Stepwise Regression
Contents
9.6. Stepwise Regression¶
In a stepwise regression, variables are added and removed from the model based on significance. You can have a forward selection stepwise which adds variables if they are statistically significant until all the variables outside the model are not significant, a backwards elimination stepwise regression which puts in all the variables and then removes those that are not statistically significant until only statistically significant ones remain, and a bidirectional elimination which both adds and removes until all the variables inside are significant AND all those outside are not significant.
import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import seaborn as sns
import statsmodels.api as sm
import warnings
warnings.filterwarnings('ignore')
df.head()
| CRASH DATE | CRASH TIME | BOROUGH | ZIP CODE | LATITUDE | LONGITUDE | LOCATION | ON STREET NAME | CROSS STREET NAME | OFF STREET NAME | ... | VEHICLE TYPE CODE 5 | lat | long | hour | date | BROOK | BRONX | MANHAT | STATEN | QUEENS | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 01/01/2022 | 7:05 | NaN | NaN | NaN | NaN | NaN | EAST 128 STREET | 3 AVENUE BRIDGE | NaN | ... | NaN | 0 | 0 | 7 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 01/01/2022 | 14:43 | NaN | NaN | 40.769993 | -73.915825 | (40.769993, -73.915825) | GRAND CENTRAL PKWY | NaN | NaN | ... | NaN | 1 | 1 | 14 | 1 | 0 | 0 | 0 | 0 | 0 |
| 2 | 01/01/2022 | 21:20 | QUEENS | 11414.0 | 40.657230 | -73.841380 | (40.65723, -73.84138) | 91 STREET | 160 AVENUE | NaN | ... | NaN | 1 | 1 | 21 | 1 | 0 | 0 | 0 | 0 | 1 |
| 3 | 01/01/2022 | 4:30 | NaN | NaN | NaN | NaN | NaN | Southern parkway | Jfk expressway | NaN | ... | NaN | 0 | 0 | 4 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 01/01/2022 | 7:57 | NaN | NaN | NaN | NaN | NaN | WESTCHESTER AVENUE | SHERIDAN EXPRESSWAY | NaN | ... | NaN | 0 | 0 | 7 | 1 | 0 | 0 | 0 | 0 | 0 |
5 rows × 38 columns
9.6.1. Data scrubbing¶
df = pd.read_csv("../data/nyc_mv_collisions_202201.csv")
zeros = []
for i in range(len(df)):
zeros.append(0)
df["lat"] = zeros
for i in range(len(df)):
if df["LATITUDE"][i] > 0 or df["LATITUDE"][i] < 0:
df["lat"][i] = 1
else:
df["lat"][i] = 0
df["long"] = zeros
for i in range(len(df)):
if df["LATITUDE"][i] > 0 or df["LONGITUDE"][i] < 0:
df["long"][i] = 1
else:
df["long"][i] = 0
df["hour"] = zeros
for i in range(len(df)):
if (df["CRASH TIME"][i])[1:2] == ':':
df["hour"][i] = str(df["CRASH TIME"][i])[0:1]
else:
df["hour"][i] = str(df["CRASH TIME"][i])[0:2]
cap = int(df["hour"][i])
df["hour"][i] = cap
df["date"] = zeros
for i in range(len(df)):
df["date"][i] = str(df["CRASH DATE"][i])[3:5]
cap = int(df["date"][i])
df["date"][i] = cap
df["BROOK"] = zeros
df["BRONX"] = zeros
df["MANHAT"] = zeros
df["STATEN"] = zeros
df["QUEENS"] = zeros
for i in range(len(df)):
if df["BOROUGH"][i] == "NaN":
cat = 1
elif df["BOROUGH"][i] == 'BROOKLYN':
df["BROOK"][i] = 1
elif df["BOROUGH"][i] == 'BRONX':
df["BRONX"][i] = 1
elif df["BOROUGH"][i] == 'MANHATTAN':
df["MANHAT"][i] = 1
elif df["BOROUGH"][i] == 'STATEN ISLAND':
df["STATEN"][i] = 1
elif df["BOROUGH"][i] == 'QUEENS':
df["QUEENS"][i] = 1
x_columns = ["date", "hour", "BROOK", "BRONX", "STATEN", "QUEENS", "MANHAT", "NUMBER OF PERSONS KILLED", "lat"]
y = df["NUMBER OF PERSONS INJURED"]
def get_stats():
x = df[x_columns]
results = sm.OLS(y, x).fit()
print(results.summary())
def rem_high():
x = df[x_columns]
results = sm.OLS(y, x).fit()
spot = 0
for i in range(len(results.pvalues)):
if results.pvalues[spot] < results.pvalues[i]:
spot = i
if results.pvalues[spot] > .05:
x_columns.pop(spot)
print(results.summary())
get_stats()
rem_high()
date hour BROOK BRONX STATEN QUEENS MANHAT \
0 1 7 0 0 0 0 0
1 1 14 0 0 0 0 0
2 1 21 0 0 0 1 0
3 1 4 0 0 0 0 0
4 1 7 0 0 0 0 0
... ... ... ... ... ... ... ...
