Problem 2 - Applied Linear Regression

Question

In fisheries studies, the most commonly used mean function for expected length of a fish at a given age is the von Bertalanffy function (von Bertalanffy, 1938; Haddon, 2001), given by $$\mathrm{E}(\text {Length} | \text {Age}=t)=L_{\infty}\left(1-\exp \left(-K\left(t-t_{0}\right)\right)\right.$$ The parameter \(L_{\infty}\) is the expected value of Length for extremely large ages, and so it is the asymptotic or upper limit to growth, and \(K\) is a growth rate parameter that determines how quickly the upper limit to growth is reached. When \(A g e=t_{0},\) the expected length of the fish is \(0,\) which allows fish to have nonzero length at birth if \(t_{0}<0\) a. The data in the file lakemary.txt give the \(A g e\) in years and Length in millimeters for a sample of 78 bluegill fish from Lake Mary, Minnesota, in 1981 (courtesy of Richard Frie). Age is determined by counting the number of rings on a scale of the fish. This is a cross-sectional data set, meaning that all the fish were measured once. Draw a scatterplot of the data b. Use nonlinear regression to fit the von Bertalanffy function to these data. To get starting values, first guess at \(L_{\infty}\) from the scatterplot to be a value larger than any of the observed values in the data. Next, divide both sides of (11.21) by the initial estimate of \(L_{\infty}\) and rearrange terms to get just \(\exp \left(-K\left(t-t_{0}\right)\) on the right of the \right. equation. Take logarithms, to get a linear mean function, and then use ous for the linear mean function to get the remaining starting values. Draw the fitted mean function on your scatterplot. c. Obtain a \(95 \%\) confidence interval for \(L_{\infty}\) using the large- sample approximation, and using the bootstrap.

Step-by-Step Solution

Answer

In this analysis, we used the von Bertalanffy function to study the relationship between the age and length of bluegill fish. We created a scatterplot, fitted the von Bertalanffy function using nonlinear regression, and calculated the 95% confidence interval for \(L_{\infty}\) employing both large-sample approximation and bootstrap method.

Step 1: a) Create a Scatterplot

To create the scatterplot, first, import the data from the provided "lakemary.txt" file. Then, plot the Age against Length, making sure to label the axes accordingly.

Step 2: b) Fitting the von Bertalanffy Function

Step 1: Initial Estimate for \(L_{\infty}\) Inspecting the scatterplot, choose an initial estimate for \(L_{\infty}\) by selecting a value larger than the largest observed Length. Step 2: Transform the von Bertalanffy Function Divide both sides of the von Bertalanffy function by the initial estimate of \(L_{\infty}\) (let's call this \(L_{\infty, init}\)) and then rearrange terms to obtain an expression for \(\exp(-K(t-t_0))\): \[ \frac{\mathrm{E}(\text{Length}|\text{ Age}=t)}{L_{\infty, init}} = 1 - \exp(-K(t-t_0)). \] Take the natural logarithm of both sides to obtain a linear mean function: \[ \ln \left(1 - \frac{\mathrm{E}(\text{Length}|\text{ Age}=t)}{L_{\infty, init}}\right) = -K (t-t_0). \] Step 3: Perform a Linear Regression Using the linear mean function from Step 2, perform a simple linear regression with \(t\) (the Age) as the independent variable and \(\ln \left(1 - \frac{\mathrm{E}(\text{Length}|\text{ Age}=t)}{L_{\infty, init}}\right)\) as the dependent variable. Use the estimates to find the initial estimates for \(K\) and \(t_0\). Step 4: Nonlinear Regression Using the initial values obtained (Initial estimate for \(L_{\infty}\), and initial estimates for \(K\) and \(t_0\) from linear regression), perform nonlinear regression to fit the von Bertalanffy function to the data. Step 5: Draw the Fitted Mean Function Overlay the fitted mean function on the scatterplot and compare with the observed data points to assess the fit.

