diff --git a/R/biokinetics.R b/R/biokinetics.R
index c6711b2..d087f02 100644
--- a/R/biokinetics.R
+++ b/R/biokinetics.R
@@ -11,8 +11,8 @@ biokinetics <- R6::R6Class(
     scale = NULL,
     priors = NULL,
     preds_sd = NULL,
-    upper_detection_limit = NULL,
-    lower_detection_limit = NULL,
+    upper_censoring_limit = NULL,
+    lower_censoring_limit = NULL,
     smallest_value = NULL,
     data = NULL,
     covariate_formula = NULL,
@@ -29,28 +29,18 @@ biokinetics <- R6::R6Class(
         stop("Model has not been fitted yet. Call 'fit' before calling this function.")
       }
     },
-    get_upper_detection_limit = function(upper_detection_limit) {
-      if (is.null(upper_detection_limit)) {
-        private$upper_detection_limit <- max(private$data$value)
+    get_upper_censoring_limit = function(upper_censoring_limit) {
+      if (is.null(upper_censoring_limit)) {
+        private$upper_censoring_limit <- max(private$data$value)
       } else {
-        max_value <- max(private$data$value)
-        if (max_value >= upper_detection_limit) {
-          warning(sprintf("Data contains values >= the upper detection limit %s. These will be censored.",
-                          upper_detection_limit))
-        }
-        private$upper_detection_limit <- upper_detection_limit
+        private$upper_censoring_limit <- upper_censoring_limit
       }
     },
-    get_lower_detection_limit = function(lower_detection_limit) {
-      if (is.null(lower_detection_limit)) {
-        private$lower_detection_limit <- min(private$data$value)
+    get_lower_censoring_limit = function(lower_censoring_limit) {
+      if (is.null(lower_censoring_limit)) {
+        private$lower_censoring_limit <- min(private$data$value)
       } else {
-        min_value <- min(private$data$value)
-        if (min_value <= lower_detection_limit) {
-          warning(sprintf("Data contains values <= the lower detection limit %s. These will be censored.",
-                          lower_detection_limit))
-        }
-        private$lower_detection_limit <- lower_detection_limit
+        private$lower_censoring_limit <- lower_censoring_limit
       }
     },
     model_matrix_with_dummy = function(data) {
@@ -204,11 +194,11 @@ biokinetics <- R6::R6Class(
       stan_data$P <- ncol(private$design_matrix)
       if (private$scale == "natural") {
         # do the same transformation as used on the data
-        stan_data$upper_detection_limit <- log2(private$upper_detection_limit / private$smallest_value)
-        stan_data$lower_detection_limit <- log2(private$lower_detection_limit / private$smallest_value)
+        stan_data$upper_censoring_limit <- log2(private$upper_censoring_limit / private$smallest_value)
+        stan_data$lower_censoring_limit <- log2(private$lower_censoring_limit / private$smallest_value)
       } else {
-        stan_data$upper_detection_limit <- private$upper_detection_limit
-        stan_data$lower_detection_limit <- private$lower_detection_limit
+        stan_data$upper_censoring_limit <- private$upper_censoring_limit
+        stan_data$lower_censoring_limit <- private$lower_censoring_limit
       }
       private$stan_input_data <- c(stan_data, private$priors)
     },
@@ -248,24 +238,24 @@ biokinetics <- R6::R6Class(
     #' @param preds_sd Standard deviation of predictor coefficients. Default 0.25.
     #' @param scale One of "log" or "natural". Default "natural". Is provided data on a log or a natural scale? If on a natural scale it
     #' will be converted to a log scale for model fitting.
-    #' @param upper_detection_limit Optional upper detection limit of the titre used. This is needed to construct a likelihood for upper censored
-    #' values, so only needs to be provided if you have such values in the dataset. If not provided, the model will default
-    #' to using the largest value in the dataset as the upper detection limit.
-    #' @param lower_detection_limit Optional lower detection limit of the titre used. This is needed to construct a likelihood for lower censored
-    #' values, so only needs to be provided if you have such values in the dataset. If not provided, the model will default
-    #' to using the smallest value in the dataset as the lower detection limit.
-    #' @param truncate_upper Logical. Whether values greater than the upper detection limit should be truncated, i.e. set to the same value as the limit. Default TRUE.
-    #' @param truncate_lower Logical. Whether values smaller than the lower detection limit should be truncated, i.e. set to the same value as the limit. Default TRUE.
+    #' @param upper_censoring_limit Optional value at which to upper censor data points. This is needed to construct a likelihood for upper censored
+    #' values, so only needs to be provided if you have such values in the dataset. If not provided, no censoring will be done.
+    #' @param lower_censoring_limit Optional value at which to lower censor data points. This is needed to construct a likelihood for lower censored
+    #' values, so only needs to be provided if you have such values in the dataset. If not provided, no censoring will be done.
+    #' @param strict_upper_limit Logical. Whether values greater than the upper censoring limit should be censored.
+    #' If FALSE, only values exactly equal to the upper censoring limit will be censored. Default TRUE.
+    #' @param strict_lower_limit Logical. Whether values smaller than the lower censoring limit should be censored.
+    #' If FALSE, only values exactly equal to the lower censoring limit will be censored. Default TRUE.
     initialize = function(priors = biokinetics_priors(),
                           data = NULL,
                           file_path = NULL,
                           covariate_formula = ~0,
                           preds_sd = 0.25,
                           scale = "natural",
-                          upper_detection_limit = NULL,
-                          lower_detection_limit = NULL,
-                          truncate_upper = TRUE,
-                          truncate_lower = TRUE) {
+                          upper_censoring_limit = NULL,
+                          lower_censoring_limit = NULL,
+                          strict_upper_limit = TRUE,
+                          strict_lower_limit = TRUE) {
       validate_priors(priors)
       private$priors <- priors
       validate_numeric(preds_sd)
@@ -291,18 +281,36 @@ biokinetics <- R6::R6Class(
       validate_required_cols(private$data)
       validate_formula_vars(private$all_formula_vars, private$data)
       logger::log_info("Preparing data for stan")
-      private$get_upper_detection_limit(upper_detection_limit)
