Zhou et al.

Satoshi Hattori, Department of Biomedical Statistics, Graduate School of
Medicine, Osaka University, Osaka, Japan.
Xiao-Hua Zhou, Beijing International Center for Mathematical Research, Peking
University, Beijing, China.

# Copas-Heckman-type sensitivity analysis for publication bias in rare-event
meta-analysis under the framework of the generalized linear mixed model

Yi Zhou  Taojun Hu  Xiao-Hua Zhou  Satoshi Hattori  Beijing International
Center for Mathematical Research, Peking University, Beijing, China
Department of Biomedical Statistics, Graduate School of Medicine, Osaka
University, Osaka, Japan  Department of Biostatistics, Peking University,
Beijing, China  Chongqing Big Data Research Institute, Peking University,
Chongqing, China  Integrated Frontier Research for Medical Science Division,
Institute for Open and Transdisciplinary Research Initiatives, Osaka
University, Osaka, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>Y. Zhou  T. Hu  X-H. Zhou  S. Hattori

(26 April 2016; 6 June 2016; 6 June 2016)

###### Abstract

[Summary] Publication bias (PB) is one of the serious issues in meta-analysis.
Many existing methods dealing with PB are based on the normal-normal (NN)
random-effects model assuming normal models in both the within-study and the
between-study levels. For rare-event meta-analysis where the data contain rare
occurrences of event, the standard NN random-effects model may perform poorly.
Instead, the generalized linear mixed effects model (GLMM) using the exact
within-study model is recommended. However, no method has been proposed for
dealing with PB in rare-event meta-analysis using the GLMM. In this paper, we
propose sensitivity analysis methods for evaluating the impact of PB on the
GLMM based on the famous Copas-Heckman-type selection model. The proposed
methods can be easily implemented with the standard software coring the
nonlinear mixed-effects model. We use a real-word example to show how the
usefulness of the proposed methods in evaluating the potential impact of PB in
meta-analysis of the log-transformed odds ratio based on the GLMM using the
non-central hypergeometric or binomial distribution as the within-study model.
An extension of the proposed method is also introduced for evaluating PB in
meta-analysis of proportion based on the GLMM with the binomial within-study
model.

, , , and (2023), Copas-Heckman-type sensitivity analysis for publication bias
in rare-event meta-analysis under the framework of the generalized linear
mixed effects model, Res Syn Meth, 2024;00:1–6.

###### keywords:

generalized linear mixed model, publication bias, sensitivity analysis, rare-
event meta-analysis

††articletype:

##  1 Introduction

In meta-analyses comparing two groups, it is often of interest whether the
events (e.g., adverse events) are more or less possible to occur in the
treatment group. This treatment effect is commonly quantified by the odds
ratio (OR) or the log-transformed OR (lnOR). When events are rare or sample
sizes of studies are small, it frequently happens that the observed numbers of
events are zero, making the OR and its standard error (SE) to be zero or
undefined. To address these issues, the continuity correction (CC) methods1
are widely applied. However, the CC methods are not on sound rationale; among
different CC methods, no single method is consistently recommended.2

Despite the inconsistency of the CC methods, the standard meta-analytical
methods, such as the inverse-variance methods including the normal-normal
random-effects model3 (hereinafter, the NN model), can be problematic. The NN
model3 comprises a two-level model. The within-study model assumes the
reported outcome to be normally distributed with the study-specific mean and
the variance of outcome. The between-study model assumes that the treatment
effect of each study is normally distributed with a common value and between-
study variance. For meta-analysis with binary outcomes (e.g., the occurrence
of event), the lnORs are often modeled using the NN model. In situations with
small study sizes or rare events, the normal approximation in the within-study
model is questionable. Additionally, the NN model assumes the independence
between lnOR estimates and their SEs, which may not be satisfied practically
and introduce bias.4, 5 Alternative to the NN model,3 the generalized linear
mixed model (GLMM) with exact within-study likelihood can be promising.4, 5, 6

Meta-analytical results are often threatened by publication bias (PB), a
phenomenon where studies with significant results are more likely to be
published. Combining only those selectively published studies may lead to
over-optimistic conclusions in meta-analyses.7 In recent decades, many
sophisticated selection-model-based methods have been proposed to
quantitatively evaluate the potential impact of PB.7, 8 Famous ones include
the sensitivity analysis methods by Copas and colleagues, such as the Copas-
Heckman-type selection model,9, 10, 11 also known as the Heckman model12, 13
in the field of econometrics; Others include the $t$-statistic-based selection
model14 and the non-parametric worst-case bounds.15 Among them, the Copas-
Heckman-type selection model has been extensively developed16, 17, 18 and
further extended to address PB in multivariate meta-analyses.19, 20, 21, 22
Most of these developments rely on the NN model or its multivariate versions.
The Copas-Heckman-type selection model could avoid the correction of zero
entries in data during the inference; Hattori and Zhou19 extended the Copas-
Heckman-type selection model for diagnostic meta-analysis using the bivariate
exact likelihood.

In rare-event meta-analysis, existing methods for PB inherit limitations of
the CC methods or the NN model. While the GLMM with the exact likelihood can
outperform the NN model, no methods are currently available for evaluating PB
in the meta-analytical results of the GLMM. With the success of Hattori and
Zhou,19 we aim to develop new methods for evaluating the impact of PB in rare-
event meta-analysis using the GLMM.

##  2 GLMM for meta-analysis of odds ratios without publication bias

Suppose that a meta-analysis contains $N$ studies with binary outcomes. Each
study has treatment and control groups; the numbers of events and subjects are
accessible. Let $y_{i0}$ and $n_{i0}$ be the number of events and subjects in
the control group, respectively; $y_{i1}$ and $n_{i1}$ are defined similarly
in the treatment group. The data of study $i$ are formulated by the
contingency matrix in Table 1. The common practice of meta-analysis is based
on the NN model, where the summary statistic such as the lnOR calculated from
Table 1 is used and the likelihood is constructed based on the normal
approximation. (The details are outlined in Appendix A.)

Exact likelihoods based on the observations $(y_{i0},y_{i1})$ are recommended
and constructed as follows. The non-central hypergeometric distribution is
proposed for the within-study likelihood, i.e., $y_{i1}\mid y_{i}$ follows the
non-central hypergeometric distribution.4 The between-study model is assumed
to be normally distributed, i.e., $\theta_{i}\sim N(\theta,\tau^{2})$, where
$\theta_{i}$ is the true lnOR in each study, $\theta$ the overall lnOR of
interest, and $\tau^{2}$ the between-study variance. This model is called the
hypergeometric-normal (HN)-GLMM with parameters $(\theta,\tau)$ estimated from
the likelihood:

$\displaystyle\prod_{i=1}^{N}\int_{-\infty}^{\infty}L_{i}(\theta_{i})\dfrac{1}{\tau}\phi\left(\dfrac{\theta_{i}-\theta}{\tau}\right)d\theta_{i}\text{
with
}L_{i}(\theta_{i})=\dfrac{\binom{n_{i1}}{y_{i1}}\binom{n_{i0}}{y_{i0}}\exp(\theta_{i}y_{i1})}{\sum_{k\in
K}\binom{n_{i1}}{k}\binom{n_{i1}}{y_{i}-k}\exp(\theta_{i}k)},$ (1)

where $\phi(.)$ denotes the density of standard normal distribution, and $K$
is the set of all possible values of $y_{i1}$ constrained by the marginal
totals in Table 1.

