Course Outline

segmentGetting Started (Don't Skip This Part)

segmentStatistics and Data Science: A Modeling Approach

segmentPART I: EXPLORING VARIATION

segmentChapter 1  Welcome to Statistics: A Modeling Approach

segmentChapter 2  Understanding Data

segmentChapter 3  Examining Distributions

segmentChapter 4  Explaining Variation

segmentPART II: MODELING VARIATION

segmentChapter 5  A Simple Model

segmentChapter 6  Quantifying Error

segmentChapter 7  Adding an Explanatory Variable to the Model

segmentChapter 8  Digging Deeper into Group Models

segmentChapter 9  Models with a Quantitative Explanatory Variable

segmentPART III: EVALUATING MODELS

segmentChapter 10  The Logic of Inference

segmentChapter 11  Model Comparison with F

segmentChapter 12  Parameter Estimation and Confidence Intervals

12.11 Confidence Interval for the Slope of a Regression Line

segmentChapter 13  What You Have Learned

segmentFinishing Up (Don't Skip This Part!)

segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
12.11 Confidence Interval for the Slope of a Regression Line
Letâ€™s go back to the regression model we fit using total Check
to predict Tip
. We can specify this model of the DGP like this:
\[Y_i=\beta_0+\beta_{1}X_i+\epsilon_i\]
Here is the output for the bestfitting Check
model using lm()
.
Call:
lm(formula = Tip ~ Check, data = TipExperiment)
Coefficients:
(Intercept) Check
18.74805 0.05074
Use the code window below to find the confidence interval for the slope of this regression line.
require(coursekata)
# we've created the Check model for you
Check_model < lm(Tip ~ Check, data = TipExperiment)
# find the confidence interval around the slope
# we've created the Check model for you
Check_model < lm(Tip ~ Check, data = TipExperiment)
# find the confidence interval around the slope
confint(Check_model)
ex() %>%
check_function("confint") %>%
check_result() %>%
check_equal()
2.5 % 97.5 %
(Intercept) 12.76280568 24.73328496
Check 0.02716385 0.07431286
The \(\beta_1\) represents the increment that is added to the predicted tip percent in the DGP for every additional dollar spent on the total check. The confidence interval of \(\beta_1\) represents the range of \(\beta_1\)s that would be likely to produce the sample \(b_1\). About 0.03 is the lowest \(\beta_1\) that would be likely to produce the sample \(b_1\) and 0.07 is the highest.
Now that we have tried confint()
, try using the resample()
function to bootstrap the 95% confidence interval for the slope of the regression line. See how your bootstrapped confidence interval compares to the results obtained by using confint()
.
require(coursekata)
# make a bootstrapped sampling distribution
sdob1_boot <
# we've added some code to visualize this distribution in a histogram
gf_histogram(~ b1, data = sdob1_boot, fill = ~middle(b1, .95), bins = 100)
# make a bootstrapped sampling distribution
sdob1_boot < do(1000) * b1(Tip ~ Check, data = resample(TipExperiment))
# we've added some code to visualize this distribution in a histogram
gf_histogram(~ b1, data = sdob1_boot, fill = ~middle(b1, .95), bins = 100)
ex() %>%
check_object("sdob1_boot") %>%
check_equal()
Here is a histogram of the bootstrapped sampling distribution we created. Yours will be a little different, of course, because it is random.
The center of the bootstrapped sampling distribution is approximately the same as the sample \(b_1\) of .05. This is what we would expect because bootstrapping assumes that the sample is representative of the DGP.
As explained previously, we can use the .025 cutoffs that separate the unlikely tails from the likely middle of the sampling distribution as a handy way to find the lower and upper bound of the 95% confidence interval. We can eyeball these cutoffs by looking at the histogram, or we can calculate them by arranging the bootstrapped sampling distribution to find the actual 26th and 975th b1
s.
require(coursekata)
# we have made a bootstrapped sampling distribution
sdob1_boot < do(1000) * b1(Tip ~ Check, data = resample(TipExperiment))
# modify the code below to arrange the sampling distribution in order by b1
sdob1_boot < arrange()
# find the 26th and 975th b1
sdob1_boot$b1[ ]
sdob1_boot$b1[ ]
# we have made a bootstrapped sampling distribution
sdob1_boot < do(1000) * b1(Tip ~ Check, data = resample(TipExperiment))
# modify the code below to arrange the sampling distribution in order by b1
sdob1_boot < arrange(sdob1_boot, b1)
# find the 26th and 975th b1
sdob1_boot$b1[26]
sdob1_boot$b1[975]
ex() %>% {
check_output_expr(., "sdob1_boot$b1[26]")
check_output_expr(., "sdob1_boot$b1[975]")
}
0.0195979449561059
0.0741166494222207
To find the confidence interval, we sorted the randomly generated \(b_1\)s from lowest to highest, and then used the 26th and 975th \(b_1\)s as the lower and upper bounds of the confidence interval. Your results will be a little different from ours because resampling is random. We got a bootstrapped confidence interval of about .02 to .07, which is close to what we got from confint()
(.027 and .074).
The bootstrapped sampling distribution of slopes in this case is not exactly symmetrical; it is a bit skewed to the right. For this reason, the center of the confidence interval will not be exactly at the sample \(b_1\). This is in contrast to the mathematical approach that assumes that the sample \(b_1\) is exactly in the middle of a perfectly symmetrical tdistribution. This difference does not mean that bootstrapping is less accurate. It might be that there is something about the distributions of Check
and Tip
that results in this asymmetry.
The important thing we want to focus on for now is that all of these methods result in approximately the same results. These similarities show us what confidence intervals mean and what they can tell us. Later, in more advanced courses, you can take up the question of why the results differ across methods when they do.