Can anyone explain the first paragragh " ... A log link function is normally used since this results in a multiplicative model structure of factors..."? Also the last paragragh "Alternatively, ...". Why this leads to identical model form in the case of the Poisson model?
a possible answer I've tried to understand the material in the context of the stuff in chapter 17, so forgive me if i'm off on a tangent. I think the log link function thing is explained by the basic maths behind logs. Say you had some relativities relating to different factors - call them a, b and c. Their combined effect would be a * b * c (lets call this y). This is clearly multiplicative, like how we normally look at relativities. Now if we chose to model the log of y, then log y = the sum of the logs of a, b and c. which is additive. In the usual set up for GLMs we look at our link function g(mu) to relate the mean of the response to the linear predictor. In the case of an identity link function g(mu) = mu. if it was a log link function, we look at g(mu) = log mu. This will turn the multiplicative model above into an additive one.
Thanks for helping out. I thought this place is hopeless. Using the motor example. Factor A (age) Young 3 Old 1 Factor B (Car grp) Low 1 High 2 The expected relativity (mu) for a young driver with a high car group is 3 times 2. Are you saying that in a normal GLM set-up, we would add up the relativities. But this would lead to 3 plus 2, which is not right. Hence we think of a way to get around it by using the log, such that the item we are interested is now log(mu), and the linear predictor is now log3 plus log2 = log6. This is then regarded as the estimate for log(mu) Is my understanding now correct?
Your understanding appears correct now. Looked at another way, if we choose our link function g to be log, then g^(-1) is the exponential function. Hence mu = exp(linear predictor). The exponential function has the property that exp[sum(A,B)] = product[exp(A), exp(B)] which results in a model where we can multiply the effects of different factors together.
