statisticalrethinkin..

110 4. LINEAR MODELSRethinking: What do parameters mean? A basic issue with interpreting model-based estimates isin knowing the meaning of parameters. ere is no consensus about what an estimate means, however,because different people take different philosophical stances towards models, probability, andprediction. e perspective in this book is a common Bayesian perspective: posterior probabilitiesof parameter values describe the relative compatibility of different states of the world with the data, accordingto the model. ese are small world (Chapter 2) numbers. So reasonable people may disagreeabout the large world meaning, and the details of those disagreements depend strongly upon context.Such disagreements are productive, because they lead to model criticism and revision, something thatgolems cannot do for themselves.4.4.3.1. Tables of estimates. With the new linear regression fit to the Kalahari data, weinspect the estimates:R code4.37precis( m4.3 )Mean StdDev 2.5% 97.5%a 113.89 1.91 110.16 117.63sigma 5.07 0.19 4.70 5.45b 0.90 0.04 0.82 0.99e first row gives the quadratic approximation for α, the second the approximation for σ,and the third approximation for β. Let’s try to make some sense of them, in this very simplemodel.Best to begin from the bottom, with b (β), because it’s the new parameter. Since β isa slope, the value 0.90 can be read as a person 1 kg heavier is expected to be 0.90 cm taller.95% of the posterior probability lies between 0.82 and 0.99. at suggests that β values closeto zero or greatly above one are highly incompatible with these data and this model. If youwere thinking that perhaps there was no relationship at all between height and weight, thenthis estimate becomes strong evidence of a positive relationship instead. But maybe youjust wanted as precise a measurement as possible of the relationship between height andweight. is estimate embodies that measurement, conditional on the model, as always. Fora different model, the measure of the relationship might be different.e estimate of α, a in the precis table, indicates that a person of weight 0 should be114cm tall. is is nonsense, since real people always have positive weight, yet it is also true.Parameters like α are “intercepts” that tell us the value of µ when all of the predictor variableshave value zero. As a consequence, the value of the intercept is frequently uninterpretablewithout also studying any β parameters.Finally, the estimate for σ, sigma, informs us of the width of the distribution of heightsaround the mean. A quick way to interpret it is to recall that about 95% of the probability ina Gaussian distribution lies between two standard deviations. So in this case, the estimatetells us that it is most likely that 95% of heights lie within 10cm (2σ) of the mean height. Butthere is also uncertainty about this, as always. So we can also take note of the fact that the95% percentile interval says that 95% of heights lie within 9.4cm to 10.9cm of the mean. Inother words, 95% of the plausible values of 2σ lie between 9.4 and 10.9.e numbers in the default precis output aren’t sufficient to describe the quadratic posteriorcompletely. For that, we also require the variance-covariance matrix. We’re interestedin correlations among parameters—we already have their variance in the table above—solet’s go straight to the correlation matrix: