Before-After Studies Part II

Before-After StudiesPart II Spring 2013

Before-After Studies Recap: we need to define the notation that will be used for performing the two tasks at hand. Let: be the expected number of target crashes of a specific entity in an after period would have been had it not been treated; is what has to be predicted. be the expected number of target crashes of a specific entity in an after period; is what has to be estimated.

Before-After Studies The effect of a treatment is judge by comparing and . The two comparisons we are usually interested are the following: the reduction in the after period of the expected number of target crashes (by kind and severity). the ratio of what was the treatment to what it would have been without the treatment; this is defined as the index of effectiveness.

Before-After Studies The estimation of the safety of a treatment is done through a 4-step process. This step is done for each entity. STEP 1: Estimate and predict . There are many ways to estimate or predict these values. Some will be shown in this course. STEP 2: Estimate and . These estimates depend on the methods chosen. Often, is assumed to be Poisson distributed, thus . If a statistical model is used: Same as for Poisson or Poisson-gamma model

Before-After Studies The estimation of the safety of a treatment is done through a 4-step process. STEP 3: Estimate and using and from STEP1 and from STEP 2. Correction factor when less than 500 observations are used

Before-After Studies The estimation of the safety of a treatment is done through a 4-step process. STEP 4: Estimate and .

Before-After Studies Accounting for change in traffic flow.

Before-After Studies Adjustment factor for change in traffic flow: Note:

Before-After Studies with Traffic Flow Factors STEP 1 & STEP 2 Estimates of Coefficients Estimates of Variances

Before-After Studies with Traffic Flow Factors STEP 3 & STEP 4

Before-After Studies Estimation of rtf CA and CB denote the derivates of “f” with respect to traffic flow Aavg and Bavg.

Before-After Studies Estimation of rtf Using the following equation Let CA and CB denote the derivatives of f:

Before-After Studies Estimation of rtf Ratio Aavg / Bavg Coefficient of variation (Std Dev / Mean)

Before-After Studies Coefficient of Variation Equation: Factors:

Before-After Studies Coefficient of Variation Example: Assume 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period on a rural long-distance highway. Now assume that the functional relationship between crashes and flow is given by . Compute and .

Before-After Studies Coefficient of Variation Example: Assume 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period on a rural long-distance highway. Now assume that the functional relationship between crashes and flow is given by . Compute and . From Table 8.7

Before-After Studies Continuing with the previous example. Now, assume that a road section has been resurfaced. In the two-year ‘before’ period, 30 wet-pavement crashes were recorded on this section. In the two-year ‘after’ period, 40 wet-pavement crashes were reported. As before, 572 vehicles were counted during a two-hour count for the before period and 637 were counted for the after period. The function relationship is still the same: . In addition, there were 50 wet-pavement days for the before period and 40 wet-pavement days for the after period. Estimate , and the standard deviation of these estimates.

Before-After Studies STEP 1: Estimate and .

Empirical Bayes Method • Premise: the safety of a site is estimated using two sources of information: • 1) information obtained from sites that have the same characteristics (reference population) • 2) information obtained from the actual site where the EB method is being applied • Reference population • Method of moments • Statistical model

Empirical Bayes Model Formulation: Note: we use previously. where Mean Dispersion parameter

Empirical Bayes Model Formulation of the variance: The EB Variance

Empirical Bayes Model If the estimate of and is available, one can estimate the coefficients and from the gamma distribution (two-parameter).

Empirical Bayes Model It can be shown that by using the Bayes theorem, we can incorporate the crashes occurring on the given site to develop a new gamma function:

Empirical Bayes Model Estimating and . Method of Moments Sample mean Sample variance K(n) = the number of crashes on each entity n= the number of entities

Empirical Bayes Model Estimating and . Method of Moments Estimated mean Estimated variance

Empirical Bayes Model Estimating and . Statistical Model Estimated mean Estimated variance

Empirical Bayes Model Before-After Study using the EB model STEP 1: Develop statistical models. Using data from the control group, develop one or several statistical models. From the model(s), estimate the dispersion parameter .

Empirical Bayes Model Before-After Study using the EB model STEP 2: Estimate and for the before period. = expected annual number of crashes for the before period = crash count during the period “t” years (labeled as tb)

Empirical Bayes Model Before-After Study using the EB model STEP 3: Estimate . For each site, use the characteristics for the after period For each site, use the characteristics for the before period

Empirical Bayes Model Before-After Study using the EB model STEP 4: Estimate the number of collision for the after period. = the number of years for the after period

Empirical Bayes Model Before-After Study using the EB model STEP 5: Estimate . (same as before) STEP 6: Estimate and .

Empirical Bayes Model Before-After Study using the EB model STEP 7: Estimate and using the output from STEP 4, STEP 5 and STEP 6.

Empirical Bayes Model Before-After Study using the EB model STEP 8: Estimate and .

Before-After Studies Part II