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Mission Imputable: Correcting for Berkson Error When Imputing a Censored Covariate
Kyle F. Grosser
The authors gratefully acknowledge the National Institute of Environmental Health Sciences grants T32ES007018 and P30ES010126 and the National Institute of Neurological Disorders and Stroke grant K01NS099343.

Department of Biostatistics, University of North Carolina
at Chapel Hill Gillings School of Global Public Health

Sarah C. Lotspeich

Department of Statistical Sciences, Wake Forest University

Tanya P. Garcia

Department of Biostatistics, University of North Carolina
at Chapel Hill Gillings School of Global Public Health

Mission Imputable: Correcting for Berkson Error When Imputing a Censored Covariate

To select outcomes for clinical trials testing experimental therapies for Huntington disease, a fatal neurodegenerative disorder, analysts model how potential outcomes change over time. Yet, subjects with Huntington disease are often observed at different levels of disease progression. To account for these differences, analysts include time to clinical diagnosis as a covariate when modeling potential outcomes, but this covariate is often censored. One popular solution is imputation, whereby we impute censored values using predictions from a model of the censored covariate given other data, then analyze the imputed dataset. However, when this imputation model is misspecified, our outcome model estimates can be biased. To address this problem, we developed a novel method, dubbed “ACE imputation.” First, we model imputed values as error-prone versions of the true covariate values. Then, we correct for these errors using semiparametric theory. Specifically, we derive an outcome model estimator that is consistent, even when the censored covariate is imputed using a misspecified imputation model. Simulation results show that ACE imputation remains empirically unbiased even if the imputation model is misspecified, unlike multiple imputation which yields $>100\%$ bias. Applying our method to a Huntington disease study pinpoints outcomes for clinical trials aimed at slowing disease progression.

Keywords: Censored data, Huntington disease, imputation correction, measurement error, semiparametric theory

§ INTRODUCTION

§.§ Statistical hurdles for clinical trials of Huntington disease

Clinical trials are now underway to test experimental therapies aimed at slowing the progression of Huntington disease, a genetically inherited disease that leads to progressive cognitive and motor impairment. No effective therapies for the disease have been developed yet; one major difficulty is finding an outcome to help assess if an experimental therapy has an effect <cit.>. Clinical trialists want to avoid selecting an outcome that changes too slowly over time, because then it would be difficult to determine if an experimental therapy actually slowed progression. This is because, for an outcome that changes slowly, the progression would not have changed much over the trial period regardless of the therapy being tested.

Therefore, an ideal outcome would be one
in which change can be easily detected over the course of the trial, but
determining such an outcome is statistically challenging. Data measuring cognitive and motor impairment over time are typically collected from individuals who are at different levels of disease progression, some more advanced than others. These differences can obscure our search to identify the best outcome to use in a clinical trial. For example, plots of the Symbol Digit Modalities Test (SDMT) scores (a measure of cognitive impairment) show little to no change over the course of the study if we do not adjust for individuals being at different levels of disease progression (Figure <ref>A). To capture disease progression, we use time to clinical diagnosis, which refers to the difference between the time at which a subject is observed and their time of clinical diagnosis (i.e., the day on which a clinician determines that a subject's motor impairment is unequivocally attributable to Huntington disease) <cit.>.
by plotting SDMT scores as a function of time to clinical diagnosis instead of simply time in the study (i.e., adjusting for disease progression), we see that cognitive impairment worsens rapidly in the time period immediately before and after diagnosis (Figure <ref>B). These findings based on the adjusted trajectories of cognitive impairment
agree with previous clinical findings, whereas those based on the unadjusted trajectories
do not <cit.>.

Symbol Digit Modality Test (SDMT) scores from uncensored (i.e., clinically diagnosed) subjects plotted against (A) time since study entry and (B) time to clinical diagnosis. Thin lines indicate change over time for individual subjects, and the thick line indicates a LOESS curve summarizing the overall trajectories.

Modeling cognitive impairment as a function of time to clinical diagnosis is a common way to adjust for individuals being at different levels of disease progression <cit.>. With data from a fully diagnosed cohort, making this adjustment to our analyses would be straightforward. However, because Huntington disease progresses slowly over time, time of clinical diagnosis is often unobserved for at least some subjects in a given cohort. Yet, Huntington disease is fully penetrant, i.e., anyone who inherits the genetic mutation should eventually meet the criteria for a diagnosis. We therefore know that, for undiagnosed subjects, the time of clinical diagnosis must lie beyond when they were last observed. This phenomenon is known as right censoring. Hence, to select outcomes for clinical trials of Huntington disease, we must accurately model the change in potential outcomes given a right-censored covariate: time of clinical diagnosis.

§.§ Statistical modeling with a censored covariate

One might be tempted to replace censored subjects' times of clinical diagnosis with their times of last observation instead (e.g., last visit), and then model the potential outcomes using these replacements. However,
these right-censored replacements will be necessarily less than the true clinical diagnosis times. As a result, such “naive” analyses tend to produce biased model estimates <cit.>.
Alternatively, we could remove the censored subjects from our data, and then model the potential outcomes using only the uncensored data.
However, it is well established that such “complete case” analyses yield parameter estimates that are less precise (i.e., less efficient) than those that would be obtained if the full dataset were available <cit.>.
Complete case analyses can also lead to biased parameter estimates and inflated type I error rates for hypothesis tests <cit.>.

Hence, we do not want to “throw away” our uncensored data as in complete case analyses, but we cannot use the data “as is” as in naive analyses. Imputation — whereby we impute censored covariate values and then analyze the imputed dataset — provides a more promising solution.

These imputed values can be generated from a model of the censored covariate given other, fully observed data (an “imputation model”) in many ways, including random draws <cit.>, conditional means <cit.>, and conditional quantiles <cit.>.

Although imputation for censored covariates is still a growing area of research, we can learn much from the massive body of literature concerning imputation for “traditional” missing data. Importantly, it is well understood that imputation for traditional missing data will yield consistent estimates of our outcome model parameters given two key assumptions: i) the data are missing at random and ii) the imputation model is correctly specified <cit.>. In particular, the missing at random assumption requires that the probability of the variables “being missing” is conditionally independent of the missing variables themselves, given the other fully observed variables.

With censored data, this assumption is immediately violated since the probability that a variable $X$ is right censored by a variable $C$ (i.e., $P(C<X)$) depends directly on $X$. For example, with right-censored data, a possibly censored variable $X$ is more likely to be censored by a variable $C$ if $X=2$ than if $X=4$, since $P(C<4)\leq P(C<2)$ (with strict inequality in general). Fortunately, <cit.> and <cit.> show (both in theory and simulation) that consistent parameter estimates can still be obtained when we use imputation to address censored covariates. These findings demonstrate that the missing at random assumption is not a necessary assumption when imputing censored covariates. Yet, all of the cited imputation approaches hinge on a model of the censored covariate; when this imputation model is misspecified, it could introduce bias into our outcome model estimates <cit.>.

§.§ Overview

In this paper, we present a novel method to consistently estimate a linear mixed effects model for a continuous outcome given a censored covariate. We build on existing imputation methods for censored covariates, which result in biased outcome model estimates when the imputation model is misspecified, by accounting for such misspecification in order to reduce that bias.

To account for possible imputation model misspecification, we model the difference between the true (but censored) covariate values and their imputed replacements using a Berkson error model <cit.>. Then, to avoid possibly misspecifying the distribution of this “imputation error” (since that could lead to bias and inefficiency), we adapt a flexible, semiparametric method developed by <cit.> that allows the imputation error to follow any distribution.
We introduce our proposed method, “active correction for error in imputation” (dubbed “ACE imputation”), in Section <ref>.
In Section <ref>, we show that ACE imputation produces an estimator that is identifiable, consistent, and asymptotically normal, even under imputation model misspecification.
In  Section <ref>, simulations show that ACE imputation outperforms competing solutions for censored covariates when the imputation model is correct and remains unbiased even when the underlying imputation model has been misspecified.
In Section <ref>, we show that applying ACE imputation to Huntington disease data

helps pinpoint outcomes
that could be used to test experimental therapies for Huntington disease. Section <ref> concludes this paper with a discussion of potential limitations and future work.

§ METHODS

To better pinpoint outcomes for clinical trials of Huntington disease (i.e., outcomes that change quickly enough so that effective therapies can be reliably identified) we must accurately model how potential outcomes progress over time as a function of time of clinical diagnosis, a right-censored covariate. In this section, we present our ACE imputation method, which estimates the parameters of a longitudinal model given a right-censored covariate. To achieve this goal, we first implement an existing solution to covariate censoring, in which we impute censored values from a Cox model. Then, because this method can produce bias when that imputation model is misspecified, we reduce this bias by adjusting for the errors that occur due to such misspecification.

§.§ Notation and longitudinal model

We longitudinally model our potential outcomes using data collected for $n$ subjects (indexed by $i$, $i=1,\ldots,n$) and $m_i$ observations per subject (indexed by $j$, $j=1,\ldots,m_i$). The outcome for subject $i$ at visit $j$ is $Y_{ij}$, observed at time $s_{ij}$. We also observe time of clinical diagnosis $X_i$, $p_a$-dimensional covariates $\bZ^a_{ij}$, and $p_b$-dimensional covariates $\bZ^b_{ij}$. To account for clustering between outcomes from the same subject, we include a $p_b$-dimensional vector of unobserved, subject-specific random effects $\bb_i$. In Huntington disease, for example, $Y_{ij}$  is the outcome being considered for a clinical trial, $s_{ij}$ is the time of observation (years) since the start of the study, $X_i$ is the time of clinical diagnosis (years) relative to the start of study, $\bZ^a_{ij}$ are baseline age, sex, education, and genetic information, and $\bb_i$ is a subject-specific random intercept (hence, $\bZ^b_{ij}$ is simply a vector of ones).

To account for clustering between $Y_{i1},\ldots,Y_{im_i}$, we employ the linear mixed model:
Y_ij =
α(s_ij - X_i) + ^T ^a_ij + _i^T ^b_ij + ϵ_ij,    _i∼f_,    ϵ_ij∼(0,σ^2).Herein, $\alpha$ is the parameter associated with $(s_{ij} - X_i)$, $\bbeta$ is the $p_a$-dimensional parameter vector associated with $\bZ^a_{ij}$, and $\sigma^2$ is the variance of the random errors $\epsilon_{ij}$.

We assume that subjects are independent of one another and that outcomes $Y_{ij}$ from the $i$th subject are conditionally independent given $\bb_i\sim f_{\bb}$. Typically with mixed models, analysts assume a distribution for $\bb$ such as a multivariate normal distribution with mean $\bzero$ and unknown covariance matrix. To promote model flexibility, we instead allow $\bb$ to follow any distribution $f_{\bb}$. We do this so to avoid possibly misspecifying $f_{\bb}$, since misspecification can lead to bias and inefficiency <cit.>.

We refer to $(s_{ij} - X_i)$ as “time to clinical diagnosis” (whereas $X_i$ on its own is time of clinical diagnosis). To illustrate, $(s_{ij} - X_i) = -2$ when subject $i$ is two years before they are clinically diagnosed and $(s_{ij} - X_i) = 2$ when they are two years after being clinically diagnosed. As discussed in Section <ref>, we use this difference rather than just $s_{ij}$ to capture the fact that some subjects are farther along in their disease progression than others.

A key challenge with the model is that $X_i$ is often right-censored; in the Huntington disease data that we analyze, diagnosis time $X_i$ is right-censored for $\approx 78\%$ of subjects. Hence, rather than observe $X_i$, we observe $W_i=\min(X_i,C_i)$ and $\Delta_i=I(X_i\leq C_i)$, where $C_i$ is a continuous, random right-censoring time and $\Delta_i$ is the censoring indicator. We refer to $C_i$ as a “right-censoring” time because when time of clinical diagnosis $X_i$ is censored by $C_i$, we know that $X_i$ falls to the right of $C_i$, i.e., $X_i > C_i$.

In summary, the observed data are $(\bY_i,\bs_i,\bZ^a_i,\bZ^b_i,W_i,\Delta_i)$ for $i=1,\ldots,n$ where  $\bY_i=(\bY_{i1},\ldots,\bY_{im_i})^T,\bs_i=(s_{i1},\ldots,s_{im_i})^T,\bZ^a_i=(\bZ^a_{i1},\ldots,\bZ^a_{im_i})^T $, and $\bZ^b_i=(\bZ^b_{i1},\ldots,\bZ^b_{im_i})^T$.
We want to estimate  $\btheta=(\alpha, \bbeta\trans, \sigma^2)\trans$ in the presence of right-censoring on $X_i$.

§.§ Imputing censored times of diagnosis

Imputation is a compelling solution to covariate censoring because it allows us to “fill in” censored times of clinical diagnosis. We can then analyze this newly complete dataset using traditional methods such as restricted maximum likelihood estimation (REML). Furthermore, by preserving data from all subjects, we increase the efficiency of our parameter estimates relative to complete case analysis. Analyzing an imputed dataset will not yield the same level of efficiency as analyzing a dataset that was never censored to begin with, but it is more efficient than discarding censored subjects outright.

Therefore, to estimate $\btheta$ in the presence of covariate censoring, we first impute censored values of $X_i$.

