# load packages
library(tidyverse) # for data wrangling
library(ggformula) # for visualizing data
library(broom) # for nicely displaying models
library(mosaic) # for shuffling
library(scales) # for pretty axis labels
library(coursekata) # highlighting middle of distributions
library(knitr) # for neatly formatted tables
library(kableExtra) # also for neatly formatted tables
# set default theme and larger font size for ggplot2
ggplot2::theme_set(ggplot2::theme_bw(base_size = 16))SLR: Confidence Intervals
Estimating the slope via bootstrapping
Prof. Eric Friedlander
Review: Hypothesis Testing for Slope
- Last class, we learned how to test whether the slope in a simple linear regression is different from zero.
- We set up null (\(H_0: \beta_1 = 0\)) and alternative (\(H_A: \beta_1 \ne 0\)) hypotheses.
- Used permutation/randomization methods to simulate the null distribution of the slope.
- Calculated p-values to assess evidence against the null hypothesis.
- Connected hypothesis testing to estimation: today, we shift focus to quantifying uncertainty in our slope estimate using confidence intervals.
Application exercise
Complete Exercises 0–2.
Computational setup
Bootstrapped Confidence Intervals: Topics
- Find range of plausible values for the slope using bootstrap confidence intervals
Exploratory data analysis
Code
heb <- read_csv(here::here("data/HEBIncome.csv")) |>
mutate(Avg_Income_K = Avg_Household_Income/1000)
gf_point(Number_Organic ~ Avg_Income_K, data = heb, alpha = 0.7) |>
gf_labs(
x = "Average Household Income (in thousands)",
y = "Number of Organic Vegetables",
) |>
gf_refine(scale_x_continuous(labels = label_dollar()))
Modeling
# A tibble: 2 × 5
term estimate std.error statistic p.value
<chr> <dbl> <dbl> <dbl> <dbl>
1 (Intercept) -14.7 9.30 -1.58 0.122
2 Avg_Income_K 0.959 0.128 7.50 0.00000000863- Intercept: HEBs in Zip Codes with an average household income of $0 are expected to have -14.72 organic vegetable options, on average.
- Slope: For each additional $1,000 in average household income, we expect the number of organic options available at nearby HEBs to increase by 0.96, on average.
From sample to population
For each additional $1,000 in average household income, we expect the number of organic options available at nearby HEBs to increase by 0.96, on average.
- What is the goal of “statistical inference”?
- What is the goal of a “hypothesis test”?
Confidence interval for the slope
Confidence interval
- Confidence interval: plausible range of values for a population parameter
- single point estimate \(\implies\) fishing in a murky lake with a spear
- confidence interval \(\implies\) fishing with a net
- We can throw a spear where we saw a fish but we will probably miss, if we toss a net in that area, we have a good chance of catching the fish
- If we report a point estimate, we probably will not hit the exact population parameter, but if we report a range of plausible values we have a good shot at capturing the parameter
- High confidence \(\implies\) wider interval (larger net)
- Remember: single CI \(\implies\) either you hit parameter or you don’t
Confidence interval for the slope
A confidence interval will allow us to make a statement like “For each $1K in average income, the model predicts the number of organic vegetables available at local supermarkets to be higher, on average, by 0.96, plus or minus X options.”
- Should X be 1? 2? 3?
- If we were to take another sample of 37 would we expect the slope calculated based on that sample to be exactly 0.96? Off by 1? 2? 3?
- The answer depends on how variable (from one sample to another sample) the sample statistic (the slope) is
- We need a way to quantify the variability of the sample statistic
Quantify the variability of the slope
for estimation
- Two approaches:
- Via simulation (what we’ll do today)
- Via mathematical models (what we’ll do soon)
- Bootstrapping to quantify the variability of the slope for the purpose of estimation:
- Generate new samples by sampling with replacement from the original sample
- Fit models to each of the new samples and estimate the slope
- Use features of the distribution of the bootstrapped slopes to construct a confidence interval
Original Sample
Bootstrap sample 1
Bootstrap sample 2
Bootstrap sample 3
Bootstrap sample 4
Bootstrap sample 5
Bootstrap samples 1 - 5
Bootstrap samples 1 - 100
Slopes of bootstrap samples
Fill in the blank: For each additional $1k in average household income, the model predicts the number of organic vegetables available to be higher, on average, by 0.96, plus or minus ___.
Slopes of bootstrap samples
Fill in the blank: For each additional $1k in average household income, the model predicts the number of organic vegetables available to be higher, on average, by 0.96, plus or minus ___.
Confidence level
How confident are you that the true slope is between 0.8 and 1.2? How about 0.9 and 1.0? How about 1.0 and 1.4?
