17.2 Hierarchical Model with varying intercept

17.2.1 Model buildings

17.2.1.1 Layer 1 Within-group: within runner

\[Y_{ij} | \beta_0, \beta_1, \sigma_j \sim N(\mu_{ij}, \sigma_j^2)\]

We added a bunch of \(j\)! So now we have runner specific mean (\(\mu_{ij}\)) and the variance with a runner (\(\sigma_j\))

\[\mu_{ij} = \beta_{0j} + \beta_1X_{ij}\] Here we are using a specific intercept for each runner (\(\beta_{oj}\)) but we are still using a global age coefficient (\(\beta_1\)).

17.2.1.2 Layer 2: Between Runners

Quizz!

Which of our current parameters (\(\beta_{0j}, \beta_1, \sigma_y\)) do we need to model in the next layer? (hint:title)

\[\beta_{0j} | \beta_{0}, \sigma_0 \overset{\text{ind}}{\sim} N(\beta_0, \sigma_0^2)\]

\(\beta_{0j}\) is our intercept for each runner and it follow a normal distribution with the global average of intercept (\(\beta_0\)) and the between-group variability (\(\sigma_0\)).

Now quiz!

For Which model parameters must we specify priors in the final layer of our hierarchical regression model?

\[\beta_{0c} \sim N(m_0, s_0^2)\]

\[\beta_1 \sim N(m_1, s_1^2)\]

\[\sigma_y \sim Exp(l_y)\]

\[\sigma_0 \sim Exp(l_0)\]

Normal hierarchical regression assumptions:

• structure of the data: conditioned on \(X_{ij}\), \(Y_{ij}\) on any group j is independant of other group k but different data point within the same group are correlated

• structure of the relationship: Linear relation

• structure of variability within groups: Within any group j at any predictor value \(X_{ij}\) the observed values of \(Y_{ij}\) will vary normally

• Structure of variability between groups

17.2.1.3 Tuning the prior

\[\beta_{0c} \sim N(100, 10^2)\] runing tine is around 80 - 120 mins

\[\beta_1 \sim N(2.5, 1^2)\] We just know that it increase and it can range from 0.5 to 4.5 mins / year (on average)

\[\sigma_y \sim Exp(0.078)\]

\[\sigma_0 \sim Exp(1)\]

Then we use weakly informative priors.

running_model_1_prior <- stan_glmer(
  net ~ age + (1 | runner),  # formula
  data = running, family = gaussian,
  prior_intercept = normal(100, 10),
  prior = normal(2.5, 1), 
  prior_aux = exponential(1, autoscale = TRUE),
  prior_covariance = decov(reg = 1, conc = 1, shape = 1, scale = 1),
  chains = 4, iter = 5000*2, seed = 84735, 
  prior_PD = TRUE) # just the prior

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running |>
  # here we just used 100 sims
  add_predicted_draws(running_model_1_prior, n = 100) |>
  ggplot(aes(x = net)) +
    geom_density(aes(x = .prediction, group = .draw)) +
    xlim(-100,300)

## Warning: 
## In add_predicted_draws(): The `n` argument is a deprecated alias for `ndraws`.
## Use the `ndraws` argument instead.
## See help("tidybayes-deprecated").

## Warning: Removed 46 rows containing non-finite values (`stat_density()`).