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170 5. MULTIVARIATE LINEAR MODELS5.7.7. m3. It is sometimes observed that the best predictor of fire risk is the presence of firefighters—States and localities with many fire fighters also have more fires. Presumably firefighters do not cause fires. Nevertheless, this is not a spurious correlation. Instead fires causefire fighters. Consider the same reversal of causal inference in the context of the divorce andmarriage data. How might a high divorce rate cause a higher marriage rate? Can you thinkof a way to evaluate this relationship, using multiple regression?5.7.8. m4. In the divorce data, States with high numbers of Mormons (members of eChurch of Jesus Christ of Latter-day Saints) have much lower divorce rates than the regressionmodels expected. Find a list of Mormon population by State and use those numbersas a predictor variable, predicting divorce rate using marriage rate, median age at marriage,and percent Mormon population (possibly standardized). You may want to consider transformationsof the raw percent Mormon variable. [I actually don’t know how this regressionturns out. I suspect getting it right requires interacting Mormon population with medianage marriage, which will have to wait until Chapter 7.]5.7.9. m5. One way to reason through multiple causation hypotheses is to imagine detailedmechanisms through which predictor variables may influence outcomes. For example, it issometimes argued that the price of gasoline (predictor variable) is positively associated withlower obesity rates (outcome variable). However, there are at least two important mechanismsby which the price of gas could reduce obesity. First, it could lead to less driving andtherefore more exercise. Second, it could lead to less driving, which leads to less eating out,which leads to less consumption of huge restaurant meals. Can you outline one or more multipleregressions that address these two mechanisms? Assume you can have any predictordata you need.HardAll three exercises below use the same data, data(foxes) (part of rethinking). 69 eurban fox (Vulpes vulpes) is a successful exploiter of human habitat. Since urban foxes movein packs and defend territories, data on habitat quality and population density is also included.e data frame has five columns:(1) group: Number of the social group the individual fox belongs to(2) avgfood: e average amount of food available in the territory(3) groupsize: e number of foxes in the social group(4) area: Size of the territory(5) weight: Body weight of the individual fox5.7.10. Bivariate entrée. Fit two bivariate Gaussian regressions, using map: (1) body weightas a linear function of territory size (area), and (2) body weight as a linear function ofgroupsize. Plot the results of these regressions, displaying the MAP regression line and the95% confidence interval of the mean. Is either variable important for predicting fox bodyweight?5.7.11. Multivariate course. Now fit a multiple linear regression with weight as the outcomeand both area and groupsize as predictor variables. Plot the predictions of the modelfor each predictor, holding the other predictor constant at its mean. What does this modelsay about the importance of each variable? Why do you get different results than you got inexercise just above?
HARD 1715.7.12. Model comparison dessert. Finally, consider the avgfood variable. Fit two moremultiple regressions: (1) body weight as an additive function of avgfood and groupsize,and (2) body weight as an additive function of all three variables, avgfood and groupsizeand area. Compare the results of these models to the previous models you’ve fit, in the firsttwo exercises.(a) Is avgfood or area a better predictor of body weight? If you had to choose one orthe other to include in a model, which would it be? Support your assessment with any tablesor plots you choose.(b) When both avgfood or area are in the same model, their effects are reduced (closerto zero) and their standard errors are larger than when they are included in separate models.Can you explain this result?
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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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86 4. LINEAR MODELSexperiment with
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88 4. LINEAR MODELS4.1.4.2. Epistem
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90 4. LINEAR MODELSe approach above
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92 4. LINEAR MODELS'data.frame': 54
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94 4. LINEAR MODELSe point isn’t
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96 4. LINEAR MODELShave the samples
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98 4. LINEAR MODELSere’s no troub
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100 4. LINEAR MODELS)Note the comma
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102 4. LINEAR MODELSpreviously obse
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104 4. LINEAR MODELS4.4. Adding a p
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106 4. LINEAR MODELS(1) What is the
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108 4. LINEAR MODELSdata(Howell1)d
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110 4. LINEAR MODELSRethinking: Wha
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112 4. LINEAR MODELSheight140 150 1
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114 4. LINEAR MODELSheight140 150 1
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116 4. LINEAR MODELSYou end up with
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118 4. LINEAR MODELS(1) Use link to
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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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