Add page on count models

topics/Count_Models.md

@@ -1,9 +1,11 @@
 ---
-title: "Count_Models"
+title: "Count Models"
 categories:
   - Statistical Methods
 ---
 
+
+
 This page discusses why count models are necessary in certain applications, and
 discusses beginning details of the Poisson, negative binomial, and hurdle models.
 
@@ -14,34 +16,178 @@ The result is a linear equation of a vector $\beta$ that describes the relations
 between each element of $X$ and the outcome $y$.
 
 $$y = X\beta$$
+
 This regression framework assumes that $y$ is a continuous variable, meaning
 that it can take any numeric value within a particular range. The plot below
-shows the relationship between the distance between home and workplace on the
-$x$ axis, and the total miles driven on all vehicle trips on the $y$ axis,
-for a sample of 5,000 reported car commuters who responded to the 2017 NHTS.
+shows the relationship between the average distance between home and workplace for
+workers in the household on the
+$x$ axis, and the household VMT on the $y$ axis,
+for households in smaller cities who responded to the 2017 NHTS.
 Both of these variables are continuous, meaning that a simple $y = X\beta$
 regression model is appropriate, though more information might need to be
 added to the model below to improve its fit and help explain outlying observations
 or control for heteroskedasticity.
 
 
-![](count_continuous-1.png)<!-- -->
+![Household VMT versus distance to work.](countmodels_continuous-1.png)<!-- -->
 
-But consider the plot below, showing the distance between home and work on the $x$ axis and
-the number of vehicles owned by the commuter's household on the $y$ axis. Because
-the number of vehicles is discrete and not continuous, the plot looks kind of
-funny. But more importantly than this, we want a model that will predict
-a discrete number of vehicles as an outcome variable, and the blue regression
-line we estimated below will predict between 2.5 and 3.2 vehicles per household;
-this isn't ideal.
+But consider the plot below, showing the same $x$ axis but with
+the number of home-based work trips produced by each household on the $y$ axis.
+Because the number of trips is discrete and not continuous, the plot looks kind
+of funny. But more importantly than this, we want a model that will predict a
+discrete number of vehicles as an outcome variable, and the blue regression line
+we estimated below will predict between 2 and 1 trips household; this
+isn't ideal.
 
 
-![](count_discrete-1.png)<!-- -->
+
+![Household work trips versus distance to work.](countmodels_discrete-1.png)<!-- -->
+
 
 ## Poisson Model
 
+A better option would be to predict a probability that each household will produce
+a certain discrete number of trips. One way to do this is with a Poisson
+regression model. In this model, an analyst predicts the mean of a Poisson
+distribution with a regression equation (instead of a line). The
+Poisson distribution is:
+
+$$P(k | \lambda_i) = \frac{\lambda_i^ke^{-\lambda_i}}{k!}$$
+
+where the probability of a discrete outcome $k$ is determined by the mean
+$\lambda_i$ of the distribution. The plot below shows how as the mean increases,
+the probability of higher outcomes increases.
+
+
+
+![Poisson distribution at different levels of mean.](countmodels_poissondist-1.png)<!-- -->
+
+A Poisson regression model allows attributes of an observation to affect the
+value of the mean. So instead of $y = X\beta$ in the linear regression, we now
+have $\lambda = X\beta$, and $\lambda$ gets put into the distribution equation
+above. In the model below, average work distance decreases the average number of
+trips, but more workers and more vehicles increases the average number.
+
+
+<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
+ <thead>
+  <tr>
+   <th style="text-align:left;">   </th>
+   <th style="text-align:center;"> Linear </th>
+   <th style="text-align:center;"> Poisson </th>
+  </tr>
+ </thead>
+<tbody>
+  <tr>
+   <td style="text-align:left;"> (Intercept) </td>
+   <td style="text-align:center;"> -0.106 </td>
+   <td style="text-align:center;"> -0.570*** </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;">  </td>
+   <td style="text-align:center;"> (-1.625) </td>
+   <td style="text-align:center;"> (-13.865) </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> avg_workdist </td>
+   <td style="text-align:center;"> -0.010*** </td>
+   <td style="text-align:center;"> -0.006*** </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;">  </td>
+   <td style="text-align:center;"> (-6.851) </td>
+   <td style="text-align:center;"> (-7.070) </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> wrkcount </td>
+   <td style="text-align:center;"> 1.168*** </td>
+   <td style="text-align:center;"> 0.665*** </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;">  </td>
+   <td style="text-align:center;"> (29.000) </td>
+   <td style="text-align:center;"> (28.801) </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> hhvehcnt </td>
+   <td style="text-align:center;"> 0.147*** </td>
+   <td style="text-align:center;"> 0.089*** </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;box-shadow: 0px 1px">  </td>
+   <td style="text-align:center;box-shadow: 0px 1px"> (5.626) </td>
+   <td style="text-align:center;box-shadow: 0px 1px"> (5.925) </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> Num.Obs. </td>
+   <td style="text-align:center;"> 5533 </td>
+   <td style="text-align:center;"> 5533 </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> R2 </td>
+   <td style="text-align:center;"> 0.185 </td>
+   <td style="text-align:center;">  </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> R2 Adj. </td>
+   <td style="text-align:center;"> 0.185 </td>
+   <td style="text-align:center;">  </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> AIC </td>
+   <td style="text-align:center;"> 19111.6 </td>
+   <td style="text-align:center;"> 17871.7 </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> BIC </td>
+   <td style="text-align:center;"> 19144.7 </td>
+   <td style="text-align:center;"> 17898.1 </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> Log.Lik. </td>
+   <td style="text-align:center;"> -9550.813 </td>
+   <td style="text-align:center;"> -8931.830 </td>
+  </tr>
+  <tr>
+   <td style="text-align:left;"> F </td>
+   <td style="text-align:center;"> 419.097 </td>
+   <td style="text-align:center;">  </td>
+  </tr>
+</tbody>
+</table>
+
+Of course, this is just an average. In a trip-based model, this average for each
+household might be sufficient. But you could also simulate a discrete choice
+for each person. The plot below shows the probability of a certain number of trips
+made by a sample of households, alongside what the predicted Poisson mean was.
+Households with a higher predicted mean have a higher probability of making more
+trips.
+
+
+
+![Poisson regression model outcomes.](countmodels_outcomes.png)<!-- -->
+
+
 
 ### Negative Binomial Model
+The Poisson model assumes that the mean and standard deviation of the distribution
+are the same. This can be a bad assumption, because it forces the distribution to
+spread out when the mean is higher. The negative binomial model relaxes this assumption,
+and might be useful in some contexts.
+
+### Hurdle Model
+The Poisson and negative binomial models assume the same distribution across
+all outcomes; this might not be desirable if the number of zeros is high or
+low for some structural reason. For example, owning zero vehicles is very different
+from owning one or two. A hurdle model breaks the distribution into two different
+components:
+
+  - A binomial model determines the probability of choosing zero versus a positive
+  number.
+  - A poisson or negative binomial model (with zero removed) determines the
+  probability of a specific positive number, conditioned on the previous model.
+
 
+## References
 
-## Hurdle Model
+ 1. [UCLA Stats](https://stats.idre.ucla.edu/r/dae/poisson-regression/)