The calculation follows equation A3 in Holder 2001 (see references below):
Arguments
- sparse_pk
A sparse_pk object from
as_sparse_pk()
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.
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.
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