Skewness‐Corrected Confidence Intervals for Predictive Values in Enrichment Studies

The positive predictive value (PPV) and negative predictive value (NPV) can be expressed as functions of disease prevalence ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow> <mml:annotation>$$ \rho $$</mml:annotation></mml:semantics> </mml:math> ) and the ratios of two binomial proportions ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:annotation>$$ \phi $$</mml:annotation></mml:semantics> </mml:math> ), where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>ϕ</mml:mi> <mml:mi>ppv</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mfrac><mml:mrow><mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mtext>specificity</mml:mtext></mml:mrow> <mml:mtext>sensitivity</mml:mtext></mml:mfrac> </mml:mrow> <mml:annotation>$$ {\phi}_{ppv}=\frac{1- specificity}{sensitivity} $$</mml:annotation></mml:semantics> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msub><mml:mi>ϕ</mml:mi> <mml:mi>npv</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mfrac><mml:mrow><mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mtext>sensitivity</mml:mtext></mml:mrow> <mml:mtext>specificity</mml:mtext></mml:mfrac> </mml:mrow> <mml:annotation>$$ {\phi}_{npv}=\frac{1- sensitivity}{specificity} $$</mml:annotation></mml:semantics> </mml:math> . In prospective studies, where the proportion of subjects with the disease in the study cohort is an unbiased estimate of the disease prevalence, the confidence intervals (CIs) of PPV and NPV can be estimated using established methods for single proportion. However, in enrichment studies, such as case-control studies, where the proportion of diseased subjects significantly differs from disease prevalence, estimating CIs for PPV and NPV remains a challenge in terms of skewness and overall coverage, especially under extreme conditions (e.g., <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>NPV</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> <mml:annotation>$$ \mathrm{NPV}=1 $$</mml:annotation></mml:semantics> </mml:math> ). In this article, we extend the method adopted by Li, where CIs for PPV and NPV were derived from those of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:annotation>$$ \phi $$</mml:annotation></mml:semantics> </mml:math> . We explored additional CI methods for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow> <mml:annotation>$$ \phi $$</mml:annotation></mml:semantics> </mml:math> , including those by Gart & Nam (GN), MoverJ, and Walter and convert their corresponding CIs for PPV and NPV. Through simulations, we compared these methods with established CI methods, Fieller, Pepe, and Delta in terms of skewness and overall coverage. While no method proves universally optimal, GN and MoverJ methods generally emerge as recommended choices.