More specifically, we employ conditional mean imputation based on a Cox model <cit.>, which adjusts for censoring and does not impose strict distributional assumptions on the time of clinical diagnosis $X_i$. First, we fit a Cox model with times of diagnosis, $X_i$, as the outcome given a vector of time-invariant covariates $\bV_i$. The covariates $\bV_i$ can be a subset of the variables in $\bZ^a_i$ or $\bZ^b_i$, or auxiliary variables.

The Cox model assumes that the hazard function for $X_i$ given $\bV_i$ is
where $\lambda_0(x)$ and  $\bgamma$ are the baseline hazard function and covariate effects, respectively. The elements of $\bgamma$ represent log-hazard ratios for the corresponding elements of $\bV_i$. After fitting this model, we replace times of clinical diagnosis $X_i$ with:

X_i   =   Δ_iX_i+(1-Δ_i)E(X_i|X_i>C_i,_i).
For uncensored subjects, $X_i$ is unchanged (i.e., $\widehat{X}_i=X_i$ for $i$ such that $\Delta_i = 1$). For censored subjects, we replace $X_i$ with its conditional mean given $X_i>C_i$ and $\bV_i$ (i.e., $\widehat{X}_i={\rm E}(X_i|X_i>C_i,\bV_i)$ for $i$ such that $\Delta_i = 0$).

<cit.> show that this conditional expectation can be approximated as $\widehat{\rm E}(X_i|X_i>C_i,\bV_i) =$
C_i + 1/2[∑_j=1^n-1I(W_(j) ≥C_i){S_0(W_(j+1))^exp(^T_i)+ S_0(W_(j))^exp(^T_i)}(W_(j+1)-W_(j))/S_0(C_i)^exp(^T_i)],
where $W_{(1)} < W_{(2)} < \dots < W_{(n)}$ are the ordered values of $W = \min(X, C)$ and $S_0(x|\bV_i)$ is the baseline survival function for $X_i$ given $\bV_i$. To compute this approximation, we estimate the log-hazard ratios $\bgamma$ using standard statistical software and the baseline cumulative hazard function $S_0(\cdot)$ using the Breslow estimate <cit.>.

§.§ Why imputation introduces errors

For imputation to yield consistent parameter estimates, we must correctly specify this Cox model. When this model is misspecified, the imputed values $\widehat{X}_i$ can be very far from the true times of clinical diagnosis, $X_i$. As we empirically show in Section <ref>, this “imputation error” (i.e., the difference between $X_i$ and $\widehat{X}_i$) can lead to serious bias.

To reduce this bias, we model the imputation error. <cit.> note that imputation error is a type of Berkson error, which arises when subjects in a group are assigned the same value for a missing variable. When subjects with similar traits are all assigned the same value (e.g., an estimated group mean), error arises because true individual values deviate from this group mean by an unobserved amount. With conditional mean imputation, we replace censored $X_i$ with $\widehat{X}_i={\rm E}(X_i|X_i>C_i,\bV_i)$. So, if two subjects $i$ and $j$ have the same traits, i.e., $(C_i, \bV_i) = (C_j, \bV_j)$, we will assign equivalent imputed values to each, i.e., $\widehat{X}_i = \widehat{X}_j$. As a result, we know that the relationship between $X_i$ and $\widehat{X}_i$ follows a Berkson error model. We represent this as
X_i = X_i + U_i
with measurement error $U_i\sim f_U$ <cit.>.
Given the definition of $\widehat{X}_i$ in Equation (<ref>), there is no imputation error when a subject's time of clinical diagnosis is uncensored (i.e., $U_i=0$ for $i$ such that $\Delta_i=1$), since we need not impute for that subject.

§.§ Correcting for errors in imputation

Since we know that error between the true (but censored) values of $X_i$ and their imputed replacements $\widehat{X}_i$ can lead to bias (Section <ref>), we adjust for this imputation error ($U_i$) to more accurately estimate $\btheta$.
To estimate $\btheta$  under the most flexible modeling assumptions, we will not make any distributional assumptions on the random effects, $\bb_i$, or imputation error, $U_i$. Instead, treating the random effects and imputation error as so-called latent variables (i.e., variables not observed), our longitudinal mixed effects model belongs in the class of generalized linear latent variable models. For such a class of models, a semiparametric method exists <cit.> for which model parameters are estimated using an intermediate quantity that plays a similar role as that of the classical sufficient and complete statistic.
This method results in parameter estimators that are consistent, efficient, and robust to misspecification.

Extending this semiparametric method to our problem, we show in Appendix <ref> that the estimator of interest, $\ACEest$, is the solution to  the estimating equation:

∑_i=1^n {_eff^full(_i,_i,_i;)}≡∑_i=1^n {Δ_i_eff(_i,_i;) + (1 - Δ_i)_eff^*(_i,_i;)}=,
where $\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)$ is the score vector for uncensored data
$\tilD_i=(\bs_i, X_i, \bZ^a_i, \bZ^b_i)$,  and
$\bS_{\rm{eff}}^*(\bY_i,\widehat{\tilD}_i;\btheta)$ is the score vector for censored data $\widehat{\tilD}_i=(\bs_i, \widehat{X}_i, \bZ^a_i, \bZ^b_i)$. The derivation of these score vectors shows that initially, constructing these score vectors requires solving a computationally slow and numerically unstable problem. However, by leveraging properties of multivariate normal distributions and linear projection theory, we derive straightforward, closed-form versions of the score vectors (see Appendix <ref>).

In Appendix <ref>, we show that $\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)=$

σ^2^aT_i(_m_i - _^b_i){_i-(_i;)}

σ^2(_i-X_i_m_i)^T(_m_i - _^b_i){_i-(_i;)}

E(_i|_i,_i)   =   _^b_i_i+(_m_i-_^b_i)(_i;)

E(_i^T_i|_i,_i)   =   σ^2(m_i-p_b)+||E(_i|_i,_i)||^2_2,
in which $\bT_i=\sigma^{-2}\bZ^{bT}_i\bY_i$,
$\bP_{\bZ^b_i}=\bZ^b_i(\bZ^{bT}_i\bZ^b_i)^{-1}\bZ^{bT}_i$, and $\bzeta(\tilD_i;\btheta)=\bZ^a_i\bbeta  + \alpha(\bS_i - X_i\bone_{m_i})$. In Appendix <ref>, we show that $\bS_{\rm{eff}}^*(\bY_i,\widehat{\tilD}_i;\btheta)=$

E(_i|_i^aug,_i)   =   _(_m_i,^b_i)_i+(_m_i-_(_m_i,^b_i))(_i;)

E(_i^T_i|_i^aug,_i)   =   σ^2(m_i-p_b)+||E(_i|_i^aug,_i)||^2_2,
in which $\bT_i^{\rm aug}=\sigma^{-2}(\bone_{m_i},\bZ^b_i)^T\bY_i$,
and $\bzeta(\widehat{\tilD}_i;\btheta)=\bZ^a_i\bbeta  + \alpha(\bS_i - \widehat{X}_i\bone_{m_i})$.

The construction of $S_{\rm eff}(\bY_i,\bD_i;\btheta)$ and $S_{\rm eff}^*(\bY_i,\wh\bD_i;\btheta)$ appears complex, but it involves computationally simple matrix algebra and, perhaps more crucially, completely avoids all terms involving both the random effects $\bb$ and the imputation error $U$. We are able to achieve this result by leveraging the properties of multivariate normal distributions and linear projection operators to show that all terms containing either $\bb$ or $U$ drop out of the efficient score vectors (Appendix <ref>). As a result, ACE imputation requires that we neither propose forms for ($f_{\bb}, f_U$) nor estimate these unknown distributions. Therefore, the proposed method is unlike traditional maximum likelihood estimation, which requires that we assume specific forms for ($f_{\bb}, f_U$), such as a multivariate normal distribution. Although maximum likelihood estimators will be more efficient than semiparametric or nonparametric methods when these distributional assumptions are correct, they will suffer from bias when these assumptions are incorrect. The proposed method is also unlike traditional semiparametric estimators, which require positing the nuisance distributions; although these methods produce consistent estimators regardless of whether the posited distributions are correct (an advantage over traditional maximum likelihood estimation), they suffer efficiency losses when the posited distributions are incorrect  <cit.>. Since our method does not require positing the nuisance distributions ($f_{\bb}, f_U$), $\ACEest$ achieves both consistency and optimal efficiency without any prior knowledge about the nuisance distributions.

§ PROPERTIES OF OUR NOVEL ESTIMATOR

§.§ Identifiability

We now discuss properties of the estimator, $\ACEest$, that solves Equation (<ref>). We begin with the identifiability of $\ACEest$:

Let $\bone_n$ denote an $n \times 1$ vector of ones and $\bI_n$ an $n \times x$ identity matrix, and define $\bP_{\bM}=\bM(\bM^T\bM)^{-1}\bM^T$ for a matrix $\bM$. If the matrix

\Big\{\Delta_i (\bZ^a_i,\bs_i-X_i\bone_{m_i})^T(\bI_m - \bP_{\bZ^b_i})(\bZ^a_i,\bs_i-X_i\bone_{m_i}) + %\\
(1 - \Delta_i)(\bZ^a_i,\bs_i-X_i\bone_{m_i})^T(\bI_m - \bP_{\bZ^{b*}_i})(\bZ^a_i,\bs_i-X_i\bone_{m_i})\Big\}

is non-singular (i.e., if $\bN$ is invertible), then $\ACEest$ is identifiable.

We prove Theorem <ref> in Appendix <ref>. The closed form of $\bN$ in Theorem <ref> makes it easy to check for identifiability for a given dataset: we can compute $\bN$ and if it is invertible, we know we can uniquely estimate $\ACEest$ in the presence of the nuisance distributions ($f_{\bb}$, $f_U$).

§.§ Consistency

Next, to establish that $\ACEest$ is a consistent estimator for $\btheta_0$, we must first show that the estimating functions in Equation (<ref>) are unbiased (i.e., that the estimating functions have mean $\bzero$). To this end, we first highlight the simple assumptions that lead to unbiased estimating functions.

Consider the following conditions:

(C1) $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$.

(C2) $\bY_i$ is conditionally independent of $C_i$ given $\widehat{\tilD}_i$.

If (C1) holds, then $\E\{\Delta_i\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)\} = \bzero$.

If (C2) holds, then  $\E\{(1-\Delta_i)\bS_{\rm{eff}}^*(\bY_i,\widehat{\tilD}_i)\} = \bzero$.

We prove Proposition <ref> in Appendix <ref>. Recall that $\tilD_i=(\bZ^a_i, \bs_i, X_i, \bZ^b_i)$ and that $\widehat{\tilD}_i=(\bZ^a_i, \bs_i, \widehat{X}_i, \bZ^b_i)$. Given these definitions, condition (C1) is equivalent to the claim that once we have observed all covariates of interest, the censoring variable itself gives us no new information about the outcome. Condition (C2) is similar to condition (C1), except that $\tilD_i=(\bZ^a_i, \bs_i, X_i, \bZ^b_i)$ has been replaced by $\widehat{\tilD}_i=(\bZ^a_i, \bs_i, \widehat{X}_i, \bZ^b_i)$ (i.e., $X_i$ has been replaced by the imputed value $\widehat{X}_i$). Whether these conditional independence assumptions are valid will depend on the model and data at hand.

Given this unbiasedness, we establish the consistency of $\ACEest$ by applying the Inverse Function Theorem for likelihood-type estimators <cit.>, which requires 1) that our estimating functions are unbiased and 2) the technical conditions outlined in Theorem <ref>:

Consider the regularity conditions:

(C3) The domain $\boldsymbol{\Theta}$ of the parameter $\btheta$ is a compact set.

(C4) The estimating equation $\sum_i\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)$ and $\E\{\sum_i\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)\}$ are sufficiently smooth functions of $\btheta$ in a neighborhood of $\btheta_0$.

(C5) The expectation of our estimating equation, $\E\{\sum_i\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)\}$, has a unique solution and each component of $\E\{{\rm sup}_{\btheta \in \Theta}|\sum_i\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)|\}$ is finite.

If conditions (C1)–(C5) hold. then $\ACEest$ is consistent for $\btheta_0$ (i.e., $\ACEest$ converges in probability to $\btheta_0$).

The consistency of $\ACEest$ is especially appealing given that it does not rely on correctly specifying the nuisance distributions, ($f_{\bb}$, $f_U$). In fact, one does not even need to specify a working model for either of these distributions, and $\ACEest$ still achieves consistency. This contrasts from existing imputation-based solutions to censored covariates <cit.>, which rely on correctly specifying the underlying imputation model.

§.§ Asymptotic normality

$\ACEest$ is not only consistent but also asymptotically normal, as shown next.

Consider the regularity conditions:

(C6) The matrix $n^{-1}\sum_{i=1}^n\frac{\partial}{\partial\btheta} \{\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)\}$ converges uniformly, in probability, to a matrix $E[\frac{\partial}{\partial\btheta}\{\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)\}]$ in a neighborhood of $\btheta_0$.

(C7) The matrix $E[\frac{\partial}{\partial\btheta}\{\bS_{\rm{eff}}^{\rm{full}}(\bY_i,\tilD_i,\widehat{\tilD}_i;\btheta)\}]$ is a bounded, smooth function of $\btheta$ in a neighborhood $\btheta_0$ and is nonsingular.