-      private$get_lower_detection_limit(lower_detection_limit)
-      if (truncate_upper) {
-        private$data[, value := ifelse(value > private$upper_detection_limit, private$upper_detection_limit, value)]
-      }
-      if (truncate_lower) {
-        private$data[, value := ifelse(value < private$lower_detection_limit, private$lower_detection_limit, value)]
+      private$get_upper_censoring_limit(upper_censoring_limit)
+      private$get_lower_censoring_limit(lower_censoring_limit)
+      max_value <- max(private$data$value)
+      min_value <- min(private$data$value)
+      values_above <- max_value > private$upper_censoring_limit
+      values_below <- min_value < private$lower_censoring_limit
+      if (strict_upper_limit) {
+        if (values_above) {
+          warning(sprintf("Data contains values above the upper censoring limit %s and these will be censored. To turn off this behaviour set strict_upper_limit to FALSE.",
+                          private$upper_censoring_limit))
+        }
+        private$data[, value := ifelse(value > private$upper_censoring_limit, private$upper_censoring_limit, value)]
+      } else if (values_above) {
+        warning(sprintf("Data contains values above the upper censoring limit %s. To treat these as censored set strict_upper_limit to TRUE.",
+                        private$upper_censoring_limit))
+      }
+      if (strict_lower_limit) {
+        if (values_below) {
+          warning(sprintf("Data contains values below the lower censoring limit %s and these will be censored. To turn off this behaviour set strict_lower_limit to FALSE.",
+                          private$lower_censoring_limit))
+        }
+        private$data[, value := ifelse(value < private$lower_censoring_limit, private$lower_censoring_limit, value)]
+      } else if (values_below) {
+        warning(sprintf("Data contains values below the lower censoring limit %s. To treat these as censored set strict_lower_limit to TRUE.",
+                        private$lower_censoring_limit))
       }
       private$data[, `:=`(obs_id = seq_len(.N),
                           time_since_last_exp = as.integer(day - last_exp_day, units = "days"),
-                          censored_lo = value == private$lower_detection_limit,
-                          censored_hi = value == private$upper_detection_limit)]
+                          censored_lo = value == private$lower_censoring_limit,
+                          censored_hi = value == private$upper_censoring_limit)]
       private$construct_design_matrix()
       private$build_covariate_lookup_table()
       private$build_pid_lookup()
@@ -332,8 +340,8 @@ biokinetics <- R6::R6Class(
            tmax = tmax,
            n_draws = n_draws,
            data = private$data,
-           upper_detection_limit = private$stan_input_data$upper_detection_limit,
-           lower_detection_limit = private$stan_input_data$lower_detection_limit)
+           upper_censoring_limit = private$stan_input_data$upper_censoring_limit,
+           lower_censoring_limit = private$stan_input_data$lower_censoring_limit)
     },
     #' @description Plot model input data with a smoothing function. Note that
     #' this plot is of the data as provided to the Stan model so is on a log scale,
@@ -344,32 +352,32 @@ biokinetics <- R6::R6Class(
       plot_sero_data(private$data,
                      tmax = tmax,
                      covariates = private$all_formula_vars,
-                     upper_detection_limit = private$stan_input_data$upper_detection_limit,
-                     lower_detection_limit = private$stan_input_data$lower_detection_limit)
+                     upper_censoring_limit = private$stan_input_data$upper_censoring_limit,
+                     lower_censoring_limit = private$stan_input_data$lower_censoring_limit)
     },
-    #' @description View the data that is passed to the stan model, for debugging purposes.
-    #' @return A list of arguments that will be passed to the stan model.
+        #' @description View the data that is passed to the stan model, for debugging purposes.
+        #' @return A list of arguments that will be passed to the stan model.
     get_stan_data = function() {
       private$stan_input_data
     },
-    #' @description View the mapping of human readable covariate names to the model variable p.
-    #' @return A data.table mapping the model variable p to human readable covariates.
+        #' @description View the mapping of human readable covariate names to the model variable p.
+        #' @return A data.table mapping the model variable p to human readable covariates.
     get_covariate_lookup_table = function() {
       private$covariate_lookup_table
     },
-    #' @description Fit the model and return CmdStanMCMC fitted model object.
-    #' @return A CmdStanMCMC fitted model object: <https://mc-stan.org/cmdstanr/reference/CmdStanMCMC.html>
-    #' @param ... Named arguments to the `sample()` method of CmdStan model.
-    #'   objects: <https://mc-stan.org/cmdstanr/reference/model-method-sample.html>
+        #' @description Fit the model and return CmdStanMCMC fitted model object.
+        #' @return A CmdStanMCMC fitted model object: <https://mc-stan.org/cmdstanr/reference/CmdStanMCMC.html>
+        #' @param ... Named arguments to the `sample()` method of CmdStan model.
+        #'   objects: <https://mc-stan.org/cmdstanr/reference/model-method-sample.html>
     fit = function(...) {
       logger::log_info("Fitting model")
       private$fitted <- private$model$sample(private$stan_input_data, ...)
       private$fitted
     },
-    #' @description Extract fitted population parameters
-    #' @return A data.table
-    #' @param n_draws Integer. Default 2000.
-    #' @param human_readable_covariates Logical. Default TRUE.
+        #' @description Extract fitted population parameters
+        #' @return A data.table
+        #' @param n_draws Integer. Default 2000.
+        #' @param human_readable_covariates Logical. Default TRUE.
     extract_population_parameters = function(n_draws = 2000,
                                              human_readable_covariates = TRUE) {
       private$check_fitted()
@@ -489,8 +497,8 @@ biokinetics <- R6::R6Class(
       attr(dt_out, "summarised") <- summarise
       attr(dt_out, "scale") <- private$scale
       attr(dt_out, "covariates") <- private$all_formula_vars
-      attr(dt_out, "upper_detection_limit") <- private$upper_detection_limit
-      attr(dt_out, "lower_detection_limit") <- private$lower_detection_limit
+      attr(dt_out, "upper_censoring_limit") <- private$upper_censoring_limit
+      attr(dt_out, "lower_censoring_limit") <- private$lower_censoring_limit
 
