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STATISTICS REVIEW Copyright © 2016 STC, UK Basic Questions What to measure? Number of crashes Effectiveness of a measure Is change due to Randomness? Treatment? Safety Evaluations Before and After Cross-Sectional Hazardous locations Multivariate crash models Terminology Population Sample Descriptive statistics Parameter vs. Statistic Data Descriptors Mean Average number of distribution Median Middle point of a distribution Mode Most frequent number in a distribution Variance Data Descriptors Example Consider the following speed measurements: 45, 46, 51, 45, 48, 39, 46, 52, 43, 44 Mean 45.9 mph Median 45.5 mph Mode 45.0 mph Variance 14.3 mph Poisson Distribution Discrete variable distribution Probability of event occurrence Models Crash data at a single site Vehicle arrivals at a point Mean = Variance Negative Binomial Distribution Discrete variable distribution Models Crash data at several sites Regression to the Mean (RTM) How to Address RTM More data from similar sites More years of data Empirical Bayes Empirical Bayes Method Conditional probability Number of expected crashes Weighted average estimate based on Crashes at site Crashes from other sites (prediction) Basic Equations N = N1 w + N2 (1-w) where: N: estimated expected crashes N1: crashes from other similar sites N2: observed crashes at site w: weight factor w = 1/[1+(N1 Y/)φ] where: Υ: timeframe in years for crash counts φ: overdispersion parameter EB Example (1/4) Road segment 1.8 mi long ADT 4,000 12 crashes last year N1 = 0.0224 (ADT0.564)crashes/mi φ = 2.05/mi How many crashes should we expect next year? EB Example (2/4) Step 1: Expected number of crashes N1 = 0.0224(40000.564)(1.8)=4.34 crashes/mi-year Step 2: Weight estimation N1 = 0.0224(40000.564)=2.41 crashes/mi w=1/[1+(2.41)(1)/2.050]=0.460 Step 3: Estimated number of crashes N = 0.460(4.34)+0.540(12)=8.48 crashes/mi-year 𝜎= 1−𝑤 𝑁 = crashes/mi-year (1 − 0.460(8.48) = ±2.14 EB Example (3/4) Assuming availability of three years of data 12, 7 and 8 Same ADT for each year How many crashes should we expect next year? EB Example (4/4) Step 1: Expected number of crashes N1 =0.0224(40000.564)(1.8)(3)=13.01 crashes 3 yrs Step 2: Weight estimation w=1/[1+(2.41)(3)/2.050]=0.220 crashes/mi Step 3: Estimated number of crashes N = 0.220(13.01)+0.780(27)=23.92 crashes 3 yrs 𝜎= 1−𝑤 𝑁 = (1 − 0.220(23.92) = ±4.32 crashes/mi-year 3 yrs EB Example Summary One year data 8.48 ± 2.14 crashes/mi-year Three years of data 23.92 ± 4.32 crashes in 3 years (23.92 ± 4.32)/(3x1.8)= 4.43 ± 0.80 crashes/mi-year Safety Performance Function Relationship of crashes to other variables Basic question What is the expected number of crashes? Forms Linear Non-linear Multi-variate Y= a+bx Y=ea+bx Y=a0+a1x1+a2x2+…+anxn Regression Basic Stats Form Y = a+ b X b slope of line; a intercept Basic tests Is b = 0? How good is the model? Is a = 0? Regression Output Model Summaryb Model 1 R R Square a .596 .355 Adjus ted R Square .333 Std. Error of the Es timate 104.27217 R2 35% of observed variation a. Predictors : (Constant), DRIVEXP b. Dependent Variable: SCORE SC=1023.5 +55.848 DR a Coefficients Model 1 (Cons tant) DRIVEXP Uns tandardized Coefficients B Std. Error 1023.500 43.620 55.848 13.751 a. Dependent Variable: SCORE Standardi zed Coefficien ts Beta .596 t 23.464 4.061 Sig. .000 .000 b & a not 0 Approach Steps for Studies (1/2) Identify area of interest Collect background information Form potential research hypotheses Gather/identify data Form a single research hypothesis to be tested with available data Perform a descriptive and investigative analysis of the data Approach Steps for Studies Perform the test(s) Compare results to past results Develop potential explanations Discuss limitations of the findings Discuss applicability of findings Determine future research (2/2)