The data set represented the amount of people crossing the manhattan bridge on march of 2017. This data set contains the amount of people crossing the manhattan bridge during certain weather conditions. The random response variable is the rate of people corssing the bridge. The predictor variables are precipitation, High temp, Low temp, and the total population size.

0.1 Introduction

First few records in the data set
Date Day HighTemp LowTemp Precipitation ManhattanBridge Total AveTemp NewPrecipitation.cat AveTemp.cat
2017-04-01 2017-04-01 46.0 37 0.00 1446 5397 41.50 0 cold weather
2017-04-02 2017-04-02 62.1 41 0.00 3943 13033 51.55 0 room temperature
2017-04-03 2017-04-03 63.0 50 0.03 4988 16325 56.50 1 room temperature
2017-04-04 2017-04-04 51.1 46 1.18 1913 6581 48.55 1 cold weather
2017-04-05 2017-04-05 63.0 46 0.00 5276 17991 54.50 0 room temperature
2017-04-06 2017-04-06 48.9 41 0.73 1324 4896 44.95 1 cold weather

Since it’s reasonable to think that the expected count of people crossing the Manahattan Bridge is proportional to the population size, we would prefer to model the crossing rate per weather condition. However, illustration purposes, we will fit the Poisson regression model with both counts and rate of people crossing the Manhattan bridge.

0.2 Poisson Regression on People Crossing the Manhattan Bridge

We first build a Poisson frequency regression model and ignore the population size.

The Poisson regression model for the amount of people crossing the Manhattan Bridge versus the different weather conditions.
Estimate Std. Error z value Pr(>|z|)
(Intercept) 8.0277179 0.0088790 904.12364 0
AveTemp.catroom temperature 0.4116338 0.0091048 45.21039 0
AveTemp.catWarm Temperature 0.7311650 0.0095863 76.27160 0
NewPrecipitation.cat1 -0.2758641 0.0062160 -44.37938 0

The above inferential table about the regression coefficients shows both Average temperature and New Precipitation are statistically significant. This means, if we look at the amount of people crossing the Manhattan bridge across the different weather conditions, there is statistical evidence to support the potential discrepancy across the different groups of weather conditions. However, However, statistical significance is not equivalent the clinical importance. Moreover, the sample size could impact the statistical significance of the variables.

The other way to look at the model is the appropriateness model. The amount of people crossing the bridge are dependent on the population size. Ignoring the population size implies the information in the sample was not effectively used.

The other way to look at the model is goodness of the model. The amount of people crossing the bridge are dependent on the population size. Ignoring the population size implies the information in the sample was not used effectively. In the next subsection, we model the bridge crossing rate that involves the population size.

0.3 Poisson Regression on Rates

The following model assesses the possible relationship between the Manhattan Bridge Crossing rate and precipitation. We also want to adjust the relationship of the different average temperatures.

The Poisson regression model for the amount of people crossing the Manhattan Bridge versus the different weather conditions.
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.2565743 0.0091374 -137.519291 0.0000000
AveTemp.catroom temperature 0.0183081 0.0092995 1.968720 0.0489853
AveTemp.catWarm Temperature 0.0322721 0.0096871 3.331435 0.0008640
NewPrecipitation.cat1 -0.0114946 0.0063000 -1.824541 0.0680703

The above table indicates that the log of the bridge crossing rate is not identical across the different weather conditions. For example, the log rates of cold weather (baseline temperature) is lower compared the other two temperature groups. Also, the baseline NewPrecipitation.cat0 (When it does not rain) has a higher log rate. The regression coefficients represent the change of log rate between the associates temperature group and the reference temperature group. The same interpretation applies to the change in log rate among the precipitation groups.

0.4 Quasi Poisson Rate Model

The Quasi Poisson regression model for the amount of people crossing the Manhattan Bridge versus the different weather conditions.
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.2565743 0.0091374 -137.519291 0.0000000
AveTemp.catroom temperature 0.0183081 0.0092995 1.968720 0.0489853
AveTemp.catWarm Temperature 0.0322721 0.0096871 3.331435 0.0008640
NewPrecipitation.cat1 -0.0114946 0.0063000 -1.824541 0.0680703

The dispersion index can be extracted from the quasi-Poisson object with the bottom code

## Warning: Unknown or uninitialised column: `cases`.
Dispersion
0

0.5 Final Working Model

The dispersion index is 0. It is slightly dispersed. Thus, we stay with the regular Poisson regression model.

The inferential tables of the Poisson regression models in the previous sections give numerical information about the potential discrepancy across the temperature and precipitation groups. Afterwards, we create a graph to visualize the relationship between the bridge crossing rate and the temperature and age groups.

First of all, every city has a trend line that reflects the relationship between the cancer rate and age. We next find the rates of combinations of precipitation and temperature groups based on our following working rate model. \[ \text{log-rate} = -1.25+0.018 \times \text{Ave.Temp.catroomtemperature} +0.032 \times \text{Ave.Temp.catwarmtemperature} -0.011 \times \text{NewPrecipitation.cat1+} \]

We use Newprecipitation as the horizontal axis and the estimated bridge crossing rates as the vertical axis to make the trend lines for each of the average temperature groups using the following code.

To make the visual representation, we calculate cancer rates of the corresponding combinations of Average temperatures and Newprecipitation group in the following calculation based on the regression equation with coefficients given in above table.

For example, exp(−1.25) gives the the crossing rate of the baseline average temperature, Avetemp.catcoldtemperature, and the baseline New precipitation group 0. exp(−1.25+.018) gives the crossing rate of baseline average temperature and the new precipitation group 1. Following the same pattern, we can find the Manhattan bridge crossing rate for each combination of the Average temperature and new precipitation group.

0.6 Conlusion of the Analysis

The regression model based on the amount of people crossing the Manhattan bridge is not appropriate because the information on the population size is an important variable in the study of the distribution becauseing including the population size in the regression model will increase the statistical significance of NewPrecipitation.

The Poisson regression model for the amount of people crossing the Manhattan Bridge versus the different weather conditions.
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.1031618 0.1101132 -19.099995 0.0000000
AveTemp.catroom temperature -0.0108203 0.0100298 -1.078815 0.2806704
AveTemp.catWarm Temperature -0.0227046 0.0119812 -1.895020 0.0580897
NewPrecipitation.cat1 0.0100578 0.0068920 1.459347 0.1444696
log(Total) 1.0902695 0.0116910 93.257076 0.0000000

The brdige corssing rate while the average temperature is cold is significantly lower compared to the other average temperature groups. It seems that there is no significant difference between room temperature and warm temperature. The reason why cold temperature has the lowest crossing rate could be because people often prefer not to go excersizing outside when it is cold outside.

There is a straight linear relationship between New precipitation and the bridgecrossingrate. The bridgecrossing rate decreases as new precipitation increases. The pattern seems consistent because when it rains while the weather gets colder, people are less likely to go outside.

Also, there is an interaction effect between the cold average temperature and room average temperature because the rate curves are intersecting eachother as precipitation increases.

Unfortunatley, This is only a small data set with not much information. All conclusions in this report are only based on our given data set.