Statistics 541: Poisson Regression

Admistrivia

• Read section 7.5 and 7.6
• Any questions about the project so far?

Poisson RV review

• e^-mumu^y/y!
• If time involved: e^-t*mu (t*mu)^y/y!
• E(Y) = mu (or t*mu)
• Var(Y) = mu (or t*mu)
• Y is approximately normal if mean is large
• So we could use weighted least squares since variance depends only on the mean which we will have to know anyway.
• Note: We don't have to estimate the variance unlike regular regression

Poisson regression model

• E(Y) = X beta
• E(Y) = t * X beta (if using time)
• E(Y) = exp^{X beta}
• E(Y) = t * exp^{X beta} (if using time)
• Called "link function"
• Y = X beta + Z where E(Z) = 0, var(Z) = X beta
• Y/E(Y) = X beta/E(Y) + Z where E(Z) = 0, var(Z) = 1
• So as long as we know E(Y) we can estimate E(Y) using regression
• Oops!
• IRLS to the rescue
• Works when mean is large and poisson looks like a normal distribution

Poisson likelihood (no regression case)

• goal is maximize: product e^-mumu^y/y!
• same as maximize: sum -mu + y log(mu) - log y!
• max at: sum (-1 + y/mu) = 0
• max at: mu-hat = sum(y)/n

Poisson likelihood (regression case)

• If mu is a function of beta, say mu = t X beta
• maximize: sum -t X beta + y log(t X beta) - log y!
• max is at: sum -t X + y /(X beta) * X = 0
• max is at: sum (Y - Y-hat)/(y-hat) * X = 0
• close to: sum (Y - Y-hat)/(sigma-hat) * X = 0
• Which is exactly weighted least squares (IRLS)
• Again, we don't have to estimate the variance

Poisson likelihood (regression with link function)

• If mu is a function of beta, say mu = exp^{t X beta}
• Usually written: ln(E(Y)) = t X beta
• maximize: sum -exp^{t X beta} + y t X beta - log y!
• max is at: sum -t X exp^{t X beta} + y t X = 0
• max is at: sum (Y - Y-hat) * t X = 0
• Since Y-hat is non-linear this isn't as easy as it appears