# Exploring the validity of the complete case analysis for regression models
with a right-censored covariate

Marissa C. Ashner
Department of Biostatistics
Unversity of North Carolina at Chapel Hill
Chapel Hill, NC 27599
<EMAIL_ADDRESS>
&Tanya P. Garcia
Department of Biostatistics
Unversity of North Carolina at Chapel Hill
Chapel Hill, NC 27599
<EMAIL_ADDRESS>

###### Abstract

Despite its drawbacks, the complete case analysis is commonly used in
regression models with missing covariates. Understanding when implementing
complete cases will lead to consistent parameter estimation is vital before
use. Here, our aim is to demonstrate when a complete case analysis is
appropriate for a nuanced type of missing covariate, the randomly right-
censored covariate. Across the censored covariate literature, different
assumptions are made to ensure a complete case analysis produces a consistent
estimator, which leads to confusion in practice. We make several contributions
to dispel this confusion. First, we summarize the language surrounding the
assumptions that lead to a consistent complete case estimator. Then, we show a
unidirectional hierarchical relationship between these assumptions, which
leads us to one sufficient assumption to consider before using a complete case
analysis. Lastly, we conduct a simulation study to illustrate the performance
of a complete case analysis with a right-censored covariate under different
censoring mechanism assumptions, and we demonstrate its use with a Huntington
disease data example.

_K_ eywords complete case $\cdot$ covariate censoring $\cdot$ missing not at
random $\cdot$ random censoring

##  1 Introduction

In the ongoing conversation about regression models with missing covariates,
complete case analyses, which remove all observations with missingness, almost
always have a seat at the table. However, being perceived as the simple,
convenient method with many drawbacks, complete case analyses typically take a
back seat. Despite its drawbacks, this convenient approach is still used by
many analysts, especially newcomers to the field of missing data.
Additionally, the complete case analysis is often used as a comparison to
novel methods or as an initial estimate for sophisticated approaches. The
continual use of the complete case analysis should earn the method a central
role in the missing covariate conversation to avoid often overlooked
properties and non-unified language surrounding the topic.

Here, we focus on clarifying when and why the complete case analysis leads to
a consistent estimator for a nuanced type of missing covariate: right-censored
covariates. For a right-censored covariate, the true value of the covariate is
unknown due to missingness but is known to be greater than some value. This
aspect gives us partial information regarding the unknown truth, which is more
than is obtained from a general missing covariate. The body of literature
surrounding the statistical handling of censored covariates mentions complete
case analyses frequently. However, there are slight variations across the
literature in terms of the assumptions made to ensure that the estimator
produced by the complete case analysis is consistent. Key differences in the
assumptions stem from the relationships assumed between the censored
covariate, the value at which it is censored, and the censoring indicator.
Each source considers one assumption, but assumptions across sources have not
been compared before, making it unclear how the assumptions are related, if at
all.

Understanding the relationship between the assumptions that drive consistency
for a complete case estimator is important for several reasons. First, it is
important for more sophisticated procedures for handling censored covariates,
like multiple imputation, which uses the partial information available from
the censored subject. The consistency of a multiple imputation estimator
depends on the correct specification of the “imputation model” [1]. For some
proposed imputation procedures, the imputation model itself builds from an
consistent initial estimator, and in practice, the complete case estimator is
often that initial estimator [2, 3, 4]. This initial use means that if the
complete case estimator is not consistent, the imputation model will be
misspecified, leading to potential bias in the final estimator. Second, the
complete case analysis is the default method for handling any type of missing
data in most software packages [5]. Many analysts will use these programs
without a second thought, so we are advocating for further awareness of when
the default is valid and how to ensure the censored data are set up properly
to handle the default. Finally, when new methods are proposed, many
statisticians use simulations to compare the performance of their method to
others, including the complete case analysis, which serves as a useful
baseline (e.g., see [6, 7]). It is crucial to be aware of the properties of
all methods considered in simulations to fully understand the results.

In this paper, we will lay a clear foundation for what assumptions drive
consistency when using a complete case analysis for regression models with a
censored covariate. We clarify that, in fact, the different assumptions are
interconnected in a hierarchical fashion and share a common property: either
the censoring value or the censoring indicator must be independent of the
outcome, given the covariates. The rest of the paper is organized as follows:
Section 2 details and condenses the censoring mechanism assumptions present in
the existing literature and proves the relationship between the mechanism
assumptions as well as the consistency of the complete case estimator under
said assumptions; Section 3 contains simulation studies conducted under many
settings to compare a complete case analysis to the analysis that would be
done if there were no censoring; Section 4 applies the complete case analysis
to a Huntington disease observational study with a censored covariate; and
Section 5 presents a discussion of our findings.

##  2 Establishing consistency of the complete case estimator

###  2.1 The complete case estimator with a right-censored covariate

We consider the regression model

$\displaystyle{Y}_{i}=m(X_{i},{\mathbf{Z}}_{i},{\boldsymbol{\beta}})+{\epsilon}_{i},\qquad
i=1,\ldots,n,$ (1)

where $Y_{i}$ is the outcome, $X_{i}$ is a right-censored covariate,
${\mathbf{Z}}_{i}$ is a $p$-dimensional vector of fully observed covariates,
${\epsilon}_{i}$ is random error, and $n$ is the number of observations. We
will use the subscript $i$ only when needed for clarity. Rather than assume a
particular distributional assumption for the random error, we assume that
$E(\epsilon|X,{\mathbf{Z}})=0$ to make our regression model flexible enough to
handle a range of random error distributions. In equation (1), $m(\cdot)$ is
any linear or nonlinear function known up to ${\boldsymbol{\beta}}$, a
$(p+1)$-dimensional parameter. Due to right-censoring, rather than observe
$X$, we observe $W=\min(X,C)$ and $\Delta=I(X\leq C)$, where $C$ is a random
variable equal to the value at which $X$ is censored.

One of our goals is to show when the complete case estimator for
${\boldsymbol{\beta}}$ is consistent; we first show how to construct the
complete case estimator. The estimators in this paper will be solutions to
estimating equations. For a full dataset (i.e., $X$ is not censored), the
estimator for ${\boldsymbol{\beta}}$ is the solution to the estimating
equation

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}})={\mathbf{0}},$
(2)

where
${\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}})={\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\{Y-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\}$
and ${\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})$ is any $(p+1)$
dimensional function. The optimal choice of
${\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})$, in terms of maximal
efficiency, is when
${\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})=\partial
m(X,{\mathbf{Z}};{\boldsymbol{\beta}})/\partial{\boldsymbol{\beta}}\times
E(\epsilon^{2}|X,{\mathbf{Z}})^{-1}$ [8]. In this paper, we will assume the
error term is independent of the covariates such that
$E(\epsilon^{2}|X,{\mathbf{Z}})=E(\epsilon^{2})$, making
${\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})=\partial
m(X,{\mathbf{Z}};{\boldsymbol{\beta}})/\partial{\boldsymbol{\beta}}$ optimal.

The complete case analysis discards all data for which $X$ is right-censored.
Mathematically, the complete case estimator, denoted by
$\widehat{\boldsymbol{\beta}}$, is the solution to

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},W_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}})={\mathbf{0}},$
(3)

where we add $\mbox{$\Delta$}_{i}$ so that there is no contribution from each
observation with a censored $X_{i}$. Even though, in general, there is no
closed form for $\widehat{\boldsymbol{\beta}}$, it can be solved using
numerical techniques in available software packages, such as nls in R; we
provide working code at [URL redacted for blinded submission]. Note that the
default for packages like nls is to delete observations with a missing
covariate, so before using these packages on censored data, we must ensure
that the censored $X$ values are coded as missing values, not as the value at
which it was censored. To show consistency of the complete case estimator,
$\widehat{\boldsymbol{\beta}}$, additional assumptions are needed. To set the
stage, we first draw connections to missing data assumptions.

###  2.2 Connections to missing data

At first glance, a right-censored covariate seems no different than a missing
covariate. Yet, even though censoring is a type of missingness, it is
different. With general missingness, we have no information about the true
covariate value. With censoring, though, we have _partial information_ : if
$X$ is right-censored, its true value must be larger than $C$. Statistical
methods for handling missing data do not adjust for such partial information
[9].

That said, methods and language used in the missing data context can serve as
a starting point for handling right-censored covariates. For example, when
dealing with missing data, it is fundamental to consider the so-called
missingness mechanisms –assumptions that explore the relationship between the
missingness variable (i.e., $\Delta$) and the data values (i.e.,
$X,Y,{\mathbf{Z}}$) –to explain why missing data occur [10, 11]. The three
most common missingness mechanisms are: (1) missing completely at random, when
the distribution of the missing data indicator is independent of all other
variables; (2) missing at random, when the distribution of the missing data
indicator depends on fully observed variables; and (3) missing not at random,
when the distribution of the missing data indicator depends on the missing
values themselves.

Models with a right-censored covariate case fall into the category of missing
not at random because the “missing data" indicator is
$\mbox{$\Delta$}=I({X}\leq{C})$, which explicitly depends on incomplete ${X}$.
Data that are missing not at random must be considered carefully, since many
existing methods require missing at random data [12]. However, an
underappreciated fact is that a complete case analysis can lead to consistent
estimators for ${\boldsymbol{\beta}}$ even when data are missing not at random
[13]. This consistency holds so long as certain assumptions about the
missingness mechanisms hold. From now on, we will refer to missingness
mechanisms as “censoring mechanisms” in the context of censored covariates,
and we clarify the censoring mechanisms that lead to consistent complete case
estimators next.

###  2.3 Censoring mechanisms for consistent complete case estimators

Unlike missingness mechanisms for missing data, the language surrounding
censoring mechanisms for censored covariates is not yet unified. Literature
for censored covariate problems use different censoring mechanism assumptions
to ensure a complete case estimator is consistent, and it is unclear which
censoring mechanism an analyst working on a new problem should consider. We
aim to unify this language by first summarizing five common censoring
mechanisms that current sources use. Those censoring mechanisms are as
follows, where the symbol $\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern 2.0mu{\scriptscriptstyle\perp}}}$
denotes independence between random variables:

  1. (C1)

Exogenous Censoring:  $E(\epsilon|\Delta,X,{\mathbf{Z}})=0$ [14, 12, 15];

  2. (C2)

Strict Exogenous Censoring:  ${\epsilon}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mbox{$\Delta$}|({X},{\mathbf{Z}})$ [14];

  3. (C3)

Conditionally Independent Censoring given $({X},{\mathbf{Z}})$:
${C}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}{Y}|({X},{\mathbf{Z}})$ [3, 4, 16, 17];

  4. (C4)

Conditionally Independent Censoring given ${\mathbf{Z}}$:
${C}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}({X},{Y})|{\mathbf{Z}}$ [18, 19];

  5. (C5)

Independent Censoring:  ${C}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}({Y},{X},{\mathbf{Z}})$ [7].

All five mechanisms have one common feature: either the censoring variable
${C}$ or the censoring indicator $\Delta$ is independent of the outcome ${Y}$
or error term $\epsilon$ in some way. There are still, however, differences
between these mechanisms. As we move down the list of censoring mechanisms
from (C1) to (C5), the statements appear to get more restrictive in that each
censoring mechanism imposes a more predetermined structure than the last.

Recall our model assumption that $E(\epsilon|X,{\mathbf{Z}})=0$. Censoring
mechanism (C1) imposes a stronger assumption on the model error such that,
rather than requiring the conditional mean of $\epsilon$ to equal 0, it
requires the conditional mean of $\epsilon$ among those where $\Delta=0$ to be
0 and the conditional mean of $\epsilon$ among those where $\Delta=1$ to be 0.
Mechanism (C2) further requires that the conditional distributions of
${\epsilon}$ and $\Delta$ are independent given $X$ and ${\mathbf{Z}}$. This
means that not only is the conditional mean of ${\epsilon}$ independent of
$\Delta$ but also the entire probability distribution of ${\epsilon}$ is
independent of $\Delta$. Censoring mechanism (C3) switches things up by
considering the distribution of the censoring variable ${C}$, rather than the
censoring indicator $\Delta$. Under mechanism (C2), we expect that the value
of binary variable $\Delta$ is independent of the conditional density of
${\epsilon}$, and under mechanism (C3), we expect that the value of continuous
variable $C$ is independent of the conditional probability density of ${Y}$.
Censoring mechanisms (C4) and (C5) are very similar to mechanism (C3) in that
they require independence of censoring variable ${C}$ and outcome ${Y}$, but
differ in that they condition on fewer variables.

At face value, it is unclear if these censoring mechanisms are formally
related to one another. Will the complete case estimator be consistent if we
assume that just one of the assumptions holds, or must we consider them all?
In fact, we show in Theorem 1 that there is indeed a hierarchical relationship
between censoring mechanisms (C1)–(C5).