When many studies are involved in meta-analysis, the estimation of HN-GLMM can
be computationally demanding.5 The following binomial distribution:
$y_{i1}\mid
y_{i}\sim\text{Bin}\left(y_{i},\dfrac{\exp\\{\log(n_{i1}/n_{i0})+\theta_{i}\\}}{1+\exp\\{\log(n_{i1}/n_{i0})+\theta_{i}\\}}\right)$
can approximate to the non-central hypergeometric distribution in the HN-GLMM
when the total number of events is much smaller than the sample sizes;4 we
call this the binomial-normal (BN)-GLMM with the likelihood:

$\displaystyle\prod_{i=1}^{N}\int_{-\infty}^{\infty}L_{i}(\theta_{i})\dfrac{1}{\tau}\phi\left(\dfrac{\theta_{i}-\theta}{\tau}\right)d\theta_{i}\text{
with
}L_{i}(\theta_{i})=\binom{y_{i}}{y_{i1}}\dfrac{\exp\\{\log(n_{i1}/n_{i0})+\theta_{i}\\}^{y_{i1}}}{[1+\exp\\{\log(n_{i1}/n_{i0})+\theta_{i}\\}]^{y_{i}}}.$
(2)

##  3 New sensitivity analysis methods for PB in the GLMM

In the presence of selective publication of studies, the data of $N$ studies
may be biased sample from the population, possibly leading to biased estimates
of parameters. Along with the idea by Copas and Shi10 for the NN model, we
construct a sensitivity analysis method, where a latent Gaussian variable
responsible for selective publication is introduced, and a conditional
likelihood given the published is constructed. For the NN model, the
conditional likelihood is obtained on the property of the joint normal
distribution of the outcome and the latent variable as summarized in Appendix
B.

For binary outcomes, an alternative construction is needed. We consider a
latent variable $Z_{i}$ with the selection equation on the samples sizes:9

$\displaystyle Z_{i}=a_{0}+a_{1}\sqrt{n_{i}}+\delta_{i}$ (3)

where $a_{0}$ and $a_{1}$ are constants, $n_{i}$ the total number of subjects
in Table 1, and $\delta_{i}\sim N(0,1)$; study $i$ is published if and only if
$Z_{i}>0$. As explained in Appendix B, the latent variable (B3) is often used
for the NN model. One application of the latent variable (3) was for PB in
diagnostic meta-analysis.19

The probability of publishing study $i$ is modeled by:

$\displaystyle P(Z_{i}>0\mid n_{i})=\Phi(a_{0}+a_{1}\sqrt{n_{i}}),$ (4)

where $\Phi(.)$ is the standard normal cumulative distribution function. Model
(4) indicates that larger studies are more likely to be published. We then
employ bivariate normal distribution with correlation coefficient $\rho$ to
present the dependence between $\delta_{i}$ and $\theta_{i}$ in the HN-GLMM
(1) or BN-GLMM (2):

$\displaystyle\binom{\theta_{i}}{\delta_{i}}$ $\displaystyle\sim
N\left(\binom{\theta}{0},\begin{bmatrix}{{\tau^{2}}}&{{\rho\tau}}\\\
{{\rho\tau}}&{{1}}\end{bmatrix}\right).$

Considering PB, the likelihood conditional on the published is constructed as

$\displaystyle\prod_{i=1}^{N}\int L_{i}(\theta_{i}\mid
Z_{i}>0,n_{i})\dfrac{1}{\tau}\phi\left(\dfrac{\theta_{i}-\theta}{\tau}\right)d\theta_{i}$
(5)

with

$\displaystyle L_{i}(\theta_{i}\mid Z_{i}>0,n_{i})$
$\displaystyle=\dfrac{P(Z_{i}>0\mid\theta_{i},n_{i})L_{i}(\theta_{i})}{P(Z_{i}>0\mid
n_{i})}$
$\displaystyle=\dfrac{P\left(a_{0}+a_{1}\sqrt{n_{i}}+\delta_{i}>0)\mid\theta_{i},n_{i}\right)L_{i}(\theta_{i})}{P(Z_{i}>0\mid
n_{i})}$
$\displaystyle=\dfrac{\Phi\left\\{\dfrac{a_{0}+a_{1}\sqrt{n_{i}}+\rho(\theta_{i}-\theta)/\tau}{\sqrt{1-\rho^{2}}}\right\\}L_{i}(\theta_{i})}{\Phi(a_{0}+a_{1}\sqrt{n_{i}})},$

where $L_{i}(\theta_{i})$ is defined in likelihood (1) or (2). The parameters
in (5) can be estimated by the maximum likelihood method. One concern is that
parameters in (5) may not be fully identified; then, a sensitivity analysis
approach can be taken as follows. Let $n_{\text{min}}$ and $n_{\text{max}}$ be
the minimum and maximum numbers of subjects among the published studies;
$P_{\text{min}}$ and $P_{\text{max}}$ denotes the probabilities of publishing
a study with $n_{\text{min}}$ and $n_{\text{max}}$ subjects (practically,
$P_{\text{min}}\leq P_{\text{max}}$). With model (4),
$P_{\text{min}}=\Phi(a_{0}+a_{1}\sqrt{n_{\text{min}}})$ and
$P_{\text{max}}=\Phi(a_{0}+a_{1}\sqrt{n_{\text{max}}})$ hold. Given the values
of $(P_{\text{min}},P_{\text{max}})$, the constants $(a_{0},a_{1})$ are
derived by

$\displaystyle
a_{1}=\dfrac{\Phi^{-1}(P_{\text{max}})-\Phi^{-1}(P_{\text{min}})}{\sqrt{n_{\text{max}}}-\sqrt{n_{\text{min}}}}\text{
and }a_{0}=\Phi^{-1}(P_{\text{max}})-a_{1}\sqrt{n_{\text{max}}}.$

Then, the parameters $(\theta,\tau,\rho)$ in the likelihood conditional on the
published (e.g., likelihood (5)) can be estimated by the ML method. On the
other hand, the number of unpublished studies is estimated by
$\sum_{i=1}^{N}\\{1-P(Z_{i}>0\mid n_{i})\\}/P(Z_{i}>0\mid n_{i})$.

##  4 Application

We revisit the meta-analysis of lnORs4 with the background and estimates
without PB summarized in Box 1. We employed the proposed methods and the
original Copas-Heckman-type selection model10 to evaluate the potential PB on
the lnOR estimated by the HN/BN-GLMM and the NN model, respectively. The
sensitivity analysis approach was used for all the methods to estimate
$(\theta,\tau,\rho)$ by the ML method, where some reasonable values,
$(P_{\text{min}},P_{\text{max}})=\\{(0.9,0.99),(0.8,0.99),\dots,(0.1,0.99)\\}$,
were fixed. The impact of PB was assessed via the changes of the estimated
lnOR. Since the Copas-Heckman-type selection model adopted selection equation
on the precision (i.e., $1/s_{i}$) of the lnOR,10 the parameter
$P_{\text{min}}$, for example, then indicates the probabilities of publishing
study with maximum SE (minimum subjects).

