91Ó°ÊÓ

The article "Increases in Steroid Binding Globulins Induced by Tamoxifen in Patients with Carcinoma of the Breast" \((J\). Endocrinol., 1978: 219-226) reports data on the effects of the drug tamoxifen on change in the level of cortisol-binding globulin (CBG) of patients during treatment. With age \(=x\) and \(\Delta \mathrm{CBG}=y\), summary values are \(n=26, \sum x_{i}=1613, \sum\left(x_{i}-\bar{x}\right)^{2}=3756.96\), \(\sum y_{i}=281.9, \quad \sum\left(y_{i}-\bar{y}\right)^{2}=465.34, \quad\) and \(\sum x_{i} y_{i}=16,731\) a. Compute a \(90 \%\) CI for the true correlation coefficient \(\rho\). b. Test \(H_{0}: \rho=-.5\) versus \(H_{\mathrm{a}}: \rho<-.5\) at level \(.05\). c. In a regression analysis of \(y\) on \(x\), what proportion of variation in change of cortisol-binding globulin level could be explained by variation in patient age within the sample? d. If you decide to perform a regression analysis with age as the dependent variable, what proportion of variation in age is explainable by variation in \(\triangle \mathrm{CBG}\) ?

Step by step solution

01

Compute the Sample Correlation Coefficient

To find the correlation coefficient \( r \), use the formula \( r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} \), where \( S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) \), \( S_{xx} = \sum (x_i - \bar{x})^2 \), and \( S_{yy} = \sum (y_i - \bar{y})^2 \). First, calculate \( S_{xy} = \sum x_i y_i - \frac{\sum x_i \sum y_i}{n} \). Plug in the values: \( S_{xy} = 16731 - \frac{1613 \times 281.9}{26} = 16731 - 17525.5379 = 119.4621 \). Then, calculate \( r = \frac{119.4621}{\sqrt{3756.96 \times 465.34}} = \frac{119.4621}{\sqrt{1747472.2164}} = \frac{119.4621}{1321.0705} = 0.0904 \).

02

Determine the Confidence Interval for \( \rho \)

Using Fisher's Z-transformation, compute \( z = 0.5 \times \ln((1+r)/(1-r)) \). For \( r = 0.0904 \), \( z = 0.5 \times \ln((1+0.0904)/(1-0.0904)) = 0.5 \times \ln(1.1992) = 0.5 \times 0.0953 = 0.04765 \). For a 90% confidence interval, the critical z-value is approximately 1.645. Compute the lower and upper bounds: \( z_{lb} = z - \frac{1.645}{\sqrt{n-3}} = 0.04765 - \frac{1.645}{\sqrt{23}} = -0.2875 \), \( z_{ub} = z + \frac{1.645}{\sqrt{23}} = 0.3828 \). Convert back to \( r \) using \( r = \frac{e^{2z} - 1}{e^{2z} + 1} \). Lower: \( r_{lb} = \frac{e^{-0.2875 \times 2} - 1}{e^{-0.2875 \times 2} + 1} = -0.280 \), Upper: \( r_{ub} = 0.365 \). Hence, the 90% CI is \((-0.280, 0.365)\).

03

Hypothesis Test for \( \rho \)

Under \( H_0: \rho = -0.5 \), use Fisher's Z-transformation to find \( z_{obs} \). Calculate \( z_{obs} = \frac{z_r - z_0}{\sqrt{\frac{1}{n-3}}} \), where \( z_0 \) is the Fisher's Z-transform of -0.5: \( z_0 = 0.5 \times \ln((1+(-0.5))/(1-(-0.5))) = -0.5493 \). Then \( z_{obs} = \frac{0.04765 - (-0.5493)}{\sqrt{\frac{1}{23}}} = 0.58695/0.2041 = 2.876 \). Compare with \( z_{critical} = -1.645 \) for 5% level: since \( 2.876 > -1.645 \), fail to reject \( H_0 \).

