Predicted lm() means of log-transformed and untransformed data not equal

Question

Why is the back-transformation of the predicted values so different from the observed when the observed are log-transformed?

Sample data:

trt=c("a","a","a","a","a","a","a","b","b","b","b","b","b","b","c","c","c","c","c","c","c")
resp=c(1,2,3,1,2,3,1,10,20,30,10,20,30,10,100,200,300,100,200,300,100)
resp=(log(resp))
observed=cbind(by(resp,trt,mean))
colnames(observed)="obs"
data1=data.frame(trt,resp)

Models: log.transformed and untransformed

model.nolog=lm(resp~trt,data=data1)
model.log=lm(log(resp)~trt,data=data1)

Predicted means

library(predictmeans)
estimated.nolog=predictmeans(model.nolog,"trt",adj="tukey")[[1]]
estimated.log=exp(predictmeans(model.log,"trt",adj="tukey")[[1]])
compare=cbind(observed,"est.nolog"=estimated.nolog,"est.log"=estimated.log)
options(digits=2)
> compare
   obs est.nolog est.log
a 0.51      0.51     1.7
b 2.81      2.81    16.7
c 5.12      5.12   166.9

Answer 1

My answer is you got mixed-up somewhere in the chain of calculations.

> trt = c("a","a","a","a","a","a","a",
          "b","b","b","b","b","b","b",
          "c","c","c","c","c","c","c")
> resp = c(1,2,3,1,2,3,1,
           10,20,30,10,20,30,10,
           100,200,300,100,200,300,100)

> model.nolog = lm(resp ~ trt)
> predict(model.nolog, newdata = data.frame(trt=c("a","b","c")))
         1          2          3 
  1.857143  18.571429 185.714286 

> model.log = lm(log(resp) ~ trt)
> exp(predict(model.log, newdata = data.frame(trt=c("a","b","c"))))
        1         2         3 
  1.66851  16.68510 166.85104 

The results differ because the transformation is nonlinear. But not by as much as you indicated.