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Lecture 3 - Poisson Regression continued In the PR model...

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Poisson Regression… continued
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In the PR model, the mean μ and variance V β are assumed/restricted to be equal…something that rarely occurs in practice (as real data almost always rejects this restriction when tested). Typically, the variance is greater than the mean a condition known as “ over-dispersion ”. Over-dispersion implies that the spread of the discrete probability distribution of integer outcomes is greater than the estimated mean outcome: Variance V > “expected value” μ (a.k.a. event density )
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The increase in the variance is represented by a constant multiple of variance-covariance matrix: n V β = θ ·{ Σ μ i x i x ' i } -1 , i=1 where θ is estimated from n θ = (1/(n-k)) · Σ [ (y i μ i ) 2 / μ i ] i=1 ^ ^ ^ ^ Note: Programs such as NCSS and SAS provide the option of using theta ( θ ) in the calculation of the variance estimates of the Poisson Regression coefficients.
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How good is the fit of the PR model? Overall model performance is measured by two different χ 2 tests: (1) Pearson’s P : n P p = Σ (y i μ i ) 2 / μ i i=1 where y i ’s are the observed outcome values and μ i ’s are the predicted or estimated values of the model (as a function of the regressors). Essentially it is the sum of squared error deflated (standardized) by the estimated values. ^ ^ ^
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Pearson’s goodness of fit test : P p is approximately distributed as χ 2 with n-k degrees of freedom (where n is the number of observations and k is the number of parameters estimated in the PR model). (Null hypothesis) H o : the overall fit of the model is good If P p > χ 2 , then we reject H o there is a “significant lack of fit” If P p < χ 2 , then we fail to reject H o there is no evidence of a lack of fit Note this test should only be used on grouped data (i.e., only on a frequency count variable).
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Pearson’s P p as a “dispersion deviation” test: Also, if P p > n-k evidence of over-dispersion; that is, the true variance exceeds the mean, which implies that E [(y i μ i ) / μ i ] > 1.0. By contrast, if P p < n-k evidence of under-dispersion; that is, the true variance is less than the mean, which implies that E [(y i μ i ) / μ i ] < 1.0. Note that in general, departures of P p from (n-k) may actually reflect “ misspecification ” of the conditional mean (suggesting model misspecification).
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(2) The Deviance D p statistic (sometimes called the G Statistic” -- not to be confused with the local SA index created by Art Getis ) is another useful “goodness -of- fit” measure. It is defined as n D p = Σ { y i · ln[y i / u i ] (y i μ i ) } i=1 and is a test statistic that is also distributed as χ 2 with n- k degrees of freedom… with a test that is carried out the same way as Pearson’s P p . ^ ^
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(3) Pseudo R-square statistic (R p 2 ) … typically expressed as a function of the log-likelihood values ( L ) of three distinct models: R p 2 = [ L U L R ] / [ L max L R ] , where L U is the log-likelihood value of the unrestricted “best - fitting” model; L R is the log-likelihood value of the “restricted” (y -mean/intercept) model; and L max
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