From 55f21757f0def883b91ff0496d340011d6aa9128 Mon Sep 17 00:00:00 2001 From: eveskew Date: Sun, 8 Mar 2020 21:06:44 -0400 Subject: [PATCH] Adding week 07 lecture and lab --- lab/lab_week_07.Rmd | 66 ++++++ lecture/lecture_week_07.R | 433 +++++++++++++++++++++++++++++++++++ lecture/lecture_week_07.pptx | Bin 0 -> 6580654 bytes 3 files changed, 499 insertions(+) create mode 100644 lab/lab_week_07.Rmd create mode 100644 lecture/lecture_week_07.R create mode 100644 lecture/lecture_week_07.pptx diff --git a/lab/lab_week_07.Rmd b/lab/lab_week_07.Rmd new file mode 100644 index 0000000..7609595 --- /dev/null +++ b/lab/lab_week_07.Rmd @@ -0,0 +1,66 @@ +--- +title: "EEEB UN3005/GR5005 \nLab - Week 07 - 09 and 11 March 2020" +author: "USE YOUR NAME HERE" +output: pdf_document +fontsize: 12pt +--- + +```{r setup, include = FALSE} + +knitr::opts_chunk$set(echo = TRUE) + +library(rethinking) +``` + + +# Multiple Regression Models + +For this week's lab, we'll be using data on foxes occupying urban England. You can access this data once you've loaded the `rethinking` package with the code `data(foxes)`. You can get a brief summary of all the variables contained in the dataset using `?foxes` or a data frame summary function of your choice. + +Note, for all models you'll be asked to construct here, assume a prior of `dnorm(0, 10)` for all intercept and slope parameters and a prior of `dunif(0, 10)` for all standard deviation parameters. Furthermore, start values of 0 for all intercept and slope parameters and 5 for all standard deviation parameters should ensure good model fits throughout these exercises. + + +## Exercise 1: Linear Regression with Territory Size as a Predictor + +Construct a linear regression model of fox body weight (`weight` variable) using territory size (`area` variable) as a predictor. + +After fitting the model, use `precis()` to display the 90% PIs for all model parameters. Plot the results of this regression model by displaying the fox body weight and territory size variables and the MAP regression line. + +What does this model suggest about the effect of territory size on fox body weight? + +```{r} + +``` + + +## Exercise 2: Linear Regression with Group Size as a Predictor + +Now, construct a linear regression model of fox body weight using fox group size (`groupsize` variable) as a predictor. + +After fitting the model, use `precis()` to display the 90% PIs for all model parameters. Plot the results of this regression model by displaying the fox body weight and group size variables and the MAP regression line. + +What does this model suggest about the effect of group size on fox body weight? + +```{r} + +``` + + +## Exercise 3: Multiple Regression with Both Predictors + +Now fit a multiple regression with fox body weight as the outcome and both territory size and group size as predictor variables. + +What does this model say about the importance of each predictor variable? Why do you get different results than you got in the exercises above? + +```{r} + +``` + + +## Exercise 4: Counterfactual Plots for a Multiple Regression + +For the multiple regression model, generate counterfactual plots that show the mean trend line and the 95% HPDI of the mean for each predictor, holding the other predictor constant at its average value. + +```{r} + +``` diff --git a/lecture/lecture_week_07.R b/lecture/lecture_week_07.R new file mode 100644 index 0000000..c818482 --- /dev/null +++ b/lecture/lecture_week_07.R @@ -0,0 +1,433 @@ + + +library(rethinking) + +#========================================================== + + +# Linear regression of milk energy data + +# Single predictor: neocortex percentage (np) + + +# Prepare milk energy data for analysis +data(milk) +d <- milk +dcc <- d[complete.cases(d), ] + +# Plot raw data +plot(kcal.per.g ~ neocortex.perc, + data = dcc, col = "cornflowerblue", pch = 19, + xlim = c(0, 100), ylim = c(0, 1)) + +# Visualize priors for intercept, slope, and standard +# deviation parameters +curve( + dnorm(x, 0, 100), + from = -300, to = 300, + main = "Prior for a (intercept)", + xlab = "a", + ylab = "Density" +) +curve( + dnorm(x, 0, 1), + from = -10, to = 10, + main = "Prior for bn (slope)", + xlab = "bn", + ylab = "Density" +) +curve( + dunif(x, 0, 1), + from = -5, to = 5, + main = "Prior for sigma", + xlab = "sigma", + ylab = "Density" +) + +# Fit linear regression using map() +m5.5 <- map( + data = dcc, # specify data to fit + alist( + kcal.per.g ~ dnorm(mu, sigma), # likelihood + mu <- a + bn*neocortex.perc, + a ~ dnorm(0, 100), # prior for the intercept + bn ~ dnorm(0, 1), # prior for the np effect + sigma ~ dunif(0, 1) # prior for the standard deviation + ) +) + +# Summarize model output +precis(m5.5, digits = 3, prob = 0.9) + +# Get posterior samples +post1 <- extract.samples(m5.5, n = 10000) + +# Plot posterior samples +dens( + post1$a, + main = "Posterior for a (intercept)", + xlab = "a", + show.HPDI = 0.9 +) +dens( + post1$bn, + main = "Posterior for bn (slope)", + xlab = "bn", + show.HPDI = 0.9 +) +# Correlation of intercept and slope +plot( + post1$a, post1$bn, + xlab = "a (intercept)", ylab = "bn (slope)", + pch = 19 +) + + +# Counterfactual plot + + +# Generate counterfactual data +summary(dcc$neocortex.perc) +np.seq <- 0:100 # sequence of np values to use +pred.data <- data.frame(neocortex.perc = np.seq) + +# Generate values for Gaussian mean +mu <- link(m5.5, data = pred.data, n = 10000) +mu.mean <- apply(mu, 2, mean) +mu.PI <- apply(mu, 2, PI, prob = 0.95) + +# Plotting counterfactuals +plot(kcal.per.g ~ neocortex.perc, + data = dcc, type = "n", + xlim = c(0, 100), ylim = c(0, 1)) +lines(np.seq, mu.mean) +shade(mu.PI, np.seq) + +#========================================================== + + +# Linear regression of milk energy data + +# Single predictor: log body mass + + +# Prepare milk energy data for analysis +dcc$log.mass <- log(dcc$mass) + +# Plot raw data +plot(kcal.per.g ~ log.mass, + data = dcc, col = "cornflowerblue", pch = 19, + xlim = c(-5, 5), ylim = c(0, 1)) + +# Fit linear regression using map() +m5.6 <- map( + data = dcc, # specify data to fit + alist( + kcal.per.g ~ dnorm(mu, sigma), + mu <- a + bm*log.mass, + a ~ dnorm(0, 100), # prior for the intercept + bm ~ dnorm(0, 1), # prior for the body mass effect + sigma ~ dunif(0, 1) # prior for the standard deviation + ) +) + +# Summarize model output +precis(m5.6, digits = 3, prob = 0.9) + +# Get posterior samples +post2 <- extract.samples(m5.6, n = 10000) + +# Plot posterior samples +dens( + post2$a, + main = "Posterior for a (intercept)", + xlab = "a", + show.HPDI = 0.9 +) +dens( + post2$bm, + main = "Posterior for bm (slope)", + xlab = "bm", + show.HPDI = 0.9 +) +# Correlation of intercept and slope +plot( + post2$a, post2$bm, + xlab = "a (intercept)", ylab = "bm (slope)", + pch = 19 +) + + +# Counterfactual plot + + +# Generate counterfactual data +summary(dcc$log.mass) +lbm.seq <- seq(from = -5, to = 5, length.out = 100) +pred.data <- data.frame(log.mass = lbm.seq) + +# Generate values for Gaussian mean +mu <- link(m5.6, data = pred.data, n = 10000) +mu.mean <- apply(mu, 2, mean) +mu.PI <- apply(mu, 2, PI, prob = 0.95) + +# Plotting counterfactuals +plot(kcal.per.g ~ log.mass, + data = dcc, type = "n", + xlim = c(-5, 5), ylim = c(0, 1)) +lines(lbm.seq, mu.mean) +shade(mu.PI, lbm.seq) + +#========================================================== + + +# Multiple regression of milk energy data + +# Two predictors: neocortex percentage and log body mass + + +# Fit linear regression using map() +m5.7 <- map( + data = dcc, # specify data to fit + alist( + kcal.per.g ~ dnorm(mu, sigma), + mu <- a + bn*neocortex.perc + bm*log.mass, + a ~ dnorm(0, 100), # prior for the intercept + bn ~ dnorm(0, 1), # prior for the np effect + bm ~ dnorm(0, 1), # prior for the body mass effect + sigma ~ dunif(0, 1) # prior for the standard deviation + ) +) + +# Summarize model output +precis(m5.7, digits = 3, prob = 0.9) + +# Get posterior samples +post3 <- extract.samples(m5.7, n = 10000) + +# Plot posterior samples +dens( + post3$a, + main = "Posterior for a (intercept)", + xlab = "a", + show.HPDI = 0.9 +) +dens( + post3$bn, + main = "Posterior for bn (slope)", + xlab = "bn", + show.HPDI = 0.9 +) +dens( + post3$bm, + main = "Posterior for bm (slope)", + xlab = "bm", + show.HPDI = 0.9 +) + + +# Compare posteriors for the slope parameters from the +# first two regressions (black lines) with the multiple +# regression (red lines) +par(mfrow = c(1, 2)) +dens( + post1$bn, + main = "Posterior for bn (slope)", + xlab = "bn", + xlim = c(-0.05, 0.08) +) +dens( + post3$bn, + col = "red", add = TRUE +) +dens( + post2$bm, + main = "Posterior for bm (slope)", + xlab = "bm", + xlim = c(-0.2, 0.05) +) +dens( + post3$bm, + col = "red", add = TRUE +) + + +# Counterfactual plots + + +# For neocortex percentage +mean.log.mass <- mean(dcc$log.mass) +pred.data <- + data.frame(neocortex.perc = np.seq, + log.mass = mean.log.mass) + +mu <- link(m5.7, data = pred.data, n = 10000) +mu.mean <- apply(mu, 2, mean) +mu.PI <- apply(mu, 2, PI, prob = 0.95) + +plot(kcal.per.g ~ neocortex.perc, + data = dcc, type = "n", + xlim = c(0, 100), ylim = c(0, 2)) +lines(np.seq, mu.mean) +shade(mu.PI, np.seq) + +# For log body mass +mean.neocortex.perc <- mean(dcc$neocortex.perc) +pred.data <- + data.frame(neocortex.perc = mean.neocortex.perc, + log.mass = lbm.seq) + +mu <- link(m5.7, data = pred.data, n = 10000) +mu.mean <- apply(mu, 2, mean) +mu.PI <- apply(mu, 2, PI, prob = 0.95) + +plot(kcal.per.g ~ log.mass, + data = dcc, type = "n", + xlim = c(-5, 5), ylim = c(0, 2)) +lines(lbm.seq, mu.mean) +shade(mu.PI, lbm.seq) + +par(mfrow = c(1, 1)) + +#========================================================== + + +# Computing predictor residuals and using these to fit +# multiple bivariate regressions + + +# Residual neocortex percentage, after accounting for +# log body mass + +plot(neocortex.perc ~ log.mass, data = dcc, + col = "cornflowerblue", pch = 19) + +# Using lm() as a shortcut to visualize best fit line +# and compute residuals +fit <- lm(neocortex.perc ~ log.mass, data = dcc) +abline(fit) +mu <- coef(fit)[1] + coef(fit)[2]*dcc$log.mass +dcc$np.residual <- resid(fit) + +for (i in 1:length(dcc$np.residual)) { + + x <- dcc$log.mass[i] # x location of line segment + y <- dcc$neocortex.perc[i] # observed endpoint of line segment + # draw the line segments + lines(c(x, x), c(mu[i], y), + lwd = 0.8, col = col.alpha("black", 0.8)) +} + +# Fit a linear regression using neocortex percentage +# RESIDUALS as a predictor of milk energy +m.np.residual <- map( + data = dcc, + alist( + kcal.per.g ~ dnorm(mu, sigma), + mu <- a + bnr*np.residual, + a ~ dnorm(0, 100), + bnr ~ dnorm(0, 1), + sigma ~ dunif(0, 1) + ) +) + + +# Residual log body mass, after accounting for neocortex +# percentage + +plot(log.mass ~ neocortex.perc, data = dcc, + col = "cornflowerblue", pch = 19) + +# Using lm() as a shortcut to visualize best fit line +# and compute residuals +fit <- lm(log.mass ~ neocortex.perc, data = dcc) +abline(fit) +mu <- coef(fit)[1] + coef(fit)[2]*dcc$neocortex.perc +dcc$lbm.residual <- resid(fit) + +for (i in 1:length(dcc$lbm.residual)) { + + x <- dcc$neocortex.perc[i] # x location of line segment + y <- dcc$log.mass[i] # observed endpoint of line segment + # draw the line segments + lines(c(x, x), c(mu[i], y), + lwd = 0.8, col = col.alpha("black", 0.8)) +} + +# Fit a linear regression using log body mass RESIDUALS +# as a predictor of milk energy +m.lbm.residual <- map( + data = dcc, + alist( + kcal.per.g ~ dnorm(mu, sigma), + mu <- a + bmr*lbm.residual, + a ~ dnorm(0, 100), + bmr ~ dnorm(0, 1), + sigma ~ dunif(0, 1) + ) +) + + +# Compare fit models using residual predictor variables +# with the multiple regression using both original +# variables +precis(m.np.residual) +precis(m.lbm.residual) +precis(m5.7) + +# The take home message is that the multiple regression +# using both of the predictor variables effectively +# controls for the values of the other predictor, giving +# us equivalent results to what we would see if we ran a +# bivariate regression with predictor residuals + +#========================================================== + + +# Linear regression with a categorical predictor variable + + +# Import Howell data +data(Howell1) +d <- Howell1 +str(d) +d$male + +# In this case, "male" is already a dummy variable, but we +# could easily create it with an "ifelse()" statement if +# it was a string variable (see book for examples) + +# Fit a linear regression using sex as a predictor of +# height +m5.15 <- map( + data = d, # specify data to fit + alist( + height ~ dnorm(mu, sigma), + mu <- a + bm*male, + a ~ dnorm(178, 100), # intercept prior + bm ~ dnorm(0, 10), # (male) dummy variable prior + sigma ~ dunif(0, 50) # standard deviation prior + ) +) + +# Summarize the fit model +precis(m5.15) + +# Generate posterior samples 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