###### Theorem 1

Consider the setup in Section 2.1 and the censoring mechanisms (C1)–(C5). Then
the following relationship holds between the mechanisms: Censoring mechanism
(C5) implies (C4), (C4) implies (C3), (C3) implies (C2), and (C2) implies
(C1). Given this hierarchical relationship, we have the following
contrapositive: if censoring mechanism (C1) does not hold, then (C2) does not
hold; if (C2) does not hold, (C3) does not hold; if (C3) does not hold, (C4)
does not hold; and if (C4) does not hold, (C5) does not hold.

The proof of Theorem 1 is in Supplementary Materials S.1. This hierarchy helps
to establish which censoring mechanism is sufficient to ensure the complete
case estimator is consistent and asymptotically normal, as described in
Theorem 2.

###### Theorem 2

Under the following regularity conditions

  1. (_R1_)

$E\\{{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}={\mathbf{0}}$
for all ${\boldsymbol{\beta}}$;

  2. (_R2_)

$E\\{\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\\}$
is nonsingular;

  3. (_R3_)

$n^{-1}\sum_{i=1}^{n}\frac{\partial{\boldsymbol{\Phi}}_{i}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}}_{0})}{\partial{\boldsymbol{\beta}}^{T}}$
converges uniformly to
$E\\{\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}}_{0})}{\partial{\boldsymbol{\beta}}^{T}}\\}$
in a neighborhood of ${\boldsymbol{\beta}}_{0}$, where
${\boldsymbol{\beta}}_{0}$ is the true parameter value,

the complete case estimator $\widehat{\boldsymbol{\beta}}$ is consistent if
$E(\epsilon|X,\Delta,{\mathbf{Z}})=0$ (i.e., we have exogenous censoring);
that is, $\widehat{{\boldsymbol{\beta}}}$ converges in probability to the true
parameter value ${\boldsymbol{\beta}}_{0}$. Moreover, the asymptotic
distribution of $\widehat{{\boldsymbol{\beta}}}$ will be such that

$\sqrt{n}(\widehat{{\boldsymbol{\beta}}}-{\boldsymbol{\beta}}_{0})\rightarrow\hbox{Normal}\\{{\mathbf{0}},{\mathbf{A}}^{-1}{\mathbf{B}}({\mathbf{A}}^{-1})^{T}\\},$

where
${\mathbf{A}}=E\\{\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})/\partial{\boldsymbol{\beta}}^{T}\\}$
and
${\mathbf{B}}=E[\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}^{T}]$.

The proof of Theorem 2 is in Supplementary Materials S.2. Theorem 2
establishes consistency when censoring mechanism (C1) holds. Corollary 2.1
will establish that the complete case estimator is consistent under _any_ one
of the censoring mechanisms (C1) to (C5).

###### Corollary 2.1

Under any of the censoring mechanisms (C1) to (C5), the complete case
estimator $\widehat{\boldsymbol{\beta}}$ is a consistent and asymptotically
normal estimator.

Corollary 2.1 follows directly from Theorems 1 and 2. The hierarchy in Theorem
1 shows that if any one of the censoring mechanisms (C2) to (C5) hold, then
(C1) holds. From Theorem 2, if mechanism (C1) holds, then the estimator
$\widehat{\boldsymbol{\beta}}$ is consistent. Therefore, combining the results
from Theorems 1 and 2 leads to the result in Corollary 2.1: any of the five
censoring mechanisms will lead to a consistent estimator for
${\boldsymbol{\beta}}$.

###  2.4 Unbiasedness of the linear complete case estimator

Consistency is a desirable property for any estimator, since it ensures that
$\widehat{\boldsymbol{\beta}}$ converges to the true parameter value,
${\boldsymbol{\beta}}_{0}$, in probability. Convergence in probability is a
large sample property meaning that the estimator will have higher probability
of being “close” to the true value as the sample size increases to infinity.
But what happens when we have a finite sample size? At what point will it be
large enough for the large sample property to start “working”? We show that
when the regression model in equation (1) is linear, then the complete case
estimator exhibits unbiasedness, which is a finite sample property.

###### Remark 1

Consider a linear regression model with a right-censored covariate where in
equation (1),
$m(X,{\mathbf{Z}};{\boldsymbol{\beta}})=\beta_{0}+\beta_{1}X+{\boldsymbol{\beta}}_{2}^{T}{\mathbf{Z}}$.
Let
${\mathbf{U}}_{i}=\begin{bmatrix}1&X_{i}&{\mathbf{Z}}_{i}^{T}\end{bmatrix}$ be
the $i$th row of the design matrix associated with this linear regression
model. If and only if exogenous censoring (C1) holds, then the complete case
estimator found by solving Equation (3), which is the least squares solution
$\widehat{{\boldsymbol{\beta}}}=(\sum_{i=1}^{n}\Delta_{i}{\mathbf{U}}_{i}^{T}{\mathbf{U}}_{i})^{-1}\sum_{i=1}^{n}\Delta_{i}{\mathbf{U}}_{i}^{T}Y_{i}$,
is not only a consistent estimator but is also unbiased (i.e.,
$E(\widehat{{\boldsymbol{\beta}}})={\boldsymbol{\beta}}_{0}$).

Remark 1 says that, for any finite sample, the expectation of the estimator
$\widehat{{\boldsymbol{\beta}}}$ will equal the true parameter
${\boldsymbol{\beta}}_{0}$. Similar to Corollary 2.1 for consistency, we have
unbiasedness when any of the censoring mechanisms (C1)–(C5) hold. The proof of
unbiasedness under exogenous censoring is similar to that used for linear
regression models when no covariate is right-censored; see for example [20].

Having this result allows us to investigate the bias that can occur when
exogenous censoring (C1) does not hold. In Supplementary Materials S.3, we
show that for a linear model with an intercept, only the parameter estimate
for the intercept (i.e., $\widehat{\beta}_{0}$) will be biased if exogenous
censoring does not hold, and the other estimates remain unbiased.
Additionally, if the intercept is removed from the model, as in
$m({X},{\mathbf{Z}};{\boldsymbol{\beta}})=\beta_{1}X+{\boldsymbol{\beta}}_{2}{\mathbf{Z}}$,
it is possible to see bias in the parameter estimates for $\beta_{1}$ and
${\boldsymbol{\beta}}_{2}$.

###  2.5 Checking the censoring mechanism assumptions

While censoring mechanisms (C1)–(C5) help to ensure consistency of the
complete case estimator, how can an analyst check when these censoring
mechanisms hold? Unfortunately, verifying any assumptions made on the
censoring mechanisms is nearly impossible based on the observed data alone,
since they all rely on incomplete ${X}$ [21].

However, we may still be able to gauge from contextual knowledge of the data
whether or not these censoring mechanisms hold. One way to visually explore
the assumptions about independence between variables is to use a directed
acyclic graph (DAG) [13]. DAGs are commonly used in the context of causal
inference and are a tool used to visualize our contextual knowledge and a
priori assumptions about the causal relationships between the variables in our
model. DAGs consist of nodes representing random variables and edges
representing the relationships between these variables. The edges have arrows,
meaning they imply a direction; an arrow from ${X}$ to ${Y}$ implies that
${X}$ causes ${Y}$. After using this framework to lay out the hypothesized
causal structure of the variables of interest, the DAG can be used to describe
marginal and conditional independence between variables. To do so, let us
define a few terms:

  1. 1.

A path between two variables on a DAG is a route that connects the two
variables by following a sequence of edges;

  2. 2.

A collider on a path is a node on the path where two arrowheads collide.

Two variables are marginally independent if, for all paths between the two,
there is at least one collider. Two variables are conditionally independent
if, for all paths between the two, there is at least one collider and/or we
condition on at least one other variable (a non-collider) on the path [22].

This concept can be best understood through an example. Consider a
longitudinal observational study of patients who tested positive for the
genetic mutation that causes Huntington disease. Let ${X}$ be the time to
clinical diagnosis (in years, which may be right-censored). $\Delta$ is the
indicator of whether or not ${X}$ is censored, and ${C}$ is the time to
censoring (in years). Let us define ${\mathbf{Z}}$ as the number of cytosine,
adenine, guanine (CAG) repeats in the patient’s huntingtin gene and their
current age, which are both factors known to be associated with impairment in
this disease [23]. We will look at two outcomes ${Y}$. First, let ${Y}$ be the
current level of apathy felt by the patient. Then, let ${Y}$ be the current
motor impairment in Huntington disease patients (called the total motor
score), where a higher score means greater impairment. We hypothesize that
some people dropped out of the study because their motor impairment prevented
them from continuing. By this we mean, a patient’s motor impairment could
eventually limit their physical ability to continue participating in the
study.

Figure 1:  Directed acyclic graphs (DAGs) that represent the hypothesized
causal relationships in the Huntington disease example for an apathy outcome
(a) and a total motor score outcome (b).

First consider the DAG in Figure 1. We draw a directed edge from ${X}$ to
${Y}$ since this is our primary association of interest, and directed edges
from ${X}$ and ${C}$ to $\Delta$, since by definition, the censoring indicator
is associated with both ${X}$ and ${C}$. Now, we draw a directed edge from
${\mathbf{Z}}$ to ${X}$, since it has been shown in previous studies that CAG
repeat length and age are predictive of disease diagnosis [23]. We also draw
an edge from ${\mathbf{Z}}$ to ${C}$, since one could argue that CAG repeat
length and age may lead to an earlier dropout . We want to determine whether
or not a complete case analysis is consistent based on this DAG. Consider the
following paths from ${C}$ to ${Y}$: (1)
${Y}\leftarrow{X}\leftarrow{\mathbf{Z}}\rightarrow{C}$; and (2)
${Y}\leftarrow{X}\rightarrow\mbox{$\Delta$}\leftarrow{C}$. If we condition on
${X}$ and ${\mathbf{Z}}$, then all paths between ${C}$ and ${Y}$ meet the
criteria for conditional independence, so ${Y}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}{C}|({X},{\mathbf{Z}})$ and censoring
mechanism (C3) holds. This means we have consistency of the complete case
estimator.

Now, consider the DAG in Figure 1. Most of the directed edges are hypothesized
to match Figure 1, but one stark distinction is the directed edge from ${Y}$
to ${C}$. We might hypothesize that the current total motor score is directly
associated with the time to censoring, since patients with a higher total
motor score may drop out sooner. We cannot condition on any variables in this
direct path, so ${C}$ is dependent on ${Y}$. We therefore think the censoring
mechanisms may not hold, meaning we may not have guaranteed consistency using
a complete case estimator.

It is important to recognize that these conclusions are not formal, since the
DAGs are made from a priori assumptions regarding subject-matter knowledge.
One could instead argue that in Figure 1, apathy is associated directly with
the time to censoring, since apathetic people may lose interest in the study.
We also note that we cannot consider exogenous censoring from DAGs, which
relies on an expectation assumption rather than an independence assumption.
However, DAGs are still helpful for visualizing the mechanisms.

##  3 Simulation Study

We conduct a simulation study to empirically demonstrate the consistency of
the complete case estimator. Namely, we generate data under six different
combinations of censoring mechanism assumptions to show that a complete case
estimator is consistent when at least mechanism (C1), i.e., exogenous
censoring, holds.

###  3.1 Data Generation and Metrics for Comparison

Our simulations use sample sizes $n=400$ and $n=1200$ and censoring rates of
$25\%$ and $75\%$. The combination of $n=1200$ and a censoring rate of $75\%$
is a similar dataset size to that of the real data example that will be shown
in Section 4. We also use a smaller sample size and smaller censoring rate to
illustrate other dataset types seen in practice.

We generate data for two regression models in equation (1): a linear model
where $m(X,Z;{\boldsymbol{\beta}})=0.5+X-2Z$ and a nonlinear logistic model
where
$m(X,Z;{\boldsymbol{\beta}})=1/[1+\exp\left\\{-5(0.005+0.01X-0.02Z)\right\\}]$.
One thousand replications are generated for each combination of sample size,
censoring rate, and mean function.

For all settings, we generate ${X}\sim\hbox{Uniform}(0,3)$ and
$Z\sim\hbox{Normal}(0,1)$. The generation of ${C}$, $\Delta$, and ${\epsilon}$
vary depending on which of the censoring mechanisms hold. The choices made for
the data generation of ${C}$, $\Delta$, and ${\epsilon}$ are described in
Table 1, and the justification for these data generation choices are described
in Supplementary Materials S.4.

Table 1: Data generation for six simulation settings, each with a different
combination of censoring mechanism assumptions. For some settings, $\Delta$ is
generated directly and $C$ is left unspecified, and for the rest, $C$ is
generated directly and $\Delta=I(X\leq C)$. Let $r$ be the censoring rate in
decimal form.

