BAIS 660 GLM Homework #2
Clifford Gammon
November 19, 2018
Introduction
This Homework is on better understanding of how to build a linear model and check the assumptions of that model. Data for this assignment can be found at: http://asayanalytics.com/bikeshare_csv. The code and analysis are property of Clifford Gammon.
Question 1
##
## Call:
## lm(formula = total.rider ~ poly(temp, 3, raw = TRUE) + as.factor(Promotion) +
## poly(windspeed, 3, raw = T) + poly(humidity, 2, raw = T) +
## as.factor(weathersit) + as.factor(mnth), data = bike_share)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3301.3 -355.1 58.5 431.3 2269.5
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.035e+03 6.668e+02 3.052 0.002360 **
## poly(temp, 3, raw = TRUE)1 -2.911e+02 8.182e+01 -3.557 0.000400 ***
## poly(temp, 3, raw = TRUE)2 3.321e+01 4.440e+00 7.480 2.21e-13 ***
## poly(temp, 3, raw = TRUE)3 -7.041e-01 7.416e-02 -9.494 < 2e-16 ***
## as.factor(Promotion)1 1.945e+03 5.528e+01 35.186 < 2e-16 ***
## poly(windspeed, 3, raw = T)1 -1.057e+02 6.173e+01 -1.713 0.087230 .
## poly(windspeed, 3, raw = T)2 5.571e+00 4.264e+00 1.306 0.191869
## poly(windspeed, 3, raw = T)3 -1.544e-01 9.000e-02 -1.716 0.086690 .
## poly(humidity, 2, raw = T)1 3.736e+01 1.398e+01 2.673 0.007689 **
## poly(humidity, 2, raw = T)2 -4.974e-01 1.146e-01 -4.338 1.64e-05 ***
## as.factor(weathersit)2 -3.310e+02 7.370e+01 -4.491 8.28e-06 ***
## as.factor(weathersit)3 -1.414e+03 2.083e+02 -6.791 2.36e-11 ***
## as.factor(mnth)2 4.744e+01 1.405e+02 0.338 0.735760
## as.factor(mnth)3 5.145e+02 1.528e+02 3.367 0.000802 ***
## as.factor(mnth)4 6.475e+02 1.701e+02 3.808 0.000153 ***
## as.factor(mnth)5 8.068e+02 1.915e+02 4.214 2.84e-05 ***
## as.factor(mnth)6 9.282e+02 2.160e+02 4.298 1.97e-05 ***
## as.factor(mnth)7 1.149e+03 2.381e+02 4.823 1.73e-06 ***
## as.factor(mnth)8 9.351e+02 2.207e+02 4.237 2.57e-05 ***
## as.factor(mnth)9 1.251e+03 1.974e+02 6.338 4.15e-10 ***
## as.factor(mnth)10 1.323e+03 1.710e+02 7.733 3.62e-14 ***
## as.factor(mnth)11 1.076e+03 1.551e+02 6.942 8.77e-12 ***
## as.factor(mnth)12 7.853e+02 1.453e+02 5.406 8.82e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 722.8 on 708 degrees of freedom
## Multiple R-squared: 0.865, Adjusted R-squared: 0.8608
## F-statistic: 206.1 on 22 and 708 DF, p-value: < 2.2e-16## bike_share$humidity bike_share$weathersit
## bike_share$temp 0.1269629 -0.12060224
## bike_share$Promotion -0.1106510 -0.04872654
## bike_share$windspeed -0.2484891 0.03951106
## bike_share$mnth
## bike_share$temp 0.220205335
## bike_share$Promotion -0.001792434
## bike_share$windspeed -0.207501752Based on the output above we can see that over 86% of the variation in our total rider population can be explained by our model. And that none of our variables are very highly correlated
Question 2
Based on the above Studentized residual QQPlot we can see that there is some curvature to our residuals. This would indicate that our assumptions of normally distributed residuals is in danger of being violated
From our crPlots we can see that the indepentant variables have been raised to the right power and that they have a fairly linear relationship. The only indepentant variable that might be some cause for concern in windspeed as it still does have some curvature to it. This is most likely caused by the potential outlier to the far left of the graph.
## GVIF Df GVIF^(1/(2*Df))
## poly(temp, 3, raw = TRUE) 20.992285 3 1.660899
## as.factor(Promotion) 1.068785 1 1.033820
## poly(windspeed, 3, raw = T) 1.380579 3 1.055221
## poly(humidity, 2, raw = T) 3.113551 2 1.328354
## as.factor(weathersit) 2.375788 2 1.241515
## as.factor(mnth) 24.303959 11 1.156073From our VIF output we can see that most of the variables are not multicollinearity. However, month has a very high collinear VIF score of over 24, this is most likely because our other variable Temp is better able to explain our total bike rider.