Under conditions (C1)–(C7),
we have that $n^{1/2}(\ACEest-\btheta_0)$ converges in distribution to $\Normal(0, {\cal V})$, where the matrix ${\cal V} = {\cal A}^{-1} {\cal B} ({\cal A}^{-1})^T$ with
A   =    {∂{_eff^full,_i,_i;_0)}/∂^T},
B=  Var{_eff^full,_i,_i;_0)}.

A proof of Theorem <ref> is provided in Appendix <ref>. This result is powerful because if conditions (C1)–(C2) (Proposition <ref>), (C3)–(C5) (Theorem <ref>), and (C6)–(C7) (Theorem <ref>) all hold, then we can perform asymptotically valid inference about our parameters of interest, $\btheta$. For example, we can construct Wald-type confidence intervals for $\btheta$ and conduct asymptotically valid hypothesis tests about $\btheta$. Moreover, this asymptotic inference can be done without specifying or estimating the nuisance distributions, ($f_{\bb}$, $f_U$). Again, this method is unlike existing methods, which require correctly specifying the imputation model.

§ SIMULATION STUDY

We next compare how ACE imputation performs vs. a competitor in simulation studies. The competitor is multiple conditional mean imputation, where we perform conditional mean imputation $M$ times, estimate $\btheta$ by applying REML to the multiple imputed datasets, and pool the $M$ sets of parameter estimates. The core difference between ACE imputation and the competitor is that our method adjusts for errors that occur when the imputation model is incorrect, whereas the competitor does not adjust for these errors.

§.§ Data generation

We simulate data for $n=1000$ subjects with $m=3$ observations each. We simulate times of clinical diagnosis $X_i$ in two ways. In the first simulation setting, we generate $X_i$ according to a Cox model with two distinct covariates; later in this setting, we impute censored values of $X_i$ using a Cox model that includes both of these covariates, so the imputation model is correctly specified. In the second simulation setting, we generate $X_i$ according to a Cox model with linear and quadratic terms for the same covariate; when we impute censored values of $X_i$ in this setting, we omit the quadratic term and hence misspecify the imputation model. The hazard functions to generate these data are:
λ(t|_i)   =   λ_0(t)exp(η_11V_i1-η_12V_i2),

λ(t|_i)   =   λ_0(t)exp(η_21V_i1-η_22 V_i1^2),
corresponding to the correctly specified imputation model and incorrectly specified imputation model settings, respectively. We set the log-hazard ratios $(\eta_{11},\eta_{12})\trans=(1,0.5)\trans$ and $(\eta_{21},\eta_{22})\trans=(1,0.25)\trans$. The covariates $V_{i1}, V_{i2}$ are independent and normally distributed with mean $0$ and variance $1$. To generate $X_i$ so that it follows the Cox models in Equations (<ref>) and (<ref>), we follow the strategy outlined by <cit.>.

We generate $Y_{ij}$ according to the following linear mixed model:
Y_ij =
βZ^a_ij + α(s_ij-X_i) + b_iZ_ij + ϵ_ij,   b_i∼f_b,    ϵ_ij∼(0,σ^2), where $\btheta=(\beta,\alpha,\sigma^2)\trans=(1,1,1)\trans$. We generate $Z^a_{ij}\sim\Normal(0,1)$, $Z_{ij}\sim\Normal(5,1)$,

$b_i\sim\Normal(0,1)$, and $s_{ij}=j-1$. Having $s_{ij}=j-1$ creates data where subsequent observations for the same subject are evenly spaced by $1$ unit of time and observations begin at time = $0$; this choice of $s_{ij}$ replicates the PREDICT-HD data, where the average time between subsequent observations is $\approx 1.18$ years. Lastly, $X_i$ was censored by a random right-censoring variable $C_i$ that we generated from an exponential distribution with varied rate parameter $\lambda_C$. Having $\lambda_C=0.125,0.5$, and $2$ led to light ($\approx 25\%$), medium ($\approx 50\%$), and heavy ($\approx 75\%$) censoring, respectively.

§.§ Model evaluation methods

For any $X_i$ that is censored, we impute the censored values using conditional mean imputation (Section <ref>). First, we impute using a model that is correctly specified, i.e., we include linear terms for both $V_{i1}$ and $V_{i2}$ in accordance with Equation (<ref>). In the second setting, however, we misspecify the imputation model by omitting the quadratic term $V_{i}^2$ from Equation (<ref>) and fit the incorrect imputation model:

which is misspecified given the true data generation mechanism in Equation (<ref>).

We simulated 1000 datasets and, for each, we estimate $\btheta$ using the following methods:

* Oracle estimator: We apply REML to the full, uncensored dataset. The result from this method is what we would obtain had the data never been censored. REML is a standard procedure for estimating linear mixed models without censored data <cit.>. Of course, this approach is not feasible in practice when times of clinical diagnosis have been censored; hence, we consider these estimates to be the “gold standard” or “Oracle” in our simulations.

* Multiple Conditional Mean Imputation (MCMI): We also apply a multiple imputation procedure that incorporates conditional mean imputation. We repeat the imputation procedure described in Section <ref> above $M=15$ times, but with $\widehat{\boldsymbol{\eta}}$ — the vector of estimated log-hazard ratios in the imputation model — instead drawn from  $\Normal\{\widehat{\boldsymbol{\eta}}, {\rm Cov}(\widehat{\boldsymbol{\eta}})\}$ <cit.>. We then pool these $M$ sets of parameter estimates using Rubin's rules for multiple imputation. We expect this method to be unbiased when the imputation model is correctly specified, but biased when the imputation model is misspecified.

* ACE imputation (ACE): We apply our proposed method, which we expect to be unbiased regardless of whether the imputation model has been correctly specified.

To evaluate the performance of these methods, we calculate the empirical bias, the empirical mean of the standard error estimates, the empirical standard deviation of the parameter estimates, and the empirical mean of the squared biases. We also report the observed coverage of the Wald-type confidence intervals with nominal 95% coverage. We do not present standard error estimates and coverage probabilities for $\sigma^2$ from REML, since the variability of $\sigma^2$ is not typically of interest with this method. Results under medium and heavy censoring are shown in Table <ref> and Table <ref>. Results under light censoring are shown in Table <ref> and Table <ref> in Appendix <ref>.

We employ both existing and new software to carry out these imputation and estimation procedures in R <cit.>. We use the  coxph() function from the survival package to estimate all Cox models <cit.>; the
condl_mean_impute() function from the imputeCensoRd package to execute conditional mean imputation  <cit.>; and the lmer() function from the lme4 package to carry out REML <cit.>. To implement ACE imputation, we developed an R package, ACEimpute; this package contains the eff_score_vector() function, which constructs the estimating equation shown in Equation (<ref>). Lastly, to solve for the root of this estimating equation, we use the m_estimate() function from the geex package <cit.>.

§.§ A correctly specified imputation model

ACE imputation yields highly accurate parameter estimates when the imputation model is correctly specified (Table <ref>). The average empirical bias from ACE imputation is $< 0.003$ for each parameter under light, medium, and heavy censoring. Since the true parameters are all 1, this finding is equivalent to an average percent bias $< 1\%$. Furthermore, we see that the average standard error estimates capture the true variability of the parameter estimates. Together, these results lead to valid inference about $\alpha$ and $\beta$; specifically, we see that

the observed coverage probability of the $95\%$ confidence intervals for $\alpha$ and $\beta$ are between 94% and 96% under each censoring rate.

Furthermore, ACE imputation outperforms the competitor, MCMI. We expected that the competitor would produce unbiased parameter estimates since conditional mean imputation (followed by ordinary least squares regression, instead of REML) is empirically unbiased in cross-sectional settings <cit.>. In our longitudinal simulations, however, MCMI estimates $\alpha$ with $\geq 20\%$ bias, on average. This approach also yields an estimate of $\sigma^2$ with bias $\geq 150\%$ under all three censoring rates. We see that MCMI does yield accurate estimates of $\beta$ (with $\approx 0$ bias on average), but these estimates of $\beta$ are more variable than those produced by ACE imputation. Based on these findings, MCMI leads to seriously biased estimates of $\alpha$ and $\sigma^2$. Fortunately, ACE imputation provides a powerful remedy to this estimation bias, while estimating $\beta$ more efficiently.

Censoring     Param.     Method     Bias     SEE     ESE     MSE     CPr

None     $\alpha$     Oracle     -0.001     0.009     0.010     0.000     0.928

$\beta$     Oracle     0.000     0.022     0.022     0.000     0.940

$\sigma^2$     Oracle     -0.002          0.032     0.001

Medium     $\alpha$     ACE     0.000     0.026     0.024     0.001     0.960

($\approx 50\%$)
MCMI     0.312     0.033     0.107     0.109     0.000

$\beta$     ACE     0.001     0.026     0.025     0.001     0.958

MCMI     0.001     0.040     0.040     0.002     0.966

$\sigma^2$     ACE     -0.000     0.036     0.037     0.001     0.940

MCMI     2.498          2.947     14.914

Heavy     $\alpha$     ACE     0.001     0.028     0.029     0.001     0.947

($\approx 75\%$)
MCMI     0.206     0.044     0.072     0.048     0.014

$\beta$     ACE     0.001     0.028     0.028     0.001     0.952

MCMI     -0.001     0.045     0.046     0.002     0.950

$\sigma^2$     ACE     -0.003     0.040     0.040     0.002     0.940

MCMI     3.356          3.426     22.989

Simulation results when the imputation model is correctly specified. Results are shown for three methods: ACE imputation,
MCMI (multiple conditional mean imputation), and the Oracle estimator. We present empirical bias, the average standard error estimate (SEE), the empirical standard error (ESE), the average mean squared errors (MSE), and the observed coverage probability (CPr) of the Wald-type confidence intervals with nominal 95% coverage.

§.§ A misspecified imputation model

Table <ref> shows the results when we omit the quadratic term from (and hence, misspecify) the imputation model. We see that the competitor, MCMI, performs much worse under this setting than when the imputation model is correct. Although this approach still estimates $\beta$ quite accurately, it now estimates $\alpha$ with $> 100\%$ bias under all three censoring rates and drastically overestimates $\sigma^2$, with an average bias $>> 100\%$. This bias is to be expected; when we misspecify our imputation model, we should anticipate that the conditional means ${\rm E}(X_i|X_i>C_i,V_{i1})$, with which we replace censored $X_i$, should be further away from the true $X_i$. In contrast, we see that imputation model misspecification does not hinder the performance of ACE imputation, which performs very similarly in both settings considered. ACE imputation produces accurate parameter estimates and reliable inference even though we have omitted the quadratic term from our imputation model.

ACE imputation thus allows us to consistently estimate the parameters of a linear mixed model with a censored covariate, even when we do not know the correct imputation model a priori, which can easily occur in biological settings <cit.>.

Censoring     Param.     Method     Bias     SEE     ESE     MSE     Coverage

None     $\alpha$     Oracle     -0.000     0.003     0.003     0.000     0.954

$\beta$     Oracle     0.001     0.022     0.022     0.000     0.953

$\sigma^2$     Oracle     -0.001          0.032     0.001

Medium     $\alpha$     ACE     0.000     0.026     0.026     0.001     0.948

($\approx 50\%$)
MCMI     1.591     0.264     6.569     45.639     0.005

$\beta$     ACE     0.001     0.026     0.026     0.001     0.949

MCMI     -0.069     0.327     1.137     1.296     0.955

$\sigma^2$     ACE     -0.002     0.037     0.038     0.001     0.943

MCMI     $\geq 100$          $\geq 100$     $\geq 100$

Heavy     $\alpha$     ACE     0.000     0.029     0.029     0.001     0.946

($\approx 75\%$)
MCMI     0.716     0.326     3.085     10.018     0.188

$\beta$     ACE     0.000     0.029     0.029     0.001     0.948

MCMI     -0.073     0.332     1.263     1.598     0.949

$\sigma^2$     ACE     -0.001     0.041     0.040     0.002     0.951

MCMI     $\geq 100$          $\geq 100$     $\geq 100$

Simulation results when the imputation model is misspecified. Results are shown for three methods: ACE imputation,
MCMI (multiple conditional mean imputation), and the Oracle estimator. We present empirical bias, the average standard error estimate (SEE), the empirical standard error (ESE), the average mean squared errors (MSE), and the observed coverage probability (CPr) of the Wald-type confidence intervals with nominal 95% coverage.

§ MODELING THE PROGRESSION OF HUNTINGTON DISEASE

Huntington disease is a fatal neurodegenerative disorder that causes impairment across motor, cognitive, and psychiatric domains. The disease is caused by repeated cytosine-adenine-guanine (CAG) mutations in the huntingtin gene and people with more than 36 CAG repeats are guaranteed to develop Huntington disease <cit.>. This creates a unique opportunity for clinicians studying the disease; with genetic testing, it is possible to recruit subjects who are guaranteed to develop Huntington disease (“gene mutation carriers”) and test therapies aimed at slowing its progression.

A key consideration when planning clinical trials of Huntington disease is selecting outcomes that change quickly enough over time, so that analysts can more easily detect differences between the treatment and placebo groups. To facilitate this selection, analysts have sought to compare possible outcomes in terms of how quickly they change in gene mutation carriers <cit.>. To quantify the speed at which these outcomes change, analysts estimate longitudinal models for these outcomes using data from untreated subjects; these “disease progression models” then give a metric for how quickly the possible outcomes progress over time in untreated subjects. Hence, by comparing these estimated disease progression models, we can easily compare how quickly these outcomes progress.

