How does lm() work when one axis is logarithmic?-CodePudding

I've tried fitting a linear equation to data with a logarithmic y axis a couple different ways. I seem to get the correct line and equation when I use stat_poly_line and stat_poly_eq. (See example with mtcars)

library(tidyverse)
library(ggpmisc)

mtcars <- mtcars %>% 
  mutate(mpg10 = 10^mpg) #add column to have something work on logarithmic scale

ggplot(mtcars, aes(x = wt, y = mpg10))  
  geom_point()  
  scale_y_continuous(trans='log10')   #make y axis logarithmic
  stat_poly_line(fullrange = TRUE, se = FALSE)  
  stat_poly_eq(aes(label = paste(..eq.label.., ..rr.label.., sep = "~~~~~")), 
               parse=TRUE,label.x.npc = "right")

But when I use lm() to fit the line I get a different slope and intercept. Even though the line drawn on the plot looks identical to that drawn with stat_poly_line. Why is the equation for the line here incorrect? Am I calling the wrong values? or using lm() incorrectly?

model <- lm(log(mtcars$mpg10) ~ mtcars$wt)

ggplot(mtcars, aes(x = wt, y = mpg10))  
  geom_point()  
  scale_y_continuous(trans='log10')  
  stat_smooth(method = "lm", fullrange = TRUE, se = FALSE)  
  labs(title = paste("R^2 = ",signif(summary(model)$adj.r.squared, 2),
                     "     ",
                     "y=",signif(model$coefficients[[1]],3 ), signif(model$coefficients[[2]], 3), "x"))

Ultimately, I'd like to be able to use nlme::lmList to record the slope and intercept of many separate lines but I can only do this if I am implementing lm correctly. It currently only seems to work for me if I don't have the logarithmic y axis.

CodePudding user response：

The regression equations are different because in the first version, stat_poly_eq is using log to the base 10. In the second version you used log(mtcars$mpg10) in your model, which is the natural log. You need to use log10 to bring it in line with the other version:

model <- lm(log10(mtcars$mpg10) ~ mtcars$wt)

ggplot(mtcars, aes(x = wt, y = mpg10))  
  geom_point()  
  scale_y_continuous(trans='log10')  
  stat_smooth(method = "lm", fullrange = TRUE, se = FALSE)  
  labs(title = paste("R^2 = ",signif(summary(model)$adj.r.squared, 2),
                     "     ",
                     "y=",signif(model$coefficients[[1]],3 ), 
                     signif(model$coefficients[[2]], 3), "x"))