95% confidence interval
- 95% bootstrapped confidence interval: bounded by the middle 95% of the bootstrap distribution
- We are 95% confident that for each additional $1K in average household income, the model predicts the number of organic vegetables options at local supermarkets to be higher, on average, by 0.81 to 1.31.
Computing the CI for the slope I
Calculate the observed slope:
Computing the CI for the slope II
Computing the CI for the slope III
Take 1000 bootstrap samples and fit models to each one:
set.seed(1120)
sampling_dist <- do(1000)*lm(Number_Organic ~ Avg_Income_K, data = resample(heb))
sampling_dist Intercept Avg_Income_K sigma r.squared F numdf dendf .row
1 -23.89764420 1.0926051 17.02901 0.6610086 68.24744 1 35 1
2 -18.63197537 1.0217747 16.98311 0.6096639 54.66631 1 35 1
3 -1.95596922 0.8281509 16.79535 0.5597411 44.49868 1 35 1
4 -15.11850120 0.9514971 16.08612 0.6550748 66.47127 1 35 1
5 -15.80808227 0.9556646 19.25506 0.4807061 32.39921 1 35 1
6 -29.34800456 1.2442774 14.77653 0.6713860 71.50793 1 35 1
7 -7.95317626 0.9156941 19.23312 0.4357960 27.03430 1 35 1
8 -3.78676018 0.9041745 19.61764 0.4907528 33.72890 1 35 1
9 -27.59917266 1.1142346 19.16121 0.6050149 53.61093 1 35 1
10 -8.75142965 0.9474722 15.98968 0.7161198 88.29143 1 35 1
11 -16.74672872 0.9625598 15.74160 0.5600201 44.54909 1 35 1
12 -16.76101106 0.9598329 13.38965 0.7330256 96.09871 1 35 1
13 -19.87983364 0.9735612 14.41448 0.7822693 125.74900 1 35 1
14 -28.09906187 1.1875464 17.86724 0.4977532 34.68686 1 35 1
15 -17.43964467 1.0142925 17.48793 0.5907531 50.52295 1 35 1
16 -13.76695220 1.0060280 17.33812 0.6363068 61.23497 1 35 1
17 -19.66228602 1.0627823 17.08761 0.6642464 69.24310 1 35 1
18 -23.09048893 1.0332131 19.08924 0.5562485 43.87297 1 35 1
19 -15.06926051 0.9511575 17.35675 0.6763953 73.15663 1 35 1
20 -34.80062542 1.3186151 17.18758 0.6089659 54.50625 1 35 1
21 -3.22117789 0.8273451 17.20797 0.5745115 47.25840 1 35 1
22 -7.37017941 0.9162991 17.77949 0.5997116 52.43696 1 35 1
23 -11.07745007 0.8825488 18.17933 0.4710274 31.16600 1 35 1
24 -22.84142498 0.9896383 16.53559 0.6680004 70.42180 1 35 1
25 -18.28010691 1.0192911 15.20394 0.6837420 75.66915 1 35 1
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30 -10.16341952 0.8969878 17.45491 0.6268723 58.80166 1 35 1
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95 -15.00770934 0.9138502 14.86631 0.6857621 76.38058 1 35 1
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865 -8.68592478 0.8513617 16.57737 0.5185374 37.69516 1 35 1
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898 -7.74429079 0.9323516 20.97416 0.4069926 24.02118 1 35 1
899 -23.56010931 1.1039502 16.94090 0.6608793 68.20810 1 35 1
900 -12.41802733 0.9586005 18.92564 0.5826416 48.86077 1 35 1
901 -25.43591947 1.0783663 15.77808 0.6651110 69.51224 1 35 1
902 -20.57580954 1.0972653 16.59956 0.5705101 46.49202 1 35 1
903 -13.83936823 0.9433811 20.86470 0.4850728 32.97078 1 35 1
904 -29.51818418 1.2454162 13.32285 0.7699026 117.10947 1 35 1
905 -17.23376085 1.0373771 17.43608 0.6755441 72.87289 1 35 1
906 -27.15753645 1.1313866 15.74260 0.5794008 48.21461 1 35 1
907 -16.44238894 0.9252070 15.65347 0.7266411 93.03683 1 35 1
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909 -3.70294889 0.8329839 17.08705 0.6379534 61.67265 1 35 1
910 -7.59751782 0.8929686 18.48349 0.6039499 53.37266 1 35 1
911 -21.24462488 1.0293133 16.37474 0.6921352 78.68626 1 35 1
912 -14.71932880 0.9512565 16.56846 0.6774941 73.52516 1 35 1
913 -31.04096738 1.1861750 17.52663 0.6923848 78.77851 1 35 1
914 -16.67394901 0.9685169 16.64744 0.6441442 63.35445 1 35 1
