1. Negative Binomial Regression
NASCAR Lead Changes

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(m) = ln(m)

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, X2 is approximately chi-square with 10-4=6 degrees of freedom (10 cells, 4 estimated parameters).
The critical value for an a=0.05 level test is
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 >>  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(m) = ln(m)

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 b:

12. Computational Aspects - III
Newton-Raphson Algorithm Steps:
Step 1: Set k*=0 (k=1) and iterate to obtain estimate of b:
Step 2: Use b’ of Step 1 and iterate to obtain estimate of k*:
Step 3: Use results from steps 1 and 2 as starting values to obtain estimates of k* and b
Step 4: Back-transform k* to get estimate of k: k=exp(k*)