Figure 1 presents the sensitivity analysis results on the lnOR estimated by
different methods with the detailed estimates of parameters in Table A1. Given
$P_{\text{max}}=0.99$ and decreasing $P_{\text{min}}$, the lnOR increased with
increasing number of unpublished studies. According to the results of the
proposed methods, the lnOR considering PB and estimated by the HN/BN-GLMM
remained significant, indicating that the meta-analytical results by the
HN/BN-GLMM were robust against potential PB. For the HN/BN-GLMM model, there
was little change of the lnOR over $P_{\text{min}}$ ($x$-axis in Figure 1). On
the other hand, for the NN model, certain change was observed possibly due to
the influence of sparsity of data.

##  5 Discussion

Publication bias has been recognized as an inevitable problem in meta-
analysis. Although many advanced selection-model-based methods are proposed
for evaluating the impact of PB on the meta-analytical results, implementing
these methods on rare-event meta-analysis can be problematic since they are
based on the NN model. The GLMM with the exact within-study likelihood is
recommended for rare-event meta-analysis;4, 5, 6 we proposed sensitivity
analysis methods to fill missingness of methods for evaluating the potential
PB in the GLMM, and the proposed methods can be easily implemented by the
standard statistical software. With the simple implementation, the proposed
methods could have wide applicability in rare-event meta-analysis. As
demonstrated in likelihood (5), our development can handle the exact
likelihoods in a unified way by setting a relevant likelihood function. We
present the additional case of rare-event meta-analysis of proportions in
single group with a real example in Appendix D. In summary, our proposal
provides practical tools for evaluating PB in rare-event meta-analysis using
the GLMM.

*Acknowledgments

This research was partly supported by Grant-in-Aid for Challenging Exploratory
Research (16K12403) and for Scientific Research(16H06299, 18H03208) from the
Ministry of Education, Science, Sports and Technology of Japan.

*Data availability statement R codes together with a sample application data set are available at https://github.com/meta2020/metaeventsa-r

*Supporting information Additional supporting information can be found online in the Supporting Information section at the end of this article.

*Highlights

*What is already known

  * •

For rare-event meta-analysis, the generalized linear mixed effects model
(GLMM) adopting the exact within-study likelihood, such as the hypergeometric
or the binomial distributions, are recommended than the standard normal-normal
random-effects model using the approximately normal likelihood.

  * •

Publication bias (PB), also known as small study effects, is one of the major
issues in meta-analysis; the existing methods for PB are developed for the NN
model and can be problematic for rare-event meta-analysis.

  * •

The Copas-Heckman-type selection model is one of the famous sensitivity
analysis methods for PB and has been widely developed for various meta-
analyses; however, this model cannot be used directly for PB in rare-event
meta-analysis using the GLMM.

*What is new

  * •

We successfully extended the Copas-Heckman-type selection model to deal with
PB in rare-event meta-analysis based on the GLMM.

  * •

This work is the first development of the Copas-Heckman-type selection model
for evaluating the impact of PB in rare-event meta-analysis using the exact
likelihood.

*Potential impact for Research Synthesis Methods readers outside the authors’ field

  * •

The proposed methods help researchers analyze the impact of PB on the meta-
analytical estimates (e.g., the log-transformed odds ratio, the logit-
proportion) using the GLMM, and then robust conclusions can be drawn from
rare-event meta-analysis.

  * •

The proposed methods provide simple tools for evaluating PB in rare-event
meta-analysis based on the GLMM, and we provided the open-source R codes for
researchers to implement the methods in practice.

## References

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Figure 1: Application: Comparison of sensitivity analysis methods for PB in
meta-analysis of lnORs.

Table 1: Example of the contingency table for data of study $i$ | Event | Non-event | Total
---|---|---|---
Treatment | $y_{i1}$ | $n_{i1}-y_{i1}$ | $n_{i1}$
Control | $y_{i0}$ | $n_{i0}-y_{i0}$ | $n_{i0}$
Total | $y_{i}$ | $n_{i}-y_{i}$ | $n_{i}$

BOX 1 Meta-analysis in the application without considering PB Background: This
is the meta-analysis of lnORs of catheter-related bloodstream infection
(CRBSI) between anti-infective-treated (AIT) central venous catheters and the
standard catheter.23 The data of this meta-analysis are presented below.

| AIT catheter | Standard catheter
---|---|---
Study | CRBSI ($y_{i1}$) | Patients ($n_{i1}$) | CRBSI ($y_{i0}$) | Patients ($n_{i0}$)
1 | 3 | 117 | 0 | 116
2 | 3 | 35 | 1 | 44
3 | 9 | 195 | 2 | 208
4 | 7 | 136 | 0 | 130
5 | 6 | 157 | 5 | 151
6 | 4 | 139 | 1 | 98
7 | 3 | 177 | 1 | 174
8 | 2 | 39 | 1 | 74
9 | 19 | 103 | 1 | 97
10 | 2 | 122 | 1 | 113
11 | 7 | 64 | 0 | 66
12 | 1 | 58 | 0 | 70
13 | 5 | 175 | 3 | 188
14 | 11 | 180 | 6 | 187
15 | 0 | 105 | 0 | 118
16 | 1 | 262 | 0 | 252
17 | 3 | 362 | 1 | 345
18 | 1 | 69 | 4 | 64

Methods: In Stijnen et al,4 the meta-analytical results by the NN model and
the HN/BN-GLMM were compared without considering selective publication of
studies. We reproduced the results and estimated the parameters by the ML
method. The optimization was implemented by the R function nlminb(); the SEs
of the parameters were estimated by the inverse of the empirical Fisher
information matrix following the ML theory. R function hcubature() was used to
calculate the integrals in the likelihoods of HN/BN-GLMM.

Results: The HN-GLMM and the BN-GLMM gained similar results with the lnOR
estimated as $-1.353\leavevmode\nobreak\ (-2.041\text{ to }-0.665)$ and
$-1.303\leavevmode\nobreak\ (-1.966\text{ to }-0.639)$, respectively, since
the total number of CRBSI was relatively small to the total number of
subjects. The NN model estimated the meta-analytical lnOR to be
$-0.955\leavevmode\nobreak\ (95\%\text{CI, }-1.501\text{ to }-0.408)$, which
was biased towards zero.4 Without accounting for selective publication, there
are certain discrepancies between the NN model and the HN/BN-GLMM model,
indicating that normal approximation in the NN model was questionable.

##  Appendix A Review of The normal-normal random-effects model without
publication bias

As mentioned in Section 2 of the main text, we present the likelihood of the
normal-normal random-effects model (hereinafter, the NN model) without
considering publication bias (PB).

