Maximum diversity weighting for biomarkers with application in HIV-1 vaccine studies

Abstract

While studying the association between risk of HIV-1 infection and vaccine-elicited immune responses in preventative HIV-1 vaccine recipients, we encountered a need to combine a collection of biomarkers in an unsupervised fashion with the goal of preserving signal diversity within that collection. Inspired by methods for weighting protein sequences from the biological sequence analysis literature, we propose novel methods for weighting biomarkers, which we call maximum diversity weights. These weights are defined as the weights that maximize measures of signal diversity within a collection of biomarkers. While the optimization problems do not admit analytical solutions, they are convex and hence can be solved efficiently using iterative search algorithms. Through Monte Carlo studies and a real data example from HIV-1 vaccine research, we show that using maximum diversity weights in association studies can lead to an increase in power over other commonly used weights such as uniform weights or principal component-based weights.

Keywords: HIV-1 vaccine development; multivariate analysis; profile hidden Markov model; unsupervised feature selection.

Figure 1.

Conceptual similarity between weighting protein sequences and weighting continuous biomarkers for maximum diversity. (a) A group of 3 amino acid sequences arranged vertically. Each sequence (seq.) is assigned a weight. The numbers 1, 2 and 3 index the positions (pos.) within a sequence. (b) A panel of 3 immune response biomarkers measured from three subjects. The biomarker values are represented by color-coded tickmarks. Each color (biomarker) is assigned a weight. The numbers 1, 2, and 3 index the subjects (subj.). The arrows represent the real lines, and the bell curves over the tickmarks illustrate kernel density estimation.

Figure 2.

The weight assigned to x₂ versus ρ when the correlation matrix is C_3,b(ρ) in Scenario I of the simulation study. Three values of the bandwidth parameter h under the maximum entropy approach are compared with the maximum variance approach. When h = nrd, the bandwidth is selected using a standard bandwidth selection procedure for density estimation [14].

Figure 3.

Simulation study Scenario II. (a) The sum of weights assigned to x₁, x₂, x₃ versus ρ when the correlation matrices are C_5,a(ρ), (b) the weight assigned to x₃ versus ρ when the correlation matrices are C_5,b(ρ).