91Ó°ÊÓ

The article "Increases in Steroid Binding Globulins Induced by Tamoxifen in Patients with Carcinoma of the Breast" \((J\). of Endocrinology, 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 \(n=26\), \(\sum x_{i}=1613, \sum\left(x_{i}-\bar{x}\right)^{2}=3756.96, \sum y_{i}=281.9\) \(\sum\left(y_{i}-\bar{y}\right)^{2}=465.34\), 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

Calculate sample means

To find the sample means, we use the given sums for age \(x\) and change in cortisol-binding globulin \(y\):\[\bar{x} = \frac{\sum x_i}{n} = \frac{1613}{26} \approx 62.04\] \[\bar{y} = \frac{\sum y_i}{n} = \frac{281.9}{26} \approx 10.84\]

02

Calculate covariance and correlation coefficient

Covariance \(S_{xy}\) is calculated using the sum of products:\[S_{xy} = \sum (x_i y_i) - n \bar{x}\bar{y} = 16731 - 26 \times 62.04 \times 10.84 \approx 1315.64\] For variance, we use \(\sum(x_i-\bar{x})^2\): \[S_{xx} = \sum (x_i - \bar{x})^2 = 3756.96\] The variance \(S_{yy}\) is given: \[S_{yy} = 465.34\]Correlation coefficient \(r\) is then:\[r = \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} = \frac{1315.64}{\sqrt{3756.96 \times 465.34}} \approx -0.31\]

03

Compute 90% confidence interval for \(\rho\)

Use Fisher's z-transformation to compute the confidence interval:\[z' = \frac{1}{2} \ln \left(\frac{1+r}{1-r}\right)\approx \frac{1}{2} \ln \left(\frac{1-0.31}{1+0.31}\right) \approx -0.320\] Standard error is \(\text{SE} = \frac{1}{\sqrt{n-3}} = \frac{1}{\sqrt{23}} \approx 0.208\)For a 90% CI, z-value is 1.645:\[ z' \pm 1.645 \times \text{SE} \approx -0.320\pm 0.342\]Back transform confidence limits for \(r\):\[\text{CI} = \left(\tanh(-0.662), \tanh(0.022)\right) \approx (-0.58, 0.02)\]

04

Perform hypothesis test for \(H_0: \rho = -0.5\)

Transform hypothesized \(\rho = -0.5\) using Fisher's transformation:\[z_{\rho} = \text{atanh}(-0.5) \approx -0.5493\] Compute test statistic using \(z'\): \[ z = \frac{-0.320 - (-0.5493)}{0.208} \approx 1.102\] Compare with critical z-value at \(0.05\) significance \(\text{one-tailed}\) \(z_c = -1.645\). Since 1.102 > -1.645, do not reject \(H_0\).

05

Determine variation explained in regression of \(y\) on \(x\)

The proportion of variation is the square of the correlation coefficient:\[ R^2 = (-0.31)^2 \approx 0.096\]Which corresponds to \(9.6\%\) of variation in \(\Delta \text{CBG}\) explained by age.

06

Variation explained if switching dependent variable

If age is the dependent variable, the explained variation remains the same due to the symmetric nature of correlation:\[ R^2 = (-0.31)^2 \approx 0.096\] Thus, \(9.6\%\) of variation in age is explained by \(\Delta \text{CBG}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Confidence Interval

In statistics, a confidence interval (CI) is a range of values that is used to estimate a population parameter. This range is constructed so that, with a specified degree of confidence, it contains the true parameter value. In the context of correlation analysis, the confidence interval is used to estimate the true correlation coefficient \(\rho\) of the population.

In our problem, we calculated a 90% confidence interval for the true correlation coefficient between age and changes in cortisol-binding globulin levels. Using Fisher's z-transformation, we found that the interval is approximately (-0.58, 0.02).

This interval tells us that we are 90% confident that the true correlation coefficient lies within this range. It's important to note that the interval includes zero, suggesting that there might be no correlation between the two variables, according to this data.

Regression Analysis

Regression analysis is a statistical method used to examine the relationship between two or more variables. In this context, we are looking at a simple linear regression where one variable \(y\) is predicted by another variable \(x\).

When conducting regression analysis on the cortisol-binding globulin levels with age as the independent variable, we used the correlation coefficient to estimate how much of the variation in globulin levels can be explained by age. This is achieved through the calculation of the coefficient of determination, which is denoted as \(R^2\).

In this exercise, the calculated \(R^2\) value is 0.096, which indicates that only 9.6% of the variance in cortisol-binding globulin levels can be accounted for by age. This suggests that other factors may play a more significant role in influencing these levels.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves testing an assumption (called the null hypothesis) against an alternate hypothesis.

In the context of this exercise, the null hypothesis \(H_0\) was that the population correlation coefficient \(\rho\) equals -0.5. The alternate hypothesis \(H_a\) suggests that \(\rho < -0.5\).

Using Fisher's transformation and our sample data, we calculated a test statistic, which we compared to predefined critical values. In this case, the test statistic did not significantly differ from the hypothesized value. This resulted in failing to reject the null hypothesis, meaning we do not have sufficient evidence to claim the population correlation is less than -0.5 with the given data.

Fisher's Z-Transformation

Fisher's Z-Transformation is an essential technique in correlation analysis that stabilizes the variance of the correlation coefficient, making it usable for constructing confidence intervals and performing hypothesis tests.

The transformation involves converting the Pearson correlation coefficient to a Z-score. This transformation is particularly useful when dealing with small sample sizes because it allows for more accurate interval estimates and hypothesis testing.

When applying Fisher's Z-transformation in our exercise, we initially transformed the sample correlation coefficient \(r\) to a Z-score to find the confidence interval for the true correlation \(\rho\) and also to perform hypothesis testing. By converting to the Z-form and back, we can interpret statistical results more reliably, especially in instances involving smaller sample sizes.