Problem 27 - Applied Linear Statistical Models

Question

In a small-scale regression study, the following data were obtained: $$\begin{array}{crrrrrr} i & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline x_{n}: & 7 & 4 & 16 & 3 & 21 & 8 \\ X_{i 2}: & 33 & 47 & 7 & 49 & 5 & 31 \\ Y_{i}: & 42 & 33 & 75 & 28 & 91 & 55 \end{array}$$ Assume that regression model ( 6.1 ) with independent normal error terms is approptiate. Using matrix methods, obtain \((a) \mathbf{b} ;(b) e_{i}(c) H ;(d) S S R ;(e) s^{2}(b) ;(f) \hat{Y}_{h}\) when \(X_{h 1}=10, X_{h 2}=30$$(g) s^{2}\left(\hat{Y}_{h}\right)\) when \(X_{h 1}=10, X_{h 2}=30\)

Step-by-Step Solution

Answer

Estimate the coefficients, compute residuals, hat matrix, sum of squares for regression, standard errors of estimates, prediction, and standard errors of prediction.

Step 1 - Define the Regression Model

The regression model can be written as: \[ Y = X \beta + \varepsilon \] where \( Y \) is the response vector, \( X \) is the matrix of predictors, \( \beta \) is the vector of coefficients, and \( \varepsilon \) is the error term.

Step 2 - Set up Matrices

Construct the matrices for the predictors \(X\), response \(Y\), and the coefficients \(\beta\): \[ X = \begin{bmatrix} 1 & x_{i1} & x_{i2} \end{bmatrix}, Y = \begin{bmatrix} 42 & 33 & 75 & 28 & 91 & 55 \end{bmatrix}, \beta = \begin{bmatrix} \beta_0 & \beta_1 & \beta_2 \end{bmatrix} \] Here, the \( X \) matrix will include a column of ones for the intercept.

Step 3 - Compute \(\mathbf{b}\)

Calculate \(\mathbf{b}\), the estimate of \(\beta\), using the formula: \[ \mathbf{b} = (X^T X)^{-1} X^T Y \] Plug in the given values to find \(\mathbf{b}\).

Step 4 - Compute Residuals \(e_i\)

Calculate the residuals using: \[ e_i = Y_i - \hat{Y}_i \] where \( \hat{Y}_i = X_i \mathbf{b} \).

Step 5 - Compute Hat Matrix \(H\)

The hat matrix \(H\) is given by: \[ H = X (X^T X)^{-1} X^T \] Compute \(H\) using the given \(X\) matrix.

Step 6 - Calculate SSR

Compute the sum of squares for regression (SSR) using the formula: \[ SSR = \mathbf{b}^T X^T Y \] This represents the variation explained by the model.

Step 7 - Calculate \(s^2(b)\)

The standard error of the estimated coefficients \(\mathbf{b}\) is given by: \[ s^2(b) = \frac{SSR}{n-p} \cdot (X^T X)^{-1} \] where \(n\) is the number of observations and \(p\) is the number of parameters.

Step 8 - Compute \(\hat{Y}_h\)

To estimate \(\hat{Y}_h\) for new predictors \(X_{h1} = 10\) and \(X_{h2} = 30\): \[ \hat{Y_h} = X_h \mathbf{b} \] where \(X_h\) includes the new predictor values.

Step 9 - Calculate \(s^2(\hat{Y}_h)\)

The standard error of \(\hat{Y}_h\) is given by: \[ s^2(\hat{Y}_h) = s^2 (1 + X_h^T (X^T X)^{-1} X_h) \] Substitute the appropriate values to compute this.

Key Concepts

regression model

In this regression exercise, we start with a fundamental concept: the regression model. A regression model allows us to understand the relationship between the dependent variable (response) and one or more independent variables (predictors). We express this relationship through the equation:
\( Y = X \beta + \varepsilon \)
Here, \( Y \) is the response vector, \( X \) is the matrix of predictors, \( \beta \) is the vector of coefficients, and \( \varepsilon \) represents the error term. The error term accounts for the difference between the observed and predicted values, assuming it's normally distributed with a mean of zero.

matrix calculation

In matrix calculations, we organize our data into matrices to simplify and solve regression problems efficiently. For this exercise, we define the predictors matrix \(X\), response vector \(Y\), and coefficients vector \(\beta\) as follows:
\[ X = \begin{bmatrix} 1 & x_{i1} & x_{i2} \ 1 & 7 & 33 \ 1 & 4 & 47 \ 1 & 16 & 7 \ 1 & 3 & 49 \ 1 & 21 & 5 \ 1 & 8 & 31 \end{bmatrix}, \ Y = \begin{bmatrix} 42 \ 33 \ 75 \ 28 \ 91 \ 55 \end{bmatrix}, \ \beta = \begin{bmatrix} \beta_0 \ \beta_1 \ \beta_2 \end{bmatrix} \ \ \]
Next, we calculate the estimated coefficients vector \(\mathbf{b}\) using the formula:
\( \mathbf{b} = (X^T X)^{-1} X^T Y \)
This formula combines the matrices to provide us with the best-fitting coefficients for our model.

residuals

Residuals measure the difference between observed and predicted values in the regression model. They give us insight into the accuracy of our model. The residuals \(e_i\) are calculated using the equation:
\( e_i = Y_i - \hat{Y}_i \)
where \(\hat{Y}_i\) is the predicted value given by \( \hat{Y}_i = X_i \mathbf{b} \). Residuals help identify patterns that might suggest a poor fit, indicating improvements or modifications needed for the model.

hat matrix

The hat matrix \(H\) is essential in regression analysis as it helps in identifying influential points and diagnosing the model fit. The hat matrix is calculated using the formula:
\( H = X (X^T X)^{-1} X^T \)
The main role of \(H\) is to project the observed values onto the predicted values. It's called the hat matrix because it 'puts a hat' on \(Y\), transforming it into \(\hat{Y}\). This matrix is instrumental in assessing how much influence each data point has on the fitted values.

sum of squares for regression

Sum of squares for regression (SSR) quantifies the total variation explained by the regression model. It is a key component in evaluating the model's goodness of fit. The SSR is calculated using:
\( SSR = \mathbf{b}^T X^T Y \)
This value represents how much of the variation in the response variable \(Y\) can be explained by the predictors \(X\). A higher SSR indicates a better-fitting model.

standard error

The standard error of the estimated coefficients provides a measure of the accuracy of the coefficient estimates. It's calculated using:
\( s^2(\mathbf{b}) = \frac{SSR}{n-p} \cdot (X^T X)^{-1} \)
Here, \(n\) is the number of observations and \(p\) is the number of parameters. The standard error aids in understanding the precision of the coefficients; smaller values indicate more reliable estimates.

predictor estimation

Predictor estimation involves using the regression model to predict new values. For given new predictors \(X_{h1}=10\) and \(X_{h2}=30\), we calculate the predicted response as:
\( \hat{Y}_h = X_h \mathbf{b} \)
where \(X_h\) includes the new predictor values. Additionally, the standard error of \(\hat{Y}_h\) is computed as:
\( s^2(\hat{Y}_h) = s^2 (1 + X_h^T (X^T X)^{-1} X_h) \)
This metric helps in estimating the variability of the predicted value, providing a confidence interval for predictions.