Confidence Intervals

Confidence Intervals

1. Definition & conceptual overview

• A confidence interval (CI) is a range of values, computed from sample data, within which the true (unknown) population parameter is expected to lie with a stated level of confidence — conventionally 95% (Swinscow 10e Ch.4, pp.40–1).
• The standard 95% CI is constructed as point estimate ± (1·96 × standard error) for large samples, exploiting the property that, for a Normally distributed estimate, 95% of the distribution lies within ±1·96 standard deviations of the mean (Swinscow 10e Ch.4, pp.40–1).
• The general form is estimate ± (critical value × standard error of the estimate); the critical value is the multiplier from the reference (sampling) distribution that cuts off the central 95% (or other chosen %) — 1·96 from the standard Normal distribution for large samples, or the appropriate Student's t value for small samples (Swinscow 10e Ch.4, p.41; Ch.7, p.63).
• The CI is an interval estimate of the parameter, in contrast to the single point estimate (the sample mean, proportion, difference, OR, etc.); it conveys the precision of that point estimate (Swinscow 10e Ch.4, p.42).
• Correct frequentist interpretation: if the sampling-and-estimation procedure were repeated many times, 95% of the CIs so constructed would contain the true population parameter — for any single computed interval we cannot know whether it is one of the 95% that captures the parameter or one of the 5% that misses it (Swinscow 10e Ch.4, p.42).
• The interval is bounded by confidence limits (the lower and upper values); the distance between them is the interval's width and indexes precision (Swinscow 10e Ch.4, p.41).
• The "100 investigators" intuition: of every 100 independently constructed 95% CIs, on average 95 will bracket the true parameter and 5 will not — but the individual investigator "doesn't know which ones" (Swinscow 10e Ch.4, p.42).