Family: poisson
Links: mu = log
Formula: Olympics_Attended ~ Total_Medals + (1 | Sport) + (1 | Team)
Data: smaller_athlete_data (Number of observations: 1408)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Multilevel Hyperparameters:
~Sport (Number of levels: 56)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.03 0.02 0.00 0.08 1.00 2524 2206
~Team (Number of levels: 194)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.02 0.02 0.00 0.07 1.00 3107 2475
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 0.23 0.03 0.17 0.28 1.00 5851 2993
Total_Medals 0.18 0.02 0.13 0.23 1.00 7035 2680
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).Models
Model Details
Data Generating Mechanism
This analysis employed a Bayesian generalized linear model to explore the relationship between the number of Olympics attended by athletes and the total number of medals they have won. The model is specified as follows: \[ \text{Olympics\_Attended} \sim \text{Poisson}(\lambda) \]
Where the log of the expected value of the Poisson distribution is modeled as:
$$ \log(\lambda) = \beta_0 + \beta_1 \times \text{Total\_Medals} + \text{Sport}_{i} + \text{Team}_{j} $$
In the formula:
Olympics_Attended is the dependent variable (the number of Olympics attended).
Total_Medals is the predictor variable (the total number of medals won).
Sport and Team are random effects for the sport and team, respectively.
Parameter Estimates
The model was fit using the brms package, and the following summarizes the key findings from the analysis:
**Intercept: Represents the baseline log-rate of the number of Olympics attended when the total number of medals is zero.
**Total Medals: Indicates how the log-rate of the number of Olympics attended changes with each additional medal won.
Random Effects: Include variations attributed to different sports and teams.