# Using meta-analytic priors to incorporate external information for study
evaluation

Thilo Welz Department of Statistics, TU Dortmund University, Fraunhofer Str.
2-4, 44227 Dortmund, Germany Eric Knop Department of Statistics, TU Dortmund
University, Fraunhofer Str. 2-4, 44227 Dortmund, Germany Department of
Statistics, TU Dortmund University, Fraunhofer Str. 2-4, 44227 Dortmund,
Germany Frank Konietschke Institut für Biometrie und Klinische Epidemiologie,
Charité - Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany
Institut für Biometrie und Klinische Epidemiologie, Charité -
Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany Jan-Hendrik
B. Hardenberg Institut für Biometrie und Klinische Epidemiologie, Charité -
Universitätsmedizin Berlin, Charitéplatz 1, 10117 Berlin, Germany Institut für
Biometrie und Klinische Epidemiologie, Charité - Universitätsmedizin Berlin,
Charitéplatz 1, 10117 Berlin, Germany Markus Pauly Department of Statistics,
TU Dortmund University, Fraunhofer Str. 2-4, 44227 Dortmund, Germany UA Ruhr,
Research Center Trustworthy Data Science and Security, 44227, Dortmund,
Germany Department of Statistics, TU Dortmund University, Fraunhofer Str. 2-4,
44227 Dortmund, Germany UA Ruhr, Research Center Trustworthy Data Science and
Security, 44227, Dortmund, Germany Christian Röver Universitätsmedizin
Göttingen, Institut für Medizinische Statistik, Humboldtallee 32, 37073
Göttingen, Germany

(December 2022)

###### Abstract

Background The COVID-19 pandemic has had a profound impact on health, everyday
life and economics around the world. An important complication that can arise
in connection with a COVID-19 infection is acute kidney injury. A recent
observational cohort study of COVID-19 patients treated at multiple sites of a
tertiary care center in Berlin, Germany identified risk factors for the
development of (severe) acute kidney injury. Since inferring results from a
single study can be tricky, we validate these findings and potentially adjust
results by including external information from other studies on acute kidney
injury and COVID-19.

Methods We synthesize the results of the main study with other trials via a
Bayesian meta-analysis. The external information is used to construct a
predictive distribution and to derive posterior estimates for the study of
interest. We focus on various important potential risk factors for acute
kidney injury development such as mechanical ventilation, use of vasopressors,
hypertension, obesity, diabetes, gender and smoking.

Results Our results show that depending on the degree of heterogeneity in the
data the estimated effect sizes may be refined considerably with inclusion of
external data. Our findings confirm that mechanical ventilation and use of
vasopressors are important risk factors for the development of acute kidney
injury in COVID-19 patients. Hypertension also appears to be a risk factor
that should not be ignored. Shrinkage weights depended to a large extent on
the estimated heterogeneity in the model.

Conclusions Our work shows how external information can be used to adjust the
results from a primary study, using a Bayesian meta-analytic approach. How
much information is borrowed from external studies will depend on the degree
of heterogeneity present in the model.

##  1 Introduction

Over the past years the global COVID-19 pandemic, caused by the SARS-CoV-2
virus, has had far reaching and damaging effects on society, both in a health
and economic sense (Padhan and Prabheesh,, 2021). One important health
complication that can arise in connection with COVID-19 is acute kidney injury
(AKI) (Batlle et al.,, 2020). Reported rates of AKI suggest the kidney could
be the second most affected organ in patients with COVID-19 (Gabarre et al.,,
2020; Hardenberg et al.,, 2021). However, the underlying mechanisms are not
yet fully explored (Siew and Birkelo,, 2020). A recent observational cohort
study of COVID-19 patients treated at multiple sites of a tertiary care center
in Berlin, Germany observed the development of severe AKI in 70 out of 223
patients (Hardenberg et al.,, 2021). Patients with severe AKI were older,
mostly male and also had more comorbidities as well as excess mortality
Hardenberg et al., (2021).

