interpreting poisson regression models with log transformation and factors/qualitative variable

Question

I have a dataset with both quantitative ($x_1,x_2, \text{and} \ x_3$) and qualitative variables ($x_4$ - 4 levels ~0,1,2,3). 3 variables ($x_1,x_2,x_3$) have been log transformed. I do not know how to interpret coefficients when its log transformed.

glm(formula = y ~ log(1 + x1) + log(1 + x2) +                   
      log(1 + x3) + factor(x4), family = "quasipoisson",data = data)

                Estimate
(Intercept)     0.20
log(1 + x1)     0.76
log(1 + x2)     -0.1
log(1 + x3)     0.25
factor(x4)1     0.4
factor(x4)2     0.45
factor(x4)3     0.57

Let's suppose, if I want $x_4$ (for levels 0,1,2,3) to vary $x_1$ from 0,1,2,...,40 how would it effect my response considering everything being equal ? In addition, how to interpret $x_1,x_2, \text{ and } x_3$ ?.

Numerical Example, I want to vary $x_3$ between 0,1,2,3,4,5,... and so on and determine its impact on y for 4 different levels in variable $x_4$:

Let's suppose I want to predict for factor 0 which is when $x_4$ at 0 when $x_3 = 5$:

$$y = exp^{(0.20+0.25*5)}$$

Let's suppose I want to predict for factor 2 which is when $x_4$ at level 1 when x3 = 5:

$$y = exp^{(0.25*5+0.45)}$$

is my interpretation correct ?

Answer 1

Interpreting the coefficient of a log-transformed variables is reasonably straightforward: it represents the predicted change in the dependent variable for a 1-log-unit change in the independent variable.

Here, the dependent variable (in the default log-link for the quasipoisson family in glm) is $log(y)$. After the transformations of the variables $x_1$ through $x_3$, they are no longer the independent variables for the regression. The regression coefficients need to be interpreted in terms of the new independent variables $log(1+x_1)$ through $log(1+x_3)$.

So for the relation of $x_1$ to $y$, with the other independent variables held constant, you have a change of 1 log unit in $(1+x_1)$ corresponding to a change of 0.76 in $log(y)$. That pesky 1 in $log(1+x_1)$ makes is hard to provide a more general direct relation between $x_1$ itself and $y$.