       dt_out
     },
diff --git a/R/plot.R b/R/plot.R
index 56346eb..cb49ca8 100644
--- a/R/plot.R
+++ b/R/plot.R
@@ -9,15 +9,15 @@
 #' @param tmax Integer. The number of time points in each simulated trajectory. Default 150.
 #' @param n_draws Integer. The number of trajectories to simulate. Default 2000.
 #' @param data Optional data.frame with columns time_since_last_exp and value. The raw data to compare to.
-#' @param upper_detection_limit Optional upper detection limit.
-#' @param lower_detection_limit Optional lower detection limit.
+#' @param upper_censoring_limit Optional upper detection limit.
+#' @param lower_censoring_limit Optional lower detection limit.
 plot.biokinetics_priors <- function(x,
                                     ...,
                                     tmax = 150,
                                     n_draws = 2000,
                                     data = NULL,
-                                    upper_detection_limit = NULL,
-                                    lower_detection_limit = NULL) {
+                                    upper_censoring_limit = NULL,
+                                    lower_censoring_limit = NULL) {
 
   # Declare variables to suppress notes when compiling package
   # https://github.com/Rdatatable/data.table/issues/850#issuecomment-259466153
@@ -57,7 +57,7 @@ plot.biokinetics_priors <- function(x,
                  aes(x = time_since_last_exp,
                      y = value))
 
