Link to the previous post : https://statinfer.com/204-1-9-issue-of-multicollinearity-in-python/

In this practice post we will build a Multiple Regression model and try to improve it by clearing the problem of multicollinearity in the model.

Practice : Multiple Regression

Dataset: Webpage_Product_Sales/Webpage_Product_Sales.csv
Build a model to predict sales using rest of the variables
Drop the less impacting variables based on p-values.
Is there any multicollinearity?
How many variables are there in the final model?
What is the R-squared of the final model?
Can you improve the model using same data and variables?

In [57]:

import pandas as pd 
Webpage_Product_Sales=pd.read_csv("datasets\\Webpage_Product_Sales\\Webpage_Product_Sales.csv")
Webpage_Product_Sales.shape

Out[57]:

(675, 12)

In [58]:

Webpage_Product_Sales.columns

Out[58]:

Index(['ID', 'DayofMonth', 'Weekday', 'Month', 'Social_Network_Ref_links',
       'Online_Ad_Paid_ref_links', 'Clicks_From_Serach_Engine',
       'Special_Discount', 'Holiday', 'Server_Down_time_Sec', 'Web_UI_Score',
       'Sales'],
      dtype='object')

In [59]:

import statsmodels.formula.api as sm
model1 = sm.ols(formula='Sales ~ Web_UI_Score+Server_Down_time_Sec+Holiday+Special_Discount+Clicks_From_Serach_Engine+Online_Ad_Paid_ref_links+Social_Network_Ref_links+Month+Weekday+DayofMonth', data=Webpage_Product_Sales)
fitted1 = model1.fit()
fitted1.summary()

Out[59]:

OLS Regression Results
Dep. Variable:	Sales	R-squared:	0.818
Model:	OLS	Adj. R-squared:	0.815
Method:	Least Squares	F-statistic:	298.4
Date:	Wed, 27 Jul 2016	Prob (F-statistic):	5.54e-238
Time:	12:45:36	Log-Likelihood:	-6456.7
No. Observations:	675	AIC:	1.294e+04
Df Residuals:	664	BIC:	1.299e+04
Df Model:	10
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[95.0% Conf. Int.]
Intercept	6545.8922	1286.240	5.089	0.000	4020.304 9071.481
Web_UI_Score	-6.2582	11.545	-0.542	0.588	-28.928 16.412
Server_Down_time_Sec	-134.0441	14.009	-9.569	0.000	-161.551 -106.537
Holiday	1.877e+04	683.077	27.477	0.000	1.74e+04 2.01e+04
Special_Discount	4718.3978	402.019	11.737	0.000	3929.016 5507.780
Clicks_From_Serach_Engine	-0.1258	0.944	-0.133	0.894	-1.980 1.728
Online_Ad_Paid_ref_links	6.1557	1.002	6.142	0.000	4.188 8.124
Social_Network_Ref_links	6.6841	0.411	16.261	0.000	5.877 7.491
Month	481.0294	41.508	11.589	0.000	399.527 562.532
Weekday	1355.2153	67.224	20.160	0.000	1223.218 1487.213
DayofMonth	47.0579	15.198	3.096	0.002	17.216 76.900

Omnibus:	40.759	Durbin-Watson:	1.356
Prob(Omnibus):	0.000	Jarque-Bera (JB):	102.136
Skew:	0.297	Prob(JB):	6.63e-23
Kurtosis:	4.811	Cond. No.	2.57e+04

In [60]:

#VIF
vif_cal(Webpage_Product_Sales,"Sales")

ID  VIF =  1.18
DayofMonth  VIF =  1.01
Weekday  VIF =  1.0
Month  VIF =  1.19
Social_Network_Ref_links  VIF =  1.02
Online_Ad_Paid_ref_links  VIF =  12.13
Clicks_From_Serach_Engine  VIF =  12.08
Special_Discount  VIF =  1.37
Holiday  VIF =  1.38
Server_Down_time_Sec  VIF =  1.02
Web_UI_Score  VIF =  1.02

In [61]:

##Dropped Clicks_From_Serach_Engine based on VIF

import statsmodels.formula.api as sm
model2 = sm.ols(formula='Sales ~ Web_UI_Score+Server_Down_time_Sec+Holiday+Special_Discount+Online_Ad_Paid_ref_links+Social_Network_Ref_links+Month+Weekday+DayofMonth', data=Webpage_Product_Sales)
fitted2 = model2.fit()
fitted2.summary()

Out[61]:

