interpreting odds ratio (OR) and relative risk ratios (RRR) for continuous variables from logistic and multinomial logit regressions

[Chiara Mussida]

Dear All,
I'm writing to ask for the interpretation of OR and RRR for the covariates age and age squared (both continuous) in terms of shape of the relationship (U-shape or inverse U-shaped) and the associated minimum or maximum.
I estimated a logit model an obtained the OR of escaping severe material deprivation (dependent dummy varible).

The findings are as follows (OR):
1.09 age
1.11 age squred
my questions are:
by looking at the OR together, assuming both are significant, how can I infer the shape of the relation, i.e. U-shape or inverse U-shape (between age and age squared)?
after determining the shape of the relation how can I calculate the corresponding minimum or maximum age?

Thanks for your help,
CM

[Andrew Musau]

A quadratic equation has the form \(y= ax^2 + bx + c\). The standard way to find the maximum or minimum (generally referred to extrema) is to set the first order derivative of the function w.r.t. \(x\) equal to zero and then solve for the value of \(x\) , i.e., find a stationary point. This is however a necessary but not sufficient condition. For a maximum, you need the second derivative to be negative whereas for a minimum, it needs to be positive. Therefore, you have that:

$$\frac{d y(x)}{d x}= \frac{d (ax^2 + bx + c)}{d x}= 2ax +b.$$

At the maximum, we require

$$2ax+ b= 0 \rightarrow x= \frac{-b}{2a}$$

provided that \(2a<0\). margins and marginsplot can help you here.

webuse lbw, clear
logit low c.age##c.age i.race, or
display "The maximum is at age= `=(-1*_b[age])/ (2*_b[age#age])'"
local max= `=(-1*_b[age])/ (2*_b[age#age])'
qui sum age
margins, at(age = (`r(min)'/`r(max)')) atmeans
marginsplot, noci recast(line) xline(`max') scheme(s1mono)

[Chiara Mussida]

Dear Andrew,
thanks for your reply: I think this works for a quadratic equation with coef estimates. Thanks also for hints for logit estimates and margins. I'm not sure this works for odds ratio and relative risk ratios.
I mean, in my estimates I had
material deprivation (0/1, dummy variable) = 1.09 age + 1.11 age squared, where these are odds ratios and not coefficient. My question is: how can I find the nature of the relation between age and age squared (U or iverse U), and what is the min or max age?

Thank you,
CM

[Clyde Schechter]

You cannot do the same calculations with odds ratios that can be done with the regression coefficients. But this is not really an obstacle. Even if you used the -logistic- command, which displays odds ratios rather than coefficients in the output, to do your regression, the _b[] vector referred to in #2 nevertheless contains the coefficients, so everything shown there still works the same way.