Step 3: c) Confidence Interval for L_{\infty}

Step 1: Large-sample approximation Use the large-sample approximation to calculate a 95% confidence interval for \(L_{\infty}\). This will involve using the standard error for the asymptotic length estimate to find the lower and upper bounds of the range. Step 2: Bootstrap Method To obtain the 95% confidence interval for \(L_{\infty}\) using bootstrap, follow these steps: 1. Resample the data with replacement, preserving the relationship between Age and Length. 2. Fit the von Bertalanffy function to the resampled data and obtain a new estimate for \(L_{\infty}\). 3. Repeat steps 1 and 2 for a large number of iterations (say, 1000 or more). 4. Sort the bootstrap estimates of \(L_{\infty}\). 5. Calculate the 2.5% and 97.5% percentiles of the sorted estimates to obtain the lower and upper bounds, respectively, of the 95% bootstrap confidence interval.

Key Concepts

Nonlinear Regression

Nonlinear regression is a statistical technique used to model complex relationships between a dependent variable and one or more independent variables. Unlike linear regression where the model predicts the outcome as a straight line, nonlinear regression can handle curves and more intricate patterns in data. This is especially useful in fields like fisheries, where growth patterns such as in the von Bertalanffy function do not follow a straight line.

In the context of fisheries studies, the relationship between the length of fish at a given age is often nonlinear and characterized by parameters which, when estimated through nonlinear regression, give insights into the growth rate and potential maximum size of the fish. Nonlinear regression uses an iterative process to estimate the parameters that minimize the difference between the observed data and the model's predictions.

To carry out nonlinear regression, starting values for the model parameters are required. These are usually obtained through a preliminary analysis, such as a visual inspection of a scatterplot or transformation of the nonlinear function into a linear one for initial parameter estimation, as in the von Bertalanffy function example.

Scatterplot of Data

A scatterplot is a type of graphical representation where two variables are plotted along two axes, with each point representing an observation in the dataset. It is particularly useful for visualizing the relationship between two quantitative variables, and it's a fundamental step for data analysis in various fields, including fisheries studies.

For example, when researchers examine the length and age of fish, as done with the bluegill fish from Lake Mary, a scatterplot can reveal trends or patterns in growth over time. This visual representation provides insights even before any complex statistical analysis and can inform the choice of an appropriate model for regression analysis.

Creating a scatterplot involves placing one variable on the x-axis (in this case, Age) and another on the y-axis (Length), and plotting points where the values intersect. From such a plot, researchers can deduce approximate values for important parameters like the asymptotic length, which are crucial for further analysis like nonlinear regression.

Confidence Interval Estimation

Confidence interval estimation provides a range of values within which the true value of a parameter likely lies. It's quantified by a confidence level, commonly set at 95%, indicating that if the experiment were repeated numerous times, the interval would contain the true parameter value in 95% of the cases.

In fisheries studies and other sciences, these intervals are valuable as they give an indication of the precision and reliability of the estimated parameters, like the asymptotic maximum size of fish (\(L_{\tiny{\text{∞}}}\)) in the von Bertalanffy function. Large-sample approximation is a method to derive a confidence interval when the sample size is large. This approach assumes that the distribution of the estimate can be approximated by a normal distribution due to the large sample size.

Besides this, non-standard techniques like bootstrapping can be used to estimate confidence intervals, which do not rely on large-sample assumptions and can be applied to complex or non-standard statistics.

Bootstrap Method

The bootstrap method is a powerful statistical tool used to estimate the distribution of a statistic, such as the mean, median, or in this case, a parameter like the asymptotic length \(L_{\text{∞}}\) of fish. This method involves resampling the original dataset with replacement many times to generate a bootstrap distribution of the statistic.

For confidence interval estimation, the bootstrap method is particularly useful when the theoretical distribution of the statistic is unknown or intractable. By using resampling techniques to create many pseudo-datasets, one can calculate the statistic for each and then assess the variability directly from the data without relying on traditional assumptions of the data distribution.

In the von Bertalanffy function exercise, we apply the bootstrap approach to estimate a 95% confidence interval of the \(L_{\text{∞}}\) parameter. This is done by taking the 2.5% and 97.5% percentiles of the bootstrap estimates and provides an interval believed to capture the true \(L_{\text{∞}}\) with a 95% certainty. It's a robust alternative to the large-sample approximation and can yield more accurate intervals, especially for smaller sample sizes or more complex models.