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literal 0 HcmV?d00001 diff --git a/topics/Count_Models.md b/topics/Count_Models.md new file mode 100644 index 0000000000..2a4b79f396 --- /dev/null +++ b/topics/Count_Models.md @@ -0,0 +1,47 @@ +--- +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. + +## Continuous versus count outcomes +Typical [regression models](Regression_Analysis) are aimed at predicting the +response of an outcome variable $y$ to a series of input variables $X = [x_1, x_2, \ldots x_p]$. +The result is a linear equation of a vector $\beta$ that describes the relationship +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. +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) + +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. + + +![](count_discrete-1.png) + +## Poisson Model + + +### Negative Binomial Model + + +## Hurdle Model