Censoring Mechanism Setting |  Censoring Variable $\Delta$ or ${C}$ |  Error Term ${\epsilon}$
---|---|---
Exogenous Censoring does not hold: (C1)–(C5) do not hold |  $\mbox{$\Delta$}\sim\hbox{Bernoulli}(1-r)$ |  $\hbox{Normal}(\mu,\sigma^{2})^{1}$ where $\mu=\frac{\sigma}{8}(1-\mbox{$\Delta$})-\frac{\sigma}{8}\frac{r}{1-r}\mbox{$\Delta$}$
Strict Exogenous Censoring does not hold: (C1) holds and (C2)–(C5) do not hold |  $\mbox{$\Delta$}=\begin{cases}1,&|{\epsilon}|\geq\text{the rth quantile of }|{\epsilon}|,\\\ 0&\text{otherwise}\end{cases}$ |  $\hbox{Normal}(0,\sigma^{2})$
Conditional Independence given (X,Z) does not hold: (C1)–(C2) hold and (C3)–(C5) do not hold |  ${C}\sim\begin{cases}\hbox{Uniform}(0,6-6r)&r\geq 0.5,\\\ \hbox{Uniform}(0,\frac{3}{2r})&\text{otherwise}\end{cases}$ |  ${\epsilon}\sim\begin{cases}\hbox{Normal}(0,0.5\sigma^{2})&C\in\\\ &\text{IQR}({C})^{2},\\\ \hbox{Normal}(0,1.5\sigma^{2})&\text{otherwise}\end{cases}$
Conditional Independence given Z does not hold: (C1)–(C3) hold and (C4)–(C5) do not hold |  ${C}\sim\hbox{Uniform}(0,X/r)$ |  $\hbox{Normal}(0,\sigma^{2})$
Independence does not hold: (C1)–(C4) hold and (C5) does not hold |  ${C}\sim\begin{cases}\hbox{Uniform}(0,6-6r)&r\geq 0.5\&\\\ &Z>\text{med}(Z_{1}\ldots,Z_{n})^{3},\\\ \hbox{Uniform}(\frac{3-6r}{2-2r},3)&r\geq 0.5\&\\\ &Z\leq\text{med}(Z_{1}\ldots,Z_{n}),\\\ \hbox{Uniform}(0,\frac{3}{2r})&r<0.5\&\\\ &Z>\text{med}(Z_{1}\ldots,Z_{n}),\\\ \hbox{Uniform}(3-6r,3)&r<0.5\&\\\ &Z\leq\text{med}(Z_{1}\ldots,Z_{n})\end{cases}$ |  $\hbox{Normal}(0,\sigma^{2})$
Independence holds: (C1)–(C5) hold |  ${C}\sim\begin{cases}\hbox{Uniform}(0,6-6r)&r\geq 0.5,\\\ \hbox{Uniform}(0,\frac{3}{2r})&\text{otherwise}\end{cases}$ |  $\hbox{Normal}(0,\sigma^{2})$
1Note that $\sigma^{2}=2$ for the linear mean model and $\sigma^{2}=0.08$ for
the logistic model.
2IQR$(\cdot)$ represents the interquartile range.
3med$(\cdot)$ represents the median.

For each setting, we estimate the regression parameters ${\boldsymbol{\beta}}$
using two methods. First, we use a complete case estimator, where only
observations with an observed ${X}$ were included in the final sample.
Secondly, we use a so-called oracle estimator, where all observations are kept
in the analysis, rather than deleting those that are censored. This method is
not possible in practice since ${X}$ is censored, but it gives us a gold
standard to compare the complete case analysis to. We compare results from the
complete case and oracle estimators in terms of empirical bias and estimated
variability. We report bias in terms of percent bias calculated as
$1/1000\sum_{i=1}^{1000}(\widehat{{\boldsymbol{\beta}}}_{i}-{\boldsymbol{\beta}})/{\boldsymbol{\beta}}\times
100,$ and the estimated variability in terms of the empirical mean of the
estimated standard error calculated as
$1/1000\sum_{i=1}^{1000}\widehat{SE}_{i}$, where $\widehat{SE}_{i}$ is
calculated from the sandwich variance estimator. The sandwich estimator is an
estimate of the asymptotic variance of $\widehat{{\boldsymbol{\beta}}}$
outlined in Theorem 2. Namely,
$\widehat{Var}(\widehat{{\boldsymbol{\beta}}})=\widehat{{\mathbf{A}}}^{-1}\widehat{{\mathbf{B}}}(\widehat{{\mathbf{A}}}^{-1})^{T}$
where
$\widehat{{\mathbf{A}}}=n^{-1}\sum_{i=1}^{n}\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};\widehat{{\boldsymbol{\beta}}})/\partial{\boldsymbol{\beta}}^{T}$,
$\widehat{{\mathbf{B}}}=n^{-1}\sum_{i=1}^{n}\\{\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};\widehat{{\boldsymbol{\beta}}})\\}\\{\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};\widehat{{\boldsymbol{\beta}}})\\}^{T}$,
and $\widehat{SE}=\sqrt{\widehat{Var}(\widehat{{\boldsymbol{\beta}}})/n}$.
Additionally, we calculate the observed coverage probability for a Wald-type
95% confidence interval for each method. Since we are using the oracle
estimator as a gold standard, we want to determine under what simulation
settings (i.e., considering sample size, censoring rate, mean function, and
censoring mechanisms) these metrics are comparable between the two methods and
under what settings they are substantially different.

###  3.2 Results

Table 2:  Simulation results for a linear model with a right-censored
covariate when data are generated under six different combinations of
censoring mechanism assumptions that correspond with those in Table 1. We
report the percent bias, estimated standard errors (SE), and 95% coverage
probabilities for all parameters in the linear model when we estimate
parameters using the complete case estimator (CC) and oracle estimator
(Oracle). Results are based on 1000 simulated datasets, each with a sample
size of 400.

| Percent Bias | Estimated SE | 95% Coverage
---|---|---|---
Censoring Rate | Method | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$
Exogenous Censoring does not hold
| CC | -13.01 | 0.43 | 0.29 | 0.16 | 0.09 | 0.08 | 0.94 | 0.94 | 0.94
25% | Oracle | -0.78 | 0.24 | 0.16 | 0.14 | 0.08 | 0.07 | 0.95 | 0.95 | 0.94
| CC | -109.01 | 0.72 | 0.38 | 0.28 | 0.16 | 0.14 | 0.50 | 0.94 | 0.94
75% | Oracle | -0.67 | 0.20 | 0.17 | 0.14 | 0.08 | 0.07 | 0.96 | 0.95 | 0.94
Strict Exogenous Censoring does not hold
| CC | 0.35 | -0.07 | -0.11 | 0.19 | 0.11 | 0.09 | 0.94 | 0.94 | 0.94
25% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  3 0.93
| CC | -0.78 | -0.16 | -0.30 | 0.48 | 0.27 | 0.24 | 0.94 | 0.95 | 0.92
75% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  3 0.93
Conditional Independence given (X,Z) does not hold
| CC | -1.38 | 0.44 | 0.18 | 0.15 | 0.09 | 0.08 | 0.96 | 0.96 | 0.94
25% | Oracle | -0.70 | 0.19 | 0.11 | 0.14 | 0.08 | 0.07 | 0.95 | 0.94 | 0.94
| CC | -2.18 | 1.34 | 0.41 | 0.24 | 0.43 | 0.14 | 0.94 | 0.94 | 0.94
75% | Oracle | -0.70 | 0.19 | 0.11 | 0.14 | 0.08 | 0.07 | 0.95 | 0.94 | 0.94
Conditional Independence given Z does not hold
| CC | 0.11 | 0.14 | -0.10 | 0.16 | 0.09 | 0.08 | 0.94 | 0.94 | 0.93
25% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  2 0.93
| CC | -0.40 | 0.25 | -0.44 | 0.28 | 0.16 | 0.14 | 0.93 | 0.94 | 0.93
75% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  2 0.93
Independence does not hold
| CC | 0.15 | 0.03 | -0.16 | 0.15 | 0.10 | 0.08 | 0.94 | 0.95 | 0.93
25% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  1 0.93
| CC | -0.50 | 0.70 | -0.45 | 0.23 | 0.24 | 0.15 | 0.93 | 0.92 | 0.92
75% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 |  1 0.93
Independence holds
| CC | 0.14 | 0.00 | -0.18 | 0.15 | 0.10 | 0.08 | 0.94 | 0.95 | 0.93
25% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 | 0.93
| CC | 0.18 | 0.37 | -0.17 | 0.24 | 0.39 | 0.14 | 0.94 | 0.94 | 0.94
75% | Oracle | 0.16 | -0.04 | -0.08 | 0.14 | 0.08 | 0.07 | 0.94 | 0.95 | 0.93

Whether the sample size was 400 (Table 2) or 1200 (Table S.4, Supplementary
Materials S.5), we only see substantial empirical bias (i.e., bias $>10\%$)
when exogenous censoring does not hold, for a linear regression model. The
bias under this setting congregates in the intercept estimate,
$\widehat{\beta}_{0}$, and gets worse with increasing censoring rates. The
amount of bias in the intercept depends on how much the mean of the error term
for $\Delta=1$ differs from $0$ (See Supplementary Materials S.3). Here,
$E({\epsilon}|\mbox{$\Delta$}=1,{X},{\mathbf{Z}})=-(\sigma/8)\times\\{r/(1-r)\\}$.
We choose $\sigma/8$ arbitrarily; if the expectation were closer to $0$, there
would be less bias, and if the difference were further than $0$, then the bias
would be even worse.

The bias for the parameter estimate associated with the censored covariate
${X}$, $\widehat{\beta}_{1}$, however, is quite low across all settings (i.e.,
under $2\%$ bias). If the main goal of this analysis was to get an unbiased
parameter estimate for $\beta_{1}$, then even a violation of exogenous
censoring is okay, as long as the model has an intercept. The importance of
the intercept is illustrated empirically in Table S.5, Supplementary Materials
S.5, where we compare the percent bias for the $n=400$ simulation with the
same simulation that has no intercept (i.e., the mean model is
$m(X,Z;{\boldsymbol{\beta}})=X-2Z$). We see that when the intercept is
removed, the bias transfers to the estimate for $\beta_{1}$ up to $27\%$,
which is undesirable.

The precision of the estimates is a different story, as seen from the standard
error estimates. Estimates from the complete case analysis increase in
variability as the censoring rate increases and/or the sample size decreases
due to a smaller effective sample size in both cases. For example, the
complete case estimator leads to standard errors as high as 0.48 when $n=400$,
compared to 0.14 for the oracle estimator, and 0.25 when $n=1200$, compared to
0.08 for the oracle estimator. Similarly, when $n=400$, the largest standard
error estimate when the censoring rate is $25\%$ is 0.19, as opposed to 0.48
for the $75\%$ censoring rate. The estimates also vary widely in precision
across the different censoring mechanism settings, where we see standard error
estimates for $\widehat{\beta}_{1}$ range from 0.16 to 0.43 when $n=400$ and
the censoring rate is $75\%$. The variation in precision is simply a
consequence of how the data were generated in each setting. This result shows
that despite the empirically unbiased complete case estimates, high censoring
rates can still lead to imprecise estimates, depending on the underlying
properties of the data. Despite the imprecision, however, we see that the
coverage rates are close to the nominal $95\%$ for the complete case estimator
in all settings where we have empirical unbiasedness.

The results from the nonlinear regression model with a logistic mean function
can be found in Supplementary Material S.5 (Tables S.6 and S.7). We see the
most substantial bias for $\widehat{\beta}_{0}$ when exogenous censoring does
not hold, ranging from $200\%$ to almost $2000\%$. For some censoring
mechanism combinations, there is more bias from the $\widehat{\beta}_{1}$
complete case estimator than from the oracle estimator. The bias could be due,
in part, to the effective sample size of the complete case estimator being too
small for consistency to “kick in”. Illustrating this concept, we see bias as
high as $24\%$ when $n=400$ and only $16\%$ when $n=1200$. In addition, the
bias in $\widehat{\beta}_{1}$ is likely due to the high variability we see in
these estimates. For example, in the “Conditional Independence given $Z$ does
not hold” setting, the standard error is seven times greater for complete case
analysis than for oracle, when considering the $\widehat{\beta}_{1}$ at
$n=400$ and censoring rate $75\%$. We compare that relative efficiency to only
three times greater in the linear case, where we see less bias.

##  4 Apathy in Huntington disease as it relates to clinical diagnosis time

###  4.1 Motivation and Data

Huntington disease is a neurodegenerative disease characterized by motor
disturbances, cognitive decline, and psychiatric symptoms [24]. Huntington
disease is fully penetrant, meaning that if someone has the causal gene
mutation, they are guaranteed to develop the disease in their lifetime.
Individuals with the gene mutation and some symptoms, but not yet displaying
enough motor disturbances to merit a clinical diagnosis, are in the
“prodromal” disease stage [23]. Treating Huntington disease patients in the
prodromal stage is important for possibly delaying diagnosis, avoiding
irreparable damage and/or slowing symptom progression [25].