OLS Regression Results
Dep. Variable:	Sales	R-squared:	0.818
Model:	OLS	Adj. R-squared:	0.815
Method:	Least Squares	F-statistic:	332.0
Date:	Wed, 27 Jul 2016	Prob (F-statistic):	2.98e-239
Time:	12:48:18	Log-Likelihood:	-6456.7
No. Observations:	675	AIC:	1.293e+04
Df Residuals:	665	BIC:	1.298e+04
Df Model:	9
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[95.0% Conf. Int.]
Intercept	6598.7469	1222.658	5.397	0.000	4198.012 8999.482
Web_UI_Score	-6.3332	11.523	-0.550	0.583	-28.959 16.293
Server_Down_time_Sec	-133.9518	13.981	-9.581	0.000	-161.405 -106.499
Holiday	1.877e+04	681.292	27.557	0.000	1.74e+04 2.01e+04
Special_Discount	4713.9295	400.323	11.775	0.000	3927.881 5499.978
Online_Ad_Paid_ref_links	6.0279	0.291	20.740	0.000	5.457 6.599
Social_Network_Ref_links	6.6872	0.410	16.307	0.000	5.882 7.492
Month	480.6876	41.398	11.611	0.000	399.401 561.974
Weekday	1355.2536	67.174	20.175	0.000	1223.355 1487.152
DayofMonth	47.0168	15.184	3.097	0.002	17.203 76.831

Omnibus:	40.826	Durbin-Watson:	1.356
Prob(Omnibus):	0.000	Jarque-Bera (JB):	102.313
Skew:	0.298	Prob(JB):	6.07e-23
Kurtosis:	4.812	Cond. No.	1.94e+04

In [62]:

#VIF for the updated model
vif_cal(Webpage_Product_Sales.drop(["Clicks_From_Serach_Engine"],axis=1),"Sales")

ID  VIF =  1.18
DayofMonth  VIF =  1.01
Weekday  VIF =  1.0
Month  VIF =  1.19
Social_Network_Ref_links  VIF =  1.01
Online_Ad_Paid_ref_links  VIF =  1.02
Special_Discount  VIF =  1.36
Holiday  VIF =  1.38
Server_Down_time_Sec  VIF =  1.02
Web_UI_Score  VIF =  1.02

In [63]:

##Drop the less impacting variables based on p-values.
##Dropped Web_UI_Score based on P-value

import statsmodels.formula.api as sm
model3 = sm.ols(formula='Sales ~ Server_Down_time_Sec+Holiday+Special_Discount+Online_Ad_Paid_ref_links+Social_Network_Ref_links+Month+Weekday+DayofMonth', data=Webpage_Product_Sales)
fitted3 = model3.fit()
fitted3.summary()

Out[63]:

OLS Regression Results
Dep. Variable:	Sales	R-squared:	0.818
Model:	OLS	Adj. R-squared:	0.816
Method:	Least Squares	F-statistic:	373.9
Date:	Wed, 27 Jul 2016	Prob (F-statistic):	1.74e-240
Time:	12:49:15	Log-Likelihood:	-6456.9
No. Observations:	675	AIC:	1.293e+04
Df Residuals:	666	BIC:	1.297e+04
Df Model:	8
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[95.0% Conf. Int.]
Intercept	6101.1539	821.286	7.429	0.000	4488.532 7713.776
Server_Down_time_Sec	-134.0717	13.972	-9.596	0.000	-161.507 -106.637
Holiday	1.874e+04	678.528	27.623	0.000	1.74e+04 2.01e+04
Special_Discount	4726.1858	399.491	11.831	0.000	3941.771 5510.600
Online_Ad_Paid_ref_links	6.0357	0.290	20.802	0.000	5.466 6.605
Social_Network_Ref_links	6.6738	0.409	16.312	0.000	5.870 7.477
Month	479.5231	41.322	11.605	0.000	398.386 560.660
Weekday	1354.4252	67.122	20.179	0.000	1222.629 1486.221
DayofMonth	46.9564	15.175	3.094	0.002	17.159 76.754

Omnibus:	41.049	Durbin-Watson:	1.352
Prob(Omnibus):	0.000	Jarque-Bera (JB):	103.243
Skew:	0.298	Prob(JB):	3.81e-23
Kurtosis:	4.821	Cond. No.	1.31e+04

In [65]:

#How many variables are there in the final model?
8

Out[65]:

In [69]:

#What is the R-squared of the final model?
fitted3.rsquared

Out[69]:

0.8178742020411971


The next post is interaction terms.
Link to the next post : https://statinfer.com/204-1-11-interaction-terms/

204.1.10 Practice : Multiple Regression with Multicollinearity

Dealing with multicollinearity.

Practice : Multiple Regression

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204.1.10 Practice : Multiple Regression with Multicollinearity

Dealing with multicollinearity.

Practice : Multiple Regression

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