## Empirical Bayesian Estimation of Censored Gaussian (Tobit) Matrices

### What is it?

An R package for denoising censored, Gaussian means with empirical Bayes $$g$$-modeling. The general model is as follows:

$\theta_i \sim_{iid} g \quad (\subseteq \mathbb{R}^p)$

$X_{ij} \mid \theta_{ij} \sim_{indep.} N(\theta_{ij}, \sigma^2)$

$L_{ij} \leq X_{ij} \leq R_{ij}$

The data is represented with matrices:

$\theta = \begin{bmatrix} \theta_{11} & \dots & \theta_{1p} \\ \theta_{21} & \dots & \theta_{2p} \\ \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{np} \\ \end{bmatrix} \qquad X = \begin{bmatrix} X_{11} & \dots & X_{1p} \\ X_{21} & \dots & X_{2p} \\ \vdots & \ddots & \vdots \\ X_{n1} & \dots & X_{np} \\ \end{bmatrix}$

$L = \begin{bmatrix} L_{11} & \dots & L_{1p} \\ L_{21} & \dots & L_{2p} \\ \vdots & \ddots & \vdots \\ L_{n1} & \dots & L_{np} \\ \end{bmatrix} \qquad R = \begin{bmatrix} R_{11} & \dots & R_{1p} \\ R_{21} & \dots & R_{2p} \\ \vdots & \ddots & \vdots \\ R_{n1} & \dots & R_{np} \\ \end{bmatrix}$

The bounds $$L_{ij}$$ and $$R_{ij}$$ are assumed to be known. When $$L_{ij} = R_{ij}$$ there is a direct (noisy) measurement of $$\theta_{ij}$$, if $$L_{ij} < R_{ij}$$ then there is a censored measurement of $$\theta_{ij}$$. This structure is commonly referred to as partially interval censored data and it allows for any combination of observed measurements and left-, right-, and interval-censored measurements.

We use a Tobit likelihood for each measurement:

$P(L, R \mid \theta) = \begin{cases} \phi_{\sigma} ( L - \theta ) & L = R \\ \Phi_{\sigma} ( R - \theta ) - \Phi_{\sigma} ( L - \theta ) & L < R \end{cases}$

where the standard Gaussian likelihood is used when there is a direct Gaussian measurement (ie $$L = X = R$$) and a Gaussian probability is used when there is a censored Gaussian measurement (ie $$L < R$$).

### What does it do?

This package provides an object ebTobit (Empirical Bayes model with Tobit likelihood) that estimates the prior, $$g$$ over a user-specified grid gr and then computes the posterior mean or $$\ell_1$$ mediod as estimates for $$\theta$$. In one dimension, the $$\ell_1$$ mediod is the median. By default gr is set using the exemplar method so the grid is the maximum likelihood estimate for each $$\theta_{ij}$$. When the censoring interval is finite, the maximum likelihood estimate for each $$\theta_{ij}$$ is $$0.5 ( L_{ij} + R_{ij} )$$

Suppose $$p = 1$$ and there is no censoring, then the basic utility is:

library(ebTobit)

# create noisy measurements
n <- 100
t <- sample(c(0, 5), size = n, replace = TRUE, prob = c(0.8, 0.2))
x <- t + stats::rnorm(n)

# fit g-model with default prior grid
res1 <- ebTobit(x)

# measure performance of estimated posterior mean
mean((t - fitted(res1))^2)

Next we can look at a more complicated example with $$p = 10$$:

library(ebTobit)

# create noisy measurements (low rank structure)
n <- 1000; p <- 10
t <- matrix(stats::rgamma(n*p, shape = 5, rate = 1), n, p)
x <- t + matrix(stats::rnorm(n*p), n, p)

# assume we can't accurately measure x < 1 but we know theta > 0
L <- ifelse(x < 1, 0, x)
R <- ifelse(x < 1, 1, x)

# fit g-model with default prior grid
res2 <- ebTobit(x)
res3 <- ebTobit(L, R)

# oberve that the censoring affects the fitted range
range(fitted(res2))
range(fitted(res3))

# fit censored data with a different grid (large and random not MLE)
res4 <- ebTobit(
L = L,
R = R,
gr = sapply(1:p, function(j) stats::runif(1e+4, min = min(L[,j]), max = max(R[,j]))),
algorithm = "EM"
)

# compute posterior mean and L1mediod given new data
# we can also predict based on partially interval-censored observations
y <- matrix(stats::rexp(5*p, rate = 0.5), 5, p)
predict(res4, y) # posterior mean
predict(res4, y, method = "L1mediod") # posterior L1-mediod

### How do install it?

Until the package is available on CRAN, it can be installed directly from GitHub:

remotes::install_github("barbehenna/ebTobit")

### Data

This R package also includes a real bile acid data.frame taken directly from Lei et al. (2018) (https://doi.org/10.1096/fj.201700055R) via https://github.com/WandeRum/GSimp (https://doi.org/10.1371/journal.pcbi.1005973). The bile acid data contains measurements of 34 bile acids for 198 patients; no missing values are present in the data. In our modeling, we assume the bile acid values are independent log-normal measurements.

data(BileAcid, package = "ebTobit") # attach the bile acid data