#  Sensitivity analysis for transportability in multi-study,
multi-outcome settings

Ngoc Q. Duong, Amy J. Pitts, Soohyun Kim, Caleb H. Miles

###### Abstract

Existing work in data fusion has covered identification of causal estimands
when integrating data from heterogeneous sources. These results typically
require additional assumptions to make valid estimation and inference.
However, there is little literature on transporting and generalizing causal
effects in multiple-outcome setting, where the primary outcome is
systematically missing on the study level but for which other outcome
variables may serve as proxies. We review an identification result developed
in ongoing work that utilizes information from these proxies to obtain more
efficient estimators and the corresponding key identification assumption. We
then introduce methods for assessing the sensitivity of this approach to the
identification assumption.

Department of Biostatistics, Mailman School of Public Health, Columbia
University

Keywords: Causal inference, Data fusion, External validity, Generalizability,
Missing data, Proxy variable

##  1 Introduction

Research in clinical medicine and public health is often concerned with
estimating the effect of some treatment in a specific target population.
However, even in a randomized clinical trial, which is considered the gold-
standard study design, ensuring external validity remains a challenge. This
can be due to a variety of reasons, including non-random sampling, overly
stringent exclusion criteria, or an ill-defined target population of interest
(Tan et al.,, 2022; Kennedy-Martin et al.,, 2015). Meta-analysis of summary
statistics is a commonly used tool to synthesize and generalize findings from
published study-level summary statistics, but tends to rely on strong, often
implausible assumptions. An alternative approach that allows for more control
over the nuances and heterogeneity across studies is to combine individual-
level data, when available, from multiple studies, each of which may contain
insufficient information to address a given scientific question by itself, but
which collectively have the power to do so. There has been a growing body of
work on generalizability and transportability methods, which can help address
the problem of external validity of the effect estimates from integrating
individual level data across studies.

Generalizability concerns the setting where the study population is a subset
of the target population of interest while transportability addresses the
setting where the study population is partially or completely external to the
target population (Degtiar and Rose,, 2023). Specifically, generalizability
typically involves extending the causal effect estimate derived from a study
as long as the covariates in the study population and the target population
have common support (Gechter,, 2015; Tipton,, 2014). On the other hand,
transportability entails extrapolating the effect estimated from a study in
which some primary outcome of interest is observed to a population represented
by a sample in which the outcome is not measured.

Existing methodologies involve directly transporting some estimated causal
effect, e.g., the average treatmemt effect (ATE), from studies where the
outcomes are observed to other studies with missing outcomes or across
heterogeneous study designs and settings (Bareinboim and Pearl,, 2016; Dong et
al.,, 2020; Pearl and Bareinboim,, 2014; Hünermund and Bareinboim,, 2019), or
to some broader target population (Dahabreh et al., 2020a, ; Dahabreh et al.,
2020b, ; Lesko et al.,, 2017; Westreich et al.,, 2017). When considering
multiple studies, it is often the case that one will observe different
outcomes at follow up. However, existing methods do not take advantage of
these other potentially correlated and informative outcome variables measured
at follow-up, which could potentially be leveraged to achieve large efficiency
gains. Existing outcome proxy-blind methods typically rely on an assumption of
homogeneous conditional potential outcome means for valid transportation of
estimation from one population to another. Sensitivity analysis strategies
have been proposed to study the extent to which the violation of these
assumptions will affect the estimations and inferences drawn (Nguyen et al.,,
2017; Dahabreh and Hernán,, 2019; Dahabreh et al.,, 2022).

In ongoing work, we have developed a new strategy to more efficiently estimate
the ATE from integrated data across multi-outcome studies, with inconsistent
availability of the primary outcome of interest at the study level. The
proposed methodology takes advantage of the availability of follow-up
measurements of potential correlates of the main outcome to yield more precise
estimate of the causal effects. In this article, we consider the key common
outcome regression (or conditional exchangeability for study selection)
assumption for transportability while leveraging these outcome proxies, which
differs slightly from the common outcome regression assumption that has been
traditionally used for transportability. We discuss the resulting bias when
this assumption is not met, and develop methodology for sensitivity analysis
to the violation of this assumption.

The remainder of the article is organized as follows. In Section 2, we discuss
identification of the average treatment effect in the multi-study, multi-
outcome setting. In Section 3, we discuss the bias incurred by violations of
the key conditional exchangeability assumption. In Section 4, we compare the
conditional exchangeability assumption in our setting with that used in
settings that do not leverage outcome proxies. In Section 5, we develop
methods for sensitivity analysis for when our assumption is violated. We
demonstrate the empirical performance of our proposed methods in a simulation
study in Section 6, and conclude with a discussion in Section 7.

##  2 Data integration for studies with primary outcome missing systematically

###  2.1 Study and data setting

In this setting, we let $A$ be the treatment indicator, $W$ be a set of
covariates that are commonly observed across studies, $Y$ be the primary
outcome variable, the set $\\{T_{1},\dots,T_{k}\\}$ be all the potential
outcome proxies measured at follow-up in any study, and $J_{s}$ be the study-
specific subset of $\\{T_{1},\dots,T_{k}\\}$ that is measured in study $s$.
Suppose there are $\mathcal{S}$ studies that are ordered such that for each
$s$ in the first $s^{*}$ studies, we observe the set of variables
$(Y,A,J_{s},W)$, while for each $s$ in the remaining $\mathcal{S}-s^{*}$
studies, only the subset $(A,J_{s},W)$ are observed. In other words, $Y$ is
systematically missing in the latter set of studies. Unlike the standard setup
in other works concerning effect transportability that only involves
$(Y,A,W)$, we introduced the use of $\mathcal{T}_{s}$, where
$\mathcal{T}_{s}\subset J_{s}$ is some user-specified subset of $J_{s}$ for
each study $s$. $\mathcal{T}_{s}$ could be chosen based on availability and
subject matter knowledge and must be chosen such that they are observed in at
least one of the studies $\\{1,2,\dots,s^{*}\\}$.

Studies can be randomized experiments or observational; however, we will not
consider scenarios in which some studies are randomized experiments and others
are observational in this work. Then the study-specific average treatment
effect and conditional average treatment effect can be written as:

$\displaystyle ATE(s)$ $\displaystyle=E(Y_{1}-Y_{0}\mid S=s)$ $\displaystyle
CATE(w,s)$ $\displaystyle=E(Y_{1}-Y_{0}\mid W=w,S=s).$

Accordingly, we can define the overall average treatment effect and
conditional average treatment effect as:

$\displaystyle ATE$ $\displaystyle=\sum^{\mathcal{S}}_{s=1}ATE(s)$
$\displaystyle CATE(w)$ $\displaystyle=E(Y_{1}-Y_{0}\mid W=w)$

where the weights can be user-specified such that $\sum_{s}\pi_{s}=1$. For
instance, one can choose $\pi_{s}=P(S=s)$, or the marginal probability of
being in each study. Alternatively, we could define $ATE=E_{Q_{W,S}}CATE(W,S)$
for a user-specified, known distribution $Q_{W,S}$ of $W$ and $S$.

Since $Y$ is not measured in $s\in\\{s^{*}+1,\dots,{\mathcal{S}}\\}$, we
cannot directly estimate the ATE and CATE using data from these studies alone.
Our purpose is to transport the ATE from the first $s^{*}$ studies where $Y$
is observed, to the remaining ${\mathcal{S}}-s^{*}$ studies while also
leveraging the information from the outcome proxy set $\mathcal{T}_{s}$ to
improve efficiency. For ease of notation, let $\sigma_{s}$ be a subset of the
first $s^{*}$ studies in which both Y and $\mathcal{T}_{s}$ are observed. We
can then use this information from the studies that form $\sigma_{s}$ to
estimate the outcome regression that will allow us to transport the causal
effects to study $s$. In this setting, we have shown in ongoing, not-yet-
published work that the ATE can be nonparametrically identified as:

$\displaystyle\Psi^{ATE}$ $\displaystyle=\sum^{s^{*}}_{s=1}\pi_{s}E\\{E(Y\mid
W,A=1,S=s)-E(Y\mid W,A=0,S=s)\mid S=s\\}$
$\displaystyle+\sum^{{\mathcal{S}}}_{s=s^{*}+1}\pi_{s}E[E\\{E(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s})\mid
W,A=1,S=s\\}$ (1)
$\displaystyle\quad\quad\quad\quad\quad-E\\{E(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s})\mid
W,A=0,S=s\\}\mid S=s].$

The terms in the first sum are simply the standard identification formula for
the (study-specific) average treatment effects when $Y$ is observed. The
second sum is identified since it only depends on the distribution of $Y$ in
the studies in $\sigma_{s}$, i.e., in which $Y$ is actually observed.