7654 31 12 0 0 0 1 0
7655 31 21 1 0 0 0 0
7656 31 9 1 0 0 0 0
7657 31 6 0 1 0 0 0
7658 31 14 0 0 0 1 0
NUMBER OF PERSONS KILLED lat
0 0 0
1 0 1
2 0 1
3 0 0
4 0 0
... ... ...
7654 0 1
7655 0 1
7656 0 1
7657 0 1
7658 0 1
[7659 rows x 9 columns]
rem_high()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.226
Model: OLS Adj. R-squared (uncentered): 0.225
Method: Least Squares F-statistic: 279.0
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 03:31:10 Log-Likelihood: -8476.1
No. Observations: 7659 AIC: 1.697e+04
Df Residuals: 7651 BIC: 1.702e+04
Df Model: 8
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
date 0.0018 0.001 2.109 0.035 0.000 0.003
hour 0.0139 0.001 11.350 0.000 0.011 0.016
BROOK -0.0278 0.022 -1.279 0.201 -0.070 0.015
STATEN -0.0293 0.053 -0.554 0.580 -0.133 0.074
QUEENS -0.0682 0.023 -2.918 0.004 -0.114 -0.022
MANHAT -0.0622 0.030 -2.084 0.037 -0.121 -0.004
NUMBER OF PERSONS KILLED 0.2820 0.173 1.633 0.103 -0.057 0.621
lat 0.2189 0.022 9.965 0.000 0.176 0.262
==============================================================================
Omnibus: 4117.098 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35735.067
Skew: 2.450 Prob(JB): 0.00
Kurtosis: 12.379 Cond. No. 452.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
9.6.2. The Bronx is removed¶
rem_high()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.226
Model: OLS Adj. R-squared (uncentered): 0.225
Method: Least Squares F-statistic: 318.8
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 03:31:12 Log-Likelihood: -8476.3
No. Observations: 7659 AIC: 1.697e+04
Df Residuals: 7652 BIC: 1.702e+04
Df Model: 7
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
date 0.0018 0.001 2.108 0.035 0.000 0.003
hour 0.0139 0.001 11.340 0.000 0.011 0.016
BROOK -0.0261 0.022 -1.214 0.225 -0.068 0.016
QUEENS -0.0666 0.023 -2.871 0.004 -0.112 -0.021
MANHAT -0.0605 0.030 -2.039 0.042 -0.119 -0.002
NUMBER OF PERSONS KILLED 0.2826 0.173 1.636 0.102 -0.056 0.621
lat 0.2175 0.022 9.970 0.000 0.175 0.260
==============================================================================
Omnibus: 4118.495 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35771.998
Skew: 2.451 Prob(JB): 0.00
Kurtosis: 12.384 Cond. No. 452.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
9.6.3. Staten island is removed¶
rem_high()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.226
Model: OLS Adj. R-squared (uncentered): 0.225
Method: Least Squares F-statistic: 371.7
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 03:31:15 Log-Likelihood: -8477.0
No. Observations: 7659 AIC: 1.697e+04
Df Residuals: 7653 BIC: 1.701e+04
Df Model: 6
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
date 0.0018 0.001 2.072 0.038 9.57e-05 0.003
hour 0.0138 0.001 11.301 0.000 0.011 0.016
QUEENS -0.0579 0.022 -2.624 0.009 -0.101 -0.015
MANHAT -0.0518 0.029 -1.798 0.072 -0.108 0.005
NUMBER OF PERSONS KILLED 0.2818 0.173 1.632 0.103 -0.057 0.620
lat 0.2098 0.021 10.049 0.000 0.169 0.251
==============================================================================
Omnibus: 4121.849 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35880.243
Skew: 2.453 Prob(JB): 0.00
Kurtosis: 12.400 Cond. No. 452.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
9.6.4. Brooklyn is removed¶
rem_high()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.225
Model: OLS Adj. R-squared (uncentered): 0.225
Method: Least Squares F-statistic: 445.4
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 03:31:18 Log-Likelihood: -8478.3
No. Observations: 7659 AIC: 1.697e+04
Df Residuals: 7654 BIC: 1.700e+04
Df Model: 5
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
date 0.0018 0.001 2.084 0.037 0.000 0.003
hour 0.0138 0.001 11.296 0.000 0.011 0.016
QUEENS -0.0578 0.022 -2.621 0.009 -0.101 -0.015
MANHAT -0.0516 0.029 -1.791 0.073 -0.108 0.005
lat 0.2102 0.021 10.071 0.000 0.169 0.251
==============================================================================
Omnibus: 4130.912 Durbin-Watson: 1.964
Prob(Omnibus): 0.000 Jarque-Bera (JB): 36165.015
Skew: 2.458 Prob(JB): 0.00
Kurtosis: 12.443 Cond. No. 77.7
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
9.6.5. Persons Killed is removed¶
rem_high()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.225
Model: OLS Adj. R-squared (uncentered): 0.225
Method: Least Squares F-statistic: 555.8
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 03:31:21 Log-Likelihood: -8479.9
No. Observations: 7659 AIC: 1.697e+04
Df Residuals: 7655 BIC: 1.700e+04
