APReferences for the Appendices

Wan, Zhang, and Frazier

Correlation Improves Group Testing

Correlation Improves Group Testing

Jiayue Wan††thanks: Equal contribution.School of Operations Research &
Information Engineering, Cornell University, NY 14850<EMAIL_ADDRESS>Yujia Zhang††footnotemark:  Center for Applied Mathematics, Cornell
University, NY 14850<EMAIL_ADDRESS>Peter I. Frazier School of Operations
Research & Information Engineering, Cornell University, NY 14850,
<EMAIL_ADDRESS>
Population-wide screening to identify and isolate infectious individuals is a
powerful tool for controlling COVID-19 and other infectious diseases. Testing
an entire population, however, requires significant resources. Group testing
can enable large-scale screening, but dilution degrades its sensitivity,
reducing its effectiveness as an infection control measure. Analysis of this
tradeoff typically assumes pooled samples are independent. Building on recent
empirical results in the literature, we argue that this assumption
significantly underestimates group testing’s true benefits. Indeed, placing
samples from a social group into the same pool correlates a pool’s samples.
Hence, a positive pool likely contains multiple positive samples, increasing a
pooled test’s sensitivity and also tending to reduce the number of pools
requiring follow-up testing. We prove that under a general correlation
structure, pooling correlated samples together (called correlated pooling)
achieves higher sensitivity and requires fewer tests per positive identified
compared to independently pooling the samples (called naive pooling) using the
same pool size within the classic two-stage Dorfman procedure. To the best of
our knowledge, our work is the first to theoretically characterize
correlation’s effect on sensitivity and test usage under models of general
correlation structure and realistic test errors. Under a 1% starting
prevalence, simulation results estimate that correlated pooling requires 12.9%
fewer tests than naive pooling to achieve infection control. Thus, we argue
that correlation is an important consideration for policy-makers designing
infection control interventions: it makes screening more attractive for
infection control and it suggests that sample collection should maximize
correlation.

COVID-19, group testing, pooled testing, infection control, screening,
polymerase chain reaction (PCR)

##  1 Introduction

The SARS-CoV-2 virus has killed over 5 million people while causing enormous
economic losses. Large-scale screening has proven effective in curbing the
virus’s spread (Mercer and Salit 2021, Xing et al. 2020, Barak et al. 2021)
through promptly identifying and isolating infected individuals and their
contacts (Cleary et al. 2021, Brault et al. 2021). Nevertheless, screening the
entire population requires a massive amount of resources. Diagnosis of SARS-
CoV-2 infection is commonly performed using polymerase chain reaction (PCR)
tests, which require chemical reagents, machine time and trained medical
personnel. The scale and complexity of this demand makes population-wide
screening hard, if not infeasible, for many countries (GRID COVID-19 Study
Group 2020).

A promising solution to this conundrum is group testing. The Dorfman
procedure, the first group testing protocol proposed in 1943 to screen
enlisted soldiers for syphilis (Dorfman 1943), pools multiple samples together
and tests each pool using a single test. Samples from a pool testing negative
are cleared, while samples from a pool testing positive receive individual
followup tests. Especially in low-prevalence settings, group testing can save
significant test resources compared to individual testing (Kim et al. 2007).

Group testing has been successfully implemented for large-scale screening in
multiple communities worldwide and yielded promising results in controlling
the spread of SARS-CoV-2. In May 2020, the city of Wuhan employed pooled
testing to screen nine million population over ten days (Fan 2020).
Furthermore, repeated screening of the entire population helps to achieve
persistent infection control. In fall 2020, Cornell University conducted
repeated surveillance testing on 5-7K Cornell students and employees per day
using pools of size five and reopened its campus safely (Lefkowitz 2020).
Going forward, repeated community-based screening may be necessary for long-
term containment of COVID-19 under low prevalence (Barak et al. 2021).

Repeated large-scale screening using group testing provides the most value
when the sensitivity, i.e., the probability of correctly identifying positive
samples, and efficiency, i.e., the number of individuals screened per PCR
test, are high. High sensitivity helps identify the positives accurately,
while high efficiency permits more frequent screening under limited resources.
Together, they enable early identification and isolation of positives as well
as quarantine of close contacts, which prevents further disease spread and
contributes to better infection control.

