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I have the following regression equation with a count variable as the dependent variable and diversity as the predictor.

glm.nb(numberOfJobApplicationsReceived ~ controlVar1 + FunctionalDiversityOfTheFirm)

The FunctionalDiversityOfTheFirm variable is computed as a Blau's index

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The variable value ranges from 0 to 0.833. The maximum value is 0.833 as I have six different functional categories. So maximum is 1 -(6*(1/6)^2).

The regression output indicates that the FunctionalDiversityOfTheFirm variable has a significant coefficient of 3.0788 (p < .05).

Since this is a negative binomial regression, I understand that this means that for 1 unit change in the FunctionalDiversityOfTheFirm variable, the numberOfJobApplicationsReceived increases by log(3.0788). But I am not sure what one unit change in the FunctionalDiversityOfTheFirm means. How do I meaningfully translate this to draw practical conclusions about increasing diversity?

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You got it backwards. The R function (from MASS) glm.nb uses a log link function as default. That means that $\DeclareMathOperator{\E}{\mathbb{E}} \E Y= e^{\beta_0 + \beta_1 x_1 + \dotsm}= e^{\beta_0} (e^{\beta_1})^{x_1} \dotsm$ so when diversity increases with 1 the expectation is multiplied with $e^{3.0788}$. But the range of diversity is less than one, so it is better to express this for some meaningful (that is, smaller) change. You are probably better off using visualization, for example try an R package like effects (on CRAN).

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