Here, we introduced a modification to how transportability has traditionally
been done by incorporating information from a set of outcomes measured at
follow-up that are correlated with the main outcome of interest.

###  2.2 Assumptions for Identification of the ATE

This derivation ATE can be nonparametrically identified given the assumptions
that are standard for identification for ATE when outcomes are all observed:

######  Assumption 1 (Positivity).

$P(A=1\mid W=w)>0$ for all $w$ with positive probability.

######  Assumption 2 (Consistency).

$Y=AY_{1}+(1-A)Y_{0}$.

######  Assumption 3 (Within-study conditional exchangeability).

$E[Y^{a}\mid W,A,S=s]=E[Y^{a}\mid W,S=s]\textrm{ for all }s.$

The validity of our estimator relies on a fourth assumption that allows for
the transportation of the effect across studies:

######  Assumption 4 (Common outcome regression (proxy-aware version)).

$E(Y\mid\mathcal{T}_{s},W,A=a,S=s)=E(Y\mid\mathcal{T}_{s},W,A=a,S\in\sigma_{s})\textrm{
for all }s.$

This is a missing at random (MAR)-type assumption, where $S$ can in a sense be
thought of as a missingness indicator, since missingness is systematic by
study.

We can also introduce a fifth assumption that is not necessary for
identification, but allows for more borrowing of information across studies,
which can help with efficiency:

######  Assumption 5 (Common distribution of outcome proxies).

$\mathcal{T}_{s}\perp S\mid W,A$ for all $s$.

This implies the distribution of $\mathcal{T}_{s}$ conditional on treatment
assignment and baseline covariates is the same across studies. Under this
additional assumption, the identification result simplifies to:

$\displaystyle ATE$
$\displaystyle=\sum^{\mathcal{S}}_{s=1}\pi_{s}E[E\\{E(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s})\mid
W,A=1\\}$
$\displaystyle\quad-E\\{E(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s})\mid
W,A=0\\}\mid S=s].$

In ongoing work, we have developed a simple substitution estimator that
involves replacing each expectation with a regression-based estimate and the
outer expectation with an empirical mean.

For the outcome proxy-blind approach, in addition to the first three standard
internal validity assumptions, Assumption 4 is replaced by a slightly
different mean outcome exchangeability assumption: across studies assumption
(exchangeability over $S$) (Dahabreh and Hernán,, 2019; Lesko et al.,, 2017):

######  Assumption 6 (Common outcome regression (proxy-blind version)).

$\displaystyle E(Y\mid W,A=a,S=s)=E(Y\mid W,A=a,S\in\sigma_{s})\textrm{ for
all }s.$

Assumption 4 differs from Assumption 6 by additionally conditioning on
$\mathcal{T}_{s}$ for each study $s$. Assumptions 4 and 5 together imply
Assumption 6. In this article, we will only consider sensitivity analysis for
the violation of Assumption 4. When Assumption 5 is violated, the ATE
estimator based on Assumption 4 (i.e., the substitution estimator based on the
identification formula (2.1)) will remain consistent.

##  3 Characterizing the bias resulting from violation of the identification
assumption

The validity of $\Psi^{ATE}$ is dependent on the key assumption 4. This
assumption requires no heterogeneity in the conditional outcome means given
treatment, covariates, and outcomes proxies between studies with and without
missing outcome ($Y$) data. This allows for transportation of the conditional
outcome means, and correspondingly, the ATE and CATE, estimable from one study
to others.

In practice, this could be a strong assumption to make while also untestable
using observed data. For instance, in previous unpublished work, we estimated
the average treatment effect of cognitive remediation (CR) therapy on Social
Behavioral Scale (SBS) score, a measure for social functioning, using
harmonized data from three trials in the NIMH Database of Cognitive Training
and Remediation Studies (DoCTRS) database. However, the degree of
effectiveness of CR, especially on functional and occupational outcomes, was
less evident and has been suggested to vary depending on the setting in which
the treatment was administered (Barlati et al.,, 2013; Combs et al.,, 2008;
McGurk et al.,, 2007; Wykes et al.,, 2007, 2011). When this assumption is
violated, the substitution estimators described in the previous section will
be biased. Therefore, we examine two strategies for sensitivity analysis in
order to examine the robustness of estimates under varying degrees of
assumption violation.

To quantify the degree of violation, let the bias functions be defined as:

$\displaystyle u(A=1,\mathcal{T}_{s},W)$
$\displaystyle=E(Y\mid\mathcal{T}_{s},W,A=1,S=s)-E(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s}),$
$\displaystyle u(A=0,\mathcal{T}_{s},W)$
$\displaystyle=E(Y\mid\mathcal{T}_{s},W,A=0,S=s)-E(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s})$
(2)

Then, equation (2.1) when assumption 4 is violated instead becomes:

$\displaystyle\mathrm{ATE}$
$\displaystyle=\sum_{s=1}^{s^{*}}\pi_{s}E\\{E(Y\mid W,A=1,S=s)-E(Y\mid
W,A=0,S=s)\mid S=s)$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{\mathcal{S}}\pi_{s}E\left[E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s}\right)\mid
W,A=1,S=s\right)\right\\}$
$\displaystyle\quad\quad\quad\quad\left.\left.-E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s}\right)\mid
W,A=0,S=s\right)\right\\}\mid S=s\right]$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{\mathcal{S}}\pi_{s}E[E\left\\{u\left(A=1,\mathcal{T}_{s},W\right)\mid
W,A=1,S=s\right\\}$
$\displaystyle\quad\quad\quad\quad-E\left\\{u\left(A=0,\mathcal{T}_{S},W\right)\mid
W,A=0,S=s\right\\}\mid S=s],$

where the last sum is not identified. Then, the study-specific bias for study
$s$ is:

$\displaystyle E\left[E\left\\{u\left(A=1,\mathcal{T}_{s},W\right)\mid
W,A=1,S=s\right\\}-E\left\\{u\left(A=0,\mathcal{T}_{s},W\right)\mid
W,A=0,S=s\right\\}\mid S=s\right]$ $\displaystyle=$
$\displaystyle\;E[\delta^{*}(W)|S=s].$ (3)

By rearranging terms, $\delta^{*}(W)$ can be alternatively written as:

$\displaystyle
E\left[E\left(Y\mid\mathcal{T}_{s},W,A=1,S=s\right)-E\left(Y\mid\mathcal{T}_{S},W,A=1,s\in\sigma_{s}\right)\mid
W,A=1,S=s\right]$
$\displaystyle\quad\quad-E\left[E\left(Y\mid\mathcal{T}_{s},W,A=0,S=s\right)-E\left(Y\mid\mathcal{T}_{s},W,A=0,s\in\sigma_{s}\right)\mid
W,A=0,S=s\right]$ $\displaystyle=E(Y\mid W,A=1,S=s)-E(Y\mid W,A=0,S=s)$
$\displaystyle\quad\quad-\\{E\left[E\left(Y\mid\mathcal{T}_{s},W,A=1,s\in\sigma_{s}\right)\mid
W,A=1,S=s\right]$
$\displaystyle\quad\quad\quad\quad-E\left[E\left(Y\mid\mathcal{T}_{s},W,A=0,s\in\sigma_{s}\right)\mid
W,A=0,S=s\right]\\}.$ (4)

The latter term cannot be simplified unless Assumption 5 holds.

##  4 Comparison with bias functions in settings without incorporation of
follow-up surrogate outcomes

In recent work, Dahabreh and Hernán, (2019) developed sensitivity analysis for
transportability considering a similar setting of two types of studies with
and without missing outcomes. In the base case, there are two studies
considered (missingness of the outcome variable denoted by a binary indicator
$S$). To describe this setting using our notation, we simply have
$\sigma_{0}=\sigma_{1}=\\{1\\}$ (i.e., study $S=1$ with the observed outcome
of interest is used to impute the conditional outcome means for study $S=0$).
Equivalently, for ease of interpretation in the base case, let $S=1$ and $S=0$
denote the study where the primary outcome of interest is observed and not
observed, respectively.

