cov_holder {PKNCA}

Calculate the covariance for two time points with sparse sampling

Description

The calculation follows equation A3 in Holder 2001 (see references below):

Usage

cov_holder(sparse_pk)

Arguments

sparse_pk

A sparse_pk object from as_sparse_pk()

Details

\hat{\sigma}_{ij} = \sum\limits_{k=1}^{r_{ij}}{\frac{\left(x_{ik} - \bar{x}_i\right)\left(x_{jk} - \bar{x}_j\right)}{\left(r_{ij} - 1\right) + \left(1 - \frac{r_{ij}}{r_i}\right)\left(1 - \frac{r_{ij}}{r_j}\right)}}

If r_{ij} = 0, then \hat{\sigma}_{ij} is defined as zero (rather than dividing by zero).

Where:

\hat{\sigma}_{ij}

The covariance of times i and j

r_i and r_j

The number of subjects (usually animals) at times i and j, respectively

r_{ij}{r_ij}

The number of subjects (usually animals) at both times i and j

x_{ik} and x_{jk}

The concentration measured for animal k at times i and j, respectively

\bar{x}_i and \bar{x}_j

The mean of the concentrations at times i and j, respectively

The Cauchy-Schwartz inequality is enforced for covariances to keep correlation coefficients between -1 and 1, inclusive, as described in equations 8 and 9 of Nedelman and Jia 1998.

Value

A matrix with one row and one column for each element of sparse_pk_attribute. The covariances are on the off diagonals, and for simplicity of use, it also calculates the variance on the diagonal elements.

References

Holder DJ. Comments on Nedelman and Jia’s Extension of Satterthwaite’s Approximation Applied to Pharmacokinetics. Journal of Biopharmaceutical Statistics. 2001;11(1-2):75-79. doi:10.1081/BIP-100104199

Nedelman JR, Jia X. An extension of Satterthwaite’s approximation applied to pharmacokinetics. Journal of Biopharmaceutical Statistics. 1998;8(2):317-328. doi:10.1080/10543409808835241