91Ó°ÊÓ

Cost-to-charge ratio (the percentage of the amount billed that represents the actual cost) for inpatient and outpatient services at 11 Oregon hospitals is shown in the following table (Oregon Department of Health Services, 2002). A scatterplot of the data is also shown. $$ \begin{array}{ccc} \hline \text { Hospital } & \begin{array}{l} \text { Outpatient } \\ \text { Care } \end{array} & \begin{array}{l} \text { Inpatient } \\ \text { Care } \end{array} \\ \hline 1 & 62 & 80 \\ 2 & 66 & 76 \\ 3 & 63 & 75 \\ 4 & 51 & 62 \\ 5 & 75 & 100 \\ 6 & 65 & 88 \\ 7 & 56 & 64 \\ 8 & 45 & 50 \\ 9 & 48 & 54 \\ 10 & 71 & 83 \\ 11 & 54 & 100 \\ \hline \end{array} $$ The least-squares regression line with \(y=\) inpatient costto-charge ratio and \(x=\) outpatient cost-to-charge ratio is \(\hat{y}=-1.1+1.29 x\). a. Is the observation for Hospital 11 an influential observation? Justify your answer. b. Is the observation for Hospital 11 an outlier? Explain. c. Is the observation for Hospital 5 an influential observation? Justify your answer. d. Is the observation for Hospital 5 an outlier? Explain.

Step by step solution

01

Identifying an Influential Observation

First, let's handle question (a) and consider Hospital 11. An influential observation is one where, if the observation were removed from the data, the estimated regression equation would change considerably. With the current regression line, calculate the predicted inpatient cost-to-charge ratio for Hospital 11 i.e., Replace \(x\) with 54 in the regression equation \(\hat{y}=-1.1+1.29x\) to find \(\hat{y}\).

02

Checking for Outliers

Now, let's deal with question (b) and see if Hospital 11 is an outlier. An outlier is an observation that departs significantly from the other observations in the dataset. Here, the data value for Hospital 11 seems large compared to the data for the other hospitals. To formally confirm whether it is an outlier, perform a residual analysis. Find the residual for Hospital 11 by subtracting the predicted ratio from the observed ratio. If this residual is much larger or smaller than the other residuals, Hospital 11 can be considered an outlier.

03

Repeat Steps 1 and 2 for Hospital 5

Repeat the above steps for Hospital 5. First, determine if Hospital 5 is an influential observation by calculating the predicted inpatient cost-to-charge ratio using the given regression line equation, and then investigating whether the removal of the Hospital 5 observation would significantly affect the regression equation. Then, test if Hospital 5 is an outlier by performing a residual analysis, comparing the size of its residual with the residuals of the other observations.

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Key Concepts

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

Statistical Regression

Statistical regression is a powerful tool used to discover the relationship between variables. Generally, it's a way to predict the value of a dependent variable (like inpatient cost-to-charge ratio) based on one or more independent variables (like outpatient cost-to-charge ratio). In the case of the Oregon hospitals, regression helps us understand the relationship between outpatient and inpatient costs.

In this scenario, a least-squares regression line is used, which is the line that minimizes the sum of the squares of the residuals—the differences between observed and predicted values. This line is represented by the equation \(\hat{y}=-1.1+1.29x\), where \(\hat{y}\) is the predicted inpatient cost-to-charge ratio and \(x\) represents the outpatient cost-to-charge ratio. By analyzing the regression, we can estimate costs, make informed decisions, and understand how the two types of costs are related among Oregon hospitals.

Influential Observation

An influential observation has a significant impact on the outcome of the regression analysis. If removing this observation causes a noticeable change in the regression equation, it is deemed as influential. To determine if a point is influential, we often look at its leverage and influence on the regression coefficients.

For example, in the exercise, Hospital 11 is tested to find out if it's an influential observation by checking how the regression line would alter without it. Since the least-squares regression line would change considerably without Hospital 11's data, it indicates that this point might be influential. Understanding which observations are influential can help analysts make more reliable predictions and ensure that the regression model truly represents the remaining data.

Outliers Detection

Outliers are data points that are substantially different from the pattern of the rest of the dataset. They may occur due to measurement errors, data-entry errors, or they could be accurate but unusual points.

Detection of outliers is critical because they can skew the results of data analysis. In the context of the hospitals' cost-to-charge ratio, an outlier would be a hospital whose inpatient or outpatient ratio is substantially different from others. In the exercise, we look at Hospital 11 and Hospital 5 to determine if they are outliers by comparing their residuals to those from other hospitals. A significantly larger or smaller residual indicates an outlier. Identifying outliers can provide insights into special cases or data anomalies and helps ensure the integrity of the regression analysis.

Residual Analysis

Residual analysis involves studying the residuals—the differences between observed and predicted values—to assess the accuracy of a regression model. It's a form of diagnostic to check the adequacy of the model and discover patterns that might suggest problems like non-linearity, heteroscedasticity, or outliers.

In the given problem, the residual for each hospital is calculated by subtracting the predicted inpatient cost-to-charge ratio from the observed ratio using the regression equation. By scrutinizing these residuals, as done for Hospital 11 and Hospital 5, we can determine if their data significantly differs from what the model predicts, signaling if they are outliers or if the model needs reconsideration to account for more complex relationships in the data.