Henry et al. (2012) homing failure formula, assumptions, and basic mathematics: a comment

[Extract] In March 2012 Henry et al. published a paper that explored whether or not the consumption of thiamethoxam via nectar could be a causal factor of Colony Collapse Disorder (CCD) in honeybees. In the first part of their report, Henry et al. (2012) measured the homing success after "ecologically relevant" thiamethoxam exposure and compared it to non-thiamethoxam exposed, control homing rates [see Guez (2013) for a critique of this aspect of their work]. In the second part of their report, they applied their homing study results to the honeybee population dynamic model devised by Khoury et al. (2011), and from that they concluded that dietary thiamethoxam intoxication may potentially contribute to CCD. 
 
Khoury et al.'s (2011) model is build upon the hypothesis that colony failure occurs when bee death rate become unsustainable at the colony level, and the salient assumption that mortality within the hive is negligible. Khoury et al.'s (2011) model allows the evolution of the honeybee hive population to be projected over time. Model outputs are dependent on the total hive population (at the start, and then at any given time), the queen's egg laying rate (L), an eclosion rate that is directly dependent upon the hive population and modulated via the parameter w (the larger w the lower the eclosion rate), and the forager mortality rate or forager homing failure (m). In Khoury et al.'s (2011) model therefore, the population growth of the colony is controlled mainly by the parameters L and w [but see Cresswell and Thompson (2012) for a critique of the choice of w], whereas population decline is dependent on m, the forager mortality rate or forager homing failure.

In March 2012 Henry et al. published a paper that explored whether or not the consumption of thiamethoxam via nectar could be a causal factor of Colony Collapse Disorder (CCD) in honeybees. In the first part of their report, Henry et al. (2012) measured the homing success after "ecologically relevant" thiamethoxam exposure and compared it to non-thiamethoxam exposed, control homing rates [see Guez (2013) for a critique of this aspect of their work]. In the second part of their report, they applied their homing study results to the honeybee population dynamic model devised by Khoury et al. (2011), and from that they concluded that dietary thiamethoxam intoxication may potentially contribute to CCD. Khoury et al.'s (2011) model is build upon the hypothesis that colony failure occurs when bee death rate become unsustainable at the colony level, and the salient assumption that mortality within the hive is negligible. Khoury et al.'s (2011) model allows the evolution of the honeybee hive population to be projected over time. Model outputs are dependent on the total hive population (at the start, and then at any given time), the queen's egg laying rate (L), an eclosion rate that is directly dependent upon the hive population and modulated via the parameter w (the larger w the lower the eclosion rate), and the forager mortality rate or forager homing failure (m). In Khoury et al.'s (2011) model therefore, the population growth of the colony is controlled mainly by the parameters L and w [but see Cresswell and Thompson (2012) for a critique of the choice of w], whereas population decline is dependent on m, the forager mortality rate or forager homing failure.
In order to model the population dynamic under dietary thiamethoxam exposure, Henry et al.'s (2012) undertaking was to calculate the homing failure due to pesticide exposure (m hf ). m hf was then used to increase the value of m for population projection under a dietary thiamethoxam exposed scenario, compared to the "normal" homing failure m postulated in the non-thiamethoxam exposed scenario. In my previous critic of Henry et al. (2012) I pointed out that the way in which Henry et al. calculated m hf was incorrect given the authors claim that m hf "[. . . ] estimates the proportion of exposed foragers that might disappear due solely to post-exposure homing failure, all other sources of mortality or homing failure set apart (natural mortality, predation, manipulation stress)." In their answer to my critic (Guez, 2013) Henry and Decourtye (2013) maintain that the formula used in Henry et al. (2012) to calculate homing failure in honeybees post pesticide exposure (m hf ) is correct. In this note I show that the calculated m hf value is largely impacted by assumptions, and that regardless of which assumption is taken, Henry et al.'s (2012) formula for calculating honeybee homing failure post-pesticide exposure is incorrect.
In their original report Henry et al. (2012) propose that: whereas Guez (2013)  We ran simulations under the hypotheses of (i) constant forager death rate with no forager exposure, and (ii) forager death rate raised by post-exposure homing failure m hf during a 30-days oilseed rape flowering period [. . . ]. In the later configuration, exposed foragers were assigned a probability of disappearance combining daily death rate and the additional mortality due solely to post-exposure homing failure.
From this statement it is clear that: (i) m hf was used by Henry et al. (2012) to raise the homing failure rate after pesticide exposition, and (ii) m hf represents the post-exposure homing failure.

