categorical Chris Parrish June 7, 2016

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1 categorical Chris Parrish June 7, 2016 Contents Waffle House 2 data exploratory data analysis Waffle Houses Marriage.s MedianAgeMarriage.s MedianAgeMarriage.s and Marriage.s model divorce rate ~ age at marriage map m model divorce rate ~ marriage rate map m model divorce rate ~ age at marriage and marriage rate map m model marriage rate ~ age at marriage map m plotting multivariate posteriors predictor residual plots counterfactual plots posterior prediction plots simulating spurious association milk 19 data map m5.5 with NA m5.5 complete cases m m legs 24 data exploratory data analysis model map parameter distributions samples parameter distributions milk 35 data exploratory data analysis model map parameter distributions plot parameter distribution

2 map plot parameter distribution map plot parameter distributions correlated predictor variables plants 46 data exploratory data analysis map parameter distributions plot parameter distributions map plot parameter distributions Dobe!Kung 52 data exploratory data analysis model map PI for male height reparametrize the model model map parameter distributions milk data map analysis unique intercepts lm 58 categorical references: - McElreath, Statistical Rethinking, chap 5, pp !Kung people - images of!kung people Waffle House library(rethinking) ## Loading required package: rstan ## Loading required package: ggplot2 ## rstan (Version , packaged: :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 2

3 ## rethinking (Version 1.58) library(ggplot2) 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",..: ## $ Loc : Factor w/ 50 levels "AK","AL","AR",..: ## $ Population : num ## $ MedianAgeMarriage: num ## $ Marriage : num ## $ Marriage.SE : num ## $ Divorce : num ## $ Divorce.SE : num ## $ WaffleHouses : int ## $ South : int ## $ Slaves1860 : int ## $ Population1860 : int ## $ PropSlaves1860 : num head(d) ## Location Loc Population MedianAgeMarriage Marriage Marriage.SE Divorce ## 1 Alabama AL ## 2 Alaska AK ## 3 Arizona AZ ## 4 Arkansas AR ## 5 California CA ## 6 Colorado CO ## Divorce.SE WaffleHouses South Slaves1860 Population1860 PropSlaves1860 ## ## ## ## ## ## 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") 3

4 14 12 Divorce rate 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") 4

5 12 Divorce 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") 5

6 12 Divorce 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") 6

7 3 2 1 Marriage.s 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

8 ## ba ## sigma plot( precis(m5.1) ) a ba sigma 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 ) 8

9 Divorce 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 ## br ## sigma plot( precis(m5.2) ) 9

10 a br sigma 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 ## br ## ba ## sigma ## R code 5.5 plot( precis(m5.3) ) 10

11 a br ba sigma 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 ## b ## sigma plot( precis(m5.4) ) 11

12 a b sigma 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) ) } 12

13 Marriage.s 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 / ] 13

14 [ 2000 / ] [ 3000 / ] [ 4000 / ] [ 5000 / ] [ 6000 / ] [ 7000 / ] [ 8000 / ] [ 9000 / ] [ / ] 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 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 ] 14

15 [ 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 / ] [ 2000 / ] [ 3000 / ] [ 4000 / ] [ 5000 / ] [ 6000 / ] [ 7000 / ] [ 8000 / ] [ 9000 / ] [ / ] 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 MedianAgeMarriage.s 15

16 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 / ] [ 2000 / ] [ 3000 / ] [ 4000 / ] [ 5000 / ] [ 6000 / ] [ 7000 / ] [ 8000 / ] [ 9000 / ] [ / ] 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 ) 16

17 Predicted divorce ## integer(0) 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" ) } 17

18 ME AR AL AK KY GA OK CO RI LA MS NH IN TN AZ SD OR VT WV NM WA MA MD KS IA OH DC NC DE MI HI TX VA MO WY IL MT FL CA NY PA WI SC NE CT UT ND MN NJ ID 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") 18

19 y x_real x_spur 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 ## ## ## Coefficients: ## Estimate Std. Error t value Pr(> t ) ## (Intercept) ## x_real e-14 ## x_spur ## ## Residual standard error: on 97 degrees of freedom ## Multiple R-squared: , Adjusted R-squared: ## F-statistic: on 2 and 97 DF, p-value: 7.398e-16 milk library(rethinking) 19

