R help - Oct 2008 - Coefficients in a polynomial glm with family poisson/binomial

Dear R-users

When running a glm polynomial model with one explanatory variable (example
Y~X+X^2), with a poisson or binomial error distribution, the predicted values
obtained from using the predict() function and those obtained from using the
coefficients from the summary table "as is" in an equation of the form
Y=INTERCEPT+ XCoef x X + XCoef x X^2, differ considerably. The former are
correct and the latter are wrong.
This does not occur using lm() or in a glm with family as normal. I conclude
that this is due to the link function, predict() having some way of back
transforming the data. But if this is so, are the estimated coefficients
wortheless in this case?
I need to get accurate coefficients (for use in another model using offset), and
have resorted to re-estimating them by running a second polynomial (lm() this
time) on the predicted values from predict() of the glm. This is clearly not a
nice way of doing things.

Could anyone please inform me of why this is happening and of a better way
around this?


Code:

glm2<-glm(FEDSTATUS1~AGE+I(AGE^2), family=binomial(link="probit"))
summary(glm2) ### first set of "wrong coefficients"

nd1<-expand.grid(AGE=c(1:70))
Pred.Fed1<-predict(glm2,nd1,type="response")
points(predict(glm2,nd1,type="response")~nd1$AGE, col=2)


AGE11<-c(11:70)
Pred<-t(rbind(Pred.Fed1,AGE11))
Pred<-as.data.frame(Pred)
model<-lm(Pred$Pred.Fed1~Pred$AGE11+I(Pred$AGE11^2))
summary(model) ### "accurate coefficients"


Thanks

Samuel Riou
University of Leeds

I don't know what you mean by XCoef x X. But your problem is (as it works if
you specify "normal" in a glm) that the functional relationship
between your
predictors, i.e. Intercept+X+X^2, and Y is not linear for a binomial or a
poisson distribution.

Generalized linear model implies that the model is linear in the predictors.
It does not mean, however, that the functional relationship between the
linear predictor and Y is linear.

E.g. Y=exp(Intercept+X+X^2) is linear in the predictor, but it is a
nonlinear function because "e" is raised to the linear predictor.
Consult
any book on generalized linear models for more help.

The estimated coefficients are typically not worthless as allow you to say
how much your Y will change with a change of delta*x at the mean of X, for
example, you just have to respect the functional form of the relationship
between X and Y. Thus, the coefficients you get are accurate. Just what you
do with them is not.

For the last part of your question, I am not sure what you are trying to do
there, but it does not sound right to me in the first place.

Cheers,
Daniel 


-------------------------
cuncta stricte discussurus
-------------------------

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Von: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] Im
Auftrag von sam_oi at yahoo.fr
Gesendet: Friday, October 10, 2008 12:30 PM
An: r-help at r-project.org
Betreff: [R] Coefficients in a polynomial glm with family poisson/binomial

Dear R-users

When running a glm polynomial model with one explanatory variable (example
Y~X+X^2), with a poisson or binomial error distribution, the predicted
values obtained from using the predict() function and those obtained from
using the coefficients from the summary table "as is" in an equation
of the
form Y=INTERCEPT+ XCoef x X + XCoef x X^2, differ considerably. The former
are correct and the latter are wrong. 
This does not occur using lm() or in a glm with family as normal. I conclude
that this is due to the link function, predict() having some way of back
transforming the data. But if this is so, are the estimated coefficients
wortheless in this case? 
I need to get accurate coefficients (for use in another model using offset),
and have resorted to re-estimating them by running a second polynomial (lm()
this time) on the predicted values from predict() of the glm. This is
clearly not a nice way of doing things. 

Could anyone please inform me of why this is happening and of a better way
around this? 


Code:

glm2<-glm(FEDSTATUS1~AGE+I(AGE^2), family=binomial(link="probit"))
summary(glm2) ### first set of "wrong coefficients"

nd1<-expand.grid(AGE=c(1:70))
Pred.Fed1<-predict(glm2,nd1,type="response")
points(predict(glm2,nd1,type="response")~nd1$AGE, col=2)


AGE11<-c(11:70)
Pred<-t(rbind(Pred.Fed1,AGE11))
Pred<-as.data.frame(Pred)
model<-lm(Pred$Pred.Fed1~Pred$AGE11+I(Pred$AGE11^2))
summary(model) ### "accurate coefficients"


Thanks

Samuel Riou
University of Leeds 





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