  Detection limits (DLs), where a variable is unable to be measured outside of
a certain range, are common in research. Most approaches to handle DLs in the
response variable implicitly make parametric assumptions on the distribution of
data outside DLs. We propose a new approach to deal with DLs based on a widely
used ordinal regression model, the cumulative probability model (CPM). The CPM
is a type of semiparametric linear transformation model. CPMs are rank-based
and can handle mixed distributions of continuous and discrete outcome
variables. These features are key for analyzing data with DLs because while
observations inside DLs are typically continuous, those outside DLs are
censored and generally put into discrete categories. With a single lower DL,
the CPM assigns values below the DL as having the lowest rank. When there are
multiple DLs, the CPM likelihood can be modified to appropriately distribute
probability mass. We demonstrate the use of CPMs with simulations and two HIV
data examples. The first example models a biomarker in which 15% of
observations are below a DL. The second uses multi-cohort data to model viral
load, where approximately 55% of observations are outside DLs which vary across
sites and over time.
