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Chapter 7: Problem 110

Treatment with statins can reduce the risk of a major cardiovascular event in people with specified risk factors. During about 2 years of follow-up in the JUPITER trial (P. M. Ridker et al. \([8]), 142\) of 8901 subjects treated with a statin had a major cardiovascular event. Estimate the 2-year risk, and give a \(95 \%\) confidence interval for this estimate.

Short Answer

Expert verified
The estimated 2-year risk is 1.59% with a 95% confidence interval of (1.34%, 1.84%).

Step by step solution

01

Calculate the Risk

To estimate the 2-year risk of a major cardiovascular event for subjects treated with statins, divide the number of events by the total number of subjects. The formula is:\[\text{Risk} = \frac{\text{Number of events}}{\text{Total number of subjects}} = \frac{142}{8901}.\]Compute this to get the risk.
02

Plug-In Risk Calculation

Using the formula from Step 1:\[\text{Risk} = \frac{142}{8901} \approx 0.0159.\]So, the estimated 2-year risk is approximately 0.0159 or 1.59%.
03

Determine the Standard Error

The standard error (SE) of a proportion is calculated using the formula for standard error: \[SE = \sqrt{\frac{p(1-p)}{n}},\]where \(p\) is the risk and \(n\) is the total number of subjects. Substituting the values:\[p = 0.0159, \quad n = 8901,\]compute the SE.
04

Plug-In Standard Error Calculation

Substitute the values into the SE formula:\[SE = \sqrt{\frac{0.0159 \times (1 - 0.0159)}{8901}} \approx 0.0013.\]
05

Compute the 95% Confidence Interval

A 95% confidence interval for a proportion can be calculated using the formula:\[CI = p \pm Z \times SE,\]where \(Z\) is the Z-score for a 95% confidence level, approximately 1.96.Plugging in the values:\[CI = 0.0159 \pm 1.96 \times 0.0013.\]Compute the bounds of the confidence interval.
06

Confidence Interval Bounds

Calculate the lower and upper bounds of the confidence interval:\[\text{Lower bound} = 0.0159 - 1.96 \times 0.0013 \approx 0.0134,\]\[\text{Upper bound} = 0.0159 + 1.96 \times 0.0013 \approx 0.0184.\]
07

Final Confidence Interval

The 95% confidence interval for the 2-year risk is:\[(0.0134, 0.0184).\]

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Key Concepts

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

Risk Estimation
Risk estimation in biostatistics helps us understand the likelihood of an event occurring within a specified timeframe. In our exercise, we estimated the 2-year risk of suffering a major cardiovascular event for individuals treated with statins. To find this risk, we use the simple formula: the number of events divided by the total number of participants. This division yields a decimal, which represents the probability of one individual experiencing the event in that period.

In our example:
  • Number of events = 142
  • Total number of participants = 8901
The calculation gives us a risk of approximately 0.0159. Transforming this decimal to a percentage, we find that about 1.59% of people experienced a major event. This estimation allows healthcare professionals to communicate risks clearly and helps individuals make informed decisions about their health interventions.
Confidence Interval
A confidence interval (CI) provides a range of values which is likely to contain an estimate's true value. In biostatistics, it conveys the uncertainty surrounding our calculated risk. For a 95% confidence interval, we believe that the true risk value will fall within this range in 95 out of 100 cases.

The confidence interval is calculated using the formula: \[CI = p \pm Z \times SE\] where:
  • \(p\) is the calculated risk
  • \(Z\) is the Z-score (1.96 for 95% CI)
  • \(SE\) is the standard error
In our scenario, the resulting confidence interval is (0.0134, 0.0184), indicating that we can be 95% confident that the true risk falls between 1.34% and 1.84%. This range helps in understanding the precision of the risk estimate, giving practitioners valuable context on the reliability of the data.
Standard Error
Standard error (SE) measures the variability or spread of a sample statistic. It is an essential component when constructing confidence intervals, giving us an idea of how much our sample proportion differs from the true population proportion. A smaller SE indicates a more precise estimate.

The formula for the standard error of a proportion is:\[SE = \sqrt{\frac{p(1-p)}{n}}\] where:
  • \(p\) is the calculated risk
  • \(n\) is the total number of subjects
In our example, we substitute \(p = 0.0159\) and \(n = 8901\) into the equation resulting in an SE of approximately 0.0013. This small value indicates that our risk estimation is relatively precise, reflecting a low variation between the calculated proportion and potential true population proportion.
Statistical Analysis
Statistical analysis is a broad field involving processes for collecting, analyzing, interpreting, and presenting data. In our exercise, statistical analysis includes calculating risk, standard error, and confidence intervals to understand the impact of a treatment.

This specific analysis lets researchers determine the effectiveness and safety of statins in reducing the likelihood of cardiovascular events in at-risk individuals. Through statistical analysis, conclusions about treatment benefits are made, guiding clinical decisions.
  • Risk and confidence intervals assess treatment effectiveness.
  • Standard error reflects the precision of our estimation.
By using statistical analysis, researchers can ensure that decisions are based on solid data, reducing biases and increasing the validity of their findings. This meticulous process contributes powerfully to advancing medical understanding and improving patient care decisions.

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