20 data ## R code 5.16 data(milk) d <- milk str(d) ## 'data.frame': 29 obs. of 8 variables: ## $ clade : Factor w/ 4 levels "Ape","New World Monkey",..: ## $ species : Factor w/ 29 levels "A palliata","alouatta seniculus",..: ## $ kcal.per.g : num ## $ perc.fat : num ## $ perc.protein : num ## $ perc.lactose : num ## $ mass : num ## $ neocortex.perc: num 55.2 NA NA NA NA... map m5.5 with NA Problem here with incomplete cases. ## R code 5.17 # m5.5 <- map( # alist( # kcal.per.g ~ dnorm( mu, sigma ), # mu <- a + bn*neocortex.perc, # a ~ dnorm( 0, 100 ), # bn ~ dnorm( 0, 1 ), # sigma ~ dunif( 0, 1 ) # ), # data=d ) ## R code 5.18 d$neocortex.perc ## [1] NA NA NA NA NA ## [12] NA NA NA NA NA ## [23] NA NA m5.5 complete cases ## R code 5.19 dcc <- d[ complete.cases(d), ] a.start <- mean(dcc$kcal.per.g) bn.start <- 0 ## R code 5.20 m5.5 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), 20

21 mu <- a + bn*neocortex.perc, a ~ dnorm( 0, 100 ), bn ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 1 ) ), data = dcc, start = list(a=a.start, bn=bn.start)) ## R code 5.21 precis( m5.5, digits=3 ) ## Mean StdDev 5.5% 94.5% ## a ## bn ## sigma ## R code 5.22 coef(m5.5)["bn"] * ( ) ## bn ## ## R code 5.23 np.seq <- 0:100 pred.data <- data.frame( neocortex.perc=np.seq ) mu <- link( m5.5, data=pred.data, n=1e4 ) ## [ 1000 / ] [ 2000 / ] [ 3000 / ] [ 4000 / ] [ 5000 / ] [ 6000 / ] [ 7000 / ] [ 8000 / ] [ 9000 / ] [ / ] mu.mean <- apply( mu, 2, mean ) mu.pi <- apply( mu, 2, PI ) plot( kcal.per.g ~ neocortex.perc, data=dcc, col=rangi2 ) lines( np.seq, mu.mean ) lines( np.seq, mu.pi[1,], lty=2 ) lines( np.seq, mu.pi[2,], lty=2 ) 21

22 kcal.per.g ## R code 5.24 dcc$log.mass <- log(dcc$mass) neocortex.perc m5.6 ## R code 5.25 m5.6 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + bm*log.mass, a ~ dnorm( 0, 100 ), bm ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 1 ) ), data=dcc ) precis(m5.6) ## Mean StdDev 5.5% 94.5% ## a ## bm ## sigma m5.7 Bad start value (1) ## R code 5.26 m5.7 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + bn*neocortex.perc + bm*log.mass, 22

23 a ~ dnorm( 0, 100 ), bn ~ dnorm( 0, 1 ), bm ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 1 ) ), data=dcc ) precis(m5.7) ## Mean StdDev 5.5% 94.5% ## a ## bn ## bm ## sigma ## R code 5.27 mean.log.mass <- mean( log(dcc$mass) ) np.seq <- 0:100 pred.data <- data.frame( neocortex.perc=np.seq, log.mass=mean.log.mass ) mu <- link( m5.7, data=pred.data, n=1e4 ) ## [ 1000 / ] [ 2000 / ] [ 3000 / ] [ 4000 / ] [ 5000 / ] [ 6000 / ] [ 7000 / ] [ 8000 / ] [ 9000 / ] [ / ] mu.mean <- apply( mu, 2, mean ) mu.pi <- apply( mu, 2, PI ) plot( kcal.per.g ~ neocortex.perc, data=dcc, type="n" ) lines( np.seq, mu.mean ) lines( np.seq, mu.pi[1,], lty=2 ) lines( np.seq, mu.pi[2,], lty=2 ) 23

