1. Negative Binomial Regression NASCAR Lead Changes 1975-1979

2. Data Description Units – 151 NASCAR races during the 1975- 1979 Seasons Response - # of Lead Changes in a Race Predictors:  # Laps in the Race  # Drivers in the Race  Track Length (Circumference, in miles)  Models:  Poisson (assumes E(Y) = V(Y))  Negative Binomial (Allows for V(Y) > E(Y))

3. Poisson Regression Random Component: Poisson Distribution for # of Lead Changes Systematic Component: Linear function with Predictors: Laps, Drivers, Trklength Link Function: log: g(  ) = ln(  )

4. Regression Coefficients – Z-tests Note: All predictors are highly significant. Holding all other factors constant: As # of laps increases, lead changes increase As # of drivers increases, lead changes increase As Track Length increases, lead changes increase

5. Testing Goodness-of-Fit Break races down into 10 groups of approximately equal size based on their fitted values The Pearson residuals are obtained by computing: Under the hypothesis that the model is adequate, X 2 is approximately chi-square with 10-4=6 degrees of freedom (10 cells, 4 estimated parameters). The critical value for an  =0.05 level test is 12.59. The data (next slide) clearly are not consistent with the model. Note that the variances within each group are orders of magnitude larger than the mean.

6. Testing Goodness-of-Fit 107.4 >> 12.59  Data are not consistent with Poisson model

7. Negative Binomial Regression Random Component: Negative Binomial Distribution for # of Lead Changes Systematic Component: Linear function with Predictors: Laps, Drivers, Trklength Link Function: log: g(  ) = ln(  )

8. Regression Coefficients – Z-tests Note that SAS and STATA estimate 1/k in this model.

9. Goodness-of-Fit Test Clearly this model fits better than Poisson Regression Model. For the negative binomial model, SD/mean is estimated to be 0.43 = sqrt(1/k). For these 10 cells, ratios range from 0.24 to 0.67, consistent with that value.

10. Computational Aspects - I k is restricted to be positive, so we estimate k* = log(k) which can take on any value. Note that software packages estimating 1/k are estimating –k* Likelihood Function: Log-Likelihood Function:

11. Computational Aspects - II Derivatives wrt k* and 

12. Computational Aspects - III Newton-Raphson Algorithm Steps: Step 1: Set k*=0 (k=1) and iterate to obtain estimate of  Step 2: Set  ’ = [1 0 0 0] and iterate to obtain estimate of k*: Step 3: Use results from steps and 2 as starting values (software packages seem to use different intercept) to obtain estimates of k* and  Step 4: Back-transform k* to get estimate of k: k=exp(k*)