Inferring results from a single study however can be tricky, especially if the
sample size is small. Researchers may be faced with one or several samples
that have been gathered without necessarily being powered to achieve a certain
pre-specified power. The overall aim for such studies may be rather
exploratory in nature. This is the present context for a recent study on AKI
in COVID-19 patients Hardenberg et al., (2021). Here, a tempting approach is
to include external information from other studies. This is also known as
borrowing of strength (Röver and Friede,, 2020). A natural way to facilitate
this is the application of Bayesian methods (Schmidli et al.,, 2014). These
allow to synthesize external studies via meta-analysis and to derive a
predictive distribution for the main effect of a new study. We refer to this
as the meta-analytic-predictive (MAP) prior (McKinney et al.,, 2021; Schmidli
et al.,, 2014). The MAP prior can be used as a prior for the new study in
order to derive a posterior distribution via Bayes’ theorem. This is
equivalent to a meta-analytic-combined (MAC) approach, where all studies
(including the new one) are meta-analyzed and a shrinkage estimate is
subsequently calculated for the new study (Röver et al.,, 2019). This
shrinkage estimate will be a weighted average between the meta-analytic
estimate of the overall effect and the study-specific estimate. Whether the
shrinkage estimate lies closer to the study-specific estimate or the meta-
analytic estimate of the overall effect depends to a large degree on the
heterogeneity present in the data. As highlighted in Röver and Friede, (2021),
it is even possible to calculate bounds for the “shrinkage weights”, i.e. the
contribution of the study of focus to its own shrinkage estimate. We follow a
MAC approach, in order to validate and potentially adjust the results in
Hardenberg et al., (2021). We do this by incorporating study results from Cai
et al., (2021), who conducted a systematic review and meta-analysis of
(potential) risk factors for AKI in adult patients with COVID-19. Whereas
Hardenberg et al., (2021) mainly considered longitudinal data and temporal
associations with disease course, we focus on baseline characteristics, due to
a lack of comparable studies with longitudinal data.

In the following Section 2, we first review the standard normal-normal
hierarchical model (NNHM) for meta-analysis in Subsection 2.1. This is
followed by a description of posterior distributions in Subsection 2.2 and a
description of the shrinkage estimates of study-specific means in Subsection
2.3. In Section 3 we introduce research on the occurrence of acute kidney
injury and possible contributing factors in COVID-19 patients. We apply the
described Bayesian methodology on data regarding acute kidney injury in
COVID-19 patients from Hardenberg et al., (2021) and Cai et al., (2021) and
present the results. We close with a discussion and an outlook in Section 4.

##  2 Methods

###  2.1 The normal-normal hierarchical model

We briefly describe the standard NNHM, which can be used to address a wide
range of problems and refer to Borenstein et al., (2021) for more details. In
the NNHM, study-level effects are modeled using normal distributions, while
heterogeneity is modeled as an additive normal variance component on a second
hierarchy level. Given $k$ studies with their respective effects $y_{i}$ and
(assumed known) sampling variances $\sigma_{i}^{2}$, with $i=1,\ldots,k$, the
goal is to infer an overall effect $\mu$. The $y_{i}$ are estimates of $\mu$,
which are assumed to be normally distributed around the $i$th study’s true
effect $\theta_{i}$: $y_{i}\sim\mathcal{N}(\theta_{i},\sigma_{i}^{2})$. In
random-effects meta-analysis the study level effects $\theta_{i}$ are assumed
to vary normally around the overall mean $\mu$ with heterogeneity variance
$\tau^{2}$, i.e. $\theta_{i}\sim\mathcal{N}(\mu,\tau^{2})$. The marginal
formulation of this model is

$y_{i}\sim\mathcal{N}(\mu,\sigma_{i}^{2}+\tau^{2}),$ (1)

which may be attractive if the study-specific parameters $\theta_{i}$ are only
of secondary interest (Borenstein et al.,, 2021).

