Notes on Poisson Regression and Some Extensions

More documents

Recommendations

Info

Moments.var(Y ) = µE(Y ) = µThe mean rateThe variance mean identityHow well does the mean/variance equivalence hold in the example data above? Obtain theobserved proportions of counts and apply the usual formula for the mean and variance of arandom variable. We find, as before, E(Y ) = 0.899 and var(Y ) = 0.860, which are close enoughfor most government work.Models and Estimation.We usually let µ depend on data through a loglinear model , so thatlog(µ i ) = x ′ iβThe log likelihood is written aslog L =n∑ {x′i β − exp(x ′ iβ) − log Γ(y i + 1) }i=1We can ignore the last term in this expression as it does not involve parameters. In other words,the log L evaluated without this term differs from the log L above by an additive constant that isindependent of the model parameters.Exercise 1: Suppose we are interested in finding the MLE of µ (i.e., without covariates). Showthat the log likelihood is simplyn∑log L = log µ y i − nµand that the MLE is ̂µ = ∑ y i /n by solving the first-order conditions for an MLE.Exercise 2: Use the second-order conditions for a maximum to show that the variance of̂µ = ̂µ 2 / ∑ y i .Exercise 3: Develop a least-squares estimator for µ by (1) devising a appropriate loss function(i.e., sum of squared deviations around the expected value), and (2) solving the normal equationfor µ.i=1Numerical Methods. (Please skip this if you are not interested in gory details underlying theblack box of most statistical programs.) The score vector and Hessian matrix areandu = X ′ (y − m)H = −X ′ MX,where m = exp(x ′ îβ) and M = diag(m)We can apply Newton-Raphson or IRLS to this problem. The NR iteration steps arêβ (t) = ̂β (t−1) [− H (t−1)] −1U (t−1) .2
IRLS uses a weighted regression of ẑẑ i = ̂η i + (y i − ̂m i )/ ̂m i ,where ̂η i = x ′ îβ and the iterative weights are ̂m i = exp(̂η i ). All quantities (except y) are updatedat each iteration.Interpretation. The exponentiated model parameters exp(β) are interpreted as multiplicativeeffects on the rate. For a given coefficient pertaining to a unit change in a variable x, valuesgreater than 1 raise the expected rate; values less than 1 lower the expected rate; values equal to1 have no impact on the expected rate.Examples. The following examples show results from models for number of children born for asample of women as a function of several covariates. A first step is to examine the data. Here weare using the GSS. Let’s determine the overall mean childbearing rate by fitting a null modelusing the glm procedure or poisson. Note the technique used to get baseline rate, or theexponentiated effect of the constant term.Null Model.. * NULL model. gen cons = 1. glm childs cons, nocons f(p)Generalized linear models No. of obs = 1501Optimization : ML Residual df = 1500Scale parameter = 1Deviance = 2430.731764 (1/df) Deviance = 1.620488Pearson = 2146.818275 (1/df) Pearson = 1.431212Variance function: V(u) = u[Poisson]Link function : g(u) = ln(u) [Log]AIC = 3.68491Log likelihood = -2764.524679 BIC = -8540.098------------------------------------------------------------------------------| OIMchilds | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------cons | .6623651 .0185344 35.74 0.000 .6260383 .6986919------------------------------------------------------------------------------. glm, eform------------------------------------------------------------------------------| OIMchilds | IRR Std. Err. z P>|z| [95% Conf. Interval]-------------+----------------------------------------------------------------cons | 1.939374 .0359452 35.74 0.000 1.870187 2.01112------------------------------------------------------------------------------3
Page 1: Notes on Poisson R
Page 5 and 6: . tab yy | Freq. Percent Cum.------
Page 7 and 8: . fitstatMeasures of Fit for poisso
Page 9 and 10: -----------------------------------
Page 11 and 12: such that E(v) = α/β and var(v) =
Page 13 and 14: Proportion0 .1 .2 .30 2 4 6 8Number
Page 15 and 16: Density0 1 2 3 4 5.8 1 1.2 1.4vhat1
Page 17 and 18: coded 0, otherwise it is coded 1 (e
Page 19 and 20: logT | (offset)--------------------
Page 21 and 22: infile v1 intr v3 d ts tf v7 cid c9

Notes on Poisson Regression and Some Extensions

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?