In the setting where the model used to impute conditional potential outcomes
does not utilize information from $\mathcal{T}_{s}$, Dahabreh and Hernán,
(2019) define:

$u(A=a,W)=E[Y\mid A=a,W,S=1]-E[Y\mid A=a,W,S=0].$

The difference between these bias functions can then be obtained as:

$\displaystyle\delta(W)$ $\displaystyle=u(A=1,W)-u(A=0,W)$
$\displaystyle=E[Y^{1}-Y^{0}\mid W,S=1]-E[Y^{1}-Y^{0}\mid W,S=0]$

This expression can be qualitatively expressed as the difference in the
conditional average treatment effects between the two studies. This
qualitative interpretation can aid in conceptualizing and thinking about more
appropriate values and range for sensitivity parameters when examining
robustness of the results. More specifically, assuming higher levels of the
outcome are preferred, if we believe the participants in studies with missing
outcomes benefit less from treatment, then true $\delta$ can be assumed to be
positive and vice versa (Dahabreh and Hernán,, 2019). Since our bias functions
are conditional on the set of proxy outcomes, the term $\delta^{*}(W)$ in (3)
unfortunately cannot be reduced further to a more interpretable statistical
entity. When we take $\mathcal{T}_{s}$ to be the empty set, the bias function
$\delta^{*}(W)$ reduces to the same expression.

##  5 Accounting for violation of the common outcome regression assumption
through sensitivity analyses

We consider two scenarios in which we assume the bias terms
$u(A=1,\mathcal{T}_{s},W)$ and $u(A=0,\mathcal{T}_{s},W)$ to be 1) constants
and 2) bounded functions of the outcome proxies and/or baseline covariates.
The first scenario involves making a stronger assumption about the bias terms.
On the other hand, the second scenario requires weaker assumptions but allow
them to be non-constant.

###  5.1 Bias functions assumed to be some fixed values

Although it might be more reasonable to assume that the bias functions are
dependent on some baseline covariates, for ease of implementation of
sensitivity analysis, one can also suppose they are constant. When
$u(A=1,\mathcal{T}_{s},W)$ and $u(A=0,\mathcal{T}_{s},W)$ are independent of
the baseline covariates $W$ and the outcome proxy set $\mathcal{T}_{s}$, the
conditional expectations of the bias functions, and in turn, the term
$\delta^{*}(W)$ in (3), reduce to:

$\delta=u_{1}-u_{0},\textrm{ where }\delta,u_{1},\textrm{ and
}u_{0}\in\mathbb{R}$ (5)

The sensitivity analysis involves correcting for the above-mentioned bias term
by adding it back to the identification formula $\Psi^{ATE}$, which relies on
the common outcome regression assumption.

$\displaystyle ATE=$
$\displaystyle\sum_{s=1}^{{s^{*}}}\pi_{s}E\left\\{E\left(Y\mid
W,A=1,S\in\sigma_{s}\right)-E\left(Y\mid W,A=0,S\in\sigma_{s}\right)\mid
S=s\right\\}$
$\displaystyle+\sum_{s=s^{*}+1}^{{{\mathcal{S}}}}\pi_{s}E\left[E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s}\right)\mid
W,A=1,S=s\right\\}\right.$
$\displaystyle-\left.E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s}\right)\mid
W,A=0,S=s\right\\}\mid
S=s\right]+\sum_{s=s^{*}+1}^{{{\mathcal{S}}}}\pi_{s}\left(u_{1}-u_{0}\right)$
$\displaystyle=$
$\displaystyle\Psi^{ATE}+\sum_{s=s^{*}+1}^{{\mathcal{S}}}\pi_{s}\left(u_{1}-u_{0}\right)$
(6)

where $u_{1}$ and $u_{0}$ are scalars.

In practice, the true bias term would be unknown. Thus, one strategy is to
propose a grid of sensitivity parameters that covers the potential range of
values in which the true bias term might fall. This grid of sensitivity
parameters can be specified using subject-matter knowledge. We can then adjust
for the bias term in the estimation step by adding back the different
sensitivity parameters to the estimated ATE using our proposed method. This
also allows for observation of the behavior of the estimated ATE as we vary
the sensitivity parameters.

###  5.2 Bounded covariate-dependent bias functions

One might also believe that the bias term is not constant at all levels of the
baseline covariates and/or the outcome proxies. When the assumption of fixed-
value bias terms is considered too strong, but the functional forms for bias
terms cannot be confidently determined from existing knowledge of the data
mechanism (as will typically be the case), one can still recover some
information about the true ATE without having to correctly specify the bias
terms. If we instead assume the bias terms to be some bounded functions, we
can compute a bound around the (naïve) ATE estimate that contains the true ATE
by varying the bounds of these functions. This provides information on how far
away the true ATE can be from the estimate obtained constrained by the bounds
of the bias term.

Identifying the bounds for the bias term can be expressed as maximizing and
minimizing the objective function:

$E[E[u(A=1,\mathcal{T}_{s},W)\mid W,A=1,S=s]-E[u(A=0,\mathcal{T}_{s},W)\mid
W,A=0,S=s]\mid S=s]$

subject to the following constraints:

$\displaystyle|u(A=1,\mathcal{T}_{s}=t_{s},W=w)|$
$\displaystyle\leq\gamma_{1}$
$\displaystyle|u(A=0,\mathcal{T}_{s}=t_{s},W=w)|$
$\displaystyle\leq\gamma_{0}$

for all $t_{s}$ and $w$, which implies $|E[u(A=1,\mathcal{T}_{s},W)\mid
W,A=1,S=s]|\leq\gamma_{1}$ and $|E[u(A=0,\mathcal{T}_{s},W)\mid
W,A=0,S=s]|\leq\gamma_{0}$ where $\gamma_{1},\gamma_{1}\in\mathbb{R}^{+}$.

Then we have $-(\gamma_{1}+\gamma_{0})\leq
u(A=1,\mathcal{T}_{s},W)-u(A=0,\mathcal{T}_{s},W)\leq\gamma_{1}+\gamma_{0}$.
If we have no reason to suspect we know more about the bounds of one bias
function than the other (as will typically be the case), we may simply choose
to specify a scalar sensitivity parameter $\gamma$ to be the maximum of
$\gamma_{1}$ and $\gamma_{2}$, in which case we have $-2\gamma\leq
u(A=1,\mathcal{T}_{s},W)-u(A=0,\mathcal{T}_{s},W)\leq 2\gamma$.

By equation (6) even though we do not know the form of the bias functions
$u(A=1,\mathcal{T}_{s},W)$ and $u(A=0,\mathcal{T}_{s},W)$, we can partially
recover the true ATE using the bounds around the naïve estimate:

$\displaystyle\Psi^{ATE}-2\max(\gamma_{1},\gamma_{0})$ $\displaystyle\leq
ATE\leq\Psi^{ATE}+2\max(\gamma_{1},\gamma_{0})$
$\displaystyle\Psi^{ATE}-2\gamma$ $\displaystyle\leq
ATE\leq\Psi^{ATE}+2\gamma$ (7)

If the bias functions are in fact bounded by some value smaller than or equal
to our specified values for the sensitivity bounds, the true ATE would fall
between $[\Psi^{ATE}-2\gamma,\Psi^{ATE}+2\gamma]$. Then, the true ATE is
partially identified without assumptions about the functional form of
$u(A=1,\mathcal{T}_{s},W)$ and $u(A=0,\mathcal{T}_{s},W)$. One can then use
the bootstrap standard error for the substitution estimator of the
identification formula (2.1) to determine the amount to add and subtract from
the upper and lower bounds, respectively, in order to produce confidence
intervals for the partial identification sets for each value of the
sensitivity parameter. Since the sensitivity bounds are a deterministic
function of the sensitivity parameter, bootstrapping need only be done once.