###  2.2 Posterior Distributions

To simplify notation we set
$\boldsymbol{y}\coloneqq(y_{1},\ldots,y_{k})^{\prime}$ for the observed effect
sizes and
$\boldsymbol{\sigma^{2}}\coloneqq(\sigma_{1}^{2},\ldots,\sigma_{k}^{2})^{\prime}$
for the respective sampling variances, pooled from our $k$ studies. Given an
improper uniform or normal prior for the effect $\mu$, the effect’s
conditional posterior distribution for a given heterogeneity $\tau^{2}$,
$p(\mu\mid\tau^{2},\boldsymbol{\sigma^{2}},\boldsymbol{y})$, is again normal
(Röver,, 2020). In case of an improper uniform prior we obtain
$\hat{\mu}=\frac{\sum_{i=1}^{k}w_{i}y_{i}}{\sum_{i=1}^{k}w_{i}}$ and
$\hat{\sigma}=1/\sqrt{\sum_{i=1}^{k}w_{i}}$ as posterior estimators for the
mean and the standard deviation, respectively, where
$w_{i}=(\sigma_{i}^{2}+\tau^{2})^{-1}$. In case of a normal prior for $\mu$
with mean $\mu_{p}$ and variance $\sigma_{p}^{2}$ we obtain posterior mean
$\hat{\mu}=\frac{\mu_{p}/\sigma_{p}^{2}+\sum_{i=1}^{k}w_{i}y_{i}}{\sigma_{p}^{-2}+\sum_{i=1}^{k}w_{i}}$
and standard deviation
$\hat{\sigma}=1/\sqrt{\sigma_{p}^{-2}+\sum_{i=1}^{k}w_{i}}$ (Gelman et al.,,
2014; Röver,, 2020).

To additionally estimate study specific means we consider shrinkage estimates.

###  2.3 Shrinkage Estimates of study-specific means

In many applications the main interest focuses on the overall mean estimate of
$\mu$. However, here we are interested in estimating the study-specific
parameter $\theta_{i}$, which is informed by both the $i$-th study estimate as
well as the estimated overall mean effect and the between-study heterogeneity.
This means we are interested in inferring the posterior distribution of the
study-specific effect $\theta_{i}$ for study $i$ with $i=1,\ldots,k$. In the
following we assume the standard NNHM model. Thus, conditional on a
heterogeneity variance of $\tau^{2}$, the posterior distribution of
$\theta_{i}$ is normal with moments given by

$\displaystyle\mathbb{E}[\theta_{i}\mid\tau^{2},\boldsymbol{\sigma^{2}},\boldsymbol{y}]$
$\displaystyle=\frac{y_{i}/\sigma_{i}^{2}+\hat{\mu}(\tau^{2})/\tau^{2}}{\sigma_{i}^{-2}+\tau^{-2}},$
(2) $\displaystyle
Var(\theta_{i}\mid\tau^{2},\boldsymbol{\sigma^{2}},\boldsymbol{y})$
$\displaystyle=\frac{1}{\sigma_{i}^{-2}+\tau^{-2}}+\left(\frac{\tau^{-2}}{\sigma_{i}^{-2}+\tau^{-2}}\hat{\sigma}\right)^{2},$
(3)

where $\hat{\mu}(\tau^{2})$ is the estimated summary effect and $\hat{\sigma}$
the estimated standard deviation of $\hat{\mu}$ (Gelman et al.,, 2014). These
formulas illustrate the pull of the $i$th study effect estimate toward the
estimated population mean. Hence the term “shrinkage estimate”. Although the
posterior distribution of $\theta_{i}$ conditional on $\tau^{2}$ is normal, in
practice the amount of heterogeneity $\tau^{2}$ is unknown and must be
estimated. The resulting posterior distribution for $\theta_{i}$ will be a
mixture distribution (Röver,, 2020), which we denote as $F_{i}$ for study $i$.
Two possibilities for constructing confidence intervals (CIs) are (a) define a
$(1-\alpha)$-CI via $[c_{i,\alpha/2},c_{i,1-\alpha/2}]$, where
$c_{i,\alpha/2}$ is the $\alpha/2$ quantile of $F_{i}$ or (b) numerically
determine a “shortest” interval, which for unimodal posteriors is equal to the
highest posterior density region (Gelman et al.,, 1995). We apply approach (b)
using the R package bayesmeta, in order to modify results from a recent study
on acute kidney injury in COVID-19 patients by Hardenberg et al., (2021).

##  3 Acute Kidney Injury and COVID-19

AKI is a relevant complication in COVID-19. Some studies have reported that
the incidence of AKI in COVID-19 patients is more than twice the incidence of
AKI in non COVID-19 patients (Cai et al.,, 2021). Furthermore, evidence
suggests that patients suffering from AKI experience substantially increased
in-hospital mortality compared with those without AKI. However, the underlying
relationships are not fully known. In order to shed some light on these
mechanisms, an observational cohort study of 223 COVID-19 patients treated at
three sites of a tertiary care center in Berlin, Germany at the Charité-
Universitätsmedizin Berlin was performed. Both descriptive statistics and
multivariable Cox regression models with time-varying covariates were used to
identify risks factors of severe AKI (Hardenberg et al.,, 2021). The latter
considered time-to-severe AKI as the primary endpoint with prior death as a
competing risk. Their study will be our main focus in the application of the
methodology described in Section 1.