-    plot <- add_limits(plot, upper_detection_limit, lower_detection_limit)
+    plot <- add_limits(plot, upper_censoring_limit, lower_censoring_limit)
   }
   plot
 }
@@ -70,13 +70,13 @@ plot.biokinetics_priors <- function(x,
 #' @param data A data.table with required columns time_since_last_exp, value and titre_type.
 #' @param tmax Integer. The number of time points in each simulated trajectory. Default 150.
 #' @param covariates Optional vector of covariate names to facet by (these must correspond to columns in the data.table)
-#' @param upper_detection_limit Optional upper detection limit.
-#' @param lower_detection_limit Optional lower detection limit.
+#' @param upper_censoring_limit Optional upper detection limit.
+#' @param lower_censoring_limit Optional lower detection limit.
 plot_sero_data <- function(data,
                            tmax = 150,
                            covariates = character(0),
-                           upper_detection_limit = NULL,
-                           lower_detection_limit = NULL) {
+                           upper_censoring_limit = NULL,
+                           lower_censoring_limit = NULL) {
   validate_required_cols(data, c("time_since_last_exp", "value", "titre_type"))
   data <- data[time_since_last_exp <= tmax,]
   # Declare variables to suppress notes when compiling package
@@ -90,7 +90,7 @@ plot_sero_data <- function(data,
     facet_wrap(eval(parse(text = facet_formula(covariates)))) +
     guides(colour = guide_legend(title = "Titre type"))
 
-  add_limits(plot, upper_detection_limit, lower_detection_limit)
+  add_limits(plot, upper_censoring_limit, lower_censoring_limit)
 }
 
 #' Plot method for "biokinetics_population_trajectories" class
@@ -104,8 +104,8 @@ plot_sero_data <- function(data,
 plot.biokinetics_population_trajectories <- function(x, ...,
                                                      data = NULL) {
   covariates <- attr(x, "covariates")
-  upper_detection_limit <- attr(x, "upper_detection_limit")
-  lower_detection_limit <- attr(x, "lower_detection_limit")
+  upper_censoring_limit <- attr(x, "upper_censoring_limit")
+  lower_censoring_limit <- attr(x, "lower_censoring_limit")
 
   # Declare variables to suppress notes when compiling package
   # https://github.com/Rdatatable/data.table/issues/850#issuecomment-259466153
@@ -139,7 +139,7 @@ plot.biokinetics_population_trajectories <- function(x, ...,
     guides(fill = guide_legend(title = "Titre type"),
            colour = "none")
   if (!is.null(data)) {
-    plot <- add_limits(plot, upper_detection_limit, lower_detection_limit)
+    plot <- add_limits(plot, upper_censoring_limit, lower_censoring_limit)
   }
   plot
 }
@@ -148,24 +148,24 @@ facet_formula <- function(covariates) {
   paste("~", paste(c("titre_type", covariates), collapse = "+"))
 }
 
