5 Confidence intervals

5.1 ⭐️Overview

This R Notebook contains a demonstration that is intended to help you develop an intuition for frequentist interpretations of 95% confidence intervals.

5.3 📦Load packages

library(dplyr, warn.conflicts = FALSE)
library(meantables)
library(freqtables)
library(ggplot2)
library(purrr)
library(tidyr)

5.4 🔢Simulate data

Generating 10,000 random participants. About half male and half female.

set.seed(123)
n <- 10000
random_data <- tibble(
  sex = sample(c("Female", "Male"), n, T)
)
random_data %>% 
  freq_table(sex)
## # A tibble: 2 × 9
##   var   cat        n n_total percent    se t_crit   lcl   ucl
##   <chr> <chr>  <int>   <int>   <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1 sex   Female  5017   10000    50.2 0.500   1.96  49.2  51.1
## 2 sex   Male    4983   10000    49.8 0.500   1.96  48.9  50.8

Assigning heights to the generated participants at random (within certain parameters).

set.seed(123)
random_data <- random_data %>% 
  mutate(
    height = if_else(
      sex == "Female",
      rnorm(n, 71.00, 4.32),
      rnorm(n, 64.30, 2.11)
    )
  )
random_data %>% 
  group_by(sex) %>% 
  mean_table(height) %>% 
  select(group_cat, mean, sd, lcl, ucl)
## # A tibble: 2 × 5
##   group_cat  mean    sd   lcl   ucl
##   <chr>     <dbl> <dbl> <dbl> <dbl>
## 1 Female     71.0  4.33  70.9  71.1
## 2 Male       64.3  2.1   64.2  64.4

5.5 Draw samples from population

Randomly select samples of 100 participants

# Demonstrate sample of data
set.seed(123)
sample_n(random_data, 10)
## # A tibble: 10 × 2
##    sex   height
##    <chr>  <dbl>
##  1 Male    65.8
##  2 Male    65.1
##  3 Male    68.4
##  4 Male    61.9
##  5 Male    64.0
##  6 Male    64.3
##  7 Male    64.3
##  8 Male    62.7
##  9 Male    61.2
## 10 Male    63.1
# Demonstrate multiple samples of data
set.seed(123)
map(1:2, ~ sample_n(random_data, 10))
## [[1]]
## # A tibble: 10 × 2
##    sex   height
##    <chr>  <dbl>
##  1 Male    65.8
##  2 Male    65.1
##  3 Male    68.4
##  4 Male    61.9
##  5 Male    64.0
##  6 Male    64.3
##  7 Male    64.3
##  8 Male    62.7
##  9 Male    61.2
## 10 Male    63.1
## 
## [[2]]
## # A tibble: 10 × 2
##    sex    height
##    <chr>   <dbl>
##  1 Female   66.9
##  2 Male     68.9
##  3 Female   74.1
##  4 Male     66.8
##  5 Female   71.2
##  6 Male     71.2
##  7 Male     64.8
##  8 Male     61.1
##  9 Male     62.8
## 10 Male     64.7
set.seed(137)
n_samples <- 100
random_samples <- tibble(
  sample = 1:n_samples,
  data = map(1:n_samples, ~ sample_n(random_data, 10))
)
# Demonstrate extracting a df from a list column
random_samples$data[[1]]
## # A tibble: 10 × 2
##    sex    height
##    <chr>   <dbl>
##  1 Male     61.5
##  2 Male     64.2
##  3 Female   74.3
##  4 Female   63.5
##  5 Male     60.0
##  6 Male     69.6
##  7 Male     63.1
##  8 Female   69.2
##  9 Female   62.6
## 10 Female   76.1
# Demonstrate passing an extracted df to the mean_table() function
random_samples$data[[1]] %>% 
  mean_table(height) %>% 
  select(mean, lcl, ucl)
## # A tibble: 1 × 3
##    mean   lcl   ucl
##   <dbl> <dbl> <dbl>
## 1  66.4  62.4  70.4

Calculate the stats of interest for each of the samples

random_samples <- random_samples %>% 
  mutate(
    mean_table = map(data, ~ mean_table(.x, height))
  )
# Demonstrate extracting a df of results from a list column
random_samples$mean_table[[1]]
## # A tibble: 1 × 9
##   response_var     n  mean    sd   sem   lcl   ucl   min   max
##   <chr>        <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 height          10  66.4  5.55  1.75  62.4  70.4  60.0  76.1

Unnest the results

random_samples %>% 
  unnest(mean_table)
## # A tibble: 100 × 11
##    sample data     response_var     n  mean    sd   sem   lcl   ucl   min   max
##     <int> <list>   <chr>        <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
##  1      1 <tibble> height          10  66.4  5.55 1.75   62.4  70.4  60.0  76.1
##  2      2 <tibble> height          10  67.2  4.79 1.52   63.8  70.7  60.9  73.9
##  3      3 <tibble> height          10  69.0  5.18 1.64   65.3  72.8  59.6  77.0
##  4      4 <tibble> height          10  70.6  5.46 1.73   66.7  74.5  62.4  77.9
##  5      5 <tibble> height          10  65.3  3.14 0.992  63.1  67.6  61.6  73.0
##  6      6 <tibble> height          10  66.2  5.91 1.87   61.9  70.4  60.1  76.7
##  7      7 <tibble> height          10  67    4.6  1.46   63.7  70.3  59.4  73.6
##  8      8 <tibble> height          10  68.5  4.87 1.54   65.0  72.0  60.1  76.8
##  9      9 <tibble> height          10  69.1  5.03 1.59   65.5  72.7  62.0  79.0
## 10     10 <tibble> height          10  68.2  5.69 1.80   64.2  72.3  59.5  76.5
## # ℹ 90 more rows

Count the number of CI’s that do not include the true population mean

mean_table(random_data, height)
## # A tibble: 1 × 9
##   response_var     n  mean    sd    sem   lcl   ucl   min   max
##   <chr>        <int> <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 height       10000  67.7  4.78 0.0478  67.6  67.8  56.0  87.6
summary <- random_samples %>% 
  unnest(mean_table) %>% 
  select(sample, mean, lcl, ucl)
summary <- summary %>% 
  mutate(contain_pop_mean = lcl <= 67.67 & 67.67 <= ucl)
summary
## # A tibble: 100 × 5
##    sample  mean   lcl   ucl contain_pop_mean
##     <int> <dbl> <dbl> <dbl> <lgl>           
##  1      1  66.4  62.4  70.4 TRUE            
##  2      2  67.2  63.8  70.7 TRUE            
##  3      3  69.0  65.3  72.8 TRUE            
##  4      4  70.6  66.7  74.5 TRUE            
##  5      5  65.3  63.1  67.6 FALSE           
##  6      6  66.2  61.9  70.4 TRUE            
##  7      7  67    63.7  70.3 TRUE            
##  8      8  68.5  65.0  72.0 TRUE            
##  9      9  69.1  65.5  72.7 TRUE            
## 10     10  68.2  64.2  72.3 TRUE            
## # ℹ 90 more rows
summary %>% 
  freq_table(contain_pop_mean)
## # A tibble: 2 × 9
##   var              cat       n n_total percent    se t_crit   lcl   ucl
##   <chr>            <chr> <int>   <int>   <dbl> <dbl>  <dbl> <dbl> <dbl>
## 1 contain_pop_mean FALSE     5     100       5  2.19   1.98  2.06  11.6
## 2 contain_pop_mean TRUE     95     100      95  2.19   1.98 88.4   97.9