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waffle Chris Parrish June 18, 2016 Contents Waffle House 1 data..................................................... 2 exploratory data analysis......................................... 2 Waffle Houses............................................. 2 Marriage.s.............................................. 3 MedianAgeMarriage.s....................................... 4 MedianAgeMarriage.s and Marriage.s.............................. 5 model divorce rate ~ age at marriage................................... 6 map m5.1............................................... 6 model divorce rate ~ marriage rate..................................... 8 map m5.2............................................... 8 model divorce rate ~ age at marriage and marriage rate......................... 9 map m5.3............................................... 9 model marriage rate ~ age at marriage.................................. 10 map m5.4............................................... 10 plotting multivariate posteriors...................................... 11 predictor residual plots........................................ 11 counterfactual plots......................................... 12 posterior prediction plots...................................... 15 simulating spurious association................................... 17 waffle reference: - McElreath, Statistical Rethinking, chap 5, pp.119-164 Waffle House library(rethinking) ## Loading required package: rstan ## Loading required package: ggplot2 ## rstan (Version 2.9.0-3, packaged: 2016-02-11 15:54:41 UTC, GitRev: 05c3d0058b6a) ## For execution on a local, multicore CPU with excess RAM we recommend calling ## rstan_options(auto_write = TRUE) ## options(mc.cores = parallel::detectcores()) ## Loading required package: parallel ## rethinking (Version 1.58) library(ggplot2) 1

data ## R code 5.1 # load data data(waffledivorce) d <- WaffleDivorce str(d) ## 'data.frame': 50 obs. of 13 variables: ## $ Location : Factor w/ 50 levels "Alabama","Alaska",..: 1 2 3 4 5 6 7 8 9 10... ## $ Loc : Factor w/ 50 levels "AK","AL","AR",..: 2 1 4 3 5 6 7 9 8 10... ## $ Population : num 4.78 0.71 6.33 2.92 37.25... ## $ MedianAgeMarriage: num 25.3 25.2 25.8 24.3 26.8 25.7 27.6 26.6 29.7 26.4... ## $ Marriage : num 20.2 26 20.3 26.4 19.1 23.5 17.1 23.1 17.7 17... ## $ Marriage.SE : num 1.27 2.93 0.98 1.7 0.39 1.24 1.06 2.89 2.53 0.58... ## $ Divorce : num 12.7 12.5 10.8 13.5 8 11.6 6.7 8.9 6.3 8.5... ## $ Divorce.SE : num 0.79 2.05 0.74 1.22 0.24 0.94 0.77 1.39 1.89 0.32... ## $ WaffleHouses : int 128 0 18 41 0 11 0 3 0 133... ## $ South : int 1 0 0 1 0 0 0 0 0 1... ## $ Slaves1860 : int 435080 0 0 111115 0 0 0 1798 0 61745... ## $ Population1860 : int 964201 0 0 435450 379994 34277 460147 112216 75080 140424... ## $ PropSlaves1860 : num 0.45 0 0 0.26 0 0 0 0.016 0 0.44... head(d) ## Location Loc Population MedianAgeMarriage Marriage Marriage.SE Divorce ## 1 Alabama AL 4.78 25.3 20.2 1.27 12.7 ## 2 Alaska AK 0.71 25.2 26.0 2.93 12.5 ## 3 Arizona AZ 6.33 25.8 20.3 0.98 10.8 ## 4 Arkansas AR 2.92 24.3 26.4 1.70 13.5 ## 5 California CA 37.25 26.8 19.1 0.39 8.0 ## 6 Colorado CO 5.03 25.7 23.5 1.24 11.6 ## Divorce.SE WaffleHouses South Slaves1860 Population1860 PropSlaves1860 ## 1 0.79 128 1 435080 964201 0.45 ## 2 2.05 0 0 0 0 0.00 ## 3 0.74 18 0 0 0 0.00 ## 4 1.22 41 1 111115 435450 0.26 ## 5 0.24 0 0 0 379994 0.00 ## 6 0.94 11 0 0 34277 0.00 exploratory data analysis Waffle Houses ggplot(d, aes(x = WaffleHouses / Population, y = Divorce)) + geom_point(shape = 20, color = "darkred") + geom_smooth(method = "lm") + labs(x = "Waffle Houses per million", y = "Divorce rate") 2

