Regression to the Mean and Empirical Bayes Interpretation

Regression to the Mean and Empirical Bayes Interpretation

Observations fluctuate around underlying performance

Crash counts vary from period to period even when exposure and conditions are stable. A site selected because of an unusually high recent count may have a lower count later without treatment. This regression-to-the-mean (RTM) tendency complicates screening and before-after evaluation. It does not guarantee a decline at every site or prove that a treatment had no effect.

Consider a curve with a long-run average near 1.5 crashes per year and a recent year with seven. A later average near 1.8 after signs were installed cannot identify how much change came from signs versus random variation or other conditions. The original selection based on the spike creates a particular risk of overcrediting the treatment.

Combine model information and site history

Empirical Bayes (EB) combines an applicable prediction with observed history to estimate long-term expected frequency. Model information reflects comparable sites and conditions; observed history adds site-specific information. The method reduces RTM bias when its assumptions, model, and data are suitable. It does not make uncertainty disappear or identify the physical cause of every excess crash.

For a simplified common formulation using cumulative counts over the same study period:

$$E=wP+(1-w)O$$

$P$ is the summed predicted count, $O$ is the observed count, and $E$ is the EB-expected count. A common weight is:

$$w=\frac{1}{1+kP}$$

$k$ is the appropriate dispersion parameter under the model's formulation. Other procedures can specify different details; use the documented method. Do not mix an annual prediction with a multi-year observed total or reuse a dispersion parameter with incompatible units.

Interpret the weight

The weight on the model is $w$, and the weight on observations is $1-w$. With a larger product $kP$, the observed history receives more weight. With a smaller product, the model receives more weight. The formula does not directly use how extreme the observed count is to choose $w$ in this simplified version.

A longer period can increase cumulative predicted counts and shift weight toward the observed history, holding other assumptions constant. This does not prove the observations are free of reporting errors or that the model is irrelevant. Data quality remains essential. A zero recorded count with incomplete capture should not be treated as reliable evidence of zero risk.

Check three worked examples

The following figures are illustrative and use the cumulative-count formulation above.

For the first case, $w=1/(1+0.5\times0.8)=0.7143$ and $E=0.7143\times0.8+0.2857\times6=2.2857$. For the second, $w=1/(1+0.35\times5.2)=0.3546$ and $E$ is about 2.49. For the third, use the full three-year prediction of 72; the expected annual mean is $54.7965/3$, about 18.27. Keep unrounded intermediate values when comparing totals.

Expected counts are estimates of means and can be fractional. They are not a prediction that a fraction of a crash will literally occur. This distinction helps explain why integer observations and decimal model estimates coexist.

Interpret excess expected frequency

An excess-expected measure compares EB-expected frequency with a suitable predicted peer baseline. If both are annual means, a simplified Potential for Safety Improvement (PSI) expression is $E-P$ in annual units. If using cumulative counts, both terms must cover the same period. Some screening definitions distinguish base-condition and site-condition predictions; follow the selected method's documentation.

A positive value identifies performance above the modeled comparison and may support priority for diagnosis. It does not prove that all the excess is treatable or caused by geometry. A negative value does not prove the site is safe or that no systemic treatment can help. The comparison inherits model and data limitations.

Recognize assumptions and uncertainty

EB needs an appropriate SPF, compatible crash definitions, relevant exposure, and a dispersion parameter suitable for the model. A model from the wrong facility type can pull an estimate toward an unsuitable peer mean. Incorrect location or severity coding can distort both history and prediction.

The method addresses a statistical problem; it does not replace field diagnosis or measure every contributing factor. Interpret the result alongside collision patterns, users, roadway conditions, and feasible treatment evidence. FHWA network-screening guidance provides context for selecting suitable performance measures.

Use the estimate in the management process

A screening estimate can identify where further investigation is valuable. A before-after evaluation can use EB to estimate what would have occurred without treatment, with after-period exposure and other relevant changes incorporated. These are related applications, but an expected-before estimate alone is not a completed treatment-effect evaluation.

For a nontechnical audience, explain that the estimate combines site experience with comparable-road evidence to reduce overreaction to an unusual year. Report uncertainty and the next diagnostic step. Avoid promising that a sophisticated method “eliminates every bias” or proves a project's future benefit.