An article in response to Grant
From www.norsemathology.org
Contents 
Preamble to Katie and Grayson:
"How to make engineers write concisely with sentences? By combining journalism with the technical report format. In a newspaper article, the paragraphs are ordered by importance, so that the reader can stop reading the article at whatever point they lose interest, knowing that the part they have read was more important than the part left unread."
"State your message in one sentence. That is your title. Write one paragraph justifying the message. That is your abstract. Circle each phrase in the abstract that needs clarification or more context. Write a paragraph or two for each such phrase. That is the body of your report. Identify each sentence in the body that needs clarification and write a paragraph or two in the appendix. Include your contact information for readers who require further detail."
 William A. Wood (email), September 8, 2005
from this article by Edward Tufte, on "PowerPoint abuse".
A Census of Cicadas Via Selective Predation by the CicadaKiller Wasp Sphecius speciosus
Grant's title: Opportunistic predation and offspring sex ratios of cicadakiller wasps (Sphecius speciosus Drury)
A story of the great cicada census
Abstract
Observations of cicadakiller wasp sphecius speciosus in Florida provide data that suggest that predation by wasps on cicadas is not opportunistic. Furthermore, size differences in two populations of wasp and in the cicada species they prey upon provide the opportunity to create a "relative census" of the four cicada species in the study area. We also estimate (simultaneously) the massconversion equation of turning cicadas into wasps.
It turns out that sex allocation ratios of wasps provide some of the tools essential for this estimate.
Introduction
The question of whether cicada predation by cicadakiller wasp sphecius speciosus is opportunistic or not has been raised in the literature, and seemingly answered in the affirmative in Peter Grant's article Opportunistic predation and offspring sex ratios of cicadakiller wasps (Sphecius speciosus Drury).
We investigate several issues from the Grant paper, in particular the two issues raised in the title:
 how are sex allocation ratios determined ("one of by male, two if by female"?), and
 whether predation on cicadas by wasps is opportunistic or not (defined most easily by the question: have we ever seen a wasp turn down a cicada?). Opportunistic predation:
 All cicadas welcome!
 No cicada too small!
In addition to issues raised in the Grant paper,
 we study the "Newberry/St. John" problem. Why are populations of sphecius speciosus in the two locations of different sizes? In particular, why are the wasps of Newberry apparently smaller than those of St. John?
Materials and Methods
Data collection
Here Jon would explain exactly how the data were obtained, what was obtained, etc.
The CicadaKiller and Cicada Data
Mathematical methods
We might explain the types of methods we used in our mathematical modeling and statistical analysis:
 Our modeling process
 Anova
permits us to distinguish whether means of samples are significantly different or not.
 Multivariate Linear Regression
permits us to determine a (linear) model for one variable as a function of one or more predictors.
 NonLinear Regression
permits us to determine a (nonlinear) model for one variable as a function of one or more predictors.
 Kernel smoothing
is a method for picking up a trend (such as a linear model) in a cloud of messy data. It smooths out the cloud, and produces a curve that should give us a better idea of how variables are related in the midst of noise.
 Leibniz's formula
translating a density for one variable into a density of another
where
M = f(L) is the function which turns a RWL into a Mass.  etc.
Results
On why wasps allocate males and females differently
Why is the question of offspring sex allocation important?
 It gets to the question of why  or whether  wasps use a "one if by male, two if by female" strategy for allocating their eggs (or even a "four if by male, seven if by female" strategy!); or how they might behave under resource limitation (all males?).
 Figuring out where the break point is in terms of "production" of males versus females.
 It helps us to understand "the psychology of the wasp".
Frank (Sex Allocation in Solitary Bees and Wasps (Vol. 146, No. 2. The American Naturalist. August 1995, and cited in Peter Grant's article) claims to have developed "....a simple theory that predicts an increasing F/M allocation ratio with increasing F/M size ratio."
Results related to Frank's paper Sex Allocation in Solitary Bees and Wasps
Wasps use small amounts of resources to make males, and larger amounts of resources to make females; but they don't do it by inspecting cicadas for weight  it's evidently a count ("one if by male, two if by female").
But females may, in hunting, be looking at their resources x as their total success in obtaining cicadas; and, based on how they're doing, they may allocate more or less to males, say.
If resources are limited, they may invest more in males (females won't find enough to produce offspring, so invest in males as they may inseminate those alpha females which manage to succeed in tough times).
It may be that a female without a single cicada goes out looking for more, doesn't find any more, and so lays a male. In a way, it's a Baysian approach to resource availability: if I'm not finding any, then there aren't any. If a wasp goes hunting and finds a second, then there are lots of resources, and I should make a female. Let the market decide whether we've got lots of resources, and hence can sustain a large number of females, and our population will swell; or there are few resources, and we should cut down on production (females).
Size variation of the cicada killer wasps in Newberry and St. John, Florida
In this study, the Eastern cicada killer, sphecius speciosus, was observed in both Newberry, Florida and St. John, Florida. However, there was a large variation in the size of the wasps from one location to the other. The wasps in St. John, Florida were significantly larger than those in Newberry (see table below, and figure).

One hypothesis to account for the size variation of the wasps is that the wasps in Newberry are smaller because Newberry is more residential. In Newberry, female wasps were found burrowing in the lawns of homeowners. These homeowners may consider the wasps a pest because the soil that is excavated forms a mound around the entrance to their nests and the male wasps have very aggressive territorial behavior. Extermination would tend towards the largest female wasps because their nests are the largest, and therefore, more noticeable.