Based on the data in Table 1 of the main text, to evaluate the occurrence of
events between two groups, the measurement of log-transformed odds ratio
(lnOR) is often reported and estimated by

$\hat{\theta}_{i}=\log\dfrac{y_{i1}/(n_{i1}-y_{i1})}{y_{i0}/(n_{i0}-y_{i0})},$

with its SE estimated by

$s_{i}=\sqrt{\dfrac{1}{y_{i1}}+\dfrac{1}{n_{i1}-y_{i1}}+\dfrac{1}{y_{i0}}+\dfrac{1}{n_{i0}-y_{i0}}}.$

If there are zero entries in the data, the continuity correction method is
usually conducted by adding 0.5 to the numbers of events and non-events before
estimating the lnOR and its SE.

In the absence of PB, the NN model3 assumes that, in the within-study level,
the distribution of the estimated lnOR $\hat{\theta}_{i}$ is modeled by

$\displaystyle\hat{\theta}_{i}\mid\theta_{i}\sim N(\theta_{i},s_{i}^{2}),$

where $\theta_{i}$ is the true lnOR in each study. Following the convention in
meta-analysis field, $s_{i}$ is regarded as known and fixed. In the between-
study level, it is assumed that

$\displaystyle\theta_{i}\sim N(\theta,\tau^{2}),$

where $\theta$ is the overall lnOR of interest, and $\tau^{2}$ is the between-
study variance. Marginally, $\hat{\theta}_{i}$ has the following normal
distribution:

$\displaystyle\hat{\theta}_{i}\sim N(\theta,s_{i}^{2}+\tau^{2}).$ (6)

In the NN model, the within-study likelihood is constructed based on the
observations $\hat{\theta}_{i}$ which approximately follow normal
distribution. Then, allowing for the existence of between-study heterogeneity,
the unknown parameters $(\theta,\tau)$ can be estimated based on the
likelihood:

$\displaystyle\prod_{i=1}^{N}\int_{-\infty}^{\infty}L_{i}(\theta_{i})\dfrac{1}{\tau}\phi\left(\dfrac{\theta_{i}-\theta}{\tau}\right)\leavevmode\nobreak\
d\theta_{i}\text{ with
}L_{i}(\theta_{i})=\dfrac{1}{\sqrt{2\pi}s_{i}}\exp\left(-\dfrac{(\hat{\theta}_{i}-\theta_{i})^{2}}{2s_{i}^{2}}\right),$
(7)

where $\phi(.)$ denotes the density of standard normal distribution.

##  Appendix B Review of The Copas-heckman-type selection model for PB in the
NN model

As mentioned in Section 3 of the main text, we review the original Copas-
Heckman-type selection model10 addressing PB in the NN model.

Based on the NN model (6), Copas and Shi10 considered the variance components
model for the outcome:

$\displaystyle\hat{\theta}_{i}=\theta+{(\sigma_{i}+\tau^{2})}^{1/2}\epsilon_{i},$

where $\epsilon_{i}\sim N(0,1)$ and
$\sigma_{i}^{2}=var(\hat{\theta}_{i}\mid\theta_{i})$ is the within-study
sampling variance. (Given large numbers of subjects in each study,
$\sigma_{i}^{2}$ can be replaced by $s_{i}^{2}$.) They introduced a latent
Gaussian variable $Z_{i}$ with the selection equation:

$\displaystyle Z_{i}=\gamma_{0}+\gamma_{1}/s_{i}+\delta_{i},$ (8)

where $\gamma_{0}$ and $\gamma_{1}$ are constants and $\delta_{i}\sim N(0,1)$;
study $i$ is published if and only if $Z_{i}>0$. A correlation was presented
to link publication with the outcome:

$\displaystyle\rho=corr(\epsilon_{i},\delta_{i}),$

When $\rho=0$, $\theta_{i}$ is independent of $\delta_{i}$, indicating that
study outcome does not influence its publication, and then there is no PB. A
probit model was employed to model the probability of selectively publishing
one study:

$\displaystyle P(Z_{i}>0\mid s_{i})=\Phi(\gamma_{0}+\gamma_{1}/s_{i}),$ (9)

where $\Phi(.)$ is the standard normal cumulative distribution function.
Probability (9) indicates that studies with small SE (large studies) are more
likely to be published.

Under selective publication of studies, the log-likelihood conditional on the
published studies was derived based on the NN model, as shown in equation (4)
in Copas and Shi:10

$\displaystyle\sum_{i=1}^{N}\log p(\hat{\theta}_{i}\mid z_{i}>0,s_{i})$
$\displaystyle=\sum_{i=1}^{N}\left\\{\log p(\hat{\theta}_{i})+\log
p(z_{i}>0\mid\hat{\theta}_{i},s_{i})-\log p(z_{i}>0\mid s_{i})\right\\}$
$\displaystyle=\sum_{i=1}^{N}\left\\{-\dfrac{1}{2}\log(\tau^{2}+\sigma_{i}^{2})-\dfrac{\hat{\theta}_{i}-\theta}{2(\tau^{2}+\sigma_{i}^{2})}-\log\Phi(\gamma_{0}+\gamma_{1}/s_{i})+\log\Phi(v_{i})\right\\}$

with

$\displaystyle
v_{i}=\dfrac{\gamma_{0}+\gamma_{1}/s_{i}+\tilde{\rho}_{i}(\hat{\theta}_{i}-\theta)/\sqrt{\tau^{2}+\sigma_{i}^{2}}}{\sqrt{1-\rho_{i}^{2}}}\text{
and }\tilde{\rho}_{i}=\dfrac{\sigma_{i}}{\sqrt{\tau^{2}+\sigma_{i}^{2}}}\rho.$

The parameters $(\theta,\tau,\rho)$ in the above likelihood can be estimated
by the maximum likelihood method; meanwhile, the number of unpublished studies
is estimated by $\sum_{i=1}^{N}\\{1-P(Z_{i}>0\mid s_{i})\\}/P(Z_{i}>0\mid
s_{i})$.

##  Appendix C The estimation results in Application

As mentioned in Section 3 of the main text, in Table 2, we presented the
estimates of parameters taking PB into account by the proposed methods and the
original Copas-Heckman-type selection model.

Table 2: Application in the main text: Summary of the estimations of different
sensitivity analysis methods