However, pooled tests face a tradeoff between sensitivity and efficiency due
to the dilution effect. The concentration of virus particles (called the viral
load) in the sample of an infected individual is diluted when it is pooled
with negative samples (Wein and Zenios 1996). The dilution effect lowers the
overall viral load in the pooled sample and hence the sensitivity. For larger
pools, the decrease in sensitivity due to dilution is stronger. On the other
hand, reducing the pool size to avoid sensitivity degradation often results in
a lower efficiency, which reduces the benefit of group testing in resource
consumption.

Past analyses of this tradeoff, such as Kim et al. (2007) and Westreich et al.
(2008), assume that samples in the same pool are independent. In practice,
however, human behavior and logistical constraints in sample collection
naturally create correlation. Specifically, if a person is infected, this
increases the likelihood that others in their immediate social circles are
infected (Vang et al. 2021, Rader et al. 2020, Lan et al. 2020). Most notably,
the transmission probability among people in the same household has been
estimated in a meta-study to be 16.6% (Madewell et al. 2020) and in some
literature as high as 44.6% (Boscolo-Rizzo et al. 2020). Moreover, in large-
scale screening, those frequenting the same testing center usually live, work,
or socialize close to each other. As a result, members of the same social
group are often placed into the same pool (Barak et al. 2021).

Intuitively, correlation should cause group testing to be more sensitive than
it would be if samples were independent. Under correlation, a pool containing
a positive sample more likely contains multiple positive samples. This
increases the viral load in the pooled sample and improves the overall
sensitivity. Recently, Barak et al. (2021) finds that the sensitivity of group
testing performed in Israel was higher than independent sampling would suggest
and conjectures that correlation was the cause. Analysis and simulation
results in Comess et al. (2021) point to the same intuition, though it assumes
a higher prevalence in correlated pools due to additional network
transmission.

Intuition also suggests that correlation should improve the efficiency of the
Dorfman procedure by concentrating positives in fewer pools. Thus, a smaller
number of pools should test positive, requiring fewer follow-up tests. This
has been observed in simulation studies (Lendle et al. 2012, Deckert et al.
2020) though current understanding is limited as recent theoretical analysis
(Augenblick et al. 2020, Lin et al. 2020) assumes that tests are error-free,
while testing errors in practice have an important effect on test utilization
in group testing. Basso et al. (2021) allows for testing errors, but assumes a
fixed sensitivity that does not depend on the pool size or the number of
positives in the pool, thus ignoring the dilution effect and the effect of
correlation on sensitivity.

Thus, while the benefits of correlation for both sensitivity and efficiency
are important, current theoretical understanding is limited. In this paper, we
address this gap in the literature. We prove that under a general correlation
structure in the population and other mild assumptions, pooling correlated
samples together (called correlated pooling) in the two-stage Dorfman
procedure achieves higher sensitivity compared to independently pooling the
samples (called naive pooling) using the same pool size. We also prove that
correlated pooling uses fewer tests per positive identified.

Notably, we show an example in which correlated pooling has strictly lower
efficiency than naive pooling, where we recall that efficiency is the number
of people screened per test. Because correlated pooling has higher
sensitivity, some pools with positive samples test positive under correlated
pooling but negative under naive pooling. This effect can dominate the
reduction in the number of positive-containing pools caused by correlation. As
a result, when the two effects are combined, correlated pooling can have more
pools testing positive than naive pooling and thus require more tests. This
stands in contrast to the theoretical results in the literature on efficiency
(Augenblick et al. 2020, Lin et al. 2020, Basso et al. 2021), which find that
correlation always improves efficiency. This discrepancy in qualitative
conclusions follows in part from the literature’s focus on a limited set of
models for testing error in which correlation does not improve sensitivity. We
argue that tests per positive identified better quantifies a procedure’s
utility for screening than efficiency, and so correlated pooling remains
attractive for infection control despite this example.

