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Chapter 12: Problem 23

A clinical trial admits subjects suffering from high cholesterol, who are then randomly assigned to take a drug or a placebo for a 12 -week study. For the population, without taking any drug, the correlation between the cholesterol readings at times 12 weeks apart is 0.70 . The mean cholesterol reading at any given time is 200 , with the same standard deviation at each time. Consider all the subjects with a cholesterol level of 300 at the start of the study, who take placebo. a. What would you predict for their mean cholesterol level at the end of the study? b. Does this suggest that placebo is effective for treating high cholesterol? Explain.

Short Answer

Expert verified
a. 270; b. No, the change is due to statistical tendency, not placebo effect.

Step by step solution

01

Identify the Known Values

For subjects taking a placebo, we know the initial cholesterol level is 300. The population mean cholesterol level is 200 with a correlation of 0.70 between initial and after 12 weeks without any treatment.
02

Apply Regression to the Mean Concept

Use the concept of regression towards the mean to predict the ending cholesterol level. Regression towards the mean suggests that extreme observations tend to be closer to the mean on a subsequent measurement, based on the correlation.
03

Calculate the Predicted Mean Cholesterol Level

The predicted end cholesterol ( y ) is given by the formula: \[y = ar{x} + r(x_0 - \bar{x})\] where \( \bar{x} = 200 \) is the mean cholesterol, \( r = 0.70 \) is the correlation, and \( x_0 = 300 \) is the starting cholesterol. Substitute to get: \[y = 200 + 0.70(300 - 200)\] \[y = 200 + 0.70 imes 100\] \[y = 200 + 70 = 270\] This predicts their cholesterol level to be 270 after 12 weeks.
04

Analyze the Result

With a predicted cholesterol level of 270, which is lower than 300 but higher than the normal mean of 200, we can assess whether the placebo had an effect.
05

Conclusion on Placebo Effect

Since the regression towards the mean suggests expected movement towards 200 naturally due to the statistical tendency rather than the effect of the placebo, a reduction from 300 to 270 does not necessarily indicate the placebo's efficacy.

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Key Concepts

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

Regression to the Mean
When participating in a clinical trial, especially a cholesterol study, we often encounter the concept called **regression to the mean**. This phenomenon occurs when an initially extreme measurement, such as a high cholesterol level, tends to move closer to the population average over time without any specific intervention.
This is especially important in scenarios like the one described in the exercise where the initial cholesterol level is high at 300, while the population average is 200. The correlation coefficient of 0.70 suggests a strong, but not perfect, relationship between initial and subsequent measurements after 12 weeks. Therefore, when considering extreme values, regression to the mean helps predict that cholesterol levels will naturally decrease, moving closer to the population mean.
  • This statistical trend often causes misunderstanding in interpreting treatment effects.
  • Key takeaway: Always consider natural fluctuations before concluding treatment efficacy.
Placebo Effect
The **placebo effect** is a fascinating aspect of clinical trials that can complicate the interpretation of results. A placebo is an inert substance with no therapeutic effect, like a sugar pill, given to some participants in a study. It is used to test whether the actual drug being tested has a genuine therapeutic effect beyond psychological benefit.
When a participant believes they are receiving treatment, even if it's just a placebo, they may experience changes in health or symptoms simply due to their expectations. This can sometimes lead to real physiological responses, making it crucial to differentiate between genuine drug effects and changes due to participant expectations.
  • Placebo effects are important as they highlight the power of perception and belief in treatment success.
  • In the case of this cholesterol study, any observed change while taking a placebo must be weighed against the expected regression to the mean behavior.
Understanding whether changes in cholesterol levels are due to this effect or just natural fluctuations is key in assessing the trial's results.
Correlation Coefficient
The **correlation coefficient** is a statistical measure that describes the extent of the relationship between two variables. In the context of our cholesterol study, this number is 0.70. This indicates a strong positive correlation between initial cholesterol levels and the readings taken 12 weeks later.
This means that high initial cholesterol levels predict higher levels at the end of the study, but not perfectly. A correlation coefficient ranges from -1 to 1, where 1 indicates a perfect positive correlation and -1 a perfect negative one. A value of 0 would mean no correlation at all.
  • In practical applications, a correlation of 0.70 shows substantial predictability; however, it's important to note it's not absolute.
  • Any deviation in cholesterol levels towards the average is influenced by this correlation, as it dictates how much regression towards the mean to expect.
Understanding this concept helps explain why, even with no medication, cholesterol levels might shift downwards towards the population mean.
Cholesterol Study
In a **cholesterol study**, subjects are often assessed to determine the effect of a treatment on blood cholesterol levels. Understanding the dynamics of cholesterol levels is paramount to clinical trial analysis, particularly when looking at the efficacy of treatments or interventions.
This specific study involves subjects starting with a high cholesterol level of 300, significantly above the population mean of 200. The intervention includes some participants receiving a placebo to evaluate if perceived treatment impacts cholesterol levels.
  • Details such as the mean cholesterol level, standard deviation, and the correlation coefficient between time points are critical.
  • Using such data allows us to predict changes and evaluate treatment success or the natural tendency towards a population norm due to statistical principles.
This approach helps differentiate between the effects of regression to the mean, placebo effects, and any genuine benefits of cholesterol-lowering interventions.

