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library(rethinking) | ||
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#========================================================== | ||
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# Load the reed frog data | ||
data(reedfrogs) | ||
d <- reedfrogs | ||
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# Make the tank cluster variable | ||
d$tank <- 1:nrow(d) | ||
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# Fit a binomial GLM predicting frog survival using a | ||
# tank-level index variable | ||
m12.1 <- map2stan( | ||
data = d, | ||
alist( | ||
surv ~ dbinom(size = density, prob = p), | ||
logit(p) <- a_tank[tank], | ||
a_tank[tank] ~ dnorm(0, 5) | ||
), | ||
chains = 4, | ||
iter = 5000 | ||
) | ||
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precis(m12.1, depth = 2, prob = 0.97) | ||
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# Fit a binomial multilevel model predicting frog survival | ||
# using a tank-level index variable | ||
m12.2 <- map2stan( | ||
data = d, | ||
alist( | ||
surv ~ dbinom(size = density, prob = p), | ||
logit(p) <- a_tank[tank], | ||
a_tank[tank] ~ dnorm(a, sigma), | ||
a ~ dnorm(0, 1), | ||
sigma ~ dcauchy(0, 1) | ||
), | ||
chains = 4, | ||
iter = 5000 | ||
) | ||
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precis(m12.2, depth = 2, prob = 0.97) | ||
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#========================================================== | ||
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# Show shrinkage of multilevel model parameter estimates | ||
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# Extract posterior samples from model m12.1 | ||
post.glm <- extract.samples(m12.1) | ||
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# Extract posterior samples from model m12.2 | ||
post.glmm <- extract.samples(m12.2) | ||
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# Note: this will not work since you're asking for more | ||
# samples than you have | ||
post.glmm <- extract.samples(m12.2, n = 20000) | ||
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# Compute median intercept for each tank and transform to | ||
# probability with "logistic()" | ||
d$propsurv.est.glm <- | ||
logistic(apply(post.glm$a_tank, 2, median)) | ||
d$propsurv.est.glmm <- | ||
logistic(apply(post.glmm$a_tank, 2, median)) | ||
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# Display raw proportions surviving in each tank | ||
plot(d$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)) | ||
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# Overlay posterior medians from the GLM... | ||
points(d$propsurv.est.glm, col = "darkred") | ||
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# Display raw proportions surviving in each tank | ||
plot(d$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)) | ||
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# Overlay posterior medians from the mulitlevel model | ||
points(d$propsurv.est.glmm) | ||
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# Mark posterior median probability across tanks | ||
abline(h = logistic(median(post.glmm$a)), lty = 2) | ||
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# 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") | ||
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# You could get a very similar plot using "coef()" to | ||
# extract posterior means for parameters | ||
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# Display raw proportions surviving in each tank | ||
plot(d$propsurv, ylim = c(0, 1), pch = 16, xaxt = "n", | ||
xlab = "tank", | ||
ylab = "proportion survival", | ||
col = rangi2) | ||
axis(1, at = c(1, 16, 32, 48), | ||
labels = c(1, 16, 32, 48)) | ||
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# Overlay posterior means from the multilevel model | ||
points(logistic(coef(m12.2)[1:48])) | ||
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# Mark posterior mean probability across tanks | ||
abline(h = logistic(mean(coef(m12.2)["a"])), lty = 2) | ||
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# Draw vertical dividers between tank densities | ||
abline(v = 16.5, lwd = 0.5) | ||
abline(v = 32.5, lwd = 0.5) | ||
text(8, 0, "small tanks") | ||
text(16+8, 0, "medium tanks") | ||
text(32+8, 0, "large tanks") | ||
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#========================================================== | ||
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# Simulating frog survival data among "ponds" | ||
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a <- 1.4 # average pond's log-odds of survival | ||
sigma <- 1.5 # survival standard deviation among ponds | ||
nponds <- 60 # number of ponds to simulate | ||
ni <- # sample size across ponds | ||
as.integer(rep(c(5, 10, 25, 35), each = 15)) | ||
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# Simulate "true" pond-level log-odds of survival | ||
a_pond <- rnorm(nponds, mean = a, sd = sigma) | ||
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# Package data into a data frame | ||
dsim <- data.frame( | ||
pond = 1:nponds, | ||
ni = ni, | ||
true_a = a_pond | ||
) | ||
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# Simulate survivors across ponds | ||
dsim$si <- rbinom(nponds, size = dsim$ni, | ||
prob = logistic(dsim$true_a)) | ||
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# 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) | ||
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# 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") | ||
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# Draw horizontal line representing the average | ||
# pond's true proportion survival | ||
abline(h = logistic(a), lty = 2) | ||
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#========================================================== | ||
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# 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 | ||
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dens(y1, col = alpha("red", 0.5), xlab = "Outcome value") | ||
dens(y2, col = alpha("green", 0.5), add = TRUE) | ||
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# 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 | ||
) | ||
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precis(m12.2.alt, depth = 2, prob = 0.97) | ||
precis(m12.2, depth = 2, prob = 0.97) |
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