Baseline adjustment when both pre- and post values are subject to left-censoring Assume $X$ is a binary treatment variable, $Y$ is a continuous variable measured pre- and post-treatment $(Y_{pre}$, $Y_{post})$, and $Z$ represents the remaining covariates. Both $Y_{pre}$ and $Y_{post}$ are subject to left-censoring.
Does anyone know an approach to analyze the effect of $X$ on $Y_{post}$ adjusted for $Y_{pre}$ and $Z$? Tobit regression (e.g.) can handle the situation where only $Y_{post}$ is censored, and various methods are suggested in the case where only $Y_{pre}$ (and/or $Z$) is censored. But what if both $Y_{pre}$ and $Y_{post}$ are subject to left-censoring?
 A: If $Y_{pre}$ and $Y_{post}$ are, for instance, lab analytes where the assay can only detect down to a lower limit of detection, you can use an EM algorithm to impute the values according to some distributional assumption.
To illustrate with a univariate case: consider first the M step. To calculate the likelihood, an order statistic result is to just input the CDF functional. For instance:
set.seed(123)
n <- 100
ypre <- rlnorm(n = n, meanlog = -3, sdlog = 3)
ypre[ypre < 1] <- 1

negloglik <- function(param, x, cens) {
  -sum(pnorm(x[cens], mean = param[1], sd=param[2], log.p = T)) -
    sum(dnorm(x[!cens], mean=param[1], sd=param[2], log = T))
}

nlm(f = negloglik, p=c(0,1), x=log(ypre), cens = ypre==1)

Gives (with lots of warnings because the nlm algo is a bit rough)
$estimate
[1] -2.509127  2.642844

Which we can then apply expectation (E-step) based on the conditional expectation
paramest <- nlm(f = negloglik, p=c(0,1), x=log(ypre), cens = ypre==1)$estimate
pcens <- mean(ypre == 1)
expval <- integrate(function(x) x*dnorm(x, mean=paramest[1], sd = paramest[2]),
  lower=-Inf, upper=qnorm(pcens))
ypre_e <- ifelse(ypre==1, exp(expval), ypre)

Now this is considerably more complicated with multivariate analyses. If we are able to assume normality for the $Y$s (or some transformation), we can use linear regression as a maximum likelihood process. We can factor the joint likelihood of Ypre, X, and Ypost into the likelihood of Ypre and of Ypost|X, Ypre. I think we can thus develop the algorithm:

*

*Initialize parameter estimates for Ypre, the regression parameters for the mean Ypost response adjusting for Ypre and treatment, and the residual standard error of Ypost

*Calculate the expected value of Ypre

*Estimate the regression using a maximum likelihood procedure with the conditional mean of Ypost given by the current regression estimates and current residual variance

*Update the regression estimates and residual variance according to the MLE.

*Repeat steps 3 and 4 until convergence.

