Understanding lincom confidence intervals

[Sam Wilding]

I'm try to get my head around interpreting confidence intervals for linear combinations of paramters (lincom command). Let's say I'm interested in whether smoking is associated with low birth weight (using the lbw dataset, see example in help logit).

webuse lbw
logit low age lwt i.race smoke ptl ht ui

This produces the following output:

var	Coef	SE	z	p	LB	UB
age	-.0271003	.03645404	-0.74	0.457	-.0985418	.0443412
lwt	-.0151508	.0069259	-2.19	0.029	-.0287253	-.0015763
black	1.262647	.5264101	2.40	0.016	.2309024	2.294392
other	.8620792	.4391532	1.96	0.050	.0013548	1.722804
smoke	.9233448	.4008266	2.30	0.021	.137739	1.708951
ptl	.5418366	.346249	1.56	0.118	-.136799	1.220472
ht	1.832518	.6916292	2.65	0.008	.4769494	3.188086
ui	.7585135	.4593768	1.65	0.099	-.1418484	1.658875
_cons	.4612239	1.20459	0.38	0.702	-1.899729	2.822176

As I understand, the logit for a non-smoker is .46, which has a 95% confidence interval of -1.9 ; 2.82 (_cons). The additional effect for being a smoker is .92, which has a 95% confidence interval of .14 ; 1.71 (smoke), and this effect is significant at the 95% level (p = .021). Thus the expected birth weight of smokers differs at the 95% level.

I want to calculate the logit and confidence interval of low birth weight for: 1) non-smokers 2) smokers, so I turn to the lincom command.

lincom _cons+smoke

This produces the following output:

var	Coef	SE	z	LB	UB
1.384569	1.155	1.20	0.231	-.8810857	3.650223

The 95% confidence interval for smokers (-88 ; 3.6) overlaps with that of non-smokers (-1.89 ; 2.82) even though the 95% confidence interval for the additional effect of smoking !=0 with 95% confidence. So p tells us whether the 'effect' overlaps with zero, but it does not take the uncertainty of the constant into account? How would you explain this in the paper format, e.g. we found that smoking has a negative effect on low birth weight, but the expected birth weight for smokers and non-smokers do not differ at the 95% level?

I have posted this on statexchange before (http://stats.stackexchange.com/quest...ntervals-stata) and the only reply I received was:
"It has to do with the uncertainty estimated in the intercept/constant. This value is unknown, but per the model should be the same for both smokers and non-smokers". Any help on how I would write-up this result would be appreciated.

Thanks,
Sam

[Bruce Weaver]

Hi Sam. If I follow, I think this short CMAJ article should address your concerns. It discusses a difference between two independent means, but the principal is the same.

HTH.

[Bruce Weaver]

Hi Sam. If I follow, I think this short CMAJ article should address your concerns. It discusses a difference between two independent means, but the principal is the same.

I meant principle, of course.

[Sam Wilding]

Hi Bruce,

Thanks for your quick reply - the article covers the exact issue I've been stuck with. Thanks for your help!

[Maarten Buis]

a similar point was made in the American Statistician: http://www.stat.columbia.edu/~gelman...ed/signif4.pdf