Df Model: 4
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
date 0.0017 0.001 2.036 0.042 6.47e-05 0.003
hour 0.0137 0.001 11.253 0.000 0.011 0.016
QUEENS -0.0515 0.022 -2.365 0.018 -0.094 -0.009
lat 0.2052 0.021 9.918 0.000 0.165 0.246
==============================================================================
Omnibus: 4134.561 Durbin-Watson: 1.965
Prob(Omnibus): 0.000 Jarque-Bera (JB): 36246.070
Skew: 2.460 Prob(JB): 0.00
Kurtosis: 12.454 Cond. No. 60.8
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
9.6.6. Manhattan is removed¶
From here we can see all variables are under the .05 threshold established previously. Our formula would come out to be
y = date * .0017 + hour * .0137 - QUEENS * .0515 + lat * .2052
x_col = []
used = 0
def crunch_num():
p_val = 1
location = 0
for i in range(len(x_columns)):
if used == 0:
x = df[x_columns[i]]
results = sm.OLS(y, x).fit()
print(p_val)
print(results.pvalues[len(x_col)])
if (p_val > results.pvalues[len(x_col)]):
p_val = results.pvalues[len(x_col)]
location = i
print(results.summary())
if (p_val <.05):
print(location)
x_col.append(x_columns[location])
x_columns.remove(x_columns[location])
crunch_num()
1
1.4323814844141276e-306
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.167
Model: OLS Adj. R-squared (uncentered): 0.167
Method: Least Squares F-statistic: 1537.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 1.43e-306
Time: 05:18:18 Log-Likelihood: -8755.8
No. Observations: 7659 AIC: 1.751e+04
Df Residuals: 7658 BIC: 1.752e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
date 0.0185 0.000 39.207 0.000 0.018 0.019
==============================================================================
Omnibus: 3858.755 Durbin-Watson: 1.810
Prob(Omnibus): 0.000 Jarque-Bera (JB): 29521.077
Skew: 2.303 Prob(JB): 0.00
Kurtosis: 11.444 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
1.4323814844141276e-306
0.0
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 2014.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:18 Log-Likelihood: -8562.1
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7658 BIC: 1.713e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0271 0.001 44.880 0.000 0.026 0.028
==============================================================================
Omnibus: 3979.018 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33128.310
Skew: 2.360 Prob(JB): 0.00
Kurtosis: 12.030 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
4.269856558800235e-90
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.052
Model: OLS Adj. R-squared (uncentered): 0.051
Method: Least Squares F-statistic: 416.0
Date: Tue, 19 Apr 2022 Prob (F-statistic): 4.27e-90
Time: 05:18:18 Log-Likelihood: -9253.7
No. Observations: 7659 AIC: 1.851e+04
Df Residuals: 7658 BIC: 1.852e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
BROOK 0.3956 0.019 20.397 0.000 0.358 0.434
==============================================================================
Omnibus: 3929.307 Durbin-Watson: 1.678
Prob(Omnibus): 0.000 Jarque-Bera (JB): 32281.384
Skew: 2.327 Prob(JB): 0.00
Kurtosis: 11.916 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
2.393436485478676e-52
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.030
Model: OLS Adj. R-squared (uncentered): 0.030
Method: Least Squares F-statistic: 235.4
Date: Tue, 19 Apr 2022 Prob (F-statistic): 2.39e-52
Time: 05:18:18 Log-Likelihood: -9340.3
No. Observations: 7659 AIC: 1.868e+04
Df Residuals: 7658 BIC: 1.869e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
BRONX 0.4082 0.027 15.342 0.000 0.356 0.460
==============================================================================
Omnibus: 3987.014 Durbin-Watson: 1.601
Prob(Omnibus): 0.000 Jarque-Bera (JB): 32899.862
Skew: 2.370 Prob(JB): 0.00
Kurtosis: 11.979 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
4.369430499033895e-11
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.006
Model: OLS Adj. R-squared (uncentered): 0.006
Method: Least Squares F-statistic: 43.57
Date: Tue, 19 Apr 2022 Prob (F-statistic): 4.37e-11
Time: 05:18:18 Log-Likelihood: -9434.6
No. Observations: 7659 AIC: 1.887e+04
Df Residuals: 7658 BIC: 1.888e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
STATEN 0.3842 0.058 6.601 0.000 0.270 0.498
==============================================================================
Omnibus: 4179.444 Durbin-Watson: 1.527
Prob(Omnibus): 0.000 Jarque-Bera (JB): 37164.558
Skew: 2.489 Prob(JB): 0.00
Kurtosis: 12.574 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