One of the most common prodromal symptoms is apathy, described as lacking
feeling, emotion, interest, and/or concern [26, 27]. Apathy is distressing not
only to the patient but also to the caretaker, since apathy presents as
stubbornness and hostility which can strain relationships [28, 29]. There is
currently no treatment for apathy symptoms in Huntington disease, despite
efforts to develop one [30]. Therefore, one step in treating apathy as early
as possible would be to characterize how it changes as a function of time to
diagnosis, or disease progression, specifically within the prodromal stage.
Many researchers have concluded that apathy does indeed appear to be greater
on average for those post-diagnosis than for those in the prodromal disease
stage [31, 32]. We are interested in determining how apathy changes over time
to see where on the disease path apathy begins to increase or worsen. Knowing
this information would enable researchers to target this time of initial
apathy increase for future treatments and interventions.

The Neurobiological Predictors of Huntington Disease (PREDICT-HD) study is a
longitudinal, observational study designed to explore symptoms of Huntington
disease in the prodromal stage [25]. In one study of apathy in the prodromal
period using PREDICT-HD data [33], a cross-sectional linear regression
analysis was performed with data from the baseline visit and apathy as the
outcome. One variable [33] included as a covariate in their model is a
commonly used measure of disease burden called the CAG-Age-Product (CAP)
score, which is a function of CAG repeat length and age [23]. The purpose of
the CAP score is to predict “time to clinical diagnosis”, or briefly, “time to
diagnosis”. Since [33] did not see a significant relationship between CAP
score and apathy while controlling for other variables, they concluded that
apathy may not be directly related to the progression of Huntington disease
during the prodromal stage. This conclusion would mean that despite the
literature arguing apathy increases across disease stages, the relationship
may be more difficult to detect within the prodromal stage or may be
confounded by other variables.

Given the longitudinal nature of PREDICT-HD, some of the participants were
given a clinical diagnosis during the course of the study, and therefore their
time to diagnosis was observed. Instead of using the CAP score to draw
conclusions about the relationship between apathy and the progression of
Huntington disease, we can use the patient’s actual observed time to
diagnosis. This approach is important because CAP scores are known to be
imprecise and do not always accurately reflect clinical diagnosis [34].
Therefore, running the analysis with the true time to diagnosis can more
accurately assess whether the lack of relationship [33] drew between apathy
and disease progression is valid. The time to diagnosis covariate, however, is
right-censored, since not everyone was given a clinical diagnosis by the time
the study was over. This case presents an opportunity to analyze these data
with a complete case estimator, which is consistent under one of the censoring
mechanism assumptions (C1) –(C5).

###  4.2 Considering Censoring Mechanism Assumptions

As discussed in Section 2.5, it is not possible to check whether censoring
mechanism assumptions hold based on the observed data alone, due to the
censoring of ${X}$. That being said, the dropout rate for PREDICT-HD was less
than $5\%$ per year, and sample size variation year to year was due to study
design considerations rather than dropouts [35]. Keeping this in mind, we can
look back at the DAG in Figure 1 to revisit our proposed relationships between
these variables. We remember from Section 2.5 that
${Y}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}{C}|({X},{\mathbf{Z}})$ as long as there is no
direct path between ${Y}$ and ${C}$. We mentioned the possibility that
apathetic people may lose interest in the study and dropout, causing a
relationship between apathy and the time to censoring. However, given such a
low dropout rate, we are confident in assuming this path does not exist in the
true DAG. Therefore, we assume censoring mechanism (C3), which will lead to a
consistent complete case estimator.

###  4.3 Analysis

We focus on analyzing individuals who were genetically confirmed to have the
gene mutation for Huntington disease (i.e., a CAG repeat length of at least
36). We also remove anyone who was given a clinical diagnosis at the first
visit because we wanted a subset of individuals in the prodromal stage at
baseline. Furthermore, we run the analysis using data from the second visit,
since some patients were diagnosed at the second visit, giving us a wider
range for the time to diagnosis variable. Finally, we remove any individuals
with missing outcome or covariate information, leaving us with an analysis set
of 781, where 212 have an observed time to diagnosis. This is a censoring rate
of about $73\%$.

The goal of this analysis is to quantify the linear association between apathy
and time to clinical diagnosis (in years), while controlling for sex,
depression, total motor score, years of education, single digit modalities
test score (SDMT), CAG repeat length, and age, similar to the model
construction by [33]. Apathy was measured using the Frontal Systems Behavior
Scale, which measures behavior associated with frontal systems of the brain
and has subscales for apathy, disinhibition, and executive function [36].
Depression was measured using the Symptom Checklist-90-R, which measures
psychological distress and has several subscales including depression [37].
Both the patient and caretaker filled out these rating scales and we use the
caretaker results. We chose a linear model to reflect the relationship between
apathy and disease progression from previous studies [31, 33]. This set of
covariates has been shown to be related to apathy, and we want to control for
them to see if time to diagnosis is directly related to apathy, rather than
through a relationship with these control variables. We want to compare our
results to the model from [33] to see if using time to diagnosis yields
different results than the CAP score. If the results agree, we will conclude
that there is no evidence for a strong, direct relationship between apathy and
disease progression during the prodromal stage. If the results are different,
there will be some evidence to suggest that apathy does change with disease
progression when the true time to diagnosis variable is used.

We use the following linear regression model to test these relationships:

$\displaystyle
Y_{i}=\beta_{0}+\beta_{1}X_{i}+\beta_{2}Z_{1i}+\beta_{3}Z_{2i}+\beta_{4}Z_{3i}+\beta_{5}Z_{4i}+\beta_{6}Z_{5i}+\beta_{7}Z_{6i}+\beta_{8}Z_{7i}+\epsilon_{i},$

where $Y_{i}$ is apathy score, $X_{i}$ is time to diagnosis in years, $Z_{1i}$
is biological sex indicator (1 if female), $Z_{2i}$ is depression score,
$Z_{3i}$ is total motor score, $Z_{4i}$ is number years of education, $Z_{5i}$
is SDMT score, $Z_{6i}$ is CAG repeat length and $Z_{7i}$ is age in years, all
at the second visit. All covariates, except time to diagnosis, were centered
and scaled.

The results from the regression (Table 3) show no evidence that our measure of
disease progression (i.e., time to diagnosis) is associated with apathy after
controlling for other variables. These results agree with the same regression
replacing time to diagnosis with CAP score (Table S.8, Supplementary Materials
S.5) and the results from the CAP score model in [33]. Therefore, our results
agree with the conclusion made from the CAP score model, even if using CAP
score may not have been a precise measure of disease progression. It is also
worth noting that the only covariate whose confidence interval does not
contain the null value in our time to diagnosis model is depression. The [33]
model and our CAP score model also saw evidence of a relationship between
apathy and total motor score, as well as cognitive control (SDMT score, here).
One reason for this result could be the lack of efficiency that comes with a
complete case analysis. Because we deleted so much data for the time to
diagnosis analysis, our standard errors are quite large, despite being
confident in having a consistent estimator. Overall, our analysis shows that
it may be difficult to detect a change in apathy scores in Huntington disease
patients during the prodromal change, despite the clear increase across
disease stage groups presented in the literature. From these results, we
cannot recommend a clear way to recruit patients for trials that would be
expected to detect a change in apathy over time.

Table 3:  Results from the PREDICT-HD study quantifying the relationship between apathy score and time to clinical diagnosis, controlling for other covariates. The parameters are estimated using a complete case analysis (n = 212). We report the parameter estimate, the estimated standard error (SE), and the Wald-type 95% confidence interval (95% CI).  Parameter | Estimate | Estimated SE |  $95\%$ CI
---|---|---|---
$\beta_{0}:$ Intercept | 15.02 | 0.96 | (13.15, 16.90)
$\beta_{1}:$ Time to Diagnosis | -0.015 | 0.15 | (-0.30, 0.27)
$\beta_{2}:$ Biological Sex | -1.38 | 0.86 | (-3.05, 0.30)
$\beta_{3}:$ Depression | 4.16 | 0.38 | (3.41, 4.91)
$\beta_{4}:$ Total Motor Score | 0.61 | 0.38 | (-0.13, 1.36)
$\beta_{5}:$ Years of Education | 0.30 | 0.46 | (-0.60, 1.21)
$\beta_{6}:$ SDMT Score | -0.24 | 0.50 | (-1.23, 0.75)
$\beta_{7}:$ CAG Repeat Length | 0.58 | 0.54 | (-0.48, 1.65)
$\beta_{8}:$ Age | -0.36 | 0.51 | (-1.36 , 0.64)

##  5 Discussion

This paper provides a unified summary of complete case estimator consistency
for censored covariates, and we recommend that these properties are taken into
consideration when using the method in any scenario. Despite its main drawback
of lack of precision due to the discarding of data, the complete case
estimator can serve as a powerful initial estimator for complex methods or as
a baseline comparison when benchmarking novel methods. Additionally, using the
complete case estimator does not require any model assumptions on the censored
covariate, meaning there is less chance of model misspecification leading to
bias.

We show here that for regression models with a right-censored covariate, the
complete case estimator is consistent when any one of the five censoring
mechanisms (C1) through (C5) hold. These assumptions revolve around the idea
that the censoring variable (in terms of ${C}$ or $\Delta$) is independent of
the outcome ${Y}$, conditional or unconditional on covariates ${X}$ and
${\mathbf{Z}}$. Moreover, a consistent estimator is not guaranteed when the
exogenous censoring assumption does not hold. Although it is impossible to
verify the assumptions made, domain knowledge and DAGs can be used to help
gain confidence. Lastly, while we presented results when a covariate is
randomly right-censored, our conclusions similarly apply when the covariate is
left-censored (i.e., $\Delta=I(X\geq C$)) or when the censoring is due to a
limit of detection (i.e., mechanisms (C1) –(C2) only because there is no
random $C$).

SUPPLEMENTARY MATERIAL

The supplementary material contains all proofs, additional simulation tables
and results for the Huntington disease example. R code for the simulation
study is provided at
https://github.com/marissaashner/Complete_Case_Censored_Covariates.

## Acknowledgments

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under grant DGE-2040435, the National Institute
of Environmental Health Sciences under grant P30ES010126, and the National
Institute of Neurological Disorders and Stroke under grant K01NS099343. The
authors thank PREDICT-HD for permission to present their data.

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Supplementary Material to Exploring the validity of the complete case analysis
for regression models with a right-censored covariate

##  S.1 Proof of Theorem 1

We prove Theorem 1 by proving four implications between the five censoring
mechanisms. Each of these proofs is labeled below, where ‘$\implies$’ means
‘implies’.

Censoring mechanism (C5) $\implies$ Censoring mechanism (C4): Assuming
censoring mechanism (C5) holds, i.e., ${C}$ is independent of
$({X},{Y},{\mathbf{Z}})$, we have that

$\displaystyle f_{{C},{\mathbf{Z}}}(c,{\mathbf{z}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{{C},{Y},{X},{\mathbf{Z}}}(c,y,x,{\mathbf{z}})dydx=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{{C}}(c)f_{{Y},{X},{\mathbf{Z}}}(y,x,{\mathbf{z}})dydx$
$\displaystyle=$ $\displaystyle
f_{{C}}(c)\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f_{{Y},{X},{\mathbf{Z}}}(y,x,z)dydx=f_{{C}}(c)f_{{\mathbf{Z}}}(z).$

The result above shows that the joint density of ${C}$ and ${\mathbf{Z}}$
equals the product of their marginal distributions, which means ${C}$ and
${\mathbf{Z}}$ are marginally independent. This marginal independence combined
with censoring mechanism (C5) further implies that

$\displaystyle f_{({C},{Y},{X})|{\mathbf{Z}}}(c,y,x|{\mathbf{z}})$
$\displaystyle=$
$\displaystyle\frac{f_{{C},{Y},{X},{\mathbf{Z}}}(c,y,x,{\mathbf{z}})}{f_{{\mathbf{Z}}}({\mathbf{z}})}$
$\displaystyle=$
$\displaystyle\frac{f_{{C}}(c)f_{{Y},{X},{\mathbf{Z}}}(y,x,{\mathbf{z}})}{f_{{\mathbf{Z}}}({\mathbf{z}})}$
$\displaystyle=$ $\displaystyle
f_{{C}}(c)f_{{Y},{X}|{\mathbf{Z}}}(y,x|{\mathbf{z}})$ $\displaystyle=$
$\displaystyle
f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y},{X}|{\mathbf{Z}}}(y,x|{\mathbf{z}}).$

The second equality above holds because we assumed censoring mechanism (C5),
and the last equality holds because ${C}$ and ${\mathbf{Z}}$ are marginally
independent. We therefore showed
$f_{({C},{Y},{X})|{\mathbf{Z}}}(c,y,x|z)=f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y},{X}|{\mathbf{Z}}}(y,x|z)$,
which proves ${C}$ is independent of $({X},{Y})$ given ${\mathbf{Z}}$.