##  6 Simulations

###  6.1 Data generating mechanism

We consider the setting of two studies, with $S$ = 1 indicating the study
where the primary outcome is available. We generate random sample draws with
sample size n = 100 for both studies. The data generating mechanism is as
follows. $W,T_{0}$ come from independent standard normal distributions, and
$T_{1}$ comes from a normal distribution with mean and variance of 1. Then

$\displaystyle T$ $\displaystyle=I(A=1)\times T_{1}+I(A=0)\times T_{0}$
$\displaystyle Y^{0}$ $\displaystyle=-4T_{0}+W+\epsilon_{0}$ $\displaystyle
Y^{1}$ $\displaystyle=4T_{1}+W+\epsilon_{1}$ $\displaystyle Y$
$\displaystyle=I(A=1)\times Y_{1}+I(A=0)\times Y_{0}$

where $\epsilon_{1},\epsilon_{0}\sim N(0,1)$.

Via these specifications, $T$ fully mediates the relationship between $A$ and
$Y$ (direct effect from $A$ to $Y$ is constrained to be 0). As a result, the
true ATE $=4$. This is also a more basic setting in which the vector T is
observed in all studies.

Due to the nature of the DoCTRS database, which is comprised of randomized
clinical trials, in our base setting, we specified the marginal probability
P(A = 1) = 0.5, representing random treatment assignment. This treatment
assignment satisfies the positivity and exchangeability assumption.

Specifically, to incorporate the difference in conditional outcome means
between the two types of studies, in studies missing the outcome, we added
constant bias terms to the counterfactual outcomes $Y_{0}$ and $Y_{1}$.
Similar to the data generating step, we preserved the observed counterfactual
outcome from the corresponding treatment assignment, which satisfies the
consistency assumption. By (5), we have:

$\displaystyle Y_{S=1}^{0}$ $\displaystyle=Y_{S=0}^{0}+u_{0}$ $\displaystyle
Y_{S=1}^{1}$ $\displaystyle=Y_{S=0}^{1}+u_{0}+\delta$ (8)

for $u_{0}\in\\{-3,0,3\\},\delta\in\\{-2,0,2\\}$.

Then the bias reduces to a single parameter $\delta$, since it is no longer a
function of $u_{0}$ when computing the ATE:

$E(Y^{1}-Y^{0}\mid S=1)=E(Y^{1}-Y^{0}\mid S=0)+\delta$ (9)

In the case where the bias term is a function of baseline covariates and
surrogate outcome, we had the following specification for the true bias:

$\begin{array}[]{l}u_{0}=b_{0}\times\sin\left(T_{s}+W\right)\\\
u1=b_{1}\times\frac{\exp\left(T_{s}+W\right)}{1+\exp\left(T_{s}+W\right)}\end{array}$

for $b_{0}\in\\{2,3,4\\}$ and $b_{1}\in\\{1,2,3\\}$.

###  6.2  Adjusting for sensitivity parameter in estimation step

In the presence of non-zero bias, when the value of the sensitivity parameter
$\delta$ is specified such that it is equal to true $\delta$, the ATE estimate
after bias adjustment tends to be closer to the true ATE after compared to
before. In addition, the corresponding 95% CIs are expected to cover the true
ATE 95% of the times. Although coverage probability can be examined more in a
more robust fashion using bootstrapped confidence intervals across all
simulations, in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the
value of the sensitivity parameter that reflects the degree of assumption
violation all but one instance, which is in line with our expectations.

Scenario 1. When the bias terms are assumed to be constants, a natural
approach would be to specify a two-dimensional grid of sensitivity parameters
for both scalars $u_{0}$ and $u_{1}$. However, by (8), it is equivalent to
specifying $u_{0}$ (or $u_{1}$) and $\delta$. In fact, since the $u_{0}$ (or
$u_{1}$) as constant terms cancel out during adjustment, it is sufficient to
specify one sensitivity parameter $\delta$ (9). We also note that $\delta$
being 0 does not necessarily imply assumption 4 is met, since the bias terms
$u_{0}$ and $u_{1}$ could cancel exactly.

To implement sensitivity analysis, we follow the steps:

  1. 1.

Specify a grid of sensitivity parameters $\delta$. The grid should be
reasonably wide to contain true $\delta$.

  2. 2.

Estimate the naïvely transported ATE using the identification result in (2.1)

  3. 3.

Sequentially add the values in the sensitivity parameter grid to the naïvely
estimated ATE, using the result in (6) to obtain the bias-corrected ATE
estimates.

We then plotted the bias-corrected estimates under different sensitivity
parameters against the true ATE. Additionally, we bootstrapped the bias-
corrected estimates to obtain the 95% confidence intervals and explore
coverage across different values of $u_{0}$ and $\delta$.

Scenario 2. When we want to make minimal assumptions about the functional form
of the bias, we can still perform sensitivity analysis on the true ATE using
the following steps:

  1. 1.

Specify a grid of sensitivity parameters called $\gamma$ that potentially
include the upper and lower bounds of the true bias functions

  2. 2.

Computed the “naïve” ATE estimate using the identification result in (2.1)

  3. 3.

Construct the upper and lower bound around the estimated ATE using (5.2) where
$\gamma$ is replaced with the sensitivity parameters.

We also plot the naïve ATE estimates and the bounds around these estimates at
each value of the sensitivity parameters. In practice, the bias functions are
of course unknown and cannot be estimated from observed data. Therefore, when
specifying the grid of sensitivity parameters, the analyst needs to employ
subject matter knowledge about the data generating mechanism to select values
of $\delta$ and $\gamma$.

We then explore the behavior of the adjusted estimators via simulations. In
the first case, we focused on the general unbiasedness of the correctly-
adjusted point estimate for both the overall ATE and ATE among studies with
missing outcomes, as well as the 95% CI coverage across degrees of assumption
violation (i.e., across values of true $u_{0}$ and $\delta$). In the second
case, we looked for correct bounding of the true ATE.

###  6.3 Simulation Results

####  6.3.1 Bias terms as constants

We examine the estimates produced by our method under the different degrees of
violation of assumption 4, before and after taking into account the specified
sensitivity parameter. Figure 1 shows the estimates (95% CI) for the true
overall ATE using our method under varying magnitudes and directions of the
bias terms from one single simulation.

Figure 1: Sensitivity-parameter-adjusted ATE estimate shown against the true
overall ATE across values of the true bias and sensitivity parameter; n=100
for each study, 95% CI constructed from 1000 bootstrap samples. When
sensitivity parameter $\delta$ = 0, the adjusted estimate corresponds to the
unadjusted estimate. Horizontal dotted line shows the true ATE given true
$\delta$; vertical dotted line indicates sensitivity parameter $\delta$ equals
true $\delta$

In the presence of non-zero bias, when the value of the sensitivity parameter
$\delta$ is specified such that it is equal to true $\delta$, the ATE estimate
after bias adjustment tends to be closer to the true ATE after compared to
before. In addition, the corresponding 95% CIs are expected to cover the true
ATE 95% of the times. Although coverage probability can be examined more in a
more robust fashion using bootstrapped confidence intervals across all
simulations, in Fig. 1, 2, and A.1-A.4, the 95% CIs covers the true ATE at the
value of the sensitivity parameter that reflects the degree of assumption
violation all but one instance, which is in line with our expectations.

Figure 2: Sensitivity-parameter-adjusted ATE estimates shown against the true
study-specific ATE in the study in which the outcome is unobserved across
values of the true bias and sensitivity parameter; n=100 for each study, 95%
CI constructed from 1000 bootstrap samples. When sensitivity parameter
$\delta$ = 0, the adjusted estimate corresponds to the unadjusted estimate.
Horizontal dotted line shows the true study-specific ATE given true $\delta$;
vertical dotted line indicates sensitivity parameter $\delta$ equals true
$\delta$

Figure 2 shows similar results for the study-specific ATE estimates in the
study with missing outcomes (before and after bias adjustment) from the same
simulated data. Compared to the results in Figure 1, after adjustment using
the correct sensitivity parameters, the 95% CIs contain the true ATE more
frequently than the CIs of the unadjusted estimates in the study with missing
primary outcome. Figure 2 also shows an example where inference is sensitive
to the violation of our assumption at a magnitude of $\delta$ between -1 and
-2 ($u_{0}=-3$, bottom left panel), between which the 95% CI changes from not
containing to zero to containing zero.

When we increased the sample size (n=200 and n=500), we saw general reductions
in the errors of these single estimates (Figures A.1, A.3). In most cases,
even when there is error in the adjusted estimates, the 95% CI bootstrap
confidence intervals provide good coverage (Figures 1, A.1, A.3). The
reduction in error and improved coverage are more pronounced when estimating
the study-specific effect in the study with missing outcomes than in the
overall ATE combining the two studies (Figures A.2, A.4).