6.945005855138339e-56
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.032
Model: OLS Adj. R-squared (uncentered): 0.032
Method: Least Squares F-statistic: 252.1
Date: Tue, 19 Apr 2022 Prob (F-statistic): 6.95e-56
Time: 05:18:18 Log-Likelihood: -9332.2
No. Observations: 7659 AIC: 1.867e+04
Df Residuals: 7658 BIC: 1.867e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
QUEENS 0.3482 0.022 15.878 0.000 0.305 0.391
==============================================================================
Omnibus: 3969.257 Durbin-Watson: 1.627
Prob(Omnibus): 0.000 Jarque-Bera (JB): 32316.549
Skew: 2.362 Prob(JB): 0.00
Kurtosis: 11.886 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
4.461967379856326e-32
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.018
Model: OLS Adj. R-squared (uncentered): 0.018
Method: Least Squares F-statistic: 140.3
Date: Tue, 19 Apr 2022 Prob (F-statistic): 4.46e-32
Time: 05:18:18 Log-Likelihood: -9386.8
No. Observations: 7659 AIC: 1.878e+04
Df Residuals: 7658 BIC: 1.878e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
MANHAT 0.3596 0.030 11.843 0.000 0.300 0.419
==============================================================================
Omnibus: 4087.592 Durbin-Watson: 1.584
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35155.863
Skew: 2.431 Prob(JB): 0.00
Kurtosis: 12.302 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
0.0018205759792527649
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.001
Model: OLS Adj. R-squared (uncentered): 0.001
Method: Least Squares F-statistic: 9.729
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00182
Time: 05:18:18 Log-Likelihood: -9451.4
No. Observations: 7659 AIC: 1.890e+04
Df Residuals: 7658 BIC: 1.891e+04
Df Model: 1
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
NUMBER OF PERSONS KILLED 0.6111 0.196 3.119 0.002 0.227 0.995
==============================================================================
Omnibus: 4194.263 Durbin-Watson: 1.510
Prob(Omnibus): 0.000 Jarque-Bera (JB): 37329.833
Skew: 2.501 Prob(JB): 0.00
Kurtosis: 12.590 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
0.0
0.0
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.210
Model: OLS Adj. R-squared (uncentered): 0.210
Method: Least Squares F-statistic: 2032.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:18 Log-Likelihood: -8555.0
No. Observations: 7659 AIC: 1.711e+04
Df Residuals: 7658 BIC: 1.712e+04
Df Model: 1
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
lat 0.3962 0.009 45.077 0.000 0.379 0.413
==============================================================================
Omnibus: 4119.911 Durbin-Watson: 1.965
Prob(Omnibus): 0.000 Jarque-Bera (JB): 35447.666
Skew: 2.456 Prob(JB): 0.00
Kurtosis: 12.324 Cond. No. 1.00
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
1
9.6.7. Start by adding Hour¶
def crunch_nums():
p_val = 1
location = 0
for i in range(len(x_columns)-len(x_col)):
x_col.append(x_columns[i])
x = df[x_col]
results = sm.OLS(y, x).fit()
print(results.summary())
x_col.pop(len(x_col)-1)
print(p_val)
print(results.pvalues[len(x_col)])
if (p_val > results.pvalues[len(x_col)]):
p_val = results.pvalues[len(x_col)]
location = i
if (p_val <.05):
print(location)
x_col.append(x_columns[location])
x_columns.remove(x_columns[int(location)])
crunch_nums()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.215
Model: OLS Adj. R-squared (uncentered): 0.215
Method: Least Squares F-statistic: 1049.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8529.0
No. Observations: 7659 AIC: 1.706e+04
Df Residuals: 7657 BIC: 1.708e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0210 0.001 21.614 0.000 0.019 0.023
date 0.0060 0.001 8.154 0.000 0.005 0.007
==============================================================================
Omnibus: 4031.194 Durbin-Watson: 1.943
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33946.805
Skew: 2.396 Prob(JB): 0.00
Kurtosis: 12.133 Cond. No. 3.01
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
1
4.0666379515353367e-16
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.209
Method: Least Squares F-statistic: 1013.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8557.5
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.713e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0263 0.001 39.070 0.000 0.025 0.028
BROOK 0.0593 0.020 3.011 0.003 0.021 0.098
==============================================================================
Omnibus: 3992.980 Durbin-Watson: 1.959
