# Contrasts Overview¶

[1]:

import numpy as np
import statsmodels.api as sm


This document is based heavily on this excellent resource from UCLA http://www.ats.ucla.edu/stat/r/library/contrast_coding.htm

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¶

[2]:

import pandas as pd

url = "https://stats.idre.ucla.edu/stat/data/hsb2.csv"

[3]:

hsb2.head(10)

[3]:

id female race ses schtyp prog read write math science socst
0 70 0 4 1 1 1 57 52 41 47 57
1 121 1 4 2 1 3 68 59 53 63 61
2 86 0 4 3 1 1 44 33 54 58 31
3 141 0 4 3 1 3 63 44 47 53 56
4 172 0 4 2 1 2 47 52 57 53 61
5 113 0 4 2 1 2 44 52 51 63 61
6 50 0 3 2 1 1 50 59 42 53 61
7 11 0 1 2 1 2 34 46 45 39 36
8 84 0 4 2 1 1 63 57 54 58 51
9 48 0 3 2 1 2 57 55 52 50 51

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)).

[4]:

hsb2.groupby("race")["write"].mean()

[4]:

race
1    46.458333
2    58.000000
3    48.200000
4    54.055172
Name: write, dtype: float64


## 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

[5]:

from patsy.contrasts import Treatment

levels = [1, 2, 3, 4]
contrast = Treatment(reference=0).code_without_intercept(levels)
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.

[6]:

hsb2.race.head(10)

[6]:

0    4
1    4
2    4
3    4
4    4
5    4
6    3
7    1
8    4
9    3
Name: race, dtype: int64

[7]:

print(contrast.matrix[hsb2.race - 1, :][:20])

[[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 1. 0.]
[0. 0. 0.]
[0. 0. 1.]
[0. 1. 0.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 1. 0.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]
[0. 0. 1.]]

[8]:

pd.get_dummies(hsb2.race.values, drop_first=False)

[8]:

1 2 3 4
0 0 0 0 1
1 0 0 0 1
2 0 0 0 1
3 0 0 0 1
4 0 0 0 1
... ... ... ... ...
195 0 1 0 0
196 0 0 0 1
197 0 0 0 1
198 0 0 0 1
199 0 0 0 1

200 rows × 4 columns

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

[9]:

from statsmodels.formula.api import ols

mod = ols("write ~ C(race, Treatment)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.107
Method:                 Least Squares   F-statistic:                     7.833
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.78e-05
Time:                        09:27:23   Log-Likelihood:                -721.77
No. Observations:                 200   AIC:                             1452.
Df Residuals:                     196   BIC:                             1465.
Df Model:                           3
Covariance Type:            nonrobust
===========================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------------------
Intercept                  46.4583      1.842     25.218      0.000      42.825      50.091
C(race, Treatment)[T.2]    11.5417      3.286      3.512      0.001       5.061      18.022
C(race, Treatment)[T.3]     1.7417      2.732      0.637      0.525      -3.647       7.131
C(race, Treatment)[T.4]     7.5968      1.989      3.820      0.000       3.675      11.519
==============================================================================
Omnibus:                       10.487   Durbin-Watson:                   1.779
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               11.031
Skew:                          -0.551   Prob(JB):                      0.00402
Kurtosis:                       2.670   Cond. No.                         8.25
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


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. Patsy does not have the Simple contrast included, but you can easily define your own contrasts. To do so, write a class that contains a code_with_intercept and a code_without_intercept method that returns a patsy.contrast.ContrastMatrix instance

[10]:

from patsy.contrasts import ContrastMatrix

def _name_levels(prefix, levels):
return ["[%s%s]" % (prefix, level) for level in levels]

class Simple(object):
def _simple_contrast(self, levels):
nlevels = len(levels)
contr = -1.0 / nlevels * np.ones((nlevels, nlevels - 1))
contr[1:][np.diag_indices(nlevels - 1)] = (nlevels - 1.0) / 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]))

[11]:

hsb2.groupby("race")["write"].mean().mean()

[11]:

51.67837643678162

[12]:

contrast = Simple().code_without_intercept(levels)
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]]

[13]:

mod = ols("write ~ C(race, Simple)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.107
Method:                 Least Squares   F-statistic:                     7.833
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.78e-05
Time:                        09:27:23   Log-Likelihood:                -721.77
No. Observations:                 200   AIC:                             1452.
Df Residuals:                     196   BIC:                             1465.
Df Model:                           3
Covariance Type:            nonrobust
===========================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------------------
Intercept                  51.6784      0.982     52.619      0.000      49.741      53.615
C(race, Simple)[Simp.1]    11.5417      3.286      3.512      0.001       5.061      18.022
C(race, Simple)[Simp.2]     1.7417      2.732      0.637      0.525      -3.647       7.131
C(race, Simple)[Simp.3]     7.5968      1.989      3.820      0.000       3.675      11.519
==============================================================================
Omnibus:                       10.487   Durbin-Watson:                   1.779
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               11.031
Skew:                          -0.551   Prob(JB):                      0.00402
Kurtosis:                       2.670   Cond. No.                         7.03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


### 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.

[14]:

from patsy.contrasts import Sum

contrast = Sum().code_without_intercept(levels)
print(contrast.matrix)

[[ 1.  0.  0.]
[ 0.  1.  0.]
[ 0.  0.  1.]
[-1. -1. -1.]]