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916 -15.81405541 0.9799170 16.47061 0.6195050 56.98543 1 35 1
917 -25.89287369 1.0942577 14.84838 0.7757780 121.09527 1 35 1
918 -6.78884358 0.8317648 17.56525 0.6178179 56.57939 1 35 1
919 -12.52672362 0.9338943 18.24587 0.6270827 58.85460 1 35 1
920 -12.11450731 0.9115059 17.61365 0.6079288 54.26951 1 35 1
921 -10.02512416 0.9323711 19.36870 0.5665153 45.74103 1 35 1
922 -11.70267292 0.8975727 17.47062 0.5697579 46.34954 1 35 1
923 -23.88046653 1.1154380 16.16483 0.5971999 51.89174 1 35 1
924 -14.58701554 0.9533674 15.42016 0.7012630 82.15993 1 35 1
925 -22.37704440 1.0397479 14.09302 0.7719361 118.46578 1 35 1
926 -9.93775929 0.8521184 16.77516 0.5428776 41.56592 1 35 1
927 -11.44307215 0.9108361 16.21526 0.5281850 39.18162 1 35 1
928 -22.29378857 1.0284844 15.19262 0.7091840 85.35102 1 35 1
929 -23.43439346 1.1096315 15.49960 0.7348978 97.02455 1 35 1
930 -23.74431783 1.1449309 14.49890 0.7611158 111.51453 1 35 1
931 -9.35667507 0.8115745 18.79985 0.5510738 42.96381 1 35 1
932 -13.66674909 0.9472573 18.05082 0.6014889 52.82690 1 35 1
933 -11.30395381 0.8207088 14.73700 0.7436162 101.51408 1 35 1
934 -21.61033628 1.0793797 19.54294 0.5527928 43.26350 1 35 1
935 -25.32823966 1.1325119 13.56794 0.7400110 99.62107 1 35 1
936 -12.75817540 0.9377911 17.85320 0.5972539 51.90338 1 35 1
937 -7.45138145 0.8630020 14.15948 0.6710016 71.38348 1 35 1
938 -3.24139882 0.8366480 19.92081 0.4727612 31.38358 1 35 1
939 -26.98568449 1.1676002 16.76762 0.6833249 75.52338 1 35 1
940 -14.16411054 0.9086287 19.36150 0.3442754 18.37607 1 35 1
941 -19.29231648 1.0759414 16.44185 0.4802740 32.34318 1 35 1
942 -15.91397111 0.9423495 14.37862 0.6841911 75.82653 1 35 1
943 -31.90377421 1.1902530 13.40969 0.7655179 114.26512 1 35 1
944 -23.76909916 1.1126138 18.39748 0.6431704 63.08604 1 35 1
945 -18.21353544 1.0717181 17.05921 0.6815048 74.89178 1 35 1
946 -16.46437986 0.9763661 17.49797 0.6240044 58.08621 1 35 1
947 -19.50915407 0.9737354 15.87747 0.7150796 87.84133 1 35 1
948 -18.30906787 0.9567812 15.21316 0.6722591 71.79167 1 35 1
949 -16.07106479 0.9966349 19.65488 0.5983318 52.13659 1 35 1
950 -27.90404822 1.1907695 13.77292 0.7727597 119.02196 1 35 1
951 13.07043112 0.6460973 17.13810 0.5057930 35.82053 1 35 1
952 -12.45366471 0.9200941 19.96182 0.4328508 26.71215 1 35 1
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970 -13.65915411 0.9225580 15.55125 0.6707169 71.29152 1 35 1
971 -12.60367536 0.9087621 17.68532 0.6287693 59.28100 1 35 1
972 -17.82927614 0.9513082 13.77138 0.7936329 134.60069 1 35 1
973 -25.27719592 1.0614644 16.05576 0.6222547 57.65502 1 35 1
974 -19.19581959 1.0131014 18.28638 0.5356230 40.36979 1 35 1
975 -6.53849900 0.8079950 18.21692 0.5725911 46.88879 1 35 1
976 -19.22818053 1.0089627 14.39348 0.7142005 87.46346 1 35 1
977 -23.87714131 1.0851655 14.99148 0.6874541 76.98353 1 35 1
978 -18.82399586 1.0280561 18.98727 0.5846990 49.27623 1 35 1
979 -6.46096019 0.8668979 17.87570 0.5960251 51.63905 1 35 1
980 -25.98361879 1.0847943 14.80378 0.7285389 93.93192 1 35 1
981 -20.60480749 0.9837385 16.43393 0.5923281 50.85336 1 35 1
982 2.08236330 0.7848888 18.41011 0.5346145 40.20647 1 35 1
983 -13.28237433 0.9812382 15.30557 0.6517487 65.50214 1 35 1
984 -1.33386466 0.8031379 16.65063 0.6267651 58.77472 1 35 1
985 -15.89238175 0.9292226 15.54902 0.6589650 67.62876 1 35 1
986 -9.02804764 0.8584256 15.10142 0.6783834 73.82521 1 35 1
987 -13.66056916 0.9569417 16.72660 0.5452032 41.95745 1 35 1
988 -1.75095434 0.8103110 21.09797 0.3591995 19.61918 1 35 1
989 -24.36577820 1.1141509 17.32097 0.5580354 44.19186 1 35 1
990 -20.53514621 1.0453352 15.98578 0.6913592 78.40042 1 35 1
991 -11.69646708 0.9138612 19.78386 0.4887573 33.46064 1 35 1