To contribute to this ongoing effort, we analyze data from PREDICT-HD, a longitudinal study of gene mutation carriers who were recruited prior to diagnosis <cit.>. Specifically, we assess seven potential outcomes that quantify Huntington disease symptoms: (1) total motor score (TMS); (2) SDMT score; (3–5) scores on the Stroop word, color, and interference tests; (6) total functional capacity (TFC); and (7) composite unified Huntington disease rating scale (cUHDRS). TMS assesses a subject's motor impairment. The SDMT and Stroop tests assess the degree of cognitive impairment. TFC assesses a subject's functional ability to perform daily tasks. The cUHDRS is a linear combination of TFC, TMS, SDMT score, and the Stroop word test score <cit.>.

For each outcome $Y_{ij}$, we fit the following linear mixed effects model:

Y_ij = β_0 + ^T _ij + α(s_ij - X_i) + b_i + ϵ_ij,
with fixed intercept $\beta_0$, fixed slopes ($\bbeta^\textrm{T}$, $\alpha$), and random intercept $b_i$. The random intercepts $b_i$ are assumed to follow the distribution $f_b$, which is left unspecified, and the random errors $\epsilon_{ij}$ are assumed to be normally distributed with mean $0$ and variance $\sigma^2$. For each subject $i$, the additional covariates $\bZ_{ij}=({\rm AGE}_i,{\rm SEX}_i,{\rm EDUCATION}_i,{\rm CAG}_i)$ are included, all measured at baseline. ${\rm AGE}_i$ is age at baseline, ${\rm SEX}_i$ is $1$ if subject $i$ is male and $0$ if subject $i$ is female, ${\rm EDUCATION}_i$ is years of education, and ${\rm CAG}_i$ is number of CAG repeats <cit.>.
The variables $s_{ij}$ and $X_i$ denote time in the study (in years) and time of clinical diagnosis (in years), respectively, such that $s_{ij} - X_i$ denotes time to clinical diagnosis. As discussed in Section <ref>, we choose to use time to clinical diagnosis ($s_{ij} - X_i$) instead of just time in the study ($s_{ij}$) to account for the fact that subjects enter the study at different levels of disease progression.

To compare how the potential outcomes progress in gene mutation carriers, we compare the estimated slopes on time to clinical diagnosis (i.e., $\hat{\alpha}$) when the model in Equation (<ref>) is fit for each outcome (cUHDRS, TMS, SDMT score, TFC, and the Stroop test scores). The outcomes are ranked based on their “standardized slopes,” calculated as the absolute values of the slopes divided by their estimated standard errors  (i.e., $|\hat{\alpha}|/{\rm SE}(\hat{\alpha})$).

Prior to analysis, we apply exclusion criteria similar to those employed by <cit.>. We require subjects to be a Huntington disease carrier (i.e., to have $\geq$ 36 CAG repeats) and not yet
be clinically diagnosed at study entry (i.e., to have a diagnostic confidence level $\leq 3$ at their first visit). We also filter to include only those visits that have complete data for all outcomes and covariates $Z_{ij}$ of interest, which removes only 25 subjects (2%) and 260 visits (4%).
With these criteria, we arrive at an analytic dataset containing 1,102 unique subjects with 5,612 total observations. Descriptive statistics are provided in Table <ref> in Appendix <ref>.

§.§ Imputing censored times of clinical diagnosis

Of the 1,102 subjects, 244 (22.1%) were clinically diagnosed during the course of study, and 858 (77.9%) were not (i.e., their times of diagnosis were right-censored). We imputed all censored times of diagnosis using the Cox-based conditional mean imputation (Section <ref>).

We use CAG-Age product, TMS, and SDMT score as our set of Cox model covariates. The CAG-Age Product is defined as $X_{i, \rm age} \times (X_{i, \rm CAG} - 34)$ <cit.> and conveys the “burden of disease” that has accumulated over a subject's life. <cit.> show that these three variables predict time of clinical diagnosis better than competing sets of covariates, and we believe that these variables are also an intuitive choice for predicting time of clinical diagnosis. It has been well established that the number of CAG repeats is negatively associated with time of clinical diagnosis <cit.>. In addition, TMS and SDMT score measure a subject's motor and cognitive capacities, respectively, which are both known to decrease as a subject approaches clinical diagnosis. With these covariates, we fit a Cox model with hazard function

where covariates $\bV_i$ include the visit 1 values of TMS, SDMT score, and CAG-Age Product for subject $i$ and $\bgamma = (\gamma_1,\gamma_2,\gamma_3)\trans$ are the corresponding log-hazard ratios. Before fitting this model, we center and scale the covariates using their sample means and standard deviations (given in Table <ref> in Appendix <ref>) to reduce possible collinearity.

With this Cox model estimate, we implement conditional mean imputation to replace values of $X_i$ with $\widehat{X}_i=\Delta_iX_i+(1-\Delta_i){\rm E}(X_i|X_i>C_i,\bV_i)$.

§.§ Ranking measures of Huntington disease impairment

Using this imputed dataset, we estimate the linear mixed effects model in Equation (<ref>) for each of the seven outcomes. We compute how quickly each potential outcome changed over time using the scaled slope $|\hat{\alpha}|/ {\rm SE}(\hat{\alpha})$ and rank them. We carry out this ranking procedure once for each of three methods: i) complete case analysis (CCA), with REML applied to uncensored subjects only, ii) multiple conditional mean imputation (described in Section <ref>), and iii) ACE imputation.

Table <ref> shows the results of these ranking procedures. All three methods agree that the cUHDRS shows the quickest progression. This finding is in contrast to that of <cit.>, who

show that TMS changes most quickly among the outcomes analyzed and is therefore the preferred outcome.

Whereas TMS measures only motor impairment, the cUHDRS is a linear combination of TFC, TMS, SDMT score, and Stroop word test score; hence, the cUHDRS is a more comprehensive measure of the impairment caused by Huntington disease. Identifying a therapy that improves subjects' cUHDRS could, therefore, more holistically improve quality of life.
This holistic improvement is a main reason why the cUHDRS is being advocated for as a primary outcome for clinical trials targeting Huntington disease in subjects who have already been diagnosed. Having analyzed data exclusively from patients who were undiagnosed at baseline, our results suggest that the cUHDRS could also be chosen as an easily detectable outcome for patients recruited in this “pre-manifest” stage.

It should be noted that since the cUHDRS is a linear combination of four other measures, it could be more prone to missing-ness for a given dataset. Given this possible concern, investigators may want to measure impairment due to Huntington disease using an outcome which measures only one symptom (e.g., SDMT score, TMS). According to our results, the best outcome that is a measure of only one symptom is SDMT score. In fact, it has been established that cognitive impairment shows up earlier than motor impairment <cit.>. Our analysis aligns with this previous finding, whereas the other methods point to motor impairment instead.

Method     Outcome     Rank     Estimate     SE     Scaled Slope     Prop. to Max.

CCA     cUHDRS     1     -0.506     0.011     44.179     1.000

TMS     2     2.304     0.062     37.292     0.844

Stroop Word     3     -2.106     0.070     30.118     0.682

SDMT     4     -1.258     0.043     29.205     0.661

Stroop Color     5     -1.608     0.060     26.629     0.603

TFC     6     -0.253     0.010     24.104     0.546

Stroop Interference     7     -0.762     0.044     17.419     0.394

[gray]0.75ACE     cUHDRS     1     -0.253     0.013     20.064     1.000

[gray]0.75    SDMT     2     -0.757     0.041     18.303     0.912

[gray]0.75    Stroop Word     3     -1.177     0.070     16.883     0.841

[gray]0.75    TMS     4     0.968     0.064     15.026     0.749

[gray]0.75    Stroop Color     5     -0.802     0.059     13.532     0.674

[gray]0.75    TFC     6     -0.102     0.009     11.466     0.571

[gray]0.75    Stroop Interference     7     -0.314     0.037     8.489     0.423

MCMI     cUHDRS     1     -0.250     0.005     45.714     1.000

TMS     2     1.068     0.024     44.250     0.968

Stroop Word     3     -1.170     0.039     29.876     0.654

SDMT     4     -0.739     0.026     28.226     0.617

TFC     5     -0.109     0.004     27.069     0.592

Stroop Color     6     -0.807     0.033     24.189     0.529

Stroop Interference     7     -0.338     0.025     13.613     0.298

CCA: complete case analysis. ACE: ACE imputation.
MCMI: multiple conditional mean imputation. cUHDRS: Composite Unified Huntington Disease Rating Scale. TMS: total motor score. SDMT: Symbol Digits Modality Test. TFC: total functional capacity. Scaled slope $= |\hat{\alpha} / {\rm SE}(\hat{\alpha})|$, a measure of how quickly the outcome changes over time until clinical diagnosis. Rank: ordering of scaled slopes estimated by the same method. Prop. to max.: proportion of scaled slope to the largest scaled slope estimated by the same method.

§.§ Calculating sample size for a clinical trial

Next, we calculate the sample size required for a hypothetical clinical trial of Huntington disease. Then, we compare the sample size estimates based on ACE imputation, conditional mean imputation, and complete case analysis. We present the sample size required to detect treatment effects in a clinical trial with SDMT score as the primary outcome. For simplicity, we assume a balanced design, such that both the treatment and placebo groups are of the same size. To investigate the treatment effect, we compare the trajectories of SDMT scores between the placebo (subscript $p$) and treatment (subscript $t$) groups, respectively, using the following linear models:
Y_p   =   β_p + α_p (s - X)

Y_t   =   β_t + α_t (s - X),
where $\beta$ denotes the intercept and $\alpha$ denotes the slope on time to clinical diagnosis, $s - X$. With these models, we aim to make inference about the quantity $\delta = \alpha_t - \alpha_p$.

Specifically, to investigate whether the treatment under study slowed cognitive impairment (as measured by SDMT scores), we define the null and alternative hypotheses $H_0: \delta=0$ and $H_A:\delta > 0$, respectively. To calculate the required sample sizes to test these hypotheses with desired power, we begin with the Z-statistic for $\delta$, $\hat{\delta} \sqrt{ n_g \{ {\rm Var} (\delta) \}^{-1}}$.
From this statistic, the per-group sample size given type I error rate $\kappa$ and power $1 - \Gamma$ is
[ √(Var(δ)){Φ^-1(1 - Γ)-Φ^-1(κ)}/-d ]^2,
where $\Phi^{-1}(\cdot)$ denotes the inverse cumulative distribution function for the distribution $\Normal(0,1)$ and $d$ is the assumed true value of $\delta$.
For simplicity, we assume $\sqrt{{\rm Var}(\delta)}=1$.

Suppose that we are specifically interested in the sample size required to detect
a 10% treatment effect with 80% power, such that $\alpha_t = (1 - 0.1)\alpha_p$, $\kappa=0.05$, and $1 - \Gamma=0.8$. Borrowing the model estimates for $\alpha$ from Section <ref> as the slopes on $s - X$ for the placebo group (i.e., using these estimates as $\alpha_p$) we can calculate this sample size based on each approach. Based on $\hat{\alpha_p}=-0.757$ from ACE imputation, we find that a per-group sample size of $1079$ would be needed for our desired test. In contrast, basing the calculations on the complete case analysis led to a smaller sample size ($\hat{\alpha_p}= -1.258$, 391 required) and MCMI led to a larger required sample size ($\hat{\alpha_p}= -0.739$, 1134 required).

Given the theoretical and empirical evidence supporting ACE imputation, we believe $1079$ to be the most reliable per-group sample size for the proposed test. Hence, using this sample size as a reference, the sample size from complete case analysis would drastically under-power our ability to detect a 10% treatment effect. Although the sample size from MCMI is much closer to that from ACE imputation, recruiting 55 more subjects per group would be both costly and potentially infeasible given the low prevalence of Huntington disease, at 12 per 100,000 people <cit.>. The corresponding power curves for all approaches for 10%, 15%, and 20% treatment effects are given in Figure <ref> in Appendix <ref>.

§ DISCUSSION

Because Huntington disease causes such multifaceted impairment, clinicians must select from many possible outcomes when designing clinical trials. To inform this selection, analysts have sought to rank possible outcomes by how rapidly they decline over time in gene mutation carriers. This ranking can only be done when we have accurate disease progression models, which rely on knowing the time at which subjects are clinically diagnosed with Huntington disease. When this time of clinical diagnosis is censored — as it often is in observational studies of Huntington disease — we can impute it, but then our outcome model estimates are sensitive to whether we correctly specified our imputation model; when we misspecify this imputation model, bias and inefficiency can result.

In this paper, we presented ACE imputation, which accurately estimates a longitudinal model given an imputed covariate, even when we use a misspecified imputation model.
We proved that this estimator can be calculated without positing distributions for the random effects $\bb$ or the imputation error $U$. Moreover, this estimator achieves consistency and asymptotic normality without prior knowledge about these nuisance distributions. Not having to estimate or postulate distributions for $\bb$ or $U$ is unlike typical semiparametric methods, which require positing nuisance distributions and which maintain consistency but suffer from inefficiency when the posited distributions are incorrect.

Although ACE imputation produces an estimator which is identifiable, consistent, and asymptotically normal, this novel method is not without limits.