14 12 Divorce rate 10 8 6 0 10 20 30 40 Waffle Houses per million Marriage.s Standardize marriage. d$marriage.s <- (d$marriage - mean(d$marriage))/sd(d$marriage) ggplot(d, aes(marriage.s, Divorce)) + geom_point(aes(x = Marriage.s, y = Divorce), shape = 20, color = "darkred") + geom_smooth(method = "lm") + labs(x = "Marriage.s", y = "Divorce") 3

12 Divorce 10 8 6 1 0 1 2 3 Marriage.s M edianagem arriage.s Standardize median age at marriage. # standardize predictor d$medianagemarriage.s <- (d$medianagemarriage-mean(d$medianagemarriage))/ sd(d$medianagemarriage) ggplot(d, aes(medianagemarriage.s, Divorce)) + geom_point(aes(x = MedianAgeMarriage.s, y = Divorce), shape = 20, color = "darkred") + geom_smooth(method = "lm") + labs(x = "MedianAgeMarriage.s", y = "Divorce") 4

12 Divorce 10 8 6 2 1 0 1 2 3 MedianAgeMarriage.s M edianagem arriage.s and M arriage.s How are marriage rate and median age at marriage related? ggplot(d, aes(medianagemarriage.s, Marriage.s)) + geom_point(shape = 20, color = "darkred") + geom_smooth(method = "lm") + labs(x = "MedianAgeMarriage.s", y = "Marriage.s") 5

3 2 1 Marriage.s 0 1 2 3 2 1 0 1 2 3 MedianAgeMarriage.s model divorce rate ~ age at marriage. D i Normal(µ i, σ) µ i = α + β A A i α Normal(10, 10) β A Normal(0, 1) σ Uniform(0, 10) map m5.1 # fit model m5.1 <- map( alist( Divorce ~ dnorm( mu, sigma ), mu <- a + ba * MedianAgeMarriage.s, a ~ dnorm( 10, 10 ), ba ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data = d ) precis(m5.1) ## Mean StdDev 5.5% 94.5% ## a 9.69 0.20 9.36 10.01 6

## ba -1.04 0.20-1.37-0.72 ## sigma 1.45 0.14 1.22 1.68 plot( precis(m5.1) ) a ba sigma 0 2 4 6 8 10 Value ## R code 5.2 # compute percentile interval of mean MAM.seq <- seq( from=-3, to=3.5, length.out=30 ) mu <- link( m5.1, data=data.frame(medianagemarriage.s=mam.seq) ) ## [ 100 / 1000 ] [ 200 / 1000 ] [ 300 / 1000 ] [ 400 / 1000 ] [ 500 / 1000 ] [ 600 / 1000 ] [ 700 / 1000 ] [ 800 / 1000 ] [ 900 / 1000 ] [ 1000 / 1000 ] mu.pi <- apply( mu, 2, PI ) # plot it all plot( Divorce ~ MedianAgeMarriage.s, data=d, col=rangi2 ) abline( m5.1 ) ## Warning in abline(m5.1): only using the first two of 3 regression ## coefficients shade( mu.pi, MAM.seq ) 7

Divorce 6 8 10 12 2 1 0 1 2 3 MedianAgeMarriage.s model divorce rate ~ marriage rate. D i Normal(µ i, σ) µ i = α + β R R i α Normal(10, 10) β R Normal(0, 1) σ Uniform(0, 10) map m5.2 ## R code 5.3 d$marriage.s <- (d$marriage - mean(d$marriage))/sd(d$marriage) m5.2 <- map( alist( Divorce ~ dnorm( mu, sigma ), mu <- a + br * Marriage.s, a ~ dnorm( 10, 10 ), br ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data = d ) precis(m5.2) ## Mean StdDev 5.5% 94.5% ## a 9.69 0.24 9.31 10.07 ## br 0.64 0.23 0.27 1.02 ## sigma 1.67 0.17 1.40 1.94 plot( precis(m5.2) ) 8

a br sigma 0 2 4 6 8 10 Value model divorce rate ~ age at marriage and marriage rate. D i Normal(µ i, σ) µ i = α + β R R i + β A A i α Normal(10, 10) β R Normal(0, 1) β A Normal(0, 1) σ Uniform(0, 10) map m5.3 ## R code 5.4 m5.3 <- map( alist( Divorce ~ dnorm( mu, sigma ), mu <- a + br*marriage.s + ba*medianagemarriage.s, a ~ dnorm( 10, 10 ), br ~ dnorm( 0, 1 ), ba ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data = d ) precis( m5.3 ) ## Mean StdDev 5.5% 94.5% ## a 9.69 0.20 9.36 10.01 ## br -0.13 0.28-0.58 0.31 ## ba -1.13 0.28-1.58-0.69 ## sigma 1.44 0.14 1.21 1.67 ## R code 5.5 plot( precis(m5.3) ) 9