Censoring mechanism (C4) $\implies$ Censoring mechanism (C3): Assuming
censoring mechanism (C4), i.e., ${C}$ is independent of $({X},{Y})$ given
${\mathbf{Z}}$, we have that

$\displaystyle f_{{C},{X}|{\mathbf{Z}}}(c,x|{\mathbf{z}})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}f_{{C},{Y},{X}|{\mathbf{Z}}}(c,y,x|{\mathbf{z}})dy=\int_{-\infty}^{\infty}f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y},{X}|{\mathbf{Z}}}(y,x|{\mathbf{z}})dy$
$\displaystyle=$ $\displaystyle
f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})\int_{-\infty}^{\infty}f_{{Y},{X}|{\mathbf{Z}}}(y,x|{\mathbf{z}})dy=f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}}).$

Having shown
$f_{{C},{X}|{\mathbf{Z}}}(c,x|{\mathbf{z}})=f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}})$,
we proved that ${C}$ and ${X}$ are independent conditional on ${\mathbf{Z}}$.
This conditional independence combined with censoring mechanism (C4) implies
that

$\displaystyle f_{{C},{Y}|{X},{\mathbf{Z}}}(c,y|{x},{\mathbf{z}})$
$\displaystyle=$
$\displaystyle\frac{f_{{C},{Y},{X}|{\mathbf{Z}}}(c,y,x|{\mathbf{z}})}{f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}})}$
$\displaystyle=$
$\displaystyle\frac{f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y},{X}|{\mathbf{Z}}}(x,y|z)}{f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}})}$
$\displaystyle=$
$\displaystyle\frac{f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y}|{X},{\mathbf{Z}}}(y|x,{\mathbf{z}})f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}})}{f_{{X}|{\mathbf{Z}}}(x|{\mathbf{z}})}$
$\displaystyle=$ $\displaystyle
f_{{C}|{\mathbf{Z}}}(c|{\mathbf{z}})f_{{Y}|{X},{\mathbf{Z}}}(y|x,{\mathbf{z}})$
$\displaystyle=$ $\displaystyle
f_{{C}|{X},{\mathbf{Z}}}(c|x,{\mathbf{z}})f_{{Y}|{X},{\mathbf{Z}}}(y|x,{\mathbf{z}}).$

The second equality holds by censoring mechanism (C4), and the last equality
holds because ${C}$ and ${X}$ are independent given ${\mathbf{Z}}$. Because we
showed that the conditional density of $({C},{Y})|{X},{\mathbf{Z}}$ is equal
to the product of the conditional densities of ${C}|{X},{\mathbf{Z}}$ and
${Y}|{X},{\mathbf{Z}}$, we proved ${C}$ is independent of ${Y}$ given
$({X},{\mathbf{Z}})$.

Censoring mechanism (C3) $\implies$ Censoring mechanism (C2): Let
$F_{V}(v)=P(V\leq v)$ denote the cumulative distribution function for a random
variable $V$. Assuming censoring mechanism (C3), i.e., ${C}$ is independent of
${Y}$ given $({X},{\mathbf{Z}})$, we have that

$\displaystyle
F_{{C},{Y}|{X},{\mathbf{Z}}}(c,y|X=x,{\mathbf{Z}}={\mathbf{z}})=F_{{C}|{X},{\mathbf{Z}}}(c|X=x,{\mathbf{Z}}={\mathbf{z}})F_{{Y}|{X},{\mathbf{Z}}}(y|X=x,{\mathbf{Z}}={\mathbf{z}}),$

for all $c,y,x,{\mathbf{z}}$. If we show that

$\displaystyle
F_{\mbox{$\Delta$},{\epsilon}|{X},{\mathbf{Z}}}(d,e|X=x,{\mathbf{Z}}={\mathbf{z}})=F_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(d|X=x,{\mathbf{Z}}={\mathbf{z}})F_{{\epsilon}|{X},{\mathbf{Z}}}(e|X=x,{\mathbf{Z}}={\mathbf{z}})$
(1)

for all $d,e,x,{\mathbf{z}}$, we will have proven that censoring mechanism
(C2) holds. We will prove equation (1) holds by considering three cases for
$d\in\mathbb{R}$:

  1. 1.

Case 1: $d<0$. Because $\Delta$ can only be $0$ or $1$ with nonzero
probability, then it immediately follows that
$F_{\mbox{$\Delta$},{\epsilon}|{X},{\mathbf{Z}}}(d,e|X=x,{\mathbf{Z}}={\mathbf{z}})=0$
and $F_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(d|X=x,{\mathbf{Z}}={\mathbf{z}})=0$.
Therefore, equation (1) holds trivially.

  2. 2.

Case 2: $0\leq d<1$. When $0\leq d<1$, the only plausible value for $\Delta$
that gives a non-zero probability is $d=0$. Therefore,

$\displaystyle
F_{\mbox{$\Delta$},{\epsilon}|{X},{\mathbf{Z}}}(d,e|X=x,{\mathbf{Z}}={\mathbf{z}})$
$\displaystyle=$ $\displaystyle P(\mbox{$\Delta$}\leq d,{\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle
P\\{\mbox{$\Delta$}=0,{Y}-m({X},{\mathbf{Z}};{\boldsymbol{\beta}})\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}}\\}$ $\displaystyle=$ $\displaystyle
P({C}\leq{X}|X=x,{\mathbf{Z}}={\mathbf{z}})P\\{{Y}\leq
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})+e|X=x,{\mathbf{Z}}={\mathbf{z}}\\}$
$\displaystyle=$ $\displaystyle
P(\mbox{$\Delta$}=0|X=x,{\mathbf{Z}}={\mathbf{z}})P({\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle
P(\mbox{$\Delta$}\leq d|X=x,{\mathbf{Z}}={\mathbf{z}})P({\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle
F_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(d|X=x,{\mathbf{Z}}={\mathbf{z}})F_{{\epsilon}|{X},{\mathbf{Z}}}(e|X=x,{\mathbf{Z}}={\mathbf{z}}),$

where the third equality holds by censoring mechanism (C3).

  3. 3.

Case 3: $d\geq 1$. When $d\geq 1$, the plausible values for $\Delta$ that
gives a non-zero probability are $d=0$ and $d=1$. Therefore,

$\displaystyle
F_{\mbox{$\Delta$},{\epsilon}|{X},{\mathbf{Z}}}(d,e|X=x,{\mathbf{Z}}={\mathbf{z}})$
$\displaystyle=$ $\displaystyle P(\mbox{$\Delta$}\leq d,{\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle
P(\mbox{$\Delta$}=0\text{ or }1,{\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle 1\times
P({\epsilon}\leq e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$
$\displaystyle P(\mbox{$\Delta$}\leq
d|X=x,{\mathbf{Z}}={\mathbf{z}})P({\epsilon}\leq
e|X=x,{\mathbf{Z}}={\mathbf{z}})$ $\displaystyle=$ $\displaystyle
F_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(d|X=x,{\mathbf{Z}}={\mathbf{z}})F_{{\epsilon}|{X},{\mathbf{Z}}}(e|X=x,{\mathbf{Z}}={\mathbf{z}}).$

Censoring mechanism (C2) $\implies$ Censoring mechanism (C1): Assuming
censoring mechanism (C2), i.e., $\Delta$ is independent of ${\epsilon}$ given
$({X},{\mathbf{Z}})$, we have

$\displaystyle
E({\epsilon}|\mbox{$\Delta$},{X},{\mathbf{Z}})=E({\epsilon}|{X},{\mathbf{Z}}).$

But because we assume $E({\epsilon}|{X},{\mathbf{Z}})={\mathbf{0}}$, we have
that censoring mechanism (C1) holds.

##  S.2 Proof of Theorem 2

Consistency:  Following the work of [38] and [8], consistency will follow so
long as we show that the following three regularity conditions hold, under the
assumption that (_R1_)–(_R3_) hold for the fully observed estimating equation
approach, as laid out in Theorem 2 in the main text.

  1. (R1enumi)

$E\\{\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},X,{\mathbf{Z}};{\boldsymbol{\beta}})\\}={\mathbf{0}}$
for all ${\boldsymbol{\beta}}$

  2. (R2enumi)

$E\left\\{\frac{\partial\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}$
is nonsingular

  3. (R3enumi)

$n^{-1}\sum_{i=1}^{n}\frac{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}$
converges uniformly to
$E\\{\frac{\partial\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},X,{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\\}$
in a neighborhood of ${\boldsymbol{\beta}}_{0}$, where
${\boldsymbol{\beta}}_{0}$ is the true parameter value.

(R1enumi): First, we show
$E\\{\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}={\mathbf{0}}$
for all ${\boldsymbol{\beta}}$. Observe that

$\displaystyle
E\\{\mbox{$\Delta$}{\boldsymbol{\Phi}}(Y,{X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}=E[\Delta_{i}{\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\{Y-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\}]$
$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}},\mbox{$\Delta$}}\\{E_{Y|{X},{\mathbf{Z}},\mbox{$\Delta$}}[\Delta_{i}{\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\{Y-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\}|{X},{\mathbf{Z}},\mbox{$\Delta$}]\\}$
$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}},\mbox{$\Delta$}}[\Delta_{i}{\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\{E_{Y|{X},{\mathbf{Z}},\mbox{$\Delta$}}(Y|{X},{\mathbf{Z}},\mbox{$\Delta$})-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})\\}]$
$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}},\mbox{$\Delta$}}(\Delta_{i}{\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})[E_{Y|{X},{\mathbf{Z}},\mbox{$\Delta$}}\\{\epsilon+m(X,{\mathbf{Z}};{\boldsymbol{\beta}})|{X},{\mathbf{Z}},\mbox{$\Delta$}\\}-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})])$
$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}},\mbox{$\Delta$}}(\Delta_{i}{\mathbf{A}}(X,{\mathbf{Z}};{\boldsymbol{\beta}})[E_{Y|{X},{\mathbf{Z}},\mbox{$\Delta$}}\\{\epsilon|{X},{\mathbf{Z}},\mbox{$\Delta$}\\}+m(X,{\mathbf{Z}};{\boldsymbol{\beta}})-m(X,{\mathbf{Z}};{\boldsymbol{\beta}})]),$

where the second equality holds by iterated expectations conditioning on
$\mbox{$\Delta$},{X},{\mathbf{Z}}$. Since we are conditioning on
$\mbox{$\Delta$},{X},{\mathbf{Z}}$, we can pull any terms dependent on these
variables out of the inner expectation in the third equality. From the last
equality, the $m(X,{\mathbf{Z}};{\boldsymbol{\beta}})$ will cancel out,
leaving the inner expectation which equals ${\mathbf{0}}$ by exogenous
censoring. Therefore,
$E\\{\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}={\mathbf{0}}$.

(R2enumi): Now, we will show
$E\left\\{\frac{\partial\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}$
is nonsingular. By regularity condition (_R2_), we have that
$E\left\\{\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}$
is nonsingular. Then,

$\displaystyle
E\left\\{\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}$
$\displaystyle=$ $\displaystyle
E\left[\frac{\partial{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})\\{{Y}-m({X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}}{\partial{\boldsymbol{\beta}}^{T}}\right]$
$\displaystyle=$ $\displaystyle
E\left[\frac{\partial{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\\{{Y}-m({X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}-{\mathbf{A}}(\cdot)\frac{\partial
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\right],$

where the second equality holds by the product rule for derivatives. Consider
the first term,
$E\left[\partial{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})/\partial{\boldsymbol{\beta}}\\{{Y}-m({X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}\right]$.
Using iterated expectations, this is equal to

$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}}}\left(E_{{Y}|{X},{\mathbf{Z}}}\left[\frac{\partial{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\\{{Y}-m({X},{\mathbf{Z}};{\boldsymbol{\beta}})\\}|{X},{\mathbf{Z}}\right]\right)$
$\displaystyle=$ $\displaystyle
E_{{X},{\mathbf{Z}}}\left\\{\frac{\partial{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}E_{{Y}|{X},{\mathbf{Z}}}\left({\epsilon}|{X},{\mathbf{Z}}\right)\right\\},$

and the inner expectation is ${\mathbf{0}}$ by the original model assumption
on the error term. This means our assumption on the full data case simplifies
to nonsingularlity of the following matrix:
$-E\left\\{{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}}){\partial
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})}/{\partial{\boldsymbol{\beta}}}\right\\}.$

We can simplify the expectation of the complete case estimating function the
same way, and the first term will also equal ${\mathbf{0}}$ via iterative
expectations and exogenous censoring. Then we want to show the nonsingularity
of
$-E\left\\{\mbox{$\Delta$}{\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}}){\partial
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})}/{\partial{\boldsymbol{\beta}}}\right\\}.$
Observe that this expectation is equal to

$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\delta{\mathbf{A}}(x,z;{\boldsymbol{\beta}})\frac{\partial
m(x,z;{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}f_{{X},{\mathbf{Z}},\mbox{$\Delta$}}(x,z,\delta)d\delta
dxdz$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\mathbf{A}}(x,z;{\boldsymbol{\beta}})\frac{\partial
m(x,z;{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}f_{{X},{\mathbf{Z}}}(x,z)\left\\{\int_{-\infty}^{\infty}\delta
f_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(\delta|x,z)d\delta\right\\}dxdz$
$\displaystyle=$ $\displaystyle
E\left\\{P(\mbox{$\Delta$}=1|{X},{\mathbf{Z}}){\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})\frac{\partial
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}}\right\\},$

where the second equality holds since
$f_{{X},{\mathbf{Z}},\mbox{$\Delta$}}(x,z,\delta)=f_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(\delta|x,z)f_{{X},{\mathbf{Z}}}(x,z)$,
and the third equality holds since $\int_{-\infty}^{\infty}\delta
f_{\mbox{$\Delta$}|{X},{\mathbf{Z}}}(\delta|x,z)d\delta=P(\mbox{$\Delta$}=1|{X},{\mathbf{Z}})$.
This can be rewritten as
$E\left\\{{\mathbf{A}}^{*}({X},{\mathbf{Z}};{\boldsymbol{\beta}}){\partial
m({X},{\mathbf{Z}};{\boldsymbol{\beta}})}/{\partial{\boldsymbol{\beta}}}\right\\}$,
where
$A^{*}({X},{\mathbf{Z}};{\boldsymbol{\beta}})=P(\mbox{$\Delta$}=1|{X},{\mathbf{Z}}){\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})$.
Since we assumed the original matrix was nonsingular for arbitrary
${\mathbf{A}}({X},{\mathbf{Z}};{\boldsymbol{\beta}})$, it is nonsingular for
$A^{*}({X},{\mathbf{Z}};{\boldsymbol{\beta}})$.

