Reverse parameters in multinomial logit regression

[Demographer]

Hello,

I have this multinomial logit model:

proc logistic data=edu.educationClean;
class Langue(ref='0')  /param=ref;
model edu3(ref='2')= Langue /link=glogit rsquare;
weight pond / norm;
where model=1;
run;

With the following output:

Analysis of Maximum Likelihood Estimates
Parameter		edu3	DF	Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq
Intercept		0	1	-0.7505	0.0134	3131.3792	<.0001
Intercept		1	1	0.1590	0.0103	236.5849	<.0001
Langue	1	0	1	-0.0533	0.0841	0.4021	0.5260
Langue	1	1	1	-0.2527	0.0677	13.9402	0.0002
Langue	2	0	1	1.5045	0.0729	426.1495	<.0001
Langue	2	1	1	0.2267	0.0773	8.6057	0.0034

In addition to this, I would like to have reverse parameters (I mean edu3=2 vs other) and their p values. I think I can do it manually (but maybe I'm wrong), for example, for langue=1, by doing -(-0.0533+-0.2527) to get the parameter for edu3=2 compared to other categories. However, with this method, I don't get p values. Is there a way to produce this automacially in the output?

[StatDave]

That implies that you want to model a logit defined as log[ Pr(Edu3=2)/(1-Pr(Edu3=2) ].  To do this, simply create a new response variable: new=(edu3=2); and fit a binary logistic model using it as the response. 

[Demographer]

That is what I tried.

data edu.educationClean;
set edu.educationClean;
HIGH=0;
if edu3=2 then HIGH=1;
run;


proc logistic data=edu.educationClean;
class Langue(ref='0')  /param=ref;
model HIGH(ref='0')= Langue /link=glogit rsquare;
weight pond / norm;
where model=1;
run;

It gives this output:

SAS Output

Analysis of Maximum Likelihood Estimates
Parameter		HIGH	DF	Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq
Intercept		1	1	-0.4974	0.00963	2667.2291	<.0001
Langue	1	1	1	0.1913	0.0616	9.6420	0.0019
Langue	2	1	1	-0.7824	0.0675	134.4464	<.0001

Which is not what I expected as parameters. From the multinomial outputs, in my mind, the parameter for edu3=2 when langue=1 should be -(-0.0533+-0.2527)=0.306. However, it is 0.1913 when using a binomial regression. Maybe something is wrong with my reasoning. 

[StatDave]

The generalized logit model you fit originaly is:

l0=log(p0/p2)=int0+lang01+lang02
l1=log(p1/p2)=int1+lang11+lang12

You now fit the model l2=log(p2/(p0+p1))=int+lang1+lang2.

When lang=1, under that first model:

l0=int0+lang01
l1=int1+lang11

And in the second model:

l2=int+lang1

Algebraically, I don't see that -(lang01+lang11) as you want to do, gets you to the lang1 parameter in the second model.

[Demographer]

So there is no way to get log(p2/(p0+p1)) directly from the multinomial logit parameters?

[Demographer]

My reasoning was the following:

In a binomial logit, when we change the reference category of the modelized variable, parameters for explainatory variables are just the negative. For example, in my binomial logit, if the reference category were 1 rather than 0, then parameter for langue=1 would be -(0.1913). So I though it was the same for a multinomial logit. Maybe, obviously, I'm wrong.