a br ba sigma 2 0 2 4 6 8 10 Value model marriage rate ~ age at marriage map m5.4 ## R code 5.6 m5.4 <- map( alist( Marriage.s ~ dnorm( mu, sigma ), mu <- a + b*medianagemarriage.s, a ~ dnorm( 0, 10 ), b ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data = d ) precis(m5.4) ## Mean StdDev 5.5% 94.5% ## a 0.00 0.10-0.16 0.16 ## b -0.71 0.10-0.87-0.56 ## sigma 0.69 0.07 0.58 0.80 plot( precis(m5.4) ) 10

a b sigma 0.5 0.0 0.5 Value plotting multivariate posteriors predictor residual plots ## R code 5.7 # compute expected value at MAP, for each State mu <- coef(m5.4)['a'] + coef(m5.4)['b']*d$medianagemarriage.s # compute residual for each State m.resid <- d$marriage.s - mu ## R code 5.8 plot( Marriage.s ~ MedianAgeMarriage.s, d, col=rangi2 ) abline( m5.4 ) ## Warning in abline(m5.4): only using the first two of 3 regression ## coefficients # loop over States for ( i in 1:length(m.resid) ) { x <- d$medianagemarriage.s[i] # x location of line segment y <- d$marriage.s[i] # observed endpoint of line segment # draw the line segment lines( c(x,x), c(mu[i],y), lwd=0.5, col=col.alpha("black",0.7) ) } 11

Marriage.s 1 0 1 2 2 1 0 1 2 3 MedianAgeMarriage.s counterfactual plots ## R code 5.9 # prepare new counterfactual data A.avg <- mean( d$medianagemarriage.s ) R.seq <- seq( from=-3, to=3, length.out=30 ) pred.data <- data.frame( Marriage.s=R.seq, MedianAgeMarriage.s=A.avg ) # compute counterfactual mean divorce (mu) mu <- link( m5.3, data=pred.data ) ## [ 100 / 1000 ] [ 200 / 1000 ] [ 300 / 1000 ] [ 400 / 1000 ] [ 500 / 1000 ] [ 600 / 1000 ] [ 700 / 1000 ] [ 800 / 1000 ] [ 900 / 1000 ] [ 1000 / 1000 ] mu.mean <- apply( mu, 2, mean ) mu.pi <- apply( mu, 2, PI ) # simulate counterfactual divorce outcomes R.sim <- sim( m5.3, data=pred.data, n=1e4 ) ## [ 1000 / 10000 ] 12

[ 2000 / 10000 ] [ 3000 / 10000 ] [ 4000 / 10000 ] [ 5000 / 10000 ] [ 6000 / 10000 ] [ 7000 / 10000 ] [ 8000 / 10000 ] [ 9000 / 10000 ] [ 10000 / 10000 ] R.PI <- apply( R.sim, 2, PI ) # display predictions, hiding raw data with type="n" plot( Divorce ~ Marriage.s, data=d, type="n" ) mtext( "MedianAgeMarriage.s = 0" ) lines( R.seq, mu.mean ) shade( mu.pi, R.seq ) shade( R.PI, R.seq ) MedianAgeMarriage.s = 0 Divorce 6 8 10 12 1 0 1 2 Marriage.s ## R code 5.10 R.avg <- mean( d$marriage.s ) A.seq <- seq( from=-3, to=3.5, length.out=30 ) pred.data2 <- data.frame( Marriage.s=R.avg, MedianAgeMarriage.s=A.seq ) mu <- link( m5.3, data=pred.data2 ) ## [ 100 / 1000 ] [ 200 / 1000 ] [ 300 / 1000 ] [ 400 / 1000 ] 13

[ 500 / 1000 ] [ 600 / 1000 ] [ 700 / 1000 ] [ 800 / 1000 ] [ 900 / 1000 ] [ 1000 / 1000 ] mu.mean <- apply( mu, 2, mean ) mu.pi <- apply( mu, 2, PI ) A.sim <- sim( m5.3, data=pred.data2, n=1e4 ) ## [ 1000 / 10000 ] [ 2000 / 10000 ] [ 3000 / 10000 ] [ 4000 / 10000 ] [ 5000 / 10000 ] [ 6000 / 10000 ] [ 7000 / 10000 ] [ 8000 / 10000 ] [ 9000 / 10000 ] [ 10000 / 10000 ] A.PI <- apply( A.sim, 2, PI ) plot( Divorce ~ MedianAgeMarriage.s, data=d, type="n" ) mtext( "Marriage.s = 0" ) lines( A.seq, mu.mean ) shade( mu.pi, A.seq ) shade( A.PI, A.seq ) Marriage.s = 0 Divorce 6 8 10 12 2 1 0 1 2 3 MedianAgeMarriage.s 14

