In the previous exercises, we compared nested models based on adjusted R-squared.

Another way to compare nested models is by using a hypothesis test called an F-test. Suppose we want to compare the following two models:

model1 = sm.OLS.from_formula('rent ~ bedrooms + size_sqft', data=rentals).fit() model2 = sm.OLS.from_formula('rent ~ bedrooms + size_sqft + building_age_yrs + has_elevator', data=rentals).fit()

Note that the second model has two more predictors than the first (building_age_yrs and has_elevator). For an F-test comparing these two models:

  • The null hypothesis is that the coefficients on building_age_yrs and has_elevator are equal to zero (they are not useful in explaining the observed variation in rent).
  • The alternative hypothesis is that least one of the coefficients is non-zero.

We can run the test in Python as follows:

from statsmodels.stats.anova import anova_lm anova_results = anova_lm(model1, model2) print(anova_results)


df_resid ssr df_diff ss_diff F Pr(>F)
0 4997.0 1.4e+10 0.0 NaN NaN NaN
1 4995.0 1.4e+10 2.0 9.2e+08 170.9 1.6e-72

The p-value (1.6e-72, which is equal to .00000..[72 total zeros]..16) is located in the bottom right corner of this table. The column name Pr(>F) means “the probability of observing an F statistic greater than observed (170.9) if the null hypothesis is true”.

Using a significance threshold of 0.05, the p-value is below the threshold. Therefore, we would conclude that either (or both) of the coefficients on building_age_yrs and has_elevator is non-zero. Thus, including at least one of these two predictors significantly improves the model.

This would lead us to choose model2 over model1. After running this test, we might also want to separately compare a model with building_age_yrs and a model with has_elevator to see whether both are necessary. We could do this with separate F-tests or adjusted R-squared.



Using the bikes data, fit a model to predict the number of bike rentals (cnt) with the following predictors: temp, hum, and windspeed. Save the fitted model as model1.


Now fit a second model with all the same predictors plus the day of the week (weekday) and “feels like” temperature (atemp). Save this fitted model as model2.


Run an F-test to compare the two models and print out the results.

NOTE: you may want to round the results using .round(2) so that they fit in the output terminal. Otherwise, you’ll need to scroll to the right in the output table to see the p-value.


Based on the F-test and a significance threshold of 0.05, which model would you choose? Indicate your answer by setting a variable named which_model equal to 1 if you would choose model1 and equal to 2 if you would choose model2.

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