I have a set of data that I think forms a power law relationship, however I am struggling to work out the equation of the relationship.
Here is a subset of the data I am working with:
df <- data.frame('x' = c(0.25, 1, 2, 2.75, 4.25, 6.5, 8, 13.25, 16.25, 19.25, 19.75, 26.5, 31, 37.75), 'y' = c(4.485605e-08, 1.430240e-08, 7.638950e-09, 6.776308e-09, 3.269885e-09, 2.609455e-09, 4.378785e-09, 2.260540e-09, 2.039074e-09, 7.119317e-10, 2.252598e-09, 1.617082e-09, 7.511261e-09, 1.519275e-09))
Ploting this, I thought that the graph resembles that of y = 1/x:
I then plotted the log-log of the graph to see if it produced a straight line (indicating a power relationship) and attempted to fit a linear model to it to find the slope and the intercept.
The slope of the line was found to be -0.6099 and the intercept -18.178. However, when I put this back into the equation to work out the value of y for a given value of x it is way off.
For example if x=0.25 then y = (10^-18.178)*(0.25^-0.601).
This calculates y to be 1.527x10^-18 when the value of y for that data point was actually 4.486x10^-8.
I thought that perhaps is it simply that the heteroscedacity of the varience violates an assumption of the linear regression such that I can't use it for anything more than calculating the slope.
Am I correct, or is there something I've missed? Why is this value so far off?