Fig. 2

Hierarchical models leverage the rest of the sample to borrow strength, gaining resilience to missing data or outliers. The dotted black lines represent the negative binomial model shown in Fig. 1 fitted to all data points, while the red line represents the predictions of the hierarchical model fitted to the only two black data points from the specified examination alongside the rest of the sample: the first and last discrete data points (above), or the first two discrete data points (below) for four representative examinations. The grey points reflect the missing discrete data over which the model is extrapolating