Question on Extened Case Study pages 5-6

[andejes]

I am missing something.

How do they get from the Bernoulli equation to the new equation for q

e^logit(q) / (1 + e^logit(q))?

Then how is the equation for logit(q) then = the regression equation minus the error term?

[HopeDoneSoon]

not because of Bernoulli, but because e^logit(q) / (1 + e^logit(q)) =

e^ln(q/(1-q))/(1+e^ ln(q/(1-q))) = (q/(1-q))/( 1/(1-q)) is an identity. Basically,they

feel it is appropriate to model ln(E[Y]/(1-E[Y])) because of the merit of the form e^s/(1+e^s), i.e., this e^s/(1+e^s) will be btw 0 and 1 for all values of s and its curvilinear property (so called tilted S shape) represents some mortality pattern well. Also ln(E[Y]/(1-E[Y])) can be any value, good for a response variable in regression.

I just happened to see these things in some textbook(not SOA book).

I don’t follow your second question. Sorry.

[andejes]

They take the common OLS regression equation and remove the error term from the end. Is this just the regression equation they think we should use?

Thanks again for the information.

[HopeDoneSoon]

the equation on pg 6 for g is the response function for logit(q). we don't have error term anymore here because we don't measure those errors on logit. Instead, we do it later on the final q and in some way other than we did with OLS. Please see pg 18-20.
Logit(q)=g=the expression is an intermediate step.
As to error term, here we mean the fitted (expected) probability vs the observed success portion (Y=1). We mean no more observed individual Y's vs their expectations as in OLS. In other words, we measure error at group data level because we have a lot repetitions of predictors(343285 exposures just in <200 groups). That is what I see from this seminar reading, in some other book we see 'deviance' definition at individual level.
I just tried hard to educate myself, so I am afraid this is the best I can give, sufficing or not.
Idea: we may need to change our paradigm a bit. Think beyond OLS a bit.