The Effect of Dependence on Confidence Intervals for a Population Proportion

Abstract

The binomial model is widely used in statistical applications. Usually, the success probability, p, and its associated con dence interval are estimated from a random sample. Thus, the observations are independent and identically distributed. Motivated by a legal case where some grand jurors could serve a second year, this article shows that when the observationsare dependent,even slightly, the coverage probabilities of the usual con dence intervals can deviate noticeably from their nominal level. Several modi ed con dence intervals that incorporate the dependence structure are proposed and examined. Our results show that the modi ed Wilson, Agresti-Coull, and Jeffreys con dence intervals perform well and can be recommended for general use.

Uncertainty and Inference for Verification Measures

“…Examination of the autocorrelation functions for the time series analyzed provides no evidence against this assumption. Miao and Gastwirth (2004) discuss intervals for the case where the observations are not independent.…”

Using the standard Wald confidence interval for a population proportion hypothesis test is a common mistake

“…It is worth noting that it has been known for some time that the standard Wald approximations for confidence intervals for a proportion does not have good properties until the sample size is very large, and there are better methods for estimating proportions; see, for example, other studies .…”

Testing Dependence Among Serially Correlated Multicategory Variables

“…In psychological or medical trials, repeated measurements of the condition of an individual give rise to serial dependencies in contingency table data (Conaway 1989) as do observations in sociological studies of individuals' decisions over time [e.g., judges' decisions categorized against the type of case and circuit (Sunstein, Schkade, Ellman, and Sawicki 2006) or selection of jurors serving on grand juries (Miao and Gastwirth 2004)]. In meteorology, forecast accuracy for categorized weather variables is routinely investigated using contingency tables (Katz and Murphy 1997;Stephenson 2000), although these variables are often highly serially correlated.…”