I am working my way through (T.J.Diccio and B.Efron, "Bootstrap Confidence Intervals", Statistical Science 1996 v11 no3 pp189-228) (link to paper), and I'm stuck even before I get to the good stuff.

In the introduction, there is a table of before/after treatment values called cd4, and the correlation \hat{\theta} is given as 0.723. The paper then states, "We can find an exact interval for \theta if we are willing to assume bivariate normality ... The exact central 90% interval is (0.47, 0.86)."

Here I stipulate that I am a stats moron. But when I key in the data and compute a confidence interval by hand in R, either using the Fisher transform or the T distribution, I don't get that interval. How does one compute the interval they are talking about?

> cd4
   subj base oneyr
1     1 2.12  2.47
2     2 4.35  4.61
3     3 3.39  5.26
4     4 2.51  3.02
5     5 4.04  6.36
6     6 5.10  5.93
7     7 3.77  3.93
8     8 3.35  4.09
9     9 4.10  4.88
10   10 3.35  3.81
11   11 4.15  4.74
12   12 3.56  3.29
13   13 3.39  5.55
14   14 1.88  2.82
15   15 2.56  4.23
16   16 2.96  3.23
17   17 2.49  2.56
18   18 3.03  4.31
19   19 2.66  4.37
20   20 3.00  2.40
> r = cor(cd4$base, cd4$oneyr)
> r
[1] 0.7231654

Fisher transform:

> se = 1/sqrt(17)
> se
[1] 0.2425356
> tanh(c(r-1.6448*se, r+1.6448*se))
[1] 0.3133382 0.8082940

T distribution:

> sr = sqrt((1-r^2)/(20-2))
> sr
[1] 0.1627936
> tc = abs(qt(0.05/2, 18))
> c(r-tc*sr, r+tc*sr)
[1] 0.3811486 1.0651821

I am working my way through (T.J. Diccio & B. Efron, "Bootstrap Confidence Intervals", Statistical Science, 1996, 11(3), 189–228), and I'm stuck even before I get to the good stuff.

In the introduction, there is a table of before/after treatment values called cd4, and the correlation $\hat\theta$ is given as 0.723. The paper then states, "We can find an exact interval for $\theta$ if we are willing to assume bivariate normality ... The exact central 90% interval is (0.47, 0.86)."

Here I stipulate that I am a stats moron. When I key in the data and compute a confidence interval by hand in R, either using the Fisher transform or the T distribution, I don't get that interval. How does one compute the interval they are talking about?

> cd4
   subj base oneyr
1     1 2.12  2.47
2     2 4.35  4.61
3     3 3.39  5.26
4     4 2.51  3.02
5     5 4.04  6.36
6     6 5.10  5.93
7     7 3.77  3.93
8     8 3.35  4.09
9     9 4.10  4.88
10   10 3.35  3.81
11   11 4.15  4.74
12   12 3.56  3.29
13   13 3.39  5.55
14   14 1.88  2.82
15   15 2.56  4.23
16   16 2.96  3.23
17   17 2.49  2.56
18   18 3.03  4.31
19   19 2.66  4.37
20   20 3.00  2.40
> r = cor(cd4$base, cd4$oneyr)
> r
[1] 0.7231654

Fisher transform:

> se = 1/sqrt(17)
> se
[1] 0.2425356
> tanh(c(r-1.6448*se, r+1.6448*se))
[1] 0.3133382 0.8082940

T distribution:

> sr = sqrt((1-r^2)/(20-2))
> sr
[1] 0.1627936
> tc = abs(qt(0.05/2, 18))
> c(r-tc*sr, r+tc*sr)
[1] 0.3811486 1.0651821