statisticalrethinkin..

5.1. SPURIOUS ASSOCIATION 133But merely comparing parameter means between different bivariate regressions is noway to decide which predictor is better. Both of these predictors could provide independentvalue, or they could be redundant, or one could eliminate the value of the other. So we’llbuild a multivariate model with the goal of measuring the partial value of each predictor.e question we want answered is:What is the predictive value of a variable, once I already know all of the otherpredictor variables?So for example once you fit a multivariate regression to predict divorce using both marriagerate and age at marriage, the model answers the questions:(1) Aer I already know marriage rate, what additional value is there in also knowingage at marriage?(2) Aer I already know age at marriage, what additional value is there in also knowingmarriage rate?e parameter estimates corresponding to each predictor are the (oen opaque) answers tothese questions. Next we’ll fit the model that asks these questions.Rethinking: “Control” is out of control. Very oen, the question just above is spoken of as “statisticalcontrol,” as in controlling for the effect of one variable while estimating the effect of another. But this issloppy language, as it implies too much. It implies a causal interpretation (“effect”), and it implies anexperimental disassociation of the predictor variables (“control”). You may be willing to make theseassumptions, but they are not part of the model, so be wary.e point here isn’t to police language. Instead, the point is to observe the distinction betweensmall world and large world interpretations. Since most people who use statistics are not statisticians,sloppy language like “control” can promote a sloppy culture of interpretation. Such cultures tend tooverestimate the power of statistical methods, so resisting them can be difficult. Disciplining yourown language may be enough. Disciplining another’s language is hard to do, without seeming like afastidious scold, as this very box must seem.5.1.1. Multivariate notation. Multivariate regression formulas look a lot like the polynomialmodels at the end of the previous chapter—they add more parameters and variables tothe definition of µ i . e strategy is straightforward:(1) Nominate the predictor variables you want in the linear model of the mean.(2) For each predictor, make a parameter that will measure its association with theoutcome.(3) Multiply the parameter by the variable and add that term to the linear model.Examples are always necessary, so here is the model the predicts divorce rate, using bothmarriage rate and age at marriage.D i ∼ Normal(µ i , σ)[likelihood]µ i = α + β m m i + β a a i [linear model]α ∼ Normal(10, 10) [prior for α]β m ∼ Normal(0, 1) [prior for β m ]β a ∼ Normal(0, 1) [prior for β a ]σ ∼ Uniform(0, 10) [prior for σ]