91Ó°ÊÓ

The authors of the paper studied a number of variables they thought might be related to bone mineral density (BMD). The accompanying data on \(x=\) weight at age 13 and \(y=\) bone mineral density at age 27 are consistent with summary quantities for women given in the paper. A simple linear regression model was used to describe the relationship between weight at age 13 and \(\mathrm{BMD}\) at age 27\. For this data: $$ \begin{array}{lll} a=0.558 & b=0.009 & n=15 \\ \mathrm{SSTo}=0.356 & \text { SSResid }=0.313 & \end{array} $$ a. What percentage of observed variation in \(\mathrm{BMD}\) at age 27 can be explained by the simple linear regression model? b. Give a point estimate of \(\sigma\) and interpret this estimate. c. Give an estimate of the average change in BMD associated with a \(1 \mathrm{~kg}\) increase in weight at age 13 . d. Compute a point estimate of the mean BMD at age 27 for women whose age 13 weight was \(60 \mathrm{~kg}\).

Step by step solution

01

Calculate R-Squared

The percentage of observed variation in BMD at age 27 that can be explained by the model is the coefficient of determination, also known as \(R^2\). It is calculated as \(1 - (SSResid/SSTo)\). So, the \(R^2\) value is \(1 - (0.313/0.356)\).

02

Point Estimate of σ (Standard Error)

To calculate the point estimate for standard error, we need to find the root of mean square error. It is given by \(\sqrt{SSResid/(n-2)}\), where \(n\) refers to the number of observations. Hence, the point estimate of \(\sigma\) is \(\sqrt{0.313/(15-2)}\). The standard error measure tells us about the average distance that the observed values fall from the regression line. Smaller value of standard error indicates most values are closer to the regression line.

03

Estimate the Average Change in BMD

The average change in BMD associated with a 1 kg increase in weight at age 13 is given by the slope of the regression line, \(b\). So, the average change in BMD for each kg of weight increase is 0.009 per kg.

04

Estimate of Mean BMD at Age 27 for Women Who Were 60 kg at Age 13

The point estimate of mean BMD at age 27 for women of weight 60 kg at age 13 can be calculated using the equation of the line which is \(y = a + bx\). Here \(y\) is the BMD at age 27, \(a\) is the intercept, \(b\) is the slope and \(x\) is the weight at 13. Substituting \(a= 0.558\), \(b=0.009\), and \(x=60\), we get the required BMD.

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Key Concepts

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

Coefficient of Determination

Understanding the coefficient of determination, often denoted as R-squared (R^2), is crucial when working with simple linear regression analysis. It quantifies the proportion of the variability in the dependent variable (in our case, bone mineral density or BMD) that can be explained by the independent variable (here, weight at age 13). An R^2 value of 1 indicates perfect correlation, while 0 means no correlation at all.

Using the formula R^2 = 1 - (SSResid/SSTo), where SSResid is the sum of squares of residuals and SSTo is the total sum of squares, we can compute the R^2 for our problem. The closer the value of R^2 is to 1, the better the regression line fits the data, implying that most of the variation in BMD can be explained by weight at age 13. Thus, this coefficient is a helpful measure of how well our model does in explaining the observations.

Standard Error

The standard error (SE) in the context of regression analysis informs us about the typical distance that the data points deviate from the actual regression line. This gives us an idea of the precision of the regression coefficient estimates. In our exercise, we calculate the point estimate of the standard error by taking the square root of the mean square error, which utilizes the sum of the squared residuals (SSResid) and the number of observations (n).

The formula for standard error would then be SE = \(\sqrt{SSResid / (n - 2)}\), reflecting the average spread of the observations around the regression line. A smaller standard error indicates that the observations are more tightly clustered around the line, meaning our model's estimates are more precise. Conversely, a larger standard error suggests wider variability around the regression line and could signify a weaker model.

Regression Line Slope

The slope of the regression line represents the average change in the dependent variable (BMD) for every one-unit increase in the independent variable (weight at age 13). In simple linear regression, the slope coefficient, denoted by b, is a statistical measure of the strength and direction of the relationship.

The formula for calculating the slope in our example is straightforward: it's the value computed in the regression analysis (in our case, 0.009). This figure means that for every additional kilogram of weight at age 13, there is an estimated average increase of 0.009 in bone mineral density at age 27. Understanding the slope is vital as it helps estimate the magnitude of the relationship between weight and BMD.

Mean Bone Mineral Density Estimation

Estimating the mean bone mineral density (BMD) for a given weight at age 13 is done using the regression line equation. Applying the equation y = a + bx, where 'y' is the predicted BMD at age 27, 'a' is the y-intercept, 'b' is the slope, and 'x' represents the weight at age 13, we can predict BMD for any given weight.

For a weight of 60 kg at age 13, substituting the known values of 'a' and 'b' into the equation gives us the estimated mean BMD for a 27-year-old woman. Notably, this estimation has practical implications in the medical field for individuals and professionals looking to understand and manage bone health.