Our results on sensitivity and test utilization (efficiency and tests per
positive found) assume that there are no false positives. While false
positives do occur, we argue that they have little impact on sensitivity and
test utilization in practice, since the specificity, the probability of
correctly declaring a negative sample as negative, is typically quite high in
reality (e.g., Public Health Ontario 2020 finds a PCR specificity of 99.99%),
making false positives quite rare.

Specificity, however, is of independent interest when screening many people,
since even a specificity as high as $99.99\%$ will create false positives once
enough people are tested. False positives waste public health resources, cause
economic losses, disrupt personal lives, and increase the risk of infection
during treatment (Gupta and Malina 1999, Healy et al. 2021). To address this,
we briefly analyze the specificity of correlated pooling implied by a
generalization of our main model allowing for false positives. We show that
correlated pooling improves specificity compared to both naive pooling and
individual testing. For a typical prevalence, pool size, and a specificity of
$99.99\%$ for a single PCR test, we estimate the specificity of the Dorfman
procedure with correlated pooling to be at least $99.999\%$. This is a ten-
fold improvement in the false-positive rate ($1-\mathrm{specificity}$)
compared to individual testing, from $10^{-4}$ to $10^{-5}$.

As a consequence of these insights, we argue that group testing is
significantly more useful for bringing an epidemic under control than is
argued in classical analyses. We consider a use case where group testing is
applied to large-scale screening and positive individuals are isolated once
identified. We think of achieving epidemic control as a situation where the
number of active infections in a population stabilizes or declines. In our
case study focused on intra-household correlation, pooling samples from the
same household together results in higher sensitivity and efficiency, both
contributing to better epidemic control. For example, at a representative
prevalence of 1%, naive pooling has a sensitivity of 77.6% and an efficiency
of 5.87, while correlated pooling has a substantially higher sensitivity of
82.6% and efficiency of 6.33 (after tuning the pool sizes separately for each
pooling strategy). Because of these improvements, correlated pooling requires
12.9% fewer tests than naive pooling to achieve epidemic control.

This difference has the potential to have a substantial impact on real-world
policy-making. As discussed earlier, within-pool correlation arises naturally
in repeated large-scale screening. Consequently, policymakers that base their
decisions on the independence assumption may undervalue group testing and
adopt overly conservative policies (e.g., imposing a full lockdown instead of
using pooled testing to control viral spread). On the other hand, if they do
account for within-pool correlation, they may conduct large-scale group
testing that fully utilizes the resources while keeping the economy open.
Furthermore, correlation that occurs naturally can be augmented in
implementation by encouraging people from the same social group to get tested
together.

To summarize, our contributions in this paper are:

  * •

We formulate a general model of correlation in pools, derived from an
asymptotic analysis of a more general population-level model of infections
spreading across a population and how pools are formed from members of the
population.

  * •

We prove that under the general model and other mild assumptions, using
correlated pooling in the two-stage Dorfman procedure achieves (1) higher
sensitivity; (2) fewer tests per positive identified compared to naive
pooling. Our work is the first to study sensitivity or efficiency
theoretically under a general correlation model.

  * •

We provide a counterexample to the claim that correlated pooling always
improves efficiency, i.e., the number of individuals screened per PCR test,
clarifying that claims in the literature that correlated pooling always
improves efficiency do not necessarily apply outside of the limited class of
previously considered models.

  * •

We conduct a case study with realistic data showing that correlation
significantly improves the effectiveness of group testing for epidemic
control.

As a consequence of these insights, we argue that classical analysis assuming
independence underrepresents the power of group testing for infection control.
Moreover, group testing should be more widely used in large-scale screening
and implemented in a way that maximizes correlation.

The rest of this paper is organized as follows: Section 2 reviews related work
in more detail. Section 3 establishes a mathematical model for a single pool
and proves our main theoretical results: that correlation improves sensitivity
and tests per positive identified, but can reduce efficiency. Section 4
supplements Section 3 by deriving our model of correlation in a single pool
model from an asymptotic analysis of a model for a larger population, showing
that our pool-level model is well-justified in a population-wide screening
context. Section 5 presents realistic models for viral loads and PCR
sensitivity and shows that under these specific models correlated pooling has
better efficiency than naive pooling. Section 6 performs a case study where
within-pool correlation is induced by household transmission. Section 7
concludes the paper and discusses future research.

