Module 7: Next Steps and Further Exploration

Congratulations on completing the core modules of this training. You have successfully built a Bayesian model to forecast the probability of a hospital trust entering OPEL-4 status. While this is a complete project, it is also a foundation that you can build upon.

Here are some ways you could extend and improve this work.

1. Automate the Process

In a real-world application, you would want this model to run automatically every day with the latest data. You could write an R script that:

Connects to a database to pull the most recent day’s data.
Appends this new data to the historical dataset.
Re-runs the Stan model.
Saves the updated 10-day forecast plot to a specific location.
Emails the plot to key stakeholders.

You could use a scheduling tool (like cron on Linux/macOS or the Task Scheduler on Windows) to run this R script at the same time every day.

2. Improve the Model

Our model is a good start, but it could be made more sophisticated.

Add More Predictors: What other data might be relevant? Ambulance handover delays? A&E waiting times? Bed occupancy rates? You could add these as new columns in your data and new parameters (beta coefficients) in your Stan model.

Time-Series Structure: Our model assumes that each day is independent. A more advanced approach would be to use a proper time-series model, such as an autoregressive (AR) model, where the state of the system yesterday influences the state of the system today. You could add a term like beta_lag * y[t-1] to the model. Here is an example of how you might modify the opel_model.stan file to include a 1-day lag.

// Stan model for predicting OPEL-4 status with a 1-day lag

data {
  int<lower=0> N; 
  int<lower=0, upper=1> y[N]; 
  vector[N] daily_admissions;
  vector[N] staff_absences;

  int<lower=0> N_future;
  vector[N_future] future_daily_admissions;
  vector[N_future] future_staff_absences;  
}

parameters {
  real alpha; 
  real beta_admissions;
  real beta_absences;
  real<lower=0> beta_lag; // Lag effect parameter (constrained to be non-negative)
}

model {
  // Priors
  alpha ~ normal(-4, 2);
  beta_admissions ~ normal(0, 0.5);
  beta_absences ~ normal(0, 0.5);
  beta_lag ~ normal(0, 1); // Prior for the lag effect

  // Likelihood with autoregressive term
  // We start from the second observation since y[t-1] is needed
  for (t in 2:N) {
    y[t] ~ bernoulli_logit(alpha + 
                           beta_admissions * daily_admissions[t] + 
                           beta_absences * staff_absences[t] + 
                           beta_lag * y[t-1]); // y[t-1] is the previous day's outcome
  }
}

generated quantities {
  vector[N_future] p_opel_4_pred;
  vector[N_future] y_pred;
  real last_y = y[N]; // Get the last observed outcome to start the forecast

  for (t in 1:N_future) {
    real linear_predictor = alpha + 
                            beta_admissions * future_daily_admissions[t] + 
                            beta_absences * future_staff_absences[t] + 
                            beta_lag * last_y; // Use the previous day's prediction

    p_opel_4_pred[t] = inv_logit(linear_predictor);
    y_pred[t] = bernoulli_rng(p_opel_4_pred[t]);
    last_y = y_pred[t]; // Update last_y for the next iteration
  }
}

Hierarchical Modeling: If you have data from multiple hospital trusts, you could build a hierarchical model. This would estimate a separate set of parameters for each trust, but it would also learn about the overall patterns across all trusts, making the individual estimates more robust.

3. Final Thoughts

The goal of this training was to introduce you to the power of probabilistic programming for real-world problems. The key takeaway is not the specific model we built, but the framework of thinking:

Start with a clear question.
Gather and prepare relevant data.
Build a model that reflects the underlying process and quantifies uncertainty.
Use the model’s output to make informed decisions.

We encourage you to experiment with the code, try adding new features, and see how it changes the results. Good luck!

This concludes the training module. We hope you found it valuable.

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