U shape relationships

[javier rodriguez]

Hello everybody,

I am investigating the association between systolic blood pressure (SBP) and incident dementia in a cox model. We know that both lower and higher SBP are possible risk factors. I would like to study this potential u shape relationship but I do not know how to do that (except creating quartiles and study the lowest and the highest vs others).

Any idea of how to assess u shape relationships? I am doing a cox model with incident dementia as the outcome and SBP as a continuous variable that of course I can categorized.

Thank you for your help!
Best regards
JR

[Clyde Schechter]

Well, it is often possible to get a very good approximation to a U-shaped relationship by using a quadratic. So if you incorporate c.SBP##c.SBP into your Cox regression, that will set up a quadratic relationship. And if you want to visualize it graphically, -margins- with a good range of values of SBP (say -at(SBP = (80(10)180))-) followed by -marginsplot- will get that for you.

Categorizing continuous variables is usually not a good idea unless there is something truly discontinuous that happens at the cutpoint(s). That is unequivocally not true of blood pressure: the relationship of BP to pretty much everything is a continuous one, with no discrete jumps at any value. So I would recommend strongly against categorizing it. The quadratic approach is much better. If, for some other reason, you actually must categorize it (some reviewer who doesn't know what he/she is talking about demands it) use a large number of bins, not just 4. That reduces the loss of information that inevitably accompanies categorizing.

[javier rodriguez]

Hi Clyde,

Thank you so much for your reply. Just to make sure my understanding is correct.

stcox sbp

In this univariate model, I get HR : 0.99 with a p-value = 0.14. I understand that having a higher sbp of 1 mmHg is associated with a lower risk of dementia of 1% but the result is not significant.

stcox c.sbp##c.sbp

In this univariate model, the HR associated with sbp is 1.01 with p-value =0.8 and the HR associated with c.sbp#c.sbp is 0.99 with p-value=0.7.

I am not sure to understand how to interpret these coefficients, I mean practically as I did for stcox sbp. I don't know how to interpret the coefficient associated with this quadratic term...I am really sorry for this question of beginner...

When after

stcox c.sbp##c.sbp

, I do

margins, at(sbpl= (80(10)180))

, I have the following results : basically the results in the margin column (I guess the HR) are decreasing from 1.3 to 0.8. So I think the interpretation is that there is no u-shape otherwise I would have something more like HR >1 then HR <1 and then again HR >1. Is that correct?

Thank you so so much!

JR

[Clyde Schechter]

In a quadratic model, you cannot sensibly interpret either the linear or quadratic term separately. You must always consider them jointly. If you want a significance test of SBP in this quadratic model you have to do it not by looking at those coefficients and their p-values but with a joint significance test:

test sbp sbp#sbp

As for the -margins- results your interpretation might be correct, but it is incomplete. First, if you are interested in a "U-shaped" relationship, why not actually look at the shape? Run the -marginsplot-. Does it look like a U?

Concerning your interpretation of the -margins- output, a few remarks. First, a U-shaped relationship could still be found with HR > 1 everywhere, or HR < 1 everywhere if the HR = 0 points are outside the 80-180 range. So what you need to look at is whether it decreases as you go up from sbp = 80 to some minimum value and then turns around and increases as sbp continues up to 180.

Another thing you should do is locate the vertex of the parabola. If you calculate

nlcom -_b[sbp]/(2*_b[c.sbp#c.sbp])

Stata will show you the value of sbp where the bottom of the U is (or the top of an inverted-U). If that value is outside your 80 to 180 range (or is inside but very close to one of the endpoints) then you do not, for practical purposes, have a U-shaped relationship. If it is close to one of the endpoints, then you do have a curvilnear relationship that might still be worth talking about, but if the nadir of the U isn't in the range of reasonably prevalent values of sbp, then you can't really call it U-shaped.

[javier rodriguez]

Thank you so much, Clyde. Your explanations are very clear and I do appreciate them. This is incredibly helpful.

Best regards,
JR