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Computational setup

# load packageslibrary(tidyverse) # for data wrangling and visualizationlibrary(tidymodels) # for modelinglibrary(usdata) # for the county_2019 datasetlibrary(scales) # for pretty axis labelslibrary(glue) # for constructing character strings# set default theme and larger font size for ggplot2ggplot2::theme_set(ggplot2::theme_minimal(base_size =16))

These data have been compiled from the 2019 American Community Survey

Uninsurance rate

High school graduation rate

Examining the relationship

The NC Labor and Economic Analysis Division (LEAD), which â€śadministers and collects data, conducts research, and publishes information on the stateâ€™s economy, labor force, educational, and workforce-related issuesâ€ť.

Suppose that an analyst working for LEAD is interested in the relationship between uninsurance and high school graduation rates in NC counties.

What type of visualization should the analyst make to examine the relationship between these two variables?

# A tibble: 100 Ă— 3
name hs_grad uninsured
<chr> <dbl> <dbl>
1 Alamance County 86.3 11.2
2 Alexander County 82.4 8.9
3 Alleghany County 77.5 11.3
4 Anson County 80.7 11.1
5 Ashe County 85.1 12.6
6 Avery County 83.6 15.9
7 Beaufort County 87.7 12
8 Bertie County 78.4 11.9
9 Bladen County 81.3 12.9
10 Brunswick County 91.3 9.8
# â€¦ with 90 more rows

Uninsurance vs. HS graduation rates

Code

ggplot(county_2019_nc,aes(x = hs_grad, y = uninsured)) +geom_point() +scale_x_continuous(labels =label_percent(scale =1, accuracy =1)) +scale_y_continuous(labels =label_percent(scale =1, accuracy =1)) +labs(x ="High school graduate", y ="Uninsured",title ="Uninsurance vs. HS graduation rates",subtitle ="North Carolina counties, 2015 - 2019" ) +geom_point(data = county_2019_nc %>%filter(name =="Durham County"), aes(x = hs_grad, y = uninsured), shape ="circle open", color ="#8F2D56", size =4, stroke =2) +geom_text(data = county_2019_nc %>%filter(name =="Durham County"), aes(x = hs_grad, y = uninsured, label = name), color ="#8F2D56", fontface ="bold", nudge_y =3, nudge_x =2)

Modeling the relationship

Code

ggplot(county_2019_nc, aes(x = hs_grad, y = uninsured)) +geom_point() +geom_smooth(method ="lm", se =FALSE, color ="#8F2D56") +scale_x_continuous(labels =label_percent(scale =1, accuracy =1)) +scale_y_continuous(labels =label_percent(scale =1, accuracy =1)) +labs(x ="High school graduate", y ="Uninsured",title ="Uninsurance vs. HS graduation rates",subtitle ="North Carolina counties, 2015 - 2019" )

Fitting the model

With fit():

nc_fit <-linear_reg() %>%set_engine("lm") %>%fit(uninsured ~ hs_grad, data = county_2019_nc)tidy(nc_fit)

What indicates a good model fit? Higher or lower \(R^2\)? Higher or lower RMSE?

R-squared

Ranges between 0 (terrible predictor) and 1 (perfect predictor)

Unitless (Having no units of measurement; such as a ratio or percentage of two numbers which have the same units.)

Calculate with rsq():

rsq(nc_aug, truth = uninsured, estimate = .fitted)

# A tibble: 1 Ă— 3
.metric .estimator .estimate
<chr> <chr> <dbl>
1 rsq standard 0.243

Interpreting R-squared

đź—łď¸Ź Vote

The \(R^2\) of the model for predicting uninsurance rate from high school graduation rate for NC counties is 24.3%. Which of the following is the correct interpretation of this value?

High school graduation rates correctly predict 24.3% of uninsurance rates in NC counties.

24.3% of the variability in uninsurance rates in NC counties can be explained by high school graduation rates.

24.3% of the variability in high school graduation rates in NC counties can be explained by uninsurance rates.

24.3% of the time uninsurance rates in NC counties can be predicted by high school graduation rates.

Alternative approach for R-squared

Alternatively, use glance() to construct a single row summary of the model fit, including \(R^2\):

Ranges between 0 (perfect predictor) and infinity (terrible predictor)

Same units as the outcome variable

Calculate with rmse():

rmse(nc_aug, truth = uninsured, estimate = .fitted)

# A tibble: 1 Ă— 3
.metric .estimator .estimate
<chr> <chr> <dbl>
1 rmse standard 2.07

The value of RMSE is not very meaningful on its own, but itâ€™s useful for comparing across models (more on this when we get to regression with multiple predictors)

Obtaining R-squared and RMSE

Use rsq() and rmse(), respectively

rsq(nc_aug, truth = uninsured, estimate = .fitted)rmse(nc_aug, truth = uninsured, estimate = .fitted)

First argument: data frame containing truth and estimate columns

Second argument: name of the column containing truth (observed outcome)

Third argument: name of the column containing estimate (predicted outcome)

Purpose of model evaluation

\(R^2\) tells us how our model is doing to predict the data we already have

But generally we are interested in prediction for a new observation, not for one that is already in our sample, i.e. out-of-sample prediction

We have a couple ways of simulating out-of-sample prediction before actually getting new data to evaluate the performance of our models

Splitting data

Spending our data

There are several steps to create a useful model: parameter estimation, model selection, performance assessment, etc.

Doing all of this on the entire data we have available leaves us with no other data to assess our choices

We can allocate specific subsets of data for different tasks, as opposed to allocating the largest possible amount to the model parameter estimation only (which is what weâ€™ve done so far)

Simulation: data splitting

Take a random sample of 10% of the data and set aside (testing data)

Fit a model on the remaining 90% of the data (training data)

Use the coefficients from this model to make predictions for the testing data

Repeat 10 times

Predictive performance

How consistent are the predictions for different testing datasets?

How consistent are the predictions for counties with high school graduation rates in the middle of the plot vs. in the edges?

Bootstrapping

Bootstrapping our data

The idea behind bootstrapping is that if a given observation exists in a sample, there may be more like it in the population

With bootstrapping, we simulate resampling from the population by resampling from the sample we observed

Bootstrap samples are the sampled with replacement from the original sample and same size as the original sample

For example, if our sample consists of the observations {A, B, C}, bootstrap samples could be {A, A, B}, {A, C, A}, {B, C, C}, {A, B, C}, etc.

Simulation: bootstrapping

Take a bootstrap sample â€“ sample with replacement from the original data, same size as the original data

Fit model to the sample and make predictions for that sample

Repeat many times

Predictive performance

How consistent are the predictions for different bootstrap datasets?

How consistent are the predictions for counties with high school graduation rates in the middle of the plot vs. in the edges?