As we show in Appendix <ref>, the proposed estimating equation may fail to converge when a column of the fixed effects design matrix belongs to the column space of the random effects design matrix for all subjects. For example, this “column space issue” can arise when we include both a fixed intercept and a random intercept in our model. While more rigorous details are shown in Appendix <ref>, we can heuristically understand this issue as follows: when, for example, we include both fixed and random intercepts, the fixed intercept is the mean of the random intercept distribution. Therefore, there is a conflict when we simultaneously attempt to estimate the fixed intercept and treat the distribution of the random intercept (of which the fixed intercept is the mean) as a nuisance parameter. Given this understanding, perhaps this column space issue could be overcome by requiring the nuisance distribution of the random effects to have mean $\bzero$ so that we can untangle the fixed effects of interest from the nuisance distribution. Future investigations into solving this issue are required.

§ SUPPLEMENTARY MATERIAL
The supplementary material contains all technical details of our theorems and propositions. An open-source R package ACEimpute and code to replicate the simulation study is found at https://github.com/Tanya-Garcia-Lab/ACEimpute.

§ EFFICIENT SCORE VECTORS TO ESTIMATE $\BTHETA$ IN THE PRESENCE OF NUISANCE PARAMETERS

To apply the semiparametric framework from <cit.>, observe that the density for the $i$th subject in our model is:
f__i,_i,mis(_i,_i;)= ∫f__i|_i,_i(_i|_i,_i;)f__i,_i,mis(_i,_i)dμ(_i)
if $\Delta_i=1$ and f__i,_i,mis(_i,_i;)= ∫f__i|_i,_i,U_i(_i|_i,_i,U_i;)f__i,_i,U_i,mis(_i,_i,U_i)dμ(_i, U_i)
if $\Delta_i=0$. Here, the subscript $_{\rm{mis}}$ is used to indicate a term that is subject to misspecification due to misspecification of the joint density for $(\tilD_i, \bb_i)$ (or $(\tilD_i, \bb_i, U_i)$). Then, using similar techniques to those in <cit.>, we have that the efficient score vector for uncensored subjects  is:

_eff, mis(_i,_i;)=_, mis(_i,_i;)-_mis{(_i,_i)|_i,_i}
where $\bS_{\rm \btheta, mis}(\bY_i,\tilD_i;\btheta)=\frac{\partial}{\partial \btheta}\log\{f_{\bY_i,\tilD_i,{\rm mis}}(\bY_i,\tilD_i;\btheta)\}$. Here, $\bh(\tilD_i,\bb_i)$ is a function with mean $\bzero$ and which satisfies
{_, mis(_i,_i;)|_i,_i}=[_mis{(_i,_i)|_i,_i}|_i,_i].
Similarly, the efficient score vector for censored subjects is

_eff, mis^*(_i,_i;)=_, mis^*(_i,_i;)-_mis{(_i,_i,U_i)|_i,_i}
where $\bS_{\rm \btheta, mis}^*(\bY_i,\widehat{\tilD}_i;\btheta)=\frac{\partial}{\partial \btheta}\log\{f_{\bY_i,\widehat{\tilD}_i,{\rm mis}}(\bY_i,\widehat{\tilD}_i;\btheta)\}$ and  $\bh(\widehat{\tilD}_i,\bb_i,U_i)$ is a function with mean $\bzero$ and which satisfies
{_, mis^*(_i,_i;)|_i,_i,U_i}=[_mis{(_i,_i,U_i)|_i,_i}|_i,_i,U_i].

Yet, as <cit.> point out, it is difficult to find functions  $\bh(\tilD_i,\bb_i)$ and $\bh(\widehat{\tilD}_i,\bb_i,U_i)$ that satisfy these conditions. In Section <ref>, we simplify the efficient score vectors given in Equation (<ref>) and Equation (<ref>). We achieve this simplification by leveraging the properties of multivariate normal distributions and linear projection theory. In so doing, we circumvent the need for the functions $\bh(\tilD_i,\bb_i)$ and $\bh(\widehat{\tilD}_i,\bb_i,U_i)$ altogether and provide straightforward, closed forms of both efficient score vectors. Moreover, we show in Appendix <ref> that these efficient score vectors can be calculated without specifying joint distributions for $(\tilD, \bb_i)$ or $(\widehat{\tilD}, \bb_i, U_i)$; this lack of specification is impactful because if we did have to specify these joint distributions, we would run the risk of misspecifying them, which would then decrease the overall efficiency of $\ACEest$, our estimator for $\btheta$.

§ SIMPLIFYING THE EFFICIENT SCORE VECTORS

§.§ Transforming the Response Vector for Uncensored Subjects

To simplify the efficient score vector for uncensored subjects, we will 1) transform $\bY_i$ for censored subjects, 2) establish five properties about the conditional distribution of $\bY_i$ given this transformation and $(\tilD_i,\bb_i)$, then 3) leverage these properties to simplify the complicated form of the efficient score vector given in Equation (<ref>) above. Specifically, we transform $\bY_i$ via $\bT_i=\sigma^{-2}\bZ^{bT}_i\bY_i$. In Proposition <ref>, we outline the five key properties pertaining to the conditional distribution of $\bY_i$ given $(\bT_i,\tilD_i,\bb_i)$:

Consider the transformed response variable denoted by $\bT_i=\sigma^{-2}\bZ^{bT}_i\bY_i$. If the random effects design matrix $\bZ^{b}_i$ is of full column rank (i.e., if ${\rm rank}(\bZ^{b}_i)=p_b$), then,

(A) $\rm{E}(\bY_i|\bT_i,\tilD_i,\bb_i)=\bzeta(\tilD_i;\btheta)+\bP_{\bZ_i^b}\{\bY_i-\bzeta(\tilD_i;\btheta)\}$, where $\bP_{\bM} = \bM(\bM^T\bM)^{-1}\bM^T$ is the orthogonal projection operator for a matrix $\bM$.

(B) Conditional on $(\bT_i,\tilD_i)$, $\bY_i$ and $\bb_i$ are independent.

(C) $\rm{E}(\bY_i^T\bY_i|\bT_i,\tilD_i,\bb_i)=\sigma^2(m_i-p_b)+||\rm{E}(\bY_i|\bT_i,\tilD_i,\bb_i)||^2_2$, where $||\cdot||^2_2$ denotes the L2 norm.

(D) For any vector-valued function $\bg(\bT_i,\tilD_i)$, if $\E\{\bg(\bT_i,\tilD_i)|\tilD_i,\bb_i\}=\bzero$, then it follows that $\bg(\bT_i,\tilD_i)=\bzero$.

(E) Given a random variable $\bR$, $\bR$ is conditionally independent of $\bT_i=\sigma^{-2}\bZ^{bT}_i\bY_i$ given $(\bY_i, \bZ_i^b)$.

To prove Proposition <ref>, we first return to the outcome model of interest:
Y_ij =
^T ^a_ij + α(s_ij - X_i) + _i^T ^b_ij + ϵ_ij

_i∼f_,    ϵ_ij∼(0,σ^2).
From this model, we can see that conditional on $(\bZ^a_{ij}, s_{ij}, X_i, \bZ^b_{ij}, \bb_i)$, the response $\bY_{ij}$ follows a univariate normal distribution, $\Normal\{\bbeta^T \bZ^a_{ij} + \alpha(S_{ij} - X_i) + \bZ^b_{ij}\bb_i, \sigma^2\}$. Recall that we assume that responses from the $i$th subject are independent conditional on that subject's random effects $\bb_i$. Given this fact and the conditional distribution of $Y_{ij}$, we know that, conditional on $(\tilD_i, \bb_i)$, the $m$-dimensional response vector $\bY_{i}$ follows a multivariate normal distribution, $\Normal_{m_i}\{\bZ^a_i\bbeta  + \alpha(\bS_i - X_i\bone_{m_i}) + \bZ^b_i\bb_i, \sigma^2\bI_{m_i}\}$, where $\bone_{m_i}$ denotes an $m_i$-vector of ones and $\bI_{m_i}$ denotes the $m_i \times m_i$ identity matrix. For notational simplicity, let $\bzeta(\tilD_i;\btheta)=\bZ^a_i\bbeta  + \alpha(\bS_i - X_i\bone_{m_i})$. We consider this to be the fixed effects component of the linear predictor for $\bY_i$.

Next, we derive the joint conditional distribution of $(\bY_i, \bT_i)$ given $(\tilD_i,\bb_i)$. Given our definition of $\bT_i=\sigma^{-2}\bZ^{bT}_i\bY_i$, it follows that $(\bY_i^T, \bT_i^T)^T = (\bI_{m_i}, \sigma^{-2}\bZ_i^b)^T \bY_i$. From this, we know that, conditional on $(\tilD_i,\bb_i)$, $(\bY_i, \bT_i)\sim\Normal_{m_i}[(\bI_{m_i}, \sigma^{-2}\bZ_i^b)^T\{\bzeta(\tilD_i;\btheta) + \bZ_i^b\bb_i\}, \sigma^2(\bI_{m_i}, \sigma^{-2}\bZ_i^b)^T(\bI_{m_i}, \sigma^{-2}\bZ_i^b)]$. Specifically,

(_i;) + _i^b_i}

(_i;) + _i^b_i

σ^-2_i^bT{(_i;) + _i^b_i}

)    =

_m_i     σ^-2_i^b

).Next, we establish the conditional distribution of $\bY_i$ given $(\bT_i, \tilD_i,\bb_i)$. Using the previous result, we know that $\bY_i$ given $(\bT_i, \tilD_i,\bb_i)$ is also normally distributed with
E(_i|_i,_i,_i)   =   (_i;)+_i^b_i+σ^2_i^b(_i^bT_i^b)^-1[_i - σ^-2_i^bT{(_i;) + _i^b_i}]

=    σ^2_i^b(_i^bT_i^b)^-1_i + (_m_i - __i^b){(_i;) + _i^b_i},
where $\bP_{\bZ_i^b} = \bZ_i^b(\bZ_i^{bT}\bZ_i^b)^{-1}\bZ_i^{bT}$ is the orthogonal projection operator onto the column space of $\bZ_i^b$. Using this, we note that  $(\bI_{m_i} - \bP_{\bZ_i^b})\bZ_i^b\bb_i=0$ since $\bP_{\bZ_i^b}\bZ_i^b = \bZ_i^b$, given our assumption that$\bZ_i^b$ is of full column rank. Thus, we conclude that
E(_i|_i,_i,_i)   =
σ^2_i^b(_i^bT_i^b)^-1_i + (_m_i - __i^b)(_i;)

=   _i^b(_i^bT_i^b)^-1^bT_i_i + (_m_i - __i^b)(_i;)

=   (_i;)+__i^b{_i-(_i;)},
and so Proposition <ref> (A) is established. Next, observe that
Cov(_i|_i,_i,_i)   =   σ^2_m_i - σ^2_i^b(_i^bT_i^b)^-1_i^bT =σ^2(_m_i - __i^b).
In summary,
_i | (_i,_i) ∼_m_i[(_i;)+__i^b{_i-(_i;)}, σ^2(_m_i - __i^b)].
Observe that this distribution does not involve the random effects $\bb_i$. We therefore conclude — because a multivariate normal distribution is completely specified by its mean and covariance — that $\bY_i$ and $\bb_i$ are conditionally independent given $(\bT_i,\tilD_i)$. This confirms Proposition <ref> (B).

Next, we note that for a random vector $\bY$ with mean $\bmu$ and variance-covariance matrix $\bSigma$, $E(\bY^T\bY) = \bmu^T\bmu + tr(\bSigma)$. Given this, it follows from the conditional distribution of $\bY_i$ given $(\bT_i, \tilD_i,\bb_i)$ that
E(_i^T_i|_i,_i,_i)   =   tr{σ^2(_m_i-__i^b)}+||E(_i|_i,_i,_i)||^2_2

=   σ^2tr(_m_i-__i^b)+||E(_i|_i,_i,_i)||^2_2

=   σ^2(m_i-p_b)+||E(_i|_i,_i,_i)||^2_2,

since $\bI_m - \bP$ is itself an orthogonal projection operator of rank $m-r$ and the trace of an orthogonal projection operator is equal to its rank. This establishes Proposition <ref> (C).

Proposition <ref> (D) follows from the following calculation, given an arbitrary function $\bg(\bT_i,\tilD_i)$ such that $\E\{\bg(\bT_i,\tilD_i)|\bb_i,\tilD_i\}=\bzero$:

=    ∫(_i,_i)f(_i|_i,_i)dμ(_i)

=    ∫(_i,_i)

(_i^bT_i^b)^-1{_i-σ^-2_i^bT((_i;) +_i^b_i)}]dμ(_i),
which follows from the conditional distribution of $(\bY_i, \bT_i)$ in Equation (<ref>). Next, define
k(_i,_i;)   =   exp[-σ^4/2{_i-σ^-2_i^bT((_i;)+_i^b_i)}^T×

(_i^bT_i^b)^-1{_i-σ^-2_i^bT((_i;) +_i^b_i)}]dμ(_i),
which is positive for all $(\bT_i,\tilD_i,\btheta)$. Hence, the fact that
= ∫k(_i,_i;)
for all $\btheta$ implies that $k(\bT_i,\tilD_i;\btheta)\bg(\bT_i,\tilD_i)=\bzero$. Yet, because $k(\bT_i,\tilD_i;\btheta)$ is positive for all $(\bT_i,\tilD_i,\btheta)$, this can only be true if $\bg(\bT_i,\tilD_i)=\bzero$. This establishes Proposition <ref> (D). Proposition <ref> (E) follows from Lemma <ref>:

Given random variables $\bR_1$, $\bR_2$, and $\bR_3$ and a vector-valued function $\bg(\cdot)$, $\bR_1$ is independent of $\bg(\bR_2, \bR_3)$ conditional on $(\bR_2, \bR_3)$.

