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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

Negative Binomial Regression

NASCAR Lead Changes 1975-1979

data description
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))
poisson regression
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)
regression coefficients z tests
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
testing goodness of fit
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.
testing goodness of fit6
Testing Goodness-of-Fit

107.4 >> 12.59  Data are not consistent with Poisson model

negative binomial regression7
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)

regression coefficients z tests8
Regression Coefficients – Z-tests

Note that SAS and STATA estimate 1/k in this model.

goodness of fit test
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.
computational aspects i
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:

computational aspects ii
Computational Aspects - II

Derivatives wrt k* and b:

computational aspects iii
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*)