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136 5. MULTIVARIATE LINEAR MODELSYou can interpret these estimates as saying:Once we know median age at marriage for a State, there is little or no additionalpredictive power in also knowing the rate of marriage in that State.Note that this does not mean that there is no value in knowing marriage rate. If you didn’thave access to age-at-marriage data, then you’d definitely find value in knowing the marriagerate.But how did the model achieve this result? To answer that question, we’ll draw somepictures.5.1.3. Plotting multivariate posteriors. Visualizing posterior distribution in simple bivariateregressions (the previous chapter) is easy. ere’s only one predictor variable, so a singlescatterplot can convey a lot of information. And so in the previous chapter we used scattersof the data. en we overlaid regression lines and intervals to both (1) visualize the size ofthe association between the predictor and outcome and (2) to get a crude sense of the abilityof the model to predict the individual observations.With multivariate regression, you’ll need more plots. ere is a huge literature detailinga variety of plotting techniques that all attempt to help one understand multiple linear regression.None of these techniques is suitable for all jobs, and most do not generalize beyondlinear regression. So the approach I take here is to instead help you compute whatever youneed from the model. I offer three types of interpretive plots:(1) Predictor residual plots. ese plots show the outcome against residual predictorvalues.(2) Counterfactual plots. ese show the implied predictions for imaginary experimentsin which the different predictor variables can be changed independently ofone another.(3) Posterior prediction plots. ese show model-based predictions against raw data,or otherwise display the error in prediction.Each of these plot types has its advantages and deficiencies, depending upon the context andthe question of interest. In the rest of this section, I show you how to manufacture each ofthese in the context of the divorce data.5.1.3.1. Predictor residual plots. A predictor variable residual is the average predictionerror when we use all of the other predictor variables to model a predictor of interest. at’sa complicated concept, so we’ll go straight to the example, where it will make sense. ebenefit of computing these things is that, once plotted against the outcome, we have a bivariateregression of sorts that has already “controlled” for all of the other predictor variables. Itjust leaves in the variation that is not expected by the model of the mean, µ, as a function ofthe other predictors.In our multivariate model of divorce rate, we have two predictors: (1) marriage rate(Marriage.s) and (2) median age at marriage (MedianAgeMarriage.s). To compute predictorresiduals for either, we just use the other predictor to model it. So for marriage rate,
5.1. SPURIOUS ASSOCIATION 137this is the model we need:m i ∼ Normal(µ i , σ)µ i = α + βa iα ∼ Normal(0, 10)β ∼ Normal(0, 1)σ ∼ Uniform(0, 10)As before, m is marriage rate and a is median age at marriage. Note that since we standardizedboth variables, we already expect the mean α to be around zero. So I’ve centered α’s priorthere, but it’s still so flat that it hardly matters.is code will fit the model:m5.4
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Statistical RethinkingA BAYESIAN CO
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4 CONTENTS5.5. Ordinary least squar
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PrefaceMasons, when they start upon
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HOW TO USE THIS BOOK 9least a minor
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HOW TO USE THIS BOOK 11than initial
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1 e Golem of PragueIn the 16th cent
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16 1. THE GOLEM OF PRAGUEconverses
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1.2. WRECKING PRAGUE 19model P 1B ,
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1.2. WRECKING PRAGUE 21e dominant r
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1.3. THREE TOOLS FOR GOLEM ENGINEER
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2 Small Worlds and Large WorldsWhen
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2.1. PROBABILITY IS JUST COUNTING 3
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2.2. COLOMBO’S FIRST BAYESIAN MOD
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W L W W W L W L W2.2. COLOMBO’S F
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2.3. COMPONENTS OF THE MODEL 41not
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2.3. COMPONENTS OF THE MODEL 43You
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2.3. COMPONENTS OF THE MODEL 45e ma
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2.4. MAKING THE MODEL GO 47other pr
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2.4. MAKING THE MODEL GO 495 points
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2.4. MAKING THE MODEL GO 51just the
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2.4. MAKING THE MODEL GO 53between
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2.6. PRACTICE 55Medium.2.6.5. m1. R
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2.6. PRACTICE 57implied by the equa
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3 Sampling the ImaginaryLots of boo
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3. SAMPLING THE IMAGINARY 61a probl
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3.2. SAMPLING TO SUMMARIZE 63plot(
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3.2. SAMPLING TO SUMMARIZE 65Densit
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3.2. SAMPLING TO SUMMARIZE 67In con
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3.2. SAMPLING TO SUMMARIZE 69Densit
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3.3. SAMPLING TO SIMULATE PREDICTIO
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3.5. PRACTICE 793.5. PracticeEasy.