Question 3
If we were to keep everything else constant, we would expect that the month of October to have the highest number of riders than any other month. For that month alone, we would expect an additional 1323 riders just due to the month alone. However, let us take a look at how this would change if we were to have an unseasonably cold October
##
## Call:
## lm(formula = total.rider ~ poly(temp, 3, raw = TRUE) + as.factor(Promotion) +
## poly(windspeed, 3, raw = T) + poly(humidity, 2, raw = T) +
## as.factor(weathersit) + as.factor(mnth) + (as.factor(mnth) +
## as.factor(weathersit) + poly(temp, 3, raw = TRUE))^3, data = bike_share)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3341.6 -307.5 21.9 365.8 1822.0
##
## Coefficients: (28 not defined because of singularities)
## Estimate
## (Intercept) 1.364e+03
## poly(temp, 3, raw = TRUE)1 -6.666e+01
## poly(temp, 3, raw = TRUE)2 1.044e+01
## poly(temp, 3, raw = TRUE)3 -1.108e-01
## as.factor(Promotion)1 1.993e+03
## poly(windspeed, 3, raw = T)1 -7.897e+01
## poly(windspeed, 3, raw = T)2 3.726e+00
## poly(windspeed, 3, raw = T)3 -1.095e-01
## poly(humidity, 2, raw = T)1 3.146e+01
## poly(humidity, 2, raw = T)2 -4.496e-01
## as.factor(weathersit)2 -3.821e+03
## as.factor(weathersit)3 6.862e+05
## as.factor(mnth)2 3.114e+02
## as.factor(mnth)3 2.374e+03
## as.factor(mnth)4 -1.701e+04
## as.factor(mnth)5 3.625e+04
## as.factor(mnth)6 -3.345e+04
## as.factor(mnth)7 6.769e+04
## as.factor(mnth)8 -4.652e+05
## as.factor(mnth)9 -1.974e+05
## as.factor(mnth)10 -2.512e+04
## as.factor(mnth)11 -2.779e+04
## as.factor(mnth)12 -2.277e+04
## as.factor(weathersit)2:as.factor(mnth)2 -1.367e+03
## as.factor(weathersit)3:as.factor(mnth)2 4.519e+03
## as.factor(weathersit)2:as.factor(mnth)3 5.797e+02
## as.factor(weathersit)3:as.factor(mnth)3 5.347e+04
## as.factor(weathersit)2:as.factor(mnth)4 9.472e+03
## as.factor(weathersit)3:as.factor(mnth)4 4.651e+04
## as.factor(weathersit)2:as.factor(mnth)5 -2.156e+05
## as.factor(weathersit)3:as.factor(mnth)5 NA
## as.factor(weathersit)2:as.factor(mnth)6 3.226e+05
## as.factor(weathersit)3:as.factor(mnth)6 NA
## as.factor(weathersit)2:as.factor(mnth)7 1.531e+06
## as.factor(weathersit)3:as.factor(mnth)7 1.445e+05
## as.factor(weathersit)2:as.factor(mnth)8 1.617e+05
## as.factor(weathersit)3:as.factor(mnth)8 NA
## as.factor(weathersit)2:as.factor(mnth)9 2.333e+05
## as.factor(weathersit)3:as.factor(mnth)9 2.596e+06
## as.factor(weathersit)2:as.factor(mnth)10 7.966e+03
## as.factor(weathersit)3:as.factor(mnth)10 -5.490e+05
## as.factor(weathersit)2:as.factor(mnth)11 1.502e+04
## as.factor(weathersit)3:as.factor(mnth)11 2.163e+04
## as.factor(weathersit)2:as.factor(mnth)12 1.732e+04
## as.factor(weathersit)3:as.factor(mnth)12 1.548e+04
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)2 -1.150e+02
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)2 1.640e+01
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)2 -5.770e-01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)3 -4.455e+02
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)3 3.252e+01
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)3 -7.109e-01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)4 2.615e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)4 -1.191e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)4 1.699e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)5 -4.514e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)5 1.988e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)5 -2.987e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)6 3.460e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)6 -1.035e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)6 8.177e-01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)7 -6.357e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)7 2.137e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)7 -2.583e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)8 4.824e+04
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)8 -1.653e+03
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)8 1.869e+01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)9 2.355e+04
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)9 -9.136e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)9 1.160e+01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)10 4.266e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)10 -2.157e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)10 3.522e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)11 5.379e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)11 -3.238e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)11 6.372e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)12 5.161e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)12 -3.666e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)12 8.578e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2 1.017e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2 -8.484e+01
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2 2.254e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3 -1.497e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3 1.014e+04
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3 -2.215e+02
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)2 9.499e+02
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)2 -1.245e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)2 4.292e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)3 -2.678e+02
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)3 2.899e+01
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)3 -8.820e-01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)3 -2.193e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)4 -1.551e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)4 8.879e+01
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)4 -1.914e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)4 -1.901e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)5 2.911e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)5 -1.279e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)5 1.806e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)6 -3.582e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)6 1.333e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)6 -1.698e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)7 -1.547e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)7 5.217e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)7 -5.914e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)8 -1.694e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)8 6.038e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)8 -7.707e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)9 -2.837e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)9 1.161e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)9 -1.628e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)9 -2.521e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)9 6.212e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)9 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)10 -2.200e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)10 1.632e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)10 -3.807e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)10 1.250e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)10 -8.777e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)10 1.973e+02