24 kcal.per.g neocortex.perc legs library(rethinking) library(ggplot2) library(cowplot) data Correlated predictors. ## R code 5.28 N <- 100 rho <- 0.7 x_pos <- rnorm( N ) x_neg <- rnorm( N, rho*x_pos, sqrt(1-rho^2) ) y <- rnorm( N, x_pos - x_neg ) d <- data.frame(y,x_pos,x_neg) Legs ## R code 5.29 N <- 100 height <- rnorm(n,10,2) leg_prop <- runif(n,0.4,0.5) leg_left <- leg_prop*height + rnorm( N, 0, 0.02 ) leg_right <- leg_prop*height + rnorm( N, 0, 0.02 ) # number of cases # correlation btw x_pos and x_neg # x_pos as Gaussian # x_neg correlated with x_pos # y equally associated with x_pos, x_neg # bind all together in data frame # number of individuals # sim total height of each # leg as proportion of height # sim left leg as proportion + error # sim right leg as proportion + error # combine into data frame 24

25 d <- data.frame(height,leg_left,leg_right) str(d) ## 'data.frame': 100 obs. of 3 variables: ## $ height : num ## $ leg_left : num ## $ leg_right: num exploratory data analysis ggplot(d, aes(leg_left, leg_right)) + geom_point(shape = 20, color = "darkred") + theme_gray() 6 5 leg_right leg_left ggplot(d, aes(leg_left, height)) + geom_point(shape = 20, color = "darkred") + geom_smooth() + theme_gray() 25

26 12.5 height leg_left model h i Normal(µ i, σ) µ i = α + β l leg l + β r leg r α Normal(10, 100) β l Normal(2, 10) β r Normal(2, 10) σ Uniform(0, 10) map Start values problem (4) ## R code 5.30 m5.8 <- map( alist( height ~ dnorm( mu, sigma ), mu <- a + bl*leg_left + br*leg_right, a ~ dnorm( 10, 100 ), bl ~ dnorm( 2, 10 ), br ~ dnorm( 2, 10 ), sigma ~ dunif( 0, 10 ) ), 26

27 data=d ) precis(m5.8) ## Mean StdDev 5.5% 94.5% ## a ## bl ## br ## sigma ## R code 5.31 plot(precis(m5.8)) a bl br sigma Value parameter distributions Plot parameter distributions # extract samples post <- extract.samples( m5.8 ) str(post) ## 'data.frame': obs. of 4 variables: ## $ a : num ## $ bl : num ## $ br : num ## $ sigma: num # plot each parameter distribution a.plot <- parameter.dist(parameter = "a", values = post$a) bl.plot <- parameter.dist(parameter = "bl", values = post$bl) 27

28 br.plot <- parameter.dist(parameter = "br", values = post$br) sigma.plot <- parameter.dist(parameter = "sigma", values = post$sigma) # display parameter distributions plot_grid(a.plot, bl.plot, br.plot, sigma.plot, labels=c("a", "bl", "br", "sigma"), ncol = 2, nrow = 2) 28

29 a 1.5 bl 0.20 density density a bl HPDI mean HPDI HPDI mean HPDI a bl br sigma density 0.10 density br sigma HPDI mean HPDI HPDI mean HPDI br sigma 29

30 samples ## R code 5.32 post <- extract.samples(m5.8) str(post) ## 'data.frame': obs. of 4 variables: ## $ a : num ## $ bl : num ## $ br : num ## $ sigma: num plot( bl ~ br, post, col=col.alpha(rangi2,0.1), pch=16 ) bl ggplot(post, aes(bl, br)) + geom_point(shape = 20, color = "darkred") + theme_gray() br 30

31 4 0 br bl ## R code 5.33 sum_blbr <- post$bl + post$br dens( sum_blbr, col=rangi2, lwd=2, xlab="sum of bl and br" ) 31

