*This post is part of our Q&A series.*

A question from graduate students in our Spring 2021 offering of the new course “Targeted Learning in Practice” at UC Berkeley:

## Question:

Hi Mark,

I have a survival analysis question. I am working with a dataset that is left- and right-truncated. I am interested in estimating the treatment-specific multivariate survival function of a time-to-event variable. For example, a study where subjects have been randomized to two different treatment groups with baseline covariates

`$W$`

, but we only observe the outcome – time at death – for a left- and right-truncated window. Is it possible to use Targeted Maximum Likelihood Estimation (TMLE) for estimating the treatment-specific multivariate survival curve?I have seen a few papers using TMLE for right-censored data, but I assume there are important considerations when working with doubly-truncated data.

Best,

C.B.

## Answer:

Hi C.B.,

Thank you for the excellent question. You are asking about estimation of
a treatment-specific survival curve when we have a time window and a subject is
only part of the sample if a particular event such as death does not occur
before the start of the window, so the sample is conditional on `$T > C_l$`

, or
we only observe units when `$T < C_l$`

, for some truncation random variable
`$C_l$`

and time until event `$T$`

.

In another post, I talk more about this problem of left-truncation and
censoring. I will refer you to that for your question as well. Either way, yes,
TMLE can be applied for any estimation problem, so it is just a matter of
establishing the identification of the full-data distribution `P_X$`

from
observing `$O = \Phi(C, X)$`

from a conditional distribution of `$T > C_l$`

,
say, thereby handling both the biased sampling due to sampling conditional on
`$T > C_l$`

as well as the more regular right-censoring, etc., making up
a censored data structure `$\Phi(C, X)$`

. For example, one might be able to
assume `$C_l$`

is independent of `$X$`

, conditional on measured variables, and
show that a conditional distribution of `$X$`

, given `$T > C_l$`

, implies the
distribution of the full-data random variable `$X$`

or a large part of it, so
that a two-stage identification, first identifying `$P_X$`

from```
$P_{X \mid
T > C_l}$
```

and then identifying `$P_{X \mid T > C_l}$`

from `$P$`

of ```
$O
= \phi(C, X)$
```

given `$T > C_l$`

. Once we have done that, we can map the target
quantity `$\Psi^F(P_X)$`

into an estimand `$\Psi(P)$`

, specify the statistical
model, and then we are ready to apply TMLE.

Best Wishes,

Mark

**P.S.**, remember to write in to our blog at ```
vanderlaan (DOT) blog [AT]
berkeley (DOT) edu
```

. Interesting questions will be answered on our blog!