# PSU STAT 544 - Quasilikelihood and GEE (22 pages)

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## Quasilikelihood and GEE

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Lecture Notes

- Pages:
- 22
- School:
- Pennsylvania State University
- Course:
- Stat 544 - Categorical Data

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1 Stat 544 Lecture 22 Quasilikelihood and GEE Quasilikelihood Suppose that we observe responses y1 y2 yN which we want to relate to covariates The ordinary linear regression model is yi N xTi 2 where xi is a vector of covariates is a vector of coe cients to be estimated and 2 is the error variance Now let s generalize this model in two ways Introduce a link function for the mean E yi i g i xTi We could also write this as E yi i 1 where the covariates and the link become part of the function i For example in a loglinear model we would have i exp xTi 2 Stat 544 Lecture 22 Allow for heteroscedasticity so that Var yi Vi 2 In many cases Vi will depend on the mean i and therefore on and possibly on additional unknown parameters e g a scale factor For example in a traditional overdispersed loglinear model we would have Vi 2 i 2 exp xTi A maximum likelihood estimate for under this model could be computed by a Fisher scoring procedure ML estimates have two nice theoretical properties they are approximately unbiased and highly e cient Interestingly the asymptotic theory underlying these properties does not really depend on the normality of yi but only on the rst two moments That is if the mean function 1 and the variance function 2 are correct but the distribution of yi is not normal the estimate of obtained by maximizing the normal loglikelihood from yi N i Vi is still asymptotically unbiased and e cient 3 Stat 544 Lecture 22 Quasi scoring If we maximize the normality based loglikelihood without assuming that the response is normally distributed the resulting estimate of is called a quasilikelihood estimate The iterative procedure for computing the quasilikelihood estimate called quasi scoring proceeds as follows First let s collect the responses and their means into vectors of length N 2 2 3 3 y1 1 6 6 7 7 6 y 7 6 7 6 2 7 6 2 7 6 7 7 y 6 7 6 6 7 6 7 6 7 4 4 5 5 yN N Also let V be the N N matrix with V1 VN on the diagonal and zeros elsewhere Finally let D be the N p matrix

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