Model Selection For A Subterranean Trophic Cascad: Root-Feeding Caterpillars And Entomopathogenic Nematodes

Ecology, Dec, 1999 by D. R. Strong, A. V. Whipple, A. L. Child, B. Dennis

[SIC.sub.kl] = -2 ln [L.sub.kl] [r.sub.kl]ln n. (8)

Here [L.sub.kl] is the value of the maximized likelihood for model [H.sub.kl], [r.sub.kl] is the number of parameters estimated in the model, and n is the sample size (in our case, n = 120 binary observations). We considered the model with the lowest SIC among the family of models to be the best model. The SIC is sometimes called the "Bayesian information criterion" or BIC, because its original derivation was based on the Bayes factor. However, it has a compelling frequentist interpretation. Selecting the model with the lowest SIC provides an asymptotically unbiased estimate of the number of parameters in the "true" model (see Bozdogan 1987). Note that, unlike the conventional approach of sorting out many hypotheses with hierarchical pairwise comparisons in which submodels alternately assume the role of null and alternative hypotheses, in the SIC approach all hypotheses are considered at the outset to be on a level playing field.

Occasionally some models have SIC values close to that of the model with the lowest value. An informal rule-of-thumb states that one can be indifferent concerning two models for which the difference of the SIC values is [less than]2 (Sakamoto et al. 1986). When the lowest SIC value is substantially lower than the rest of the field, one can be more confident that the selected model provides a better description of the data than all the other models considered.

For each model [H.sub.kl], we also calculated the likelihood ratio goodness-of-fit statistic:

[[G.sup.2].sub.kl] = 2 [[sigma].sub.i] [[sigma].sub.j] [y.sub.ij] ln[[y.sub.ij]/([15p.sub.ij])]

2 [[sigma].sub.i] [[sigma].sub.j] (15 - [y.sub.ij])ln{(15 - [y.sub.ij])/[15(1 - [p.sub.ij])]}. (9)

Here [p.sub.ij] is Eq. 3 with the ML estimates of [[lambda].sub.1], [[lambda].sub.2], [[beta].sub.1], and [[beta].sub.2] substituted, as calculated under model [H.sub.kl]. The goodness-of-fit statistic is algebraically equivalent to Eq. 7, with [L.sub.null] corresponding to the maximized likelihood of the fitted model [H.sub.kl], and [L.sub.alt] corresponding to Eq. 5 with [y.sub.ij]/15 substituted for [p.sub.ij]. The goodness-of-fit test contrasts the fitted model (null hypothesis) with a fully saturated model in which each of the eight binomial distributions gets its own value of [p.sub.ij] that is functionally independent of the other [p.sub.ij]'s. If the null model fits, the calculated likelihood ratio statistic is an outcome from an approximate chi-square distribution with degrees of freedom equal to 8 minus the number of parameters estimated in the null model.

For comparison to the one-hit model, we performed parallel analyses with a logistic model, which takes the survival probability to be

[p.sub.ij] = [e.sup.-([a.sub.i] [b.sub.i][x.sub.j])]/1 [e.sup.-([a.sub.i] [b.sub.i][x.sub.j])] (11)

where [a.sub.1], [a.sub.2], [b.sub.1], and [b.sub.2] are parameters (compare with Eq. 3). Like the one-hit model, the logistic uses a product-binomial likelihood function (Eq. 5) and requires iterative maximization for calculating ML estimates. Unlike the one-hit model, the logistic function has an inflection point (declining S-shape). Different values of the parameters [a.sub.1], [a.sub.2], [b.sub.1], and [b.sub.2] yield a series of hypotheses parallel to those of the one-hit model in Table 1. We calculated ML estimates for the logistic hypotheses directly using the Nelder-Mead algorithm, just as we did for the hypotheses in the one-hit model family.