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)11 -3.100e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)11 2.146e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)11 -4.964e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)11 -4.014e+02
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)12 -5.151e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)12 4.457e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)12 -1.188e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)12 NA
## Std. Error
## (Intercept) 1.536e+03
## poly(temp, 3, raw = TRUE)1 5.214e+02
## poly(temp, 3, raw = TRUE)2 5.917e+01
## poly(temp, 3, raw = TRUE)3 2.081e+00
## as.factor(Promotion)1 5.773e+01
## poly(windspeed, 3, raw = T)1 6.280e+01
## poly(windspeed, 3, raw = T)2 4.336e+00
## poly(windspeed, 3, raw = T)3 9.133e-02
## poly(humidity, 2, raw = T)1 1.675e+01
## poly(humidity, 2, raw = T)2 1.342e-01
## as.factor(weathersit)2 3.659e+03
## as.factor(weathersit)3 6.666e+05
## as.factor(mnth)2 3.128e+03
## as.factor(mnth)3 6.106e+03
## as.factor(mnth)4 2.378e+04
## as.factor(mnth)5 4.146e+04
## as.factor(mnth)6 8.528e+04
## as.factor(mnth)7 3.019e+05
## as.factor(mnth)8 5.394e+05
## as.factor(mnth)9 1.159e+05
## as.factor(mnth)10 2.353e+04
## as.factor(mnth)11 1.466e+04
## as.factor(mnth)12 1.511e+04
## as.factor(weathersit)2:as.factor(mnth)2 7.825e+03
## as.factor(weathersit)3:as.factor(mnth)2 5.577e+03
## as.factor(weathersit)2:as.factor(mnth)3 1.276e+04
## as.factor(weathersit)3:as.factor(mnth)3 4.368e+04
## as.factor(weathersit)2:as.factor(mnth)4 3.190e+04
## as.factor(weathersit)3:as.factor(mnth)4 6.626e+04
## as.factor(weathersit)2:as.factor(mnth)5 8.587e+04
## as.factor(weathersit)3:as.factor(mnth)5 NA
## as.factor(weathersit)2:as.factor(mnth)6 1.538e+05
## as.factor(weathersit)3:as.factor(mnth)6 NA
## as.factor(weathersit)2:as.factor(mnth)7 2.304e+06
## as.factor(weathersit)3:as.factor(mnth)7 1.475e+05
## as.factor(weathersit)2:as.factor(mnth)8 1.186e+06
## as.factor(weathersit)3:as.factor(mnth)8 NA
## as.factor(weathersit)2:as.factor(mnth)9 1.344e+05
## as.factor(weathersit)3:as.factor(mnth)9 2.379e+06
## as.factor(weathersit)2:as.factor(mnth)10 2.933e+04
## as.factor(weathersit)3:as.factor(mnth)10 6.679e+05
## as.factor(weathersit)2:as.factor(mnth)11 3.673e+04
## as.factor(weathersit)3:as.factor(mnth)11 2.527e+04
## as.factor(weathersit)2:as.factor(mnth)12 2.501e+04
## as.factor(weathersit)3:as.factor(mnth)12 1.814e+04
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)2 8.801e+02
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)2 8.194e+01
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)2 2.520e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)3 1.291e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)3 9.596e+01
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)3 2.598e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)4 3.590e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)4 1.846e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)4 3.512e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)5 5.033e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)5 2.085e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)5 3.364e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)6 9.070e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)6 3.244e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)6 4.268e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)7 2.908e+04
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)7 9.332e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)7 1.014e+01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)8 5.571e+04
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)8 1.914e+03
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)8 2.195e+01
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)9 1.381e+04
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)9 5.475e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)9 7.413e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)10 3.850e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)10 2.113e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)10 4.106e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)11 2.929e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)11 1.950e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)11 4.427e+00
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)12 3.414e+03
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)12 2.539e+02
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)12 6.233e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2 1.144e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2 1.151e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2 3.684e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3 1.454e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3 9.845e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3 2.151e+02
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)2 2.298e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)2 2.152e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)2 6.428e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)3 2.565e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)3 1.807e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)3 4.585e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)3 1.412e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)4 5.130e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)4 2.834e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)4 5.732e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)4 2.856e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)5 1.158e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)5 5.260e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)5 8.425e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)6 1.683e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)6 6.175e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)6 8.138e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)7 2.310e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)7 7.699e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)7 8.530e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)8 1.243e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)8 4.334e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)8 5.039e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)9 1.614e+04
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)9 6.486e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)9 9.154e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)9 2.331e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)9 5.801e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)9 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)10 4.786e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)10 2.685e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)10 5.549e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)10 1.456e+05