-add_limits <- function(plot, upper_detection_limit, lower_detection_limit) {
-  if (!is.null(lower_detection_limit)) {
+add_limits <- function(plot, upper_censoring_limit, lower_censoring_limit) {
+  if (!is.null(lower_censoring_limit)) {
     plot <- plot +
-      geom_hline(yintercept = lower_detection_limit,
+      geom_hline(yintercept = lower_censoring_limit,
                  linetype = 'dotted') +
       annotate("text", x = 1,
-               y = lower_detection_limit,
+               y = lower_censoring_limit,
                label = "Lower detection limit",
                vjust = -0.5,
                hjust = 0,
                size = 3)
   }
-  if (!is.null(upper_detection_limit)) {
+  if (!is.null(upper_censoring_limit)) {
     plot <- plot +
-      geom_hline(yintercept = upper_detection_limit,
+      geom_hline(yintercept = upper_censoring_limit,
                  linetype = 'dotted') +
       annotate("text", x = 1,
-               y = upper_detection_limit,
+               y = upper_censoring_limit,
                label = "Upper detection limit",
                vjust = -0.5,
                hjust = 0,
diff --git a/man/biokinetics.Rd b/man/biokinetics.Rd
index 7e3b79f..7d6a65c 100644
--- a/man/biokinetics.Rd
+++ b/man/biokinetics.Rd
@@ -36,10 +36,10 @@ Initialise the kinetics model.
   covariate_formula = ~0,
   preds_sd = 0.25,
   scale = "natural",
-  upper_detection_limit = NULL,
-  lower_detection_limit = NULL,
-  truncate_upper = TRUE,
-  truncate_lower = TRUE
+  upper_censoring_limit = NULL,
+  lower_censoring_limit = NULL,
+  strict_upper_limit = TRUE,
+  strict_lower_limit = TRUE
 )}\if{html}{\out{</div>}}
 }
 
@@ -61,17 +61,17 @@ will be treated as categorical variables. Default ~0.}
 \item{\code{scale}}{One of "log" or "natural". Default "natural". Is provided data on a log or a natural scale? If on a natural scale it
 will be converted to a log scale for model fitting.}
 
-\item{\code{upper_detection_limit}}{Optional upper detection limit of the titre used. This is needed to construct a likelihood for upper censored
-values, so only needs to be provided if you have such values in the dataset. If not provided, the model will default
-to using the largest value in the dataset as the upper detection limit.}
+\item{\code{upper_censoring_limit}}{Optional value at which to upper censor data points. This is needed to construct a likelihood for upper censored
+values, so only needs to be provided if you have such values in the dataset. If not provided, no censoring will be done.}
 
-\item{\code{lower_detection_limit}}{Optional lower detection limit of the titre used. This is needed to construct a likelihood for lower censored
-values, so only needs to be provided if you have such values in the dataset. If not provided, the model will default
-to using the smallest value in the dataset as the lower detection limit.}
+\item{\code{lower_censoring_limit}}{Optional value at which to lower censor data points. This is needed to construct a likelihood for lower censored
+values, so only needs to be provided if you have such values in the dataset. If not provided, no censoring will be done.}
 
-\item{\code{truncate_upper}}{Logical. Whether values greater than the upper detection limit should be truncated, i.e. set to the same value as the limit. Default TRUE.}
+\item{\code{strict_upper_limit}}{Logical. Whether values greater than the upper censoring limit should be censored.
+If FALSE, only values exactly equal to the upper censoring limit will be censored. Default TRUE.}
 
