.cens: Mark a censoring indicator in a model formula

Description

.cens() marks a variable as a censoring indicator rather than a treatment, requesting inverse probability of censoring weights (IPCW). It is most often used on the left side of a formula supplied to weightit() or in the formula.list of weightitMSM(), as in .cens(C) ~ x1 + x2. It can also be called directly to tag a variable for use with the lower-level interfaces weightit.fit() and get_w_from_ps().

Usage

.cens(x)

Value

x coerced to a 0/1 numeric vector of class treat with a "treat.type" attribute of "censoring". Missing values are preserved; any value other than 0 or 1 throws an error.

Inside a formula the marker is stripped before the formula is processed, so .cens() is not actually evaluated there and the treatment name remains that of the indicator itself (e.g., C rather than .cens(C)).

Arguments

x: a censoring indicator: a numeric variable taking only the values 0 (still under observation) and 1 (censored), a logical variable, or a factor with levels 0/1 or FALSE/TRUE.

Note on the coding convention

The survival convention is used: 1 means the unit is censored (drops out of observation) and 0 means it remains under observation. This is the opposite of an "observed" or "event" indicator. When a censoring model is supplied to weightit(), the returned propensity score is \(P(C = 1 | X)\), the probability of being censored.

Assessing balance

cobalt::bal.tab() cannot be used directly on the output of a censoring model. For a point-treatment weightit object it errors because every censored unit has a weight of 0, leaving one "treatment group" with no weight; for a weightitMSM object it errors because censoring leaves missing values in the later treatments, which bal.tab() does not allow.

Because the target of a censoring model is the full at-risk sample rather than another treatment group, balance is assessed by comparing the weighted uncensored units against that sample. This can be done by stacking the two:

u <- which(W$treat == 0)
bal.tab(rbind(covs[u, ], covs),
        treat = c(rep(0L, length(u)), rep(1L, nrow(covs))),
        weights = c(W$weights[u], rep(1, nrow(covs))),
        estimand = "ATT", s.d.denom = "treated")

The "Control" group is then the weighted uncensored sample and the "Treated" group the full at-risk sample, so Diff.Adj is the quantity the weights are designed to zero out. For a weightitMSM object, use the at.risk component, which has one column per time point, to restrict each model to the units that were under observation when it was fit:

ar <- W$at.risk[, "A_3"]
bal.tab(A_3 ~ X1_2 + X2_2 + A_2, data = data[ar, ],
        weights = W$weights[ar], s.weights = W$s.weights[ar])

Details

Censoring is treated as its own treatment type, distinct from binary, multi-category, and continuous treatments. Weights are estimated only for the units still under observation, and are those that make the covariate distribution of the uncensored units match that of the full at-risk sample. Formally, for \(e(X) = P(C = 1 | X)\), the weights are \((1 - e(X))^{-1}\) for units with \(C = 0\) and exactly 0 for units with \(C = 1\).

Because only one group is weighted, the estimation problem is smaller and better conditioned than the corresponding binary-treatment problem, which would additionally solve for weights among the censored units. This matters most when few units are censored.

Because censored units receive a weight of exactly 0, they contribute nothing to a weighted outcome model, and glm_weightit(), lm_weightit(), multinom_weightit(), ordinal_weightit(), and coxph_weightit() all tolerate missing values in the model variables for those units. This is what makes it possible to fit an outcome model whose outcome is unobserved after censoring. See the Censoring weights (IPCW) section of weightit() for details.

No estimand or focal argument applies to censoring models; supplying one produces a warning and it is ignored. subclass cannot be used. Not all methods support censoring weights; see the treat_type component of .weightit_methods to check.

The right side of the formula may be empty, as in .cens(C) ~ 1, requesting a marginal censoring model in which censoring is assumed independent of the covariates. The weights are then \(1/P(C = 0)\) for the units still under observation and 0 for the censored units, whatever method is supplied; see Empty model formulas in Details at weightit(). Nothing else changes, so a marginal censoring model still composes with by, stabilize, M-estimation, and, in weightitMSM(), the risk sets and the missing values permitted after censoring.

Use outside a formula

Calling .cens() directly returns a tagged indicator that routes the lower-level interfaces to the censoring path:

# Equivalent to weightit(.cens(C) ~ x1 + x2, ...)
weightit.fit(covs, treat = .cens(C), method = "glm")
# Censoring weights from a set of censoring probabilities
get_w_from_ps(ps, treat = .cens(C))

Tagging is the only way to request censoring weights from these interfaces; an untagged 0/1 vector is treated as an ordinary binary treatment.

Examples

Run this code

data("msmdata")

# A censoring indicator: 1 = lost to follow-up after time 2
set.seed(1234)
msmdata$C_2 <- rbinom(nrow(msmdata), 1,
                      prob = plogis(-2 + 0.2 * msmdata$X1_1))

# Inverse probability of censoring weights
(Wc <- weightit(.cens(C_2) ~ X1_1 + X2_1 + A_1,
                data = msmdata, method = "glm"))

# Censored units have a weight of 0
all(Wc$weights[msmdata$C_2 == 1] == 0)

# The weights are the inverse probability of remaining
# under observation
all.equal(Wc$weights[msmdata$C_2 == 0],
          1 / (1 - Wc$ps[msmdata$C_2 == 0]))

# `.cens()` can also tag a variable directly for use with
# the lower-level interfaces
C <- .cens(msmdata$C_2)

all.equal(get_w_from_ps(Wc$ps, treat = C),
          Wc$weights, check.attributes = FALSE)

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