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)10 9.855e+03
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)10 2.153e+02
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)11 6.960e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)11 4.383e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)11 9.300e+00
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)11 7.731e+02
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)12 5.776e+03
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)12 4.379e+02
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)12 1.093e+01
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)12 NA
## t value
## (Intercept) 0.888
## poly(temp, 3, raw = TRUE)1 -0.128
## poly(temp, 3, raw = TRUE)2 0.177
## poly(temp, 3, raw = TRUE)3 -0.053
## as.factor(Promotion)1 34.519
## poly(windspeed, 3, raw = T)1 -1.257
## poly(windspeed, 3, raw = T)2 0.859
## poly(windspeed, 3, raw = T)3 -1.199
## poly(humidity, 2, raw = T)1 1.879
## poly(humidity, 2, raw = T)2 -3.350
## as.factor(weathersit)2 -1.044
## as.factor(weathersit)3 1.029
## as.factor(mnth)2 0.100
## as.factor(mnth)3 0.389
## as.factor(mnth)4 -0.715
## as.factor(mnth)5 0.874
## as.factor(mnth)6 -0.392
## as.factor(mnth)7 0.224
## as.factor(mnth)8 -0.862
## as.factor(mnth)9 -1.703
## as.factor(mnth)10 -1.067
## as.factor(mnth)11 -1.896
## as.factor(mnth)12 -1.507
## as.factor(weathersit)2:as.factor(mnth)2 -0.175
## as.factor(weathersit)3:as.factor(mnth)2 0.810
## as.factor(weathersit)2:as.factor(mnth)3 0.045
## as.factor(weathersit)3:as.factor(mnth)3 1.224
## as.factor(weathersit)2:as.factor(mnth)4 0.297
## as.factor(weathersit)3:as.factor(mnth)4 0.702
## as.factor(weathersit)2:as.factor(mnth)5 -2.510
## as.factor(weathersit)3:as.factor(mnth)5 NA
## as.factor(weathersit)2:as.factor(mnth)6 2.097
## as.factor(weathersit)3:as.factor(mnth)6 NA
## as.factor(weathersit)2:as.factor(mnth)7 0.665
## as.factor(weathersit)3:as.factor(mnth)7 0.979
## as.factor(weathersit)2:as.factor(mnth)8 0.136
## as.factor(weathersit)3:as.factor(mnth)8 NA
## as.factor(weathersit)2:as.factor(mnth)9 1.736
## as.factor(weathersit)3:as.factor(mnth)9 1.091
## as.factor(weathersit)2:as.factor(mnth)10 0.272
## as.factor(weathersit)3:as.factor(mnth)10 -0.822
## as.factor(weathersit)2:as.factor(mnth)11 0.409
## as.factor(weathersit)3:as.factor(mnth)11 0.856
## as.factor(weathersit)2:as.factor(mnth)12 0.693
## as.factor(weathersit)3:as.factor(mnth)12 0.853
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)2 -0.131
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)2 0.200
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)2 -0.229
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)3 -0.345
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)3 0.339
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)3 -0.274
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)4 0.728
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)4 -0.645
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)4 0.484
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)5 -0.897
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)5 0.953
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)5 -0.888
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)6 0.381
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)6 -0.319
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)6 0.192
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)7 -0.219
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)7 0.229
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)7 -0.255
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)8 0.866
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)8 -0.863
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)8 0.851
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)9 1.706
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)9 -1.669
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)9 1.564
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)10 1.108
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)10 -1.021
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)10 0.858
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)11 1.837
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)11 -1.660
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)11 1.440
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)12 1.512
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)12 -1.444
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)12 1.376
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2 0.889
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2 -0.737
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2 0.612
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3 -1.030
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3 1.030
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3 -1.030
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)2 0.413
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)2 -0.579
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)2 0.668
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)3 -0.104
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)3 0.160
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)3 -0.192
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)3 -1.553
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)4 -0.302
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)4 0.313
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)4 -0.334
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)4 -0.666
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)5 2.513
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)5 -2.431
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)5 2.144
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)6 -2.129
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)6 2.158
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)6 -2.086
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)7 -0.670
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)7 0.678
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)7 -0.693
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)8 -0.136
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)8 0.139
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)8 -0.153
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)9 -1.758
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)9 1.790
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)9 -1.778
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)9 -1.082
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)9 1.071
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)9 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)10 -0.460
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)10 0.608
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)10 -0.686
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)10 0.859
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)10 -0.891
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)10 0.917
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)11 -0.445
