# Formulas: Fitting models using R-style formulas¶

Link to Notebook GitHub

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:

In [ ]:
from __future__ import print_function
import numpy as np
import statsmodels.api as sm


#### Import convention¶

You can import explicitly from statsmodels.formula.api

In [ ]:
from statsmodels.formula.api import ols


Alternatively, you can just use the formula namespace of the main statsmodels.api.

In [ ]:
sm.formula.ols


Or you can use the following conventioin

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


These names are just a convenient way to get access to each model's from_formula classmethod. See, for instance

In [ ]:
sm.OLS.from_formula


All of the lower case models accept formula and data 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. data takes a pandas data frame or any other data structure that defines a __getitem__ for variable names like a structured array or a dictionary of variables.

dir(sm.formula) 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:

In [ ]:
dta = sm.datasets.get_rdataset("Guerry", "HistData", cache=True)

In [ ]:
df = dta.data[['Lottery', 'Literacy', 'Wealth', 'Region']].dropna()


Fit the model:

In [ ]:
mod = ols(formula='Lottery ~ Literacy + Wealth + Region', data=df)
res = mod.fit()
print(res.summary())


## 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:

In [ ]:
res = ols(formula='Lottery ~ Literacy + Wealth + C(Region)', data=df).fit()
print(res.params)

                            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:                Mon, 20 Jul 2015   Prob (F-statistic):           1.07e-05
Time:                        17:43:27   Log-Likelihood:                -375.30
No. Observations:                  85   AIC:                             764.6
Df Residuals:                      78   BIC:                             781.7
Df Model:                           6
Covariance Type:            nonrobust
===============================================================================
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.
==============================================================================

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


Patsy's mode advanced features for categorical variables are discussed in: 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:

In [ ]:
res = ols(formula='Lottery ~ Literacy + Wealth + C(Region) -1 ', 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
dtype: float64


### Multiplicative interactions¶

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

In [ ]:
res1 = ols(formula='Lottery ~ Literacy : Wealth - 1', data=df).fit()
res2 = ols(formula='Lottery ~ Literacy * Wealth - 1', data=df).fit()
print(res1.params, '\n')
print(res2.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
dtype: float64


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:

In [ ]:
res = smf.ols(formula='Lottery ~ np.log(Literacy)', data=df).fit()
print(res.params)

Literacy:Wealth    0.018176
dtype: float64

Literacy           0.427386
Wealth             1.080987
Literacy:Wealth   -0.013609
dtype: float64


Define a custom function:

In [ ]:
def log_plus_1(x):
return np.log(x) + 1.
res = smf.ols(formula='Lottery ~ log_plus_1(Literacy)', data=df).fit()
print(res.params)

Intercept           115.609119
np.log(Literacy)    -20.393959
dtype: float64


Any function that is in the calling namespace is available to the formula.

## 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:

In [ ]:
import patsy
f = 'Lottery ~ Literacy * Wealth'
y,X = patsy.dmatrices(f, df, return_type='dataframe')
print(y[:5])
print(X[:5])

Intercept               136.003079
log_plus_1(Literacy)    -20.393959
dtype: float64


To generate pandas data frames:

In [ ]:
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

In [ ]:
print(sm.OLS(y, X).fit().summary())

   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