##  2 Related Work

###  2.1 Group Testing and the Dilution Effect

Group testing was proposed by Dorfman (1943) to screen enlisted soldiers for
syphilis during World War II. The Dorfman procedure combines multiple samples
together and tests the pooled samples, so that samples in a pool testing
negative are cleared and samples in a pool testing positive are tested
individually for identification. This significantly increases efficiency by
screening multiple individuals with a single test. Since then, many
generalizations of the Dorfman procedure have been developed and studied
theoretically. Group testing is also widely applied in the surveillance and
control of infectious diseases. For a review of recent theory and
applications, see Kim et al. (2007) and Aprahamian et al. (2019).

The COVID-19 pandemic has raised further interests in applying group testing
to infection control and population-wide surveillance (Mercer and Salit 2021).
Multiple studies have used modeling and simulation to explore the value that
group testing offers in scaling up testing, such as Cleary et al. (2021),
Brault et al. (2021), Pilcher et al. (2020), Eberhardt et al. (2020), and
Mutesa et al. (2020). A thorough review of the literature in using group
testing for COVID-19 mitigation is provided by Yu et al. (2021).

Despite the extent to which group testing can increase testing capacity, it
may have a negative impact on sensitivity. First, the inherent error rate of
individual assays, along with the design of the pooling protocol, may lead to
failure to detect some positive samples (Graff and Roeloffs 1972, Kim et al.
2007, Westreich et al. 2008). Moreover, sensitivity may decrease due to the
dilution effect, that is, a pool dominated by negative samples may test
negative, causing its positive members to be missed. The dilution effect was
first modeled by Hwang (1976) and subsequently explored by Wein and Zenios
(1996), Zenios and Wein (1998), Weusten et al. (2002), Nguyen et al. (2019)
for HIV detection and by Hung and Swallow (1999) for prevalence estimation.

Many studies have assessed the dilution effect in SARS-CoV-2 tests from both
mathematical and empirical perspectives. Pilcher et al. (2020) assumes a
temporal viral load progression in infected individuals, which, together with
the detection limit of PCR tests, defines a “window of detection”; under this
setting, pooling is equivalent to raising the detection limit of the test and
shortening the effective window of detection. Brault et al. (2021) proposes a
similar quantification of decrease in sensitivity due to dilution based on a
mathematical model for PCR. Some experimental studies (Yelin et al. 2020,
Lohse et al. 2020) evidence that pooling up to around 30 samples does not
result in a loss of sensitivity, while Bateman et al. (2020) observes an
increasing deterioration of sensitivity in pooling 5, 10, and 50 samples.

###  2.2 Correlation in Group Testing

Most of the aforementioned literature assumes that the infection statuses of
the samples within a pool, whether binary or not, are independent from each
other. However, as we described in the introduction, correlation between
samples is often present in reality and can potentially be leveraged for our
advantage to combat the dilution effect.

One important cause of correlation is transmission within households. The
secondary attack rate (SAR), i.e., the probability that an infectious person
in a household infects another given household member, is significant for many
infectious diseases (Carcione et al. 2011, Whalen et al. 2011, Odaira et al.
2009, Meningococcal Disease Surveillance Group 1976, Glynn et al. 2018). For
SARS-CoV-2, a meta-analysis (Madewell et al. 2020) of 40 studies finds an
average SAR of 16.6% and a 95% confidence interval of 14.0%-19.3%. Beyond
household transmission, correlation in infection statuses among members of the
same social group has also been observed among college students belonging to
the same fraternity or sorority (Vang et al. 2021), people living in the same
neighborhood (Rader et al. 2020), and co-workers (Lan et al. 2020).