-\item{\code{truncate_lower}}{Logical. Whether values smaller than the lower detection limit should be truncated, i.e. set to the same value as the limit. Default TRUE.}
+\item{\code{strict_lower_limit}}{Logical. Whether values smaller than the lower censoring limit should be censored.
+If FALSE, only values exactly equal to the lower censoring limit will be censored. Default TRUE.}
 }
 \if{html}{\out{</div>}}
 }
diff --git a/man/plot.biokinetics_priors.Rd b/man/plot.biokinetics_priors.Rd
index 56cb964..4db8eba 100644
--- a/man/plot.biokinetics_priors.Rd
+++ b/man/plot.biokinetics_priors.Rd
@@ -11,8 +11,8 @@ and optionally compare to a dataset.}
   tmax = 150,
   n_draws = 2000,
   data = NULL,
-  upper_detection_limit = NULL,
-  lower_detection_limit = NULL
+  upper_censoring_limit = NULL,
+  lower_censoring_limit = NULL
 )
 }
 \arguments{
@@ -26,9 +26,9 @@ and optionally compare to a dataset.}
 
 \item{data}{Optional data.frame with columns time_since_last_exp and value. The raw data to compare to.}
 
-\item{upper_detection_limit}{Optional upper detection limit.}
+\item{upper_censoring_limit}{Optional upper detection limit.}
 
-\item{lower_detection_limit}{Optional lower detection limit.}
+\item{lower_censoring_limit}{Optional lower detection limit.}
 }
 \value{
 A ggplot2 object.
diff --git a/man/plot_sero_data.Rd b/man/plot_sero_data.Rd
index ec83395..6986bc0 100644
--- a/man/plot_sero_data.Rd
+++ b/man/plot_sero_data.Rd
@@ -8,8 +8,8 @@ plot_sero_data(
   data,
   tmax = 150,
   covariates = character(0),
-  upper_detection_limit = NULL,
-  lower_detection_limit = NULL
+  upper_censoring_limit = NULL,
+  lower_censoring_limit = NULL
 )
 }
 \arguments{
@@ -19,9 +19,9 @@ plot_sero_data(
 
 \item{covariates}{Optional vector of covariate names to facet by (these must correspond to columns in the data.table)}
 
-\item{upper_detection_limit}{Optional upper detection limit.}
+\item{upper_censoring_limit}{Optional upper detection limit.}
 
-\item{lower_detection_limit}{Optional lower detection limit.}
+\item{lower_censoring_limit}{Optional lower detection limit.}
 }
 \value{
 A ggplot2 object.
diff --git a/src/stan/antibody_kinetics_main.stan b/src/stan/antibody_kinetics_main.stan
index 48b1e4e..2d6cb38 100644
--- a/src/stan/antibody_kinetics_main.stan
+++ b/src/stan/antibody_kinetics_main.stan
@@ -29,8 +29,8 @@ data {
   int N; // Number of observations
   int N_events; // Number of exposure events
   int K; // Number of titre types
-  real upper_detection_limit; // Upper detection limit
-  real lower_detection_limit; // Lower detection limit
+  real upper_censoring_limit; // Upper detection limit
+  real lower_censoring_limit; // Lower detection limit
   array[N] int<lower=1, upper=K> titre_type; // Titre type for each observation
   vector[N] t; // Time for each observation
   vector[N] value; // Observed titre values
@@ -148,10 +148,10 @@ model {
   value[uncens_idx] ~ normal(mu[uncens_idx], sigma);
 
   // Likelihood for observations at lower limit
-  target += normal_lcdf(lower_detection_limit | mu[cens_lo_idx], sigma);
+  target += normal_lcdf(lower_censoring_limit | mu[cens_lo_idx], sigma);
 
   // Censoring at upper limit
-  target += normal_lccdf(upper_detection_limit | mu[cens_hi_idx], sigma);
+  target += normal_lccdf(upper_censoring_limit | mu[cens_hi_idx], sigma);
   
