Beyond the reaction progress variable: the meaning and significance of isotopic incorporation data

Abstract

Ecologists conduct isotopic incorporation experiments to determine the residence time of various stable isotopes in animal tissues. These experiments permit determining the time window through which isotopic ecologists perceive the course of diet changes, and therefore the scale of the inferences that we can make from isotopic data. Until recently, the results of these experiments were analyzed using first-order, one-compartment models. Cerling et al. (Oecologia 151:175–189, 2007) proposed an approach they named the reaction progress variable to: (1) determine how many compartments are needed to describe a pattern of istopic incorporation, and (2) to estimate the size and rate constant of each pool. We elaborate on the approach described by Cerling et al. (Oecologia 151:175–189, 2007) by providing a way to estimate average retention times for an isotope in a tissue (and its associate error) for multi-compartment models. We also qualify the interpretation of the parameters in multi-compartment models by showing that many possible mechanisms yield models with the same functional form. Multi-compartment models are phenomenological, rather than mechanistic descriptions, of incorporation data. Finally, we propose the use of information theoretic criteria to assess the number of compartments that must be included in models of isotopic incorporation.

Appendix 1 A variance estimate for \( {\hat{\bar{\tau}}} \)

We derive our results for the k-compartment model using the parameterization:

$$ \theta = (p_1 \cdots p_{k - 1} \;\tau _1 \;\tau _2 \cdots \tau _k )^T $$

Estimation of this form with 2k − 1, instead of 2k parameters results in a non-singular estimate of its variance covariance, V, but estimates should incorporate the restriction that forces 0 ≤ p _i  < 1 for all i, and $\sum_i {p_i = 1.}$ For the k compartment, the average retention time is:

$$ \bar \tau = \sum_{i = 1}^k {p_i \tau _i } = p_1 \tau _1 + \cdots + p_{k - 1} \tau _{k - 1} + \left( {1 - \sum_{i = 1}^{k - 1} {p_i } } \right)\tau _k $$

Applying the multivariate delta method, we pre- and post-multiply V by the vector of first partial derivatives:

$$ s^2 = {\text{Var}}(\bar \tau ) = \left\langle {\frac{{\delta \bar \tau }}{{\delta \theta }}} \right\rangle ^T V\left\langle {\frac{{\delta \bar \tau }}{{\delta \theta }}} \right\rangle . $$

The dimensions of V force this result to be a scalar. The components of the partial derivative vectors are:

$$ \begin{gathered} \frac{{\delta \bar \tau }}{{\delta p_i }} = \hat \tau _i - \hat \tau _k \quad i = 1, \ldots ,k - 1 \\ \frac{{\delta \bar \tau }}{{\delta p_i }} = p_i \quad i = 1, \ldots ,k - 1 \\ \frac{{\delta \bar \tau }}{{\delta p_i }} = \left( {1 - \mathop {\sum_{i = 1}^{k - 1} {\hat p_i } }\limits_{}^{} } \right) = \hat p_k \\ \end{gathered} $$

In the case of k = 2, the variance reduces to:

$$ (\hat \tau _1 - \hat \tau _2 \quad \hat p\quad 1 - \hat p)V\left( {\begin{array}{*{20}c} {\hat \tau _1 - \hat \tau _2 } \\ {\hat p} \\ {1 - \hat p} \\ \end{array} } \right). $$

Most non-linear regression programs include V in their output.