Relatively little research explores group testing of correlated samples. Barak
et al. (2021), a large-scale observational study, observes that individuals in
the same social groups are often pooled together in large-scale screening. It
discovers that samples with low viral load, each of which would have likely
been missed if it were the only positive sample in the pool, were detected
when pooled with high-viral-load samples. This led to higher sensitivity than
would have occurred with pools containing independently infected samples.
Augenblick et al. (2020) uses a simple example with pairwise correlation and
perfect test accuracy to illustrate reduced test consumption in the Dorfman
procedure. Lendle et al. (2012) uses simulations to show that correlation
improve the efficiency of hierarchical and matrix-based group testing. Deckert
et al. (2020) uses simulation to show that pooling individuals with similar
prevalence levels reduces costs. Building on these observations, Lin et al.
(2020) models sample collection at a testing site as a regenerative process
and calculates the cost efficiency of different testing protocols, while
assuming perfect test accuracy. Basso et al. (2021) models a constant pairwise
correlation in infections using a Beta-Binomial distribution for the number of
positives in a pool, and demonstrates that this correlation improves
efficiency. The paper does not study the effect of correlation on sensitivity:
it assumes a fixed sensitivity for pooled tests and does not model the effect
of pool size or correlation on sensitivity.

The closest paper in the literature to ours is Comess et al. (2021), which is
qualitatively motivated by similar considerations but makes theoretical
contributions that are different in nature. There are two major distinctions
between our work and Comess et al. (2021).

First, Comess et al. (2021) considers a specific model of correlation in which
all participants in a pool are close contacts of each other and infections are
acquired in a community infection stage followed by homogeneous secondary
infections within the pool. As a result, the prevalence in the correlated pool
is higher than that in a naive pool (which only assumes community infection).
Hence, the model in Comess et al. (2021) is best suited to understanding the
joint effect of increasing secondary transmission while pooling related
samples together. We argue, however, that the choice of pooling strategy
should be based on a comparison of their properties while holding the
population’s prevalence steady. This is the approach we take in our paper.

Second, Comess et al. (2021) theoretically studies a different but related
metric of test consumption. A theoretical result therein (Observation 5)
defines an efficiency metric that assumes 100% sensitivity of the pooled test
and shows that the metric is identical for both pooling methods. This metric,
though theoretically tractable, does not fully capture the difference in test
consumption in practice, as is reported in their simulation results.
Nevertheless, much of the intuition described in Comess et al. (2021) is
consistent with our results. In particular, Observation 5 claims that the
sensitivity is no worse under correlated pooling than under naive pooling. We
prove a similar result in our Theorem 3.2. In addition, the simulated
efficiencies in Figures 6 and 8 of Comess et al. (2021), though not discussed
by the authors, indicate that correlated pooling can have lower efficiency
than naive pooling, which we demonstrate is possible in Section 3.4.

Beyond viral testing, group testing with correlation has been studied in the
signal processing community. For example, graph structures may induce
correlation among nodes and edges (Ganesan et al. 2017) or impose constraints
on pool formulation (Cheraghchi et al. 2012).

##  3 Theoretical Results

In this section, we build a mathematical model for a single pool with and
without correlation among samples. We show that within-pool correlation in the
classical two-stage Dorfman procedure (Dorfman 1943) results in higher test
sensitivity and lower or comparable test consumption for identifying
positives, which offers great value to epidemic mitigation given limited
testing capacity in a global pandemic. In Section 4, we will discuss how this
single-pool model is well-justified in the context of large-scale screening.

###  3.1 Model Setup

We study the performance of pools with and without correlation among its
samples under the Dorfman procedure, the most widely studied and adopted group
testing protocol.

##### Dorfman procedure:

In the first stage of the Dorfman procedure, samples are placed into non-
overlapping, uniformly-sized pools and each pool is tested. In the second
stage, samples from pools testing positive in the first stage are tested
individually. A positive sample is correctly declared positive if and only if
its pool tests positive and it tests positive in the followup individual test.
Each pooled test or individual test is performed using a polymerase chain
reaction (PCR) test.

We consider a pooled test with its pool size equal to $n$ and the prevalence
in the pool equal to $\alpha$. That is, each sample is positive with marginal
probability $\alpha$.