USING THE m hf FORMULA PUT FORWARD BY Henry et al. (2012) AND Henry and Decourtye (2013)
If we use the equation put forward by Henry et al. (2012) (Equation 1), m hf is expressed as a proportional decrease in homing success in the treatment group relative to the control, as highlighted in the comment of Henry and Decourtye (2013). In the case of the upper bound m hf determined in Experiment 2, Henry et al. (2012) found that there was 0.316 less homing success in the treatment group than in the control group. In other words, there was 31.6% less homing success with treatment. However, since the parameter m of Khoury et al. (2011) is intended to represent the attrition of foragers with time, what is needed is the proportional increase or decrease in homing failure of the treatment given the control, not the decrease in homing success of the treatment given the control. As counterintuitive as it may be, these two values (i.e., the proportional increase in homing failure and the proportional decrease in homing success) are not numerically equivalent.
Taking Experiment 2 in Henry et al. (2012) as an example, homing success was reported as 0.83 and 0.57 individuals.day −1 in the respective control and treatment groups, meaning that 0.316 less individuals.day −1 returned to their hive in the treatment group relative to the control. Indeed, [treatment homing success] = 0.83 − (0.83 × 0.316) = 0.57 individuals.day −1 . However, these homing success rates also mean that 0.17 and 0.43 individuals.day −1 failed to return in the respective control and treatment groups ([homing failure] = 1 − [homing success]). This translates into a 1.55 fold (155%) additional increase in homing failure rate relative to the control (0.17 individuals.day −1 ), and represents a treatment homing failure of 0.43 individuals.day −1 :

USING THE m hf FORMULA PUT FORWARD BY Guez (2013)
The homing failure of the control group reflects normal attrition or mortality and to calculate m hf , we assume that pesticide exposure increases homing failure by a set amount and is not proportional to any "normal" homing failure. If so, the m parameter of the Khoury

ASSUMPTIONS AND CONSEQUENCES
The choice of Equations 2 or 3 as the basis for the calculation of m hf is solely dependent on the assumptions taken. If we choose to use Equation 2, we assume that the homing failure attributable to the exposure of a given dose of pesticide is a fixed value regardless of the "normal" homing failure [assumption (a)]. In this case we assume that most of the homing failure observed in the control is due to natural predation. However, if we choose Equation 3 we assume that the homing failure attributable to pesticide exposure is not only a fixed value 2 Cresswell and Thomson implementation is available at is available online from the Exeter Research and Institutional Content archive (ERIC) at http://hdl.handle.net/10036/3648. To run simulation under assumption (a) [m pesticide = m non-exposure + (m non-exposure × m hf )] one need to modify how the parameter m is calculated. To this end the formula contained in cell B3 should be changed to "= background + (B16 × background)." in function of the dose of pesticide, but is proportional to the level of the "normal" homing failure [assumption (b)]. For example, we would assume that most of the natural homing failure is due to an aging population of foragers which are more susceptible to the effects of pesticides in comparison to young foragers. We favor assumption (a), although others may disagree.
Nonetheless, and notwithstanding Guez's (2013) previous critic of Henry et al. (2012), if we use the correct formula as set forth herein, choosing assumptions (a) or (b) has only a minimal impact on Khoury et al.'s (2011) model projection, as exemplified by the green and red curves in Figure 1. However, the difference in model projections that are obtained under assumptions (a) and (b) can become quite significant if the assumed "normal homing failure" (here 0.154 individuals.day −1 ) is far removed from the experimentally determined control homing probability. For example, let us imagine that due to a heightened experimental stress the control homing probability is only 0.7 individuals.day −1 and the treatment homing probability is 40% less (i.e., 0.42 individuals.day −1 ).
If we use assumption (a), after 30 days of exposure we project more than 4000 less individuals in the hive than if we use assumption (b) (see Figure 1, continuous and dotted black lines). Thus, our assumptions are not without consequence on our model projections, highlighting the need for researchers to explicit all assumptions, and to describe exactly how such assumptions would impact upon outputs within the model used. This is particularly important for models projections that claim to have direct ecological significance, such as the one presented by Henry et al. (2012). Without this crucial information and an understanding of the consequences of various assumptions on model projection, model end points cannot be accurately interpreted and thus cannot be used for accurate or meaningful decision making in the field.

CONCLUSION
This contribution highlights the influence of assumptions on model projections and the importance of making explicit not only any assumptions, but also how these influence outputs within a given model. It also highlights that the formula used to calculate m hf in Henry et al. (2012) (Equation 1), is conceptually flawed since it calculates the proportional decrease in post exposure homing success given the control instead of the proportional increase in post exposure homing failure. These two values are not numerically equivalent and thus, even without taking into account Guez's (2013) critique, the model projections presented by Henry et al. (2012) are erroneous and cannot be used as a realistic evaluation of the potential impact of dietary thiamethoxam exposure on foraging honeybees. Furthermore, if as Henry et al. (2012) claimed m hf was intended to solely represent the postexposure homing failure: "all other sources of mortality or homing failure set apart (natural mortality, predation, manipulation stress)," m hf should have been calculated using Equation 3 as previously described in Guez (2013). Nonetheless, and regardless of the assumptions made by Henry et al. (2012) when calculating m hf , the m hf value presented by Henry et al. (2012) is erroneous as the equation used to calculate it is conceptually flawed. Thus, Henry et al.'s (2012) model projection should not be used as the basis for any meaningful regulatory decision making about the potential risks posed by dietary thiamethoxan intoxication in honeybees.