From d3307c84bff14486eb345cf90adf9f79120a61c1 Mon Sep 17 00:00:00 2001 From: Ariel Rokem Date: Sat, 20 Apr 2024 21:35:07 -0700 Subject: [PATCH] Adds more images/content to stats lecture. --- slides/04-stats-with-big-data.qmd | 41 +++++++--- slides/code/stats_demo.r | 94 +++++++++++------------ slides/images/loftus_comparison.png | Bin 0 -> 291414 bytes slides/images/loftus_lowry.png | Bin 0 -> 67961 bytes slides/images/loftus_plot_with_error.png | Bin 0 -> 464919 bytes slides/images/loftus_plots.png | Bin 0 -> 443458 bytes 6 files changed, 75 insertions(+), 60 deletions(-) create mode 100644 slides/images/loftus_comparison.png create mode 100644 slides/images/loftus_lowry.png create mode 100644 slides/images/loftus_plot_with_error.png create mode 100644 slides/images/loftus_plots.png diff --git a/slides/04-stats-with-big-data.qmd b/slides/04-stats-with-big-data.qmd index f7ff8fb..2d672fc 100644 --- a/slides/04-stats-with-big-data.qmd +++ b/slides/04-stats-with-big-data.qmd @@ -69,10 +69,38 @@ Data can be "big" in two different ways: # Plotting your data +![](./images/loftus_plots.png) +# Using confidence intervals to assess results + +![](./images/loftus_plot_with_error.png) + +# Using confidence intervals to assess results + +![](./images/loftus_comparison.png) + +# Using confidence intervals to assess results + +![](./images/loftus_lowry.png) + +# Explicit models + +- For example, the model that Dr. Loeb used. +- Analogies from physics are useful, because physical law provides mathematical formulations of relationships. +- Or "planned comparisons": + - Come up with a quantitative hypothesis + - Assign weights for each condition (to sum to 0) + - Compute the correlation between weights and conditional means. # Computing to the rescue +::: {.fragment} + +![](./images/computers_in_1983.png) + +::: + + # Resampling methods - Jackknife @@ -136,15 +164,6 @@ Invented by Bradley Efron - And other complex procedures. - Efron argues that this is the natural procedure Fisher et al. would have preferred in the 20's if they had computers. -# When Efron talks about computers - -::: {.fragment} - -He's talking about this: - -![](./images/computers_in_1983.png) - -::: # Demo @@ -173,14 +192,12 @@ by Tim Hesterberg. # The curse of dimensionality -What about large $p$ +What about large $p$? - There be dragons - # Data is sparser in higher dimensions - # The distance between points increases rapidly # When $p$ > $n$ multi-colinearity exists diff --git a/slides/code/stats_demo.r b/slides/code/stats_demo.r index a93c743..e933424 100644 --- a/slides/code/stats_demo.r +++ b/slides/code/stats_demo.r @@ -5,6 +5,7 @@ # and helped correct string formatting issues and debug broadcasting errors. library(boot) +library(ggplot2) # Function to calculate the mean: calc_mean <- function(data, indices) { @@ -93,36 +94,67 @@ cat(paste( "Jackknife bias: ", jackknife_bias[ii], "\n", sep = "")) -# Next, bootstrapping +###################################################################### +# Bootstrapping # Use the boot function to perform bootstrapping boot_results <- boot(data, calc_mean, R = 10000) -# Calculate the standard error -std_error <- sd(boot_results$t) -print(paste("Standard Error of the Mean: ", std_error)) +sorted_boot_results <- sort(boot_results$t) + +# Plot sorted results with ggplot +ggplot(data.frame(sorted_boot_results), + aes(x=seq_along(sorted_boot_results), + y = sorted_boot_results)) + + geom_line() + +# Calculate standard error (central 