Censored correlated cytokine concentrations: multivariate Tobit regression using clustered variance estimation

Interest in cytokines as markers for the function of the immune system is increasing. Methods quantifying cytokine concentrations are often subject to detection limits, which lead to non-detectable observations and censored distributions. When distributions are skewed, geometric mean ratios (GMRs) can be used to describe the relative concentration between two cytokines, and the GMR ratio (GMRR) can be used to compare two groups. The problem is how to estimate GMRRs from censored distributions.We evaluated methods, including simple deletion and substitution, in simulated and real data. One method applies Tobit directly to the censored difference between the two cytokine log-concentrations (Diff). However, censoring is correlated to the outcome and is therefore not independent. The correlation increases as the correlation between the two log-concentrations decreases. We propose a Tobit stacking method that uses clustered variance-covariance estimation allowing homogeneous (Stackc) or inhomogeneous (Stackh) variances. We compare it with direct estimation of the bivariate Tobit likelihood function (Bitobit) and multiple imputation. We assess sensitivity to inhomogeneity and non-normality. Simulations show that deletion and substitution are empirically biased and that Diff has an empirical bias, which increases as the correlation between the log-concentrations decreases. Estimates from multiple imputation, Stackh and Bitobit are almost identical. The estimates exhibit small empirical bias for both homogeneous and inhomogeneous normal distributions. For skewed mixture and heavy-tailed distributions, they perform reasonably well if censoring is less than 30%. We recommend these methods to estimate GMRRs. At least one of the methods is available in Stata, R or SAS.