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\title{

  List of responses to the comments for the author of:
  A multinomial generalized linear mixed model for clustered competing
  risks data

}

\begin{document}
\maketitle

\section*{Associate Editor}

The manuscript has been read by two independent reviewers and myself. It
proposes a GLMM approach to handle clustered competing risks with a
within-cluster dependence structure.

As you can see below, the manuscript has been positively evaluated, but
there are still some issues that must be addressed. In particular, the
authors should specify the differences with the existing literature and,
especially, with the work by Cederkivst et.al. (2019) that seems to have
proposed a very similar model.

\subsection*{Author's response}

We thanks the positive evaluation and clarified in the paper the
differences between our work and \cite{SCHEIKE}, these clarifications
were made in the introduction and discussion sections.  Furthermore, we
restate our main goal to let clear that our contribution relies mainly
on computational aspects of the model proposed by \cite{SCHEIKE}.

In summary, the differences are: we follow \cite{SCHEIKE} and work with
the Cumulative Incidence Function (CIF) specification proposed by
them. However, there, to be able to model and infer the CIF parameters
and within-cluster dependence, an elaborated composite likelihood
approach was proposed combined with an Adaptative Gaussian Quadrature
(AGQ) scheme to marginalize the composite likelihood. Extra steps were
needed to be able to quantify the uncertainty around the parameters. We,
on the other hand, modeled the same CIF specification via a simpler and
known formulation - a generalized linear mixed model (GLMM)
formulation. Along with the simpler formulation which also provides a
full and proper likelihood function, we used a Laplace approximation
scheme in contrast to the used AGQ. To make this simpler framework
feasible, we took advantage of state-of-the-art numerical algorithms
implementations and computational libraries. Basically, is the same CIF
specification but with different likelihood functions/formulations and
parameter estimation routines/schemes.

\section*{Reviewer #1}

This is an interesting topic, and the authors reports an extended
simulation study while proposing an estimation methods for competing
risks with clustered data following the proposal by Cedervisk et al
(2019). The author should specify with more detail differences, if they
are present, between their proposal and the original one not only from
the computational side: they declare to propose a new model but it seems
the one proposed by Cedervisk et al.

I suggest adding some short motivations could add applicative values,
examples on the usage of this class of models that could help readers to
figure out in a clearer way the possible applications.

\subsection*{Author's response}

We thanks for the insightful comments of the reviewer. We use the same
CIF specification of \cite{SCHEIKE} but with a different likelihood
formulation, a full likelihood approach instead of a composite one for
parameter estimation. Laplace approximation instead an expensive AGQ,
taking advantage of state-of-the-art computational libraries and
algorithm implementations. We restate our main goal to let clear that we
adopted the same model specification proposed by \cite{SCHEIKE},
however, we use a different parameter estimation procedure. Please, see
Introduction section.


\subsection*{Reviewer #1}

The simulations results underline some situation that can be
problematic. This is very interesting because in the original proposal
by Cedervisk et al only a real data set and a single simulation study
were considered, I suggest stressing this point and add some comparisons
with the previous proposal.  Especially for problematic situations, have
they tried to consider the original computational method proposed by
Cedervisk et al?  Can they give more insight for simulation cases where
the latent effects didn't work well?

\subsection*{Author's response}

We access \cite{SCHEIKE}'s code through GitHub. Running their code we've
not been able to reproduce their results. By adapting it to perform some
extra simulation studies like the ones that we did with our model, at
most scenarios convergence has not been reached. Given that and the
facts that 1) our likelihood functions, formulations, and estimation
procedures are very different and not directly comparable; 2) our model
runs in parallel and its running time is fast considering its complexity
and sample size, we end up choosing not to make any statement about
these differences in the results section. We emphasize that our focus
were to propose a new estimation procedure and not to compare two
different estimation procedures. We completly agree that the comparison
is an interesting topic, however, it is beyound the goal of our
paper. We included it as a topic for future work.

\subsection*{Reviewer #1}

Minor

Page 8 row 18, please specify better what is yijt
Page 3 row 52 and page 29 row 1 is Vaupel not Valpel

\subsection*{Author's response}

The clarification and the correction were made.

\section*{Reviewer #2}

This paper proposes a multinomial mixed model (similar to a pattern
mixture model) for the cumulative incidence functions of clustered
multivariate competing risks data. Inference is based on the
(Laplace-approximated) maximum likelihood. The methods are rigorously
developed and could be useful in practice. I have a few minor comments
to help the authors improve the paper.

1. In simple language, what is the main difference of the proposed
approach with Cederkvist et al. (2019)?

\subsection*{Author's response}

We thanks fot the insightful comments. We use the same CIF specification
of \cite{SCHEIKE} but with a different likelihood formulation (a full
likelihood approach instead of a composite one) and parameter estimation
routine (a Laplace approximation instead an expensive AGQ), taking
advantage of state-of-the-art computational libraries and algorithm
implementations. We restate our main goal to let our contribution
clearer.


\subsection*{Reviewer #2}

2. Recently, Ahn et al. (2022) has proposed a robuts approach to
semiparametric regression of multivariate clustered competing risks
data. Can you comment on the similarities and differences with the
proposed method (a parametric one if I understand it correctly)?

\subsection*{Author's response}

We thanks for the literature recommendation. The model proposed by
\cite{ahnetal22} is also a valid approach to handling clustered
competing risk data. The mentioned approach is similar to ours in the
sense that it is not necessary to model the censoring distribution. The
differences lie in the model category and parameter estimation and
uncertainty quantification. We have a parametric approach based on a
proper likelihood function instead a semiparametric one based on a
pseudo-likelihood. With a standard likelihood formulation based on GLMM,
we can take advantage of all well-established inferential
frameworks. With the semiparametric pseudo-likelihood formulation a
nonparametric estimator was needed to be proposed together with a
sandwich covariance estimator extra computations.

\subsection*{Reviewer #2}

The Laplace-approximated maximum likelihood looks similar to the EM
algorithm with the latent variables \(u_{i}\) treated as missing
data. Can you clarify the difference?

\subsection*{Author's response}

EM algorithms are a very interesting maximization routine since they can
be applied virtually to any model with latent components, but they
present the caveats of being slow, in the majority of the cases, and
that each model requests a specific EM formulation. Here, with our model
being easily written in the exponential family framework, a very general
and immediate Laplace approximation is easily obtained. This practical
formulation together with state-of-the-art Laplace approximation
implementation and the automatic-differentiation tool makes it a better
deal than the EM algorithm for our model specification.

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