(R3enumi): Finally, we will show
$n^{-1}\sum_{i=1}^{n}{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}})}/{\partial{\boldsymbol{\beta}}^{T}}$
converge uniformly to
$E\left\\{\frac{\partial\mbox{$\Delta$}{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}$
in a neighborhood of ${\boldsymbol{\beta}}_{0}$, where
${\boldsymbol{\beta}}_{0}$ is the true parameter value. As in [8], uniform
convergence would be satisfied if sample paths of
$\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}})/\partial{\boldsymbol{\beta}}^{T}$
are continuous in ${\boldsymbol{\beta}}$ about ${\boldsymbol{\beta}}_{0}$
almost surely and

$\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|\leq
g({Y},{X},{\mathbf{Z}},{C}),$

where $E\\{g({Y},{X},{\mathbf{Z}},{C})\\}<\infty$ and
$\mathcal{N}({\boldsymbol{\beta}}_{0})$ denotes a neighborhood in
${\boldsymbol{\beta}}$ about ${\boldsymbol{\beta}}_{0}$.

Remember our assumption that uniform convergence holds for the full data
setting. More specifically, using this last definition, this means that sample
paths of $\partial{\boldsymbol{\Phi}}/\partial{\boldsymbol{\beta}}^{T}$ are
continuous in ${\boldsymbol{\beta}}$ about ${\boldsymbol{\beta}}_{0}$ almost
surely and

$\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|\leq
g^{*}({Y},{X},{\mathbf{Z}}),$

where $E\\{g^{*}({Y},{X},{\mathbf{Z}})\\}<\infty$ and
$\mathcal{N}({\boldsymbol{\beta}}_{0})$ denotes a neighborhood in
${\boldsymbol{\beta}}$ about ${\boldsymbol{\beta}}_{0}$. We now want to show
that uniform convergence holds when we replace ${\boldsymbol{\Phi}}$ with
$\mbox{$\Delta$}{\boldsymbol{\Phi}}$. Consider

$\displaystyle g^{*}({Y},{X},{\mathbf{Z}})$ $\displaystyle\geq$
$\displaystyle\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|$
$\displaystyle\geq$
$\displaystyle\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\Delta\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|,$

where the second inequality holds since $\Delta$ can only take on values $0$
or $1$. Since $\Delta$ is a parameter that is not a function of
${\boldsymbol{\beta}}$, we can move it inside the derivative as such:

$\displaystyle\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\Delta\partial{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|=\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\partial\Delta{\boldsymbol{\Phi}}({Y},{X},{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|.$

Let $g({Y},{X},{\mathbf{Z}},{C})=g^{*}({Y},{X},{\mathbf{Z}})$. Then we’ve
shown

$\sup_{{\boldsymbol{\beta}}\in\mathcal{N}({\boldsymbol{\beta}}_{0})}\left|\frac{\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}})}{\partial{\boldsymbol{\beta}}^{T}}\right|\leq
g({Y},{X},{\mathbf{Z}},{C}).$

Asymptotic Normality:  Now, we want to show that the asymptotic distribution
of $\widehat{{\boldsymbol{\beta}}}$ is

$\sqrt{n}(\widehat{{\boldsymbol{\beta}}}-{\boldsymbol{\beta}}_{0})\rightarrow\hbox{Normal}({\mathbf{0}},{\mathbf{A}}^{-1}{\mathbf{B}}({\mathbf{A}}^{-1})^{T}),$

where
${\mathbf{A}}=E\\{\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})/\partial{\boldsymbol{\beta}}^{T}\\}$
and
${\mathbf{B}}=E[\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}^{T}]$.
This proof will follow similarly to that of [8]. First, we take the first
order Taylor expansion of the estimating equation about
${\boldsymbol{\beta}}_{0}$:

$\displaystyle{\mathbf{0}}=\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};\widehat{{\boldsymbol{\beta}}})=\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}_{0})+\left\\{\sum_{i=1}^{n}\frac{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}^{*})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}(\widehat{{\boldsymbol{\beta}}}-{\boldsymbol{\beta}}_{0}),$

where ${\boldsymbol{\beta}}^{*}$ is an intermediate value between
$\widehat{{\boldsymbol{\beta}}}$ and ${\boldsymbol{\beta}}_{0}$. Given our
nonsingularity assumption, we then isolate the difference between
$\widehat{{\boldsymbol{\beta}}}$ and ${\boldsymbol{\beta}}_{0}$ such that

$\displaystyle\sqrt{n}(\widehat{{\boldsymbol{\beta}}}-{\boldsymbol{\beta}}_{0})=-\left\\{\frac{1}{n}\sum_{i=1}^{n}\frac{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}^{*})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}^{-1}\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}_{0}).$

Now, note that, by the law of large numbers,

$\displaystyle-\left\\{\frac{1}{n}\sum_{i=1}^{n}\frac{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}^{*})}{\partial{\boldsymbol{\beta}}^{T}}\right\\}^{-1}\rightarrow_{p}-\left[E\left\\{\frac{\partial\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}_{0}}{\partial{\boldsymbol{\beta}}}\right\\}\right],$

and by the Central Limit Theorem,

$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}_{0})$
$\displaystyle=$
$\displaystyle\sqrt{n}\times(\frac{1}{n}\sum_{i=1}^{n}\Delta_{i}{\boldsymbol{\Phi}}(Y_{i},X_{i},{\mathbf{Z}}_{i};{\boldsymbol{\beta}}_{0})-{\mathbf{0}})$
$\displaystyle\rightarrow_{d}$
$\displaystyle\hbox{Normal}({\mathbf{0}},Var\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}),$

where
$Var\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}=E[\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}^{T}]$.
Finally, by Slutsky’s Theorem,

$\displaystyle\sqrt{n}(\widehat{{\boldsymbol{\beta}}}-{\boldsymbol{\beta}}_{0})\rightarrow_{d}\hbox{Normal}({\mathbf{0}},{\mathbf{A}}^{-1}{\mathbf{B}}({\mathbf{A}}^{-1})^{T}),$

where
${\mathbf{A}}=E\\{\partial\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})/\partial{\boldsymbol{\beta}}^{T}\\}$
and
${\mathbf{B}}=E[\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}\\{\Delta{\boldsymbol{\Phi}}(Y,X,{\mathbf{Z}};{\boldsymbol{\beta}}_{0})\\}^{T}]$.

##  S.3 Investigation of Bias for Linear Model

In this section, we explore the bias that can occur for a complete case
analysis with a linear model if exogenous censoring (C1) does not hold. This
follows the discussion from Section 2.4. Without loss of generality, we will
show the simplest case where there is an intercept but no other ${\mathbf{Z}}$
covariates such that $m(\cdot)=\beta_{0}+\beta_{1}X$. The estimating equation
from Section 2.4 will have two components:

  1. 1.

$\frac{1}{n}\sum_{i=1}^{n}\Delta_{i}(Y_{i}-\beta_{0}-\beta_{1}X_{i})=0$ and

  2. 2.

$\frac{1}{n}\sum_{i=1}^{n}\Delta_{i}X_{i}(Y_{i}-\beta_{0}-\beta_{1}X_{i})=0$.

Considering the first component, we can isolate $\beta_{0}$ to get
$\widehat{\beta_{0}}=\sum_{i=1}^{n}\Delta_{i}(Y_{i}-\widehat{\beta}_{1}X_{i})/\sum_{i=1}^{n}\Delta_{i}$.
Then, plugging $\widehat{\beta}_{0}$ in for $\beta_{0}$ in the second
component, we can isolate $\beta_{1}$ to get
$\widehat{\beta}_{1}=\sum_{i=1}^{n}\Delta_{i}X_{i}(Y_{i}-\sum_{i=1}^{n}\Delta_{i}Y_{i}/\sum_{i=1}^{n}\Delta_{i})/\\{\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}-(\sum_{i=1}^{n}\Delta_{i}X_{i})^{2}/\sum_{i=1}^{n}\Delta_{i}\\}$.

We will show that $\widehat{\beta}_{1}$ is unbiased, even if exogenous
censoring is broken, by showing the expectation is equal to $\beta_{1}$.

$\displaystyle E(\widehat{\beta}_{1})$ $\displaystyle=$ $\displaystyle
E\left\\{\frac{\sum_{i=1}^{n}\Delta_{i}X_{i}(Y_{i}-\sum_{i=1}^{n}\Delta_{i}Y_{i}/\sum_{i=1}^{n}\Delta_{i})}{\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}-(\sum_{i=1}^{n}\Delta_{i}X_{i})^{2}/\sum_{i=1}^{n}\Delta_{i}}\right\\}$
$\displaystyle=$ $\displaystyle
E_{X,\Delta}\left[\frac{\sum_{i=1}^{n}\Delta_{i}X_{i}\\{E(Y_{i}|X_{i},\Delta_{i})-\sum_{i=1}^{n}\Delta_{i}E(Y_{i}|X_{i},\Delta_{i})/\sum_{i=1}^{n}\Delta_{i}\\}}{\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}-(\sum_{i=1}^{n}\Delta_{i}X_{i})^{2}/\sum_{i=1}^{n}\Delta_{i}}\right].$

Note that
$E(Y_{i}|X_{i},\Delta_{i})=E(\epsilon_{i}|X_{i},\Delta_{i})+\beta_{0}+\beta_{1}X_{i}=E(\epsilon_{i}|\Delta_{i})+\beta_{0}+\beta_{1}X_{i}$
by our model assumptions in Section 2.1. Now, plugging back in to the
numerator of the expectation, we have

$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left\\{E(Y_{i}|X_{i},\Delta_{i})-\sum_{i=1}^{n}\Delta_{i}E(Y_{i}|X_{i},\Delta_{i})/\sum_{i=1}^{n}\Delta_{i}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left[E(\epsilon_{i}|\Delta_{i})+\beta_{0}+\beta_{1}X_{i}-\sum_{i=1}^{n}\Delta_{i}\left\\{E(\epsilon_{i}|\Delta_{i})+\beta_{0}+\beta_{1}X_{i}\right\\}/\sum_{i=1}^{n}\Delta_{i}\right].$

Considering the terms with $\beta_{0}$, we have

$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left(\beta_{0}-\sum_{i=1}^{n}\Delta_{i}\beta_{0}/\sum_{i=1}^{n}\Delta_{i}\right)=\sum_{i=1}^{n}\Delta_{i}X_{i}(\beta_{0}-\beta_{0})=0.$

Considering the terms with $\beta_{1}$, we have

$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left(\beta_{1}X_{i}-\sum_{i=1}^{n}\Delta_{i}X_{i}\beta_{1}/\sum_{i=1}^{n}\Delta_{i}\right)=\beta_{1}\left\\{\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}-\left(\sum_{i=1}^{n}\Delta_{i}X_{i}\right)^{2}/\sum_{i=1}^{n}\Delta_{i}\right\\},$

which will cancel out with the denominator from above leaving us with
$\beta_{1}$. Finally, considering the constant terms (i.e., no parameters), we
have

$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left\\{E(\epsilon_{i}|\Delta_{i})-\sum_{i=1}^{n}\Delta_{i}E(\epsilon_{i}|\Delta_{i})/\sum_{i=1}^{n}\Delta_{i}\right\\}.$

If $\Delta_{i}=0$, this whole term is $0$ due to the first $\Delta_{i}$ in the
summation. If $\Delta_{i}=1$, then $E(\epsilon_{i}|\Delta_{i}=1)=c$ where $c$
is a non-zero constant. Then the above expression is equal to

