4.2 centering Chris Parrish July 2, 2016
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1 4.2 centering Chris Parrish July 2, 2016 Contents centering and standardizing 1 centering data model fit centering by subtracting the mean model fit using a conventional centering point model fit centering by subtracting the mean and dividing by 2 sd model fit centering and standardizing reference: - ARM chapter 04, github library(rstan) rstan_options(auto_write = TRUE) options(mc.cores = parallel::detectcores()) library(ggplot2) centering and standardizing centering data # Data source("kidiq.data.r", echo = TRUE) > N <- 434 > kid_score <- c(65, 98, 85, 83, 115, 98, 69, 106, 102, + 95, 91, 58, 84, 78, 102, 110, 102, 99, 105, 101, 102, 115, + 100, 87, 99, 96, 72,... [TRUNCATED] > mom_hs <- c(1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1,... [TRUNCATED] 1
2 > mom_iq <- c( , , , , , , , +... [TRUNCATED] > mom_work <- c(4, 4, 4, 3, 4, 1, 4, 3, 1, 1, 1, 4, + 4, 4, 2, 1, 3, 3, 4, 3, 4, 2, 2, 4, 4, 4, 3, 4, 2, 2, 1, + 2, 3, 2, 3, 4, 4, 4, 1, 4,... [TRUNCATED] model kidiq_interaction.stan data { int<lower=0> N; vector[n] kid_score; vector[n] mom_hs; vector[n] mom_iq; transformed data { vector[n] inter; inter = mom_hs.* mom_iq; // interaction parameters { vector[4] beta; real<lower=0> sigma; model { kid_score ~ normal(beta[1] + beta[2] * mom_hs + beta[3] * mom_iq + beta[4] * inter, sigma); fit # Model: kid_score ~ mom_hs + mom_iq + mom_hs:mom_iq data.list = c("n", "kid_score", "mom_hs", "mom_iq") kidiq_interaction <- stan(file='kidiq_interaction.stan', data=data.list, iter=1000, chains=4) plot(kidiq_interaction) ci_level: 0.8 (80% intervals) outer_level: 0.95 (95% intervals) 2
3 beta[1] beta[2] beta[3] beta[4] sigma 0 50 pairs(kidiq_interaction) beta[1] 0 60 beta[2] beta[3] beta[4] sigma 1476 lp print(kidiq_interaction, pars = c("beta", "sigma", "lp ")) Inference for Stan model: kidiq_interaction. 4 chains, each with iter=1000; warmup=500; thin=1; post-warmup draws per chain=500, total post-warmup draws=2000. beta[1] beta[2] beta[3] beta[4] mean se_mean sd % % % %
4 sigma lp % n_eff Rhat beta[1] beta[2] beta[3] beta[4] sigma lp Samples were drawn using NUTS(diag_e) at Wed Jul 6 00:34: For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1). The estimated Bayesian Fraction of Missing Information is a measure of the efficiency of the sampler with values close to 1 being ideal. For each chain, these estimates are centering by subtracting the mean model kidiq_interaction_c.stan data { int<lower=0> N; vector[n] kid_score; vector[n] mom_hs; vector[n] mom_iq; transformed data { vector[n] c_mom_hs; vector[n] c_mom_iq; vector[n] inter; c_mom_hs = mom_hs - mean(mom_hs); c_mom_iq = mom_iq - mean(mom_iq); inter = c_mom_hs.* c_mom_iq; // centered predictors parameters { vector[4] beta; real<lower=0> sigma; model { kid_score ~ normal(beta[1] + beta[2] * c_mom_hs + beta[3] * c_mom_iq + beta[4] * inter, sigma); 4
5 fit # Centering by subtracting the mean kidiq_interaction_c <- stan(file='kidiq_interaction_c.stan', data=data.list, iter=1000, chains=4) print(kidiq_interaction_c) Inference for Stan model: kidiq_interaction_c. 4 chains, each with iter=1000; warmup=500; thin=1; post-warmup draws per chain=500, total post-warmup draws=2000. mean se_mean sd 2.5% 25% 50% 75% 97.5% beta[1] beta[2] beta[3] beta[4] sigma lp n_eff Rhat beta[1] beta[2] beta[3] beta[4] sigma lp Samples were drawn using NUTS(diag_e) at Wed Jul 6 00:34: For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1). The estimated Bayesian Fraction of Missing Information is a measure of the efficiency of the sampler with values close to 1 being ideal. For each chain, these estimates are using a conventional centering point model kidiq_interaction_c2.stan data { int<lower=0> N; vector[n] kid_score; vector[n] mom_hs; vector[n] mom_iq; transformed data { vector[n] c2_mom_hs; vector[n] c2_mom_iq; vector[n] inter; c2_mom_hs = mom_hs - 0.5; c2_mom_iq = mom_iq - 100; // centering on reference points 5
6 inter = c2_mom_hs.* c2_mom_iq; parameters { vector[4] beta; real<lower=0> sigma; model { kid_score ~ normal(beta[1] + beta[2] * c2_mom_hs + beta[3] * c2_mom_iq + beta[4] * inter, sigma); fit # Using a conventional centering point: # c2_mom_hs <- mom_hs # c2_mom_iq <- mom_iq kidiq_interaction_c2 <- stan(file='kidiq_interaction_c2.stan', data=data.list, iter=1000, chains=4) print(kidiq_interaction_c2) Inference for Stan model: kidiq_interaction_c2. 4 chains, each with iter=1000; warmup=500; thin=1; post-warmup draws per chain=500, total post-warmup draws=2000. mean se_mean sd 2.5% 25% 50% 75% 97.5% beta[1] beta[2] beta[3] beta[4] sigma lp n_eff Rhat beta[1] beta[2] beta[3] beta[4] sigma lp Samples were drawn using NUTS(diag_e) at Wed Jul 6 00:34: For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1). The estimated Bayesian Fraction of Missing Information is a measure of the efficiency of the sampler with values close to 1 being ideal. For each chain, these estimates are
7 centering by subtracting the mean and dividing by 2 sd model kidiq_interaction_z.stan data { int<lower=0> N; vector[n] kid_score; vector[n] mom_hs; vector[n] mom_iq; transformed data { // standardizing vector[n] z_mom_hs; vector[n] z_mom_iq; vector[n] inter; z_mom_hs = (mom_hs - mean(mom_hs)) / (2 * sd(mom_hs)); z_mom_iq = (mom_iq - mean(mom_iq)) / (2 * sd(mom_iq)); inter = z_mom_hs.* z_mom_iq; parameters { vector[4] beta; real<lower=0> sigma; model { kid_score ~ normal(beta[1] + beta[2] * z_mom_hs + beta[3] * z_mom_iq + beta[4] * inter, sigma); fit # Centering by subtracting the mean and dividing by 2 sd kidiq_interaction_z <- stan(file='kidiq_interaction_z.stan', data=data.list, iter=1000, chains=4) print(kidiq_interaction_z) Inference for Stan model: kidiq_interaction_z. 4 chains, each with iter=1000; warmup=500; thin=1; post-warmup draws per chain=500, total post-warmup draws=2000. mean se_mean sd 2.5% 25% 50% 75% 97.5% beta[1] beta[2] beta[3] beta[4] sigma lp n_eff Rhat beta[1] beta[2] beta[3] beta[4] sigma
8 lp Samples were drawn using NUTS(diag_e) at Wed Jul 6 00:34: For each parameter, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat=1). The estimated Bayesian Fraction of Missing Information is a measure of the efficiency of the sampler with values close to 1 being ideal. For each chain, these estimates are
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