32 Density Start problem (2) sum of bl and br ## R code 5.34 m5.9 <- map( alist( height ~ dnorm( mu, sigma ), mu <- a + bl*leg_left, a ~ dnorm( 10, 100 ), bl ~ dnorm( 2, 10 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis(m5.9) ## Mean StdDev 5.5% 94.5% ## a ## bl ## sigma parameter distributions Plot parameter distributions # extract samples post <- extract.samples( m5.9 ) str(post) ## 'data.frame': obs. of 3 variables: ## $ a : num

33 ## $ bl : num ## $ sigma: num # plot each parameter distribution a.plot <- parameter.dist(parameter = "a", values = post$a) bl.plot <- parameter.dist(parameter = "bl", values = post$bl) sigma.plot <- parameter.dist(parameter = "sigma", values = post$sigma) # display parameter distributions plot_grid(a.plot, bl.plot, sigma.plot, labels=c("a", "bl", "sigma"), ncol = 2, nrow = 2) 33

34 a 1.5 bl 6 density 1.0 density a bl HPDI mean HPDI HPDI mean HPDI sigma a bl 7.5 density sigma HPDI mean HPDI sigma 34

35 milk library(rethinking) library(ggplot2) library(cowplot) theme_set(theme_gray()) data ## R code 5.35 data(milk) d <- milk str(d) ## 'data.frame': 29 obs. of 8 variables: ## $ clade : Factor w/ 4 levels "Ape","New World Monkey",..: ## $ species : Factor w/ 29 levels "A palliata","alouatta seniculus",..: ## $ kcal.per.g : num ## $ perc.fat : num ## $ perc.protein : num ## $ perc.lactose : num ## $ mass : num ## $ neocortex.perc: num 55.2 NA NA NA NA... exploratory data analysis Two predictor variables are associated. ggplot(d, aes(perc.fat, kcal.per.g)) + geom_point(shape = 20, color = "darkred") + geom_smooth() 35

36 kcal.per.g perc.fat ggplot(d, aes(perc.lactose, kcal.per.g)) + geom_point(shape = 20, color = "darkred") + geom_smooth() 36

37 kcal.per.g perc.lactose ggplot(d, aes(perc.fat, perc.lactose)) + geom_point(shape = 20, color = "darkred") + geom_smooth() 37

38 80 perc.lactose perc.fat model E i Normal(µ i, σ) µ i = α + β f perc.fat + β l perc.lactose α Normal(0.6, 10) β f Normal(0, 1) β l Normal(0, 1) σ Uniform(0, 10) map ## R code 5.36 # kcal.per.g regressed on perc.fat m5.10 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + bf*perc.fat, a ~ dnorm( 0.6, 10 ), bf ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis( m5.10, digits=3 ) 38

39 ## Mean StdDev 5.5% 94.5% ## a ## bf ## sigma plot(precis( m5.10, digits=3 )) a bf sigma Value parameter distributions plot parameter distribution # extract samples post <- extract.samples( m5.10 ) str(post) ## 'data.frame': obs. of 3 variables: ## $ a : num ## $ bf : num ## $ sigma: num # plot parameter distribution bf.plot <- parameter.dist(parameter = "bf", values = post$bf) bf.plot 39

40 density bf HPDI mean HPDI bf map Start value (1) # kcal.per.g regressed on perc.lactose m5.11 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + bl*perc.lactose, a ~ dnorm( 0.6, 10 ), bl ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis( m5.11, digits=3 ) ## Mean StdDev 5.5% 94.5% ## a ## bl ## sigma plot(precis( m5.11, digits=3 )) 40

41 a bl sigma Value plot parameter distribution # extract samples post <- extract.samples( m5.11 ) str(post) ## 'data.frame': obs. of 3 variables: ## $ a : num ## $ bl : num ## $ sigma: num # plot parameter distribution bl.plot <- parameter.dist(parameter = "bl", values = post$bl) bl.plot 41

42 density bl HPDI mean HPDI bl map ## R code 5.37 m5.12 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + bf*perc.fat + bl*perc.lactose, a ~ dnorm( 0.6, 10 ), bf ~ dnorm( 0, 1 ), bl ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis( m5.12, digits=3 ) ## Mean StdDev 5.5% 94.5% ## a ## bf ## bl ## sigma plot(precis( m5.12, digits=3 )) 42