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)11 0.490
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)11 -0.534
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)11 -0.519
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)12 -0.892
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)12 1.018
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)12 -1.087
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)12 NA
## Pr(>|t|)
## (Intercept) 0.374825
## poly(temp, 3, raw = TRUE)1 0.898317
## poly(temp, 3, raw = TRUE)2 0.859942
## poly(temp, 3, raw = TRUE)3 0.957545
## as.factor(Promotion)1 < 2e-16
## poly(windspeed, 3, raw = T)1 0.209070
## poly(windspeed, 3, raw = T)2 0.390535
## poly(windspeed, 3, raw = T)3 0.231187
## poly(humidity, 2, raw = T)1 0.060776
## poly(humidity, 2, raw = T)2 0.000857
## as.factor(weathersit)2 0.296728
## as.factor(weathersit)3 0.303709
## as.factor(mnth)2 0.920726
## as.factor(mnth)3 0.697594
## as.factor(mnth)4 0.474797
## as.factor(mnth)5 0.382313
## as.factor(mnth)6 0.695046
## as.factor(mnth)7 0.822675
## as.factor(mnth)8 0.388756
## as.factor(mnth)9 0.089081
## as.factor(mnth)10 0.286267
## as.factor(mnth)11 0.058459
## as.factor(mnth)12 0.132261
## as.factor(weathersit)2:as.factor(mnth)2 0.861431
## as.factor(weathersit)3:as.factor(mnth)2 0.418075
## as.factor(weathersit)2:as.factor(mnth)3 0.963766
## as.factor(weathersit)3:as.factor(mnth)3 0.221419
## as.factor(weathersit)2:as.factor(mnth)4 0.766636
## as.factor(weathersit)3:as.factor(mnth)4 0.482986
## as.factor(weathersit)2:as.factor(mnth)5 0.012323
## as.factor(weathersit)3:as.factor(mnth)5 NA
## as.factor(weathersit)2:as.factor(mnth)6 0.036400
## as.factor(weathersit)3:as.factor(mnth)6 NA
## as.factor(weathersit)2:as.factor(mnth)7 0.506473
## as.factor(weathersit)3:as.factor(mnth)7 0.327738
## as.factor(weathersit)2:as.factor(mnth)8 0.891603
## as.factor(weathersit)3:as.factor(mnth)8 NA
## as.factor(weathersit)2:as.factor(mnth)9 0.083081
## as.factor(weathersit)3:as.factor(mnth)9 0.275607
## as.factor(weathersit)2:as.factor(mnth)10 0.786014
## as.factor(weathersit)3:as.factor(mnth)10 0.411412
## as.factor(weathersit)2:as.factor(mnth)11 0.682709
## as.factor(weathersit)3:as.factor(mnth)11 0.392242
## as.factor(weathersit)2:as.factor(mnth)12 0.488818
## as.factor(weathersit)3:as.factor(mnth)12 0.393808
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)2 0.896071
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)2 0.841440
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)2 0.819008
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)3 0.730117
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)3 0.734820
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)3 0.784453
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)4 0.466711
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)4 0.519234
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)4 0.628808
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)5 0.370051
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)5 0.340869
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)5 0.374999
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)6 0.703006
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)6 0.749878
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)6 0.848110
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)7 0.827037
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)7 0.818941
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)7 0.798986
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)8 0.386880
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)8 0.388358
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)8 0.394915
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)9 0.088594
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)9 0.095696
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)9 0.118265
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)10 0.268270
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)10 0.307716
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)10 0.391392
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)11 0.066739
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)11 0.097438
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)11 0.150516
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)12 0.131115
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)12 0.149317
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)12 0.169232
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2 0.374291
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2 0.461376
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2 0.540886
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3 0.303652
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3 0.303261
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3 0.303604
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)2 0.679532
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)2 0.563040
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)2 0.504584
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)2 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)3 0.916895
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)3 0.872582
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)3 0.847521
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)3 0.120863
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)3 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)4 0.762470
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)4 0.754173
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)4 0.738547
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)4 0.505926
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)4 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)5 0.012226
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)5 0.015351
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)5 0.032416
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)5 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)6 0.033664
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)6 0.031286
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)6 0.037383
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)6 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)7 0.503340
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)7 0.498232
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)7 0.488319
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)7 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)8 0.891621
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)8 0.889245
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)8 0.878508
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)8 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)9 0.079326
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)9 0.074022
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)9 0.075911
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)9 0.279808
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)9 0.284666
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)9 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)10 0.645969