[15]:

mod = ols("write ~ C(race, Sum)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.107
Method:                 Least Squares   F-statistic:                     7.833
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.78e-05
Time:                        09:27:23   Log-Likelihood:                -721.77
No. Observations:                 200   AIC:                             1452.
Df Residuals:                     196   BIC:                             1465.
Df Model:                           3
Covariance Type:            nonrobust
=====================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------------
Intercept            51.6784      0.982     52.619      0.000      49.741      53.615
C(race, Sum)[S.1]    -5.2200      1.631     -3.200      0.002      -8.437      -2.003
C(race, Sum)[S.2]     6.3216      2.160      2.926      0.004       2.061      10.582
C(race, Sum)[S.3]    -3.4784      1.732     -2.008      0.046      -6.895      -0.062
==============================================================================
Omnibus:                       10.487   Durbin-Watson:                   1.779
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               11.031
Skew:                          -0.551   Prob(JB):                      0.00402
Kurtosis:                       2.670   Cond. No.                         6.72
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


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.

[16]:

hsb2.groupby("race")["write"].mean().mean()

[16]:

51.67837643678162


### 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.

[17]:

from patsy.contrasts import Diff

contrast = Diff().code_without_intercept(levels)
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]]

[18]:

mod = ols("write ~ C(race, Diff)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.107
Method:                 Least Squares   F-statistic:                     7.833
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.78e-05
Time:                        09:27:23   Log-Likelihood:                -721.77
No. Observations:                 200   AIC:                             1452.
Df Residuals:                     196   BIC:                             1465.
Df Model:                           3
Covariance Type:            nonrobust
======================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept             51.6784      0.982     52.619      0.000      49.741      53.615
C(race, Diff)[D.1]    11.5417      3.286      3.512      0.001       5.061      18.022
C(race, Diff)[D.2]    -9.8000      3.388     -2.893      0.004     -16.481      -3.119
C(race, Diff)[D.3]     5.8552      2.153      2.720      0.007       1.610      10.101
==============================================================================
Omnibus:                       10.487   Durbin-Watson:                   1.779
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               11.031
Skew:                          -0.551   Prob(JB):                      0.00402
Kurtosis:                       2.670   Cond. No.                         8.30
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


For example, here the coefficient on level 1 is the mean of write at level 2 compared with the mean at level 1. Ie.,

[19]:

res.params["C(race, Diff)[D.1]"]
hsb2.groupby("race").mean()["write"][2] - hsb2.groupby("race").mean()["write"][1]

[19]:

11.541666666666664


### 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:

[20]:

from patsy.contrasts import Helmert

contrast = Helmert().code_without_intercept(levels)
print(contrast.matrix)

[[-1. -1. -1.]
[ 1. -1. -1.]
[ 0.  2. -1.]
[ 0.  0.  3.]]

[21]:

mod = ols("write ~ C(race, Helmert)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.107
Method:                 Least Squares   F-statistic:                     7.833
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.78e-05
Time:                        09:27:24   Log-Likelihood:                -721.77
No. Observations:                 200   AIC:                             1452.
Df Residuals:                     196   BIC:                             1465.
Df Model:                           3
Covariance Type:            nonrobust
=========================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-----------------------------------------------------------------------------------------
Intercept                51.6784      0.982     52.619      0.000      49.741      53.615
C(race, Helmert)[H.2]     5.7708      1.643      3.512      0.001       2.530       9.011
C(race, Helmert)[H.3]    -1.3431      0.867     -1.548      0.123      -3.054       0.368
C(race, Helmert)[H.4]     0.7923      0.372      2.130      0.034       0.059       1.526
==============================================================================
Omnibus:                       10.487   Durbin-Watson:                   1.779
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               11.031
Skew:                          -0.551   Prob(JB):                      0.00402
Kurtosis:                       2.670   Cond. No.                         7.26
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


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

[22]:

grouped = hsb2.groupby("race")
grouped.mean()["write"][4] - grouped.mean()["write"][:3].mean()

[22]:

3.169061302681982


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.

[23]:

k = 4
1.0 / k * (grouped.mean()["write"][k] - grouped.mean()["write"][: k - 1].mean())
k = 3
1.0 / k * (grouped.mean()["write"][k] - grouped.mean()["write"][: k - 1].mean())

[23]:

-1.3430555555555561


### 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.

[24]:

hsb2["readcat"] = np.asarray(pd.cut(hsb2.read, bins=4))

[24]:

readcat
(27.952, 40.0]    42.772727
(40.0, 52.0]      49.978495
(52.0, 64.0]      56.563636
(64.0, 76.0]      61.833333
Name: write, dtype: float64

[25]:

from patsy.contrasts import Poly

contrast = Poly().code_without_intercept(levels)
print(contrast.matrix)

[[-0.67082039  0.5        -0.2236068 ]
[-0.2236068  -0.5         0.67082039]
[ 0.2236068  -0.5        -0.67082039]
[ 0.67082039  0.5         0.2236068 ]]

[26]:

mod = ols("write ~ C(readcat, Poly)", data=hsb2)
res = mod.fit()
print(res.summary())

                            OLS Regression Results
==============================================================================
Dep. Variable:                  write   R-squared:                       0.346
Method:                 Least Squares   F-statistic:                     34.51
Date:                Tue, 03 Aug 2021   Prob (F-statistic):           5.95e-18
Time:                        09:27:24   Log-Likelihood:                -690.69
No. Observations:                 200   AIC:                             1389.
Df Residuals:                     196   BIC:                             1403.
Df Model:                           3
Covariance Type:            nonrobust
==============================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
----------------------------------------------------------------------------------------------
Intercept                     52.7870      0.634     83.268      0.000      51.537      54.037
C(readcat, Poly).Linear       14.2587      1.484      9.607      0.000      11.332      17.186

As you can see, readcat has a significant linear effect on the dependent variable write but the quadratic and cubic effects are insignificant.