   // Covariate-level mean priors, parameterised from previous studies
   t0_pop ~ normal(mu_t0, sigma_t0);
diff --git a/tests/testthat/test-data.R b/tests/testthat/test-data.R
index 2a7f005..4878b00 100644
--- a/tests/testthat/test-data.R
+++ b/tests/testthat/test-data.R
@@ -19,7 +19,7 @@ test_that("Can construct stan data", {
   expect_equal(names(stan_data), c("N", "N_events", "id", "value",
                                    "titre_type", "preds_sd", "K", "N_uncens", "N_lo",
                                    "N_hi", "uncens_idx", "cens_lo_idx",
-                                   "cens_hi_idx", "t", "X", "P", "upper_detection_limit", "lower_detection_limit", "mu_t0",
+                                   "cens_hi_idx", "t", "X", "P", "upper_censoring_limit", "lower_censoring_limit", "mu_t0",
                                    "mu_tp", "mu_ts", "mu_m1", "mu_m2", "mu_m3",
                                    "sigma_t0", "sigma_tp", "sigma_ts", "sigma_m1", "sigma_m2",
                                    "sigma_m3"))
@@ -32,8 +32,8 @@ test_that("Can construct stan data", {
 test_that("Data above/below limits is censored", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
   suppressWarnings({ mod <- biokinetics$new(data = dat,
-                                            lower_detection_limit = 10,
-                                            upper_detection_limit = 2500) })
+                                            lower_censoring_limit = 10,
+                                            upper_censoring_limit = 2500) })
   stan_data <- mod$get_stan_data()
   expect_equal(stan_data$N_uncens, dat[value > 10 & value < 2500, .N])
   expect_equal(stan_data$N_lo, dat[value <= 10, .N])
@@ -72,47 +72,68 @@ test_that("Log scale data is passed directly to stan", {
 
 test_that("Highest value is used as default upper limit", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
-  mod <- biokinetics$new(data = dat, lower_detection_limit = 2)
+  mod <- biokinetics$new(data = dat, lower_censoring_limit = 2)
   stan_data <- mod$get_stan_data()
-  expect_equal(stan_data$upper_detection_limit, log2(max(dat$value) / min(dat$value)))
-  expect_equal(stan_data$lower_detection_limit, log2(2 / min(dat$value)))
+  expect_equal(stan_data$upper_censoring_limit, log2(max(dat$value) / min(dat$value)))
+  expect_equal(stan_data$lower_censoring_limit, log2(2 / min(dat$value)))
 })
 
-test_that("Warns if data contains values above the upper limit", {
+test_that("Warns if data contains values above the upper limit and limit is strict", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
-  expect_warning({ mod <- biokinetics$new(data = dat, upper_detection_limit = 10) },
-                 "Data contains values >= the upper detection limit 10. These will be censored.")
+  expect_warning({ mod <- biokinetics$new(data = dat, upper_censoring_limit = 10) },
+                 "Data contains values above the upper censoring limit 10 and these will be censored. To turn off this behaviour set strict_upper_limit to FALSE.")
   stan_data <- mod$get_stan_data()
-  expect_equal(stan_data$upper_detection_limit, log2(10 / min(dat$value)))
-  expect_equal(stan_data$lower_detection_limit, 0)
+  expect_equal(stan_data$upper_censoring_limit, log2(10 / min(dat$value)))
+  expect_equal(stan_data$lower_censoring_limit, 0)
+  expect_true(all(stan_data$value <= stan_data$upper_censoring_limit))
+})
+
+test_that("Warns if data contains values above the upper limit and limit is not strict", {
+  dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
+  expect_warning({ mod <- biokinetics$new(data = dat, upper_censoring_limit = 10, strict_upper_limit = FALSE) },
+                 "Data contains values above the upper censoring limit 10. To treat these as censored set strict_upper_limit to TRUE.")
+  stan_data <- mod$get_stan_data()
+  expect_equal(stan_data$upper_censoring_limit, log2(10 / min(dat$value)))
+  expect_equal(stan_data$lower_censoring_limit, 0)
+  expect_false(all(stan_data$value <= stan_data$upper_censoring_limit))
 })
 
 test_that("Smallest value is used as default lower limit", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
-  mod <- biokinetics$new(data = dat, upper_detection_limit = 3000)
+  mod <- biokinetics$new(data = dat, upper_censoring_limit = 3000)
   stan_data <- mod$get_stan_data()
-  expect_equal(stan_data$upper_detection_limit, log2(3000 / min(dat$value)))
-  expect_equal(stan_data$lower_detection_limit, 0)
+  expect_equal(stan_data$upper_censoring_limit, log2(3000 / min(dat$value)))
+  expect_equal(stan_data$lower_censoring_limit, 0)
 })
 