43 a bf bl sigma Value plot parameter distributions # extract samples post <- extract.samples( m5.12 ) str(post) ## 'data.frame': obs. of 4 variables: ## $ a : num ## $ bf : num ## $ bl : num ## $ sigma: num # plot each parameter distribution a.plot <- parameter.dist(parameter = "a", values = post$a) bf.plot <- parameter.dist(parameter = "bf", values = post$bf) bl.plot <- parameter.dist(parameter = "bl", values = post$bl) sigma.plot <- parameter.dist(parameter = "sigma", values = post$sigma) # display parameter distributions plot_grid(a.plot, bf.plot, bl.plot, sigma.plot, labels=c("a", "bf", "bl", "sigma"), ncol = 2, nrow = 2) 43

44 a 2.0 bf density 1.0 density a bf HPDI mean HPDI HPDI mean HPDI bl a bf sigma density 100 density bl sigma HPDI mean HPDI HPDI mean HPDI bl sigma 44

45 correlated predictor variables perc.f at and perc.lactose are inversely related ## R code 5.38 pairs( ~ kcal.per.g + perc.fat + perc.lactose, data=d, col=rangi2 ) kcal.per.g perc.fat perc.lactose ## R code 5.39 cor( d$perc.fat, d$perc.lactose ) ## [1] Simulation ## R code 5.40 data(milk) d <- milk sim.coll <- function( r=0.9 ) { d$x <- rnorm( nrow(d), mean=r*d$perc.fat, sd=sqrt( (1-r^2)*var(d$perc.fat) ) ) m <- lm( kcal.per.g ~ perc.fat + x, data=d ) sqrt( diag( vcov(m) ) )[2] # stddev of parameter } rep.sim.coll <- function( r=0.9, n=100 ) { stddev <- replicate( n, sim.coll(r) ) mean(stddev) } r.seq <- seq(from=0,to=0.99,by=0.01) stddev <- sapply( r.seq, function(z) rep.sim.coll(r=z,n=100) ) plot( stddev ~ r.seq, type="l", col=rangi2, lwd=2, xlab="correlation" ) 45

46 stddev correlation plants Post-treatment bias library(rethinking) library(ggplot2) library(cowplot) theme_set(theme_gray()) data ## R code 5.41 # number of plants N <- 100 # simulate initial heights h0 <- rnorm(n,10,2) # assign treatments and simulate fungus and growth treatment <- rep( 0:1, each=n/2 ) fungus <- rbinom( N, size=1, prob=0.5 - treatment*0.4 ) h1 <- h0 + rnorm(n, 5-3*fungus) # compose a clean data frame d <- data.frame( h0=h0, h1=h1, treatment=treatment, fungus=fungus ) 46

47 exploratory data analysis d2 <- d d2$treatment <- factor(d2$treatment) str(d2) ## 'data.frame': 100 obs. of 4 variables: ## $ h0 : num ## $ h1 : num ## $ treatment: Factor w/ 2 levels "0","1": ## $ fungus : int pairs(d2, col="darkred") h h1 treatment fungus ggplot(d2, aes(h0, h1, color = treatment)) + geom_point(shape = 20) + geom_smooth(method = "lm")

48 17.5 h treatment h0 map ## R code 5.42 m5.13 <- map( alist( h1 ~ dnorm(mu,sigma), mu <- a + bh*h0 + bt*treatment + bf*fungus, a ~ dnorm(0,100), c(bh,bt,bf) ~ dnorm(0,10), sigma ~ dunif(0,10) ), data=d, start = list(a = 0)) precis(m5.13) ## Mean StdDev 5.5% 94.5% ## a ## bh ## bt ## bf ## sigma

