Making It Fit by Leaf and Murphy

Part III – Minimized Sum of Squared Residuals

Dr. Latimer knew that some lines fit the data better than others. During his research, he noticed that the differences in the line and the points should be minimized, but he was puzzled about how this could be done. He also knew that in order to determine the best fit curve, he would need an independent criterion to determine the fit of line to data. Could he use the process of “minimizing the sum of squared residual” as a criterion?

He knew from his previous work (in Part II) that the equation for the straight line, if it were fitted to the length-at-age data, would be:

L_i = mt_i + b + ε_i

and is similar to:

L_i = L_∞(1 − e^−kti) + ε_i

The parameters in both curves are on one side of the equation and define the expected portion of the line. The observed data is on the left side of the equation.

Questions

1. We could present the equation to express the residual error term as the difference of the observed and expected terms. Rewrite the equation below to solve for ε.

   L_i = mt_i + b + ε_i

   ε_i =

2. How could we express the von Bertalanffy growth function if we wanted to have the residual error value be equal to the difference in the observed and expected values?

   L_i = L_∞(1 − e^−kti) + ε_i

   ε_i =

3. Taking this a step further; what is the equation of ε² for each of the equations?

   von Bertalanffy ε_i² =

   Linear ε_i² =

4. How would these calculations work for a sample data set? Fill in the expected lengths for each of the given equations:

   Age (t)	Observed Length (Lt)	L∞ = 26, k = 0.6
   1	7.2
   2	21
   3	27.5

5. Using the “expected” values that you calculated above for the von Bertalanffy growth function, fill in the table below:

   Age (t)	Observed Length (Lt)	Observed − Expected Length	(Observed − Expected Length)2
   1	7.2
   2	21
   3	27.5

6. What is the sum of all of the elements in the (observed − expected)² column (this will be the “minimized sum of squared residuals”)? Use some alternative values for the k parameter and compare the values of the sum of the squared residuals. Which model do you think fits the data better? Why?

7. How can the poor fitting model be made to fit the data better?

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