###  3.1 Risk factors for AKI development in COVID-19 Patients

Cai et al., (2021) conducted a systematic review and meta-analysis of
(potential) risk factors for AKI in adult patients with COVID-19. They
extracted their data based on a systematic literature search in PubMed,
EMBASE, Web of Science, the Cochrane Library, CNKI, VIP and WanFang data
between December 1, 2019 and January 30, 2021 (Cai et al.,, 2021). We use
their data and results to add to the findings in Hardenberg et al., (2021).
For this analysis we additionally follow Hardenberg et al., (2021) with
respect to variable selection, i.e. with respect to selecting the main
potential risk factors for AKI development. Their results are based on a
competing risks model with time-dependent covariates. AKI was diagnosed
according to KDIGO guidelines in all considered studies. We will consider the
following (potential) risk factors for the development of AKI in COVID-19
patients: obesity (BMI $>30$), gender, smoking, hypertension, diabetes,
mechanical ventilation (MV) and the intake of vasopressors. The latter two
were deemed especially important risk factors in Hardenberg et al., (2021).
For all (binary) risk factors, we compare the AKI prevalence (any of the three
stages vs. none) in the corresponding patient subgroups for each study, as in
Cai et al., (2021). We note that this is in contrast to the study by
Hardenberg et al., (2021), who considered severe AKI, i.e. stage 3 AKI, as
primary event of interest. However, the studies collected by Cai et al.,
(2021) all compared any stage AKI with none. The effect measures are log-odds
ratios (ORs).

###  3.2 Results

In the following Figures 1–3 (5–8 in the Appendix) we show forest plots for
each risk factor separately. Displayed are the study-specific effects and
their shrinkage estimates, as well as their corresponding $95\%$ CIs. In each
instance the final data row, with shrinkage estimate highlighted in red,
represents the main study of interest by Hardenberg et al., (2021).
Additionally the estimated population means and prediction intervals are
provided along with an estimate of $\tau$, the square root of the between-
study heterogeneity variance (also with a CI). We will now consider different
potential risk factors for the development of AKI in COVID-19 patients
successively. Here, $\hat{\mu}$ describes the meta-analytic summary estimate
of the effect of the respective risk factor.

Mechanical ventilation (MV) is a statistically significant potential risk
factor with a study specific estimate of 3.488 with a $95\%$ CI of
[2.686,4.289] and the shrinkage estimate is 3.271 with a $95\%$ CI of
[2.529,4.017]. MV was pointed out as one of the main risk factors for AKI in
COVID-19 patients in Hardenberg et al., (2021). The $\hat{\mu}$ refers to the
estimated mean log-OR across all studies. Meta-analysis yields an estimated
log-OR of $\hat{\mu}=2.215$ and $95\%$ CI for $\mu$ of [1.799,2.630]. The
estimated heterogeneity is $\hat{\tau}\approx 0.920$.

The use of vasopressors is also a statistically significant potential risk
factor with a study specific estimate of 5.095 with a $95\%$ CI of
[4.076,6.113]. The respective shrinkage estimate is 4.665 with a $95\%$ CI of
[3.676,5.659]. Like MV, vasopressors were pointed out as one of the main risk
factors for AKI in COVID-19 patients in Hardenberg et al., (2021), based on a
competing risks model with time dependent covariates. Based on a meta-
analysis, the estimated log-OR was $\hat{\mu}=2.295$ with a $95\%$ CI for
$\mu$ of [1.558,3.015]. The estimated heterogeneity is $\hat{\tau}\approx
1.250$.

Moreover, hypertension is also a statistically significant risk factor with a
study specific estimate for the study by Hardenberg et al., (2021) of 1.352
and a corresponding $95\%$ CI of [0.795,1.909]. The respective shrinkage
estimate is 1.053 with a $95\%$ CI of [0.601,1.519]. Meta-analysis yields an
estimated log-OR of $\hat{\mu}=0.687$ with a $95\%$ CI for $\mu$ of
[0.535,0.855]. The estimated heterogeneity is $\hat{\tau}\approx 0.330$.