04

Proportion of Variation Explained in \( \Delta CBG \)

Use the coefficient of determination \( r^2 \) for regression of \( y \) on \( x \): \( r^2 = (0.0904)^2 = 0.00817 \) or \( 0.817\% \). This implies 0.817% of the variation in \( \Delta CBG \) level is explained by age variation.

05

Proportion of Variation Explained in Age

Since the coefficient of determination is the same irrespective of which variable is considered dependent, \( r^2 \) remains 0.00817 or 0.817\%, meaning 0.817% of the variation in age is explained by \( \Delta CBG \) variation.

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Key Concepts

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

Regression Analysis

Regression analysis allows us to explore relationships between two or more variables. In this particular exercise, we are analyzing the relationship between age and changes in cortisol-binding globulin (CBG) levels in patients. By using regression analysis, we can determine how much of the variation in one variable is explained by another.

The process typically involves finding the line that best fits the data, known as the regression line. The equation for a simple linear regression is given by:\[ y = b_0 + b_1x \]where:

• \(y\) is the dependent variable, in this case, the change in CBG levels.
• \(x\) is the independent variable, which is age.
• \(b_0\) is the y-intercept.
• \(b_1\) is the slope of the line, known as the regression coefficient.

By analyzing the slope, we can understand how a change in age is associated with a change in CBG levels. However, this relationship is also tested using other statistical measures, such as correlation coefficients and coefficients of determination.

Confidence Interval

Confidence intervals provide a range of values that are likely to contain the true population parameter. In this exercise, a 90% confidence interval for the correlation coefficient is calculated. This interval suggests that we can be 90% confident that the true correlation coefficient \(\rho\) lies within this range.

To construct a confidence interval for the correlation coefficient, we often use Fisher's Z-transformation because correlation coefficients are not normally distributed. The Z-transformation helps translate correlation coefficients into a form that is approximately normally distributed, facilitating interval estimation. We compute it as follows:\[z = 0.5 \times \ln\left(\frac{1+r}{1-r}\right)\]Using the Z value, the interval can be adjusted by critical values from the normal distribution:

• Lower bound: \( z - \frac{Z_{critical}}{\sqrt{n-3}} \)
• Upper bound: \( z + \frac{Z_{critical}}{\sqrt{n-3}} \)

After transforming back to the correlation coefficient, the interval estimates give us insights about the reliability and variation of our sample estimate.

Coefficient of Determination

The coefficient of determination, denoted as \(r^2\), quantifies the proportion of variation in the dependent variable that is predictable from the independent variable. In this scenario, \(r^2\) tells us what percentage of the variation in change of CBG levels is explained by age variation within the dataset.

It’s calculated by squaring the sample correlation coefficient \(r\), as follows:\[r^2 = (r)^2\]For our data, \(r^2\) was found to be 0.00817, or 0.817%. This implies that a very small proportion of the variability in CBG changes is accounted for by age. In other words, age is not a strong explanatory variable for the changes seen in CBG levels under these circumstances. Typically, a larger \(r^2\) value means better explanatory power.

Hypothesis Testing

Hypothesis testing involves making assumptions, called hypotheses, about a population parameter and then using sample data to test these assumptions. In the context of this exercise, we are testing a hypothesis about the population correlation coefficient \(\rho\).

The null hypothesis \((H_0)\) is that the correlation coefficient equals a specified value, here \(-0.5\). This hypothesis is tested against an alternative, which suggests that \(\rho\) is less than \(-0.5\).

To conduct this hypothesis test, Fisher's Z-transformation is used again. The observed Z-value is calculated and compared to the critical Z-value (determined by the significance level, \(\alpha = 0.05\)). If the observed value exceeds the critical value, the null hypothesis is not rejected. In this case:

• Calculate Fisher's Z-transform for the hypothesis value and comprehension through observed sample data.
• Compare calculated Z-score to critical values for decision making.

As the observed Z-value was greater than \(-1.645\), the null hypothesis was not rejected, indicating no significant evidence of \(\rho\) being less than \(-0.5\). This step verifies the assumed relationship's strength and significance.