# Follow instructions at link below to install Rtools # https://github.com/stan-dev/rstan/wiki/Installing-RStan-on-Windows#toolchain # Install brms package and all required packages install.package("brms") # Load brms package and all required packages library(brms) # Read autism data, and examine autism <- read.csv("X:\\autism.csv", h=T) dim(autism) autism[1:5,] # Remove NAs autism <- autism[!is.na(autism\$vsae),] dim(autism) # lme() approach from Chapter 6 (see prep syntax on book web page) model6.1.fit <- lme(vsae ~ age.2 + I(age.2^2) + sicdegp2.f + + age.2:sicdegp2.f + I(age.2^2):sicdegp2.f, + random = ~ age.2 + I(age.2^2), method="REML", + data = autism.grouped) # Note lack of convergence Error in lme.formula(vsae ~ age.2 + I(age.2^2) + sicdegp2.f + age.2:sicdegp2.f + : nlminb problem, convergence error code = 1 message = iteration limit reached without convergence (10) # lmer() approach from Chapter 6 model6.1.fit <- lmer(vsae ~ age.2 + I(age.2^2) + sicdegp2.f + + age.2*sicdegp2.f + I(age.2^2)*sicdegp2.f + (age.2 + I(age.2^2) | childid), + REML = T, data = autism.updated) # NOTE: NO WARNING MESSAGE!! summary(model6.1.fit) Linear mixed model fit by REML ['lmerMod'] Formula: vsae ~ age.2 + I(age.2^2) + sicdegp2.f + age.2 * sicdegp2.f + I(age.2^2) * sicdegp2.f + (age.2 + I(age.2^2) | childid) Data: autism.updated REML criterion at convergence: 4608 Scaled residuals: Min 1Q Median 3Q Max -4.3015 -0.3905 -0.0392 0.2837 6.8667 Random effects: Groups Name Variance Std.Dev. Corr childid (Intercept) 1.4332 1.1972 age.2 14.2647 3.7769 0.06 I(age.2^2) 0.1562 0.3952 0.89 -0.40 Residual 37.6414 6.1353 Number of obs: 610, groups: childid, 158 Fixed effects: Estimate Std. Error t value (Intercept) 13.78643 0.82251 16.762 age.2 5.59928 0.78666 7.118 I(age.2^2) 0.20417 0.08537 2.392 sicdegp2.f1 -5.43483 1.11108 -4.891 sicdegp2.f2 -4.03533 1.04597 -3.858 age.2:sicdegp2.f1 -3.28366 1.08196 -3.035 age.2:sicdegp2.f2 -2.75204 1.01795 -2.704 I(age.2^2):sicdegp2.f1 -0.13881 0.11833 -1.173 I(age.2^2):sicdegp2.f2 -0.13396 0.11023 -1.215 # Fit example normal model from Chapter 6 using Bayesian approach, with specified priors for fixed effects, standard deviations of random effects, and correlation of random effects fit1 <- brm(vsae ~ age.2 + I(age.2^2) + sicdegp2.f + age.2 * sicdegp2.f + I(age.2^2) * sicdegp2.f + (age.2 + I(age.2^2) | childid), data = autism.updated, family = gaussian("identity"), prior = c(set_prior("normal(0,10)", class = "b"), set_prior("cauchy(0,2)", class = "sd"), set_prior("lkj(2)", class = "cor")), warmup = 500, iter = 2000, chains = 2) # Analyze the results based on draws from the relevant posterior distributions summary(fit1, waic=T) Family: gaussian Links: mu = identity; sigma = identity Formula: vsae ~ age.2 + I(age.2^2) + sicdegp2.f + age.2 * sicdegp2.f + I(age.2^2) * sicdegp2.f + (age.2 + I(age.2^2) | childid) Data: autism.updated (Number of observations: 610) Samples: 2 chains, each with iter = 2000; warmup = 500; thin = 1; total post-warmup samples = 3000 ICs: LOO = NA; WAIC = 4219.42; R2 = NA Group-Level Effects: ~childid (Number of levels: 158) Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat sd(Intercept) 0.90 0.52 0.05 1.95 172 1.01 sd(age.2) 3.75 0.36 3.07 4.49 835 1.00 sd(Iage.2E2) 0.38 0.04 0.30 0.45 810 1.00 cor(Intercept,age.2) 0.05 0.30 -0.49 0.65 101 1.01 cor(Intercept,Iage.2E2) 0.44 0.35 -0.46 0.88 35 1.03 cor(age.2,Iage.2E2) -0.31 0.12 -0.52 -0.07 744 1.00 Population-Level Effects: Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat Intercept 13.74 0.83 12.12 15.41 2025 1.00 age.2 5.59 0.77 4.11 7.04 952 1.00 Iage.2E2 0.20 0.08 0.05 0.36 952 1.00 sicdegp2.f1 -5.36 1.11 -7.53 -3.23 3000 1.00 sicdegp2.f2 -3.99 1.09 -6.19 -1.91 3000 1.00 age.2:sicdegp2.f1 -3.31 1.07 -5.38 -1.24 1067 1.00 age.2:sicdegp2.f2 -2.71 1.02 -4.70 -0.67 1131 1.00 Iage.2E2:sicdegp2.f1 -0.13 0.12 -0.36 0.10 977 1.00 Iage.2E2:sicdegp2.f2 -0.13 0.10 -0.34 0.07 1022 1.00 Family Specific Parameters: Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat sigma 6.21 0.25 5.76 6.74 1140 1.00 Samples were drawn using sampling(NUTS). For each parameter, Eff.Sample is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence, Rhat = 1). # The results are quite similar to those from the frequentist approach, without any convergence issues! # Plot the chains and simulated draws from the posterior distributions plot(fit1) # Plot marginal effects (need to create dummies first) autism\$gp2[autism\$sicdegp == 2] <- 1 autism\$gp2[autism\$sicdegp != 2] <- 0 autism\$gp3[autism\$sicdegp == 3] <- 1 autism\$gp3[autism\$sicdegp != 3] <- 0 fit1 <- brm(vsae ~ age + gp2 + gp3 + (1 + age | childid), data = autism, family = gaussian("identity"), prior = c(set_prior("normal(0,10)", class = "b"), set_prior("cauchy(0,2)", class = "sd"), set_prior("lkj(2)", class = "cor")), warmup = 500, iter = 2000, chains = 2) summary(fit1, waic = T) plot(marginal_effects(fit1), points = TRUE, ask = TRUE) # Fit model removing the correlation between the random intercepts and random slopes fit2 <- brm(formula = vsae ~ age + factor(sicdegp) + (1 + age || childid), data = autism, family = gaussian("identity"), prior = c(set_prior("normal(0,10)", class = "b"), set_prior("cauchy(0,2)", class = "sd")), warmup = 500, iter = 2000, chains = 2) summary(fit2, waic = T) # Compare fits of two models LOO(fit1, fit2) # Fit a model with heterogeneous variance of conditional residuals across groups fit3 <- brm(bf(vsae ~ age + factor(sicdegp) + (1 | childid), sigma ~ factor(sicdegp)), data = autism, family = gaussian("identity"), prior = c(set_prior("normal(0,10)", class = "b"), set_prior("cauchy(0,2)", class = "sd")), warmup = 500, iter = 2000, chains = 2) summary(fit3, waic = T) # LATEST GITHUB VERSION (3/5/18) # Fit a model with heterogeneous variance components for random effects across groups fit <- brm(count ~ Trt + (1|gr(patient, by = Trt)), data = epilepsy) summary(fit)