We have studied a Bayesian simple linear regression model. We have also introduced some basic principles of hierarchical models. Now we’ll combine elements to formulate Bayesian hierarchical linear regression models.
This handout corresponds to Chapters 15-17 of (Johnson, Ott, and Dogucu 2022). This handout only contains a brief introduction; see (Johnson, Ott, and Dogucu 2022) for a much more thorough analysis.
We’ll consider the cherry_blossom_sample data in the bayesrules package which contains the net running times (in minutes) and age for participants in the annual 10-mile Cherry Blossom race held in Washington, D.C. Each runner in the sample is in their 50s or 60s.
library(bayesrules)
data(cherry_blossom_sample)
running <- cherry_blossom_sample |>
select(runner, age, net) |>
na.omit()
head(running)# A tibble: 6 × 3
runner age net
<fct> <int> <dbl>
1 1 53 84.0
2 1 54 74.3
3 1 55 75.2
4 1 56 74.2
5 1 57 74.2
6 2 52 82.7Suppose we fit a Bayesian simple regression model of net on age. What do the slope and intercept parameters represent in context? What might be a reasonable prior? (You can assume that the age variable has been centered by subtracting the sample mean.)
Fit the model from the previous part (see the results in Section 17.1 of (Johnson, Ott, and Dogucu 2022)). Do the results seem reasonable? (Hint: consider the posterior distribution of the slope.)
Each of the runners in the sample has run the race in multiple years. What assumption of the simple linear regression model is violated?
Formulate a Bayesian hierarchical linear model which assumes the intercept (but not slope) varies by runner. What do the parameters represent in context?
Fit the model from the previous part (see the results in Section 17.2.4 of (Johnson, Ott, and Dogucu 2022)). What are some features of the analysis you would explore? Consider the posterior distribution of the slope associated with age; does the posterior distribution based on this hierarchical model seem more reasonable than the one based on the simple linear regression model?
Formulate a Bayesian hierarchical linear model which assumes both the intercept and the slope vary by runner. What do the parameters represent in context?
Fit the model from the previous part (see the results in Section 17.3.3 of (Johnson, Ott, and Dogucu 2022)). What are some features of the analysis you would explore?
Compare the two hierarchical regression models. What are some similarities? What are some differences?
library(brms)Loading required package: RcppLoading 'brms' package (version 2.21.0). Useful instructions
can be found by typing help('brms'). A more detailed introduction
to the package is available through vignette('brms_overview').
Attaching package: 'brms'The following object is masked from 'package:stats':
arlibrary(bayesplot)This is bayesplot version 1.11.1- Online documentation and vignettes at mc-stan.org/bayesplot- bayesplot theme set to bayesplot::theme_default() * Does _not_ affect other ggplot2 plots * See ?bayesplot_theme_set for details on theme setting
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rhatlibrary(tidybayes)
Attaching package: 'tidybayes'The following objects are masked from 'package:brms':
dstudent_t, pstudent_t, qstudent_t, rstudent_tfit_lm <- brm(data = running,
family = gaussian(),
net ~ 1 + age)Compiling Stan program...Start sampling
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Chain 4: summary(fit_lm) Family: gaussian
Links: mu = identity; sigma = identity
Formula: net ~ 1 + age
Data: running (Number of observations: 185)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 75.16 24.94 26.40 124.04 1.00 4648 2589
age 0.27 0.45 -0.62 1.15 1.00 4612 2572
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 14.08 0.76 12.63 15.67 1.00 4149 2923
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).fit_hier1 <- brm(data = running,
family = gaussian(),
net ~ 1 + age + (1 | runner))Compiling Stan program...Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#bulk-essWarning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-esssummary(fit_hier1) Family: gaussian
Links: mu = identity; sigma = identity
Formula: net ~ 1 + age + (1 | runner)
Data: running (Number of observations: 185)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Multilevel Hyperparameters:
~runner (Number of levels: 36)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 13.57 1.79 10.68 17.59 1.01 321 479
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 21.57 12.67 -2.91 46.42 1.00 1415 2428
age 1.24 0.23 0.80 1.68 1.00 1751 2731
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 5.24 0.31 4.67 5.88 1.00 2247 2631
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).fit_hier2 <- brm(data = running,
family = gaussian(),
net ~ 1 + age + (1 + age | runner))Compiling Stan program...Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
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Chain 4: Warning: There were 116 divergent transitions after warmup. See
https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
to find out why this is a problem and how to eliminate them.Warning: Examine the pairs() plot to diagnose sampling problemsWarning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#bulk-essWarning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-esssummary(fit_hier2)Warning: There were 116 divergent transitions after warmup. Increasing
adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup Family: gaussian
Links: mu = identity; sigma = identity
Formula: net ~ 1 + age + (1 + age | runner)
Data: running (Number of observations: 185)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Multilevel Hyperparameters:
~runner (Number of levels: 36)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 8.15 5.96 0.36 22.67 1.00 2164 2000
sd(age) 0.26 0.11 0.05 0.53 1.02 172 544
cor(Intercept,age) -0.24 0.55 -0.97 0.88 1.02 144 335
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 21.42 12.32 -2.75 44.98 1.00 5629 2472
age 1.25 0.23 0.82 1.70 1.00 5226 2426
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 5.18 0.31 4.63 5.82 1.00 4186 3057
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).library(shinystan)
launch_shinystan(fit_hier2)