To prove Lemma <ref>, observe:
P{_1 ≤_1, (_2,_3) ≤| _2 = _2, _3 = _3}

=   P(_1 ≤_1 | _2 = _2, _3 = _3)I{(_2,_3) ≤}

=   P(_1 ≤_1 | _2 = _2, _3 = _3)P{(_2,_3) ≤| _2 = _2, _3 = _3}.

Proposition <ref> (E) follows directly from Lemma <ref>, since $\bT_i = \sigma^{-2}\bZ_i^{bT}\bY_i$ is a function of $(\bY_i, \bZ_i^b)$. Hence, Proposition <ref> is proven in full.

§.§ Transforming the Response Vector for Censored Subjects

Similarly, we will transform the response vector for censored subjects, establish five properties about the conditional distribution of the response given this transformation, then leverage these properties to simplify the complicated form of the efficient score vector given in Equation (<ref>). These five properties are outlined in Proposition <ref>:

Consider the transformed response variable $\bT_i^{\rm aug}=\sigma^{-2}(\bone_{m_i},\bZ_i^b)^T\bY_i$. If the matrix $(\bone_{m_i},\bZ_i^b)$ is of full column rank (i.e., if ${\rm rank}\{(\bone_{m_i},\bZ_i^b)\}=p_b+1$), then,

$\rm{P1}^*$ $\rm{E}(\bY_i|\bT_i^{\rm aug},\widehat{\tilD}_i,\bb_i,U_i)=\bzeta(\widehat{\tilD}_i;\btheta)+\bP_{(\bone_{m_i},\bZ_i^b)}\{\bY_i-\bzeta(\widehat{\tilD}_i;\btheta)\}$, where the matrix $\bP_{\bM} = \bM(\bM^T\bM)^{-1}\bM^T$ is the orthogonal projection operator for a matrix $\bM$.

$\rm{P2}^*$ Conditional on $(\bT_i^{\rm aug},\widehat{\tilD}_i)$, $\bY_i$ and $(\bb_i,U_i)$ are independent.

${\rm P3}^*$ ${\rm E}(\bY_i^T\bY_i|\bT_i^{\rm aug},\widehat{\tilD}_i,\bb_i,U_i)=\sigma^2(m_i-p_b)+||\rm{E}(\bY_i|\bT_i^{\rm aug},\widehat{\tilD}_i,\bb_i,U_i)||^2_2$, where $||\cdot||^2_2$ denotes the L2 norm.

${\rm P4}^*$ Given any function $\bg(\bT_i^{\rm aug},\widehat{\tilD}_i)$, if $\E\{\bg(\bT_i^{\rm aug},\widehat{\tilD}_i)|\widehat{\tilD}_i,\bb_i,U_i\}=\bzero$, then it follows that $\bg(\bT_i^{\rm aug},\widehat{\tilD}_i)=\bzero$.

${\rm P5}^*$ Given any random variable $\bR$, $\bR$ is conditionally independent of $\bT_i^{\rm aug}$ given $(\bY_i, \bZ_i^b)$.

We begin the proof of Proposition <ref> but omit the rest because it follows analogously from the proof of Proposition <ref>. Recall that for censored subjects, the outcome model is equivalent to
Y_ij =
^T ^a_ij + α(s_ij - X_i - U_i) + _i^T ^b_ij + ϵ_ij

_i∼f_,    U_i∼f_U,     ϵ_ij∼(0,σ^2)
since the imputed conditional mean $\widehat{X}_i$ is related to the censored $X_i$ through a Berkson error model, $X_i = \widehat{X}_i + U_i$. Then, conditional on the variables $(\widehat{\tilD}_i, \bb_i,U_i)$, the $m_i$-dimensional response vector $\bY_{i}$ follows a multivariate normal distribution $\Normal_{m_i}\{\bZ^a_i\bbeta  + \alpha(\bS_i - \widehat{X}_i\bone_{m_i} - U_i\bone_{m_i}) + \bZ^b_i\bb_i, \sigma^2\bI_{m_i}\}$. To simplify the mean of this multivariate normal distribution, we reuse the notation $\bzeta(\widehat{\tilD}_i;\btheta)=\bZ^a_i\bbeta+\alpha(\bs_i - \widehat{X}_i\bone_{m_i})$. With this notation, the conditional mean of $\bY_i$ given $(\widehat{\tilD}_i, \bb_i,U_i)$ is now $\bzeta(\widehat{\tilD}_i;\btheta)+(\bone_{m_i},\bZ^b_i)(-\alpha U_i, \bb_i^T)^T$.

Similar to Section <ref>, we propose transforming the response variables for censored subjects via $\bT_i^{\rm aug}=\sigma^{-2}(\bone_{m_i},\bZ^b_i)^T\bY_i$. This is the same transformation used for the response vector for uncensored subjects, except that $\bZ_i^b$ has been augmented by a vector of ones (hence the “aug” superscript). From here, the proof of Proposition <ref> follows in paralell from that of Proposition <ref>, with $\tilD_i$ replaced by $\widehat{\tilD}_i$, $\bZ_i^b$ replaced by $(\bone_{m_i},\bZ^b_i)$, $\bb_i$ replaced by $(\bb_i, U_i)$, and $\bT_i$ replaced by $\bT^{\rm aug}_i$.

§.§ Finding a Closed Form for the Efficient Score Vectors

To simplify the efficient score vector for uncensored subjects (Equation (<ref>)), first recall that the original form of this efficient score vector requires that we find a function $\bh(\tilD_i,\bb_i)$ that satisfies
{_, mis(_i,_i;)|_i,_i}=[_mis{(_i,_i)|_i,_i}|_i,_i],
where $\bS_{\btheta,\rm{mis}}(\bY_i,\tilD_i)=\frac{\partial}{\partial \btheta}\log f_{\bY_i,\tilD_i,\rm{mis}}(\bY_i,\tilD_i;\btheta)$. Observe that
{_, mis(_i,_i;)|_i,_i}
=E[{_, mis(_i,_i;)|_i,_i,_i}|_i,_i]
by the law of total expectation. Furthermore,
by Proposition <ref> (E). Therefore, our condition for $\bh(\tilD_i,\bb_{i})$ becomes
[{_, mis(_i,_i;)|_i,_i,_i}|_i,_i]=
We then leverage Proposition <ref> (B) to arrive at the following two results:
{_, mis(_i,_i;)|_i,_i,_i}   =   {_, mis(_i,_i;)|_i,_i}

_mis{(_i,_i)|_i,_i,_i}   =
Combining these results, the efficient score must therefore satisfy
[{_, mis(_i,_i;)|_i,_i}|_i,_i]=[_mis{(_i,_i)|_i,_i}|_i,_i].
Together with Proposition <ref> (D), this implies that
{_, mis(_i,_i;)|_i,_i}=_mis{(_i,_i)|_i,_i},
which further implies that

=   _mis{(_i,_i)|_i,_i}={_, mis(_i,_i;)|_i,_i}
by Proposition <ref> (E). The efficient score vector can therefore be written as
_eff,mis(_i,_i;)   =
_, mis(_i,_i;)-_mis{(_i,_i)|_i,_i}

=   _, mis(_i,_i;)-{_, mis(_i,_i;)|_i,_i}.
Importantly, this is a closed form solution. We can simplify the first term in Equation (<ref>):
_, mis(_i,_i;)   =   _mis{_(_i,_i,_i;)|_i,_i},
where the “model” score vector $\bS_{\btheta}(\bY_i,\tilD_i,\bb_i;\btheta)=\frac{\partial}{\partial\btheta} \log f_{\bY_i|\tilD_i,\bb_i}(\bY_i|\tilD_i,\bb_i)$, which is not subject to misspecification. We can also simplify the second term in Equation (<ref>):
{_, mis(_i,_i;)|_i,_i}   =   [_mis{_(_i,_i,_i;)|_i,_i}|_i,_i]

=   ∫{∫_(_i,_i,_i;)f_mis(_i|_i,_i)dμ(_i)} ×

f(_i|_i,_i) dμ(_i)

=   ∫_(_i,_i,_i;)f_mis(_i,_i|_i,_i)dμ(_i)dμ(_i)

=   _mis{_(_i,_i,_i;)|_i,_i}.
This leads to the following
_eff, mis(_i,_i;)   =   _mis{_(_i,_i,_i;)|_i,_i}-

Mirroring the work above, it can be shown that this efficient score vector for censored subjects is equivalent to
_eff, mis^*(_i,_i;)   =   _mis{_^*(_i,_i,_i,U_i;)|_i,_i}-

§ CALCULATING THE EFFICIENT SCORE VECTORS

§.§ Uncensored Subjects

Recall from Section (<ref>) that, conditional on $(\tilD_i, \bb_i)$, the response vector corresponding to a single individual, $\bY_i$, follows a $\Normal_{m_i}(\bZ_i^a\bbeta  + \alpha(\bs_i - X_i\bone_{m_i}) + \bZ_i^b\bb_i, \sigma^2\bI_{m_i})$ distribution. Therefore, we know that the density for all observations from a single individual, $f_{\bY_i|\tilD_i,\bb_i}(\bY_i|\tilD_i,\bb_i;\btheta)=$:
(2πσ^2)^-m_i/2exp[-||_i-{_i^a+ α(_i - X_m_i) + _i^b_i}||^2_2/2σ^2]
Therefore, the log-likelihood $\log\{f_{\bY_i|\tilD_i,\bb_i}(\cdot)\}=$

-m_i/2log(2πσ^2)-||_i-{_i^a+ α(_i - X_i_m_i) + _i^b_i}||^2_2/2σ^2

§.§.§ Efficient score vector for $\bbeta$

From the log-likelihood in Equation (<ref>), we obtain
_(_i,_i,_i;)   ≡   ∂/∂log{f__i|_i,_i(·)}

=   _i^aT[_i-{_i^a+ α(_i - X_i_m_i) + _i^b_i}]/σ^2

=   _i^aT_i-_1(_i,_i)/σ^2,
where $\bk_1(\tilD_i,\bb_i)=\bZ_i^{aT}\{\bZ_i^a\bbeta  + \alpha(\bs_i - X_i\bone_{m_i}) + \bZ_i^b\bb_i\}$. By Proposition <ref> parts (B) and (E), we have:
This implies that the efficient score vector for $\bbeta$, $\bS_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)$, is equivalent to:
=   _mis{_(_i,_i,_i;)|_i,_i}-_mis{_(_i,_i,_i;)|_i,_i}

=   σ^-2[_mis{_i^aT_i-_1(_i,_i)|_i,_i}-

=   σ^-2{_mis(_i^aT_i|_i,_i)-_mis(_i^aT_i|_i,_i)}

=   _i^aT{_i-_mis(_i|_i,_i)}/σ^2.
Recall that the $_{\rm{mis}}$ subscript denotes expectations that are subject to possible misspecification. Here, because $\E_{\rm{mis}}(\bY_i|\bT_i,\tilD_i)$  depends only on the posited outcome model – which we assume to be correct – this expectation is not subject to misspecification; hence, we drop the $_{\rm{mis}}$ subscript as follows:
where, by Proposition <ref>, $\rm{E}(\bY_i|\bT_i,\tilD_i,\bb_i)=\bzeta(\tilD_i;\btheta)+\bP_{\bZ_i^b}\{\bY_i-\bzeta(\tilD_i;\btheta)\}$ where $\bP_{\bM} = \bM(\bM^T\bM)^{-1}\bM^T$ for a matrix $\bM$, $\bT_i = \sigma^{-2}\bZ_i^{bT}\bY_i$, and $\bzeta(\tilD_i;\btheta)=\bZ_i^a\bbeta  + \alpha(\bS_i - X_i\bone_{m_i})$. With this in mind, we can further simplify $\bS_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)$.
_eff,(_i,_i;)   =   _i^aT{_i-(_i;)-__i^b{_i-(_i;)}/σ^2

=   _i^aT(_m_i - __i^b){_i-(_i;)}/σ^2.

§.§.§ Efficient score for $\alpha$

The derivation of the efficient score for $\alpha$, which we denote by $\bS_{\rm{eff},\alpha}(\bY_i,\tilD_i;\btheta)$, is analogous to the derivation of $\bS_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)$.
S_eff,α(_i,_i;)   =   (_i-X_i_m_i)^T{_i-(_i|_i,_i)}/σ^2

=   (_i-X_i_m_i)^T(_m_i-__i^b){_i-(_i;)}/σ^2.

