## Modelling

There is a variety of graphical, mathematical and physical models for estimating the age of insects from their development, taking temperature into account. Isomegalen and isomorphen diagrams (Grassberger and Reiter 2001; Midgley and Villet 2009b; Reiter 1984; Richards et al. 2009b) provide a simple, empirical graphical model (Fig. 7.6). Thermal accumulation models use a linear regression model as the basis for a simple, mathematical way of accumulating development at varying temperatures across an insect's whole developmental period (Higley and Haskell 2001; Higley et al. 1986). More sophisticated statistical models are becoming available (Byrd and Allen 2001; (Ieno et al. 2010; Oliveira-Costa et al. 2010)). The most popular physical model is a dead pig, which has been used with success (Schoenly et al. 1991; Schoenly et al. 1996; Schoenly et al. 2005; Schoenly et al. 2007; Shahid et al. 2003), as has minced meat (Faucherre et al. 1999; Turner and Wiltshire 1999).

Potential estimate biases in models arise from their assumptions, which all models have (Higley et al. 1986), and the degree to which a model's assumptions are violated biases its accuracy. For example, thermal accumulation models that assume a straight line relationship between growth rate and temperature perform poorly when temperatures fluctuate predominantly to one side of the optimal developmental temperature because of a systematic bias known as the Kauffman effect (Higley and Haskell 2001; Worner 1992); most current models assume that air temperature in the shade is representative of the temperature experienced by larvae; and dead animals are assumed to be representative of corpses. Only in some cases have these assumptions been validated as unbiased (e.g. (Schoenly et al. 2005; Schoenly et al. 2007; Tarone and Foran 2006; VanLaerhoven 2008)). The omission of a variable from a model is formally equivalent to assuming that it carries no weight.

1st ecdysis 2nd ecdysis pupariation

Time from hatching (hours)

Fig. 7.6 A combined isomegalen and isomorphen diagram, illustrating how to estimate the age of a larvae assuming that it measured 12.2 mm in length just before pupariation and experienced sequential daily environmental mean temperatures (going backwards from the date of discovery) of 19°C, 20°C, 24°C, 24°C, 20°C, 17°C, and 18°C. The length of each horizontal arrow is 24 h, and the ends of the arrows are joined by starting on the right and moving along the length contour to each new daily temperature. In this example, the PMI . is estimated to be 6.4 days. If retrospective temperature data are available at a better temporal resolution (e.g. hourly), the length of the arrows can be adjusted to this precision

Sometimes it is not even clear if variables are 'real': Ames and Turner (2003) suggested that 'The whole concept of Developmental] T[hreshold] T[emperature] and its input to the [thermal summation model] calculation needs an urgent review', but Trudgill et al. (2005) gave it theoretical support. At present many experimentally determined errors are reported in chronological time (usually hours) rather than in physiological time (degree-hours). While this can be expected to change as more data are gathered, those same data may fuel the creation of statistical Mixed Models (Ieno et al. 2010) that estimate time directly, without the problematic incorporation of a developmental threshold temperature.

Because each model has its own inventory of assumptions, the wide variety of models of development precludes a more detailed discussion of their estimate biases here. Users should make the assumptions of their preferred model explicit and validate them as rigorously as possible, consulting a mathematician or biometrician if necessary.

Sampling bias occurs in models when, for example, parameter values for one species are substituted for unknown values of another species. This cannot be done reliably even when the species are very close relatives such as Calliphora vicina and C. vomitoria (Ireland and Turner 2005; Kaneshrajah and Turner 2004) or

Chrysomya chloropyga and C. putoria (Richards et al. 2009b). It has also proved unwise to substitute data derived from different geographical areas, although some of the geographical effects might be predictable (Richards and Villet 2009; Richards et al. 2008).

The estimate precision of models can also be manipulated to some degree. For instance, more precise thermal summation parameters (K and D0) can be derived using the regression model developed by (Ikemoto and Takai 2000) than by more traditional methods (Higley and Haskell 2001; Trudgill et al. 2005). It is also important to avoid rounding off values prematurely in the computations. Again, it is best to consult a statistician about how to improve the estimate precision of a particular model.

The measurement precision of models of development often hinges on the ensuring that the process is sampled often enough in time (Richards and Villet 2008; Richards and Villet 2009). Measurements that exceed the precision of the required estimate by one significant figure (i.e. an order of magnitude) are desirable, and can be achieved by ensuring that the resolution of the measurements is about 10% of the likely value of the estimate (Richards and Villet 2008; Richards and Villet 2009). For example, if a developmental process takes 30 h on average, samples should be taken every 3 h.

Sampling precision can be improved by enlarging a sample, although the improvement is generally inversely proportional to the square root of the sample size. General Additive Models require more replication than regression models but making better use of it (Ieno et al. 2010). As a general guideline, samples of about 30 individuals representing each combination of factors in an experiment is highly desirable (due in part to the Central Limit Theorem), but not often practical.

Guidelines for managing sample and measurement precision in experimental calibrations of development for model building has been published by (Richards and Villet 2008). The same guidelines should be applied to sampling in case studies to keep levels of precision commensurate with experimental data.

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