Spline Smoothing for Bivariate Data with Applications to Association between Hormones

In this paper penalized weighted least-squares is used to jointly estimate nonparametric functions from contemporaneously correlated data. Under conditions generally encountered in practice, it is shown that these joint estimates have smaller posterior variances than those of marginal estimates and are therefore more efficient. We describe three methods: generalized maximum likelihood (GML), generalized cross validation (GCV) and leaving-out-one-pair cross validation (CV) to estimate the smoothing parameters, the weighting parameter and the correlation parameter simultaneously. Based on simulations we conclude that the GML method has smaller mean-square errors for the nonparametric functions and the parameters and needs less computational time than the other methods. Also, it does not overfit data when the sample size is small. Our research is motivated by and is applied to the problem of estimating associations between hormones. We find that the circadian rhythms of the hormones ACTH and cortisol have similar patterns and that cortisol lags ACTH.