Week 9 HW STA 321

0.1 Data Description

First few records in the data set

X	Date	Day	HighTemp	LowTemp	Precipitation	WilliamsburgBridge	Total
1	42826	42826	46.0	37	0.00	1915	5397
2	42827	42827	62.1	41	0.00	4207	13033
3	42828	42828	63.0	50	0.03	5178	16325
4	42829	42829	51.1	46	1.18	2279	6581
5	42830	42830	63.0	46	0.00	5711	17991
6	42831	42831	48.9	41	0.73	1739	4896

0.2 Research Question

In this assignment we are looking at how many cyclists are entering and leaving Queens, Manhattan, and Brooklyn via the East River Bridges on a 24 hour basis. We are specifically looking at the Williamsburg Bridge.

0.3 Poisson Regression on Bike Count

We first build a Poisson frequency regression model.

The Poisson regression model for the Williamsburg Bridge counts of bikes versus predictor variables.

	Estimate	Std. Error	z value	Pr(>|z|)
(Intercept)	92.5585282	15.8473148	5.840644	0e+00
Date	-0.0019898	0.0003702	-5.375607	1e-07
HighTemp	0.0124176	0.0005280	23.516474	0e+00
LowTemp	0.0090757	0.0007776	11.671588	0e+00
Precipitation	-0.8475197	0.0153209	-55.317891	0e+00

The above inferential table about the regression coefficients indicates all variables are significant. This means, if we look at Williamsburg bridge bike count across all predictor variables, there is statistical evidence to support the potential discrepancy across the variables.

0.4 Poisson Regression on Rates

The following model assesses the potential relationship between Williamsburg bridge bike count rates of people entering and leaving the bridge. This is the primary interest of the model.

Poisson regression on the rate of the the Williamsburg Bridge bikers leaving and entering from this bridge.

	Estimate	Std. Error	z value	Pr(>|z|)
(Intercept)	16.2190289	15.7531962	1.029571	0.3032116
Date	-0.0004033	0.0003679	-1.096057	0.2730539
HighTemp	-0.0016855	0.0005412	-3.114497	0.0018426
LowTemp	0.0010453	0.0007805	1.339286	0.1804777
Precipitation	0.0473201	0.0147457	3.209079	0.0013316

The above table indicates that the log of Williamsburg bridge rates is not identical across date, temps, and precipitation. The log rates of precipitation were higher than all the other predictor variables. LowTemp has the lowest log rate. The regression coefficients represent the change of log rate between the count of bikes passing on the bridge each day.

FALSE 
FALSE Call:
FALSE glm(formula = WilliamsburgBridge ~ Date + HighTemp + LowTemp + 
FALSE     Precipitation, family = quasipoisson, data = bikes, offset = log(Total))
FALSE 
FALSE Coefficients:
FALSE                 Estimate Std. Error t value Pr(>|t|)
FALSE (Intercept)   16.2190289 31.5869867   0.513    0.612
FALSE Date          -0.0004033  0.0007377  -0.547    0.589
FALSE HighTemp      -0.0016855  0.0010851  -1.553    0.133
FALSE LowTemp        0.0010453  0.0015650   0.668    0.510
FALSE Precipitation  0.0473201  0.0295668   1.600    0.122
FALSE 
FALSE (Dispersion parameter for quasipoisson family taken to be 4.02049)
FALSE 
FALSE     Null deviance: 151.051  on 29  degrees of freedom
FALSE Residual deviance:  99.642  on 25  degrees of freedom
FALSE AIC: NA
FALSE 
FALSE Number of Fisher Scoring iterations: 3

0.5 Discussions and Conclusions

The regression model based on the bike count is appropriate since the information on the total count is an important variable. Including the temperature of the days will decrease the significance of the other variables so we will not put the temperature variables in the final model. See the following output of the fitted Poisson regression model of count adjusted by total count.

The Poisson regression model for the counts of bikers onnthe Williamsburg Bridge cases versus the date, precipitation, and total biker count.

	Estimate	Std. Error	z value	Pr(>|z|)
(Intercept)	21.9017937	15.8197149	1.3844620	0.1662170
Date	-0.0005253	0.0003692	-1.4225474	0.1548674
HighTemp	-0.0009698	0.0005706	-1.6995988	0.0892064
LowTemp	0.0014775	0.0007879	1.8753115	0.0607499
Precipitation	-0.0014458	0.0193091	-0.0748743	0.9403148
log(Total)	0.9459994	0.0137645	68.7272667	0.0000000

We can see from the above output the adding total count to the model changes the p-values associated with all predictor variables. There is a strong correlation between the total bike count and the Williamsburg bridge count. On the other hand, adding the total has increased some of the other variables p-values like precipitation.

Looking at the final model, it would be best to take precipitation out since it has a very high p-value and could be brining other variables p-values up with it.

This is a small data set with limited information. All conclusions in this report are only based on the given data set.