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)10 0.543420
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)10 0.492905
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)10 0.390915
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)10 0.373503
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)10 0.359720
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)11 0.656241
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)11 0.624543
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)11 0.593701
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)11 0.603835
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)11 NA
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)12 0.372800
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)12 0.309140
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)12 0.277463
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)12 NA
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)12 NA
##
## (Intercept)
## poly(temp, 3, raw = TRUE)1
## poly(temp, 3, raw = TRUE)2
## poly(temp, 3, raw = TRUE)3
## as.factor(Promotion)1 ***
## poly(windspeed, 3, raw = T)1
## poly(windspeed, 3, raw = T)2
## poly(windspeed, 3, raw = T)3
## poly(humidity, 2, raw = T)1 .
## poly(humidity, 2, raw = T)2 ***
## as.factor(weathersit)2
## as.factor(weathersit)3
## as.factor(mnth)2
## as.factor(mnth)3
## as.factor(mnth)4
## as.factor(mnth)5
## as.factor(mnth)6
## as.factor(mnth)7
## as.factor(mnth)8
## as.factor(mnth)9 .
## as.factor(mnth)10
## as.factor(mnth)11 .
## as.factor(mnth)12
## as.factor(weathersit)2:as.factor(mnth)2
## as.factor(weathersit)3:as.factor(mnth)2
## as.factor(weathersit)2:as.factor(mnth)3
## as.factor(weathersit)3:as.factor(mnth)3
## as.factor(weathersit)2:as.factor(mnth)4
## as.factor(weathersit)3:as.factor(mnth)4
## as.factor(weathersit)2:as.factor(mnth)5 *
## as.factor(weathersit)3:as.factor(mnth)5
## as.factor(weathersit)2:as.factor(mnth)6 *
## as.factor(weathersit)3:as.factor(mnth)6
## as.factor(weathersit)2:as.factor(mnth)7
## as.factor(weathersit)3:as.factor(mnth)7
## as.factor(weathersit)2:as.factor(mnth)8
## as.factor(weathersit)3:as.factor(mnth)8
## as.factor(weathersit)2:as.factor(mnth)9 .
## as.factor(weathersit)3:as.factor(mnth)9
## as.factor(weathersit)2:as.factor(mnth)10
## as.factor(weathersit)3:as.factor(mnth)10
## as.factor(weathersit)2:as.factor(mnth)11
## as.factor(weathersit)3:as.factor(mnth)11
## as.factor(weathersit)2:as.factor(mnth)12
## as.factor(weathersit)3:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)9 .
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)9 .
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)9
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)11 .
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)11 .
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)1:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)2:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)3:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)2
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)3
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)4
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)5 *
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)5 *
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)5 *
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)5
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)6 *
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)6 *
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)6 *
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)6
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)7
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)8
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)9 .
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)9 .
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)9 .
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)9
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)9
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)9
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)10
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)11
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)2:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)2:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)2:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)1:as.factor(weathersit)3:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)2:as.factor(weathersit)3:as.factor(mnth)12
## poly(temp, 3, raw = TRUE)3:as.factor(weathersit)3:as.factor(mnth)12
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 668.7 on 609 degrees of freedom
## Multiple R-squared: 0.9006, Adjusted R-squared: 0.8808
## F-statistic: 45.6 on 121 and 609 DF, p-value: < 2.2e-16While the above output can be hard to read, suffice it to say that if the weather got colder in October the number of riders in October would take drastic decrease (as much as 2200 riders). However, this model does explain over 88% of the variation in total riders.
Question 4
To understand how our promotion play an impact in total riders let us once remind ourselves of the original model:
##
## Call:
## lm(formula = total.rider ~ poly(temp, 3, raw = TRUE) + as.factor(Promotion) +
## poly(windspeed, 3, raw = T) + poly(humidity, 2, raw = T) +
## as.factor(weathersit) + as.factor(mnth), data = bike_share)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3301.3 -355.1 58.5 431.3 2269.5
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.035e+03 6.668e+02 3.052 0.002360 **
