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More Age Estimation In the article "Bigger Teeth for Longer Life? Longevity and Molar Height in Two Roe Deer Populations" (Biology Letters [June, 2007\(]\) vol. 3 no. 3 \(268-270)\), researchers developed a model to predict the tooth height (in \(\mathrm{mm}\) ), \(y\), of roe deer based on their age, \(x_{1}\), gender, \(x_{2}(0=\) female \(, 1=\) male \(),\) and location, \(x_{3}\) (Trois Fontaines deer, which have a shorter life expectancy, and Chizé, which have a longer life expectancy, \(x_{3}=0\) for Trois Fontaines, \(x_{3}=1\) for Chizé). The model is $$ \hat{y}=7.790-0.382 x_{1}-0.587 x_{2}-0.925 x_{3}+0.091 x_{2} x_{3} $$ (a) What is the expected tooth length of a female roe deer who is 12 years old and lives in Trois Fontaines? (b) What is the expected tooth length of a male roe deer who is 8 years old and lives in Chizé? (c) What is the interaction term? What does the coefficient of the interaction term imply about tooth length?

Step by step solution

01

Understanding the Model

The model provided is \[ \hat{y} = 7.790 - 0.382x_1 - 0.587x_2 - 0.925x_3 + 0.091x_2x_3 \]. To find the expected tooth length, substitute the values of age \(x_1\), gender \(x_2\), and location \(x_3\).

02

Solve Part (a)

For a female roe deer who is 12 years old and lives in Trois Fontaines: \[ x_1 = 12, \, x_2 = 0, \, x_3 = 0 \]. Substitute these values into the model: \[ \hat{y} = 7.790 - 0.382 \times 12 - 0.587 \times 0 - 0.925 \times 0 + 0.091 \times 0 \times 0 \]. Calculate: \[ \hat{y} = 7.790 - 4.584 = 3.206 \]. Thus, the expected tooth length is 3.206 mm.

03

Solve Part (b)

For a male roe deer who is 8 years old and lives in Chizé: \[ x_1 = 8, \, x_2 = 1, \, x_3 = 1 \]. Substitute these values into the model: \[ \hat{y} = 7.790 - 0.382 \times 8 - 0.587 \times 1 - 0.925 \times 1 + 0.091 \times 1 \times 1 \]. Calculate: \[ \hat{y} = 7.790 - 3.056 - 0.587 - 0.925 + 0.091 = 3.313 \]. Thus, the expected tooth length is 3.313 mm.

04

Solve Part (c)

The interaction term is \( 0.091x_2x_3 \). This term accounts for the interaction between gender and location. The coefficient 0.091 implies that for every unit increase in the interaction between being male (\(x_2 = 1\)) and living in Chizé (\(x_3 = 1\)), the expected tooth length increases by 0.091 mm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Regression Model

A linear regression model is a statistical method used to approximate the relationship between a dependent variable and one or more independent variables. In the given exercise, the dependent variable is the tooth height of roe deer, denoted as \(y\). The independent variables are age (\(x_1\)), gender (\(x_2\)), and location (\(x_3\)).

The general form of a linear regression model can be written as: \[ \hat{y} = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n \] where:

• \( \hat{y} \) is the predicted value of the dependent variable,
• \( \beta_0 \) is the y-intercept,
• \( \beta_1, \beta_2, ..., \beta_n \) are the regression coefficients for the independent variables.

In this exercise, the model is given by: \[ \hat{y} = 7.790 - 0.382x_1 - 0.587x_2 - 0.925x_3 + 0.091x_2x_3 \].
This model can be used to predict the tooth height of a roe deer by substituting specific values for \(x_1\), \(x_2\), and \(x_3\).

Interaction Term

An interaction term in a regression model allows us to examine whether the effect of one independent variable on the dependent variable depends on the value of another independent variable.

In this exercise, the interaction term is \(0.091x_2x_3\). This term is a product of two independent variables: gender (\(x_2\)) and location (\(x_3\)). It helps us understand how the combined impact of gender and location affects tooth height.

The coefficient of the interaction term (0.091) implies that for every unit increase in the combined effect of a deer being male (\(x_2 = 1\)) and living in Chizé (\(x_3 = 1\)), the predicted tooth height increases by 0.091 mm.

This is important because it shows that the effect of gender on tooth height is different depending on the location of the deer. Such insights can be valuable in understanding the underlying biological or environmental interactions.

Variable Substitution

Variable substitution is a crucial step in the linear regression model when we want to predict the value of the dependent variable for specific cases.

To find the expected tooth length for a roe deer, you would substitute the values of the independent variables (age, gender, location) into the regression equation.

For example, to predict the tooth length for a female roe deer who is 12 years old and lives in Trois Fontaines, we use \(x_1 = 12\), \(x_2 = 0\), and \(x_3 = 0\). Substituting these values into the model:

\[ \hat{y} = 7.790 - 0.382(12) - 0.587(0) - 0.925(0) + 0.091(0)(0) = 3.206 \]

Similarly, for a male roe deer who is 8 years old and lives in Chizé, use \(x_1 = 8\), \(x_2 = 1\), and \(x_3 = 1\). Substituting these values:

\[ \hat{y} = 7.790 - 0.382(8) - 0.587(1) - 0.925(1) + 0.091(1)(1) = 3.313 \]
Therefore, variable substitution aids in deriving specific predictions from a statistical model.

Regression Coefficients

Regression coefficients are values that multiply the independent variables in a regression equation. They indicate the strength and direction of the relationship between the independent variables and the dependent variable.

In the given model:

• The coefficient for age \( -0.382 \) suggests that each additional year of age decreases the expected tooth height by 0.382 mm.
  
• The coefficient for gender \( -0.587 \) implies that being male decreases the expected tooth height by 0.587 mm compared to being female (holding other factors constant).
  
• The coefficient for location \( -0.925 \) indicates that living in Chizé (compared to Trois Fontaines) decreases the expected tooth height by 0.925 mm.
  
• The coefficient for the interaction term \[ x_2 x_3 \] \(0.091\) indicates the combined effect of being male and living in Chizé increases the expected tooth height by 0.091 mm.

Therefore, regression coefficients explain how much and in what direction the dependent variable (tooth height) changes with each one-unit change in an independent variable.