Question

A series of patients with bacterial wound infections were treated with the antibiotic Cefotaxime. Bacteriologic response (disappearance of the bacteria from the wound) was considered "satisfactory" in \(84 \%\) of the patients. Determine the standard error of \(\widetilde{P},\) the Wilson-adjusted observed proportion of "satisfactory" responses, if the series contained (a) 50 patients of whom 42 were considered "satisfactory." (b) 200 patients of whom 168 were considered "satisfactory."

Short answer

For 50 patients, the Wilson-adjusted observed proportion is 0.867 and the standard error is 0.046. For 200 patients, the Wilson-adjusted observed proportion is 0.84 and the standard error is 0.023.

Step-by-step solution

Understand the Wilson Score Interval

The Wilson Score Interval is a way to calculate a confidence interval for a proportion. This interval is more accurate than the normal approximation interval when the number of trials is not very large and the proportion is not close to 0 or 1. The adjusted observed proportion is calculated using the formula for the Wilson score interval without continuity correction.

Calculate the Wilson-adjusted proportion for 50 patients

For 50 patients with 42 satisfactory responses, calculate the Wilson-adjusted proportion using the formula: \(\widetilde{P} = \frac{p + \frac{z^2}{2n}}{1+\frac{z^2}{n}}\). Here, \(p = \frac{x}{n}\), where \(x\) is the number of satisfactory cases, \(n\) is the total number of patients, and \(z\) is the z-value for the desired confidence level (typically 1.96 for 95% confidence).

Calculate the standard error for 50 patients

The Wilson standard error for the adjusted proportion is given by: \(SE_{\widetilde{P}} = \sqrt{\frac{\widetilde{P}(1 - \widetilde{P})}{n + z^2}}\), where \(\widetilde{P}\) is as calculated in Step 2.

Repeat the calculations for 200 patients

Repeat the Wilson-adjusted proportion and standard error calculation with 200 patients and 168 satisfactory responses using the formulas provided in Steps 2 and 3. Make sure to use the same z-value unless otherwise specified.

Key concepts

Confidence Interval for Proportion

Understanding a confidence interval for a proportion is essential in statistics as it provides a range where we expect a proportion to fall in with a certain level of confidence. For instance, if a medicine shows a 95% confidence interval from 40% to 60% efficacy, it indicates that there's a 95% chance the true efficacy of the medicine is between 40% and 60%.

In the context of the antibiotic Cefotaxime and its bacteriologic response rate, the confidence interval helps determine the range within which we can expect the true satisfaction rate to be, given our sample data. This is vital for clinicians and researchers to understand the effectiveness of the treatment. The Wilson-adjusted observed proportion is particularly useful for creating a more accurate confidence interval, especially when sample sizes are small or the observed proportion is near 0 or 1.

Bacteriologic Response Rate

What is Bacteriologic Response Rate?

Bacteriologic response rate is a term used to describe the percentage of patients in a study or treatment group for whom a treatment is considered successful, based on the eradication of bacteria. In the case of Cefotaxime treating bacterial wound infections, the response rate is the percentage of patients whose infections were successfully treated by the drug.

Clinically, it's a crucial measure of an antibiotic's efficacy. To say that an 84% bacteriologic response rate is 'satisfactory' means that a vast majority of the patients benefitted from the medication, which supports its continued use and potentially guides dosage and treatment duration.

Standard Error Calculation

Importance of Calculating Standard Error

The standard error (SE) measures the precision of an estimated proportion. It is essential for statisticians and researchers to understand the variability expected in their estimates. In simpler terms, a smaller SE indicates more precision, whereas a larger SE suggests more variation and less confidence in the estimate.

For the Wilson-adjusted proportion, the SE is calculated to assess how much this adjusted proportion is likely to vary from the true proportion. It's a critical step in forming a valid confidence interval. The standard error is especially significant in medical studies like the one involving Cefotaxime, as it directly influences the reliability of the conclusion about the drug's effectiveness.