We also ran 1000 simulations under the same data generating mechanism and
obtained the unadjusted and sensitivity-parameter-adjusted estimates for each
simulation. We then showed the mean and 2.5th and 97.5th quantiles of these
estimates under each combination of the true bias values. We can see that when
averaged across 1000 simulations, the adjusted estimates closely approximate
the true ATE (Figures 3, 4) when the true value of $\delta$ is used for the
sensitivity parameter.

Figure 3: Sensitivity-parameter-adjusted ATE estimates shown against the true
overall ATE across values of the true bias sensitivity parameter; mean, 2.5th
and 97.5th quantiles obtained from 1000 simulations. When sensitivity
parameter $\delta$ = 0, the adjusted estimate corresponds to the unadjusted
estimate. Horizontal dotted line shows the true overall ATE given true
$\delta$; vertical dotted line indicates sensitivity parameter $\delta$ equals
true $\delta$ Figure 4: Sensitivity-parameter-adjusted ATE estimates shown
against the true ATE in the study with missing outcome across values of the
true bias and sensitivity parameter; mean, 2.5th and 97.5th quantiles obtained
from 1000 simulations. When sensitivity parameter $\delta$ = 0, the adjusted
estimate corresponds to the unadjusted estimate. Horizontal dotted line shows
the true study-specific ATE given true $\delta$; vertical dotted line
indicates sensitivity parameter $\delta$ equals true $\delta$

When approximate sensitivity parameters $\delta$ are used
($\delta\in\\{-1,1\\}$ when true $\delta\in\\{-2,2\\}$), the middle 95% values
of adjusted estimates also cover the true ATE whereas those of unadjusted
estimates do not (Figure 4).

Figure 5 compares the errors in the estimates and sensitivity of associated
inferences between the outcome proxy-blind method of Dahabreh et al., 2020b ;
Lesko et al., (2017) and our proposed method across 1000 simulations.

Figure 5: Sensitivity-parameter-adjusted ATE estimates obtained from our
proposed method and the outcome proxy-blind method; mean, 2.5th and 97.5th
quantiles obtained from 1000 simulations. When sensitivity parameter $\delta$
= 0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal
dotted line shows the true overall ATE given true $\delta$; vertical dotted
line indicates sensitivity parameter $\delta$ equals true $\delta$

The distributions of the estimates from both methods are centered on the true
parameter. However, the estimates tend to be more precise when we utilize the
information from the outcome proxy (as demonstrated through the narrower
2.5th-97.5th quantile range). The efficiency gains have implications for the
sensitivity analysis, since resulting inferences are not as sensitive given
analogous magnitude in violation of the identification assumption 4.

Assumption 4 implies both $u_{0}$ and $u_{1}$ equal 0. As a result, the true
$\delta$ also equals 0. This suggests transportation of the conditional
potential outcome means, and in turn, the conditional average treatment
effects, can be done without incurring bias (vertical middle panes, figure 3).
We also observed that, when $\delta$ is 0, regardless of the values of $u_{0}$
(and $u_{1}$), there is also no bias (vertical middle panes, figure 3) in the
unadjusted estimator. In both cases, no bias correction would be necessary,
and incorporating a non-zero $\delta$ sensitivity parameter will actually
introduce bias to the estimate.

####  6.3.2 Bias terms as bounded functions

When the sensitivity parameter $\gamma$ is greater or equal to
$\max\\{\gamma_{0},\gamma_{1}\\}$ for the true function bounds $\gamma_{0}$
and $\gamma_{1}$, the bounds always include the true ATE when the bias
functions are bounded by $\gamma_{0}$ and $\gamma_{1}$ (Figure 6).

Figure 6: ATE estimates with sensitivity bounds shown against the true overall
ATE across values of the true bias and sensitivity parameter. When sensitivity
parameter $\gamma$ = 0, the bounds collapse to a point estimate. Blue
horizontal dotted line shows the true study-specific ATE given true bias
functions

Although this approach requires minimal assumptions about the bias functional
form, it can also be conservative since the true bias functions are unlikely
to evaluate to the bounds across the domain of the functions. For instance,
the bottom three panels of Figure 6 show that when the sensitivity parameter
$\gamma$ is greater than or equal to max(true $\gamma_{0}$, true
$\gamma_{1}$), while the bounds on the estimate contain the true ATE, they
also contains the null value zero as well. On the other hand, these bounds do
not rely on an assumption of constant bias functions, which we may often have
no reason to believe. Here, we demonstrated through simulations that
sensitivity analysis with relaxed and more credible assumptions can still
provide helpful information about the parameter of interest. However, when the
bounds are too narrow or too wide, sensitivity analysis using bounded bias
functions might not be accurate (i.e., not containing the true parameter) or
useful (i.e., containing the null value when the truth is non-null),
respectively.

##  7 Discussion

In this paper, we discussed a data integrative method that utilizes
information from available proxies of the outcome of interest measured at
follow-up for efficiency gains. We then presented two sensitivity analysis
strategies specific to this approach for causal effect transportation when the
identification assumption is violated. Our modification to the identification
of the ATE in (2.1) allows for more efficient estimators given sufficiently
strong outcome proxies. As a result, our bias functions also have similar, yet
distinct interpretations than the bias functions of Dahabreh and Hernán,
(2019).

When the bias terms are assumed to be constants, we can obtain different bias-
adjusted point estimates based on our specification of the sensitivity
parameters. Additionally, via obtaining the 95% bootstrap confidence interval
for the bias-adjusted estimates, we can examine the robustness of inferences
made using our method under varying magnitudes of assumption violation.
Specifically, beyond certain values of the sensitivity parameters, the 95% CI
will cross the null value 0. These are the degrees of violation that can
affect inferences (where the 95% CI suggest a change from significant results
to non-significant results).

We also proposed sensitivity analysis using bounded bias functions as an
alternative when one believes the assumption of a fixed-value bias term is too
strong. This approach allows for inferences with minimal assumptions about the
unobserved bias functions but can still provide useful information about the
parameter of interest. Due to fewer assumptions being made, the results are
more conservative and robust, hence more reasonable and credible.
Specifically, although we are unable to obtain a point estimate, sensitivity
analysis using bounded bias functions can still be informative in the sense of
providing information about the general direction of the parameter of interest
(beneficial or harmful). This method is generally more conservative if the
bounds on the functions are not close to their extreme values, if the bias
functions are generally not close to their extreme values, or if there is a
large difference between the extrema of the two bias functions.

Correct specification of the bias functions would allow for more precise and
informative estimation of the true ATE. However, since they are generally
unknown and non-estimable from observed data, sensitivity analysis will
typically be the realistic course of action.

When conducting sensitivity analysis, the analyst can start off by specifying
a wide grid of the sensitivity parameter and examining the behaviors of the
point estimates and 95% CI (first approach) as well as bounds around the
estimates (second approach). They can then search for the “critical”
sensitivity parameters that still suggest rejection of the null hypothesis,
i.e., the 95% CI (in the first case) and bounds around the estimate (in the
second case) that do not contain 0. It can be determined if greater bias is
plausible by using background knowledge of the data generating mechanism or
further hypothesizing about such mechanism. If there is little or no evidence
that the true bias functions exceed these critical sensitivity parameters, one
can be more comfortable in concluding that the observed effect and associated
inferences are robust to violation of the transportability assumption (Ding
and VanderWeele,, 2016; Cornfield et al.,, 1959).

