--- title: "Confidence Intervals with Bootstrapping" subtitle: | | Chapter 12 | Math 115 ##author: "Yurk" format: revealjs: theme: beige df-print: paged editor: visual --- ## Sampling Distribution ```{r} #| include: false #| echo: false library(tidyverse) library(infer) library(openintro) ``` - A **sampling distribution** is the distribution we would obtain if we could select samples of the same sample size again and again from the same population, calculating the value of the statistic of interest each time - Much of inferential statistics is based on being able to approximate sampling distributions ------------------------------------------------------------------------ - We rarely have the ability to select many samples from the same population (if we did we would usually just select a larger sample!) - However, we can *make up* a population and repeatedly sample from it to test different statistical ideas ## Candidate X ```{r} #| include: false #| echo: false true_prop_yes <- 0.6 set.seed(12345) all_polls <- tibble(poll = rep(1:1000, each = 30), vote = sample(c("no","yes"), size = 30000, replace = TRUE, prob = c(1 - true_prop_yes, true_prop_yes))) all_props <- all_polls |> group_by(poll) |> summarise(prop_yes = mean(vote == "yes")) ``` - Let us assume that 60% of US voters support Candidate X for president (so, population parameter value p = 0.6) - We repeatedly selected random samples of 30 voters from the theoretical population (and will do it 1,000 times) and calculated the proportion of supporters for each sample(statistic: $\hat{p}$) - What does the sampling distribution look like? ------------------------------------------------------------------------ - Here several random samples from the population ```{r, message=TRUE} #| include: true #| echo: false true_prop_yes <- 0.6 set.seed(12345) ten_polls <- tibble(poll = rep(1:10, each = 30), vote = sample(c("no","yes"), size = 300, replace = TRUE, prob = c(1 - true_prop_yes, true_prop_yes))) ten_props <- ten_polls |> group_by(poll) |> summarise(count_yes=sum(vote=="yes"),prop_yes = mean(vote == "yes")) ten_props ``` ------------------------------------------------------------------------ - We can begin constructing a dot plot of sample proportions ```{r} #| include: true #| echo: false #| fig-cap: "Sampling distribution. Proportions for 10 samples of 30 from a population." ten_props |> ggplot(aes(x = prop_yes)) + geom_dotplot(shape = 21, size = 4, fill = "steelblue", color = "black") + scale_y_continuous(NULL, breaks = NULL) + labs(x = "") + theme_minimal() ``` ------------------------------------------------------------------------ - Let's generate 100 proportions. What can we say about the shape and the center? ```{r, message=TRUE} #| include: true #| echo: false true_prop_yes <- 0.6 set.seed(12345) hun_polls <- tibble(poll = rep(1:100, each = 30), vote = sample(c("no","yes"), size = 3000, replace = TRUE, prob = c(1 - true_prop_yes, true_prop_yes))) hun_props <- hun_polls |> group_by(poll) |> summarise(count_yes=sum(vote=="yes"),prop_yes = mean(vote == "yes")) ``` ```{r} #| include: true #| echo: false #| fig-cap: "Sampling distribution. Proportions for 10 samples of 30 from a population." hun_props |> ggplot(aes(x = prop_yes)) + geom_dotplot(shape = 21, size = 4, fill = "steelblue", color = "black") + scale_y_continuous(NULL, breaks = NULL) + labs(x = "") + theme_minimal() ``` ------------------------------------------------------------------------ Let us generate 1000 such proportions ::: {style="font-size: 30px"} ```{r} #| include: true #| echo: false #| fig-cap: "Sampling distribution. Proportions for 1,000 samples of 30 from a population with 60% (dashed vertical line) support for Candidate X." all_props |> ggplot(aes(x = prop_yes)) + geom_dotplot(dotsize = 0.1, fill = "steelblue", color = "steelblue") + geom_vline(xintercept = true_prop_yes, color = "red", linetype = "dashed",size=2) + scale_y_continuous(NULL, breaks = NULL) + labs(x = "proportion of Candidate X supporters (samples of 30)") + theme_minimal() ``` ::: ## Standard Error ```{r} #| include: true #| echo: false se_p <- all_props |> summarize(se = sd(prop_yes)) |> pull() ``` - We