Returning to the Zambia data set of Section 9.3.1, there is interest in assessing the association between stage of cervical lesions and condom use, after controlling for other variables. Stage of cervical lesions and condom use are both ordered categorical variables. The unadjusted Spearman's rank correlation is -0.057 (p-value = 0.50). Using the methods described above, Spearman's partial rank correlation was estimated to be -0.037 (95% confidence interval [CI] -0.196, 0.123; p-value = 0.65), adjusted for age, age^{2}, CD4, education, and marital status.

Biomarker Study of Metabolomics

Returning to the biomarker study of Section 9.3.2, there is interest in assessing the correlation between plasma levels of various metabolites to better understand how these molecules interact among persons infected with HIV who have been on long-term ART. There were 21 primary biomarkers measured on 70 HIV-infected patients. Data were complete except for a single patient who was missing a measurement of oral glucose insulin sensitivity (OGIS) 120. The distributions of the biomarkers are quite heterogenous (data not shown), many are right skewed (e.g., 2-hydroxybutyric acid highlighted in Section 9.3.2), some have several patients with values below assay detection limits, and pairwise associations are not expected to be linear. The biomarkers' scales vary and there is little interest in obtaining interpretable regression coefficients. For these reasons, Spearman's rank correlations between biomarkers would be ideal because of their robustness and their single number summary of the strength of association on a common scale between -1 and 1. However, other variables could be associated with various biomarkers that we would like to control for, including age, sex, race, BMI, CD4 cell count, smoking status, and ART duration. Hence, we also computed Spearman's partial rank correlation using the correlation of PSRs from models that adjusted for those variables. This was done by fitting a model for each biomarker using a semiparametric transformation model with the covariates listed above (with ART duration log transformed) and estimating via orm with a logit link. (Results were very similar when using a complementary log-log link, and are not shown.) As illustrated in Section 9.3.2, some of these models may have benefited from including nonlinear relationships between covariates

FIGURE 9.4

Heatmap showing the pairwise Spearman's rank correlations between 21 biomarkers. The upper-left correlations are unadjusted, the lower right correlations are partial correlations adjusted for age, sex, race, BMI, CD4, smoking status, and ART duration. Shades denote the strength of correlations with those closer to -1 and 1 being darker.

192 Quantitative Methods for HIV/AIDS Research

and biomarkers using splines; however, with only 70 patients, overfitting could be an issue. Also, these estimates are meant to be a first pass that could lead to further investigation, perhaps fine-tuning model fit using diagnostics as done in Section 9.3.2.

Figure 9.4 shows Spearman's rank correlation for all pairs of biomarkers; the quantities to the upper left of the diagonal are unadjusted, the quantities to the lower right of the diagonal are adjusted using the correlation of PSRs. Shading indicates stronger correlation and black boxes are drawn around those correlations that are significantly different from 0 (i.e., p-values < 0.05). The figure demonstrates that many of the correlations are reduced after adjusting for these additional variables. For example, homeostasis model assessment (HOMA)-2 insulin sensitivity appears to be correlated (positively or negatively) with many of the other biomarkers, with 15 of the 20 unadjusted pairwise correlations significantly different from 0. However, after controlling for covariates, the correlations generally weakened with point estimates closer to 0 and only 5 of 20 adjusted pairwise correlations significantly different from 0. A similar reduction in correlation is observed for the lactate biomarker. Although correlations generally weakened after controlling for other variables, this was not always the case. For example, the rank correlation between hemoglobin A1C and glutamic acid increased from 0.22 to 0.29 after controlling for covariates, and the rank correlation between alpha-ketogluterate and pyruvate increased from 0.22 to 0.31 in the presence of covariates.