$\displaystyle\sum_{i=1}^{n}\Delta_{i}X_{i}\left(c-\sum_{i=1}^{n}\Delta_{i}c/\sum_{i=1}^{n}\Delta_{i}\right)=\sum_{i=1}^{n}\Delta_{i}X_{i}\left(c-c\right)=0.$

Now, we want to show that $\widehat{\beta}_{0}$ may be biased when exogenous
censoring doesn’t hold. We show the expectation may not equal $\beta_{0}$ as
follows:

$\displaystyle E(\widehat{\beta}_{0})$ $\displaystyle=$ $\displaystyle
E_{X,\Delta}\left[E_{Y|X,\Delta}\left\\{\sum_{i=1}^{n}\Delta_{i}(Y_{i}-\widehat{\beta}_{1}X_{i})/\sum_{i=1}^{n}\Delta_{i}\right\\}\right]$
$\displaystyle=$ $\displaystyle
E\left(1/\sum_{i=1}^{n}\Delta_{i}\left[\left\\{\sum_{i=1}^{n}\Delta_{i}E(Y_{i}|X_{i},\Delta_{i})\right\\}-\sum_{i=1}^{n}E\left(\widehat{\beta}_{1}|X_{i},\Delta_{i}\right)\Delta_{i}X_{i}\right]\right)$
$\displaystyle=$ $\displaystyle
E\left(1/\sum_{i=1}^{n}\Delta_{i}\left[\left\\{\sum_{i=1}^{n}\Delta_{i}E(\epsilon_{i}|\Delta_{i})+\Delta_{i}\beta_{0}+\Delta_{i}\beta_{1}X_{i}\right\\}-\beta_{1}\sum_{i=1}^{n}\Delta_{i}X_{i}\right]\right)$
$\displaystyle=$ $\displaystyle\beta_{0}+E(\epsilon|\Delta=1),$

which is not equal to $\beta_{0}$ unless $E(\epsilon|\Delta=1)=0$. If
exogenous censoring does not hold, then
$E(\epsilon|\Delta=1)=E\\{E(\epsilon|\Delta=1,X)\\}$ is not necessarily $0$,
since $E(\epsilon|\Delta=1,X)\neq 0$.

Finally, we consider what happens when there is no intercept, such that
$m(\cdot)=\beta_{1}X$. The estimating equation from Section 2.4 will only have
one component; namely, the second component from above, setting $\beta_{0}=0$.
If we isolate $\beta_{1}$, we get
$\widehat{\beta}_{1}=\sum_{i=1}^{n}\Delta_{i}X_{i}Y_{i}/\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}$.
To investigate potential bias, we consider

$\displaystyle E(\widehat{\beta}_{1})$ $\displaystyle=$ $\displaystyle
E\left(\sum_{i=1}^{n}\Delta_{i}X_{i}Y_{i}/\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}\right)$
$\displaystyle=$ $\displaystyle
E\left\\{\sum_{i=1}^{n}\Delta_{i}X_{i}E(Y_{i}|\Delta_{i},X_{i})/\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}\right\\}$
$\displaystyle=$ $\displaystyle
E\left[\sum_{i=1}^{n}\Delta_{i}X_{i}\\{E(\epsilon_{i}|\Delta_{i})+\beta_{1}X_{i}\\}/\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}\right]$
$\displaystyle=$
$\displaystyle\beta_{1}+E\left\\{\sum_{i=1}^{n}\Delta_{i}X_{i}E(\epsilon_{i}|\Delta_{i})/\sum_{i=1}^{n}\Delta_{i}X_{i}^{2}\right\\},$

which is equal to $\beta_{1}$ when exogenous censoring holds (i.e.,
$E(\epsilon_{i}|\Delta_{i})=0$), but may not be equal to $\beta_{1}$
otherwise.

##  S.4 Justification for Simulation Settings

The following give justification to the data generation done for each
simulation setting as shown in Table 1. Each setting was chosen to satisfy (i)
the respective combination of censoring mechanism assumptions and (ii) the
chosen censoring rate, denoted as $r$.

  1. (a.)

Exogenous Censoring does not hold: Mechanism (C1)–(C5) do not hold.

    1. (i.)

Mechanism (C1), exogenous censoring, does not hold because
${{\mu}}=E({\epsilon}|\mbox{$\Delta$},{X},{\mathbf{Z}})\neq{\mathbf{0}}$ for
both values of $\Delta$. Specifically, ${{\mu}}=\sigma/8$ when
$\mbox{$\Delta$}=0$ and ${{\mu}}=-\sigma/8\times r/1-r$ when
$\mbox{$\Delta$}=1$.

Even though exogenous censoring does not hold, the model errors must have
conditional mean zero, i.e., $E({\epsilon}|{X},{\mathbf{Z}})={\mathbf{0}}$. To
see that the errors have conditional mean zero, note that

$\displaystyle E({\epsilon}|{X},{\mathbf{Z}})$ $\displaystyle=$ $\displaystyle
E({\epsilon}|\mbox{$\Delta$}=1,{X},{\mathbf{Z}})P(\mbox{$\Delta$}=1|{X},{\mathbf{Z}})$
$\displaystyle+$ $\displaystyle
E({\epsilon}|\mbox{$\Delta$}=0,{X},{\mathbf{Z}})P(\mbox{$\Delta$}=0|{X},{\mathbf{Z}})$
$\displaystyle=$
$\displaystyle\frac{-\sigma}{8}\frac{r}{1-r}\times(1-r)+\frac{\sigma}{8}\times
r={\mathbf{0}}.$

    2. (ii.)

Since $\mbox{$\Delta$}\sim\hbox{Bernoulli}(1-r)$, this implies the probability
of being observed is $1-r$, satisfying the desired censoring rate.

  2. (b.)

Strict Exogenous Censoring does not hold: Mechanism (C1) holds and (C2)–(C5)
do not hold.

    1. (i.)

Mechanism (C2), strict exogenous censoring, does not hold by design since
$\Delta$ is directly generated dependent on ${\epsilon}$ and independent of
covariates $({X},{\mathbf{Z}})$. Therefore, the two are not independent given
$({X},{\mathbf{Z}})$. This implies mechanisms (C3)–(C5) also do not hold.
However, exogenous censoring still holds because (1) ${\epsilon}$ is
marginally generated from a normal distribution with mean ${\mathbf{0}}$, and
(2) $\Delta$ is generated based on the absolute value of $\epsilon$, so the
expectation of $\epsilon$ given either value of $\Delta$ will still be $0$.

    2. (ii.)

We define $\mbox{$\Delta$}=0$ when the absolute value of ${\epsilon}$ is less
than to the $r$th quantile of $|{\epsilon}|$. The proportion of observations
that are less than the $r$th quantile is exactly $r$, by definition of a
quantile. This ensures that $r$ of the observations will have
$\mbox{$\Delta$}=0$, satisfying the desired censoring rate.

  3. (c.)

Conditional Independence given (X,Z) does not hold: Mechanism (C1)–(C2) hold
and (C3)–(C5) do not hold.

    1. (i.)

In this setting, ${\epsilon}$ is generated to be dependent on ${C}$,
independent of covariates $({X},{\mathbf{Z}})$, so mechanisms (C3)–(C5) do not
hold. Particularly, ${\epsilon}$ is more variable at the extremes of ${C}$.
However, this says little about the relationship between $\Delta$ and
${\epsilon}$. If $\Delta_{i}=1$, then we know $X_{i}\leq C_{i}$, but even
given $X_{i}$, this tells us nothing about whether or not
$C_{i}\in\text{IQR}(C)$ and therefore tells us nothing about $\epsilon_{i}$.
Therefore, $\Delta$ is independent of ${\epsilon}$ given $({X},{\mathbf{Z}})$,
so censoring mechanisms (C1)–(C2) do hold.

    2. (ii.)

Now to see that we have generated the desired censoring rate, consider the
following. Remember that ${X}\sim\hbox{Uniform}(0,3)$. We consider $r<0.5$ and
$r>=0.5$ separately, since for $r<0.5$, $\max(C)>\max(X)$ and vice versa for
$r>0.5$. If $r<0.5$, we define ${C}\sim\hbox{Uniform}(0,3/(2r))$. Then

$\displaystyle P(\mbox{$\Delta$}=0)$ $\displaystyle=$ $\displaystyle
P({C}\leq{X})$ $\displaystyle=$
$\displaystyle\int_{0}^{3}\int_{0}^{x}\frac{2r}{3\times 3}dcdx$
$\displaystyle=$ $\displaystyle\int_{0}^{3}\frac{2r\times x}{9}dx$
$\displaystyle=$ $\displaystyle\frac{r}{9}(3^{2}-0^{2})=r.$

And if the censoring rate is $\geq 0.5$, then
${C}\sim\hbox{Uniform}(0,(6-6r))$ and

$\displaystyle P(\mbox{$\Delta$}=1)$ $\displaystyle=$ $\displaystyle
P({X}\leq{C})$ $\displaystyle=$
$\displaystyle\int_{0}^{6-6r}\int_{0}^{c}\frac{1}{3(6-6r)}dxdc$
$\displaystyle=$ $\displaystyle\int_{0}^{6-6r}\frac{c}{3(6-6r)}dc$
$\displaystyle=$
$\displaystyle\frac{1}{6(6-6r)}((6-6r)^{2}-0^{2})=(6-6r)/6=1-r.$

  4. (d.)

Conditional Independence given Z does not hold: Mechanism (C1)–(C3) hold and
(C4)–(C5) do not hold.

    1. (i.)

${C}$ is generated directly dependent on ${X}$, so Mechanism (C4),
conditionally independent censoring given ${\mathbf{Z}}$, and therefore
mechanism (C5), independent censoring, cannot hold. However, given ${X}$ and
${\mathbf{Z}}$, ${C}$ is independent of ${Y}$, so mechanism (C3) and therefore
(C1)–(C2) all hold.

    2. (ii.)

To see that we have generated the desired censoring rate with
${C}\sim\hbox{Uniform}(0,{X}/r)$,

$\displaystyle P(\mbox{$\Delta$}=0)$ $\displaystyle=$ $\displaystyle
P({C}\leq{X})$ $\displaystyle=$
$\displaystyle\int_{0}^{3}\int_{0}^{x}\frac{r}{3x}dcdx$ $\displaystyle=$
$\displaystyle\int_{0}^{3}\frac{r}{3}dx=\frac{r}{3}(3-0)=r.$

  5. (e.)

Independence does not hold: Mechanism (C1)–(C4) hold and (C5) does not hold.

    1. (i.)

${C}$’s generation is directly dependent on ${\mathbf{Z}}$, so mechanism (C5),
independent censoring, cannot hold. However ${C}$ is independent of ${X}$ and
${Y}$ given ${\mathbf{Z}}$, so mechanism (C4) and therefore (C1)–(C3) all
hold.

    2. (ii.)

To see that we have generated the desired censoring rate, we consider the
generation of ${C}$. When $r\geq 0.5$, 50% of the observations (those with
$Z>\text{med}({\mathbf{Z}})$ will be generated from
${C}\sim\hbox{Uniform}(0,6-6r)$, which has censoring rate $r$ as shown in
(c.ii.). The other 50% (when $Z\leq\text{med}({\mathbf{Z}})$) will be
generated from ${C}\sim\hbox{Uniform}(\frac{3-6r}{2-2r},3)$. This also has
censoring rate $r$ as seen below:

$\displaystyle P(\mbox{$\Delta$}=0)$ $\displaystyle=$ $\displaystyle
P({C}\leq{X})$ $\displaystyle=$
$\displaystyle\int_{0}^{3}\int_{\frac{3-6r}{2-2r}}^{3}\frac{1}{3\times(3-\frac{3-6r}{2-2r})}dcdx$
$\displaystyle=$
$\displaystyle\int_{0}^{3}\frac{x-\frac{3-6r}{2-2r}}{3(3-\frac{3-6r}{2-2r})}dx$
$\displaystyle=$
$\displaystyle\frac{(3-\frac{3-6r}{2-2r})^{2}-(\frac{3-6r}{2-2r})^{2}}{2\times
3(3-\frac{3-6r}{2-2r})}$ $\displaystyle=$
$\displaystyle\frac{9-6\frac{3-6r}{2-2r}}{2\times 3(3-\frac{3-6r}{2-2r})}$
$\displaystyle=$
$\displaystyle\frac{3-2\frac{3-6r}{2-2r}}{6-2\frac{3-6r}{2-2r}}$
$\displaystyle=$
$\displaystyle\frac{3(2-2r)-2(3-6r)}{6(2-2r)-2(3-6r)}=\frac{6r}{6}=r.$

Similarly, when $r<0.5$, half of the observations will be generated from
${C}\sim\hbox{Uniform}(0,\frac{3}{2r})$, which has censoring rate $r$ as shown
in (c.ii). The other half will be generated from
${C}\sim\hbox{Uniform}(3-6r,3)$. This also has censoring rate $r$:

$\displaystyle P(\mbox{$\Delta$}=1)$ $\displaystyle=$ $\displaystyle
P({X}\leq{C})$ $\displaystyle=$
$\displaystyle\int_{3-6r}^{3}\int_{0}^{c}\frac{1}{3(6r)}dxdc$ $\displaystyle=$
$\displaystyle\int_{3-6r}^{3}\frac{c}{3(6r)}dc$ $\displaystyle=$
$\displaystyle\frac{3^{2}-(3-6r)^{2}}{2*3(6r)}=\frac{-36r^{2}+36r}{36r}=1-r.$

  6. (f.)