| Proposed method (HN-GLMM) | Proposed method (BN-GLMM) | Copas-Heckman-type selection model
---|---|---|---
$(P_{\text{min}},P_{\text{max}})$ | $M$ |  $\theta$ (95% CI) |  $\tau$ (SE) |  $\rho$ (SE) |  $\theta$ (95% CI) |  $\tau$ (SE) |  $\rho$ (SE) | $M$ |  $\theta$ (95% CI) |  $\tau$ (SE) |  $\rho$ (SE)
| 0 | -1.35 (-2.04, -0.67) | 0.83 (0.38) |  | -1.30 (-1.97, -0.64) | 0.78 (0.38) |  | 0 | -0.95 (-1.41, -0.49) | 0.00 (0.81) |
(0.9, 0.99) | 1 | -1.34 (-2.11, -0.56) | 0.84 (0.39) | -0.20 (2.24) | -1.29 (-2.04, -0.54) | 0.78 (0.39) | -0.20 (2.29) | 1 | -0.90 (-1.38, -0.42) | 0.00 (0.66) | -0.54 (0.61)
(0.8, 0.99) | 2 | -1.32 (-2.13, -0.51) | 0.84 (0.39) | -0.22 (1.66) | -1.28 (-2.06, -0.49) | 0.78 (0.39) | -0.22 (1.71) | 2 | -0.85 (-1.34, -0.37) | 0.00 (0.58) | -0.60 (0.46)
(0.7, 0.99) | 3 | -1.31 (-2.15, -0.47) | 0.84 (0.39) | -0.23 (1.39) | -1.26 (-2.08, -0.45) | 0.78 (0.39) | -0.23 (1.44) | 4 | -0.81 (-1.31, -0.31) | 0.00 (0.53) | -0.63 (0.40)
(0.6, 0.99) | 4 | -1.30 (-2.17, -0.42) | 0.84 (0.39) | -0.24 (1.21) | -1.25 (-2.09, -0.41) | 0.78 (0.39) | -0.23 (1.26) | 5 | -0.76 (-1.29, -0.24) | 0.00 (0.49) | -0.63 (0.38)
(0.5, 0.99) | 6 | -1.29 (-2.19, -0.38) | 0.84 (0.39) | -0.24 (1.08) | -1.24 (-2.11, -0.37) | 0.78 (0.39) | -0.24 (1.12) | 8 | -0.73 (-1.28, -0.17) | 0.00 (0.47) | -0.61 (0.36)
(0.4, 0.99) | 8 | -1.27 (-2.21, -0.34) | 0.84 (0.39) | -0.24 (0.97) | -1.23 (-2.13, -0.32) | 0.78 (0.39) | -0.24 (1.01) | 11 | -0.70 (-1.29, -0.12) | 0.00 (0.46) | -0.56 (0.35)
(0.3, 0.99) | 11 | -1.26 (-2.23, -0.29) | 0.84 (0.39) | -0.23 (0.87) | -1.21 (-2.15, -0.28) | 0.78 (0.39) | -0.23 (0.91) | 16 | -0.69 (-1.29, -0.08) | 0.00 (0.45) | -0.50 (0.32)
(0.2, 0.99) | 17 | -1.24 (-2.25, -0.23) | 0.84 (0.39) | -0.23 (0.78) | -1.20 (-2.17, -0.23) | 0.78 (0.39) | -0.23 (0.81) | 24 | -0.68 (-1.29, -0.06) | 0.00 (0.44) | -0.43 (0.29)
(0.1, 0.99) | 32 | -1.22 (-2.28, -0.16) | 0.84 (0.39) | -0.22 (0.67) | -1.18 (-2.20, -0.16) | 0.78 (0.39) | -0.22 (0.70) | 47 | -0.67 (-1.29, -0.05) | 0.00 (0.44) | -0.35 (0.24)

  * •

$M$ indicates the number of potentially unpublished studies; CI indicates the
confidence interval; SE indicates the standard error.

##  Appendix D New sensitivity analysis for PB in rare-event meta-analysis of
proportions

In Section 2-4 of the main text, we described the developed sensitivity
analysis method for evaluating the impact of PB in rare-event meta-analysis of
lnORs. As mentioned in Section 5, we introduce our sensitivity analysis method
for PB in the rare-event meta-analysis of proportions.

###  D.1 Likelihood conditional on the published studies

Suppose we are conducting the meta-analysis of proportions of event, and the
parameter of interest is the proportion $\pi$. Each study
$i\leavevmode\nobreak\ (i=1,\dots,N)$ reports the number of events $y_{i}$ and
the total number of subjects $n_{i}$. In the absence of PB, the generalized
linear mixed model (GLMM) works with the log odds
$\theta_{i}=\text{logit}(\pi_{i})=\log\dfrac{\pi_{i}}{1-\pi_{i}}$ as the
measurement of effect, where $\pi_{i}$ denotes the proportion of event in each
study. The GLMM uses the binomial distribution
$y_{i}\sim\text{Binomial}(n_{i},\pi_{i})$ as the within-study model.4 The
likelihood is written as

$\displaystyle\prod_{i=1}^{N}\int_{-\infty}^{\infty}L_{i}(\theta_{i})\dfrac{1}{\tau}\phi\left(\dfrac{\theta_{i}-\theta}{\tau}\right)d\theta_{i}\text{
with
}L_{i}(\theta_{i})=\binom{n_{i}}{y_{i}}\dfrac{\exp(\theta_{i})^{y_{i}}}{\\{1+\exp(\theta_{i})\\}^{n_{i}}}.$
(10)

To evaluate the impact of PB, we follow the approach in Section 3 of the main
text to construct the likelihood conditional on the published studies. The
likelihood (5) in the main text can be applied by replacing the within-study
likelihood $L_{i}(\theta_{i})$ with the one in likelihood (10).

###  D.2 Application

As illustration of the proposed method in rare-event meta-analysis of
proportions, we adopted the same data in Box 2 of the main text but only
analyze data in the AIT catheter group; the parameter of interest is the
logit-transformed proportion of CRBSI. Without considering PB, the logit-
proportion of events was estimated to be $-4.812\leavevmode\nobreak\
(\text{95\% CI:}-5.508\text{ to }-4.116)$ by the likelihood (10) of GLMM and
$-4.238\leavevmode\nobreak\ (-4.793\text{ to }-3.682)$ by the likelihood (7)
NN model.3 The same scenarios of sensitivity parameters in Section 4 of the
main text were considered to evaluate PB on the estimates of logit-transformed
proportion.

We presented the comparison of the proposed method and the Copas-Heckman-type
selection model in Figure 2. In the presence of potentially unpublished
studies, the estimated proportion still remained significantly different from
zero in both methods. The detailed estimated results of the proposed
sensitivity analysis method and the Copas-Heckman-type selection model are in
Table 3. For the proposed method, there was little change of the logit-
transformed proportion over $P_{\text{min}}$ ($x$-axis in Figure 2). On the
other hand, for the NN model, certain change was observed possibly due to the
influence of sparsity of data.

Figure 2: Comparison of the sensitivity analysis methods for PB in meta-
analysis of proportions. Table 3: Summary of the estimations of different
sensitivity analysis methods

