Grafen & Hails: Modern Statistics for the Life Sciences
03.06.03 Robert D. Arbeit, Clinical Research, Paratek Pharmaceuticals, Boston, MA, USA
Chapter 04, page 61
Q:I am unable to reconcile Fig 4.3 with its legend or with Fig 4.2.
I think Fig 4.3(a) is an plot of residuals of mean AMA (i.e., y = [mean AMA - individ AMA]) by residuals of mean HGHT (i.e., x = [mean AMA - individ AMA]). These are correlated. Neither however is a residual relative to predicted (i.e., to one of the regression lines shown in Fig 4.2). If so, then the legend might be clearer as "Taller than average people are better than average at maths."
Fig 4.3 (b) appears to be a plot of residuals of mean AMA (i.e., y = [mean AMA - individ AMA]) by R1 from Fig 4.2 (a) (i.e., x = [predicated HGHT - obs HGHT]). If so, then the legend might be written as "When the relation between age and height is considered, height isn't important. People who are taller or shorter than expected for their age do not tend to be better or worse than average at maths."
I really appreciate your providing this web site for feedback and clarification.
A:The text in the book doesn't quite make clear what we wanted to say. The data in Fig 4.3, parts (a) and (b), is hypothetical. Let's set out clearly here what we meant, and then look at the question again. Before the paragraph that begins at the bottom of page 60, we have already established that R1 is the residual from the regression of AMA on YEARS, and that R2 is the residual from the regression of HGHT on YEARS. The paragraph should then read
"By looking at the relationship between these two sets of residuals, we will actually be asking the question 'Are children who are taller than expected for their age also better than average for their age at mental arithmetic. So a positive correlation between R1 and R2 would suggest that this was the case (and the plot would look like the hypothetical results shown in Fig 4.3(a)), whereas no relationship would indicate that after YEARS has been taken into account, HGHT is not an important predictor of AMA (and the plot would look like the hypothetical results shown in Fig 4.3(b))."
I guess that Robert was understandably thinking the plots were based on the same data as in Figure 4.2. Then I agree its very puzzling. After all, both (a) and (b) of Fig 4.3 claim to be plots of R1 against R2, and as they're different, they can't both be right!! We will make a note to clarify this in the next printing.
Thanks for pointing this out, Robert.