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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!

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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.

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    $\begingroup$ Note however that because this is a multilevel Poisson regression model (i.e., presumably random effects have been used), the interpretation of the regression coefficients will be conditional on the random effects. For more information on this, check this discussion for mixed effects logistic regression (the same issues hold also for mixed effects Poisson regression): stats.stackexchange.com/questions/365907/… $\endgroup$ – Dimitris Rizopoulos Sep 24 '18 at 5:04

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