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How to compute adjusted R-squared (in Python, using statsmodels)

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Task

If we have fit a multivariate linear model, how can we compute the Adjusted $R^2$ for that model, to measure its goodness of fit?

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Solution

We assume you have already fit a multivariate linear model to some data, as in the code below. (If you’re unfamiliar with how to do so, see how to fit a multivariate linear model.) The data shown below is fake, and we assume you will replace it with your own real data if you use this code.

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import pandas as pd
import statsmodels.api as sm
df = pd.DataFrame( {
    'x1':[2, 7, 4, 3, 11, 18, 6, 15, 9, 12],
    'x2':[4, 6, 10, 1, 18, 11, 8, 20, 16, 13],
    'x3':[11, 16, 20, 6, 14, 8, 5, 23, 13, 10],
    'y':[24, 60, 32, 29, 90, 45, 130, 76, 100, 120]
} )
xs = df[['x1', 'x2', 'x3']]
y = df['y']
xs = sm.add_constant(xs)
model = sm.OLS(y, xs).fit()

You can get a lot of information about your model from its summary.

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model.summary()

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/opt/conda/lib/python3.11/site-packages/scipy/stats/_stats_py.py:1806: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=10
  warnings.warn("kurtosistest only valid for n>=20 ... continuing "
OLS Regression Results
Dep. Variable: y R-squared: 0.594
Model: OLS Adj. R-squared: 0.390
Method: Least Squares F-statistic: 2.921
Date: Mon, 24 Jul 2023 Prob (F-statistic): 0.122
Time: 17:47:21 Log-Likelihood: -45.689
No. Observations: 10 AIC: 99.38
Df Residuals: 6 BIC: 100.6
Df Model: 3
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
const 77.2443 27.366 2.823 0.030 10.282 144.206
x1 -2.7009 2.855 -0.946 0.381 -9.686 4.284
x2 7.2989 2.875 2.539 0.044 0.265 14.333
x3 -4.8607 2.187 -2.223 0.068 -10.211 0.490
Omnibus: 2.691 Durbin-Watson: 2.123
Prob(Omnibus): 0.260 Jarque-Bera (JB): 1.251
Skew: 0.524 Prob(JB): 0.535
Kurtosis: 1.620 Cond. No. 58.2



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

In particular, that printout contains the Adjusted $R^2$ value; it is the second value in the right-hand column, near the top.

You can also obtain it directly, as follows:

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model.rsquared_adj

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0.390392407508503

In this case, the Adjusted $R^2$ is $0.3904$.

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

Contributed by Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)