Confidence interval/standard error of predictor consisting of 2 variables- conditional PROC LOGISTIC

[Danaus]

Dear all,

In a matched case-control study, the outcome is case (0/1), the predictor variable is the continuous substance1. The best model had two fractional polynomials, i.e., substance1 squared (named substance1_2) and substance1 raised to the power of 3 (named substance1_3).

I ran PROC LOGISTIC with model case = substance1_2 substance1_3.

I managed to calculate the odds ratio for a change in substance1 from 15 to 5 units. In order to calculate the 90% CI, I will need the standard error. How can I calculate the standard error for this OR? I believe that the UNITS or ODDSRATIO statement may be helpful, but I don't know how to do it (https://documentation.sas.com/?docsetId=statug&docsetTarget=statug_logistic_syntax36.htm&docsetVersi...).

It would be great if someone can help me!

* Example (real dataset is much longer);
DATA dataset;
	INPUT case matchgroup substance1 substance1_2 substance1_3;
	DATALINES;
	1 1 2 4 8
	0 2 5 25 125
	0 3 10 100 1000
	1 4 20 400 8000
	0 1 15 225 3375
	0 2	8	64	512
	0 3	3	9	27
	0 4	17	289	4913
	1 1	12	144	1728
RUN;


title Univariate logistic regression on dataset - substance1_2 substance1_3;
proc logistic data=dataset outest=betas_substance1;
  		strata matchgroup;
       	model case(event='1')= substance1_2 substance1_3 / alpha=0.1;
run;

* Intermediate step: Regression coefficients in output betas_substance1 were renamed to beta_substance1_2 and beta_substance1_3
....;

DATA dataset;
	SET dataset;

	* log odds and odds ratio;
	log_odds_substance1 = beta_substance1_2*(15**2-5**2) + beta_substance1_3*(15**3-5**3);
	OR_substance1 = exp(log_odds_substance1);

	* 90% CI;
	Standard_error_substance1 = ????;
	LCI_OR_substance1 = exp(log_odds_substance1 - Standard_error_substance1*1.645);
	UCI_OR_substance1 = exp(log_odds_substance1 + Standard_error_substance1*1.645);
RUN;

[PaigeMiller]

According to the PROC LOGISTIC documentation

The CLODDS= option in the MODEL statement of PROC LOGISTIC is what you want

[Danaus]

Thank you. However, it says that the CLODDS= option does not work with the STRATA Statement, and I need the STRATA statement in my regression procedure (since it is a matched case-control study). Any other idea?

[Rick_SAS]

I think you misread the doc. It says that the CLODDS=PL option is not supported when you use the STRATA statement, but the CLODDS=WALD option is supported and generates the CIs.

[Danaus]

Thank you, you're right.

However, using CLODDS does not solve the problem, or I don't know how to do it (see what I did below). I get odds ratios for substance1_2 and substance1_3.

But what I am looking for is the odds ratio of substance1 (not _2 or _3) for the difference 15 to 5 (I already did that, see my first post), and I need to calculate the 90% CI for this odds ratio (for which I need the standard error). Will the covariance help to find the standard error? If so, how would I go about it (I just know that you can output it with covout)?

I appreciate any help!

title Univariate logistic regression on dataset - substance1_2 substance1_3;
proc logistic data=dataset outest=betas_substance1;
  	strata matchgroup;
       	model case(event='1')= substance1_2 substance1_3 / clodds = wald alpha=0.1;
run;

[PaigeMiller]

Then you have the wrong model. If you want information about substance_1 and not about substance1_2, you need to redefine the model and then the CLODDS= does give you what you want.

[Danaus]

Thanks for your answer.
I'm afraid that I cannot phrase my question statistically clearly. I'll give it another try:

I calculated ONE odds ratio for the function "beta2*substance1_2  + beta3*substance1_3" and need the corresponding 90% CI. I want to know how much higher is the odds to be a case for a unit of 15 compared to a unit of 5 (in the untransformed variable substance1) if the function that describes the relationship between substance1 and being a case is not a straight line, namely beta2*substance1²  + beta3*substance1³.

It would be great to hear your ideas.

[Rick_SAS · Accepted solution]

These references, which might help:

Estimating the odds ratio for X in a logistic model containing a polynomial or spline of X

Estimating an odds ratio for a variable involved in an interaction

With respect to your conjecture that you ought to be able to use the UNITS statement to specify the difference between 5 and 10, here is some UNTESTED code that you might try to modify. The model does not converge on your small sample data. I tried adding a few more observations, but finally gave up. Maybe someone can use this as a starting point for discussion.

DATA dataset;
	INPUT Y matchgroup substance1;
   x = substance1;
   substance1_2 = substance1**2;
   substance1_3 = substance1**3;
	DATALINES;
	0 1 2
	0 2 5
	0 3 10
	1 4 20
   1 5 15
   1 6 11
   0 7 8
	0 1 3
	1 2 8
	1 3 18
	0 4 15
   1 5 16
   1 6 13
   1 7 11
;

/* UNTESTED */
title Univariate logistic regression on dataset - substance1_2 substance1_3;
proc logistic data=dataset outest=betas_substance1;
   strata matchgroup;
   effect poly = polynomial(x / degree=3);
   model Y(event='1')= poly / alpha=0.1;
   oddsratio x / at(x=5);
   units x = 10;
run;

[Danaus]

Thank you very much! I find your post a very useful starting point.

Your suggestion with

 effect poly = polynomial(x / degree=3);

works well if the model is

 model Y(event='1')= substance1 substance1*substance1 substance1*substance1*substance;

My model is different, though, and in fact, I also have other models which include, e.g., the log-transformation.

I think you reference  http://support.sas.com/kb/35/189.html with the code

      proc logistic;
         model y = x x*x;
         contrast 'Log OddsRatio for X' x 1 x*x 7 / estimate=exp;
         estimate 'Log OddsRatio for X' x 1 x*x 7 / exp cl;
       run;

may be useful there. Do you think it will work for different types of transformations in the same model (e.g. ³ and log-transformation)?

If I can find the solution, I'll post it.

[Danaus]

The code you can find here did the trick  http://support.sas.com/kb/35/189.html . It works for different transformations. Thank you!

      proc logistic;
         model y = x x*x;
         contrast 'Log OddsRatio for X' x 1 x*x 7 / estimate=exp;
         estimate 'Log OddsRatio for X' x 1 x*x 7 / exp cl;
       RUN;