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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 ?

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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 $y$.

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  • $\begingroup$ Thank you @EdM, I have added 1 to x1 to avoid taking log(0) if that makes sense. $\endgroup$ – forecaster Sep 6 '15 at 19:14
  • $\begingroup$ @forecaster : That is what I thought you had done. $\endgroup$ – EdM Sep 6 '15 at 19:18
  • $\begingroup$ Thank you, how would you interpret factors? So for instance how would you determine the impact of x4 at lev ps 0,1,2,3? $\endgroup$ – forecaster Sep 6 '15 at 20:03
  • $\begingroup$ The default in R is for the Intercept to correspond to the reference levels of all categorical factors. The reference level of $x_4$ here is level 0. So as an example, for level 2 of $x_4$, $log(y)$ is 0.45 units higher than it is for level 0 of $x_4$. $\endgroup$ – EdM Sep 6 '15 at 20:27
  • $\begingroup$ Thanks again, I have edited by question with a numerical example, can you please let me know if this correct. I'm confused because this is a log-log regression with qualitative variable. I really appreciate your response. $\endgroup$ – forecaster Sep 6 '15 at 21:34

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