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##  Appendix A Additional figures

Figure A.1: Sensitivity-parameter-adjusted ATE estimates shown against the
true overall ATE across values of the true bias and sensitivity parameter;
n=200 for each study, 95% CI constructed from 1000 bootstrap samples. When
sensitivity parameter $\delta$ = 0, the adjusted estimate corresponds to the
unadjusted estimate. Horizontal dotted line shows the true overall ATE given
true $\delta$; vertical dotted line indicates sensitivity parameter $\delta$
equals true $\delta$ Figure A.2: Sensitivity-parameter-adjusted ATE estimates
shown against the true ATE in the study with missing outcome across values of
the true bias and sensitivity parameter; n=200 for each study, 95% CI
constructed from 1000 bootstrap samples. When sensitivity parameter $\delta$ =
0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal
dotted line shows the true study-specific ATE given true $\delta$; vertical
dotted line indicates sensitivity parameter $\delta$ equals true $\delta$
Figure A.3: Sensitivity-parameter-adjusted ATE estimates shown against the
true overall ATE across values of the true bias and sensitivity parameter;
n=500 for each study, 95% CI constructed from 1000 bootstrap samples. When
sensitivity parameter $\delta$ = 0, the adjusted estimate corresponds to the
unadjusted estimate. Horizontal dotted line shows the true overall ATE given
true $\delta$; vertical dotted line indicates sensitivity parameter $\delta$
equals true $\delta$ Figure A.4: Sensitivity-parameter-adjusted ATE estimates
shown against the true ATE in the study with missing outcome across values of
the true bias and sensitivity parameter; n=500 for each study, 95% CI
constructed from 1000 bootstrap samples. When sensitivity parameter $\delta$ =
0, the adjusted estimate corresponds to the unadjusted estimate. Horizontal
dotted line shows the true study-specific ATE given true $\delta$; vertical
dotted line indicates sensitivity parameter $\delta$ equals true $\delta$

##  Appendix B Derivation of the sensitivity analysis formula when Assumption
4 is violated

$\displaystyle\mathrm{ATE}$
$\displaystyle=\sum_{s=1}^{\mathrm{s^{*}}}\pi_{s}E\\{E(Y\mid
W,A=1,S=s)-E(Y\mid W,A=0,S=s)\mid S=s\\}$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{S}\pi_{s}E\left[E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=1,S=s\right)\mid
W,A=1,S=s\right\\}\right.$
$\displaystyle\quad\quad\quad\quad-\left.E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=0,S=s\right)\mid
W,A=0,S=s\right\\}\mid S=s\right]$
$\displaystyle=\sum_{s=1}^{s^{*}}\pi_{s}E\\{E(Y\mid W,A=1,S=s)-E(Y\mid
W,A=0,S=s)\mid S=s\\}$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{{\mathcal{S}}}\pi_{s}E\left[E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s}\right)+u\left(A=1,\mathcal{T}_{s},W\right)\mid
W,A=1,S=s\right\\}\right.$
$\displaystyle\quad\quad\quad\quad-\left.E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s}\right)+u\left(A=0,\mathcal{T}_{s},W\right)\mid
W,A=0,S=s\right\\}\mid S=s\right]$
$\displaystyle=\sum_{s=1}^{s^{*}}\pi_{s}E\\{E(Y\mid W,A=1,S=s)-E(Y\mid
W,A=0,S=s)\mid S=s)$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{{\mathcal{S}}}\pi_{s}E\left[E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=1,S\in\sigma_{s}\right)\mid
W,A=1,S=s\right)\right\\}$
$\displaystyle\quad\quad\quad\quad\left.\left.-E\left\\{E\left(Y\mid\mathcal{T}_{s},W,A=0,S\in\sigma_{s}\right)\mid
W,A=0,S=s\right)\right\\}\mid S=s\right]$
$\displaystyle\quad\quad+\sum_{s=s^{*}+1}^{{\mathcal{S}}}\pi_{s}E[E\left\\{u\left(A=1,\mathcal{T}_{s},W\right)\mid
W,A=1,S=s\right\\}$
$\displaystyle\quad\quad\quad\quad-E\left\\{u\left(A=0,\mathcal{T}_{s},W\right)\mid
W,A=0,S=s\right\\}\mid S=s]$


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# Unexpectedly enhanced $\bm{\alpha}$-particle preformation in 48Ti probed by
the $\bm{(p,p\alpha)}$ reaction

Yasutaka Taniguchi<EMAIL_ADDRESS>Department of Information
Engineering, National Institute of Technology (KOSEN), Kagawa College, Mitoyo,
Kagawa 769-1192, Japan  Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan  Kazuki Yoshida  Advanced Science Research
Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  Research
Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan
Yohei Chiba  Department of Physics, Osaka City University, Osaka 558-8585,
Japan  Nambu Yoichiro Institute of Theoretical and Experimental Physics
(NITEP), Osaka City University, Osaka 558-8585, Japan  Research Center for
Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan  Yoshiko
Kanada-En’yo  Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki
567-0047, Japan  Masaaki Kimura  Department of Physics, Hokkaido University,
Sapporo 060-0810, Japan  Nuclear Reaction Data Centre, Hokkaido University,
Sapporo 060-0810, Japan  Research Center for Nuclear Physics (RCNP), Osaka
University, Ibaraki 567-0047, Japan  Kazuyuki Ogata  Research Center for
Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan  Department
of Physics, Osaka City University, Osaka 558-8585, Japan  Nambu Yoichiro
Institute of Theoretical and Experimental Physics (NITEP), Osaka City
University, Osaka 558-8585, Japan

###### Abstract

The formation of $\alpha$ particle on nuclear surface has been a fundamental
problem since the early age of nuclear physics. It strongly affects the
$\alpha$ decay lifetime of heavy and superheavy elements, level scheme of
light nuclei, and the synthesis of the elements in stars. However, the
$\alpha$-particle formation in medium-mass nuclei has been poorly known
despite its importance. Here, based on the ${}^{48}{\rm
Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction analysis, we report that the
$\alpha$-particle formation in a medium-mass nucleus ${}^{48}{\rm Ti}$ is much
stronger than that expected from a mean-field approximation, and the estimated
average distance between $\alpha$ particle and the residue is as large as 4.5
fm. This new result poses a challenge of describing four nucleon correlations
by microscopic nuclear models.

###### pacs:

—Introduction. Since Gamow explained the $\alpha$ decay as the quantum
tunneling of $\alpha$ particle out of an atomic nucleus [1], the formation of
$\alpha$ particle in nuclei has been a fundamental subject for understanding
the structure and decay of nuclei [2, 3, 4, 5]. It has been considered that
$\alpha$ particles are formed at a low-density nuclear surface with a certain
probability, which is called the preformation factor or the $\alpha$-particle
preformation probability. It determines the lifetime of heavy and superheavy
nuclei, and its empirical values have often been estimated from the $\alpha$
decay lifetime. For instance, the very short lifetime of ${}^{108}{\rm Xe}$
and ${}^{104}{\rm Te}$ were recently measured [6, 7], and the enhancement of
the $\alpha$-particle preformation probability beyond proton-rich nucleus
${}^{100}{\rm Sn}$ has been discussed [8, 9, 10].

It is also well known that the $\alpha$-particle preformation manifests itself
in light nuclei as $\alpha$ clustering [11, 12] and is closely related to the
synthesis of elements in stars [13, 14]. Because it exhibits the unique
excitation spectra, $\alpha$ clustering has been identified in many light
nuclei [15, 16]. Compared to heavy or light mass nuclei, the $\alpha$-particle
preformation in medium-mass nuclei has been poorly known. Generally, it is
believed that $\alpha$-particle preformation is hindered in medium-mass nuclei
because of the largely negative $\alpha$-decay $Q$-values. The deep binding
energies of these nuclei also lead to the dominance of the mean-field dynamics
over the four nucleon correlation preventing $\alpha$-particle formation.
However, such hindrance of $\alpha$-particle preformation has never been
quantitatively confirmed by experiment due to the lack of reliable measure for
the $\alpha$-particle preformation.

The proton-induced $\alpha$-knockout reaction $(p,p\alpha)$ has been expected
as the sensitive probe for the $\alpha$-particle preformation [17, 18, 19, 20,
21]. Due to the strong absorption effect, the $\alpha$ particle kicked by the
projectile proton cannot get out from the interior of the target nucleus.
Consequently, the reaction is only sensitive to the $\alpha$ particles formed
on the surface of the target nucleus. Several experiments have been conducted
to measure the $\alpha$-particle preformation probability in light-medium mass
nuclei. Carey et al. reported a systematic measurement of the $(p,p\alpha)$
reactions with various target nuclei from ${}^{16}{\rm O}$ to ${}^{66}{\rm
Zn}$ [19]. However, due to the lack of quantitative analysis, the absolute
value of the $\alpha$-particle preformation probabilities deduced from the
cross sections have large uncertainty.