| Proposed method (BN-GLMM) | Copas-Heckman-type selection model
---|---|---
$(P_{\text{min}},P_{\text{max}})$ | $M$ |  $\theta$ (95% CI) |  $\tau$ (SE) |  $\rho$ (SE) | $M$ |  $\theta$ (95% CI) |  $\tau$ (SE) |  $\rho$ (SE)
| 0 | -4.81 (-5.51, -4.12) | 0.91 (0.33) |  | 0 | -4.24 (-4.79, -3.68) | 0.58 (0.26) |
(0.9, 0.99) | 1 | -4.88 (-5.59, -4.17) | 0.94 (0.34) | 1.00 (NaN) | 1 | -4.14 (-4.70, -3.58) | 0.53 (0.26) | -0.71 (0.46)
(0.8, 0.99) | 2 | -4.93 (-5.65, -4.20) | 0.96 (0.35) | 1.00 (NaN) | 3 | -4.05 (-4.60, -3.49) | 0.48 (0.27) | -0.82 (0.29)
(0.7, 0.99) | 3 | -4.98 (-5.72, -4.24) | 0.98 (0.36) | 1.00 (NaN) | 5 | -3.95 (-4.49, -3.40) | 0.41 (0.28) | -0.88 (0.20)
(0.6, 0.99) | 4 | -5.03 (-5.79, -4.27) | 1.00 (0.36) | 1.00 (NaN) | 7 | -3.84 (-4.37, -3.31) | 0.34 (0.29) | -0.93 (8192.00)
(0.5, 0.99) | 6 | -5.09 (-5.88, -4.30) | 1.02 (0.37) | 1.00 (NaN) | 10 | -3.71 (-4.26, -3.16) | 0.25 (0.35) | -0.97 (8192.00)
(0.4, 0.99) | 8 | -5.16 (-5.99, -4.33) | 1.03 (0.38) | 1.00 (NaN) | 15 | -3.62 (-3.66, -3.58) | 0.00 (0.04) | -1.00 (8192.00)
(0.3, 0.99) | 11 | -5.15 (-6.54, -3.77) | 0.99 (0.45) | 0.83 (1.21) | 22 | -3.39 (-3.52, -3.27) | 0.00 (0.07) | -1.00 (8192.00)
(0.2, 0.99) | 17 | -5.15 (-6.28, -4.02) | 0.96 (0.37) | 0.69 (0.73) | 36 | -3.41 (-3.87, -2.95) | 0.00 (0.29) | -0.84 (0.11)
(0.1, 0.99) | 32 | -5.16 (-6.28, -4.04) | 0.93 (0.34) | 0.57 (0.60) | 76 | -3.43 (-3.88, -2.98) | 0.00 (0.30) | -0.71 (0.11)

  * •

$M$ indicates the number of potentially unpublished studies; CI indicates the
confidence interval; SE indicates the standard error; NaN indicates “not a
value”, an undefined value.


# Choi matrices revisited

Seung-Hyeok Kye  Department of Mathematics and Institute of Mathematics, Seoul
National University, Seoul 151-742, Korea  kye at snu.ac.kr

###### Abstract.

A linear map between matrix algebras corresponds to the Choi matrix in the
tensor product of two matrix algebras, whose definition depends on the matrix
units. Paulsen and Shultz [J. Math. Phys. 54 (2013), 072201] considered the
question if one can replace matrix units by another basis of matrix algebras
in the definition of Choi matrix to retain the correspondence between complete
positivity of maps and positivity of Choi matrices, and gave a sufficient
condition on basis under which this is true. In this note, we provide
necessary and sufficient conditions, to see that the Paulsen–Shultz condition
is also necessary.

###### Key words and phrases:

Jamiołkowski–Choi isomorphism, Choi matrix, matrix units, completely positive
maps

###### 1991 Mathematics Subject Classification:

15A30, 81P15, 46L05, 46L07

partially supported by NRF-2020R1A2C1A01004587, Korea

For a given linear map $\phi$ from the algebra $M_{n}$ of all $n\times n$
complex matrices into $M_{m}$ of $m\times m$ matrices, the Choi matrix ${\rm
C}_{\phi}$ is given by

(1) ${\rm C}_{\phi}=\sum_{i,j=1}^{n}e_{ij}\otimes\phi(e_{ij})\in M_{n}\otimes
M_{m},$

with the standard matrix units $\\{e_{ij}\\}$ of $M_{n}$. This had been
defined in 1967 by de Pillis [2] who had shown that $\phi$ preserves
Hermitianity if and only if ${\rm C}_{\phi}$ is Hermitian. During the
seventies, two results appeared which are now fundamental tools in current
quantum information theory; Jamiołkowski [4] gave a condition on ${\rm
C}_{\phi}$ under which $\phi$ is a positive linear map, which is now called
block positivity; Choi [1] showed that $\phi$ is completely positive if and
only if ${\rm C}_{\phi}$ is positive (semi-definite). The isomorphism
$\phi\mapsto{\rm C}_{\phi}$ is now called the Jamiołkowski–Choi isomorphism.

Paulsen and Shultz [6] asked what happens when we replace matrix units in (1)
by another basis ${\mathcal{B}}=\\{b_{ij}\\}$ of $M_{n}$. More precisely, they
considered

(2) ${\rm
C}^{\mathcal{B}}_{\phi}=\sum_{i,j=1}^{n}b_{ij}\otimes\phi(b_{ij})=({\text{\rm
id}}_{n}\otimes\phi)\left(\textstyle\sum_{i,j}b_{ij}\otimes b_{ij}\right)\in
M_{n}\otimes M_{m}$

and asked when the following statement

(3) $\phi:M_{n}\to M_{m}\ {\text{\rm is completely positive}}\
\Longleftrightarrow\ {\rm C}^{\mathcal{B}}_{\phi}\in M_{n}\otimes M_{m}\
{\text{\rm is positive}}$

is still true. For this purpose, they considered the map $C_{\mathcal{B}}$
which sends matrix units to ${\mathcal{B}}$; the $n^{2}\times n^{2}$ matrix
$[C_{\mathcal{B}}]$ which represents the map $C_{\mathcal{B}}$ with respect to
the matrix units with a fixed order; the transpose $[C_{\mathcal{B}}]^{T}$ of
the matrix $[C_{\mathcal{B}}]$; and finally the linear map
$M_{\mathcal{B}}:M_{n}\to M_{n}$ given by the $n^{2}\times n^{2}$ matrix
$[C_{\mathcal{B}}][C_{\mathcal{B}}]^{T}$. For a given $s\in M_{n}$, we define
the map $\operatorname{Ad}_{s}:M_{n}\to M_{n}$ by

$\operatorname{Ad}_{s}(x)=s^{*}xs,\qquad x\in M_{n}.$

Paulsen and Shultz [6] showed that if $M_{\mathcal{B}}=\operatorname{Ad}_{s}$
for some $s\in M_{n}$ then the statement (3) holds for every $m=1,2,\dots$. We
show that the converse is also true.

We note that the Choi matrix may be written by

(4) ${\rm C}_{\phi}=({\text{\rm
id}}_{n}\otimes\phi)\left(\textstyle\sum_{i,j}e_{ij}\otimes
e_{ij}\right)=({\text{\rm id}}_{n}\otimes\phi)(|\omega\rangle\langle\omega|),$

where
$|\omega\rangle=\sum_{i=1}^{n}|i\rangle|i\rangle\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}$.
Very recently, the author [5] considered the question what happens when we
replace $|\omega\rangle$ in (4) by another vector. He began with a nonsingular
matrix $s=\sum_{i,j=1}^{n}s_{ij}{|i\rangle\langle j|}\in M_{n}$ to associate
the vector $\langle\tilde{s}|=\sum_{i,j=1}^{n}s_{ij}\langle i|\langle
j|\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}$, and defined

(5) ${\rm C}^{s}_{\phi}=({\text{\rm
id}}_{n}\otimes\phi)(|\tilde{s}\rangle\langle\tilde{s}|),$

to show that $\phi$ is completely positive if and only if ${\rm C}^{s}_{\phi}$
is positive. We show that non-singularity of $s$ is necessary to retain the
correspondence between completely positivity of $\phi$ and positivity of ${\rm
C}^{s}_{\phi}$.

