Notes

Source paper citation: Tian, Y., Li, C., Tu, S., James, N. T., Harrell, FrankE., & Shepherd, BryanE. (2024). Addressing Multiple Detection Limits with Semiparametric Cumulative Probability Models. Journal of the American Statistical Association, 119(546), 864–874. https://doi.org/10.1080/01621459.2024.2315667

• Main idea: a new approach to deal with multiple Detection Limits (DLs) based on a widely used ordinal regression model, the cumulative probability model (CPM)

• Feature of CPM

  • rank-based: easy to add label to censored data below or above DLs, without specifying value and distributions beyond DLs

  • semiparametric: only assumption is the link function $G$: $G[F(y

    X)]=\alpha(y)-\beta’X$

  • able to handle mixed distributions of continues and discrete outcome variables, through selecting proper anchor points

• Cumulative Probability Models (CPM)

  Outcome is modeled as $Y=H(\beta’X+\epsilon)$, where $H$ is monotonically increasing transformation, $X$ is vector of covariates, $\epsilon$ follows a known distribution with CDF $F_\epsilon$.

  Let $G=F_\epsilon^{-1}$, $\alpha=H^{-1}$, then \(G[F(y|X)]=\alpha(y)-\beta'X\)

  which is CPM.

• Nonparametric likelihood

  Data: ${(y_i,x_i): i=1,…,n}$

  Nonparametric likelihood is \(\prod_{i=1}^{n}[F(y_i|x_i)-F(y_i^-|x_i)]\) where $F(y_i^-|x_i)=\lim_{t\rightarrow y_i^{-}F(t|x_i)}$.

  Using the right continuity of CDF, the nonparametric likelihood can be derived by replacing $y_i^-$ in $F(y_i^-

  x_i)$ with the next smallest anchor point.