91Ó°ÊÓ

The data from a patient satisfaction survey in a hospital are shown next. $$ \begin{array}{cccccc} \text { Obser- } & & & & & \text { Satis- } \\ \text { vation } & \text { Age } & \text { Severity } & \text { Surg-Med } & \text { Anxiety } & \text { faction } \\ 1 & 55 & 50 & 0 & 2.1 & 68 \\ 2 & 46 & 24 & 1 & 2.8 & 77 \\ 3 & 30 & 46 & 1 & 3.3 & 96 \\ 4 & 35 & 48 & 1 & 4.5 & 80 \\ 5 & 59 & 58 & 0 & 2.0 & 43 \\ 6 & 61 & 60 & 0 & 5.1 & 44 \\ 7 & 74 & 65 & 1 & 5.5 & 26 \\ 8 & 38 & 42 & 1 & 3.2 & 88 \\ 9 & 27 & 42 & 0 & 3.1 & 75 \\ 10 & 51 & 50 & 1 & 2.4 & 57 \\ 11 & 53 & 38 & 1 & 2.2 & 56 \\ 12 & 41 & 30 & 0 & 2.1 & 88 \\ 13 & 37 & 31 & 0 & 1.9 & 88 \\ 14 & 24 & 34 & 0 & 3.1 & 102 \\ 15 & 42 & 30 & 0 & 3.0 & 88 \\ 16 & 50 & 48 & 1 & 4.2 & 70 \\ 17 & 58 & 61 & 1 & 4.6 & 52 \\ 18 & 60 & 71 & 1 & 5.3 & 43 \\ 19 & 62 & 62 & 0 & 7.2 & 46 \\ 20 & 68 & 38 & 0 & 7.8 & 56 \\ 21 & 70 & 41 & 1 & 7.0 & 59 \\ 22 & 79 & 66 & 1 & 6.2 & 26 \\ 23 & 63 & 31 & 1 & 4.1 & 52 \\ 24 & 39 & 42 & 0 & 3.5 & 83 \\ 25 & 49 & 40 & 1 & 2.1 & 75 \end{array} $$ The regressor variables are the patient's age, an illness severity index (higher values indicate greater severity), an indicator variable denoting whether the patient is a medical patient (0) or a surgical patient (1), and an anxiety index (higher values indicate greater anxiety). a. Fit a multiple linear regression model to the satisfaction response using age, illness severity, and the anxiety index as the regressors. b. Estimate \(\sigma^{2}\). c. Find the standard errors of the regression coefficients. d. Are all of the model parameters estimated with nearly the same precision? Why or why not?

Step by step solution

01

Define the Regression Model

We need to fit a multiple linear regression model where satisfaction is the dependent variable and age, illness severity, and the anxiety index are the independent variables. This can be expressed as: \[ \text{Satisfaction} = \beta_0 + \beta_1 \cdot \text{Age} + \beta_2 \cdot \text{Severity} + \beta_3 \cdot \text{Anxiety} + \epsilon \] where \( \beta_0 \) is the intercept, \( \beta_1, \beta_2, \beta_3 \) are the coefficients for the predictors, and \( \epsilon \) is the error term.

02

Fit the Regression Model

Using statistical software or a calculator that performs regression analysis, input the observations of the dependent variable (Satisfaction) and independent variables (Age, Severity, Anxiety). Compute the regression coefficients (\(\beta_0, \beta_1, \beta_2, \beta_3\)).

03

Calculate Residuals and Estimate Variance \(\sigma^2\)

The variance \(\sigma^2\) can be estimated using the Mean Squared Error (MSE) from the residuals. Calculate the residual sum of squares (RSS): \[ \text{RSS} = \sum (\text{Observed Satisfaction} - \text{Predicted Satisfaction})^2 \] Estimate \(\sigma^2\) as: \[ \sigma^2 \approx \frac{\text{RSS}}{n-p} \] where \(n\) is the number of observations and \(p\) is the number of parameters (including the intercept).

04

Compute Standard Errors of Coefficients

Using the regression software, obtain the standard errors of each coefficient (\(\beta_1, \beta_2, \beta_3\)). These are derived from the diagonal elements of the covariance matrix of the parameter estimates, which are related to the variance estimate \(\sigma^2\).

05

Analyze Precision of Parameter Estimates

Compare the standard errors of the coefficients. If they are of similar magnitude, then the parameters are estimated with similar precision. Model precision can be affected by factors such as multicollinearity and the variance of the estimates.

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Key Concepts

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

Regression Coefficients

In the context of multiple linear regression, regression coefficients represent the weights assigned to each independent variable in a predictive model. These coefficients are calculated during model fitting to best describe the relationship between the independent variables and the target variable, in this case, patient satisfaction.

To find the coefficients, we utilize the least squares method which minimizes the difference between observed and predicted values. Mathematically, the model is expressed as:

• \[ \text{Satisfaction} = \beta_0 + \beta_1 \cdot \text{Age} + \beta_2 \cdot \text{Severity} + \beta_3 \cdot \text{Anxiety}\]

Here, \( \beta_0 \) is the intercept, and \( \beta_1, \beta_2, \beta_3 \) are the regression coefficients for "Age," "Illness Severity," and "Anxiety" respectively.

These coefficients indicate how much the satisfaction score is expected to change as the predictor variables vary. For instance, a higher \( \beta_1 \) suggests that age has a substantial influence on satisfaction, holding other factors constant. Using statistical software helps in calculating these coefficients efficiently based on the given data set, ensuring accuracy in the model.

Variance Estimation

Estimating variance is crucial in understanding the spread or dispersion of the residuals, which are the differences between observed and predicted values in regression. In multiple linear regression, the variance of the residuals is denoted as \( \sigma^2 \). This variance helps assess the accuracy of the predicted model.

The Mean Squared Error (MSE) is used as an estimator for the variance \( \sigma^2 \), often calculated after determining the residual sum of squares (RSS). The formula involved is:

• \[ \sigma^2 \approx \frac{\text{RSS}}{n-p}\]

where \( n \) represents the number of observations and \( p \) denotes the number of coefficients, including the intercept.

By assessing the value of \( \sigma^2 \), we can infer how well the model fits the data. A smaller variance indicates better performance of the regression model as it suggests that the residuals are tightly clustered around the predicted line, implying higher accuracy.

Residual Sum of Squares

Residual Sum of Squares (RSS) serves as a measure of the discrepancy between the actual data points and the model's predictions. It evaluates how well a multiple linear regression model fits the data.

The RSS is computed by summing up the squared differences between observed satisfaction scores and those predicted by the regression model. The formula is expressed as:

• \[ \text{RSS} = \sum (\text{Observed Satisfaction} - \text{Predicted Satisfaction})^2\]

A large RSS value indicates the model fails to capture the underlying data pattern due to greater discrepancies, while a smaller RSS means that the model predictions closely align with the observed values.

Understanding RSS is vital for evaluating model performance, especially when deciding if additional predictor variables are necessary or if existing ones should be adjusted to enhance the model's predictive capability.