91Ó°ÊÓ

In an article in Statistics and Computing ['An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis" (1991, pp. \(119-128)\) ] Carlin and Gelfand investigated the age \((x)\) and length \((y)\) of 27 captured dugongs (sea cows). $$\begin{aligned}x=& 1.0,1.5,1.5,1.5,2.5,4.0,5.0,5.0,7.0,8.0,8.5,9.0,9.5, \\\& 9.5,10.0,12.0,12.0,13.0,13.0,14.5,15.5,15.5,16.5, \\ & 17.0,22.5,29.0,31.5 \\\y=& 1.80,1.85,1.87,1.77,2.02,2.27,2.15,2.26,2.47,2.19, \\\& 2.26,2.40,2.39,2.41,2.50,2.32,2.32,2.43,2.47,2.56, \\ & 2.65,2.47,2.64,2.56,2.70,2.72,2.57\end{aligned}$$ (a) Find the least squares estimates of the slope and the intercept in the simple linear regression model. Find an estimate of \(\sigma^{2}\) (b) Estimate the mean length of dugongs at age 11 . (c) Obtain the fitted values \(\hat{y}_{i}\) that correspond to each observed value \(y_{i} .\) Plot \(\hat{y}_{i}\) versus \(y_{i},\) and comment on what this plot would look like if the linear relationship between length and age were perfectly deterministic (no error). Does this plot indicate that age is a reasonable choice of regressor variable in this model?

Step by step solution

01

Calculate the Means

Compute the means of age \( x \) and length \( y \) using the given data. \( \overline{x} = \frac{1.0 + 1.5 + ... + 31.5}{27} \) and \( \overline{y} = \frac{1.80 + 1.85 + ... + 2.57}{27} \). Calculate these to find \( \overline{x} \approx 11.796\) and \( \overline{y} \approx 2.383 \).

02

Calculate the Slope and Intercept

Use the formulas for the slope \( b \) and intercept \( a \): \( b = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2} \) and \( a = \overline{y} - b\overline{x} \). Substitute the values to find \( b \approx 0.0208 \) and \( a \approx 2.138 \).

03

Estimate \( \sigma^2 \)

The variance \( \sigma^2 \) is estimated using \( \hat{\sigma}^2 = \frac{1}{n-2} \sum (y_i - \hat{y}_i)^2 \), where \( \hat{y}_i = a + b x_i \). Calculate for each data point and find \( \hat{\sigma}^2 \approx 0.023 \).

04

Estimate Mean Length at Age 11

Substitute \( x = 11 \) in the equation \( \hat{y} = a + bx \) to estimate the mean length. \( \hat{y} = 2.138 + 0.0208 \times 11 \approx 2.368 \).

05

Calculate Fitted Values \( \hat{y}_i \)

For every \( x_i \), compute \( \hat{y}_i = a + b x_i \). Calculate the corresponding \( \hat{y}_i \) using the formula for each \( x_i \).

06

Plot \( \hat{y}_i \) versus \( y_i \)

Create a scatter plot of the observed values \( y_i \) vs. the predicted values \( \hat{y}_i \). If the linear relationship were perfectly deterministic, points would lie on a line \( y = x \). Compare the plot to assess the fit; a reasonable scatter around the line suggests the model is appropriate, indicating age as a suitable regressor.

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Key Concepts

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

Least Squares Estimates

The least squares estimates are a fundamental concept in linear regression. They help find the best-fitting line through a set of points. In this problem, we are estimating a simple linear regression model linking age and length of dugongs.

To start, the means of age (\( x \)) and length (\( y \)) are calculated. We look for the slope (\( b \)) and the y-intercept (\( a \)) of the line using the formulas:

• \( \ b = \frac{\sum (x_i - \overline{x})(y_i - \overline{y})}{\sum (x_i - \overline{x})^2}\)
• \( a = \overline{y} - b\overline{x}\)

The slope (\( b \)) tells us how much \( y \) (length) changes for each unit change in \( x \) (age). The y-intercept (\( a \)) is where the line crosses the y-axis.

Afterward, the variance, denoted as \( \sigma^2 \), is also estimated to understand the spread of the data around the regression line. This gives insights into the accuracy of our predictions.

Bayesian Analysis

Bayesian analysis provides a way to update the probability estimates for a hypothesis as more evidence or data becomes available. It is widely used in statistics and includes working with probabilities to forecast outcomes.

In relation to linear regression, Bayesian analysis can be used to refine our model parameters, such as slope and intercept, based on prior knowledge and observed data. This approach can be particularly useful when dealing with uncertainties or in cases where the sample size is small.
By incorporating prior distributions with observed data, the Bayesian framework updates our beliefs about the model. This is different from traditional methods that only rely on the data at hand. For dugongs, one might start with a prior belief about the relationship between age and length and use the collected data to update these beliefs to get more accurate parameter estimates.

Linear Relationship Assessment

Assessing the linear relationship between two quantitative variables involves checking how well a straight line can describe the relationship between them. In simpler terms, we verify if larger or smaller values of one variable systematically correspond to larger or smaller values of the other.

In this exercise, plotting \( \hat{y}_{i} \) (predicted values) against actual lengths (\( y_{i} \)) helps to visualize this relationship. Ideally, if the relationship is perfect and deterministic, the points should form a diagonal line \( y = x \).

• A perfect match would indicate a perfectly linear relationship.
• A scatter of points suggests some variability or error in the model.
• If the scatter forms no clear pattern, it may indicate that age is not a strong predictor of length or more complex models may be necessary.

This graphical assessment helps us judge if age is a reasonable choice for the regressor variable or if a better-fitting model is needed.