Independence holds: Mechanisms (1-5) hold.

    1. (i.)

${C}$ is generated independently of ${X}$, ${Y}$, and ${\mathbf{Z}}$, so
mechanism (C5) holds by design. This implies mechanisms (C1)–(C4) also hold.

    2. (ii.)

For justification how the choice of parameters satisfies the chosen censoring
rate, see (c.ii.).

##  S.5 Supplementary Tables

Table S.4:  Simulation results for a linear model with a right-censored
covariate when data are generated under six different combinations of
censoring mechanism assumptions. We report the percent bias, estimated
standard errors (SE), and 95% coverage probabilities for all parameters in the
linear model when we estimate parameters using the complete case estimator
(CC) and oracle estimator (Oracle). Results are based on 1000 simulated
datasets, each with a sample size of 1200.

| Percent Bias | Estimated SE | 95% Coverage
---|---|---|---
Censoring Rate | Method | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$
Exogenous Censoring does not hold
| CC | -11.14 | -0.21 | -0.10 | 0.09 | 0.05 | 0.05 | 0.91 | 0.94 | 0.94
25% | Oracle | 0.45 | -0.08 | -0.10 | 0.08 | 0.05 | 0.04 | 0.93 | 0.92 | 0.94
| CC | -104.33 | -0.37 | -0.24 | 0.16 | 0.09 | 0.08 | 0.11 | 0.95 | 0.94
75% | Oracle | 0.47 | -0.10 | -0.09 | 0.08 | 0.05 | 0.04 | 0.94 | 0.92 | 0.94
Strict Exogenous Censoring does not hold
| CC | -1.19 | 0.29 | -0.06 | 0.11 | 0.06 | 0.05 | 0.96 | 0.95 | 0.95
25% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  3 0.95
| CC | -3.34 | 0.92 | 0.22 | 0.28 | 0.16 | 0.14 | 0.94 | 0.95 | 0.94
75% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  3 0.95
Conditional Independence given (X,Z) does not hold
| CC | 0.63 | -0.18 | -0.14 | 0.09 | 0.05 | 0.05 | 0.95 | 0.94 | 0.95
25% | Oracle | 0.53 | -0.08 | -0.11 | 0.08 | 0.05 | 0.04 | 0.93 | 0.93 | 0.95
| CC | 0.11 | 0.58 | 0.02 | 0.14 | 0.25 | 0.08 | 0.95 | 0.94 | 0.95
75% | Oracle | 0.53 | -0.08 | -0.11 | 0.08 | 0.05 | 0.04 | 0.93 | 0.93 | 0.95
Conditional Independence given Z does not hold
| CC | -0.95 | 0.22 | -0.06 | 0.09 | 0.05 | 0.05 | 0.96 | 0.94 | 0.95
25% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  2 0.95
| CC | -1.67 | 0.29 | 0.03 | 0.16 | 0.09 | 0.08 | 0.96 | 0.94 | 0.95
75% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  2 0.95
Independence does not hold
| CC | -0.72 | 0.06 | -0.09 | 0.09 | 0.06 | 0.05 | 0.95 | 0.95 | 0.94
25% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  1 0.95
| CC | -1.40 | 0.62 | -0.20 | 0.13 | 0.14 | 0.09 | 0.94 | 0.93 | 0.93
75% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 |  1 0.95
Independence holds
| CC | -1.00 | 0.23 | -0.06 | 0.09 | 0.06 | 0.05 | 0.95 | 0.94 | 0.95
25% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 | 0.95
| CC | -1.17 | 0.46 | -0.11 | 0.14 | 0.23 | 0.08 | 0.95 | 0.94 | 0.96
75% | Oracle | -0.95 | 0.22 | -0.04 | 0.08 | 0.05 | 0.04 | 0.95 | 0.94 | 0.95

Table S.5:  Simulation results for a linear mean model with an intercept versus without an intercept. We report the percent bias for all parameters in the linear model when we estimate parameters using the complete case estimator (CC) and oracle estimator (Oracle). Results are based on 1000 simulated datasets, each with a sample size of 400. | Percent Bias
---|---
Censoring Rate | Method | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$
Intercept Present
| CC | -13.01 | 0.43 | 0.29
0.25 | Oracle | -0.78 | 0.24 | 0.16
| CC | -109.01 | 0.72 | 0.38
0.75 | Oracle | -0.67 | 0.20 | 0.17
No Intercept Present
| CC | NA | -3.10 | -0.03
0.25 | Oracle | NA | -0.13 | -0.01
| CC | NA | -26.76 | 0.21
0.75 | Oracle | NA | -0.09 | -0.04
Table S.6:  Simulation results for a logistic model with a right-censored
covariate when data are generated under six different combinations of
censoring mechanism assumptions. We report the percent bias, estimated
standard errors (SE), and 95% coverage probabilities for all parameters in the
linear model when we estimate parameters using the complete case estimator
(CC) and oracle estimator (Oracle). Results are based on 1000 simulated
datasets, each with a sample size of 400.

| Percent Bias | Estimated SE | 95% Coverage
---|---|---|---
Censoring Rate | Method | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$
Exogenous Censoring does not hold
| CC | -209.53 | 7.24 | 5.04 | 0.03 | 0.02 | 0.01 | 0.94 | 0.94 | 0.94
25% | Oracle | -12.75 | 4.30 | 2.98 | 0.02 | 0.01 | 0.01 | 0.95 | 0.95 | 0.94
| CC | -1789.51 | 15.83 | 10.89 | 0.05 | 0.03 | 0.02 | 0.51 | 0.95 | 0.94
75% | Oracle | -10.87 | 3.54 | 3.16 | 0.02 | 0.01 | 0.01 | 0.96 | 0.95 | 0.94
Strict Exogenous Censoring does not hold
| CC | 5.73 | -0.71 | -1.07 | 0.03 | 0.02 | 0.02 | 0.94 | 0.94 | 0.94
25% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  3 0.94
| CC | -14.33 | 1.46 | -0.19 | 0.08 | 0.05 | 0.04 | 0.95 | 0.96 | 0.94
75% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  3 0.94
Conditional Independence given (X,Z) does not hold
| CC | -22.46 | 7.63 | 3.41 | 0.02 | 0.02 | 0.01 | 0.96 | 0.96 | 0.94
25% | Oracle | -11.37 | 3.35 | 2.21 | 0.02 | 0.01 | 0.01 | 0.95 | 0.94 | 0.94
| CC | -35.03 | 23.61 | 8.52 | 0.04 | 0.07 | 0.02 | 0.95 | 0.94 | 0.94
75% | Oracle | -11.37 | 3.35 | 2.21 | 0.02 | 0.01 | 0.01 | 0.95 | 0.94 | 0.94
Conditional Independence given Z does not hold
| CC | 1.92 | 2.62 | -1.06 | 0.03 | 0.02 | 0.01 | 0.94 | 0.94 | 0.93
25% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  2 0.94
| CC | -6.60 | 5.23 | -5.61 | 0.05 | 0.03 | 0.02 | 0.94 | 0.94 | 0.93
75% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  2 0.94
Independence does not hold
| CC | 2.51 | 0.88 | -2.09 | 0.02 | 0.02 | 0.01 | 0.94 | 0.95 | 0.93
25% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  1 0.94
| CC | -8.00 | 13.40 | -5.54 | 0.04 | 0.04 | 0.02 | 0.93 | 0.93 | 0.93
75% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 |  1 0.94
Independence holds
| CC | 2.34 | 0.35 | -2.49 | 0.02 | 0.02 | 0.01 | 0.94 | 0.95 | 0.93
25% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 | 0.94
| CC | 3.12 | 8.15 | -0.93 | 0.04 | 0.06 | 0.02 | 0.95 | 0.94 | 0.94
75% | Oracle | 2.65 | -0.39 | -0.93 | 0.02 | 0.01 | 0.01 | 0.94 | 0.95 | 0.94

Table S.7:  Simulation results for a logistic model with a right-censored
covariate when data are generated under six different combinations of
censoring mechanism assumptions. We report the percent bias, estimated
standard errors (SE), and 95% coverage probabilities for all parameters in the
linear model when we estimate parameters using the complete case estimator
(CC) and oracle estimator (Oracle). Results are based on 1000 simulated
datasets, each with a sample size of 1200.

| Percent Bias | Estimated SE | 95% Coverage
---|---|---|---
Censoring Rate | Method | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$ | $\widehat{\beta}_{0}$ | $\widehat{\beta}_{1}$ | $\widehat{\beta}_{2}$
Exogenous Censoring does not hold
| CC | -178.68 | -3.33 | -1.63 | 0.02 | 0.01 | 0.01 | 0.91 | 0.94 | 0.94
25% | Oracle | 7.18 | -1.19 | -1.47 | 0.01 | 0.01 | 0.01 | 0.93 | 0.92 | 0.94
| CC | -1697.33 | -3.60 | -1.10 | 0.03 | 0.02 | 0.01 | 0.12 | 0.95 | 0.95
75% | Oracle | 7.51 | -1.45 | -1.28 | 0.01 | 0.01 | 0.01 | 0.94 | 0.92 | 0.94
Strict Exogenous Censoring does not hold
| CC | -19.06 | 4.86 | -0.79 | 0.02 | 0.01 | 0.01 | 0.96 | 0.95 | 0.95
25% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  3 0.94
| CC | -53.55 | 16.10 | 5.29 | 0.05 | 0.03 | 0.02 | 0.95 | 0.95 | 0.95
75% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  3 0.94
Conditional Independence given (X,Z) does not hold
| CC | 10.03 | -2.75 | -2.03 | 0.01 | 0.01 | 0.01 | 0.94 | 0.94 | 0.95
25% | Oracle | 8.49 | -1.07 | -1.65 | 0.01 | 0.01 | 0.01 | 0.93 | 0.93 | 0.95
| CC | 2.11 | 9.86 | 0.89 | 0.02 | 0.04 | 0.01 | 0.95 | 0.94 | 0.95
75% | Oracle | 8.49 | -1.07 | -1.65 | 0.01 | 0.01 | 0.01 | 0.93 | 0.93 | 0.95
Conditional Independence given Z does not hold
| CC | -15.23 | 3.60 | -0.81 | 0.02 | 0.01 | 0.01 | 0.96 | 0.94 | 0.95
25% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  2 0.94
| CC | -26.86 | 5.14 | 1.08 | 0.03 | 0.02 | 0.01 | 0.96 | 0.94 | 0.95
75% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  2 0.94
Independence does not hold
| CC | -11.66 | 1.05 | -1.31 | 0.01 | 0.01 | 0.01 | 0.95 | 0.95 | 0.94
25% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  1 0.94
| CC | -22.67 | 10.92 | -2.66 | 0.02 | 0.02 | 0.01 | 0.94 | 0.93 | 0.93
75% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 |  1 0.94
Independence holds
| CC | -16.07 | 3.79 | -0.87 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 | 0.95
25% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 | 0.94
| CC | -19.06 | 8.16 | -1.28 | 0.02 | 0.04 | 0.01 | 0.95 | 0.95 | 0.96
75% | Oracle | -15.25 | 3.56 | -0.53 | 0.01 | 0.01 | 0.01 | 0.95 | 0.94 | 0.94

Table S.8:  Results from the PREDICT-HD study quantifying the relationship between apathy score and CAP score, controlling for other covariates. The goal of this analysis was to replicate the linear regression from [33]. The parameters are estimated using the full dataset (n=781). We report the parameter estimate, the estimated standard error (SE), and the Wald-type 95% confidence interval (95% CI). Parameter | Estimate | Estimated SE |  $95\%$ CI
---|---|---|---
$\beta_{0}:$ Intercept | 12.83 | 0.29 | (12.25, 13.40)
$\beta_{1}:$ CAP Score | 0.23 | 0.50 | (-0.75, 1.23)
$\beta_{2}:$ Biological Sex | -0.49 | 0.33 | (-1.14, 0.16)
$\beta_{3}:$ Depression | 3.59 | 0.23 | (3.13, 4.04)
$\beta_{4}:$ Total Motor Score | 0.45 | 0.23 | (0.020, 0.89)
$\beta_{5}:$ Years of Education | 0.065 | 0.16 | (-0.26, 0.39)
$\beta_{6}:$ SDMT Score | -0.41 | 0.20 | (-0.80, -0.016)
$\beta_{7}:$ CAG Repeat Length | -0.079 | 0.52 | (-1.11, 0.95)
$\beta_{8}:$ Age | -0.43 | 0.45 | (-1.31, 0.44)