multipleDL {multipleDL}

CPMs for multiple detection limits

Description

This function build the CPM for multiple detection limits (DLs).

Usage

multipleDL(formula, data, delta_lower = NULL, delta_upper = NULL, link)

Arguments

formula

an R formula object

data

a data frame including response data and covariates

delta_lower

(optional) indicators of lower DLs censoring (1: observed; 0:censored). If not specified, treat as observed.

delta_upper

(optional) indicators of upper DLs censoring(1: observed; 0:censored). If not specified, treat as observed.

link

the link function (probit, logit, loglog, cloglog)

Details

When there are multiple DLs, we appropriately modify the CPM likelihood. If a value is below a lower DL, set the censored value as the lower DL and set the lower DL indicator delta_lower to be 0. Similarly, if a value is above an upper DL, set the censored value as the upper DL and set the upper DL indicator delta_upper to be 0. This function also works when there is only a single lower and/or upper DL.

Conditional quantiles and CDFs and corresponding 95% confidence intervals can be calculated from the model fit.

Value

A list containing the following components:

coef

a numeric vector of estimated coeffiencts

var

covariance matrix of estimated coeffiencts

yunique

a numeric vector of unique response values

kint

number of alphas (intercept terms)

p

number of betas (regression coeffiencts)

fam

a list of functions associated with the specified link function

x

the design matrix

log_likelihood

the log-likelihood

References

Tian, Y., Li, C., Tu, S., James, N. T., Harrell, F. E., & Shepherd, B. E. (2022). Addressing Detection Limits with Semiparametric Cumulative Probability Models. arXiv preprint arXiv:2207.02815.

Stan Development Team (2020). RSroxygen2::roxygenize()tan: the R interface to Stan. R package version 2.19.3. https://mc-stan.org

Harrell, F. (2020). rms: Regression modeling strategies. R package version 6.1.0. https://CRAN.R-project.org/package=rms

See Also

cdf_dl, quantile_dl

Examples

## Multiple DLs
## generate a small example data: 3 sites with different lower and upper DLs
## lower DLs: site 1: - 0.2; site 2: 0.3; site 3: no lower DL
## upper DLs: site 1: no upper DL; site 2: 4; site 3: 3.5
## each site includes 100 subjects
n <- 100
x <- rnorm(n * 3)
e <- rnorm(n * 3)
y <- exp(x + e)
no_dl <- 1e6
data <- data.frame(y = y, x = x, subset = rep(c(1, 2, 3), each=n))
data$dl_l <- ifelse(data$subset == 1, 0.2, ifelse(data$subset == 2, 0.3, -no_dl))
data$dl_u <- ifelse(data$subset == 1, no_dl, ifelse(data$subset == 2, 4, 3.5))
data$delta_l <- ifelse(data$y >= data$dl_l, 1, 0)
data$delta_u <- ifelse(data$y <= data$dl_u, 1, 0)
data$z <- ifelse(data$delta_l == 0, data$dl_l, ifelse(data$delta_u == 0, data$dl_u, data$y))
# model
mod <- multipleDL(formula = z ~ x, data = data,
                 delta_lower = data$delta_l, delta_upper = data$delta_u, link='probit')
# new data
new.data <- data.frame(x = c(0, 1))
conditional_median <- quantile_dl(mod, new.data, probs = 0.5)
conditional_cdf <- cdf_dl(mod, new.data, at.y = 1.5) # P(y <= 1.5 | new.data)


## Single DL: lower DL at 0.5
n <- 100
x <- rnorm(n)
e <- rnorm(n)
y <- exp(x + e)
lower_dl <- 0.5
data <- data.frame(y = y, x = x)
data$delta_lower <- ifelse(data$y >= lower_dl, 1, 0)
data$z <- ifelse(data$delta_lower == 0, lower_dl, data$y)
mod <- multipleDL(formula = z ~ x, data = data,
                  delta_lower = data$delta_l, link='probit')