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134 5. MULTIVARIATE LINEAR MODELSYou can use whatever symbols you like for the parameters and variables, but here I’ve chosenm for marriage rate and a for age at marriage, reusing these symbols as subscripts for thecorresponding parameters. But feel free to use whichever symbols reduce the load on yourown memory.So what does it mean to assume µ i = α + β m m i + β a a i ? It means that the expectedoutcome for any State with marriage rate m i and median age at marriage a i is the sum of threeindependent terms. e first term is a constant, α. Every State gets this. e second term isthe product of the marriage rate, m i , and the coefficient, β m , that measures the associationbetween marriage rate and divorce rate. e third term is similar, but for the associationwith median age at marriage instead.If you are like most people, this is still pretty mysterious. So it might help to read the+ symbols as “or” and then say: a State’s divorce rate can be a function of either its marriagerate or its median age at marriage. e “or” indicates independent associations, which maybe purely statistical or rather causal.Overthinking: Compact notation and the design matrix. Oen, linear models are written using acompact form like:n∑µ i = α + β j x jiwhere j is an index over predictor variables and n is the number of predictor variables. e samemodel may also be abbreviated:j=1µ i = α + β 1 x 1i + β 2 x 2i + ... + β n x niBoth of these forms may be read as the mean is a modeled as the sum of an intercept and an additivecombination of the products of parameters and predictors. Even more compactly, using matrix notation:m = Xbwhere m is a vector of predicted means, one for each row in the data, b is a (column) vector of parameters,one for each predictor variable, and X is a matrix. is matrix is called a design matrix. It hasas many rows as the data, and as many columns as there are predictors plus one. So X is basically adata frame, but with an extra first column. e extra column is filled with 1s. ese 1s are multipliedby the first parameter, which is the intercept, and so return the unmodified intercept. When X ismatrix-multiplied by b, you get the predicted means. In R notation, this operation is X %*% b.We’re not going to use the design matrix approach in this book. And in general you don’t needto. But it’s good to recognize it, and sometimes it can save you a lot of work. For example, for linearregressions, there is a nice matrix formula for the maximum likelihood (or least squares) estimates.Most statistical soware exploits that formula, which requires using a design matrix.5.1.2. Fitting the model. To fit this model to the divorce data, we just expand the linearmodel. Here’s the model definition again, now with the code on the righthand side:D i ∼ Normal(µ i , σ)Divorce ~ dnorm(mu,sigma)µ i = α + β m m i + β a a i mu
5.1. SPURIOUS ASSOCIATION 135abmbasigma-2 0 2 4 6 8 10EstimateFIGURE 5.3. Quadratic approximate posterior for model m5.3. Points showeach posterior mode, and solid black line segments show percentile intervals.And here is the map fitting code:m5.3
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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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6.2. INFORMATION THEORY AND MODEL P
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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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