Question

This exercise requires the use of a computer package. The authors of the article "Absolute Versus per Unit Body Length Speed of Prey as an Estimator of Vulnerability to Predation" (Animal Behaviour [1999]: \(347-\) 352) found that the speed of a prey (twips/s) and the length of a prey (twips \(\times 100\) ) are good predictors of the time (s) required to catch the prey. (A twip is a measure of distance used by programmers.) Data were collected in an experiment where subjects were asked to "catch" an animal of prey moving across his or her computer screen by clicking on it with the mouse. The investigators varied the length of the prey and the speed with which the prey moved across the screen. The following data are consistent with summary values and a graph given in the article. Each value represents the average catch time over all subjects. The order of the various speed-length combinations was randomized for each subject. $$ \begin{array}{ccc} \begin{array}{c} \text { Prey } \\ \text { Length } \end{array} & \begin{array}{l} \text { Prey } \\ \text { Speed } \end{array} & \begin{array}{l} \text { Catch } \\ \text { Time } \end{array} \\ \hline 7 & 20 & 1.10 \\ 6 & 20 & 1.20 \\ 5 & 20 & 1.23 \\ 4 & 20 & 1.40 \\ 3 & 20 & 1.50 \\ 3 & 40 & 1.40 \\ 4 & 40 & 1.36 \\ 6 & 40 & 1.30 \\ 7 & 40 & 1.28 \\ 7 & 80 & 1.40 \\ 6 & 60 & 1.38 \\ 5 & 80 & 1.40 \\ 7 & 100 & 1.43 \\ 6 & 100 & 1.43 \\ 7 & 120 & 1.70 \\ 5 & 80 & 1.50 \\ 3 & 80 & 1.40 \\ 6 & 100 & 1.50 \\ 3 & 120 & 1.90 \\ & & \\ \hline \end{array} $$ a. Fit a multiple regression model for predicting catch time using prey length and speed as predictors. b. Predict the catch time for an animal of prey whose length is 6 and whose speed is 50 . c. Is the multiple regression model useful for predicting catch time? Test the relevant hypotheses using \(\alpha=.05\). d. The authors of the article suggest that a simple linear regression model with the single predictor \(x=\frac{\text { length }}{\text { speed }}\) might be a better model for predicting catch time. Calculate the \(x\) values and use them to fit this linear regression model. e. Which of the two models considered (the multiple regression model from Part (a) or the simple linear regression model from Part (d)) would you recommend for predicting catch time? Justify your choice.

Short answer

This problem involves fitting two types of regression models and choosing the better of the two based on R-squared values. The models are to predict the catch time of prey based on the prey's length and speed. Hypothesis testing also shows if the predictor variables significantly affect the outcome variable.

Step-by-step solution

Fitting the Multiple Regression Model

The first step is to fit a multiple regression model using length and speed of the prey as the variables. In multiple regression, we use two or more variables to predict the outcome. Use a statistical package to fit the model: 'Catch time ~ Length + Speed' where catch time is the dependent variable, and length and speed are the independent variables.

Predicting the Catch Time

The next task is to predict the catch time for a prey with a length of 6 and a speed of 50. You can do this by inputting these values into the regression model obtained in the previous step. This gives a calculated catch time.

Hypothesis testing

This part requires checking if the model is useful for predicting catch time. Set up two hypotheses: the null hypothesis (H0) which assumes that the predictors (length and speed) have no effect on the catch time, and the alternative hypothesis (H1) which assumes that at least one of the predictors affects the catch time. Perform an F-test with a significance level of 0.05. If the p-value is less than 0.05, reject the null hypothesis and conclude that the model is useful.

Fit the Simple Linear Regression Model

Now consider the alternative model proposed by the authors, where single predictor x is 'length/speed'. Calculate these x values for each data point and fit a simple linear regression model: 'Catch time ~ x'.

Model recommendation

The final step involves choosing between the two models (the multiple regression model and the simple linear regression model) based on how well they fit the data. Compare the R-squared values of both models. A higher R-squared value indicates a better fit for the data.