{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture 3

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

This preview shows pages 1–9. Sign up to view the full content.

Poisson Regression… continued

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 )
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. ^ ^ ^
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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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).
(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 . ^ ^

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
(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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern