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364 15. MISSING DATA AND OTHER OPPORTUNITIESMarriage rate estimated - observed-1.0 -0.5 0.00.5 1.5 2.5 3.5Marriage rate standard errorDivorce rate (posterior)4 6 8 10 12 1415 20 25 30Marriage rate (posterior)FIGURE 15.3. Le: Shrinkage for predictor marriage rate. Notice thatshrinkage is not balanced, but rather that the model believes the observedvalues tended to be overestimates. Right: Shrinkage of both the outcome,divorce rate, and marriage rate. Solid points are the posterior means. Openpoints are observed. Lines connect pairs of points for the same State.divorce_obs=d$Divorce,divorce_sd=d$Divorce.SE,marriage_obs=d$Marriage,marriage_sd=d$Marriage.SE,MAM=d$MedianAgeMarriage)start
15.2. MISSING DATA 365information in divorce rate to help us improve estimates of marriage rate. In contrast, sincethe relationship between divorce and median age at marriage is strong, there’s a lot of informationin age at marriage to help us improve estimates of divorce rate. at’s why divorceestimates shrink more than marriage rate estimates do.Of course, that information in age at marriage may be illusory, as we haven’t incorporateduncertainty about age at marriage. If we had those data, in the form of standard errorsor better yet a distribution of ages at marriage in each State, then we could blend that informationinto the model using the same approach as above.e big take home point for this section is that when you have a distribution of values,don’t reduce it down to a single value to use in a regression. Instead, use the entire distribution.15.2. Missing dataWith measurement error, the insight is to realize that any datum can be replaced by a distributionthat reflects uncertainty. at distribution comprises a prior that can be updatedby the model, improving estimates of data while simultaneously estimating coefficients. Informationactually flows among the cases, through the regression model, allowing each measurementto borrow strength from the others.Sometimes however data are simply missing—no measurement is available at all. Atfirst, this seems like a lost cause. What can be done when there is no measurement at all, noteven one with error? A lot can be done, because just like information flows among cases inthe measurement error example, it can also flow from present data to missing data, as longas we are willing to make a model of the whole variable.15.2.1. Baldini’s example here. Case of the giant banker?15.2.2. Imputing neocortex. Primate milk example. Back in Chapter 5, we used data(milk)to illustrate masking, using both neocortex percent and body mass to predict milk energy.One aspect of those data are 12 missing values in the neocortex.perc column. We useda COMPLETE-CASE analysis back then, dropping those 12 cases and the 12 good body massand milk energy values in them. at le us with only 17 cases to work with.Now we’ll see how to use Bayesian IMPUTATION to conserve and use the informationin the other columns, while producing estimates for the missing values. e estimates wederive for the missing values won’t be particularly crisp—they will have wide posteriors. Butthey will be honest, given the assumptions built into the model.What we’re going to do is build an example of MCAR (Missing Completely At Random)imputation. With MCAR, we assume that the location of the missing values is completelyrandom with respect to those values and all other values in the data. is is the simplest andeasiest to understand form of imputation. So it makes sense to start here.e trick to all imputation is the simultaneously model the predictor variable that hasmissing values together with the outcome variable. e not-missing values will produceestimates that comprise a prior for the missing values, and these will then be updated by therelationship between the predictor and outcome. But we have to conceive of the predictornow as a mixed vector of data and parameters. In our case, the variable with missing valuesis neocortex percent. Call it N.N = [0.55, N 2 , N 3 , N 4 , 0.65, 0.65, ..., 0.76, 0.75].
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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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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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Page 338 and 339: 338 13. MULTILEVEL MODELSrepeat obs
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Page 382 and 383: 382 ENDNOTES12. For an autopsy of t
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Page 388 and 389: 388 ENDNOTES113. Hurlbert (1984) is
Page 390 and 391: 390 BibliographyFrank, S. A. (2011)
Page 392 and 393: 392 BibliographyProulx, S. R. and A
Page 395 and 396: IndexAkaike (1973), 380, 383Akaike
Page 397: INDEX 397non-identifiability, 153Oc
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