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234 7. INTERACTIONSbs -38.91 34.94sigma 57.34 46.23bws NA -39.5nobs 27 27Now consider these estimates and try to figure out what the models are telling us about theinfluence of water and shade on the blooms. First, consider the intercepts a, α. e estimateof the intercept changes a lot from one model to the next, from 53 to −84. What do thesevalues mean? Remember, the intercept is the expected value of the outcome when all of thepredictor variables take the value zero. In this case, neither of the predictor variables evertakes the value zero within the data. As a result, these intercept estimates are very hard tointerpret. is is a very common issue, and most of the regression examples so far in thisbook have suffered from it.Now, consider the slope parameters. In the main-effect-only model, m7.6, the MAPvalue for the main effect of water is positive and the main effect for shade is negative. Takea look at the standard deviations and intervals in precis(m7.6) to verify that both posteriordistributions are reliably on one side of zero. You might infer that these posterior distributionssuggest that water increases blooms while shade reduces them. For every additionallevel of soil moisture, blooms increase by 76, on average. For every addition unit of shade,blooms decrease by 42, on average. ose sound reasonable.But the analogous posterior distributions from the interaction model, m7.7, are quitedifferent. First, assure yourself that the interaction model is indeed a much better model:R code7.22compare(m7.6,m7.7)DIC pD dDIC weightm7.7 293.41 4.87 0.00 0.99m7.6 303.88 4.29 10.47 0.01is is a slam dunk for m7.7. So let’s seriously consider the posterior distribution from m7.7.Now both main effects are positive, but the new interaction posterior mean is negative. Areyou to conclude now that the main effect of shade is to help the tulips? And the negativeinteraction itself implies that as shade increases, water has a reduced impact on blooms. Butreduced by how much?Sampling from the posterior now and plotting the model’s predictions would help immenselywith interpretation. And that’s the course I want to encourage and provide codefor. In general, I don’t think it’s ever safe to interpret interactions without plotting them.But for the moment, let’s instead look at the value of centering the predictor variables andre-estimating the models.7.3.3. Center and re-estimate. To center a variable means to create a new variable that containsthe same information as the original, but has a new mean of zero. For example, tomake centered versions of shade and water, just subtract the mean of the original fromeach value:R code7.23d$shade.c
7.3. CONTINUOUS INTERACTIONS 235e new centered variable has the same variance as the original, but now has a mean of zero.In this case, centering recodes the levels of water and shade so that instead of their rangingfrom 1 to 3, they now range from −1 to 1.Centering in this analysis will do two things for us. First, it’ll fix our previous problemwith maximum iterations. Second, it’ll make the estimates easier to interpret. Let’sre-estimate the two regression models, but now using the new centered variables shade.cand water.c. I’m also going to remove the optim fixes from before, because now we don’tneed them.m7.8
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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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116 4. LINEAR MODELSYou end up with
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118 4. LINEAR MODELS(1) Use link to
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122 4. LINEAR MODELSRethinking: Lin
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126 4. LINEAR MODELS4.7.1. e1. In t
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5 Multivariate Linear ModelsOne of
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5.1. SPURIOUS ASSOCIATION 131Divorc
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5.1. SPURIOUS ASSOCIATION 133But me
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5.1. SPURIOUS ASSOCIATION 135abmbas
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5.1. SPURIOUS ASSOCIATION 137this i
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5.1. SPURIOUS ASSOCIATION 139slower
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5.1. SPURIOUS ASSOCIATION 141Median
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5.1. SPURIOUS ASSOCIATION 143(a)(b)
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5.2. MASKED RELATIONSHIP 145N
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5.2. MASKED RELATIONSHIP 147a ~ dno
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5.2. MASKED RELATIONSHIP 149dcc$log
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5.3. WHEN ADDING VARIABLES HURTS 15
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5.4. CATEGORICAL VARIABLES 161$ wei
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5.4. CATEGORICAL VARIABLES 1635.4.2
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5.4. CATEGORICAL VARIABLES 165# sam
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168 5. MULTIVARIATE LINEAR MODELS5.
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170 5. MULTIVARIATE LINEAR MODELS5.
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6 Model Selection, Comparison, and
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6.1. THE PROBLEM WITH PARAMETERS 17
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284 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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