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Hormone replacement therapy (HRT) is thought to increase the risk of breast cancer. The accompanying data on \(x=\) percent of women using HRT and \(y=\) breast cancer incidence (cases per 100,000 women) for a region in Germany for 5 years appeared in the paper "Decline in Breast Cancer Incidence after Decrease in Utilisation of Hormone Replacement Therapy" (Epidemiology [2008]: \(427-430\) ). The authors of the paper used a simple linear regression model to describe the relationship between HRT use and breast cancer incidence. \begin{tabular}{|cc|} \hline HRT Use & Breast Cancer Incidence \\ \hline 46.30 & 103.3 \\ 40.60 & 105.0 \\ 39.50 & 100.0 \\ 36.60 & 93.8 \\ 30.00 & 83.5 \\ \hline \end{tabular} a. What is the equation of the estimated regression line? b. What is the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use? c. What would you predict the breast cancer incidence to be in a year when HRT use was \(40 \% ?\) d. Should you use this regression model to predict breast cancer incidence for a year when HRT use was \(20 \%\) ? Explain. e. Calculate and interpret the value of \(r^{2}\). f. Calculate and interpret the value of \(s_{e}\).

Step by step solution

01

Calculate the Regression Line

Using the given data, we can create a simple linear regression model: \( y = a + bx \), where x is the percent of women using HRT and y is the breast cancer incidence (cases per 100,000 women). First, we need to calculate the sums and mean values of x and y: \( \bar{x} = \dfrac{1}{n}\sum_{i=1}^n x_i \) \( \bar{y} = \dfrac{1}{n}\sum_{i=1}^n y_i \) Then calculate the slope (b) and the y-intercept (a) of the regression line using the formulas: \( b = \dfrac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} \) \( a = \bar{y} - b\bar{x} \) We now plug in the given data and find the values.

02

Calculate b

Using the formula for slope: \( b = \dfrac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2} \) We need the values of \(饾懃_i\) and \( y_i \) to calculate b.

03

Calculate a

Now we can find the value of a using the formula: \( a = \bar{y} - b\bar{x} \) With the values for a and b, we can write the equation of the estimated regression line.

04

Estimated Average Change in Breast Cancer Incidence

The slope of the regression line, b, gives the estimated average change in breast cancer incidence associated with a 1 percentage point increase in HRT use.

05

Predict Breast Cancer Incidence

To predict the breast cancer incidence for a year when HRT use was 40%, we can plug the value into the regression line equation: \( y = a + bx \)

06

Discuss Validity of the Model

We need to consider whether the regression model is suitable for predicting breast cancer incidence for a year when HRT use was 20%. We should look at the range of the data points in the dataset and determine if it is appropriate to use the model for prediction outside this range.

07

Calculate and Interpret the Value of r虏

The coefficient of determination (r虏) can be calculated using the formula: \( r^2 = \dfrac{(\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}))^2}{\sum_{i=1}^n (x_i - \bar{x})^2\sum_{i=1}^n (y_i - \bar{y})^2} \) r虏 is a value between 0 and 1 that measures the proportion of variance in the dependent variable (y) that is predictable from the independent variable (x). A higher value of r虏 indicates a stronger relationship between the variables.

08

Calculate and Interpret the Value of s鈧�

The residual standard error (s鈧�) can be calculated as: \( s_e = \sqrt{\dfrac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-2}} \) Where \( \hat{y}_i \) represents the predicted values for y. The residual standard error measures the average distance between the actual values of y and the predicted values from the regression model. A smaller s鈧� value indicates that the model's predictions are more accurate. Now, you can follow these steps to solve the exercise and calculate the required values.

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Key Concepts

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

Coefficient of Determination

The Coefficient of Determination, denoted as \(r^2\), is an essential statistical measure in linear regression. It tells us how well the independent variable, such as the percentage of women using Hormone Replacement Therapy (HRT), explains the variance in the dependent variable, which in this case is the breast cancer incidence. A value of \(r^2\) close to 1 signifies a strong relationship, indicating that most of the variance in breast cancer incidence can be explained by changes in HRT usage. Conversely, a value near 0 suggests a weak relationship, where changes in HRT usage do little to explain variations in breast cancer incidence.

To calculate \(r^2\), we use the formula:\[r^2 = \frac{(\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}))^2}{\sum_{i=1}^n (x_i - \bar{x})^2\sum_{i=1}^n (y_i - \bar{y})^2}\]
This means \(r^2\) is derived by dividing the regression sum of squares by the total sum of squares, reflecting the proportion of total variability in \(y\) that is accounted for by the linear relationship with \(x\). Understanding \(r^2\) helps assess how effective the regression model might be at prediction or understanding the dynamics between the variables.

Slope Calculation

The slope in a linear regression model represents the estimated average change in the dependent variable, here breast cancer incidence, for each one-unit change in the independent variable, which is the percentage of women using HRT. The slope is a key part of the regression equation \(y = a + bx\), where \(b\) is the slope. It essentially tells us how much we can expect \(y\) to increase or decrease when \(x\) changes by one percentage point.

To find the slope \(b\), use the formula:\[b = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}\]
This formula calculates the covariance of \(x\) and \(y\) over the variance of \(x\). A positive \(b\) indicates that as HRT usage increases, breast cancer incidence is also likely to increase. Conversely, a negative \(b\) suggests that higher HRT usage is linked to a decrease in breast cancer incidence. The slope is crucial for making predictions and understanding the direction of the relationship between the variables.

Residual Standard Error

Residual Standard Error, denoted as \(s_e\), is a measure that quantifies the variability or spread of the observed values around the regression line. It indicates how far the predicted breast cancer incidence values deviate from the actual observed values. A smaller \(s_e\) suggests that the regression line fits the data points more closely.

The formula for calculating \(s_e\) is:\[s_e = \sqrt{\dfrac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{n-2}}\]
Here, \(y_i\) is the actual incidence, \(\hat{y}_i\) is the predicted incidence, and \(n\) is the number of data points. The \(n-2\) in the denominator reflects the degrees of freedom, accounting for two parameters estimated (slope and intercept) in the regression equation. This measure helps us understand the accuracy of our model; smaller errors mean the model's predictions are closely aligned with the actual data. Thus, by minimizing \(s_e\), we improve the model's reliability in predicting breast cancer incidence based on HRT usage.