More generally, we will consider the question what happens when we replace the
matrix $\sum_{i,j}e_{ij}\otimes e_{ij}$ in (4) by another matrix $\Sigma\in
M_{n}\otimes M_{n}$, to define

(6) ${\rm C}^{\Sigma}_{\phi}=({\text{\rm id}}_{n}\otimes\phi)(\Sigma)\in
M_{n}\otimes M_{n}.$

We abuse notations in (2), (5) and (6). For a given basis
${\mathcal{B}}=\\{b_{ij}\\}$, we write
$\Sigma_{\mathcal{B}}=\sum_{i,j}b_{ij}\otimes b_{ij}$ then ${\rm
C}^{\mathcal{B}}_{\phi}$ in (2) is nothing but ${\rm
C}^{\Sigma_{\mathcal{B}}}_{\phi}$ defined in (6). The similar comment works
for (5) and (6).

If we take the linear map $\sigma:M_{n}\to M_{n}$ with $\Sigma={\rm
C}_{\sigma}$ then we have the relation

(7) ${\rm C}^{\Sigma}_{\phi}=({\text{\rm id}}_{n}\otimes\phi)({\rm
C}_{\sigma})={\rm C}_{\phi\circ\sigma},$

which plays an important role in quantum information theory. See [3, Section
4]. Now, we will say that $\Sigma\in M_{n}\otimes M_{n}$ satisfies the Choi
correspondence when complete positivity of a map $\phi:M_{n}\to M_{m}$ is
equivalent to positivity of ${\rm C}^{\Sigma}_{\phi}\in M_{n}\otimes M_{m}$
for every $m=1,2,\dots.$ We see that $\Sigma={\rm C}_{\sigma}$ satisfies the
Choi correspondence if and only if the following

(8) $\phi\in{{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{m}]\ \Longleftrightarrow\
\phi\circ\sigma\in{{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{m}]$

holds for every $m=1,2,\dots$ by (7), where
${{\mathbb{C}}{\mathbb{P}}}={{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{m}]$ denotes
the convex cone of all completely positive maps from $M_{n}$ into $M_{m}$.

Following [3], we use the bilinear pairing $\langle
X,Y\rangle=\operatorname{Tr}(XY^{\text{\rm t}})$ for matrices, and define the
bilinear pairing $\langle\phi,\psi\rangle$ of two linear maps
$\phi,\psi:M_{n}\to M_{m}$ by $\langle\phi,\psi\rangle=\langle{\rm
C}_{\phi},{\rm C}_{\psi}\rangle$. The map $\phi^{*}:M_{m}\to M_{n}$ is also
defined by $\langle\phi^{*}(x),y\rangle=\langle x,\phi(y)\rangle$. For a
closed convex cone $K$ of linear maps, we also define $K^{\circ}$ as the
convex cone consisting of all linear maps $\phi$ satisfying
$\langle\phi,\psi\rangle\geq 0$ for every $\psi\in K$. It is well known that
${{\mathbb{C}}{\mathbb{P}}}^{\circ}={{\mathbb{C}}{\mathbb{P}}}$. Then, we see
that $\phi\circ\sigma\in{{\mathbb{C}}{\mathbb{P}}}$ holds in (8) if and only
if $\langle\phi\circ\sigma,\tau\rangle\geq 0$ for every
$\tau\in{{\mathbb{C}}{\mathbb{P}}}$ if and only if
$\langle\phi,\tau\circ\sigma^{*}\rangle\geq 0$ for every
$\tau\in{{\mathbb{C}}{\mathbb{P}}}$ if and only if
$\phi\in({{\mathbb{C}}{\mathbb{P}}}\circ\\{\sigma^{*}\\})^{\circ}$. Since
${{\mathbb{C}}{\mathbb{P}}}^{\circ}={{\mathbb{C}}{\mathbb{P}}}$, we see that
(8) holds if and only if
${{\mathbb{C}}{\mathbb{P}}}={{\mathbb{C}}{\mathbb{P}}}\circ\\{\sigma^{*}\\}$
if and only if the following

(9)
${{\mathbb{C}}{\mathbb{P}}}[M_{m},M_{n}]=\\{\sigma\\}\circ{{\mathbb{C}}{\mathbb{P}}}[M_{m},M_{n}]$

holds for every $m=1,2,\dots$. Recall that a linear map $\sigma:M_{n}\to
M_{n}$ is called a complete order isomorphism if $\sigma$ is a bijection and
both $\sigma$ and $\sigma^{-1}$ are completely positive. When $m=n$, the
identity map belongs to ${{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{n}]$ in the
condition (9), and so we conclude that $\sigma$ is a complete order
isomorphism. Conversely, if $\sigma$ is a complete order isomorphism then (9)
holds for every $m=1,2,\dots$. A typical example of complete order isomorphism
is the map $\operatorname{Ad}_{s}:M_{A}\to M_{A}$ for a nonsingular $s\in
M_{A}$, with the inverse map $\operatorname{Ad}_{s^{-1}}$.

Theorem 1. Suppose that $\Sigma\in M_{n}\otimes M_{n}$ with $\Sigma={\rm
C}_{\sigma}$ for a map $\sigma:M_{n}\to M_{n}$. Then the following are
equivalent:

  1. (i)

$\Sigma$ satisfies the Choi correspondence,

  2. (ii)

$\phi\in{{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{m}]$ if and only if
$\phi\circ\sigma\in{{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{m}]$ for every
$m=1,2,\dots$,

  3. (iii)

${{\mathbb{C}}{\mathbb{P}}}[M_{m},M_{n}]=\\{\sigma\\}\circ{{\mathbb{C}}{\mathbb{P}}}[M_{m},M_{n}]$
for every $m=1,2,\dots$,

  4. (iv)

$\sigma$ is a complete order isomorphism,

  5. (v)

$\sigma=\operatorname{Ad}_{s}$ for a nonsingular matrix $s\in M_{n}$,

  6. (vi)

$\Sigma$ is a positive rank one matrix whose range vector has the full Schmidt
rank,

  7. (vii)

for any basis $\\{|\xi_{i}\rangle\\}$ of $\mathbb{C}^{n}$, there exists a
basis $\\{|\eta_{i}\rangle\\}$ of $\mathbb{C}^{n}$ such that
$\Sigma=\sum_{i,j=1}^{n}|\xi_{i}\rangle\langle\xi_{j}|\otimes|\eta_{i}\rangle\langle\eta_{j}|$,

  8. (viii)

$\Sigma=\sum_{i,j=1}^{n}|\xi_{i}\rangle\langle\xi_{j}|\otimes|\eta_{i}\rangle\langle\eta_{j}|$
for some bases $\\{|\xi_{i}\rangle\\}$ and $\\{|\eta_{i}\rangle\\}$ of
$\mathbb{C}^{n}$.

In [5], we have considered the case when $\Sigma$ is a positive rank one
matrix with the range vector
$|\tilde{s}\rangle\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}$, and showed that if
$s$ is non-singular then $\Sigma$ satisfies the Choi correspondence. The above
theorem tells us that the converse also holds.