We operate in the asymptotic regime of $\alpha\to 0$ in our subsequent
analyses. We focus on this regime for two reasons. First, pooling is most
preferable under low prevalence (Kim et al. 2007, Pilcher et al. 2020). Like
most of the literature studying group testing, we continue to focus on this
regime. Second, when screening is applied promptly and regularly since the
early stage of the epidemic, the population-level prevalence will likely be
consistently low. Despite this focus, we will show in Section 6 that the
benefit offered by within-pool correlation is robust, even when the prevalence
is as high as 10%.

We consider two pooling strategies, naive pooling and correlated pooling,
where viral loads of the samples in the same pool are independent under naive
pooling and may be correlated under correlated pooling. Let
$\\{V_{i}\\}_{i=1}^{n}$ denote the viral loads of the samples in a pool of
size $n$. Hereafter, we use the compact notation $V_{1:n}$ to denote
$V_{1},\cdots,V_{n}$. If individual $i$ is infected, then $V_{i}>0$;
otherwise, $V_{i}=0$111In reality, it is possible that the viral load varies
across different body parts of the same individual and sampling practice can
induce further noise in the sample viral load. Here we conflate individual
viral load and sample viral load, assuming homogeneity of viral load within
the same individual and no loss/noise in sampling. Hence, $V_{i}>0$ is a
surrogate for whether an individual is infected or not.. Let
$S=\sum_{i=1}^{n}\mathbbm{1}\\{V_{i}>0\\}$ denote the number of infected
individuals in the pool.

Let $\mathbb{P}_{i,\alpha}$ and $\mathbb{E}_{i,\alpha}$ denote the probability
and expectation operators, respectively, under pooling method $i$ (where $i=0$
is naive pooling, and $i=1$ is correlated pooling) and prevalence level
$\alpha$. We drop the subscripts when the probability or expectation does not
depend on the pooling method or the prevalence level.

We state two assumptions and one condition necessary for proving the
theoretical results at the pool level. Assumptions 3.1 and 3.1 are always
assumed, whereas Condition 1 is assumed when explicitly stated. Section 4 will
show that Assumptions 3.1, 3.1 and Condition 1 hold in a natural population-
level model of pooling.

The viral loads $V_{i}$ of individuals in a pool are identically distributed
and this distribution is the same in naive and correlated pools.

In the naive pool, the viral loads of individuals are independent.

###### Condition 1

In the correlated pool, conditioning on the presence of at least one positive
in the pool creates a strictly positive probability that at least two
individuals in the same pool are infected, even in the limit as prevalence
approaches 0. Mathematically, $\lim_{\alpha\to 0}\mathbb{P}_{1,\alpha}(S>1\mid
S>0)>0$.

Condition 1 is motivated by the within-pool correlation arising from pooling
members of the same social group together, as described in Section 1. It
describes the main feature that differentiates the two pooling strategies: as
prevalence approaches zero, the probability that a positive-containing pool
contains more than one positive samples diminishes for naive pools (under
Assumption 3.1) but persists for correlated pools (under Condition 1).

Having defined the two pooling strategies, we now model the test outcomes.

##### Test outcomes:

We first model the result of a single individual PCR test. Given an input
sample with viral load $v$, we assume a PCR test returns a positive result
with probability $p(v):\mathbb{R}_{\geq 0}\to[0,1]$ and a negative result with
probability $1-p(v)$. We refer to $p(v)$ as the sensitivity function. Here we
assume $p(0)=0$, i.e., no false positives; later in Section 6.2.2, we argue
that a small individual test FPR, e.g., 0.01% (Public Health Ontario 2020),
implies an FPR of correlated pooling low enough for its deployment in repeated
large-scale screening. We further assume that $p(v)>0$ for $v>0$, $p$ is
monotone increasing in $v$, and that the result of a PCR test, whether
individual or pooled, is conditionally independent from any other PCR test
given its sample viral load.

