To compute the control and treatment averages, the \(\mathrm{R}\) command tapply( \(\log (y)\), treat, mean)

returns the pair of averages 3.360, 3.085, which is not the pair reported in the text. The alternative \(\log\)-scale computation

tapply(tapply(log (y), rat, mean), trt, mean)

tapply(tapply(log (y), rat, mean), trt, var)

returns the numbers \(3.372,3.113\) for means, and \(0.174,0.200\) for variances. Under what circumstances do the two mean calculations return the same pair of averages? Explain the difference between the factors trt and treat.