992 -19.24995281 0.9703982 18.28733 0.6112481 55.03172 1 35 1
993 -21.77521595 1.0630031 15.98053 0.7297112 94.49110 1 35 1
994 -15.55509328 0.9509214 15.62132 0.6436974 63.23110 1 35 1
995 -8.94917238 0.8877049 20.99288 0.4931599 34.05532 1 35 1
996 -9.58270334 0.8441816 19.20633 0.5565237 43.92191 1 35 1
997 -32.08346988 1.1882285 16.42322 0.5930917 51.01448 1 35 1
998 -21.91700762 1.0148209 16.78853 0.5645647 45.37933 1 35 1
999 -11.80915579 0.9348738 17.54611 0.6018152 52.89888 1 35 1
1000 -14.23039581 0.8967288 14.84229 0.7686021 116.25463 1 35 1
.index
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3 3
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5 5
6 6
7 7
8 8
9 9
10 10
11 11
12 12
13 13
14 14
15 15
16 16
17 17
18 18
19 19
20 20
21 21
22 22
23 23
24 24
25 25
26 26
27 27
28 28
29 29
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1000 1000Computing the CI for the slope IV
Take 1000 bootstrap samples and fit models to each one:
Computing the CI for the slope IV
Percentile method: Compute the 95% CI as the middle 95% of the bootstrap distribution:
name lower upper level method estimate
1 Intercept -32.2579851 0.05931893 0.95 percentile -14.7188219
2 Avg_Income_K 0.7848576 1.23204508 0.95 percentile 0.9590903
3 sigma 13.4093720 20.34317313 0.95 percentile 17.4629182
4 r.squared 0.4229805 0.77245910 0.95 percentile 0.6167334
5 F 25.6565300 118.81850829 0.95 percentile 56.3202550Complete Exercises 3-6.
Precision vs. accuracy
If we want to be very certain that we capture the population parameter, should we use a wider or a narrower interval? What drawbacks are associated with using a wider interval?
Precision vs. accuracy
How can we get best of both worlds – high precision and high accuracy?
Changing confidence level
How would you modify the following code to calculate a 90% confidence interval? How would you modify it for a 99% confidence interval?
name lower upper level method estimate
1 Intercept -32.2579851 0.05931893 0.95 percentile -14.7188219
2 Avg_Income_K 0.7848576 1.23204508 0.95 percentile 0.9590903
3 sigma 13.4093720 20.34317313 0.95 percentile 17.4629182
4 r.squared 0.4229805 0.77245910 0.95 percentile 0.6167334
5 F 25.6565300 118.81850829 0.95 percentile 56.3202550Changing confidence level
name lower upper level method estimate
1 Intercept -29.8859878 -2.405684 0.9 percentile -14.7188219
2 Avg_Income_K 0.8041053 1.181688 0.9 percentile 0.9590903
3 sigma 13.9657928 19.785630 0.9 percentile 17.4629182
4 r.squared 0.4712049 0.748848 0.9 percentile 0.6167334
5 F 31.1882126 104.357979 0.9 percentile 56.3202550 name lower upper level method estimate
1 Intercept -36.0560144 6.7885121 0.99 percentile -14.7188219
2 Avg_Income_K 0.6927970 1.3126013 0.99 percentile 0.9590903
3 sigma 12.1411724 21.6191230 0.99 percentile 17.4629182
4 r.squared 0.3463309 0.7984601 0.99 percentile 0.6167334
5 F 18.5439118 138.6650262 0.99 percentile 56.3202550Finish Activity.
Recap: Bootstrapped Confidence Intervals
- Bootstrapping allows us to estimate the variability of a statistic (like the slope) by resampling from our observed data.
- A confidence interval provides a plausible range for the population parameter, not a guarantee.
- The width of the interval depends on the variability in the data and the chosen confidence level.
- Higher confidence levels yield wider intervals; lower confidence levels yield narrower intervals.
- Simulation-based (bootstrap) and mathematical (theoretical) approaches both have value—understand when to use each.
Key takeaway: Bootstrapped confidence intervals help us quantify uncertainty in our estimates, making our conclusions more robust and transparent.