posterior prediction plots ## R code 5.11 # call link without specifying new data # so it uses original data mu <- link( m5.3 ) ## [ 100 / 1000 ] [ 200 / 1000 ] [ 300 / 1000 ] [ 400 / 1000 ] [ 500 / 1000 ] [ 600 / 1000 ] [ 700 / 1000 ] [ 800 / 1000 ] [ 900 / 1000 ] [ 1000 / 1000 ] # summarize samples across cases mu.mean <- apply( mu, 2, mean ) mu.pi <- apply( mu, 2, PI ) # simulate observations # again no new data, so uses original data divorce.sim <- sim( m5.3, n=1e4 ) ## [ 1000 / 10000 ] [ 2000 / 10000 ] [ 3000 / 10000 ] [ 4000 / 10000 ] [ 5000 / 10000 ] [ 6000 / 10000 ] [ 7000 / 10000 ] [ 8000 / 10000 ] [ 9000 / 10000 ] [ 10000 / 10000 ] divorce.pi <- apply( divorce.sim, 2, PI ) ## R code 5.12 plot( mu.mean ~ d$divorce, col=rangi2, ylim=range(mu.pi), xlab="observed divorce", ylab="predicted divorce" ) abline( a=0, b=1, lty=2 ) for ( i in 1:nrow(d) ) lines( rep(d$divorce[i],2), c(mu.pi[1,i],mu.pi[2,i]), col=rangi2 ) ## R code 5.13 identify( x=d$divorce, y=mu.mean, labels=d$loc, cex=0.8 ) 15

Predicted divorce 6 8 10 12 ## integer(0) 6 8 10 12 Observed divorce ## R code 5.14 # compute residuals divorce.resid <- d$divorce - mu.mean # get ordering by divorce rate o <- order(divorce.resid) # make the plot dotchart( divorce.resid[o], labels=d$loc[o], xlim=c(-6,5), cex=0.6 ) abline( v=0, col=col.alpha("black",0.2) ) for ( i in 1:nrow(d) ) { j <- o[i] # which State in order lines( d$divorce[j]-c(mu.pi[1,j],mu.pi[2,j]), rep(i,2) ) points( d$divorce[j]-c(divorce.pi[1,j],divorce.pi[2,j]), rep(i,2), pch=3, cex=0.6, col="gray" ) } 16

ME AR AL AK KY OK GA CO RI LA MS IN NH TN AZ SD OR VT WV NM WA MD MA KS IA OH NC DC DE MI TX VA HI MO WY IL MT FL CA NY PA WI SC NE UT CT ND MN NJ ID 6 4 2 0 2 4 simulating spurious association ## R code 5.15 N <- 100 # number of cases x_real <- rnorm( N ) # x_real as Gaussian with mean 0 and stddev 1 x_spur <- rnorm( N, x_real ) # x_spur as Gaussian with mean=x_real y <- rnorm( N, x_real ) # y as Gaussian with mean=x_real d <- data.frame(y,x_real,x_spur) # bind all together in data frame pairs(~ y + x_real + x_spur, data=d, col="darkred") 17

2 1 0 1 2 y 3 1 1 3 2 0 1 2 x_real x_spur 2 0 1 2 3 3 2 1 0 1 2 3 2 1 0 1 2 3 demo.lm <- lm(y ~ x_real + x_spur, data=d) options(show.signif.stars=false) summary(demo.lm) ## ## Call: ## lm(formula = y ~ x_real + x_spur, data = d) ## ## Residuals: ## Min 1Q Median 3Q Max ## -2.5484-0.5838 0.1744 0.6885 2.3940 ## ## Coefficients: ## Estimate Std. Error t value Pr(> t ) ## (Intercept) -0.003954 0.103260-0.038 0.970 ## x_real 0.864603 0.159877 5.408 4.58e-07 ## x_spur 0.059178 0.098300 0.602 0.549 ## ## Residual standard error: 1.025 on 97 degrees of freedom ## Multiple R-squared: 0.368, Adjusted R-squared: 0.355 ## F-statistic: 28.24 on 2 and 97 DF, p-value: 2.158e-10 18

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