Question

This exercise requires the use of a computer package. The article "Movement and Habitat Use by Lake Whitefish During Spawning in a Boreal Lake: Integrating Acoustic Telemetry and Geographic Information Systems" (Transactions of the American Fisheries Society [1999]:\(939-952\) ) included the accompanying data on 17 fish caught in two consecutive years. $$ \begin{array}{ccccc} \text { Year } & \begin{array}{l} \text { Fish } \\ \text { Number } \end{array} & \begin{array}{l} \text { Weight } \\ (\mathrm{g}) \end{array} & \begin{array}{l} \text { Length } \\ (\mathrm{mm}) \end{array} & \begin{array}{l} \text { Age } \\ \text { (years) } \end{array} \\ \hline \text { Year 1 } & 1 & 776 & 410 & 9 \\ & 2 & 580 & 368 & 11 \\ & 3 & 539 & 357 & 15 \\ & 4 & 648 & 373 & 12 \\ & 5 & 538 & 361 & 9 \\ & 6 & 891 & 385 & 9 \\ & 7 & 673 & 380 & 10 \\ & 8 & 783 & 400 & 12 \\ \text { Year 2 } & 9 & 571 & 407 & 12 \\ & 10 & 627 & 410 & 13 \\ & 11 & 727 & 421 & 12 \\ & 12 & 867 & 446 & 19 \\ & 13 & 1042 & 478 & 19 \\ & 14 & 804 & 441 & 18 \\ & 15 & 832 & 454 & 12 \\ & 16 & 764 & 440 & 12 \\ & 17 & 727 & 427 & 12 \\ \hline \end{array} $$ a. Fit a multiple regression model to describe the relationship between weight and the predictors length and age. b. Carry out the model utility test to determine whether the predictors length and age, together, are useful for predicting weight.

Short answer

The relationship between weight and the predictors, length and age is given by the estimated multiple regression equation generated using the statistical software. The F-test (model utility test) provides a p-value that, if less than a predetermined significance level, indicates the predictors length and age are useful in predicting weight.

Step-by-step solution

Prepare the Data

Input the given data into the statistical software package. Ensure the data for Year, Fish Number, Weight (g), Length (mm) and Age (years) is correctly inputted for each of the 17 fish.

Generate Multiple Regression Model

From the statistical software interface, select a tool to construct a multiple regression model. Define Weight as the dependent variable and Length and Age as predictors (independent variables). The software will generate a regression equation in the form \(Weight = \beta_0 + \beta_1 * Length + \beta_2 * Age + \varepsilon\), where \(\beta_0, \beta_1\) and \(\beta_2\) are the coefficients to be estimated, and \(\varepsilon\) is the error term.

Interpret Regression Model

Check the output from the statistical software. Along with the coefficients for the predictors (Length and Age), the output will include the p-values associated with these coefficients. These p-values will help determine the statistical significance of each predictor.

Conduct Model Utility Test

Perform the Model utility test also known as F-test. The F-statistic and corresponding p-value will be contained in the output from the regression model. The null hypothesis of this test is that all regression coefficients are equal to zero, which implies that the predictors have no effect on the dependent variable. If the p-value is less than the chosen significance level (commonly 0.05), you can reject the null hypothesis and conclude that the predictors are helpful in estimating the weight.

Key concepts

Statistical Software Package

A statistical software package is a powerful tool that aids researchers and data analysts in performing complex calculations and generating models with ease. These packages come with a variety of statistical functions that can perform everything from basic descriptive statistics to advanced multivariate analysis.

In the context of our exercise, the use of such a software package allows the user to input data about fish (weight, length, and age) and employ multiple regression analysis to study the relationship between these variables. With user-friendly interfaces and the ability to handle large datasets efficiently, these packages make it possible to quickly run analyses and interpret results, which is crucial for making informed decisions based on data.

Model Utility Test

The model utility test examines whether a set of predictors in a regression model contributes to explaining the variability in the dependent variable. In simpler terms, it tests if our predictors (length and age) actually have a significant relationship with the outcome, which in this case is the weight of the fish.

Conducting this test is vital as it helps confirm if the multiple regression model we've constructed is statistically significant and useful for predicting outcomes, beyond what can be expected by chance. The test typically involves an overall F-test, which we will discuss next.

F-test

The F-test in regression is used to assess the joint effect of all independent variables on the dependent variable. It compares a model with no predictors - meaning it assumes that the dependent variable is constant - to a model with the specified independent variables.

In our scenario, it tests the null hypothesis that both length and age do not significantly predict the weight of the fish. If the p-value associated with the F-test is small (usually less than 0.05), we have evidence against the null hypothesis, indicating that our model provides a better fit than the no-predictor model.

Dependent and Independent Variables

In any regression model, identifying the dependent and independent variables is crucial for a correct setup. The dependent variable, also known as the response variable, is what we aim to predict or explain. In the fish data set, this is the weight of each fish.

Independent variables, on the other hand, are the predictors we use to estimate the dependent variable. In our case, these are the length and age of the fish. Carefully distinguishing between these variables is key as it shapes the structure and interpretation of the regression model.

Regression Equation

A regression equation is the mathematical representation of the relationship between the dependent variable and one or more independent variables. It is the cornerstone of the multiple regression model, providing a formula that can be used to make predictions.

The standard regression equation takes the form of \(Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \varepsilon\), where \beta_0 is the intercept, \beta_1, \beta_2, ... are the coefficients representing the impact of each independent variable, X_1, X_2,... are the independent variables, and \varepsilon is the error term, accounting for the variance not explained by the model.

P-value

The p-value plays a critical role in hypothesis testing and inferential statistics. It denotes the probability of obtaining results at least as extreme as the observed ones, given that the null hypothesis is true. If the p-value is lower than a pre-determined significance level (such as 0.05), it suggests that the observed data are unlikely under the null hypothesis, and thus, we may reject the null hypothesis as false.

In the context of regression, a p-value associated with an independent variable indicates whether that variable is a significant predictor of the dependent variable. Smaller p-values for coefficients imply stronger evidence against the null hypothesis, suggesting that these variables have a meaningful impact on the prediction.

Null Hypothesis

The null hypothesis is a statement used as a starting point for statistical testing. It proposes that there is no effect or difference, or in the case of regression, that the independent variables have no impact on the dependent variable.

Specifically, in our fish weight example, the null hypothesis for the F-test would state that the length and age of fish do not help in predicting their weight. Through hypothesis testing, we seek evidence to either reject or fail to reject the null hypothesis, which helps us to infer the validity of our regression model.