Recently, it has been shown that the distorted wave impulse approximation
(DWIA) with reliable optical potentials realizes an accurate description of
the $(p,p\alpha)$ reaction [22]. Taking well-known light-mass $\alpha$
clustered nucleus ${}^{20}{\rm Ne}$ as an example, it was demonstrated that
the $\alpha$-particle preformation probability is quantitatively evaluated.
The new analysis showed that the $\alpha$-particle preformation probability of
${}^{20}{\rm Ne}$ is smaller than that estimated by Carey et al. by a factor
of two. Among the nuclei studied by Carey et al., ${}^{48}{\rm Ti}$ is the
only one except for ${}^{20}{\rm Ne}$, for which the optical potentials
between a proton, $\alpha$ particle, the residue (${}^{44}\mathrm{Ca}$), and
the target nucleus (${}^{48}{\rm Ti}$) have already been known accurately [23,
24, 25]. Furthermore, the residue ${}^{44}\mathrm{Ca}$ is a magic stable
nucleus as an inert core, and hence, the enhancement of the $\alpha$-particle
preformation can be expected. Therefore, the DWIA analysis of the ${}^{48}{\rm
Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction must shed new insight into the
$\alpha$-particle preformation in medium-mass nuclei.

—DWIA framework. The DWIA framework [26, 27, 28, 22] has been adopted to
describe the ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction. Within the
factorization approximation, the triple differential cross section is given
as,

$\displaystyle\frac{\mathrm{d}^{3}\sigma}{\mathrm{d}T_{p}\mathrm{d}\Omega_{p}\mathrm{d}\Omega_{\alpha}}=C_{0}F_{\rm
kin}\frac{\mathrm{d}\sigma_{p\alpha}}{\mathrm{d}\Omega_{p\alpha}}\left|\bar{T}\right|^{2},$
(1)

where $T_{p}$, $\Omega_{p}$ and $\Omega_{\alpha}$ are the kinetic energy of
the emitted proton, the solid angles of the proton and $\alpha$ particles,
respectively. $C_{0}F_{\rm kin}$ is the kinematical factor, and
$\mathrm{d}\sigma_{p\alpha}/\mathrm{d}\Omega_{p\alpha}$ is the $p$-$\alpha$
differential cross section at the $p$-$\alpha$ relative momentum of the
$(p,p\alpha)$ reaction kinematics. The detail of this approximation is given
in Refs. [26, 27], and the validity of this approximation has been tested and
confirmed [26]. The reduced transition matrix element $\bar{T}$ is defined as,

$\displaystyle\bar{T}$ $\displaystyle=\int
d^{3}R\,F(\bm{R})y(R)Y_{00}(\hat{\bm{R}}),$ (2) $\displaystyle F(\bm{R})$
$\displaystyle=\chi_{p}^{*(-)}(\bm{R})\chi_{\alpha}^{*(-)}(\bm{R})\chi_{p}^{(+)}(\bm{R})e^{-i\bm{k}_{0}\cdot\bm{R}/12},$
(3)

where $\bm{k}_{0}$ is the momentum of the incident proton. Equation (2) shows
the sensitivity of the cross section to the $\alpha$-particle preformation
because it depends on the probability amplitude of the $\alpha$-particle
preformation $y(R)$. The other ingredients of the analysis are the optical
potentials for the $p$-${}^{48}{\rm Ti}$, $p$-${}^{44}{\rm Ca}$, and
$\alpha$-${}^{44}{\rm Ca}$ scattering, which are used to describe the
distorted waves $\chi_{p}^{(\pm)}(\bm{R})$ and $\chi_{\alpha}^{(-)}(\bm{R})$;
the superscripts $(+)$ and $(-)$ indicate outgoing and incoming boundary
conditions, respectively. It was shown that the use of the accurate optical
potentials is essential for the precise description of the cross sections and
the evaluation of $\alpha$-particle preformation. In the present work, the
EDAD1 optical potential [24, 25] with Dirac phenomenology has been adopted to
the $p$-${}^{48}\mathrm{Ti}$ and $p$-${}^{44}\mathrm{Ca}$ distorted waves.
This potential reproduces the proton-nucleus elastic scattering with various
stable targets from ${}^{12}{\rm C}$ to ${}^{208}{\rm Pb}$ in a wide energy
range from 20 MeV to 1 GeV. For the $\alpha$-${}^{44}\mathrm{Ca}$ distorted
wave, we applied the optical potential proposed by Delbar et al. [23], which
reproduces the elastic differential cross sections from 24.1 to 100 MeV very
accurately. All these optical potentials cover the required energy range for
the analysis of the ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction.

—The $\alpha$-particle preformation probability. The probability amplitude for
$\alpha$-particle preformation, called the reduced width amplitude (RWA), is
defined as,

$\displaystyle
y(R)=\sqrt{\frac{48!}{4!~{}44!}}\Braket{\delta(r-R)\Phi_{\alpha}\Phi_{\rm
Ca}Y_{00}(\hat{r})}{\Phi_{\rm Ti}}/R^{2},$ (4)

where $\Phi_{\alpha}$, $\Phi_{\rm Ca}$, and $\Phi_{\rm Ti}$ denote the ground
state wave functions of the $\alpha$ particle, the residue
(${}^{44}\mathrm{Ca}$), and the target nucleus (${}^{48}{\rm Ti}$),
respectively. In this work, the $\alpha$ is assumed to have a $(0s)^{4}$
configuration, and the wave functions of ${}^{44}{\rm Ca}$ and ${}^{48}{\rm
Ti}$ are described by using the antisymmetrized molecular dynamics (AMD) [29,
30, 31]. The parity-projected AMD wave function is given as,

$\displaystyle\Psi$
$\displaystyle=(1+P_{x})/2\times\mathcal{A}\Set{\varphi_{1}\varphi_{2}...\varphi_{A}},$
(5) $\displaystyle\varphi_{i}$
$\displaystyle=\prod_{\sigma=x,y,z}\exp\Set{-\nu_{\sigma}\left(r_{\sigma}-Z_{i\sigma}\right)^{2}}$
$\displaystyle\times(\alpha_{i}\ket{\uparrow}+\beta_{i}\ket{\downarrow})\times(\ket{p}\mbox{
or }\ket{n}),$ (6)

where $P_{x}$ is the parity operator, $\mathcal{A}$ is the antisymmetrizer and
$\varphi_{i}$ is the nucleon wave packet. The centroid of a nucleon wave
packet is a complex vector $\bm{Z}_{i}$, in which the real (imaginary) part
describes the mean position (momentum) of a nucleon. The parameters of the
model wave function are the centroids $\bm{Z}_{i}$, the spin directions
$\alpha_{i}$ and $\beta_{i}$, and the Gaussian widths $\nu_{x}$, $\nu_{y}$,
and $\nu_{z}$. The wave function of ${}^{44}{\rm Ca}$ is calculated within the
mean-field approximation, i.e., the parameters are optimized to minimize the
intrinsic energy $E=\braket{\Psi}{H}{\Psi}/\braket{\Psi}{\Psi}$. Here, the
Hamiltonian consists of the nucleon kinetic energies, the effective nucleon-
nucleon interaction, and the Coulomb interaction. As an effective nucleon-
nucleon interaction, we have used Gogny D1S density functional [32] that
reasonably reproduces the fundamental nuclear properties. After the energy
minimization, the ${}^{44}{\rm Ca}$ wave function is projected to
$J^{\pi}=0^{+}$ to calculate the RWA [Eq. (4)] using the Laplace expansion
method [33].

---
Figure 1: Density distributions of the wave functions from which the RWAs have
been calculated. (a) The mean-field solution. (b)–(f) The
$\alpha+{}^{44}\mathrm{Ca}$ system with various inter-nuclear distance $d$.
Figure 2: (a) The RWA calculated from the wave functions shown in Fig. 1. (b)
The TMD of the ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction at
$T_{p}=63$ MeV. The TMD obtained from the mean-field solution and the $d=3.0$
fm wave function are multiplied by a factor of 10 and 5, respectively. The
arrow indicates the sum of the charge radii of $\alpha$ and ${}^{44}{\rm Ca}$,
which approximately corresponds to the nuclear surface.

