I used logistic regression to model a binary dependent variable (depression diagnosis) while age (continous variable) and education level (categorical variable) were independent variables. I have obtained two models for men and women separately.
I'd like to know if a change in age (or in education level) makes greater difference for men or for women. I am sure that I cannot just compare coefficents of said models.
I could run a regression model considering dummy variables of sex and look for interactions, but I am curious if there is another way of comparing two sexes using these models.
I think there are several ways to interpret your description of the logistic regression. Let $X$ be binary random variable describing whether a particular person has depression. Are you assuming that $$P(X=1 | a= A, e =E) = 1/(1+\exp(-\beta_e a + \alpha_e))$$ (Each education level has it's own simple logistic regression), $$P(X=1 | a= A, e =E) = 1/(1+\exp(-\beta a + \alpha_e))$$ (Each education level shifts the logistic curve to the left or right), $$P(X=1 | a= A, e =E) = 1/(1+\exp(-\sum_{i=0}^n \beta_{ei} a^i))$$ (Each education level has it's own polynomial logistic regression), or $$P(X=1 | a= A, e =E) = 1/(1+\exp(-\sum_{i=0}^n \beta_{i} a^i+ \alpha_e))$$ (There is one polynomial logistic regression which is shifted left or right by the education level)?
In any of the above cases, I would be tempted to measure the effect of changing education level from education level e=1 to e=2 by $$\mathbb E[ \mathrm{abs}(X-Y) | E=1] - \mathbb E[ \mathrm{abs}(X-Y) | E=2].$$ Note that $\mathrm{abs}(X-Y)=0$ iff a randomly chosen male and a randomly chosen female are either both depressed or both happy and $\mathrm{abs}(X-Y)=1$ otherwise.
The expression $\mathbb E[ \mathrm{abs}(X-Y) | E=1]$ can be interpreted as the probability that a randomly chosen man with education level 1 and a randomly chosen woman with education level 1 will not have the same depression status.
If you wanted just one number that measures the effect of changing education level, you might try $$\sum_{i,j} P(E=i)P(E=j)(\mathbb E[ \mathrm{abs}(X-Y) | E=i] - \mathbb E[ \mathrm{abs}(X-Y) | E=j])^2.$$
Similarly, I would measure the effect of changing age from age $a_1$ to age $a_2$ by $$\mathbb E[ \mathrm{abs}(X-Y) | A=a_1] - \mathbb E[ \mathrm{abs}(X-Y) | A=a_2]$$ and if you wanted just one number $$\int_{a_1}\int_{a_2} f_A(a_1) f_A(a_2)( \mathbb E[ \mathrm{abs}(X-Y) | A=a_1] - \mathbb E[ Abs[X-Y] | A=a_2])^2\, da_1\, da_2.$$
There are many reasonable ways to measure the effect of changing age or educational level.