We assume that the size of the wasps in St. John would be the same size as the wasps in Newberry if extermination does not occur. If we take the size distribution of female wasps in St. John and remove the upper half from the population, the average female wasp becomes smaller. The maximum size cicada that a female wasp can carry will become smaller, and therefore, the wasp's offspring will be provisioned with smaller cicadas. It is known that the provision size for wasp offspring is directly proportional to the size an adult wasp can become. So a reduction in the maximum size cicada that a wasp can carry will result in an overall shift in the size distribution of wasps similar to those in Newberry.
Andy here: Now let's think about how to show if the shift is what we'd expect. If we truncate the female population so that they can't take the largest prey, then the offspring of the population will be smaller. How much smaller? Frank has some models for that in his paper on Sex Allocation. We use the model for the population with the largest removed  a truncated normal  then eliminate the big cicadas from the buffet. We use the model that Katie came up with as the resources by size (changing everything into mass), then figure out what the size distribution of the next generation should be  in mass  and reinterpret as RWL.
We might do this for a few generations, and see what happens....
Discussion
Acknowledgements
References
Web Resources
Paper Resources
 Frank, Steven A. Sex Allocation in Solitary Bees and Wasps. The American Naturalist (August, 1995), 146(2), 316323. (and cited in Grant's paper. Frank has other papers available at his website, http://stevefrank.org/reprintspdf/)
 Frank, Steven A. Individual and Population Sex Allocation Patterns. Theoretical Population Biology (February, 1987). 31(1), 4774. Frank's early paper provides background on Sex Allocation in Solitary Bees and Wasps.
 Grant, Peter. Opportunistic predation and offspring sex ratios of cicadakiller wasps (Sphecius speciosus Drury), Ecological Entomology (2006), 31, 539–547.
Appendices
Cicada Statistics
Species  nh  do  tg  tb 
N  87  14  23  14 
Mean Mass  340  419  1899  2761 
SD Mass  49.25  67.81  321.61  452.09 
Mean RWL  26.5  27.9  43.3  52.9 
SD RWL  1.14  1.10  1.59  1.95 
Allometric Models for RightWingLength (RWL) and Mass
Animal/Species  RWLtoMass equation conversion, x = RWL(mm), y = mass(mg) 
Cicada/Ne.h.  y = 0.1861 * x^{2.29} 
Cicada/D.o.  y = 0.0044 * x^{3.44} 
Cicada/T.g.  y = 0.0044 * x^{3.44} 
Cicada/T.b.  y = 0.0044 * x^{3.36} 
Combined Cicada Species  y = 0.00902 * x^{3.37} 
Wasp  y = 0.0152 * x^{3.27} 

Mass Vs. Right Wing Length NonLinear Model for Wasps
Mass Vs. Right Wing Length NonLinear Model for Cicadas
The cicadas in this study were measured by their wet mass (mg) and right wing length (mm). While it would be useful to have a nonlinear model that fit the right wing length vs. the mass, the variation between the four different cicada species resulted in an unsuccessful fit. 
Looking at the picture above, most of the data points with a right wing length between 4050 mm, fall above the nonlinear model. Most of the data points with a right wing length over 50 mm fall below the nonlinear model. As a result, the next step in trying to fit a nonlinear model was to look at each of the four species separately and try to fit a nonlinear model to each cicada species. Although the speciesspecific models had a significantly lower rsquared value for three out of four of the models, the results were much more suitable for the data. 
With the TG species taken as the standard, we obtained the following results (using this lisp program with the data in this file), and based on backwards stepwise regression with indicators for each species:
Linear Regression: Estimate SE Prob Constant 5.41798 (0.305496) 0.00000 LNRWL 3.43885 (8.469368E2) 0.00000 SPECIES_NE.H. 3.73649 (0.966062) 0.00017 SPECIESNEH*LNRWL 1.15068 (0.292194) 0.00013 SPECIESTB*LNRWL 7.939483E2 (1.180710E2) 0.00000 R Squared: 0.982698 Sigma hat: 0.111653 Number of cases: 138 Degrees of freedom: 133
We interpret this as follows: the DO and TG species have the same regression equation, while the NH requires both a different constant and slope, and the TB species requires only a different slope.
Normality has Implications for the Choice of Mass Versus Right Wing Length
One of our objectives was to create a model that would allow a wasp's right wing length to be a predictor of its mass. Using nonlinear regression, Katie created the power model y = 0.015 * x^{3.27}. It is not surprising that the power is near 3. An object's mass is the product of its density and its volume, which has three dimensions. For a box with equal sides x (a cube), we could predict its mass using the power model y = density * x^{3}. Although we have a model that transforms wasp right wing length to mass, we may want to use right wing length when using statistics because it appears to be normally distributed.
For example, I simulated a normal distribution by generating random normal data in StatCrunch with a mean of 4 and standard deviation of 1. Below is the histogram with a binwidth of 0.52.
I transformed the normal data using the model y = x^{3}. Below is the histogram with a binwidth of 30.
The right tail is longer, therefore, this distribution is rightskewed. Here, the mean is greater than median which is greater than the mode. The transformed data is no longer normally distributed (if it were, the mean, median and mode would all be equal). This demonstrates why we would want use a linear measurement, i.e. right wing length, which appears to be normally distributed when performing statistical tests.
Data
Description: Wasp right wing lengths by sex and location. SJ f are the female wasps in St. Johns, SJ m are the male wasps in St. Johns, NB f are the female wasps in Newberry, and NB m are the male wasps in Newberry. 
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Description: These data files contain cicada species, wet mass, and RWL. 
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Description: Simulated normal data and the simulated normal data cubed. 1000 rows. 
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Description: The waspRWL vs Mass data was used to create a scatter plot. A nonlinear equation was fit to the data which provides a allometric relation between the wasp RWL and mass. 
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Description: The wasp RWL and cicada RWL were used to create a scatter plot. Normal smoothing was performed on scatter plot with bidwidth 3.4. 
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