poly)", data=hsb2)
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-53-1b3036f0e66a> in <module>()
----> 1 mod = ols("write ~ C(readcat, Poly)", data=hsb2)
NameError: name 'hsb2' is not defined
In [54]: res = mod.fit()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-54-deef2687e692> in <module>()
----> 1 res = mod.fit()
NameError: name 'mod' is not defined
In [55]: print res.summary()
---------------------------------------------------------------------------
NameError Traceback (most recent call last)
<ipython-input-55-8b05a733a78f> 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 [56]: from patsy.contrasts import ContrastMatrix
In [57]: 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
....:
In [59]: 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))
....: