9.4.17 Confidence Intervals

Interpret and Explains Confidence Intervals For: Means, Proportions, Differences Between Means and Differences Between Proportions

Confidence intervals (CIs) are used to estimate the range of values within which the true value of a parameter, such as a mean or a proportion, is expected to lie with a certain level of confidence. CIs are essential in statistical inference and are often used in clinical research to estimate the precision of estimates.

Note: Do not memorize these equations, just be aware of them.

Interpreting CIs involves understanding the level of confidence, the sample size, and the width of the interval. A 95% CI means that if the study were repeated multiple times, the true value would be expected to lie within the CI in 95% of these repetitions. A wider interval indicates more uncertainty, whereas a narrower interval indicates more precision.

For means, CIs are calculated using the formula:

Where X̄ is the sample mean, s is the sample standard deviation, n is the sample size, and t is the t-statistic obtained from the t-distribution table at the desired level of confidence and degrees of freedom.

For proportions, CIs are calculated using the formula:

Where p is the sample proportion, n is the sample size, z is the z-score obtained from the standard normal distribution table at the desired level of confidence.

For the difference between means, CIs are calculated using the formula:

Where X̄1 and X̄2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

For the difference between proportions, CIs are calculated using the formula:

Where p1 and p2 are the sample proportions, and n1 and n2 are the sample sizes.

It is important to note that CIs can be used to determine whether the null hypothesis is rejected or not. If the CI for the difference between means or proportions includes zero, this suggests that the difference is not statistically significant and the null hypothesis cannot be rejected. If the CI does not include zero, this suggests that the difference is statistically significant and the null hypothesis can be rejected.

Clinical examples of CIs include estimating the mean blood pressure of a population, the proportion of patients with a certain disease, the difference in blood pressure between two treatment groups, or the difference in mortality rates between two surgical techniques.

A summary table of interpreting confidence intervals:

Type of Interval	Interpretation
Confidence Interval for Mean	We are X% confident that the true population mean falls within this interval.
Confidence Interval for Proportion	We are X% confident that the true population proportion falls within this interval.
Confidence Interval for Difference in Means	We are X% confident that the true difference in population means falls within this interval.
Confidence Interval for Difference in Proportions	We are X% confident that the true difference in population proportions falls within this interval.

Note: X represents the level of confidence (usually 95%).

References:

1. Altman, D. G., & Bland, J. M. (2005). Statistics notes: How to obtain the confidence interval from a P value. Bmj, 331(7521), 209-209.
2. Kirkwood, B. R., & Sterne, J. A. C. (2003). Essential medical statistics. Wiley.
3. Rosner, B. (2010). Fundamentals of biostatistics. Cengage Learning.