## poly(temp, 3, raw = TRUE)1 -2.911e+02 8.182e+01 -3.557 0.000400 ***
## poly(temp, 3, raw = TRUE)2 3.321e+01 4.440e+00 7.480 2.21e-13 ***
## poly(temp, 3, raw = TRUE)3 -7.041e-01 7.416e-02 -9.494 < 2e-16 ***
## as.factor(Promotion)1 1.945e+03 5.528e+01 35.186 < 2e-16 ***
## poly(windspeed, 3, raw = T)1 -1.057e+02 6.173e+01 -1.713 0.087230 .
## poly(windspeed, 3, raw = T)2 5.571e+00 4.264e+00 1.306 0.191869
## poly(windspeed, 3, raw = T)3 -1.544e-01 9.000e-02 -1.716 0.086690 .
## poly(humidity, 2, raw = T)1 3.736e+01 1.398e+01 2.673 0.007689 **
## poly(humidity, 2, raw = T)2 -4.974e-01 1.146e-01 -4.338 1.64e-05 ***
## as.factor(weathersit)2 -3.310e+02 7.370e+01 -4.491 8.28e-06 ***
## as.factor(weathersit)3 -1.414e+03 2.083e+02 -6.791 2.36e-11 ***
## as.factor(mnth)2 4.744e+01 1.405e+02 0.338 0.735760
## as.factor(mnth)3 5.145e+02 1.528e+02 3.367 0.000802 ***
## as.factor(mnth)4 6.475e+02 1.701e+02 3.808 0.000153 ***
## as.factor(mnth)5 8.068e+02 1.915e+02 4.214 2.84e-05 ***
## as.factor(mnth)6 9.282e+02 2.160e+02 4.298 1.97e-05 ***
## as.factor(mnth)7 1.149e+03 2.381e+02 4.823 1.73e-06 ***
## as.factor(mnth)8 9.351e+02 2.207e+02 4.237 2.57e-05 ***
## as.factor(mnth)9 1.251e+03 1.974e+02 6.338 4.15e-10 ***
## as.factor(mnth)10 1.323e+03 1.710e+02 7.733 3.62e-14 ***
## as.factor(mnth)11 1.076e+03 1.551e+02 6.942 8.77e-12 ***
## as.factor(mnth)12 7.853e+02 1.453e+02 5.406 8.82e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 722.8 on 708 degrees of freedom
## Multiple R-squared: 0.865, Adjusted R-squared: 0.8608
## F-statistic: 206.1 on 22 and 708 DF, p-value: < 2.2e-16Based upon the about above we can see that we do have a statistically significant increase in riders when a promotion is present. To put that into lay terms, we can say: On days that have a promotion, holding all other potential impacts constant (i.e. weather, temperature, humidity, etc.) we expect, on average, to have an additional 1945 riders using the BikeShare service.
Question 5
To understand how the promotions effect each type of rider, let us redo our model. However, instead of looking at total riders, let us run the model once for casual riders and once for registered riders.
Our first model will be for registered riders. Also for this analysis we will remove the mnth variable, as we already know that it is highly correlated with our other factors and we will add in the working day variable (this variable flags if the day is a working day or a holiday):
##
## Call:
## lm(formula = registered ~ poly(temp, 3, raw = TRUE) + as.factor(Promotion) +
## poly(windspeed, 3, raw = T) + poly(humidity, 2, raw = T) +
## as.factor(weathersit) + as.factor(workingday), data = bike_share)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2877.4 -382.8 -5.4 401.9 1657.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -169.56037 588.07791 -0.288 0.7732
## poly(temp, 3, raw = TRUE)1 -33.59695 62.72418 -0.536 0.5924
## poly(temp, 3, raw = TRUE)2 16.59063 3.36877 4.925 1.05e-06 ***
## poly(temp, 3, raw = TRUE)3 -0.40699 0.05655 -7.197 1.56e-12 ***
## as.factor(Promotion)1 1649.72243 50.11341 32.920 < 2e-16 ***
## poly(windspeed, 3, raw = T)1 -110.27432 55.65467 -1.981 0.0479 *
## poly(windspeed, 3, raw = T)2 5.21468 3.85067 1.354 0.1761
## poly(windspeed, 3, raw = T)3 -0.12155 0.08134 -1.494 0.1355
## poly(humidity, 2, raw = T)1 50.62143 12.49519 4.051 5.65e-05 ***
## poly(humidity, 2, raw = T)2 -0.52496 0.10267 -5.113 4.07e-07 ***
## as.factor(weathersit)2 -295.82239 66.68532 -4.436 1.06e-05 ***
## as.factor(weathersit)3 -978.28127 187.89765 -5.206 2.52e-07 ***
## as.factor(workingday)1 925.62085 52.93694 17.485 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 661.2 on 718 degrees of freedom
## Multiple R-squared: 0.8234, Adjusted R-squared: 0.8204
## F-statistic: 278.9 on 12 and 718 DF, p-value: < 2.2e-16
## GVIF Df GVIF^(1/(2*Df))
## poly(temp, 3, raw = TRUE) 1.273859 3 1.041167
## as.factor(Promotion) 1.049821 1 1.024608
## poly(windspeed, 3, raw = T) 1.264785 3 1.039927
## poly(humidity, 2, raw = T) 2.726180 2 1.284957
## as.factor(weathersit) 2.260061 2 1.226112
## as.factor(workingday) 1.012821 1 1.006390
## bike_share$humidity bike_share$weathersit
## bike_share$temp 0.1269629 -0.12060224
## bike_share$Promotion -0.1106510 -0.04872654
## bike_share$windspeed -0.2484891 0.03951106
## bike_share$workingday
## bike_share$temp 0.052659810
## bike_share$Promotion -0.002012621
## bike_share$windspeed -0.018796487Based upon the output above we can see that once again the presence of a promotion does have a statistically significant impact to our riders. For our registered riders, keeping all other factors constant, we would expect, on average, ridership to increase by 1650. The output also shoes that over 82% of the variation in registered ridership is explained by our model. We can also see by the crPlots, correlation and the QQPlot that our model doesn’t have high collinearity and that the linear assumptions of our model are not violated. The VIF also shows that we don’t have a high variable inflation.
