Formulas: 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¶
[1]:
import numpy as np # noqa:F401 needed in namespace for patsy
import statsmodels.api as sm
Import convention¶
You can import explicitly from statsmodels.formula.api
[2]:
from statsmodels.formula.api import ols
Alternatively, you can just use the formula namespace of the main statsmodels.api.
[3]:
sm.formula.ols
[3]:
<bound method Model.from_formula of <class 'statsmodels.regression.linear_model.OLS'>>
Or you can use the following convention
[4]:
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
[5]:
sm.OLS.from_formula
[5]:
<bound method Model.from_formula of <class 'statsmodels.regression.linear_model.OLS'>>
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:
[6]:
dta = sm.datasets.get_rdataset("Guerry", "HistData", cache=True)
[7]:
df = dta.data[["Lottery", "Literacy", "Wealth", "Region"]].dropna()
df.head()
[7]:
| 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:
[8]:
mod = 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: Mon, 23 Dec 2024 Prob (F-statistic): 1.07e-05
Time: 12:09:05 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| [0.025 0.975]
-------------------------------------------------------------------------------
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.
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
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:
[9]:
res = 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
dtype: float64
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:
[10]:
res = 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
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:
[11]:
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)
Literacy:Wealth 0.018176
dtype: float64
Literacy 0.427386
Wealth 1.080987
Literacy:Wealth -0.013609
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:
[12]:
res = smf.ols(formula="Lottery ~ np.log(Literacy)", data=df).fit()
print(res.params)
Intercept 115.609119
np.log(Literacy) -20.393959
dtype: float64
Define a custom function:
[13]:
def log_plus_1(x):
return np.log(x) + 1.0
res = smf.ols(formula="Lottery ~ log_plus_1(Literacy)", data=df).fit()
print(res.params)
Intercept 136.003079
log_plus_1(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:
[14]:
import patsy
f = "Lottery ~ Literacy * Wealth"
y, X = patsy.dmatrices(f, df, return_type="matrix")
print(y[:5])
print(X[:5])
[[41.]
[38.]
[66.]
[80.]
[79.]]
[[1.000e+00 3.700e+01 7.300e+01 2.701e+03]
[1.000e+00 5.100e+01 2.200e+01 1.122e+03]
[1.000e+00 1.300e+01 6.100e+01 7.930e+02]
[1.000e+00 4.600e+01 7.600e+01 3.496e+03]
[1.000e+00 6.900e+01 8.300e+01 5.727e+03]]
To generate pandas data frames:
[15]:
f = "Lottery ~ Literacy * Wealth"
y, X = patsy.dmatrices(f, df, return_type="dataframe")
print(y[:5])
print(X[:5])
Lottery
0 41.0
1 38.0
2 66.0
3 80.0
4 79.0
Intercept Literacy Wealth Literacy:Wealth
0 1.0 37.0 73.0 2701.0
1 1.0 51.0 22.0 1122.0
2 1.0 13.0 61.0 793.0
3 1.0 46.0 76.0 3496.0
4 1.0 69.0 83.0 5727.0
[16]:
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: Mon, 23 Dec 2024 Prob (F-statistic): 1.32e-06
Time: 12:09:05 Log-Likelihood: -377.13
No. Observations: 85 AIC: 762.3
Df Residuals: 81 BIC: 772.0
Df Model: 3
Covariance Type: nonrobust
===================================================================================
coef std err t P>|t| [0.025 0.975]
-----------------------------------------------------------------------------------
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
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.4e+04. This might indicate that there are
strong multicollinearity or other numerical problems.