68% of the distribution ): + +ci <- quantile(sorted_boot_results, c(0.16, 0.84)) +std_err <- 0.5 * (ci[2] - ci[1]) +cat(paste( + "Standard Error of the Mean: ", std_error, "\n")) + +# Calculate standard error (std of the distribution): +std_err <- sd(sorted_boot_results) +cat(paste( + "Standard Error of the Mean: ", std_error, "\n")) +# Calculate 95% CI: +ci <- quantile(sorted_boot_results, c(0.025, 0.975)) -# Define the non-linear model function -nonlinear_model <- function(x, a, tau) { - 1 - a * exp(x/tau) +cat(paste( + "95% Confidence Interval: ", ci[1], ci[2], "\n")) + +# Define a non-linear model function +nonlinear_model <- function(x, alpha, beta, kappa) { + alpha - (alpha - beta) * exp(- x / kappa) } + # Generate some sample data set.seed(123) -x <- seq(-10, 10, length.out = 100) -y <- nonlinear_model(x, 2, 3) + rnorm(length(x), mean = 0, sd = 2) +x <- seq(1, 100, length.out = 100) +y <- nonlinear_model(x, 100, 10, 20) + rnorm(length(x), mean = 0, sd = 2) + +ggplot(data.frame(x = x, y = y), aes(x = x, y = y)) + + geom_line() # Define the function to estimate the parameters using the non-linear model estimate_parameters <- function(data, indices) { x <- data$X[indices] y <- data$Y[indices] - fit <- nls(y ~ nonlinear_model(x, a, tau), start = list(a = 1, tau = 1)) + fit <- nls( + y ~ nonlinear_model(x, alpha, beta, kappa), + start = list( + alpha = 100, beta = 10, kappa = 20)) coef(fit) } # Perform bootstrap estimation -boot_results <- boot(data = data.frame(X = x, Y = y), statistic = estimate_parameters, R = 1000) +boot_results <- boot( + data = data.frame(X = x, Y = y), + statistic = estimate_parameters, R = 10000) # Get the bootstrap estimates of the parameters bootstrap_estimates <- boot_results$t @@ -130,42 +162,8 @@ bootstrap_estimates <- boot_results$t # Calculate the variance of the bootstrap estimates variance_estimates <- apply(bootstrap_estimates, 2, var) -# Print the variance estimates -print(variance_estimates) - -# Plot the bootstrap distribution of the parameters with ggplot: -library(ggplot2) -bootstrap_df <- data.frame(a = bootstrap_estimates[,1], tau = bootstrap_estimates[,2]) -ggplot(bootstrap_df, aes(x = a)) + - geom_histogram(binwidth = 0.1, fill = "lightblue", color = "black") + - labs(title = "Bootstrap Distribution of Parameter a") + - theme_minimal() - - -ggplot(bootstrap_df, aes(x = tau)) + - geom_histogram(binwidth = 0.1, fill = "lightblue", color = "black") + - labs(title = "Bootstrap Distribution of Parameter tau") + - theme_minimal() - # Calculate the confidence intervals for the parameters -conf_intervals <- boot.ci(boot_results[, 1], type = "bca", index = 1:2) +conf_intervals <- boot.ci(boot_results, + conf=0.95, + type = "prec") print(conf_intervals) - -# Jacknife example -# Generate a non-Gaussian distribution -set.seed(123) -data <- rexp(1000, rate = 0.5) - - -# Initialize a vector to store the jackknife estimates -jackknife_estimates <- numeric(length(data)) - - -# Calculate with bootstrap: -# Define a function to calculate the mean -calc_mean <- function(data, indices) { - return(mean(data[indices])) -} - -# Use the boot function to perform bootstrapping -boot_results <- boot(data, calc_mean, R = 1000) diff --git a/slides/images/loftus_comparison.png b/slides/images/loftus_comparison.png new file mode 100644 index 0000000000000000000000000000000000000000..c472400dc1132536824d75d87c692b3ff6505886 GIT binary patch literal 291414 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