91Ó°ÊÓ

The data from a patient satisfaction survey in a hospital are in Table E12-1. $$ \begin{array}{cccccc} \hline \begin{array}{l} \text { Obser- } \\ \text { vation } \end{array} & \text { Age } & \text { Severity } & \text { Surg-Med } & \text { Anxiety } & \begin{array}{c} \text { Satis- } \\ \text { faction } \end{array} \\ \hline 1 & 55 & 50 & 0 & 2.1 & 68 \\ 2 & 46 & 24 & 1 & 2.8 & 77 \\ 3 & 30 & 46 & 1 & 3.3 & 96 \\ 4 & 35 & 48 & 1 & 4.5 & 80 \\ 5 & 59 & 58 & 0 & 2.0 & 43 \\ 6 & 61 & 60 & 0 & 5.1 & 44 \\ 7 & 74 & 65 & 1 & 5.5 & 26 \\ 8 & 38 & 42 & 1 & 3.2 & 88 \\ 9 & 27 & 42 & 0 & 3.1 & 75 \\ 10 & 51 & 50 & 1 & 2.4 & 57 \\ 11 & 53 & 38 & 1 & 2.2 & 56 \\ 12 & 41 & 30 & 0 & 2.1 & 88 \\ 13 & 37 & 31 & 0 & 1.9 & 88 \\ 14 & 24 & 34 & 0 & 3.1 & 102 \\ 15 & 42 & 30 & 0 & 3.0 & 88 \\ 16 & 50 & 48 & 1 & 4.2 & 70 \\ 17 & 58 & 61 & 1 & 4.6 & 52 \\ 18 & 60 & 71 & 1 & 5.3 & 43 \\ 19 & 62 & 62 & 0 & 7.2 & 46 \\ 20 & 68 & 38 & 0 & 7.8 & 56 \\ 21 & 70 & 41 & 1 & 7.0 & 59 \\ 22 & 79 & 66 & 1 & 6.2 & 26 \\ 23 & 63 & 31 & 1 & 4.1 & 52 \\ 24 & 39 & 42 & 0 & 3.5 & 83 \\ 25 & 49 & 40 & 1 & 2.1 & 75 \\ \hline \end{array} $$ 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

Organize the data

Compile the data from Table E12-1 into a coherent format. You should have columns for Age, Severity, Anxiety, and Satisfaction that will be used in fitting the regression model.

02

Set up the regression model

Identify the dependent variable (Satisfaction) and the independent variables (Age, Severity, Anxiety) for the multiple linear regression. The regression model will have the form: \[Y = \beta_0 + \beta_1 \times \text{Age} + \beta_2 \times \text{Severity} + \beta_3 \times \text{Anxiety} + \epsilon,\]where \(Y\) is the Satisfaction score, \(\epsilon\) is the error term, and \(\beta_0, \beta_1, \beta_2, \beta_3\) are the coefficients to be estimated.

03

Fit the multiple regression model

Use statistical software to fit the regression model. Input the organized data into the software and run the regression analysis to obtain estimates of the model coefficients \(\beta_0, \beta_1, \beta_2, \beta_3\).

04

Estimate the variance (\(\sigma^2\))

To estimate \(\sigma^2\), calculate the residual sum of squares (RSS) from the regression output: \[\sigma^2 = \frac{\text{RSS}}{n - p},\]where \(n\) is the number of observations, and \(p\) is the number of parameters estimated (including the intercept).

05

Find the standard errors of the regression coefficients

From the regression analysis output, extract the standard error for each estimated regression coefficient (\(\beta_0, \beta_1, \beta_2, \beta_3\)). These standard errors quantify the uncertainty around each coefficient estimate.

06

Evaluate precision of parameter estimates

Compare the standard errors obtained for each regression coefficient. If they are similar, then the parameters are estimated with similar precision. If they differ significantly, the precision of estimation varies among the coefficients. Analyze whether the differences are conceptually or statistically significant.

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Key Concepts

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

Regression Coefficients

In multiple linear regression, the regression coefficients are key elements that carry significant meaning. These coefficients, often denoted as \( \beta_0, \beta_1, \beta_2, \ldots \), represent the relationship between each independent variable and the dependent variable.

• \(\beta_0\): This is the intercept, which is the expected value of the satisfaction score when all other predictor variables are zero.
• \(\beta_1\), \(\beta_2\), and \(\beta_3\): These are the slope coefficients that denote the change in the satisfaction score for a one-unit increase in the corresponding independent variables (Age, Severity, Anxiety), holding other variables constant.

Understanding these coefficients enables us to interpret how different factors such as patient's age or their anxiety level, impact patient satisfaction in a hospital setting. When fitting a regression model, these coefficients are estimated based on the sample data, using techniques like ordinary least squares (OLS). For example, in this scenario, a positive \(\beta_1\) indicates that as the patient's age increases, the satisfaction score is expected to increase, assuming other factors remain constant.

Standard Error

The standard error of a regression coefficient is a critical metric that measures the accuracy of a coefficient estimate. It reflects how much the estimated coefficient would vary across different samples from the same population.

• A small standard error suggests that the coefficient is estimated with high precision, implying that it would likely be similar in other samples.
• A large standard error indicates higher variability and less precision in the estimate.

To find the standard errors for each coefficient, we rely on the results from the regression analysis. The computation involves the estimated variance of the error term (\(\epsilon\)) and the variability in the independent variables. Standard errors are essential for hypothesis testing and constructing confidence intervals for the coefficients. For instance, if a standard error is small relative to the coefficient's magnitude, it suggests a narrower confidence interval, leading to more confidence in the stability of the coefficient's role in the model.

Variance Estimation

Estimating variance in the context of multiple linear regression allows us to quantify the extent to which our model's predictions are dispersed around the true values.

• The estimation of variance (\(\sigma^2\)) is performed using the residual sum of squares (RSS), which measures the discrepancy between the observed and predicted satisfaction scores.
• The formula is \(\sigma^2 = \frac{\text{RSS}}{n - p}\), where \(n\) is the number of observations and \(p\) is the number of parameters (including the intercept) estimated by the model.

This estimate provides insight into the overall fit of the model: a smaller variance suggests that the model is a good fit with predictions closely aligning with actual data points. On the other hand, a large estimated variance can point towards other potential influencing factors that the model might not be capturing effectively. Variance estimation is vital not only for evaluating the model accuracy but also for further statistical tests and constructing prediction and confidence intervals.