can summarize the sampling distribution with a mean and standard deviation - The mean of this distribution is the true proportion of yes votes for Candidate X (0.6) - The standard deviation of a statistic's sampling distribution has a special name, the **standard error** (SE) - The standard error for the proportion of yes votes for Candidate X is SE = `r round(se_p,3)`. ## A single sample ```{r} #| include: false #| echo: false one_poll <- tibble(vote = sample(c(rep("yes", 21), rep("no", 9)))) ``` ::: {style="font-size: 30px"} A comparison of the process of sampling from the estimate infinite population and resampling with replacement from the original sample.(Fugure 12.1 from IMS2) ::: ![](https://openintrostat.github.io/ims/images/boot1prop1.png) ------------------------------------------------------------------------ - Realistically, we don't have the entire population to take samples from. We only have one sample and want to use it to construct the sampling distribution - Suppose you work for Candidate X's campaign and want to estimate the proportion of US voters that support Candidate X - You conduct a poll in which you collect one sample of 30 voters ------------------------------------------------------------------------ Results of the poll ```{r} #| include: true #| echo: true one_poll ``` ------------------------------------------------------------------------ - You find that 21 out of 30 (i.e. 70%) support Candidate X - This gives you a **point estimate** for the proportion of voters who support Candidate X (0.7) - However, we know that there will be variability from sample to sample, creating uncertainty in our estimate - We can express that uncertainty by making an **interval estimate** instead - E.g., we might estimate Candidate X's support to be between 0.55 and 0.85 based on how much the statistic is expected to vary - We can find such interval with a high level of confidence ------------------------------------------------------------------------ - Calculating a **95% confidence interval** is one way to find such an estimate - To calculate one from a single sample, we need to approximate the sampling distribution - We can use a randomization-based approach called **bootstrapping** ## Bootstrapping - In practice, our single sample is the best approximation we have of the population - We can simulate repeated random sampling from the population by randomly sampling from the sample *with replacement* - We select bootstrap samples that are the same size as the original sample - Variability tends to be very close to the variability in the sampling distribution ------------------------------------------------------------------------ ```{=html}
Original Sample (n=30): 21 Yes (70%) | 9 No (30%)
Click "Generate 1 Sample" to see bootstrap sampling step-by-step
Current Bootstrap Sample: Click "Generate 1 Sample" to start
Click "Generate 1 Sample" to see bootstrap sampling in action!
Samples Generated: 0
Current Proportion: -
Bootstrap SE: -
True SE: `r round(se_p,3)`
Bootstrap Distribution:
Generate samples to see the distribution emerge...
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Bootstrap Sample Proportion
``` ------------------------------------------------------------------------ ## ![](https://openintrostat.github.io/ims/images/boot1propboth.png) - Let's select 1,000 bootstrapped sampled using the results from our one poll ```{r} #| include: true #| echo: true one_poll_boot <- one_poll |> specify(response = vote, success = "yes") |> generate(reps = 1000, type = "bootstrap") |> calculate(stat = "prop") glimpse(one_poll_boot) ``` ------------------------------------------------------------------------ ```{r} #| include: true #| echo: false #| fig-cap: "Bootstrapped sample proportions from 1000 samples. (Sample proportion 0.7)" one_poll_boot |> ggplot(aes(x = stat)) + geom_dotplot(dotsize = 0.1,color="lightpink") + scale_y_continuous(NULL, breaks = NULL) + geom_vline(xintercept = 0.7, color = "blue",linetype = "dashed", size=2) + labs(x = "") + theme_minimal() ``` ## Bootstrapping to Estimate Standard Error - We can calculate the standard error using the