statisticalrethinkin..

106 4. LINEAR MODELS(1) What is the expected height, when x i = 0? e parameter α answers this question.For this reason, α is oen called the intercept.(2) What is the change in expected height, when x i changes by 1 unit? e parameterβ answers this question.Jointly these two parameters, together with x, ask the golem to find a line that relates x to h,a line that passes though α when x i = 0 and has slope β. at is a task that golems are verygood at. It’s up to you, though, to be sure it’s a good question.Rethinking: Nothing special or natural about linear models. Note that there’s nothing special aboutthe linear model, really. You can choose a different relationship between α and β and µ. For example,the following is a perfectly legitimate definition for µ i :µ i = α exp(−βx i ).is does not define a linear regression, but it does define a regression model. e linear relationshipwe are using instead is conventional, but nothing requires that you use it. It is very common in somefields, like ecology and demography, to use functional forms for µ that come from theory, rather thanthe geocentrism of linear models.Overthinking: Units and regression models. Readers who had a traditional training in physicalsciences will know how to carry units through equations of this kind. For their benefit, here’s themodel again (omitting priors for brevity), now with units of each symbol added.h i cm ∼ Normal(µ i cm, σcm)µ i cm = αcm + β cmkg x ikgSo you can see that β must have units of cm/kg in order for the mean µ i to have units cm. One of thefacts that labeling with units clears up is that a parameter like β is a kind of rate.4.4.1.3. Priors. e remaining lines in the model define priors for the parameters to beestimated: α, β, and σ. All of these are weak priors, leading to inferences that will echo non-Bayesian methods of model fitting, such as maximum likelihood. But as always, you shouldplay around with the priors—plotting them, changing them and refitting the model—to geta sense of their influence.You’ve seen priors for α and σ before, although α was called µ back then. I’ve widenedthe prior for α, since as you’ll see it is common for the intercept in a linear model to swinga long way from the mean of the outcome variable. e flat prior here with a huge standarddeviation will allow it to move wherever it needs to.e prior for β deserves explanation. Why have a Gaussian prior with mean zero? isprior places just as much probability below zero as it does above zero, and when β = 0,weight has no relationship to height. So many people see it as conservative assumption. Andsuch a prior will pull probability mass towards zero, leading to more conservative estimatesthan a perfectly flat prior will. So many people find such a prior appealing for that reason.But note that a Gaussian prior with standard deviation of 10 is still very weak, so theamount of conservatism it induces will be very small. As you make the standard deviation inthis prior smaller, the amount of shrinkage towards zero increases and your model producesmore and more conservative estimates about the relationship between height and weight.