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

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)

• Are you using the coefficients without regard to their standard errors? If so it may not be right if a large coefficient estimate has a high standard error. – Michael R. Chernick Jun 23 '17 at 23:55
• Linear regression in sklearn.linear_models does not have any way of getting the standard error of each coefficient as far as I know. – MAA Jun 24 '17 at 0:33
• Regardless, what would a high standard error say about the coefficient? high variability in the estimate and hence less reliable? – MAA Jun 24 '17 at 0:34
• If you would know the standard error and the coefficients then you could easily calculate the $p$-values. You could calculate the standard errors yourself through some matrix calculations. However, just selecting the significant variables is not a good model selection method. Are you aiming at prediction or statistical inference? – dietervdf Jun 24 '17 at 16:36
• "In linear regression, in order to improve the model, we have to figure out the most significant features." This is not correct. Statistical significance and p-values are not a tools meant to be used for feature selection. They are, at best, used in rule of thumb approaches when the environment does not support a better way, or the scientist does not know any better way. Much more appropriate is ridge or lasso regularization combined with dilligent use of cross validation. – Matthew Drury Jun 25 '17 at 3:53

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).
• @MAA Oh, got it. Perhaps sklearn.feature_selection.f_regression is similar to what you want. Updated answer. – Ami Tavory Jun 25 '17 at 3:47