For feature selection in linear regression model, can I use coefficient estimates?

Question

I'm doing Analytics Edge course from EDX. The course is using R while I'm using Python.

In linear regression, in order to improve the model, we have to figure out the most significant features.

The course is using the summary function in R to look at the dots signifying the importance of the feature and the p-values. No such thing exists in sklearn.

So I'm using coefficients to see the most significant features. But I'm not sure I should trust coefficients to select the most significant features (even though for this problem, they are in agreement)

So is coefficients from linearRegression in sklearn reliable in determining the significance of the features? Are p-values themselves reliable in detecting the significant features?

(I know of statsmodels and do not wish to use)

Answer 1

To begin with, just to put the issue aside: clearly, if the features are not normalized to 0 mean and unit variance, it's easy to build cases where the coefficient means very little. In general, if you take a feature and multiply it by $\alpha$, a regressor will divide the coefficient by $\alpha$, for example.

Even when the variables are all normalized, large coefficients can mean very little. Say that $x$ is some hidden feature somewhat correlated with $y$, and $z$ and $w$ are observed featured which are slightly noisy versions of $x$, the regression matrix will be not very well defined, and you could get large-magnitude coefficients for $z$ and $w$ (perhaps with opposite signs). Regularization is usually used precisely to avoid this.

Perhaps sklearn.feature_selection.f_regression is similar to what you're looking for. It summarizes, for each individual feature, both the f-score and the p-value. Alternatively, for any regression scheme, a "black box" approach could be to build the model for all features except $x$, and assess its performance (using cross validation). You could then rank the features based on the performance.

Feature importance is a bit trick to define. In the above two schemes, if $x_i$ is the $i$the resulting "most important" feature, it does not necessarily mean that using $x_1, \ldots x_{i - 1}$, it is indeed the next most important one (perhaps its information is already contained in the preceding ones).

Answer 2

From my point of view, you can select features based on coefficients in the case of using Ridge regression rather than simple linear regression. I agree that a naive linear regression method will generate extremely large coefficients if any two features are highly correlated. However, when you adopt the ridge regression model, this problem will disappear.

In fact, it is rather intuitive to select features based on coefficients. If we slightly change the value of one feature, the more response value changes, the more important features are. In fact, this idea is nearly identical to the permutation feature importance, which is widely used as a black-box feature importance analysis approach.

For example, in the following code fragment, I give an example to show coefficient values of features in the regression model and their corresponding permutation feature importance. From the results, it is clear that feature coefficients are basically identical to the permutation feature importance of those features. Thus, in conclusion, it is reasonable to use the coefficients of the ridge regression model as a rough approximation of feature importance.

import numpy as np
from sklearn.datasets import load_diabetes
from sklearn.inspection import permutation_importance
from sklearn.linear_model import Ridge
from sklearn.model_selection import train_test_split

diabetes = load_diabetes()
X_train, X_val, y_train, y_val = train_test_split(
    diabetes.data, diabetes.target, random_state=0)

model = Ridge(alpha=1e-2).fit(X_train, y_train)
model.score(X_val, y_val)
for i in np.abs(model.coef_).argsort()[::-1][:5]:
    print(diabetes.feature_names[i], np.abs(model.coef_[i]))

r = permutation_importance(model, X_val, y_val,
                           n_repeats=30,
                           random_state=0)

for i in r.importances_mean.argsort()[::-1]:
    if r.importances_mean[i] - 2 * r.importances_std[i] > 0:
        print(f"{diabetes.feature_names[i]:<8}"
              f"{r.importances_mean[i]:.3f}"
              f" +/- {r.importances_std[i]:.3f}")

bmi 592.2534291944405
s5 580.078063710006
bp 297.2581037274451
s1 252.42469967919644
sex 203.43588499880846
s5      0.204 +/- 0.050
bmi     0.176 +/- 0.048
bp      0.088 +/- 0.033
sex     0.056 +/- 0.023

Finally, it should be noted that both the mentioned methods are fragile to the multicollinearity problem. Thus, in that case, we need to eliminate highly correlated features first, and then we can directly use the model coefficients as feature importance.