To test model goodness-of-fit, we calculated residuals on the logarithmic scale. This was done for each state variable from each treatment population in the "estimation" data sets by substituting the ML estimates into the conditional expected values (2.15). Differences of the logstate variables from their estimated conditional expected values are the log-residuals. We subjected these log-residuals to various diagnostic tests for autocorrelation and normality, as described in Section 2.6.

The results of the residual analyses for those SS and RR data sets designated for parameter estimation only appear in Appendix B. The residuals are the estimated outcomes of the noise variables £it in the stochastic LPA model (2.12) and should display approximately the assumed statistical properties of these noise variables. In constructing these tables of statistics, each individual time series of residuals, representing one state variable (L, P, or A) from one population, was tested separately for departure from normality and autocorrelation (first- and second-order).

Significant first-order autocorrelation is present in none of the SS series and in just one (3%) of the 36 RR residual times series. We find significant second-order autocorrelation in just two (6%) of the SS series and two of the RR series. Departure from a normal distribution is seen in six (17%) of the SS series and 12 (33%) of the RR series. These normality departures can be traced to a small number of oudiers; most of the observations in each series are well described by a normal distribution. Notice that these results compare favorably with the same analyses performed in Chapter 2 on the cos strain of T. castaneum (in which, for the twelve time series in that data, there are no first-order autocorrelated series (0%), two second-order autocorrelated series (17%), and three nonnormal series (25%)).

To test the predictive capability of the parameterized model we also performed a "prediction fit" using the data sets that we set aside and did not use in either the parameter estimation or the goodness-of-fit tests. That is to say, we repeat the diagnostic residual tests for autocorrelation and normality on the validation data sets using the same parameter estimates as appear in Tables 3.1 and 3.2. The results appear in Appendix B.

The prediction errors analyzed in Tables A. 16 and A. 17 are standardized (centered at their mean and divided by their standard deviation) in order to separate autocorrelation and normality evaluation from prediction bias evaluation. As is seen in these tables, no significant first-order or second-order autocorrelation is present in any of the times series in the validation data sets. Departure from a normal distribution is seen in just 8 (22%) of the SS series and 14 (39%) of the 36 RR series. Once again, a small number of identifiable outliers cause these departures. These results compare favorably with the same analyses performed in Chapter 2 on the cos strain of T. castaneum (in which, for the 27 time series in that data, there is one first-order autocorrelated series (4%), two second-order autocorrelated series (7%), and 10 nonnormal series (37%)).

It turns out that prediction bias is negligibly small. A prediction bias is indicated if the prediction errors are centered at a nonzero value. A systematic tendency for the model to underpredict or overpredict would be the cause. We tested each series of prediction errors for whether or not it arises from a distribution with a mean of zero (i-test). The result is that only two of the 36 SS time series show significantly nonzero means (in replicate 17, test statistic T = -2.83 and probability value P = 0.02 for adults and in replicate 18, T = -3.03 and P = 0.01 for adults). For the 36 RR time series only three show significantly nonzero means (in replicate 1, T - -2.46 and P = 0.03 for pupae; in replicate 8, T = 2.40 and P = 0.04 for larvae; and in replicate 11, T = 3.07 and P = 0.01 for pupae). On the original scale the prediction biases of the two SS series are only about three adults and seven adults, respectively, while those for the three RR series are about two pupae, 13 larvae, and six pupae, respectively.

Our residual analyses suggest that most of the systematic, predictable variability in the data is accounted for by the deterministic LPA model (3.1) and that the stochastic LPA model (2.12) adequately describes the remaining unpredictable variability. Plots of the both the estimation and validation data time series together with each one-step model prediction visually support this conclusion. Such plots of all life-cycle stages for all treatments in both SS and RR experiments are given in [44]. A sample of these plots appear in Figs. 3.4 and 3.5.

Replicate 4 Replícatelo Replicate 16 Replicate 22

Replicate 4 Replícatelo Replicate 16 Replicate 22

12 16 20 24 28 32 36 Week

12 16 20 24 28 32 36 12 16 20 24 28 32 36 Week Week

FIGURE 3.5 I Census data (open circles connected bylines) and one-step predictions (solid circles) for all four replicates of the fia = 0.5 treatment for the RR strain in the bifurcation experiment. The control treatments for the adult death rate began at week 12.

12 16 20 24 28 32 36 Week

12 16 20 24 28 32 36 12 16 20 24 28 32 36 Week Week

FIGURE 3.5 I Census data (open circles connected bylines) and one-step predictions (solid circles) for all four replicates of the fia = 0.5 treatment for the RR strain in the bifurcation experiment. The control treatments for the adult death rate began at week 12.

The statistical analyses described here confirm the accuracy of the LPA model with regard to its description of the data obtained from the bifurcation experiment. On this basis we conclude that the beetle populations do display the asymptotic dynamics predicted by the deterministic LPA model as the adult death rate /xfl is changed. In the next section we take a closer look at these dynamics and the predicted sequence of bifurcations.

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