How can I find the p-value (significance) of each coefficient?
lm = sklearn.linear_model.LinearRegression()
lm.fit(x,y)Solution:
This is kind of overkill but let’s give it a go. First lets use statsmodel to find out what the p-values should be
import pandas as pd
import numpy as np
from sklearn import datasets, linear_model
from sklearn.linear_model import LinearRegression
import statsmodels.api as sm
from scipy import statsdiabetes = datasets.load_diabetes()
X = diabetes.data
y = diabetes.targetX2 = sm.add_constant(X)
est = sm.OLS(y, X2)
est2 = est.fit()
print(est2.summary())
and we get
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.518
Model: OLS Adj. R-squared: 0.507
Method: Least Squares F-statistic: 46.27
Date: Wed, 08 Mar 2017 Prob (F-statistic): 3.83e-62
Time: 10:08:24 Log-Likelihood: -2386.0
No. Observations: 442 AIC: 4794.
Df Residuals: 431 BIC: 4839.
Df Model: 10
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 152.1335 2.576 59.061 0.000 147.071 157.196
x1 -10.0122 59.749 -0.168 0.867 -127.448 107.424
x2 -239.8191 61.222 -3.917 0.000 -360.151 -119.488
x3 519.8398 66.534 7.813 0.000 389.069 650.610
x4 324.3904 65.422 4.958 0.000 195.805 452.976
x5 -792.1842 416.684 -1.901 0.058 -1611.169 26.801
x6 476.7458 339.035 1.406 0.160 -189.621 1143.113
x7 101.0446 212.533 0.475 0.635 -316.685 518.774
x8 177.0642 161.476 1.097 0.273 -140.313 494.442
x9 751.2793 171.902 4.370 0.000 413.409 1089.150
x10 67.6254 65.984 1.025 0.306 -62.065 197.316
==============================================================================
Omnibus: 1.506 Durbin-Watson: 2.029
Prob(Omnibus): 0.471 Jarque-Bera (JB): 1.404
Skew: 0.017 Prob(JB): 0.496
Kurtosis: 2.726 Cond. No. 227.
==============================================================================Ok, let’s reproduce this. It is kind of overkill as we are almost reproducing a linear regression analysis using Matrix Algebra. But what the heck.
lm = LinearRegression()
lm.fit(X,y)
params = np.append(lm.intercept_,lm.coef_)
predictions = lm.predict(X)newX = pd.DataFrame({"Constant":np.ones(len(X))}).join(pd.DataFrame(X))
MSE = (sum((y-predictions)**2))/(len(newX)-len(newX.columns))# Note if you don't want to use a DataFrame replace the two lines above with
# newX = np.append(np.ones((len(X),1)), X, axis=1)
# MSE = (sum((y-predictions)**2))/(len(newX)-len(newX[0]))var_b = MSE*(np.linalg.inv(np.dot(newX.T,newX)).diagonal())
sd_b = np.sqrt(var_b)
ts_b = params/ sd_bp_values =[2*(1-stats.t.cdf(np.abs(i),(len(newX)-len(newX[0])))) for i in ts_b]sd_b = np.round(sd_b,3)
ts_b = np.round(ts_b,3)
p_values = np.round(p_values,3)
params = np.round(params,4)myDF3 = pd.DataFrame()
myDF3["Coefficients"],myDF3["Standard Errors"],myDF3["t values"],myDF3["Probabilities"] = [params,sd_b,ts_b,p_values]
print(myDF3)
And this gives us.
Coefficients Standard Errors t values Probabilities
0 152.1335 2.576 59.061 0.000
1 -10.0122 59.749 -0.168 0.867
2 -239.8191 61.222 -3.917 0.000
3 519.8398 66.534 7.813 0.000
4 324.3904 65.422 4.958 0.000
5 -792.1842 416.684 -1.901 0.058
6 476.7458 339.035 1.406 0.160
7 101.0446 212.533 0.475 0.635
8 177.0642 161.476 1.097 0.273
9 751.2793 171.902 4.370 0.000
10 67.6254 65.984 1.025 0.306So we can reproduce the values from statsmodel.
