The census data obtained from the two bifurcation experiments described in Section 3.1 appear in Appendix B. In [44] we performed the reparameterization and validation of the LPA model separately for each experiment (i.e., for the SS and RR strain data sets separately). We began by dividing each data set in half by picking at random two populations from each treatment (Tables 3.1 and 3.2) to use for parameter estimation. The remaining 12 populations were withheld for model evaluation.

The log-likelihood function (2.17) can be adjusted to accommodate the experimental design as follows. Combine the time series from the 12 "estimation" populations into one log-likelihood. Each time-step for each population, representing a transition from wf to wf+i, contributes a In p(wt+i | Wf) term to the log-likelihood. For the experimentally manipulated populations, the value of iia in the term In p (w,+i | wf) is fixed at the corresponding experimental value (either 0.04, 0.27, 0.50, 0.73, or 0.96). The value of ixa in the control populations is estimated directly from the census counts of adults at time t and dead adults at time t + 1 (product-binomial likelihood). The resulting estimate of fia is included in the time series log-likelihood as a constant. The control populations and experimental populations are given different variance-covariance matrices (Ec and E7 , respectively) in the log-likelihood, since it is expected that the experimental treatments alter the variability of the adult stage because of the manipulation of adult numbers.

The starting time t = 0 in the model was set at week 12 when the manipulations of the adult death rate commenced. Therefore, the data time series for each strain used in the analyses is 24 weeks in length and contains 12 one-step transitions. Although 12 observations would not normally constitute an adequate sample size for time-series analysis, in our experiment all 12 populations from the estimation half of the data were used to estimate a common set of parameters. The effective sample size was therefore 12 x 12 = 144 transitions in the log-likelihood function.

The log-likelihood function contains a total of 17 unknown parameters: b, cei, cea, cpa, in and six parameters each in the matrices Ec and £r.

TABLE 3.1 I Maximum Likelihood Parameter Estimates for the SS Genetic Strain Obtained from Data in Appendix B (replicates 7,19,8,14, 9,15,4,22, 5, 23,12,24).fl From [44].


ML estimate

95% confidence interval b

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