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33490.382
Skew: 2.368 Prob(JB): 0.00
Kurtosis: 12.084 Cond. No. 32.6
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.0026154978758844666
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.209
Method: Least Squares F-statistic: 1012.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8558.0
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.713e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0266 0.001 41.659 0.000 0.025 0.028
BRONX 0.0722 0.025 2.847 0.004 0.022 0.122
==============================================================================
Omnibus: 3981.961 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33171.456
Skew: 2.362 Prob(JB): 0.00
Kurtosis: 12.035 Cond. No. 41.9
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.004426633639098556
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8561.8
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0271 0.001 44.270 0.000 0.026 0.028
STATEN 0.0367 0.053 0.699 0.484 -0.066 0.140
==============================================================================
Omnibus: 3980.742 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33169.872
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.036 Cond. No. 86.9
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.4843669202067119
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8561.9
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0270 0.001 41.304 0.000 0.026 0.028
QUEENS 0.0114 0.021 0.532 0.594 -0.031 0.053
==============================================================================
Omnibus: 3980.405 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33143.808
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.031 Cond. No. 35.5
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.5944459717669641
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8562.0
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0271 0.001 42.898 0.000 0.026 0.028
MANHAT 0.0115 0.028 0.405 0.686 -0.044 0.067
==============================================================================
Omnibus: 3980.428 Durbin-Watson: 1.958
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33157.904
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.034 Cond. No. 47.0
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.6858402170710972
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1009.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:18:31 Log-Likelihood: -8560.3
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
hour 0.0271 0.001 44.790 0.000 0.026 0.028
NUMBER OF PERSONS KILLED 0.3296 0.175 1.888 0.059 -0.013 0.672
==============================================================================
Omnibus: 3969.378 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 32855.187
Skew: 2.355 Prob(JB): 0.00
Kurtosis: 11.987 Cond. No. 289.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.059032381034271476
0
9.6.8. Add Date¶
crunch_nums()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.215
Model: OLS Adj. R-squared (uncentered): 0.215
Method: Least Squares F-statistic: 1049.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8529.0
No. Observations: 7659 AIC: 1.706e+04
Df Residuals: 7657 BIC: 1.708e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0210 0.001 21.614 0.000 0.019 0.023
date 0.0060 0.001 8.154 0.000 0.005 0.007
==============================================================================
Omnibus: 4031.194 Durbin-Watson: 1.943
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33946.805
Skew: 2.396 Prob(JB): 0.00
Kurtosis: 12.133 Cond. No. 3.01
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
1
4.0666379515353367e-16
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.209
Method: Least Squares F-statistic: 1013.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8557.5
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.713e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0263 0.001 39.070 0.000 0.025 0.028
BROOK 0.0593 0.020 3.011 0.003 0.021 0.098
==============================================================================
Omnibus: 3992.980 Durbin-Watson: 1.959
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33490.382
Skew: 2.368 Prob(JB): 0.00
Kurtosis: 12.084 Cond. No. 32.6
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.0026154978758844666
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.209
Method: Least Squares F-statistic: 1012.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8558.0