49 parameter distributions plot parameter distributions # extract samples post <- extract.samples( m5.13 ) str(post) ## 'data.frame': obs. of 5 variables: ## $ a : num ## $ bh : num ## $ bt : num ## $ bf : num ## $ sigma: num # plot each parameter distribution bh.plot <- parameter.dist(parameter = "bh", values = post$bh) bt.plot <- parameter.dist(parameter = "bt", values = post$bt) # display parameter distributions plot_grid(bh.plot, bt.plot, labels=c("bh", "bt"), ncol = 2, nrow = 1) bh 8 bt density 4 density bh bt HPDI mean HPDI HPDI mean HPDI bh bt map ## R code 5.43 m5.14 <- map( alist( h1 ~ dnorm(mu,sigma), 49

50 mu <- a + bh*h0 + bt*treatment, a ~ dnorm(0,100), c(bh,bt) ~ dnorm(0,10), sigma ~ dunif(0,10) ), data=d, start = list(a = 0)) precis(m5.14) ## Mean StdDev 5.5% 94.5% ## a ## bh ## bt ## sigma plot parameter distributions # extract samples post <- extract.samples( m5.14 ) str(post) ## 'data.frame': obs. of 4 variables: ## $ a : num ## $ bh : num ## $ bt : num ## $ sigma: num # plot each parameter distribution a.plot <- parameter.dist(parameter = "a", values = post$a) bh.plot <- parameter.dist(parameter = "bh", values = post$bh) bt.plot <- parameter.dist(parameter = "bt", values = post$bt) sigma.plot <- parameter.dist(parameter = "sigma", values = post$sigma) # display parameter distributions plot_grid(a.plot, bh.plot, bt.plot, sigma.plot, labels=c("a", "bh", "bt", "sigma"), ncol = 2, nrow = 2) 50

51 a 0.4 bh density 0.2 density a bh HPDI mean HPDI HPDI mean HPDI bt a sigma bh density 0.5 density bt sigma HPDI mean HPDI HPDI mean HPDI bt sigma 51

52 Dobe!Kung library(rethinking) library(ggplot2) library(cowplot) theme_set(theme_gray()) data ## R code 5.44 data(howell1) d <- Howell1 str(d) ## 'data.frame': 544 obs. of 4 variables: ## $ height: num ## $ weight: num ## $ age : num ## $ male : int exploratory data analysis d2 <- d d2$gender <- factor(d2$male, labels = c("female", "male")) str(d2) ## 'data.frame': 544 obs. of 5 variables: ## $ height: num ## $ weight: num ## $ age : num ## $ male : int ## $ gender: Factor w/ 2 levels "female","male": ggplot(d2, aes(weight, height, color = gender)) + geom_point(shape = 20) + scale_color_manual(values = c("skyblue", "sienna")) 52

53 150 height gender female male weight model h i Normal(µ i, σ) µ i = α + β m m i α Normal(178, 100) β m Normal(0, 10) σ Uniform(0, 50) map ## R code 5.45 m5.15 <- map( alist( height ~ dnorm( mu, sigma ), mu <- a + bm*male, a ~ dnorm( 178, 100 ), bm ~ dnorm( 0, 10 ), sigma ~ dunif( 0, 50 ) ), data=d, start = list(a = mean(d$height))) precis(m5.15) 53

54 ## Mean StdDev 5.5% 94.5% ## a ## bm ## sigma PI for male height ## R code 5.46 post <- extract.samples(m5.15) mu.male <- post$a + post$bm PI(mu.male) ## 5% 94% ## reparametrize the model model h i Normal(µ i, σ) µ i = β f f i + β m m i β f Normal(178, 100) β m Normal(178, 100) σ Uniform(0, 50) map ## R code 5.47 m5.15b <- map( alist( height ~ dnorm( mu, sigma ), mu <- af*(1-male) + am*male, af ~ dnorm( 178, 100 ), am ~ dnorm( 178, 100 ), sigma ~ dunif( 0, 50 ) ), data=d, start = list(af = mean(d$height), am = mean(d$height))) precis(m5.15b) ## Mean StdDev 5.5% 94.5% ## af ## am ## sigma

