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Fitting models using R-style formulas

Since version 0.5.0, statsmodels allows users to fit statistical models using R-style formulas. Internally, statsmodels uses the patsy package to convert formulas and data to the matrices that are used in model fitting. The formula framework is quite powerful; this tutorial only scratches the surface. A full description of the formula language can be found in the patsy docs:

Loading modules and functions

import statsmodels.formula.api as sm
import numpy as np
import pandas

Notice that we called statsmodels.formula.api instead of the usual statsmodels.api. The formula.api hosts many of the same functions found in api (e.g. OLS, GLM), but it also holds lower case counterparts for most of these models. In general, lower case models accept formula and df arguments, whereas upper case ones take endog and exog design matrices. formula accepts a string which describes the model in terms of a patsy formula. df takes a pandas data frame.

dir(sm) will print a list of available models.

Formula-compatible models have the following generic call signature: (formula, data, subset=None, *args, **kwargs)

OLS regression using formulas

To begin, we fit the linear model described on the Getting Started page. Download the data, subset columns, and list-wise delete to remove missing observations:

url = "http://vincentarelbundock.github.com/Rdatasets/csv/HistData/Guerry.csv"
df = pandas.read_csv(url)
df = df[['Lottery', 'Literacy', 'Wealth', 'Region']].dropna()
df.head()
   Lottery  Literacy  Wealth Region
0       41        37      73      E
1       38        51      22      N
2       66        13      61      C
3       80        46      76      E
4       79        69      83      E

Fit the model:

mod = sm.ols(formula='Lottery ~ Literacy + Wealth + Region', data=df)
res = mod.fit()
print res.summary()
                            OLS Regression Results
==============================================================================
Dep. Variable:                Lottery   R-squared:                       0.338
Model:                            OLS   Adj. R-squared:                  0.287
Method:                 Least Squares   F-statistic:                     6.636
Date:                Sun, 13 Jan 2013   Prob (F-statistic):           1.07e-05
Time:                        10:38:36   Log-Likelihood:                -375.30
No. Observations:                  85   AIC:                             764.6
Df Residuals:                      78   BIC:                             781.7
Df Model:                           6
===============================================================================
                  coef    std err          t      P>|t|      [95.0% Conf. Int.]
-------------------------------------------------------------------------------
Intercept      38.6517      9.456      4.087      0.000        19.826    57.478
Region[T.E]   -15.4278      9.727     -1.586      0.117       -34.793     3.938
Region[T.N]   -10.0170      9.260     -1.082      0.283       -28.453     8.419
Region[T.S]    -4.5483      7.279     -0.625      0.534       -19.039     9.943
Region[T.W]   -10.0913      7.196     -1.402      0.165       -24.418     4.235
Literacy       -0.1858      0.210     -0.886      0.378        -0.603     0.232
Wealth          0.4515      0.103      4.390      0.000         0.247     0.656
==============================================================================
Omnibus:                        3.049   Durbin-Watson:                   1.785
Prob(Omnibus):                  0.218   Jarque-Bera (JB):                2.694
Skew:                          -0.340   Prob(JB):                        0.260
Kurtosis:                       2.454   Cond. No.                         371.
==============================================================================

Categorical variables

Looking at the summary printed above, notice that patsy determined that elements of Region were text strings, so it treated Region as a categorical variable. patsy‘s default is also to include an intercept, so we automatically dropped one of the Region categories.

If Region had been an integer variable that we wanted to treat explicitly as categorical, we could have done so by using the C() operator:

res = sm.ols(formula='Lottery ~ Literacy + Wealth + C(Region)', data=df).fit()
print res.params
Intercept         38.651655
C(Region)[T.E]   -15.427785
C(Region)[T.N]   -10.016961
C(Region)[T.S]    -4.548257
C(Region)[T.W]   -10.091276
Literacy          -0.185819
Wealth             0.451475

Examples more advanced features patsy‘s categorical variables function can be found here: Patsy: Contrast Coding Systems for categorical variables

Operators

We have already seen that “~” separates the left-hand side of the model from the right-hand side, and that “+” adds new columns to the design matrix.

