Exact interval estimation for three parameters subject to false positive misclassification

• Published in: Stat (International Statistical Institute) Vol. 13; no. 3
• Main Authors: Lu, Shuiyun, Wang, Weizhen, Xie, Tianfa
• Format: Journal Article
• Language: English
• Published: The Hague Wiley Subscription Services, Inc 01.09.2024
• Subjects: Binary data; coverage probability; Data analysis; double sampling; Intervals; Parameter estimation; the h$$ h $$‐function method; total interval length
• ISSN: 2049-1573; 2049-1573
• DOI: 10.1002/sta4.717

Summary Binary data subject to one type of misclassification exist in various fields. It is collected in a double‐sampling scheme that includes a gold standard test and a fallible test. The main parameter of interest for this type of data is the positive probability p$$ p $$ of the gold standard test. Existing intervals are unreliable because the given nominal level 1−α$$ 1-\alpha $$ is not achieved. In this paper, we construct an exact interval by inverting the E+M score tests and improve it by the general h$$ h $$‐function method. We find that the total length of the improved interval is shorter than the exact intervals that are also the improved intervals when we apply the h$$ h $$‐function to several existing approximate intervals, including the score and Bayesian intervals. Therefore, it is recommended for practice. We are also interested in two other parameters: p∗$$ {p}^{\ast } $$—the difference between the two positive rates for the fallible and gold standard tests—and ξ$$ \xi $$—the false positive rate for the fallible test. To the best of our knowledge, the research on these two parameters is limited. For p∗$$ {p}^{\ast } $$, we find that any interval for p$$ p $$ can be converted to an interval for p∗$$ {p}^{\ast } $$. So, the interval converted from the aforementioned recommended interval for p$$ p $$ is recommended for inferring p∗$$ {p}^{\ast } $$. For ξ$$ \xi $$, the improved interval by the h$$ h $$‐function method over the E+M score interval is derived. We use an example to illustrate how the intervals are computed and provide a real data analysis.