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3.5. PRACTICE 813.5.17. Predict sec
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84 4. LINEAR MODELSFIGURE 4.1. e Pt
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6.2. INFORMATION THEORY AND MODEL P
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6.3. AKAIKE INFORMATION CRITERION 1
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6.4. DEVIANCE INFORMATION CRITERION
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6.5. USING AIC 203helps guard again
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6.5. USING AIC 205compare( m6.11 ,
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6.5. USING AIC 207e attitude this b
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6.5. USING AIC 209kcal.per.g0.5 0.7
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6.7. PRACTICE 211Consider by analog
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6.7. PRACTICE 2136.7.3. e deviance
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216 7. INTERACTIONSFIGURE 7.1. TOP:
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218 7. INTERACTIONSlog(rgdppc_2000)
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220 7. INTERACTIONSird, we may want
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222 7. INTERACTIONSlog GDP year 200
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226 7. INTERACTIONSInteraction mode
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228 7. INTERACTIONSthis model and t
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232 7. INTERACTIONSe main effect li
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234 7. INTERACTIONSbs -38.91 34.94s
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236 7. INTERACTIONSe primary reason
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238 7. INTERACTIONSNow for the plot
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240 7. INTERACTIONS7.4. Higher-orde
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8 Markov Chain Monte Carlo Estimati
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8.1. GOOD KING MARKOV AND HIS ISLAN
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8.2. MARKOV CHAIN MONTE CARLO 247fo
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8.2. MARKOV CHAIN MONTE CARLO 249No
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8.3. EASY HMC: MAP2STAN 251We’re
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8.3. EASY HMC: MAP2STAN 253$ cont_a
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8.3. EASY HMC: MAP2STAN 255mcmcpair
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8.3. EASY HMC: MAP2STAN 257FIGURE 8
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8.4. CARE AND FEEDING OF YOUR MARKO
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8.6. PRACTICE 265And since the chai
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268 9. BIG ENTROPY AND THE GENERALI
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270 9. BIG ENTROPY AND THE GENERALI
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10 Distance and DurationA curious t
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10.2. GAMMA 275can be quite complic
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278 11. COUNTING AND CLASSIFICATION
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304 12. MONSTERS AND MIXTUREShere?
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306 12. MONSTERS AND MIXTURESdensit
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308 12. MONSTERS AND MIXTURESfootbr
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310 12. MONSTERS AND MIXTURESWhy di
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312 12. MONSTERS AND MIXTURESm11.3
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314 12. MONSTERS AND MIXTURESResult
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316 12. MONSTERS AND MIXTURESanswer
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318 12. MONSTERS AND MIXTURESDensit
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320 12. MONSTERS AND MIXTURESBut we
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322 12. MONSTERS AND MIXTURESdisper
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324 12. MONSTERS AND MIXTURESe resu
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326 12. MONSTERS AND MIXTURESDensit
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328 12. MONSTERS AND MIXTURESR code
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330 12. MONSTERS AND MIXTURESPoisso
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332 12. MONSTERS AND MIXTURESthe ga
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334 12. MONSTERS AND MIXTURESdata=d
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336 12. MONSTERS AND MIXTURESma2 ~
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338 13. MULTILEVEL MODELSrepeat obs
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340 13. MULTILEVEL MODELSSo how do
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342 13. MULTILEVEL MODELSprobabilit
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344 13. MULTILEVEL MODELSa conseque
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346 13. MULTILEVEL MODELSCompute th
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348 13. MULTILEVEL MODELSabsolute e
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350 13. MULTILEVEL MODELS)sigma_act
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14 Multilevel Models II: Slopes14.1
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14.1. EVERYTHING CAN VARY AND PROBA
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14.1. EVERYTHING CAN VARY AND PROBA
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360 15. MISSING DATA AND OTHER OPPO
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16 Writing Statistics379
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382 ENDNOTES12. For an autopsy of t
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384 ENDNOTES45. Fisher (1925), in C
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386 ENDNOTES77. See two famous edit
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388 ENDNOTES113. Hurlbert (1984) is
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390 BibliographyFrank, S. A. (2011)
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392 BibliographyProulx, S. R. and A
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IndexAkaike (1973), 380, 383Akaike
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INDEX 397non-identifiability, 153Oc
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