Improving the fit of bivariate smoothing splines when estimating longitudinal immunological and virological markers in HIV patients with individual antiretroviral treatment strategies.

Abstract

CD4+ lymphocyte count and HIV RNA plasma viral load are longitudinally monitored in patients with HIV infection. Because data collection intervals may be unequally spaced and these markers experience high within-patient variability, they may be smoothed before use in subsequent models. Estimation strategies must be able to accommodate the drastic changes in viral load which may occur when an individual's treatment strategy is updated. Because these treatment changes are not regimented, these dynamics cannot be modelled using standard methods. We propose univariate and bivariate cubic smoothing splines to fit CD4+ count and viral load over time. The method is developed using state space equations, and the Kalman filter is used to calculate the log-likelihood. Non-linear optimization is used to obtain the maximum likelihood estimates. A modification of the Kalman filter allows non-informative or diffuse priors at the initial observation. Since treatment changes are expected to alter the shape of the curve, we further extend the Kalman filter to permit greater flexibility in the smoothing spline at these time points. The method produces smoothed estimates of the viral load and CD4+ count curves over time.