I have a dataset. y: the dependent variable (representing a ratio between the number of objects bought with the given money & the total number of objects bought) x: the independent variable (representing the amount of money given).
My goals for this analysis is trying to determine whether there is a significant relationship between x & y. Hence mainly testing the following hypothesis:
- Whether x=0 will imply y=0 (or for the very least, not significantly different from 0)
- This is because we would expect if there is no money,nothing will be bought. But of course, since people can share the given money with another person, there is a possibility of y not being 0)
- Whether x & y are significantly related to one another
To achieve this, I am trying to determine the best transformation of x and y to fit the best linear model in R. So, the final model I got is $\sqrt y$ against ln(x). This is the only model that I've tried so far where the residuals assumptions are satisfied, no patterns in the residuals vs fitted plot. I am not sure if it is because the small dataset that I have is causing the problem.
When I fit the model in R, I obtain the following for the coefficients:
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.319615 0.028743 11.12 2.93e-10 *** x 0.150139 0.009959 15.08 9.76e-13 *** ---
The problem with this model is I can't test for hypothesis 1 as ln(x) is basically undefined at 0. Also, my x involves values between 0 to 1 which cannot be explained by the model. I am thinking of refitting the model for smaller values of x.
Would be very thankful if anyone could provide some help.