We highlight the differing amounts of heterogeneity in the models for the
respective risk factors. For MV, vasopressors and hypertension, $\tau$ was
estimated at 0.920, 1.250 and 0.330 respectively. Therefore, for MV and
vasopressors the shrinkage estimates are very close to the study-specific
estimates by Hardenberg et al., (2021), as shown in Figures 1 and 2. For
hypertension, shown in Figure 3, where the estimated heterogeneity is
considerably smaller, the summary estimate of $\mu$ has a much larger weight,
leading to a more substantial difference between the shrinkage and study-
specific estimate. This also becomes evident when one considers the numerator
in expression (2). Röver and Friede, (2021) showed that a study’s minimum
contribution to its own shrinkage estimate is its fixed-effect weight, i.e.
the study weight for $\tau^{2}=0$ (the fixed-effects model). The study’s
contribution then increases monotonically, as $\tau^{2}$ increases. For
$\tau^{2}\rightarrow\infty$, the shrinkage estimate of the $i$th study
converges to the study’s effect estimate $y_{i}$ with $i=1,\ldots,k$.

For the forest plots showing the results of gender, smoking, diabetes and
obesity, we refer to the Appendix. These were non-significant potential risk
factors in the competing risks model by Hardenberg et al., (2021), see their
Figure 1. As a reminder to the reader, the effect sizes in the time-to-event
competing risks model by Hardenberg et al., (2021) are cause-specific hazard
ratios. “The cause‐specific hazard ratio denotes the relative change in the
instantaneous rate of the occurrence of the primary event in subjects who are
currently event‐free” (Austin and Fine,, 2017). On the other hand, for the
more simple group comparisons we consider here, the effects are log-odds
ratios. Therefore, there are some important differences that should be kept in
mind. For one, the cause-specific hazard ratios incorporate a time component,
which the ORs do not. Furthermore, the ORs are unadjusted effects, as we are
dealing with simple group comparisons. The cause-specific hazard ratios
however, are adjusted effects as they are part of a model that includes
various other covariates. Even though gender, smoking, diabetes and obesity
were non-significant risk factors respectively in the competing risks model by
Hardenberg et al., (2021) with regards to the cause-specific hazard ratios,
the estimated overall ORs for gender, smoking and diabetes were in fact
statistically significant.

Smoking is a statistically significant potential risk factor with
$\hat{\mu}=0.244$ and a $95\%$ CI for $\mu$ of [0.011,0.557]. The study
specific estimate for Hardenberg et al., (2021) is 0.903 with a $95\%$ CI of
[0.189,1.616]. The corresponding shrinkage estimate is 0.441 with a $95\%$ CI
of [0.031,1.041]. The estimated heterogeneity is $\hat{\tau}\approx 0.210$.

Diabetes is a statistically significant potential risk factor for AKI in
COVID-19 patients with an estimated overall effect of $\hat{\mu}=0.548$ and a
95% CI for $\mu$ of [0.480,0.610]. The study-specific log odds ratio from
Hardenberg et al., (2021) is 0.694 with a 95% CI of [0.049,1.340]. The
corresponding shrinkage estimate is 0.555 with a 95% CI of [0.389,0.718]. The
estimated heterogeneity is very small with $\hat{\tau}\approx 0.045$.

Overall, gender is a statistically significant potential risk factor for AKI
in COVID-19 patients with $\hat{\mu}=0.432$ and a $95\%$ CI for the main
effect $\mu$ of [0.304,0.567]. The study-specific data from Hardenberg et al.,
(2021) yielded an estimated log odds ratio 0.634 with a corresponding $95\%$
CI of [0.073,1.195]. The respective shrinkage estimate is 0.527 with a $95\%$
CI of [0.134,0.927]. The estimated heterogeneity is $\hat{\tau}\approx 0.270$.

Obesity was a special case, as the study by Hardenberg et al., (2021)
presented a considerable outlier compared with the other considered studies
(cf. Figure 6). Even though the overall OR of $\hat{\mu}=0.283$ was not
statistically significant with a 95% CI of [-0.113,0.718], the study-specific
estimate of the OR for obesity was 1.955 with a 95% CI of [1.263,2.648]. The
resulting shrinkage estimate is 1.383 with a 95% CI of [0.567,2.178]. The
estimated heterogeneity is $\hat{\tau}\approx 0.520$.