 test_that("Warns if data contains values below the lower limit", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
   expect_warning({ mod <- biokinetics$new(data = dat,
-                                          lower_detection_limit = 10) },
-    "Data contains values <= the lower detection limit 10. These will be censored.")
+                                          lower_censoring_limit = 10) },
+    "Data contains values below the lower censoring limit 10 and these will be censored. To turn off this behaviour set strict_lower_limit to FALSE.")
+  stan_data <- mod$get_stan_data()
+  expect_equal(stan_data$lower_censoring_limit, log2(10/min(dat$value)))
+  expect_true(all(stan_data$value >= stan_data$lower_censoring_limit))
+})
+
+test_that("Warns if data contains values below the lower limit and limit is not strict", {
+  dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
+  expect_warning({ mod <- biokinetics$new(data = dat, lower_censoring_limit = 10, strict_lower_limit = FALSE) },
+                 "Data contains values below the lower censoring limit 10. To treat these as censored set strict_lower_limit to TRUE.")
   stan_data <- mod$get_stan_data()
-  expect_equal(stan_data$lower_detection_limit, log2(10/min(dat$value)))
+  expect_equal(stan_data$lower_censoring_limit, log2(10/min(dat$value)))
+  expect_false(all(stan_data$value <= stan_data$lower_censoring_limit))
 })
 
-test_that("Detection limits are passed to stan without transformation if using log data", {
+test_that("Censoring limits are passed to stan without transformation if using log data", {
   dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
   mod <- biokinetics$new(data = convert_log2_scale(dat,
                                                    smallest_value = 2,
                                                    vars_to_transform = "value"),
                          scale = "log",
-                         lower_detection_limit = 1,
-                         upper_detection_limit = 15)
+                         lower_censoring_limit = 1,
+                         upper_censoring_limit = 15)
   stan_data <- mod$get_stan_data()
-  expect_equal(stan_data$upper_detection_limit, 15)
-  expect_equal(stan_data$lower_detection_limit, 1)
+  expect_equal(stan_data$upper_censoring_limit, 15)
+  expect_equal(stan_data$lower_censoring_limit, 1)
 })
diff --git a/tests/testthat/test-plots.R b/tests/testthat/test-plots.R
index ec00126..1808e51 100644
--- a/tests/testthat/test-plots.R
+++ b/tests/testthat/test-plots.R
@@ -42,15 +42,15 @@ test_that("Prior predictions from model are the same", {
 test_that("Can plot input data", {
   data <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
   mod <- biokinetics$new(data = data,
-                         lower_detection_limit = 40,
-                         truncate_lower = FALSE)
+                         lower_censoring_limit = 40,
+                         strict_lower_limit = FALSE)
   suppressWarnings({plot <- mod$plot_model_inputs()})
   vdiffr::expect_doppelganger("inputdata", plot)
 
   mod <- biokinetics$new(data = data,
-                         lower_detection_limit = 40,
+                         lower_censoring_limit = 40,
                          covariate_formula = ~0 + infection_history,
-                         truncate_lower = FALSE)
+                         strict_lower_limit = FALSE)
   suppressWarnings({plot <- mod$plot_model_inputs()})
   vdiffr::expect_doppelganger("inputdata_covariates", plot)
 })
diff --git a/vignettes/biokinetics.Rmd b/vignettes/biokinetics.Rmd
index c97a926..9d735f5 100644
--- a/vignettes/biokinetics.Rmd
+++ b/vignettes/biokinetics.Rmd
@@ -28,7 +28,7 @@ The `fit` method then has the same function signature as the underlying [cmdstan
 ```{r, warning=FALSE, message=FALSE, results="hide"}
 dat <- data.table::fread(system.file("delta_full.rds", package = "epikinetics"))
 mod <- epikinetics::biokinetics$new(data = dat,
-                                    lower_detection_limit = 40,
+                                    lower_censoring_limit = 40,
                                     covariate_formula = ~0 + infection_history)
 delta <- mod$fit(parallel_chains = 4,
                  iter_warmup = 50,