From 758e590a7fa3281fb34b406de9bbdd1637589494 Mon Sep 17 00:00:00 2001 From: eveskew Date: Sun, 3 May 2020 20:19:51 -0400 Subject: [PATCH] Adding week 15 material --- lecture/lecture_week_15.R | 206 +++++++++++++++++++++++++++++++++++ lecture/lecture_week_15.pptx | Bin 0 -> 4998294 bytes 2 files changed, 206 insertions(+) create mode 100644 lecture/lecture_week_15.R create mode 100644 lecture/lecture_week_15.pptx diff --git a/lecture/lecture_week_15.R b/lecture/lecture_week_15.R new file mode 100644 index 0000000..06115bc --- /dev/null +++ b/lecture/lecture_week_15.R @@ -0,0 +1,206 @@ + + +library(rethinking) + +#========================================================== + + +# Load the reed frog data +data(reedfrogs) +d <- reedfrogs + +# Make the tank cluster variable +d$tank <- 1:nrow(d) + +# Fit a binomial GLM predicting frog survival using a +# tank-level index variable +m12.1 <- map2stan( + data = d, + alist( + surv ~ dbinom(size = density, prob = p), + logit(p) <- a_tank[tank], + a_tank[tank] ~ dnorm(0, 5) + ), + chains = 4, + iter = 5000 +) + +precis(m12.1, depth = 2, prob = 0.97) + +# Fit a binomial multilevel model predicting frog survival +# using a tank-level index variable +m12.2 <- map2stan( + data = d, + alist( + surv ~ dbinom(size = density, prob = p), + logit(p) <- a_tank[tank], + a_tank[tank] ~ dnorm(a, sigma), + a ~ dnorm(0, 1), + sigma ~ dcauchy(0, 1) + ), + chains = 4, + iter = 5000 +) + +precis(m12.2, depth = 2, prob = 0.97) + +#========================================================== + + +# Show shrinkage of multilevel model parameter estimates + + +# Extract posterior samples from model m12.1 +post.glm <- extract.samples(m12.1) + +# Extract posterior samples from model m12.2 +post.glmm <- extract.samples(m12.2) + +# Note: this will not work since you're asking for more +# samples than you have +post.glmm <- extract.samples(m12.2, n = 20000) + +# Compute median intercept for each tank and transform to +# probability with "logistic()" +d$propsurv.est.glm <- + logistic(apply(post.glm$a_tank, 2, median)) +d$propsurv.est.glmm <- + logistic(apply(post.glmm$a_tank, 2, median)) + + +# Display raw proportions surviving in each tank +plot(d$propsurv, ylim = c(0, 1), pch = 16, xaxt = "n", + xlab = "tank", + ylab = "proportion survival", + col = rangi2) +axis(1, at = c(1, 16, 32, 48), + labels = c(1, 16, 32, 48)) + +# Overlay posterior medians from the GLM... +points(d$propsurv.est.glm, col = "darkred") + + +# Display raw proportions surviving in each tank +plot(d$propsurv, ylim = c(0, 1), pch = 16, xaxt = "n", + xlab = "tank", + ylab = "proportion survival", + col = rangi2) +axis(1, at = c(1, 16, 32, 48), + labels = c(1, 16, 32, 48)) + +# Overlay posterior medians from the mulitlevel model +points(d$propsurv.est.glmm) + +# Mark posterior median probability across tanks +abline(h = logistic(median(post.glmm$a)), lty = 2) + +# Draw vertical dividers between tank densities +abline(v = 16.5, lwd = 0.5) +abline(v = 32.5, lwd = 0.5) +text(8, 0, "small tanks") +text(16+8, 0, "medium tanks") +text(32+8, 0, "large tanks") + + +# You could get a very similar plot using "coef()" to +# extract posterior means for parameters + +# Display raw proportions surviving in each tank +plot(d$propsurv, ylim = c(0, 1), pch = 16, xaxt = "n", + xlab = "tank", + ylab = "proportion survival", + col = rangi2) +axis(1, at = c(1, 16, 32, 48), + labels = c(1, 16, 32, 48)) + +# Overlay posterior means from the multilevel model +points(logistic(coef(m12.2)[1:48])) + +# Mark posterior mean probability across tanks +abline(h = logistic(mean(coef(m12.2)["a"])), lty = 2) + +# Draw vertical dividers between tank densities +abline(v = 16.5, lwd = 0.5) +abline(v = 32.5, lwd = 0.5) +text(8, 0, "small tanks") +text(16+8, 0, "medium tanks") +text(32+8, 0, "large tanks") + +#========================================================== + + +# Simulating frog survival data among "ponds" + +a <- 1.4 # average pond's log-odds of survival +sigma <- 1.5 # survival standard deviation among ponds +nponds <- 60 # number of ponds to simulate +ni <- # sample size across ponds + as.integer(rep(c(5, 10, 25, 35), each = 15)) + +# Simulate "true" pond-level log-odds of survival +a_pond <- rnorm(nponds, mean = a, sd = sigma) + +# Package data into a data frame +dsim <- data.frame( + pond = 1:nponds, + ni = ni, + true_a = a_pond +) + +# Simulate survivors across ponds +dsim$si <- rbinom(nponds, size = dsim$ni, + prob = logistic(dsim$true_a)) + + +# Plot observed survival across ponds +plot(1:nponds, dsim$si/dsim$ni, ylim = c(0, 1), + xlab = "pond", + ylab = "proportion survival") +# Add on true pond-level survival probabilities +points(logistic(dsim$true_a), col = "red", pch = 20) + +# Draw vertical dividers between pond densities +abline(v = 15.5, lwd = 0.5) +abline(v = 30.5, lwd = 0.5) +abline(v = 45.5, lwd = 0.5) +text(7.5, 0, "tiny ponds") +text(15+7.5, 0, "small ponds") +text(30+7.5, 0, "medium ponds") +text(45+7.5, 0, "large ponds") + +# Draw horizontal line representing the average +# pond's true proportion survival +abline(h = logistic(a), lty = 2) + +#========================================================== + + +# Demonstrate that taking the mean out of the Gaussian +# distribution and treating it as a constant gives +# equivalent results +y1 <- rnorm(10000, 10, 1) # mean inside the Gaussian +y2 <- 10 + rnorm(10000, 0, 1) # mean as a constant + +dens(y1, col = alpha("red", 0.5), xlab = "Outcome value") +dens(y2, col = alpha("green", 0.5), add = TRUE) + + +# Fit a binomial multilevel model predicting frog survival +# using a tank-level index variable, moving the mean out +# of the Gaussian distribution and treating it as a +# constant +m12.2.alt <- map2stan( + data = d, + alist( + surv ~ dbinom(density, p), + logit(p) <- a + a_tank[tank], + a_tank[tank] ~ dnorm(0, sigma), + a ~ dnorm(0, 1), + sigma ~ dcauchy(0, 1) + ), + chains = 4, + iter = 5000 +) + +precis(m12.2.alt, depth = 2, prob = 0.97) +precis(m12.2, depth 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