The objective of a study by Anton et al. (A-5) was to investigate the correlation structure of multiple measures of HIV burden in blood and tissue samples. They measured HIV burden four ways. Two measurements were derived from blood samples, and two measurements were made on rectal tissue.

The two blood measures were based on HIV DNA assays and a second co-culture assay that was a modification of the first measure. The third and fourth measurements were quantitations of HIV-1 DNA and RNA from rectal biopsy tissue. The table below gives data on HIV levels from these measurements for 34 subjects.

HIV DNA Blood (Y)

HIV Co-Culture Blood (X1)

HIV DNA Rectal Tissue (X2)

HIV RNA Rectal Tissue (X3)

115 .38 899 56 86 1.65 167 158 19 .16 73 152 6 .08 146 35 23 .02 82 60 147 1.98 2483 1993 27 .15 404 30 140 .25 2438 72 345 .55 780 12 92 .22 517 5 85 .09 346 5 24 .17 82 12 109 .41 1285 5 5 .02 380 5 95 .84 628 32 46 .02 451 5 25 .64 159 5 187 .20 1335 121 5 .04 30 5 47 .02 13 30 118 .24 5 5 112 .72 625 83 79 .45 719 70 52 .23 309 167 52 .06 27 29 7 .37 199 5 13 .13 510 42 80 .24 271 15 86 .96 273 45 26 .29 534 71 53 .25 473 264 185 .28 2932 108 30 .19 658 33 9 .03 103 5 76 .21 2339 5

(a) Compute the simple correlation coefficients between all possible pairs of variables.

(b) Compute the multiple correlation coefficient among the four variables. Test the overall correlation for significance.

(c) Calculate the partial correlations between HIV DNA blood and each one of the other variables while controlling for the other two. (These are called second-order partial correlation coefficients.)

(d) Calculate the partial correlation between HIV co-culture blood and HIV DNA, controlling for the other two variables.

(e) Calculate the partial correlation between HIV co-culture blood and HIV RNA, controlling for the other two variables.

(f) Calculate the partial correlations between HIV DNA and HIV RNA, controlling for the other two variables.