We define the following variables for the outcomes of a two-stage Dorfman
procedure with pool size $n$. For a pooled test with viral loads
$V_{1},\cdots,V_{n}$ in the input samples, we assume pooling leads to a
dilution factor equal to the pool size222Though assuming a dilution factor of
$n$ here, our theoretical results are easily generalizable to other dilution
factors.. Hence, the pooled test returns positive with probability
$p(\bar{V}_{n})$ where $\bar{V}_{n}=\frac{1}{n}\sum_{j=1}^{n}V_{j}$. Let
$Y=\text{Ber}\left(p\left(\bar{V}_{n}\right)\right)$ denote the outcome of the
pooled test in the first stage. Let $W_{j}=\text{Ber}\left(p(V_{j})\right)$
denote what the outcome of the individual test for sample $j$ with viral load
$V_{j}$ will be, if it is performed. Let $D=\sum_{j=1}^{n}YW_{j}$ denote the
number of positives identified in the pool, i.e., the number of positive
samples that test positive in the second stage given the pool tests positive
in the first stage. We note that the conditional independence assumption above
implies that the pooled test and individual tests are conditionally
independent given the viral loads in the participating samples, i.e.
$Y{\perp\\!\\!\\!\\!\perp}W_{j}\mid V_{1:n},\ j=1,\cdots,n$.

###  3.2 Test Sensitivity

The sensitivity of a group testing protocol is critical for epidemic control.
As discussed earlier, pooled tests face a loss of sensitivity due to the
positive sample(s) getting diluted in the pool. Here, we examine the
sensitivity of the two-stage Dorfman procedure with (i.e., correlated pooling)
and without (i.e., naive pooling) correlation among samples. To achieve this,
we first define a metric for the sensitivity of a group testing protocol.

###### Definition 3.1

Let $\beta_{0,\alpha}$ and $\beta_{1,\alpha}$ denote the overall false
negative rate, or the fraction of positive samples that are falsely declared
negative in the two-stage Dorfman procedure under prevalence $\alpha$ in the
naive and correlated pools, respectively333Our goal here is to evaluate a
pooling strategy when screening an entire population using many pools.
Therefore, rather than $1-\mathbb{E}_{i,\alpha}\left[\frac{D}{S}\right]$,
which is the expected false negative rate of a single pool, we focus on
$1-\frac{\mathbb{E}_{i,\alpha}[D]}{\mathbb{E}_{i,\alpha}[S]}$, which we argue
is the right metric for population-wide false negative rate. Indeed, in the
asymptotic regime in the population-wide screening context described in
Section 4, under mild assumptions, the number of positives found per pool
converges in probability to $\mathbb{E}_{i,\alpha}[D]$ and the number of
infected individuals per pool converges in probability to
$\mathbb{E}_{i,\alpha}[S]$. Thus, by the continuous mapping theorem, as long
as $\mathbb{E}_{i,\alpha}[S]>0$, the fraction of positives found converges in
probability to $\frac{\mathbb{E}_{i,\alpha}[D]}{\mathbb{E}_{i,\alpha}[S]}$.
Other metrics in Section 3 are defined with the same logic.. That is,
$\beta_{i,\alpha}=1-\dfrac{\mathbb{E}_{i,\alpha}[D]}{\mathbb{E}_{i,\alpha}[S]}.$

Under this metric, we present our main result in Theorem 3.2. We show that
under a general class of sensitivity functions, the two-stage Dorfman
procedure using correlated pooling achieves a lower overall false negative
rate than naive pooling in the low-prevalence setting.

###### Theorem 3.2

If $p(v)$ is monotone increasing in $v$, $\lim_{\alpha\rightarrow
0^{+}}\beta_{0,\alpha}\geq\lim_{\alpha\rightarrow 0^{+}}\beta_{1,\alpha}$. If,
in addition, $p(v)$ is strictly monotone increasing and Condition 1 holds,
then the inequality is strict.

Here we provide a proof sketch of Theorem 3.2. A complete proof is given in
Appendix 8.1.

###### Proof 3.3

Proof sketch of Theorem 3.2.

For both $i=0,1$, we can show that the overall false negative rate is given by
$\beta_{i,\alpha}=1-\mathbb{E}_{i,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0\right]$.