So far, we have seen that the statements (i), (ii), (iii) and (iv) are
equivalent. It was shown in [6] that (iv) implies (v). We show here in another
way. Suppose that $\sigma=\sum_{i\in I}\operatorname{Ad}_{s_{i}}$ is a
complete order isomorphism with the completely positive inverse
$\sigma^{-1}=\sum_{j\in J}\operatorname{Ad}_{t_{j}}$. Then we have
$\sum_{i,j}\operatorname{Ad}_{t_{j}s_{i}}=\sum_{i,j}\operatorname{Ad}_{s_{i}}\circ\operatorname{Ad}_{t_{j}}={\text{\rm
id}}$. Because the identity map generates an extreme ray of the convex cone
${{\mathbb{C}}{\mathbb{P}}}[M_{n},M_{n}]$, there exist positive numbers
$\alpha_{(i,j)}$ such that
$\alpha_{(i,j)}\operatorname{Ad}_{t_{j}s_{i}}={\text{\rm id}}$ for $(i,j)\in
I\times J$, and so, we have $\sqrt{\alpha_{(i,j)}}z_{(i,j)}{t_{j}s_{i}}=I$ for
every $(i,j)\in I\times J$, with complex numbers $z_{(i,j)}$ of modulus one.
Therefore, we see that all $s_{i}$ coincide up to scalar multiplications.
Furthermore, the invertibility of $\operatorname{Ad}_{s}$ implies non-
singularity of $s$. This shows that (iv) and (v) are equivalent. By the
relation ${\rm C}_{\operatorname{Ad}_{s}}=|\tilde{s}\rangle\langle\tilde{s}|$
in [3], we also have (v) $\Longleftrightarrow$ (vi). For a given basis
$\\{|\xi_{i}\rangle\\}$ of $\mathbb{C}^{n}$, every vector in
$\mathbb{C}^{n}\otimes\mathbb{C}^{n}$ is of the form
$\sum_{i=1}^{n}|\xi_{i}\rangle\otimes|\eta_{i}\rangle$ for a family
$\\{|\eta_{i}\rangle\\}$ of vectors, which has the full Schmidt rank if and
only if $\\{|\eta_{i}\rangle\\}$ is a basis. This shows the direction (vi)
$\Longrightarrow$ (vii). The remaining implications (vii) $\Longrightarrow$
(viii) $\Longrightarrow$ (vi) are clear.

We note that $\phi\mapsto{\rm C}^{\Sigma}_{\phi}$ is a bijection if and only
if $\phi\mapsto\phi\circ\sigma$ is a bijection between the linear mapping
space $L(M_{n},M_{m})$ if and only if the map $\sigma\in L(M_{n},M_{n})$ is
invertible. When $\sigma=\operatorname{Ad}_{s}$, we also see that $\sigma$ is
invertible if and only if $s$ is nonsingular. Therefore, we see that the Choi
correspondence of $\Sigma$ implies that $\phi\mapsto{\rm C}^{\Sigma}_{\phi}$
is a bijection.

Considering the condition in [6], we suppose that ${\mathcal{B}}=\\{b_{j}:j\in
J\\}$ is an ordered family of $n\times n$ matrices with the cardinality
$n^{2}$, and define the linear map $C_{\mathcal{B}}:M_{n}\to M_{n}$ by
$b_{j}=C_{\mathcal{B}}(e_{j})=\sum_{i\in J}x_{ij}e_{i}$ for $j\in J$, with an
ordered matrix unit $\\{e_{j}:j\in J\\}$. By the relation
$C_{\mathcal{B}}^{T}(e_{j})=\sum_{i\in J}x_{ji}e_{i}$, we have

$M_{\mathcal{B}}(e_{j})=C_{\mathcal{B}}C_{\mathcal{B}}^{T}(e_{j})=\sum_{i\in
J}x_{ji}C_{\mathcal{B}}(e_{i})=\sum_{i\in J}x_{ji}\sum_{k\in J}x_{ki}e_{k}.$

Therefore, it follows that

$\displaystyle\sum_{j\in J}e_{j}\otimes M_{\mathcal{B}}(e_{j})$
$\displaystyle=\sum_{j\in J}\sum_{i\in J}\sum_{k\in J}x_{ji}x_{ki}e_{j}\otimes
e_{k}$ $\displaystyle=\sum_{i\in J}\left(\sum_{j\in
J}x_{ji}e_{j}\right)\otimes\left(\sum_{k\in J}x_{ki}e_{k}\right)=\sum_{i\in
J}b_{i}\otimes b_{i}.$

Let ${\mathcal{B}}=\\{b_{ij}:i,j=1,2,\dots,n\\}$ be a family of matrices in
$M_{n}$. If we put

$\Sigma=\sum_{i,j=1}^{n}b_{ij}\otimes b_{ij}=\sum_{i,j=1}^{n}e_{ij}\otimes
M_{\mathcal{B}}(e_{ij})={\rm C}_{M_{\mathcal{B}}},$

then ${\rm C}^{\Sigma}_{\phi}$ in (6) is nothing but ${\rm
C}^{\mathcal{B}}_{\phi}$ defined in (2). Therefore, the statement (3) holds
for every $m=1,2,\dots$ if and only if $M_{\mathcal{B}}=\operatorname{Ad}_{s}$
for a nonsingular $s\in M_{n}$ if and only if $\sum_{i,j=1}^{n}b_{ij}\otimes
b_{ij}=\sum_{i,j=1}^{n}|\xi_{i}\rangle\langle\xi_{j}|\otimes|\eta_{i}\rangle\langle\eta_{j}|$
for some bases $\\{|\xi_{i}\rangle\\}$ and $\\{|\eta_{i}\rangle\\}$ if and
only if there exists a basis $\\{|\zeta_{i}\rangle\\}$ such that
$b_{ij}=|\zeta_{i}\rangle\langle\zeta_{j}|$ for each $i,j=1,2,\dots,n$. In
case when $\\{b_{ij}\\}$ is a basis of $M_{n}$, the matrix $s$ in this
discussion is automatically non-singular, by the invertibility of
$M_{\mathcal{B}}$. We summarize as follows:

Theorem 2. For a basis ${\mathcal{B}}=\\{b_{ij}\\}$ of $M_{n}$, the following
are equivalent:

  1. (i)

$\phi:M_{n}\to M_{m}$ is completely positive if and only if ${\rm
C}^{\mathcal{B}}_{\phi}\in M_{n}\otimes M_{m}$ is positive,

  2. (ii)

$M_{\mathcal{B}}=\operatorname{Ad}_{s}$ for a nonsingular matrix $s\in M_{n}$,

  3. (iii)

$M_{\mathcal{B}}=\operatorname{Ad}_{s}$ for a matrix $s\in M_{n}$,

  4. (iv)

there exists a basis $\\{|\zeta_{i}\rangle\\}$ of $\mathbb{C}^{n}$ such that
$b_{ij}=|\zeta_{i}\rangle\langle\zeta_{j}|$ for $i,j=1,2,\dots,n$.

The author is grateful to Kyung Hoon Han for helpful coments on the draft.

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