55 parameter distributions # extract samples post <- extract.samples( m5.15b ) str(post) ## 'data.frame': obs. of 3 variables: ## $ af : num ## $ am : num ## $ sigma: num # plot each parameter distribution af.plot <- parameter.dist(parameter = "af", values = post$af) am.plot <- parameter.dist(parameter = "am", values = post$am) # display parameter distributions plot_grid(af.plot, am.plot, labels=c("af", "am"), ncol = 2, nrow = 1) af 0.25 am density density af am HPDI mean HPDI HPDI mean HPDI af am milk data ## R code 5.48 data(milk) d <- milk unique(d$clade) ## [1] Strepsirrhine New World Monkey Old World Monkey Ape 55

56 ## Levels: Ape New World Monkey Old World Monkey Strepsirrhine ## R code 5.49 ( d$clade.nwm <- ifelse( d$clade=="new World Monkey", 1, 0 ) ) ## [1] ## R code 5.50 d$clade.owm <- ifelse( d$clade=="old World Monkey", 1, 0 ) d$clade.s <- ifelse( d$clade=="strepsirrhine", 1, 0 ) map ## R code 5.51 m5.16 <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a + b.nwm*clade.nwm + b.owm*clade.owm + b.s*clade.s, a ~ dnorm( 0.6, 10 ), b.nwm ~ dnorm( 0, 1 ), b.owm ~ dnorm( 0, 1 ), b.s ~ dnorm( 0, 1 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis(m5.16) ## Mean StdDev 5.5% 94.5% ## a ## b.nwm ## b.owm ## b.s ## sigma analysis ## R code 5.52 # sample posterior post <- extract.samples(m5.16) # compute averages for each category mu.ape <- post$a mu.nwm <- post$a + post$b.nwm mu.owm <- post$a + post$b.owm mu.s <- post$a + post$b.s # summarize using precis precis( data.frame(mu.ape,mu.nwm,mu.owm,mu.s) ) ## Mean StdDev ## mu.ape ## mu.nwm ## mu.owm

57 ## mu.s ## R code 5.53 diff.nwm.owm <- mu.nwm - mu.owm quantile( diff.nwm.owm, probs=c(0.025,0.5,0.975) ) ## 2.5% 50% 97.5% ## unique intercepts ## R code 5.54 ( d$clade_id <- coerce_index(d$clade) ) ## [1] ## R code 5.55 m5.16_alt <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a[clade_id], a[clade_id] ~ dnorm( 0.6, 10 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis( m5.16_alt, depth=2 ) ## Mean StdDev 5.5% 94.5% ## a[1] ## a[2] ## a[3] ## a[4] ## sigma ## R code 5.55 m5.16_alt <- map( alist( kcal.per.g ~ dnorm( mu, sigma ), mu <- a[clade_id], a[clade_id] ~ dnorm( 0.6, 10 ), sigma ~ dunif( 0, 10 ) ), data=d ) precis( m5.16_alt, depth=2 ) ## Mean StdDev 5.5% 94.5% ## a[1] ## a[2] ## a[3] ## a[4] ## sigma

58 lm # ## R code 5.56 # m5.17 <- lm( y ~ 1 + x, data=d ) # m5.18 <- lm( y ~ 1 + x + z + w, data=d ) # # ## R code 5.57 # m5.17 <- lm( y ~ 1 + x, data=d ) # m5.19 <- lm( y ~ x, data=d ) # # ## R code 5.58 # m5.20 <- lm( y ~ 0 + x, data=d ) # m5.21 <- lm( y ~ x - 1, data=d ) # # ## R code 5.59 # m5.22 <- lm( y ~ 1 + as.factor(season), data=d ) # # ## R code 5.60 # d$x2 <- d$x^2 # d$x3 <- d$x^3 # m5.23 <- lm( y ~ 1 + x + x2 + x3, data=d ) # # ## R code 5.61 # m5.24 <- lm( y ~ 1 + x + I(x^2) + I(x^3), data=d ) ## R code 5.62 data(cars) glimmer( dist ~ speed, data=cars ) ## alist( ## dist ~ dnorm( mu, sigma ), ## mu <- Intercept + ## b_speed*speed, ## Intercept ~ dnorm(0,10), ## b_speed ~ dnorm(0,10), ## sigma ~ dcauchy(0,2) ## ) 58