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Most popular questions from this chapter

For a class of 100 students, the teacher takes the 10 students who performed poorest on the midterm exam and enrolls them in a special tutoring program. Both the midterm and final have a class mean of 70 with standard deviation 10 , and the correlation is 0.50 between the two exam scores. The mean for the specially tutored students increases from 50 on the midterm to 60 on the final. Can we conclude that the tutoring program was successful? Explain, identifying the response and explanatory variables and the role of regression toward the mean.

All students who attend Lake Wobegon College must take the math and verbal SAT exams. Both exams have a mean of 500 and a standard deviation of 100 . The regression equation relating \(y=\) math SAT score and \(x=\) verbal SAT score is \(\hat{y}=250+0.5 x\) a. Find the predicted math SAT score for a student who has the mean verbal SAT score of \(500 .\) (Note: At the \(x\) value equal to \(\bar{x},\) the predicted value of \(y\) equals \(\bar{y} .)\) b. Show how to find the correlation. Interpret its value as a standardized slope. (Hint: Both standard deviations are equal.) c. Find \(r^{2}\) and interpret its value.

The table shows the approximate U.S. population size (in millions) at 10 -year intervals beginning in 1900 . Let \(x\) denote the number of decades since 1900 . That is, 1900 is \(x=0,1910\) is \(x=1,\) and so forth. The exponential regression model fitted to \(y=\) population size and \(x\) gives \(\hat{y}=81.14 \times 1.1339^{x}\) a. Show that the predicted population sizes are 81.14 million in 1900 and 323.3 million in 2010 . b. Explain how to interpret the value 1.1339 in the prediction equation. c. The correlation equals 0.98 between the log of the population size and the year number. What does this suggest about whether or not the exponential regression model is appropriate for these data?

For prediction intervals, an important inference assumption is a constant residual standard deviation of \(y\) values at different \(x\) values. In practice, the residual standard deviation often tends to be larger when \(\mu_{y}\) is larger. a. Sketch a hypothetical scatterplot for which this happens, using observations for the previous year on \(x=\) family income and \(y=\) amount donated to charity. b. Explain why a \(95 \%\) prediction interval would not work well at very small or at very large \(x\) values.

Although the slope does not measure association, it is useful for comparing effects for two variables that have the same units. For the Internet Use data file of 33 nations on the text \(\mathrm{CD},\) let \(x=\mathrm{GDP}\) (thousands of dollars per capita). For predicting \(y=\) percentage Internet penetration (the percentage of adult users), the prediction equation is \(\hat{y}=0.1239+0.0157 x\). For predicting \(y=\) percentage Facebook penetration, the prediction equation is \(\hat{y}=0.081+0.0075 x\) a. Explain how to interpret the two slopes. b. For these nations, explain why a one-unit increase in GDP has a slightly greater impact on the percentage using Facebook than on the percentage using the Internet.
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