Finally, we present the relative shrinkage weights of the study Hardenberg et
al., (2021) for each potential risk factor respectively in Table 1. The
shrinkage weights lie between 0.856 for vasopressors and 0.054 for diabetes.
Here a value of one would mean the shrinkage estimate is exactly equal to the
study-specific estimate and a value of zero would mean the shrinkage estimate
is exactly equal to the meta-analytic summary estimate $\hat{\mu}$. We remind
the reader that as the heterogeneity variance $\tau^{2}$ increases
(decreases), the shrinkage weight of the target study also increases
(decreases).

Table 1: Shrinkage weights of the Hardenberg et al., (2021) study for each
potential risk factor. Contribution of Hardenberg et al., (2021) to shrinkage
estimate
---
Risk factor | Shrinkage weight
Mechanical Ventilation | 0.837
Vasopressors | 0.856
Hypertension | 0.566
Obesity | 0.685
Diabetes | 0.054
Gender | 0.485
Smoking | 0.328

Figure 1: Forest plot and shrinkage estimates for the risk factor mechanical
ventilation.

Figure 2: Forest plot and shrinkage estimates for the risk factor
vasopressors.

Figure 3: Forest plot and shrinkage estimates for the risk factor
hypertension.

##  4 Discussion

In this manuscript we demonstrated a Bayesian approach that allows researchers
to construct predictive prior distributions based on meta-analyses, yielding
posterior distributions and shrinkage estimates for a study of primary
interest, which are quantified in the light of external evidence. This
approach can for example be easily implemented in R using the bayesmeta
package (Röver,, 2020). We apply the discussed approach for a study by
Hardenberg et al., (2021), who analyze potential risk factors for the
development of severe AKI in COVID-19 patients. We expand on their evidence
with a meta-analysis on AKI in COVID-19 by Cai et al., (2021). Although
Hardenberg et al., (2021) focus on time-to-event data and implement competing
risks models, due to unavailability of comparable studies, we focus on more
simple effects as published by Cai et al., (2021). The results confirm that
both mechanical ventilation and use of vasopressors are important risk factors
for the development of AKI in COVID-19 patients. Hypertension also appears to
be a risk factor for AKI development that should not be ignored. Furthermore,
our analysis of ORs supports that smoking, diabetes and male gender are
additional risk factors for AKI development in COVID-19 patients that should
be taken into account.

As demonstrated in our analysis, there is a clear connection between the
amount of heterogeneity in the meta-analytic model and the amount of borrowing
of strength that takes place from external (relative to the primary study of
interest) information. This becomes evident, when the relative shrinkage
weights for each potential risk factor from Table 1 are compared with the
respective heterogeneity variances. Mechanical ventilation and vasopressors,
which have rather large estimated heterogeneity variances with
$\hat{\tau}^{2}$ equal to 0.846 and 1.563 respectively, have the largest
shrinkage weights with 0.837 and 0.856. Diabetes on the other hand has a
heterogeneity of only $\hat{\tau}^{2}\approx 0.002$ with a corresponding
shrinkage weight of 0.054.

A limitation of this work is that we only applied the presented Bayesian
approach for the normal-normal hierarchical model as we did not have access to
or information on competing risks or covariates in other comparable
studies/analyses. Thus, a more sophisticated analysis, e.g. in the context of
a meta-regression, was not feasible. The present study was facilitated by
having access to the original data from Hardenberg et al., (2021), as this
allowed us to compute estimates corresponding to those discussed in the meta-
analyses by Cai et al., (2021). Analyses may be more difficult if researchers
do not have access to the original data from their primary study of interest.

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## Competing interests

The authors declare that they have no competing interests.

## Author’s contributions

TW performed the data analysis and took charge of manuscript writeup. EK
assisted in the research phase and data wrangling. FK was helpful in early
discussions and with setting up contacts and obtaining relevant data. JH
conducted the main study of focus on acute kidney injury in COVID-19 patients
and provided medical expertise. MP provided helpful feedback during conception
and manuscript writeup. CR advised on the Bayesian methodology and
implementation in R. All authors read and approved the final manuscript.

## Funding

The work of Christian Röver, Thilo Welz and Markus Pauly was supported by a VW
Corona Crisis and Beyond Grant. The funders had no role in data collection,
analysis or preparation of the manuscript.

## Availability of data and materials

The R code will be made publicly available at the repository
https://osf.io/4rz98/, pending publication. All relevant data are quoted in
figures/tables.