§.§.§ Efficient score for $\sigma^2$

From the log-likelihood in Equation (<ref>), we obtain
_σ^2(_i,_i,_i;)   ≡   ∂/∂σ^2log{ f__i|_i,_i(·)}

=   ||_i-{(_i;) + _i^b_i}||^2_2/2σ^4-m_i/2σ^2

=   _i^T_i/2σ^4-_i^T(_i;)/σ^4-_i^T_i^b_i/σ^4+||(_i;)+_i^b_i||^2_2/2σ^4-m_i/2σ^2

=   _i^T_i/2σ^4-_i^T(_i;)/σ^4-_i^T_i/σ^2+k_2(_i,_i)/2σ^2,
where $k_2(\tilD_i,\bb_i)=\sigma^{-2}||\bzeta(\tilD_i;\btheta_i)+\bZ_i^b\bb_i||^2_2-m_i$. By Proposition <ref> parts (B) and (E), we have:
Again by Proposition <ref> parts (B) and (E), $\E_{\rm{mis}}\{\bT_i^T\bb_i|\bY_i,\tilD_i\}=$

=   _i^T_mis(_i|_i,_i)=_mis(_i^T_i|_i,_i).
It then follows that the efficient score for $\sigma^2$, which we denote by $S_{\rm{eff},\sigma^2}(\bY_i,\tilD_i;\btheta)$, is:
S_eff,σ^2(_i,_i;)   =   _mis{_σ^2(_i,_i,_i;)|_i,_i}-

=   _mis{_i^T_i/2σ^4-_i^T(_i;)/σ^4|_i,_i}-

=   1/σ^4[1/2{_i^T_i-_mis(_i^T_i|_i,_i)}-

where we can again disregard the subscript $_{\rm{mis}}$ in these conditional expectations because the conditional distribution of $\bY_i$ given $(\bT_i,\tilD_i)$ is not subject to misspecification.

§.§.§ Efficient score vector for $\btheta$

Importantly, we have shown that each component of $\bS_{\rm{eff, mis}}(\bY_i,\tilD_i;\btheta)$ is not subject to mis-specification. To emphasize this point, we simply refer to the efficient score vector for $\btheta=(\bbeta, \alpha, \sigma^2)^T$ (corresponding to uncensored subjects) as $\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)$. Combining these results, we find that

σ^2_i^aT(_m_i - __i^b){_i-(_i;)}

σ^2(_i-X_i_m_i)^T(_m_i - __i^b){_i-(_i;)}

Recall from Proposition <ref> parts (A) and (C) that
E(_i|_i,_i,_i)   =   (_i;)+__i^b{_i-(_i;)}

E(_i^T_i|_i,_i)   =   σ^2(m_i-p_b)+||E(_i|_i,_i)||^2_2,
where $\bP_{\bM} = \bM(\bM^T\bM)^{-1}\bM^T$ for a matrix $\bM$, $\bT_i = \sigma^{-2}\bZ_i^{bT}\bY_i$, and $\bzeta(\tilD_i;\btheta)=\bZ_i^a\bbeta  + \alpha(\bS_i - X_i\bone_{m_i})$.

§.§ Censored Subjects

Recall from Section (<ref>) that, conditional on $(\widehat{\tilD}_i, \bb_i,U_i)$, the response vector $\bY_i$, from one censored subject, follows a multivariate normal distribution $\Normal_{m_i}\{\bZ^a_i\bbeta  + \alpha(\bS_i - \widehat{X}_i\bone_{m_i} - U_i\bone_{m_i}) + \bZ^b_i\bb_i, \sigma^2\bI_{m_i}\}$. Therefore, we know that the density for all observations from a single individual, $f_{\bY_i|\widehat{\tilD}_i,\bb_i,U_i}(\bY_i|\widehat{\tilD}_i,\bb_i,U_i;\btheta)=$
(2πσ^2)^-m_i/2exp[-||_i-{_i^a+ α(_i - X_i_m_i) + (_m_i,_i^b)(-αU_i, _i^T)^T}||^2_2/2σ^2].
Therefore, the log-likelihood $\log\{f_{\bY_i|\widehat{\tilD_i},\bb_i,U_i}(\bY_i|\widehat{\tilD_i},\bb_i,U_i;\btheta)\}=$

-m_i/2log(2πσ^2)-||_i-{_i^a+ α(_i - X_i_m_i) + (_m_i,_i^b)(-αU_i, _i^T)^T}||^2_2/2σ^2.

§.§.§ Efficient score vector for $\bbeta$

The derivation of the efficient score vector for $\bbeta$ corresponding to censored subjects mirrors that for uncensored subjects (Appendix <ref>). From this, we find that:
=   _i^aT(_m_i - _(_m_i,^b_i)){_i-(_i;)}/σ^2.

§.§.§ Efficient score for $\alpha$

Next, we consider the efficient score for $\alpha$ corresponding to censored subjects.
From the log-likelihood in Equation (<ref>), we find that:
_α(_i,_i,_i,U_i;)   ≡   ∂/∂αlog{ f__i|_i,_i,U_i(_i|_i,_i,U_i)}

=   {_i-(X_i+U_i)_m_i}^T[_i-{(_i;) + (_m_i,_i^b)(-αU_i, _i^T)^T}]/σ^2

=   {_i-(X_i+U_i)_m_i}^T_i-_3(_i,_i,U_i)/σ^2,
where $\bk_3(\widehat{\tilD}_i,\bb_i,U_i)=\{\bS_i-(\widehat{X}_i+U_i)\bone_{m_i}\}^T\{\bzeta(\widehat{\tilD}_i;\btheta) + (\bone_{m_i},\bZ_i^b)(-\alpha U_i, \bb_i^T)^T\}$. Then, by again leveraging Proposition <ref> parts (B) and (E), we have:
Next, we consider the term $\sigma^{-2}U_i\bone_{m_i}^T\bY_i$. Since we defined $\bT_i^{\rm aug}= \sigma^{-2}(\bone_{m_i},\bZ^b_i)^T\bY_i$, it follows that $\sigma^{-2}U_i\bone_{m_i}^T\bY_i=U_iT_{i1}^{\rm aug}$, where $T^{\rm aug}_{i1}$ denotes the first element of $\bT_i^{\rm aug}$. Then, it follows from Proposition <ref> parts (B) and (E) that $E(U_iT_{i1}^{\rm aug}|\bY_i,\widehat{\tilD}_i)=$

=   T_i1^augE(U_i|_i^aug,_i)=E(U_iT_i1^aug|_i^aug,_i).
Therefore, the conditional expectations of $\bk_3(\widehat{\tilD}_i,\bb_i,U_i)$ and $U_iT_{i1}^{\rm aug}$ drop out when calculating the efficient score for $\alpha$. As a result, we find that

§.§.§ Efficient score for $\sigma^2$

The derivation of the efficient score for $\sigma^2$ corresponding to censored subjects mirrors the derivation shown in Appendix (<ref>). Hence, we find that:
=   1/σ^4[1/2{_i^T_i-(_i^T_i|_i^aug,_i)}-

§.§.§ Efficient score vector for $\btheta$

Combining these results, we find that

=   [

σ^2_i^aT(_m_i - _(_m_i,^b_i)){_i-(_i;)}

σ^2(_i-X_i_m_i)^T(_m_i - _(_m_i,^b_i)){_i-(_i;)}

Recall from Proposition <ref> parts (A) and (C) that
E(_i|_i^aug,_i,_i,U_i)   =   (_i;)+_(_m_i,_i^b){_i-(_i;)}

E(_i^T_i|_i^aug,_i)   =   σ^2(m_i-p_b)+||E(_i|_i^aug,_i)||^2_2,
where $\bP_{\bM} = \bM(\bM^T\bM)^{-1}\bM^T$ for a matrix $\bM$, $\bT_i^{\rm aug} = \sigma^{-2}(\bone_{m_i}, \bZ^b)^T\bY_i$, and $\bzeta(\widehat{\tilD}_i;\btheta)=\bZ_i^a\bbeta  + \alpha(\bS_i - \widehat{X}_i\bone_{m_i})$.

§.§ The Full Estimating Equation

We have now derived the efficient score vectors for $\btheta$ corresponding to both uncensored and censored subjects. Combining these vectors gives us the total estimating equation:

∑_i=1^n {Δ_i_eff(_i,_i;) + (1 - Δ_i)_eff^*(_i,_i;)}=,
where the censoring indicator $\Delta_i$ is $1$ if subject $i$ is not censored and $0$ otherwise. The efficient score vectors $\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)$ and $\bS_{\rm{eff}}^*(\bY_i,\widehat{\tilD}_i;\btheta)$ are given in Equation (<ref>) and Equation (<ref>), respectively.

§ PROOF OF THEOREM <REF>

We now prove that $\ACEest$ is identifiable, i.e., that Theorem <ref> in the main text is true.
We begin by considering the first $p + 1$ elements of this estimating equation (i.e., the elements corresponding to $(\bbeta, \alpha)$) in Equation (<ref>):
=      ∑_i=1^n[Δ_i σ^2(^a_i,_i-X_i_m_i)^T(_m - _^b_i){_i-(^a_i,_i-X_i_m_i)(^T, α)^T}+

(1 - Δ_i)σ^2(^a_i,_i-X_i_m_i)^T(_m - _(_m_i,^b_i)){_i-(^a_i,_i-X_i_m_i)(^T, α)^T}].
Moving all terms without $(\bbeta, \alpha)$ to the left-hand side, we find:
∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _(_m_i,^b_i))}_i

=   ∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)(^a_i,_i-X_i_m_i)+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _(_m_i,^b_i))(^a_i,_i-X_i_m_i)}(^T, α)^T.
From this, we obtain the following regression coefficient estimates:
(^T,α̂)^T   =   [∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)(^a_i,_i-X_i_m_i)+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _(_m_i,^b_i))(^a_i,_i-X_i_m_i)}]^-1×

[∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)_i+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _^b_i)_i}]

=   _i^-1[∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)_i+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _^b_i)_i}],
assuming that the matrix
_i   =   ∑_i=1^n{Δ_i (^a_i,_i-X_i_m_i)^T(_m - _^b_i)(^a_i,_i-X_i_m_i)+

(1 - Δ_i)(^a_i,_i-X_i_m_i)^T(_m - _(_m_i,^b_i))(^a_i,_i-X_i_m_i)}
is invertible.
Next, we consider the $(p+2)$th element of the estimating equation (i.e., the element corresponding to $\sigma^2$):
0   =   ∑_i=1^n(Δ_i[1/2{_i^T_i-(_i^T_i|_i,_i)}-^T(_i;){_i-(_i|_i,_i)}]+

=   ∑_i=1^n(Δ_i[1/2{_i^T_i-σ^2(m_i-q)+||E(_i|_i,_i)||^2_2}

From this, we see that the estimate of $\sigma^2$ is given by:
σ̂^̂2̂   =   [∑_i=1^n{Δ_i(m_i-q)+(1-Δ_i)(m_i-q)}]^-1 ×

With this, we have established the identifiability of $\btheta$.

§ CONSISTENCY

We now turn our attention to the consistency of our novel estimator, $\ACEest$ (Theorem <ref>). This theorem requires that we first prove that our full estimating function has mean $\bzero$ (Proposition <ref>). Yet we begin by proving Lemma <ref>, which establishes an important relationship between assumptions (C1) and (C2) outlined in Proposition <ref> and the narrower assumptions required to prove that the full estimating function has mean $\bzero$.

§.§ Relationship between conditional independence claims

Consider the transformations $\bT_i=\sigma^2\bZ_i^{bT}\bY_i$ and $\bT_i^{\rm aug}=\sigma^{-2}(\bone_{m_i},\bZ^b_i)^T\bY_i$.

(A) If $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$, then it must also be true that $\bY_i$ is conditionally independent of $C_i$ given $(\bT_i, \tilD_i)$.

(B) If $\bY_i$ is conditionally independent of $C_i$ given $\widehat{\tilD}_i$, then it must also be true that $\bY_i$ is conditionally independent of $C_i$ given $(\bT_i^{\rm aug}, \widehat{\tilD}_i)$.

We prove Lemma <ref> (A) and omit the proof of Lemma <ref> (B), since the proof of the latter is incredibly similar to that of the former (with $(\tilD_i, \bT_i)$ replaced by $(\widehat{\tilD}_i, \bT_i^{\rm aug})$). We first show that the premise in Lemma <ref> (A) is equivalent to:
f__i| _i, C_i(_i|_i,C_i) = f__i| _i(_i|_i).
From this and Bayes theorem, we observe the following:
f__i, _i| C_i(_i,_i|C_i)/f__i| C_i(_i|C_i) = f__i, _i(_i,_i)/f_ _i(_i).
Then, since $\bT_i$ is a function of $\bY_i$ and $\tilD_i$, it follows that

f__i, _i, _i| C_i(_i,_i, _i|C_i)/f__i| C_i(_i|C_i) = f__i, _i, _i(_i,_i,_i)/f_ _i(_i).
We now apply Bayes theorem to modify the denominator on the left hand side of Equation (<ref>):
f__i,_i|C_i(_i,_i|C_i)/f_ _i|C_i(_i|C_i)=f__i|_i,C_i(_i|_i,C_i)

⇒    f_ _i|C_i(_i|C_i)=f__i,_i|C_i(_i,_i|C_i)/f__i|_i,C_i(_i|_i,C_i).
Then, because $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$ per our assumption in Lemma <ref> (A), we know that $\bT_i$ is also conditionally independent of $C_i$ given $\tilD_i$ since $\bT_i$ is a function of $\bY_i$ and $\tilD_i$. Therefore, $f_{\bT_i|\tilD_i,C_i}(\bT_i|\tilD_i,C_i)=f_{\bT_i|\tilD_i}(\bT_i|\tilD_i)$ and
f_ _i|C_i(_i|C_i)=f__i,_i|C_i(_i,_i|C_i)/f__i|_i(_i|_i).
We now apply Bayes theorem to modify the denominator on the right hand side of Equation (<ref>):
f__i,_i(_i,_i)/f_ _i(_i)=f__i|_i(_i|_i)

⇒   f_ _i(_i)=f__i,_i(_i,_i)/f__i|_i(_i|_i).
With these two modifications to the denominators of Equation (<ref>), we obtain:
f__i, _i, _i| C_i(_i,_i, _i|C_i)f__i|_i(_i|_i)/f__i,_i|C_i(_i,_i|C_i) = f__i, _i, _i(_i,_i,_i)f__i|_i(_i|_i)/f__i,_i(_i,_i).
Then, we see that the conditional density $f_{\bT_i|\tilD_i}(\bT_i|\tilD_i)$ cancels from both sides to yield:
f__i, _i, _i| C_i(_i,_i, _i|C_i)/f__i,_i|C_i(_i,_i|C_i) = f__i, _i, _i(_i,_i,_i)/f__i,_i(_i,_i),
which is equivalent to the statement that $f_{\bY_i|\tilD_i,\bT_i,C_i}(\bY_i|\tilD_i,\bT_i,C_i)=f_{\bY_i|\tilD_i,\bT_i}(\bY_i,\tilD_i,\bT_i)$ and hence that $\bY_i$ is conditionally independent of $C_i$ given $(\tilD_i, \bT_i)$. This confirms Lemma <ref> (A); the proof of Lemma <ref> (B) proceeds analogously and so is omitted for brevity.

