What is Allometry? continued…

Developing a Model Using Non-Linear Regression

Now that we have found a power function through the manual conversion of a linear regression, let’s try to create a non-linear regression that fits the data points most closely. Similar to before we will be solving for a power function. We will start with the formula

M = kma+b

Where M is the claw mass, m is mass, k is the constant of proportionality, a is a constant exponent and b is the intercept. We can use Excel Solver to determine an equation that has the least difference2, between measured claw mass and expected claw mass, resulting in the best fit equation. We begin with simple numbers:

k b a Mass (g) Claw Mass (g) Expected Claw Mass (g) Difference Difference^2
100 100 0.05 4.74516 0.303208 208.0967267 207.793519 43178.1464
10.4439 1.15915 212.4457732 211.286623 44642.03716
14.1628 1.24838 214.1714341 212.923054 45336.22695
18.8073 1.65723 215.80207 214.14484 45858.0125
23.2829 2.65709 217.0447272 214.387637 45962.05897
29.0201 3.14769 218.3409211 215.193231 46308.12671
33.2396 2.82348 219.1469116 216.323432 46795.82705
37.577 5.27659 219.8798264 214.603236 46054.54906
42.7777 7.44482 220.6593195 213.214499 45460.42278
49.6655 5.50366 221.5633726 216.059713 46681.79943
sum= 456277.207

 

Table 4: Expected Claw mass values (g) vs. actual values (g) using formula M = kma+b with variables k=100, b=100, and a=0.05.

Looking at the difference sum we can conclude that these values are not a good fit. To find the best fit we can use Excel Solver. These are the values that will produce a minimum value for the sum of the difference squared of expected claw mass and actual claw mass.  

k b a Mass (g) Claw Mass (g) Expected Claw Mass (g) Difference Difference^2
0.045036389 0 1.280381226 4.74516 0.303208 0.33069107 0.02748307 0.000755319
10.4439 1.15915 0.908021728 -0.2511283 0.063065409
14.1628 1.24838 1.341137439 0.09275744 0.008603942
18.8073 1.65723 1.92835606 0.27112606 0.07350934
23.2829 2.65709 2.534499239 -0.1225908 0.015028495
29.0201 3.14769 3.36028241 0.21259241 0.045195533
33.2396 2.82348 3.998186665 1.17470667 1.379935749
37.577 5.27659 4.678043514 -0.5985465 0.358257896
42.7777 7.44482 5.522601866 -1.9222181 3.694922556
49.6655 5.50366 6.685906539 1.18224654 1.397706879
sum= 7.03698112

 

Table 4: Expected Claw mass values (g) vs. actual values (g), using formula M = kma+b after using Excel Solver resulting in variables k=0.045036389, b=0, and a=1.280381226.

Using the results Excel Solver calculated we can now put into equation form.

y = 0.045036389x1.280381226

Now to make it in terms of RD we just add our p constant.

y = 0.045036389px1.280381226

We can compare this to the equation we derived from our linear regression.

y = .04557px1.2698

Non-Linear Linear
Difference^2 Difference^2
0.0007553 0.000672536
0.0630654 0.069114543
0.0086039 0.005052793
0.0735093 0.054907784
0.0150285 0.031177036
0.0451955 0.017787397
1.3799357 1.155282315
0.3582579 0.520287539
3.6949226 4.303614145
1.3977069 0.975366204
sum= 7.0369811 7.133262291

 

Table 5: Squared differences of linear and nonlinear derived formula, between expected claw mass and actual claw mass. Also includes sum of squared differences.

Comparing the sums of the squared differences show how similar they are which is expected because our results very closely resembled a linear equation. There was a slight concavity meaning a power function would fit best.

Testing Model

Now that we have derived a simple model we can begin adding more parameters. The first model we should check are bounds. Although our model seems like a fit compared to our experimental values we need to determine when a fiddler crab will stop fitting our model. Our equation is a power function so the incremental increases will not be the same, so we can use the ratio of mass of the crab to the claw mass to determine when our model is not viable. The smaller the ratio the more likely our model doesn’t fit. We can make an assumption that the male fiddler crab will not be able walk with a claw that 33% of its body weight or a ratio of 3. We can plot points to determine when the model we have will not fit.

Mass (g) Claw Mass (g) Total Mass (g) Percentage of Mass
1 0.04557 1.04557 4.358388248
10 0.848162425 10.84816242 7.81848936
100 15.7862519 115.7862519 13.63396055
200 38.06509492 238.0650949 15.98936414
500 121.851573 621.851573 19.59496097
1000 293.8184265 1293.818426 22.70940191
1600 533.6672877 2133.667288 25.01173874
2500 940.5529537 3440.552954 27.33726137
3700 1547.325126 5247.325126 29.48788361
6790 3344.931805 10134.9318 33.00398926

Table 6: Claw mass from derived equation and percentage of total mass.

We can develop a model that will tell us how big a claw size can be for each individual fish before tipping size based on our assumption that the claw cannot be 33% of the total weight. We set claw size = .(1/3)(mass + claw size) and solve for claw size:

claw size = (3/2*mass)/3

Allometry is a useful tool that can be used in virtually any field. The ability to develop patterns no matter how many data points you have, or how different they may seem, is a very useful tool. Although we have only done basic model building here, once we have created the model, we can easily expand upon it. For example, in the first case we can add a lower bound as well by just developing an equation for claw mass that will not let us know when it becomes too small. Deriving proportional models is a useful tool.  

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