Having run a multi-level model for a Poisson-distributed dependent variable, coefficients for my predictors are in natural logarithm form (link function is specified as log). I would like to convert these back into original scale values. I thought that this could be relatively straightforward, e.g., for an estimate (coefficient) of 0.10, e^0.10 = 1.105 (i.e., a one unit increase in predictor x results in an increase of 1.105 in my DV). But when I look at negative values I question whether this makes sense, or whether I am instead getting the odds ratio? E.g., for one of my predictors, which I know negatively predicts my DV (clear negative relationship in the raw data) the estimate was -0.07, e^(-0.07) = 0.93, and 0.93 is clearly not a negative value, but it could be the odds ratio? Or do you simply transform the absolute value and then add the negative back after, i.e., e^0.07 = 1.07 -> -1.07 (and interpret as a one unit increase in the predictor results in a decrease in the DV of 1.07)? Thank you in advance for any help!
A negative coefficient on the log-rate scale corresponds to a rate ratio (relative reduction in the rate) with one unit increase in the covariate (or the ratio between rates for a factor level relative to the reference level).
So, your example of $\exp(-0.07)=0.93$ means that for every one unit increase in the relevant covariate the rate is lower by a factor of 0.93. E.g. if the rate is 1.0 for a covariate value of 1.0, it is 0.93 for a covariate value of 2.0.