Interpretation of transformed variables in regression I did a regression with a cube root transformation for the target. The question is, how should I interpret the betas.
Variables:


*

*cnt (target): Number of rented bikes (on a daily basis).

*temp: Temperature (normalized 0 to 1). But this is not really important in this case.  
Example: Model without any transformation:
fit = lm(cnt~temp, data = d_ss)

summary(fit)

Coefficients:

             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -0.0356     3.4827   -0.01    0.992    
temp        381.2949     6.5344   58.35   <2e-16 ***

Interpretation: If temperature increases by one unit the amount of rented bikes increases by 381.29.
--> y = -0.04 + 381.29*x
Example: Cube root transformation
fit_cr = lm(cnt^(1/3)~temp, data = d_ss)

summary(fit_cr)

Coefficients:

              Estimate Std. Error t value Pr(>|t|)   
(Intercept)  2.85580    0.03880   73.61   <2e-16 ***
temp         4.38866    0.07279   60.29   <2e-16 ***

Interpretation: If temperature increases by one unit the amount of rented bikes increases by 84.60 (4.38^3).
--> y^1/3 = 2.86 + 4.39*x
--> y = 2.86^3 + 4.39^3*x
--> y = 23.27 + 84.60*x
Is this the right way? 
The problem is, if I use different or more complex transformations like this:
fit_comp = lm(cnt^(1/3)~log(temp) + sqrt(variable2) + variable3 , data = d_ss)

Then the retransformations are quite complicated. How can I handle these ones?
 A: It is misleading to call your outcome (or target) variable amount of rented bikes, since the amount terminology implies that the outcome variable is continuous.
In fact, your outcome variable is a count variable so the more appropriate definition would be number of rented bikes.
The appropriate regression model for a count variable is Poisson regression (though you may need to worry about over- or under- dispersion being an issue; or perhaps you may need to worry about an excessive number of 0 counts in your data).
Using Poisson regression would preclude transforming the outcome variable and ending up with a model whose interpretation may be challenging. 
The R command for fitting a Poisson regression model is:
glm(cnt ~ temp, family = poisson, data = d_ss) 

This model essentially says that:
$log$(expected value of cnt) = $\beta_0 + \beta_1*temp$.
If you exponentiate the estimated value of $\beta_1$ you will find the multiplicative factor $f$ by which the expected value of cnt changes when temp increases by 1 unit. For example, if $f=1.2$, then the expected value of cnt increases by 20% for each 1-unit increase in temp, since (f-1)x100%= 20%. On the other hand, if $f=0.8$, then the expected value of cnt decreases by 20% for each 1-unit increase in temp, since (f-1)x100% = -20%. Here, f is the exponentiated value of the estimated coefficient of temp in the Poisson regression model.
You don't explain wherher your counts of rented bikes are collected over the same time period (e.g., a day, a month) or over different time periods. If the counts are collected over differing time periods, the glm() function has an offset option which allows you to adjust your modelling results accordingly.
