statisticalrethinkin..

15.2. MISSING DATA 369kcal per gram0.5 0.6 0.7 0.8 0.9neocortex proportion0.55 0.65 0.750.55 0.60 0.65 0.70 0.75 0.80neocortex proportion-2 -1 0 1 2 3 4log(mass)FIGURE 15.4. Le: Inferred relationship between milk energy (vertical) andneocortex proportion (horizontal), with imputed values shown in blue. Blueline segments are 90% confidence intervals. Right: Inferred relationshipbetween the two predictors, neocortex proportion and log mass. Imputedvalues again shown in blue.So by using all the cases, the strength of the inferred relationships has diminished. is mightmake you sad, but ask yourself whether you would want your colleagues to use all the data,even if it meant inferring a weaker relationship. en apply that same standard to yourself.Let’s do some plotting to visualize what’s happened here. FIGURE 15.4 displays both theinferred relationship between milk energy and neocortex (le) and the relationship betweenthe two predictors (right). Both plots show imputed neocortex values in blue, with 90% confidenceintervals shown by the blue line segments. Although there’s a lot of uncertainty inthe imputed values—hey, Bayesian inference isn’t magic, just logic—they do show a gentletilt towards the regression line. is has happened because the observed values provideinformation that guides the estimation of the missing values.e righthand plot shows the inferred relationship between the predictors. We alreadyknow that these two predictors are positively associated—that’s what creates the maskingproblem. But notice here that the imputed values do not shown an upward slope. eydo not, because the imputation model—the first regression with neocortex (observed andmissing) as the outcome—assumed no relationship. So odds are we can improve this modelby changing the imputation model to estimate the relationship between the two predictors.Let’s do that now. e notion is to change the imputation line of the model from:toN i ∼ Normal(ν, σ N )N i ∼ Normal(ν i , σ N )ν i = α N + γ M log M i