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.713e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0266 0.001 41.659 0.000 0.025 0.028
BRONX 0.0722 0.025 2.847 0.004 0.022 0.122
==============================================================================
Omnibus: 3981.961 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33171.456
Skew: 2.362 Prob(JB): 0.00
Kurtosis: 12.035 Cond. No. 41.9
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.004426633639098556
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8561.8
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0271 0.001 44.270 0.000 0.026 0.028
STATEN 0.0367 0.053 0.699 0.484 -0.066 0.140
==============================================================================
Omnibus: 3980.742 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33169.872
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.036 Cond. No. 86.9
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.4843669202067119
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8561.9
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0270 0.001 41.304 0.000 0.026 0.028
QUEENS 0.0114 0.021 0.532 0.594 -0.031 0.053
==============================================================================
Omnibus: 3980.405 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33143.808
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.031 Cond. No. 35.5
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.5944459717669641
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.208
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1007.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8562.0
No. Observations: 7659 AIC: 1.713e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0271 0.001 42.898 0.000 0.026 0.028
MANHAT 0.0115 0.028 0.405 0.686 -0.044 0.067
==============================================================================
Omnibus: 3980.428 Durbin-Watson: 1.958
Prob(Omnibus): 0.000 Jarque-Bera (JB): 33157.904
Skew: 2.361 Prob(JB): 0.00
Kurtosis: 12.034 Cond. No. 47.0
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.6858402170710972
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.209
Model: OLS Adj. R-squared (uncentered): 0.208
Method: Least Squares F-statistic: 1009.
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:12:14 Log-Likelihood: -8560.3
No. Observations: 7659 AIC: 1.712e+04
Df Residuals: 7657 BIC: 1.714e+04
Df Model: 2
Covariance Type: nonrobust
============================================================================================
coef std err t P>|t| [0.025 0.975]
--------------------------------------------------------------------------------------------
hour 0.0271 0.001 44.790 0.000 0.026 0.028
NUMBER OF PERSONS KILLED 0.3296 0.175 1.888 0.059 -0.013 0.672
==============================================================================
Omnibus: 3969.378 Durbin-Watson: 1.957
Prob(Omnibus): 0.000 Jarque-Bera (JB): 32855.187
Skew: 2.355 Prob(JB): 0.00
Kurtosis: 11.987 Cond. No. 289.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
4.0666379515353367e-16
0.059032381034271476
9.6.9. Add Bronx¶
crunch_nums()
OLS Regression Results
==============================================================================================
Dep. Variable: NUMBER OF PERSONS INJURED R-squared (uncentered): 0.216
Model: OLS Adj. R-squared (uncentered): 0.216
Method: Least Squares F-statistic: 421.9
Date: Tue, 19 Apr 2022 Prob (F-statistic): 0.00
Time: 05:19:25 Log-Likelihood: -8524.1
No. Observations: 7659 AIC: 1.706e+04
Df Residuals: 7654 BIC: 1.709e+04
Df Model: 5
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
hour 0.0202 0.001 20.088 0.000 0.018 0.022
date 0.0056 0.001 7.367 0.000 0.004 0.007
BRONX 0.0624 0.026 2.412 0.016 0.012 0.113
BROOK 0.0472 0.020 2.333 0.020 0.008 0.087
STATEN 0.0417 0.053 0.793 0.428 -0.061 0.145
==============================================================================
Omnibus: 4043.897 Durbin-Watson: 1.945
Prob(Omnibus): 0.000 Jarque-Bera (JB): 34284.761
Skew: 2.403 Prob(JB): 0.00
Kurtosis: 12.184 Cond. No. 137.
==============================================================================
Notes:
[1] R² is computed without centering (uncentered) since the model does not contain a constant.
[2] Standard Errors assume that the covariance matrix of the errors is correctly specified.
1
0.42787080095123353
9.6.10. Add Brooklyn¶
In the new model we have Hour, Date, Bronx, and Brooklyn. A very different outcome than we saw with the other style of selection