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336 12. MONSTERS AND MIXTURESma2 ~ dnorm(0,10),pa ~ dnorm(0,10),pa2 ~ dnorm(0,10),s ~ dcauchy(0,2.5)) ,data=d2 ,start=list(ah=mean(log(d2$harvest)),af=0,ma=0,ma2=0,pa=0,pa2=0,s=1) ,constraints=list(s="lower=0") )precis(m12.13stan)Mean StdDev lower 0.95 upper 0.95ah 1.95 0.01 1.93 1.96af -0.16 0.02 -0.21 -0.12ma 0.00 0.01 -0.02 0.01ma2 -0.05 0.01 -0.07 -0.04pa -0.01 0.02 -0.04 0.03pa2 0.26 0.02 0.23 0.29s 3.01 0.05 2.89 3.10From just the numbers, we can say that harvests decline with age (negative ma2), while failuresincrease with age (positive pa2). For a more detailed view, we’ll need to compute posteriorpredictions.
13 Multilevel ModelsIntro idea: Case of Clive Wearing, as described by his wife Deborah in book ForeverToday. Wearing is a conductor and pianist who acquired anterograde amnesia aer a caseof Herpes simplex encephalitis. He can still play the piano and conduct, but he cannot formnew long-term memories. He is always forgetting recent events.Statistical models that are not multilevel are similar, in the sense that as they move fromone cluster (individual, group) in the data to another, trying to estimate parameters for eachcluster, they forget everything about the previous clusters. ey behave this way, becausethe model assumptions force them to, assuming that cluster intercepts and slopes containno joint information. is would be like forgetting you had ever been in a coffee shop, eachtime you go to the coffee shop. Coffee shops do vary, but information about one does helpyour expectations about the others. As it is in life, it is in statistics: anterograde amnesia isbad for learning about the world. We want models that instead use all of the information insavvy ways. Multilevel models do that.13.1. What are multilevel models good for?e basic reason to use a multilevel model is that your data are clustered somehow. Byclustered, I mean observations (cases) within different groups differ in their averages, resultingin correlations within groups. As a result, describing these differences can improveprediction and inference. ere are two common forms of clustering: (1) repeat observationsof the same entities and (2) observations of separate entities that are associated temporally,spatial or causally. In my mind, these are really the same thing, but it’s worth providingexamples of each.Repeat observations arise when we observe the same individual (person, tadpole, chimpanzee)multiple times. is may arise because we use the same individuals in multiple treatmentsof an experiment (within subjects design). Or it may arise because some individualssimply appear more oen by chance. Another common way to realize repeat observationsis with a time series, in which each individual appears once at each time step, but manytimes within the data. In all of these contexts, heterogeneity among individuals will lead toa particular pattern of variation between observations: variation will tend to cluster, if onlya little, by individual.e entities that exhibit clustering could just as easily be days of the week, cities, orstreams. We might observe individuals only once, but many individuals on common daysand in common places. Observations of all individuals on Monday, for example, may cluster,because of unobserved causal influences on that day. Likewise, observations of all individualsfrom a common stream or other location may also cluster. In this way, we actually have337
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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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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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Page 338 and 339: 338 13. MULTILEVEL MODELSrepeat obs
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Page 355 and 356: 14.1. EVERYTHING CAN VARY AND PROBA
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Page 360 and 361: 360 15. MISSING DATA AND OTHER OPPO
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Page 382 and 383: 382 ENDNOTES12. For an autopsy of t
Page 384 and 385: 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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