bootstrapped proportions ```{r} #| include: true #| echo: false se_p_boot <- one_poll_boot |> summarize(se_boot = sd(stat)) |> pull() ``` - We can estimate the standard error using the standard deviation of the bootstrap distribution - Using the bootstrap distribution we get a standard error estimate of $SE_{boot}=$ `r round(se_p_boot,3)` - Compare this with the standard error we measured earlier using the actual sampling distribution $SE=$ `r round(se_p,3)` ## Bootstrap 95% Confidence Interval - The **95 % bootstrap percentile confidence interval** for a parameter $p$ is obtained by calculating the 2.5% and 97.5% percentiles for the bootstrapped statistics. - We say **we are 95% confident that the value of the true proportion of yes votes is between 0.533 and 0.867** - Does the interval include 0.6? ```{r} #| include: false #| echo: false bounds95CI <- one_poll_boot |> summarize(lower = quantile(stat, 0.025), upper = quantile(stat, 0.975)) bounds95CI ``` ------------------------------------------------------------------------ - The confidence interval reflects what we think are **plausible** values for the parameter - For example, it is *plausible* that 80% of people plan to vote for Candidate X. - It is *not plausible* that 50% (or less) of people plan to vote for Candidate X. - This would be good news for Candidate X. ## Other Confidence Levels ```{r} #| include: false #| echo: false one_poll_boot |> summarize(lower = quantile(stat, 0.005), upper = quantile(stat, 0.995)) ``` ::: {style="font-size: 30px"} - A 99% CI is between 0.05% and 99.9% percentiles of the bootstrap distribution. It is the interval between 0.499 and 0.9 - 99% CI is *larger* than 95% CI - It needs to be wider for us to be more confident that it contains the value of the parameter - Both intervals have the same center at ***0.7*** which is the sample proportion of the original sample ::: ## Properties of Confidence Intervals - The confidence interval will contain the observed value of the statistic. (at or near the center of the interval) ![](https://math.hope.edu/bekmetjev/219/F25/Slides/ciform.png) ## Properties of Confidence Intervals - Larger sample sizes result in narrower confidence intervals (we are more confident that the parameter is close to the point estimate if the point estimate comes from a larger sample) - If we were to repeatedly sample from the population and calculate a 95% confidence interval from each sample, about 95% of the confidence intervals would contain the true value of the parameter ## Two density plots - Let's place the population density plot and the bootstrap density plot together - As you can see, the true value of the parameter (0.6) is inside of the 95% CI ```{r} #| include: true #| echo: false both_polls = data.frame(prop=c(all_props$prop_yes, one_poll_boot$stat),type=rep(c("population","bootstrap"),c(length(all_props$prop_yes), length(one_poll_boot$stat)))) both_polls |> ggplot(aes(x = prop, fill = type)) + geom_density(alpha = 0.3) + geom_vline(xintercept = bounds95CI$lower, color = "orange",linetype = "dashed")+ geom_vline(xintercept = bounds95CI$upper, color = "orange",linetype = "dashed")+ theme(axis.text.x = element_text(size=12)) + scale_x_continuous(breaks=c(bounds95CI$lower,true_prop_yes, 0.7, bounds95CI$upper), labels=c("95% LB",true_prop_yes, 0.7, "95% UB"))+ theme(text = element_text(size = 8)) ``` ## Why Do Bootstrap Confidence Intervals Work? ![Illustration of sampling distribution and bootstrap distribution. From IMS2 Tutorial 4.4.](https://openintro.shinyapps.io/ims-04-foundations-04/_w_f147826e893e45b7b9167bc146189afe/04-04-lesson_files/figure-html/unnamed-chunk-5-1.png) - Bootstrap distribution has approximately same SE as sampling distribution ------------------------------------------------------------------------ - In sampling distribution 95% of values are within about 1.96 SE of the true value - 95% bootstrap CI includes bootstrapped values within about 1.96 SE of the observed value of the statistic - Compare - 95 bootsrap CI based on the percentiles: $$(0.533,0.867)$$ - 95% CI based on $1.96SE$: $$(0.7-1.96 \cdot 0.0822, 0.7+1.960 \cdot .0822)=(0.539, 0.861)$$