Now let us take a look at the casual rider model. Again for this model we will remove the mnth factor due to is collinearity and add in working day:
##
## Call:
## lm(formula = casual ~ poly(temp, 3, raw = TRUE) + as.factor(Promotion) +
## poly(windspeed, 3, raw = T) + poly(humidity, 2, raw = T) +
## as.factor(weathersit) + as.factor(workingday), data = bike_share)
##
## Residuals:
## Min 1Q Median 3Q Max
## -971.58 -223.26 -35.37 177.28 1555.50
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1192.36107 312.08730 3.821 0.000145 ***
## poly(temp, 3, raw = TRUE)1 -93.38705 33.28712 -2.806 0.005160 **
## poly(temp, 3, raw = TRUE)2 11.51025 1.78778 6.438 2.21e-10 ***
## poly(temp, 3, raw = TRUE)3 -0.24175 0.03001 -8.055 3.31e-15 ***
## as.factor(Promotion)1 258.20551 26.59471 9.709 < 2e-16 ***
## poly(windspeed, 3, raw = T)1 -66.97410 29.53540 -2.268 0.023651 *
## poly(windspeed, 3, raw = T)2 4.40675 2.04351 2.156 0.031379 *
## poly(windspeed, 3, raw = T)3 -0.10986 0.04316 -2.545 0.011128 *
## poly(humidity, 2, raw = T)1 4.13414 6.63108 0.623 0.533187
## poly(humidity, 2, raw = T)2 -0.10402 0.05449 -1.909 0.056646 .
## as.factor(weathersit)2 -53.27834 35.38926 -1.505 0.132637
## as.factor(weathersit)3 -188.40453 99.71548 -1.889 0.059238 .
## as.factor(workingday)1 -816.51129 28.09312 -29.064 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 350.9 on 718 degrees of freedom
## Multiple R-squared: 0.7431, Adjusted R-squared: 0.7388
## F-statistic: 173.1 on 12 and 718 DF, p-value: < 2.2e-16
## GVIF Df GVIF^(1/(2*Df))
## poly(temp, 3, raw = TRUE) 1.273859 3 1.041167
## as.factor(Promotion) 1.049821 1 1.024608
## poly(windspeed, 3, raw = T) 1.264785 3 1.039927
## poly(humidity, 2, raw = T) 2.726180 2 1.284957
## as.factor(weathersit) 2.260061 2 1.226112
## as.factor(workingday) 1.012821 1 1.006390
## bike_share$humidity bike_share$weathersit
## bike_share$temp 0.1269629 -0.12060224
## bike_share$Promotion -0.1106510 -0.04872654
## bike_share$windspeed -0.2484891 0.03951106
## bike_share$workingday
## bike_share$temp 0.052659810
## bike_share$Promotion -0.002012621
## bike_share$windspeed -0.018796487Again we see from our output that the presence of a promotion does have a statistically significant impact to overall ridership. However, the impact of the promotion is much smaller to our casual riders vs. our registered riders. On average, holding all other factors constant, we would expect our casual ridership to increase by 258 riders with the presence of a promotion. We can also see that almost 74% of casual rider variation can be explained by our model. We can also see by the crPlots, correlation and the QQPlot that our model doesn’t have high collinearity and that the linear assumptions of our model are not horribly violated (although there are some outliers that need to be investigated). Our VIF output also shows that we have very small variable inflation.
Question 6
While we have been able to build a great model that will help us predict total ridership based upon the presence of a promotion, temperature, windspeed, humidity and the weather situation (i.e. raining, snowing, etc.) we still do not full understand the full impact that the promotions have to the BiekShare Company.
In order to better answer this question, we would need access to some more data. The first type of data that we need to get is the conversation rate (or how many casual riders made the switch to registered riders after participating in a promotion). We would also need to understand the profitability of this promotion (if every promotion cost more than is taken in and conversion rate is low, than offering a promotion is not a good financial investment in the long run).
Neighborhood specific data would be helpful also, as this would allow us to understand if this type of promotion works for all riders, or are there other types of promotions that would work better in other areas.
Maintenance cost and theft loss are two other data types that would be good to have. We currently assume that promotions do not cause any significant increase cost in repairs or stolen bicycles. With this data we could validate that claim.