0.1 Case Study

First few records in the data set
Date Day HighTemp LowTemp Precipitation ManhattanBridge Total precip.cat LowTemp.cat
2017-04-01 2017-04-01 46.0 37 0.00 1446 5397 no chance of rain cold weather
2017-04-02 2017-04-02 62.1 41 0.00 3943 13033 no chance of rain cold weather
2017-04-03 2017-04-03 63.0 50 0.03 4988 16325 low chance of rain cold weather
2017-04-04 2017-04-04 51.1 46 1.18 1913 6581 high chance of rain cold weather
2017-04-05 2017-04-05 63.0 46 0.00 5276 17991 no chance of rain cold weather
2017-04-06 2017-04-06 48.9 41 0.73 1324 4896 high chance of rain cold weather

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.

0.2 Poisson Regression on People Crossing the Manhattan Bridge

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) 8.4478529 0.0043059 1961.92671 0
LowTemp.catroom temperature 0.0740337 0.0065113 11.37005 0
LowTemp.catWarm Temperature 0.2961534 0.0075902 39.01773 0
precip.catlow chance of rain -0.2328171 0.0062917 -37.00372 0
precip.cathigh chance of rain -1.1042111 0.0153005 -72.16843 0

The above table shows that Lowtemp.cat and precipt.cat are statistically significant. This means, there could statistical evidence to support the discrepancy across different weathers. This could mean that people are less likely to go out walking when their is poor weather condition. Moreover, the sample size could impact the statistical significance of these variables.

We will also look at the appropriateness model. The number of people crossing the bridge are dependent on the population size. Not involving the population size shows that the information in the sample was not used effectively. In the next step, we model the bridge crossing rates that involve our population size. Also, we can look at the goodness of the model

0.3 Poission Regression on Rates

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) -1.2302797 0.0043250 -284.4570983 0.0000000
LowTemp.catroom temperature -0.0145432 0.0065068 -2.2350785 0.0254122
LowTemp.catWarm Temperature 0.0044086 0.0075847 0.5812476 0.5610736
precip.catlow chance of rain -0.0104900 0.0062771 -1.6711714 0.0946878
precip.cathigh chance of rain -0.0666685 0.0153059 -4.3557543 0.0000133

The Tables shows that the log rates are not identical across different weather conditions. From the table, we can see the different low temperatures during the night time have different log rates. More, the log of the Manhattan bridge crossing rate is not identical across the weather conditions. To emphasize, the log rates of cold temperature is higher compared to room temperature and warm temperature but lower compared to warm temperaure. Moreover, the log rates of having a 0 percent chance of rain is higher compared to having any chance of rain.

0.4 Graphical Compairson

The following calculation came from the regression equation with coefficients given in the above table. For example, exp(−1.23) gives us the Manhattanbridge crossing rate of the baseline temperature, Cold Temperature, and the baseline precipitation group no chance of rain. exp(−1.23-.014) gives the brdigecrossing rate of room temperature and precipitation group low chance of rain. Following this similar pattern, you can find the Manhattan bridge crossing rate for each combination of temperature and chance of precipitation.

0.5 Conclusion

The regression model based on the amount of people crossing the Manhattan bridge is not appropriate since we cannot use the information on the population size. However, we can include the population size in the regression model to improve the model performance.

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) 7.2376684 0.0139410 519.164464 0.00e+00
LowTemp.catroom temperature -0.0380950 0.0066225 -5.752347 0.00e+00
LowTemp.catWarm Temperature -0.0549800 0.0083718 -6.567262 0.00e+00
precip.catlow chance of rain 0.0281976 0.0069053 4.083452 4.44e-05
precip.cathigh chance of rain -0.3066471 0.0176916 -17.332945 0.00e+00
Total 0.0000727 0.0000008 94.502107 0.00e+00

It seems that the Manhattan bridgecrossing rate in Cold temp is significantly higher compared to room temperature and warm temperature. Also, there does not seem to be much of a difference between room temperature and warm temperature. The reason why cold temp has the highest rate of bridge crossing need more investigation and extra information.

There is a negative linear relationship between Lowtemp and the Manahattan bridgege crossing rate. The manahattan bridge crossing rate decreases as LowTemp increases. However, This is a small data set with limited information. All conclusions in this report are only based on our given data set.