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I'm building a count model using a Poisson regression. When I run my regression model without an interaction term, both of my main study variables (X1 and X2) show a positive sign. However, when I add an interaction term for these two variables, both signs turn negative. Multicollinearity is not a problem in my model, so the change in signs appears to be real (it also makes theoretical sense). Since the interaction term is positive, I interpret this as showing that the positive moderation of X2 on X1 (or viceversa...) only happens at "relatively high values" of X1 and X2.. Would this interpretation be correct? How can I show at exactly what high values of X1 and X2 does the interaction becomes positive?

Here's a summary of my results: 

Coefficients:
                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)       1.924e+00  1.227e+00   1.567 0.117070     
X1               -1.697e+00  6.921e-01  -2.452 0.014197 *  
X2               -1.172e+00  5.950e-01  -1.971 0.048777 *  
X1:X2             1.014e+00  3.110e-01   3.260 0.001113 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Is there a way to show numerically at what values of X1 and X2 the interaction becomes positive?

Charlie Glez
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1 Answers1

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It is actually convenient to report the exponentiated coefficients. You would then cat the following interpretation.

When both $x_1$ and $x_2$ are 0 then we expect the outcome to be $\exp(1.924)\approx 6.8$. For a unit increase in $x_1$ while $x_2$ is 0, the outcome decreases by a ratio of $\exp(-1.697)\approx 0.18$ or $(0.18-1)\times 100\%=-82\%$. Similarly, the outcome changes by $-69\%$ for a unit increase in $x_2$ while $x_1$ is 0. The effect of $x_1$ changes by a factor $\exp(1.014)\approx 2.76$ or $(2.76-1)\times100\%=176\%$ for a unit increase in $x_2$. The effect of $x_2$ also increase a $176\%$ for a unit increase in $x_1$.

The effect (in terms of raw coefficients) of $x_1$ is: $-1.697 + 1.014 \times x_2$. So if you want to know when that effect becomes positive, you need to solve the equation $-1.697 + 1.014 \times x_2 = 0$, so $x_2 = \frac{1.697}{1.014}\approx1.67$. Similarly, the effect of $x_2$ becomes positive when $x_1\approx1.16$.

Maarten Buis
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  • According to Cameron and Trivedi, 2005, p.124, this explanation is only appropriate when $\beta_{j}$ is close to 0, is this right or am I missing something? – Jason Goal Dec 14 '22 at 07:54
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    I don't have Cameron and Trivendi with me here, so I can't check what they say. My guess is that they talk about a linear regression with a logarithmic transformed dependent variable, i.e. it models E(log(y)). The difference with poisson regression is that that use a logarithmic link function, i.e. it models log(E(y)). – Maarten Buis Dec 14 '22 at 12:02
  • Thanks for the reply. From what I read, they are talking about log(E[y|x]) – Jason Goal Dec 14 '22 at 23:13