## Appendix

## Time-to-acute-kidney-injury-development in COVID-19 patients

Hardenberg et al., (2021) implemented multivariable Cox proportional hazards
regression models with time-varying covariates. More specifically, they fit
two competing risks models with time-varying covariables: a main model with 11
covariates and a reduced model with 8 covariates. The main event of interest
was time to severe AKI (AKI stage 3) with prior death as the competing risk.
The respective cumulative incidence functions are shown in Figure 4(a),
similar to Supplemental Figure 2 in Hardenberg et al., (2021). The results are
based on $n=223$ patients, 70 of which experienced severe AKI during hospital
admission with 9 patients dying without prior severe AKI (the competing
event). Additionally, Figure 4(b) shows the cumulative incidence functions
stratified by gender.

(a) Cumulative incidence of AKI stage 3 and death without prior AKI stage 3 in
COVID-19 patients.

(b) Cumulative incidence of AKI stage 3 and death without prior AKI stage 3 in
COVID-19 patients stratified by gender.

Hardenberg et al., (2021) used these competing risks models to estimate cause-
specific hazard ratios for potential risk factors for the development of
severe AKI, along with $95\%$ CIs. Their results indicate that MV and a need
for vasopressors are especially strong potential risk factors for AKI
development in COVID-19 patients. For MV and based on the main model, we
obtain an estimated cause-specific hazard ratio of 8.17 with a $95\%$ CI of
[1.95,34.25]. For vasopressors, the main model yields a hazard ratio of 3.23
with a $95\%$ CI of [1.28,8.15]. Further notable potential risk factors are a
high leucocyte (white blood cells) count, procalcitonin (a precursor of the
hormone calcitonin) increase and arguably hypertension. For hypertension the
estimated cause-specific hazard ratio was 1.86 with a $95\%$ CI of [0.93,3.72]
based on the main model and a hazard ratio of 2.19 with a $95\%$ CI of
[1.12,4.29] for the reduced model. The cause-specific hazard for leucocyte
counts was estimated at 1.67 for the main model with a $95\%$ CI for
[1.00,2.79] and the hazard ratio for procalcitonin was estimated at 1.82 with
a $95\%$ CI of [1.11,2.97]. Data regarding leucocyte counts and procalcitonin
were not available in the meta-analyses by Cai et al., (2021). The results
above are summarized in Table 2.

Table 2: Cause-specific hazard ratios and confidence intervals based on a competing risks model from Hardenberg et al., (2021). Risk Factor | Hazard Ratio | 95% CI
---|---|---
Mechanical Ventilation | $8.17$ | [1.95, |  $34.25$ ]
Vasopressors | $3.23$ | [1.28, |  $8.15$ ]
Hypertension | $1.86$ | [0.93, |  $3.27$ ]
Leucocyte count | $1.67$ | [1.00, |  $2.79$ ]
Procalcitonin | $1.82$ | [1.11, |  $2.97$ ]

In Section 3, we considered the potential impacts of mechanical ventilation,
vasopressors and hypertension on AKI development from an alternative
viewpoint, i.e. non time-to-event data. There we analyzed simple log odds
ratios comparing the occurrence of AKI in respective subgroups of COVID-19
patients. In this chapter we consider time-to-event data and cause-specific
hazard ratios in the context of competing risks models. In both settings, MV,
vasopressors and hypertension are statistically significant and meaningful
(based on effect size) risk factors for AKI development in COVID-19 patients.
The log odds ratios tell us that there are more AKI injury cases in those
patient subgroups that received MV, vasopressors or that have hypertension
compared to those that did not. The cause-specific hazard ratios calculated
from the competing risks models additionally show that MV, vasopressors and
the presence of hypertension increase the hazard of severe AKI in COVID-19
patients.

Unfortunately we were unable to find other studies that analyzed a Cox
proportional hazards regression with competing risks that consider time-to-AKI
in COVID-19 patients with prior death as a competing event. Therefore we could
not apply the Bayesian approach outlined in Section 1 for the competing risks
setting.

Figure 5: Forest plot and shrinkage estimates for the risk factor smoking.

Figure 6: Forest plot and shrinkage estimates for the risk factor obesity.

Figure 7: Forest plot and shrinkage estimates for the risk factor diabetes.

Figure 8: Forest plot and shrinkage estimates for the risk factor gender.