Removing variables

The “-” sign can be used to remove columns/variables. For instance, we can remove the intercept from a model by:

res = sm.ols(formula='Lottery ~ Literacy + Wealth + C(Region) -1 ', data=df).fit()
print res.params
C(Region)[C]    38.651655
C(Region)[E]    23.223870
C(Region)[N]    28.634694
C(Region)[S]    34.103399
C(Region)[W]    28.560379
Literacy        -0.185819
Wealth           0.451475

Multiplicative interactions

”:” adds a new column to the design matrix with the product of the other two columns. “*” will also include the individual columns that were multiplied together:

res1 = sm.ols(formula='Lottery ~ Literacy : Wealth - 1', data=df).fit()
res2 = sm.ols(formula='Lottery ~ Literacy * Wealth - 1', data=df).fit()
print res1.params, '\n'
print res2.params
Literacy:Wealth    0.018176

Literacy           0.427386
Wealth             1.080987
Literacy:Wealth   -0.013609

Many other things are possible with operators. Please consult the patsy docs to learn more.

Functions

You can apply vectorized functions to the variables in your model:

res = sm.ols(formula='Lottery ~ np.log(Literacy)', data=df).fit()
print res.params
Intercept           115.609119
np.log(Literacy)    -20.393959

Define a custom function:

def log_plus_1(x):
    return np.log(x) + 1.
res = sm.ols(formula='Lottery ~ log_plus_1(Literacy)', data=df).fit()
print res.params
Intercept               136.003079
log_plus_1(Literacy)    -20.393959

Using formulas with models that do not (yet) support them

Even if a given statsmodels function does not support formulas, you can still use patsy‘s formula language to produce design matrices. Those matrices can then be fed to the fitting function as endog and exog arguments.

To generate numpy arrays:

import patsy
f = 'Lottery ~ Literacy * Wealth'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
print y[:5]
print X[:5]
   Lottery
0       41
1       38
2       66
3       80
4       79
   Intercept  Literacy  Wealth  Literacy:Wealth
0          1        37      73             2701
1          1        51      22             1122
2          1        13      61              793
3          1        46      76             3496
4          1        69      83             5727

To generate pandas data frames:

f = 'Lottery ~ Literacy * Wealth'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
print y[:5]
print X[:5]
   Lottery
0       41
1       38
2       66
3       80
4       79
   Intercept  Literacy  Wealth  Literacy:Wealth
0          1        37      73             2701
1          1        51      22             1122
2          1        13      61              793
3          1        46      76             3496
4          1        69      83             5727
print sm.OLS(y, X).fit().summary()
                            OLS Regression Results
==============================================================================
Dep. Variable:                Lottery   R-squared:                       0.309
Model:                            OLS   Adj. R-squared:                  0.283
Method:                 Least Squares   F-statistic:                     12.06
Date:                Sun, 13 Jan 2013   Prob (F-statistic):           1.32e-06
Time:                        10:38:36   Log-Likelihood:                -377.13
No. Observations:                  85   AIC:                             762.3
Df Residuals:                      81   BIC:                             772.0
Df Model:                           3
===================================================================================
                      coef    std err          t      P>|t|      [95.0% Conf. Int.]
-----------------------------------------------------------------------------------
Intercept          38.6348     15.825      2.441      0.017         7.149    70.121
Literacy           -0.3522      0.334     -1.056      0.294        -1.016     0.312
Wealth              0.4364      0.283      1.544      0.126        -0.126     0.999
Literacy:Wealth    -0.0005      0.006     -0.085      0.933        -0.013     0.012
==============================================================================
Omnibus:                        4.447   Durbin-Watson:                   1.953
Prob(Omnibus):                  0.108   Jarque-Bera (JB):                3.228
Skew:                          -0.332   Prob(JB):                        0.199
Kurtosis:                       2.314   Cond. No.                     1.40e+04
==============================================================================

The condition number is large, 1.4e+04. This might indicate that there are
strong multicollinearity or other numerical problems.