§.§ Proof of Proposition <ref>

We prove Proposition <ref> (A) and omit the proof of Proposition <ref> (B), since the proof of the latter follows analogously from that of the former (with $(\tilD_i, \bT_i)$ replaced by $(\widehat{\tilD}_i, \bT_i^{\rm aug})$).

To prove Proposition <ref> (A), we begin by applying the law of total expectation to find:
{Δ_i_eff(_i,_i;)}   =   [{Δ_i_eff(_i,_i;)|_i,_i,C_i}]

=   {I(X_i ≤C_i)_eff(_i,_i;)|_i,_i,C_i}

=   I(X_i ≤C_i){_eff(_i,_i;)|_i,_i,C_i}
since, by definition, $\tilD_i=(\bZ^a_i, \bs_i, X_i, \bZ^b_i)$ includes $X_i$. Hence, to show the desired result that $\E\{\Delta_i\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)\} = \bzero$, it suffices to show that $\E\{\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta)|\bT_i,\tilD_i,C_i\}=\bzero$. We accomplish this in a component-wise fashion. Recall from Section <ref> that:
{_eff,(_i,_i)|_i,_i,C_i}   =   [^aT_i{_i-(_i|_i,_i)}|_i,_i,C_i]/σ^2

=   ^aT_i{(_i|_i,_i,C_i)-(_i|_i,_i)}/σ^2.
Per Lemma <ref>, our assumption that $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$ implies the conditional independence of $\bY_i$ and $C_i$ given $(\bT_i,\tilD_i)$. It therefore follows that $E(\bY_i|\bT_i,\tilD_i,C_i)=E(\bY_i|\bT_i,\tilD_i)$ and hence that $E\{\bS_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)|\bT_i,\tilD_i,C_i\}=\bzero$.

The proof corresponding to $S_{\rm{eff},\alpha}(\bY_i,\tilD_i;\btheta)$ proceeds similarly. From Section <ref>, we know that for uncensored subjects, the efficient score vector corresponding to $\alpha$ is:
S_eff,α(_i,_i;)   =   (_i-X_i_m_i)^T{_i-(_i|_i,_i)}/σ^2.
As a result, we observe that:
{_eff,α(_i,_i)|_i,_i,C_i}   =   (_i-X_i_m_i)^T{(_i|_i,_i,C_i)-(_i|_i,_i)}/σ^2.
Again, we know from our assumption — that $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$ — and Lemma <ref> that $\E\{\bS_{\rm{eff,\alpha}}(\bY_i,\tilD_i)|\bT_i,\tilD_i,C_i\}=0$.

Next, recall from Section <ref> that:
From this, we know that
σ^4{_eff,σ^2(_i,_i)|_i,_i,C_i}   =   1/2{(_i^T_i|_i,_i,C_i)-(_i^T_i|_i,_i)}-

which, similarly, is $0$ because we assume $\bY_i$ is conditionally independent of $C_i$ given $\tilD_i$. This establishes that each component of $\bS_{\rm eff}(\bY_i,\tilD_i)$ has mean $\bzero$, confirming Proposition <ref> (A). As noted, the proof of  Proposition <ref> (B) proceeds in parallel (with $(\tilD_i, \bT_i)$ replaced by $(\widehat{\tilD}_i, \bT_i^{\rm aug})$).

§ PROOF OF THEOREM <REF>

Again, we use $\bS_{\rm{eff}}^{\rm{full}}(Y_i,\tilD_i,\widehat{\tilD}_i;\btheta) = \Delta_i\bS_{\rm{eff}}(\bY_i,\tilD_i;\btheta) + (1 - \Delta_i)\bS_{\rm{eff}}^*(\bY_i,\widehat{\tilD}_i;\btheta)$ to denote the full estimating function. Since $\E\left\{\bS_{\rm{eff}}^{\rm{full}}(Y_i,\tilD_i,\widehat{\tilD}_i;\btheta)\right\}=\bzero$ under conditions (C1) and (C2) in Proposition <ref>, we can use Taylor's theorem to expand around $\btheta_0$ as follows:
=    ∑_i=1^n {_eff^full(Y_i,_i,_i;)}

=    n^-1/2∑_i=1^n {_eff^full(Y_i,_i,_i;_0)} +

n^-1∑_i=1^n {∂/∂^T _eff^full(Y_i,_i,_i;^*)}n^1/2(-_0),
where $\btheta^*$ lies on the line connecting $\ACEest$ and $\btheta_0$. Then, by C6,
=    n^-1/2∑_i=1^n {_eff^full(Y_i,_i,_i;_0)} +

{∂/∂^T _eff^full(Y_i,_i,_i;_0)}n^1/2(-_0) + o_p(1).
n^1/2(-_0)   =   [{∂/∂^T _eff^full(Y_i,_i,_i;_0)}]^-1×

n^-1/2∑_i=1^n _eff^full(Y_i,_i,_i;_0)  + o_p(1)
by C7. Hence, it follows directly from the central limit theorem that $n^{1/2}(\ACEest-\btheta_0)$ converges in distribution to $\Normal_{p_a+2}(0, {\cal V})$, where ${\cal V} = {\cal A}^{-1} {\cal B} ({\cal A}^{-1})^T$ with
A^-1   =    [{∂/∂^T _eff^full(Y_i,_i,_i;_0)}]^-1

B   =     {_eff^full(Y_i,_i,_i;_0)}.

§ LIMITATION

We now demonstrate a limitation with the proposed estimating equation:

Let $C(\bM)$ denote the column space of a matrix $\bM$ and let $(\bZ^a_i)_k$ denote the $k$th column of $\bZ^a_i$.

A If $(\bZ^a_i)_k$ belongs to $C(\bZ^b_i)$, then the $k$th element of $S_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)$ will be $0$. Similarly, if $(\bs_i-X_i\bone_{m_i})$ belongs to $C(\bZ^b_i)$, then $S_{\rm{eff},\alpha}(\bY_i,\tilD_i;\btheta)$ will be $0$.

B If $(\bZ^a_i)_k$ belongs to $C\{(\bone_{m_i},\bZ_i^b)\}$, then the $k$th element of $S_{\rm{eff},\bbeta}^*(\bY_i,\widehat{\tilD}_i;\btheta)$ will be $0$. Similarly, if $(\bs_i-\widehat{X}_i\bone_{m_i})$ belongs to $C\{(\bone_{m_i},\bZ_i^b)\}$, then $S_{\rm{eff},\alpha}^*(\bY_i,\widehat{\tilD}_i;\btheta)$ will be $0$.

Lemma <ref> follows quickly from the definitions of the efficient score vectors given in Section <ref>. To prove this, first suppose that $(\bZ^a_i)_k$ belongs to the column space of $\bZ_i^b$. Then, because $\bP_{\bZ_i^b}$ is the orthogonal projection operator onto the column space of $\bZ_i^b$, we know that $\bI_{m_i}(\bZ^a_i)_k=(\bZ^a_i)_k =\bP_{\bZ^b_i}(\bZ^a_i)_k$. This implies that $(\bI_{m_i} - \bP_{\bZ^b_i})(\bZ^a_i)_k = \bzero$ and, equivalently, that $(\bZ^a_i)_k^T(\bI_{m_i} - \bP_{\bZ^b_i}) = \bzero^T$. As a result, $\bZ^{aT}_i(\bI_{m_i} - \bP_{\bZ^b_i})$ has $k$th row equal to $\bzero$ and hence, the $k$th element of $S_{\rm{eff},\bbeta}(\bY_i,\tilD_i;\btheta)$ will be 0. Proposition <ref> — an immediate consequence of Lemma <ref> — describes the broader situation in which this poses a computational challenge.

Let $k$ be a fixed integer between $1$ and $p_a$. Suppose that $(\bZ^a_i)_k$ belongs to $C(\bZ^b_i)$ for all $i$ such that $\Delta_i=1$. Further suppose that $(\bZ^a_i)_k$ belongs to $C\{(\bone_{m_i},\bZ_i^b)\}$ for all $i$ such that $\Delta_i=0$. Then, the $k$th element of the total estimating equation given in Equation (<ref>) will be $0$.

In less generality, if $\bZ^a_i$ and $\bZ^b_i$ “share” any columns for all $i$, the estimating equation may fail to converge. This can happen if, for example, we include both a fixed and a random intercept in the outcome model of interest.

§ SUPPLEMENTAL TABLES

Censoring     Parameter     Method     Bias     SEE     ESE     MSE     CPr

Light     $\alpha$     ACE     -0.000     0.022     0.022     0.000     0.950

MCMI     0.173     0.023     0.127     0.046     0.070

$\beta$     ACE     0.000     0.024     0.024     0.001     0.942

MCMI     0.002     0.036     0.038     0.001     0.951

$\sigma^2$     ACE     -0.003     0.034     0.034     0.001     0.940

MCMI     1.940          2.979     12.629

Simulation results under light censoring when the imputation model is correctly specified. ACE: proposed estimating equation applied to the imputed dataset. MCMI: multiple conditional mean imputation. We present empirical bias, the average standard error estimates (SEE), the empirical standard error (ESE), the average mean squared errors (MSE), and the observed coverage probability (CPr) of the Wald-type confidence intervals with nominal 95% coverage.

Censoring     Parameter     Method     Bias     SEE     ESE     MSE     Coverage

Light     $\alpha$     ACE     0.000     0.022     0.022     0.000     0.946

MCMI     1.872     0.175     10.429     112.154     0.000

$\beta$     ACE     0.001     0.024     0.024     0.001     0.948

MCMI     -0.071     0.316     1.289     1.665     0.953

$\sigma^2$     ACE     -0.002     0.034     0.035     0.001     0.944

MCMI     $\geq 100$          $\geq 100$     $\geq 100$

Simulation results under light censoring when the imputation model is mis-specified. ACE: proposed estimating equation applied to the imputed dataset. MCMI: multiple conditional mean imputation. We present empirical bias, the average standard error estimates (SEE), the empirical standard error (ESE), the average mean squared errors (MSE), and the observed coverage probability (CPr) of the Wald-type confidence intervals with nominal 95% coverage.

Censored ($\Delta_i$ = 0)     Uncensored ($\Delta_i$ = 1)

n     858     244

Age     38.82 (10.43)     43.23 (10.26)

Sex = Male (%)     311 (36.2)     83 (34.0)

Education     14.64 (2.64)     14.11 (2.55)

CAG     42.22 (2.62)     43.48 (2.81)

CAG-Age Product     304.68 (78.74)     389.54 (72.88)

cUHDRS     16.88 (1.78)     15.35 (1.94)

Total Motor Score     3.79 (4.27)     8.36 (6.71)

SDMT Score     52.47 (11.14)     44.57 (10.56)

Total Functional Capacity     12.83 (0.75)     12.70 (0.80)

Stroop Color Test Score     79.20 (13.69)     70.95 (13.40)

Stroop Word Test Score     101.01 (17.28)     91.62 (16.38)

Stroop Interference Test Score     46.45 (10.25)     39.55 (9.16)

$\widehat{X}_i$     11.26 (1.35)     4.50 (2.78)

Summary of model covariates ($\bZ^a_{ij}$) and potential outcomes ($\bY_i$) at visit 1. CAG: number of cytosine-adenine-guanine repeats. cUHDRS: composite Unified Huntington Disease Rating Scale. SDMT: Symbol Digits Modality Test.  $\widehat{X}_i=\Delta_iX_i+(1-\Delta_i){\rm E}(X_i|X_i>C_i,\bV_i)$. Continuous variables summarized by mean (standard deviation). Binary variables summarized by count (percent).

Power to detect 10%, 15%, and 20% treatment effects for given per-group sample sizes with type 1 error rate $0.05$. Results are based on estimated slope in placebo group from complete case analysis (CCA), multiple conditional mean imputation (MCMI), and ACE imputation (ACE).