Intepretation of quadratic effect in logistic regression

[Miracle]

Dear SAS forumers,

Can I please ask how do I interpret the age effect from the estimates produced by the code below?

Thank you very much.

Parameter	Estimate	Standard Error	P-value§
Intercept	-2.5487	0.6559	0.0001
Age	0.062	0.0267	0.0204
Age in quadratic	-0.001	0.0003	0.0032

proc genmod data=dset2 plots=all desc;
	weight rimwgt;
	class gpwgt / param=glm;
	model Q352=age age2 ....... / dist=bin link = logit type3;
	estimate 'HD2 vs HD1' gpwgt  -1 1 0 0 / exp;      
        estimate 'HD3 vs HD1' gpwgt  -1 0 1 0 / exp;  
        estimate 'HD4 vs HD1' gpwgt  -1 0 0 1 / exp;  
	contrast  'Linear trend p-value' gpwgt -3 -1 1 3;
run;

[Miracle]

Hi,

I have googled and found that if the coefficient on age is positive and the coefficient on age^2 is negative, that suggests that age has a positive effect on Q352 until a turning point is reached, e.g. -0.062/(2*-0.001)=31. Beyond that value, age has a negative impact on Q352. 

Is 31 the age in years? If so, can I say as one gets older, he/she is more likely to have Q352 until a turning point is reached, eg -0.062/(2*-0.001)=31. After 31 years old, he/she will be less ikely to have Q352? Is it correct? 

Thank you very much

[SteveDenham]

I agree with that interpretation.  The only caveat I have is that you should probably get the parameter estimates into a data set, and calculate from there.  Rounding errors may be critical in this interpretation.  Also, try calculating this at the confidence bounds of the parameters to get a feeling for how much variability there may be in the inflection point (it's easier than calculating the standard error of the inflection point using the delta method).

Steve Denham

[Miracle]

Hi Steve.

Thank you for your reply.

Parameter  DF  Estimate  Standard Error  Wald 95% Confidence Limits  Wald Chi-Square  Pr > ChiSq
Intercept   1	-2.5487	  0.6559	  -3.8342   -1.2633	       15.1	         0.0001				
age	    1	 0.062	  0.0267	   0.0096    0.1143	        5.37	         0.0204		
age2	    1	-0.001	  0.0003	  -0.0016   -0.0003	        8.67	         0.0032	

 Inflection point and its confidence bounds as you suggested via -age/(2*age2),

                                                    inflection_
   Label           age               age2              point
Estimate       0.0619612101    -0.00095217047629       32.537
LowerWaldCL    0.0095774949    -0.00158594223929        3.019
UpperWaldCL    0.1143449253    -0.00031839871329      179.562

What does this indicate? It seems like there is a lot of variability in this inflection point with such wide confidence bounds(from 3 to 180 years old? We only have have individuals aged between 12 and 80 in our data). Is it an indication of dropping age2 in spite of being significant?

Another question is if 32.5years old is the turning point ignoring the large variability, is it correct to intepret the estimate from the first table as one year increase in age will increase Q352 by exp(0.062)=6.4% [between exp(0.0096) =.96% and exp(0.1143)=12%]. However, it will be exp(-.001)=10% [between exp(-0.0016)=16% and exp(-0.0003)=3%] less likely to have Q352 as every one year increase in age after 32.5 years old.

By the way @SteveDenham, would you be able to provide me the reference to the delta method so I can have a read?

Your insight is greatly appreciated.

[SteveDenham]

Not surprising that the "error" bounds are so wide--after all, it is a quadratic.  But that is why the delta method was worked out.  Here is a great source to get started (although the coding is in R):

http://www.ats.ucla.edu/stat/r/faq/deltamethod.htm

As far as interpreting, remember that the linear and quadratic terms are correlated, and so you must address both at once when looking at single-unit changes.  It is not like two separate regressors, where you can look at a unit change in one, and assume the other is unchanged.

Steve Denham