The wave function of ${}^{48}{\rm Ti}$ is also calculated in the same manner.
The obtained wave function, i.e., the mean-field solution for ${}^{48}{\rm
Ti}$, is shown in Fig. 1 (a). It has an almost spherical shape and does not
clearly show the $\alpha$-particle preformation. Indeed, the RWA calculated
from this mean-field solution [Fig. 2 (a)] has only a small peak at $R=4.8$
fm, and as discussed later, it is too small to reproduce the observed cross
section. Therefore, we artificially generate the test wave functions of
${}^{48}{\rm Ti}$ that exhibit prominent $\alpha$-particle preformation. For
this purpose, we introduce an approximate inter-nuclear distance $d$ [34, 35],

$\displaystyle d=\left|\frac{1}{4}\sum_{i=1,...,4}{\rm
Re}\bm{Z}_{i}-\frac{1}{44}\sum_{i=5,...,48}{\rm Re}\bm{Z}_{j}\right|,$ (7)

where the first and second terms correspond to the center-of-mass of $\alpha$
and ${}^{44}{\rm Ca}$, respectively. We perform the energy variation with the
constraint on the value of $d$ and obtain the wave functions which mimic the
$\alpha$-particle preformation with various inter-nuclear distance as shown in
Fig. 1 (b)–(f). The RWAs calculated from these wave functions shown in Fig. 2
(a) have prominent peaks that become higher and move outward with the increase
of $d$. Note that the RWAs are strongly suppressed in the interior of the
residual nucleus ($R\lesssim 5$ fm) due to the Pauli principle. Consequently,
the peak position is not necessarily the same as the value of $d$.

Figure 3: Triple differential cross section of the ${}^{48}{\rm
Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction obtained by the DWIA calculations using
the RWAs shown in Fig. 2 (a) compared with the experiment [19]. The incident
proton energy, the emitted angles of proton and $\alpha$ are set to
$E_{p}=101.5$ MeV, $\theta_{p}=-70.0^{\circ}$ and
$\theta_{\alpha}=45.0^{\circ}$, respectively.
Figure 4: (a) Binding energy (b) charge radii, and (c) E2 transition matrix of
${}^{44}{\rm Ti}$ calculated from the mean-field solution and
$\alpha+{}^{44}\mathrm{Ca}$ wave functions in comparison with the experimental
data [36, 37, 38]. The binding energy is given relative to the
$\alpha$+${}^{44}{\rm Ca}$ decay threshold.

—Results and Discussions. Figure 3 shows the triple differential cross
sections of the ${}^{48}\mathrm{Ti}(p,p\alpha){}^{44}\mathrm{Ca}$ reaction
obtained by the DWIA calculations using the RWAs shown in Fig. 2 (a). The
cross sections are plotted as functions of the outgoing proton energy. The
incident proton energy, the emitted angles of the proton and $\alpha$ particle
are set to the same values as in the experiment by Carey et al. [19].
Unexpectedly, it is found that the mean-field solution does not reproduce the
observed cross section at all. It underestimates the cross section in three
orders of magnitude, which cannot be explained by the uncertainty of the
optical potentials used in DWIA analysis or the density functional (Gogny D1S)
used to calculate the mean-field solution. Consequently, we conclude that the
$\alpha$ preformation probability is much larger than that described by the
mean-field solution.

To estimate the degree of $\alpha$-particle preformation, we have also tested
the RWAs obtained from the $\alpha+{}^{44}\mathrm{Ca}$ wave functions with
various inter-nuclear distances. Figure 3 shows that these RWAs yield much
larger cross sections than the mean-field solution, and the cross section
increases by approximately one order of magnitude for every 1 fm increase of
the inter-nuclear distance. It is found that the RWA obtained from the
$\alpha+{}^{44}\mathrm{Ca}$ wave function with $d=4.5$ fm gives the most
plausible description of the observed cross section. The peripherality of the
$(p,p\alpha)$ reaction is confirmed from the real part of the transition
matrix density (TMD) [27] that is defined as,

$\displaystyle\delta(R)=\bar{T}^{*}\int
d\hat{\bm{R}}~{}R^{2}F(\bm{R})y(R)Y_{00}(\hat{\bm{R}}).$ (8)

Note that the integral of TMD over the distance is equal to the square of the
transition matrix $T$, and hence, $\delta(R)$ gives a hint at which distance
$R$ the reaction takes places. As shown in Fig. 2 (b), TMD is negligible in
the interior region ($R\lesssim 5$ fm) due to the strong absorption of an
$\alpha$ particle and small RWA. It explains why the cross section with the
mean-field solution is smaller in order of magnitude than that with the
$\alpha+{}^{44}\mathrm{Ca}$ wave functions. We also note that the peak
position ($T_{p}\sim 63$ MeV) and width of the cross section are approximately
determined by the kinematical condition (recoil-less condition for the residue
${}^{44}{\rm Ca}$) and the momentum distribution of the RWA, respectively.

Although the $\alpha+{}^{44}\mathrm{Ca}$ wave function with $d=4.5$ fm gives
the best result for the ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction,
its validity should be verified from different perspectives. Firstly, it must
be noted that the binding energies of the $\alpha+{}^{44}\mathrm{Ca}$ wave
functions are much smaller than that of the mean-field solution because of the
artificial constraint imposed on the inter-nuclear distance [Eq. (7)]. Figure
4 (a) shows that the binding energy of the $\alpha+{}^{44}\mathrm{Ca}$ wave
function rapidly decreases as the inter-nuclear distance increases. At $d=$4.5
fm, it underestimates the experimental value [36] by approximately 10 MeV and
yields the positive $Q$-value of the $\alpha$ decay, whereas the mean-field
solution gives reasonable binding energy and $Q$-value. Panels (b) and (c)
show the charge radius and the reduced matrix elements for the E2 transition
from the ground state to the $2^{+}_{1}$ state, respectively. As expected,
both the charge radius and E2 transition matrix elements increase with the
inter-nuclear distance. Although the $\alpha+{}^{44}\mathrm{Ca}$ wave function
gives reasonable values at $d=2.0$–2.5 fm, it overestimates the observed
values [37, 38] at $d=4.5$ fm. In short, the $\alpha+{}^{44}\mathrm{Ca}$ wave
function can describe the ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction,
but it fails to reproduce the fundamental structural properties. On the
contrary, the mean-field solution offers a better description of the energy,
radius, and E2 transition but fails in the $\alpha$ knockout reaction. From
these results, we can deduce that the ground state wave function should be an
admixture of the mean-field solution and the $\alpha+{}^{44}\mathrm{Ca}$ type
wave functions. The mean-field solution should be the dominant component of
the ground state due to its large binding energy, but the contamination of the
$\alpha+{}^{44}\mathrm{Ca}$ wave function is indispensable to explain the
observed large $\alpha$ knockout cross section.

—Summary. The ${}^{48}{\rm Ti}(p,p\alpha)^{44}{\rm Ca}$ reaction has been
studied to investigate the $\alpha$-particle preformation in a medium-mass
nucleus ${}^{48}{\rm Ti}$. The DWIA analysis using accurate optical potentials
offers a reliable and quantitative description of the $\alpha$-knockout
reaction, and it has revealed that the $\alpha$-particle preformation in
${}^{48}{\rm Ti}$ is unexpectedly enhanced. It has been shown that the mean-
field solution underestimates the cross section in orders of magnitude, and
one must assume the $\alpha+{}^{44}\mathrm{Ca}$ wave function whose the inter-
nuclear distance is as large as $d=4.5$ fm to reproduce the observed cross
section. However, the $\alpha+{}^{44}\mathrm{Ca}$ wave function fails to
explain other basic properties of ${}^{48}{\rm Ti}$, which are reasonably
described by the mean-field approximation. Hence, we conclude that the ground
state is an admixture of the mean-field and $\alpha+{}^{44}\mathrm{Ca}$
configurations. This new insight requests the systematic analysis of the
$(p,p\alpha)$ reactions to reveal the universality of the $\alpha$-particle
preformation and poses a challenge to the microscopic nuclear models for
describing $\alpha$-particle preformation in medium-mass nuclei.

###### Acknowledgements.

This work was supported by the COREnet program at the RCNP, Osaka University,
the Hattori Hokokai Foundation Grant-in-Aid for Technological and Engineering
Research, and JSPS KAKENHI Grant Nos. JP16K05352, JP18H05407, JP18K03617, and
JP20K14475. Numerical calculations were performed using Oakforest-PACS at the
Center for Computational Sciences, University of Tsukuba, and XC40 at Yukawa
Institute for Theoretical Physics, Kyoto University.

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