Testing for Serial Homogeneity and Pooled Correlation for Longitudinally Measured Biomarkers.

Abstract

Salivary biomarkers play an important role in predicting oral disease status along with oral bacterial pathogens. Thus, our work is motivated by a study that longitudinally measured periodontal biomarkers and levels of bacterial pathogens in the oral cavity with the intent of testing whether the correlation between each biomarker and each pathogen is homogeneous over time.

We first developed both frequentist and Bayesian approaches for testing for serial homogeneity of correlation coefficients. We proposed two Wald tests and an F-test based on the asymptotic distributions of sample correlation coefficients. We found that the Wald test based on Fisher's Z-transformation and the F-test have nominal sizes when the data fit our assumed model, while the other Wald test has a more inflated size in small samples. The Wald test based on Fisher's Z-transformation is generally robust to mis-specified models and heavier tailed data.

We then applied the concepts of Bayesian credible intervals and Bayesian posterior predictive p-values. We decomposed the variance/covariance matrix of the data to standard deviation elements and correlation elements and ran a Metropolis-Hastings algorithm within Gibbs with a set of parameters being updated at one time. Our simulation results showed that Bayesian tests provide an alternative way of testing homogeneity of serial correlations.

Under an assumption of homogeneity, we then developed a Mantel-Haenszel-type estimator of the pooled correlation coefficient and its asymptotic variance estimate as the sample size goes to infinity. Through simulations, we found that our proposed Mantel-Haenszel estimator is very close to the true common correlation, and that the variance estimator also performs well even with a small sample size. In addition, the variance estimator remains robust to model mis-specification.

When applied to actual data, we found some significant, time-invariant correlation did exist between MMP-8 and MMP-9 and some red complex pathogens. These results are supported by published clinical research and demonstrate the utility of our methods for providing guidance to investigators as to which biomarker/pathogen pairs might best describe disease severity over time.