# Since incidence rate ratios are always positive, how does one determine the sign of the effect?

I want the results from a Poisson regression to be comparable to previous OLS and Tobit regressions. To do this, I am utilizing the irr (incidence rate ratio) option in Stata. However, since all the estimated coefficients are being exponentiated, all irr coefficients are positive. Given that, how does one determine the sign of each estimated coefficient? Would it be kosher to just assume that the negative or positive sign from running the same Poisson estimation without the irr option is still applicable?

With linear regression and Tobit (assuming you are looking at the latent variable) the coefficients measure the difference in expected (latent) outocome for two groups who are one unit of your explanatory variable appart. So a positive sign for that coefficient means that the expected outcome increases with your explanatory variable and a negative sign means that your expected outcome decreases with your explanatory variable.

With models using a log-link function (e.g. poisson regression) the exponentiated coefficients measure the ratio of expected outcomes for two groups who are one unit of your explanatory variable appart. So a ratio larger than 1 means that the expected outcome increases with your explanatory variable and a ratio less than 1 means that the expected outcome decreases with your explanatory variable.

So if you want to know the "sign" of an incidence risk ratio you look at whether it is larger or smaller than 1. This is equivalent to looking at whether the raw coefficients are larger or smaller than 0 ($\exp(0)=1$).