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Published byPierce Reynolds Modified over 2 years ago

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Spring 2013

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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.

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

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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

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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

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

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Before-After Studies Accounting for change in traffic flow.

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Before-After Studies Accounting for change in traffic flow.

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

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

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

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Before-After Studies Estimation of r tf C A and C B denote the derivates of “f” with respect to traffic flow A avg and B avg.

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Before-After Studies Estimation of r tf Using the following equation Let C A and C B denote the derivatives of f:

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Before-After Studies Estimation of r tf Coefficient of variation (Std Dev / Mean) Ratio A avg / B avg

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Before-After Studies Coefficient of Variation Factors: Equation:

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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.

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Before-After Studies Coefficient of Variation From Table 8.7 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.

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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.

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Before-After Studies STEP 1: Estimate and.

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Before-After Studies STEP 2: Estimate and.

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Before-After Studies STEP 4: Estimate and.

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Before-After Studies STEP 4: Estimate and.

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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

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Formulation: where Dispersion parameter Mean Note: we use previously.

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Formulation of the variance: The EB Variance

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

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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:

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

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

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

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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.

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

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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

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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

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

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

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

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