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Negative Binomial Regression

Negative Binomial Regression. NASCAR Lead Changes 1975-1979. 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:

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Negative Binomial Regression

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  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(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 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(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: Set b’ = [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 b Step 4: Back-transform k* to get estimate of k: k=exp(k*)

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