For naive pooling, the $V_{i}$’s are i.i.d. As $\alpha\to 0^{+}$, the
probability that a positive pool contains multiple positive samples vanishes,
and we can show that $\lim_{\alpha\to
0^{+}}\beta_{0,\alpha}=1-\mathbb{E}\left[p\left(\frac{1}{n}V_{1}\right)p(V_{1})\mid
V_{1}>0\right]$.

For correlated pooling, a positive pool contains multiple positives with non-
negligible probability, so we can write
$\beta_{1,\alpha}=1-\sum_{\ell=1}^{n}A_{\ell}\cdot\mathbb{P}_{1,\alpha}(S=\ell\mid
S>0)$, where
$A_{\ell}\overset{\Delta}{=}\mathbb{E}_{1,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]$. When $\ell=1$,
$A_{1}=\mathbb{E}[p(\frac{1}{n}V_{1})p(V_{1})\mid V_{1}>0]=1-\lim_{\alpha\to
0^{+}}\beta_{0,\alpha}$. When $\ell\geq 2$, we have
$\bar{V}_{n}>\frac{1}{n}V_{1}$ because there exists at least one other sample
with positive viral load. Assuming $p(v)$ is a monotone increasing function in
$v$, we obtain $p(\bar{V}_{n})\geq p(\frac{1}{n}V_{1})$, which, combined with
$p(V_{1})>0$ given $V_{1}>0$, implies that $A_{\ell}\geq A_{1}$. Therefore,
taking $\alpha\to 0^{+}$ gives $\lim_{\alpha\to
0^{+}}\beta_{1,\alpha}\leq\lim_{\alpha\to 0^{+}}\beta_{0,\alpha}$. The
inequality is strict if $p(v)$ is strictly increasing in $v$ and Condition 1
holds.

###  3.3 Test Consumption

In addition to accuracy, test consumption is another key consideration. In
large-scale screening with limited resources, the ability to screen many
individuals with few tests significantly expands the testing capacity, which
translates to better epidemic mitigation (Mercer and Salit 2021).

In this section, we investigate the test consumption of the two-stage Dorfman
procedure under naive and correlated pooling. One metric commonly used in
literature is efficiency, i.e., the number of individuals screened per PCR
test (Kim et al. 2007, Westreich et al. 2008). However, a higher efficiency
does not necessarily indicate better epidemic control. In reality, it is
through the identification and isolation of positive cases that screening most
directly mitigates the epidemic spread. Thus, in lieu of the standard metric
we examine the test consumption of a testing protocol by the number of
positive cases identified per PCR test consumed, as it better captures the
value of a testing protocol in epidemic control.

Recall that in a size-$n$ pool, $n$ followup individual tests are performed if
the pool tests positive, i.e., $Y=1$; no followup individual tests are
performed otherwise. We now formally define the metric we use for test
consumption.

###### Definition 3.4

Let $\gamma_{0,\alpha}$ and $\gamma_{1,\alpha}$ denote the expected number of
positive cases identified per PCR test consumed under prevalence $\alpha$ in
the naive and correlated pools, respectively. That is,

$\gamma_{i,\alpha}=\frac{\mathbb{E}_{i,\alpha}[D]}{1+n\mathbb{E}_{i,\alpha}[Y]},\quad
i=0,1.$ (1)

Note that $\gamma_{i,\alpha}$ differs from efficiency in the numerator. The
numerator of efficiency is the pool size $n$, while the numerator of
$\gamma_{i,\alpha}$ is $\mathbb{E}_{i,\alpha}[D]$, the number of positive
cases identified in the pool. Given
$\mathbb{E}_{i,\alpha}[D]=n\alpha(1-\beta_{i,\alpha})$, it follows that
$\gamma_{i,\alpha}$ is directly proportional to efficiency and sensitivity
(i.e., $1-\beta_{i,\alpha}$) of the pooling strategy. We will see in Section
6.3 that $\gamma_{i,\alpha}$ is the key metric for measuring the effectiveness
of a group testing protocol in epidemic control.

To understand the behavior of $\gamma_{i,\alpha}$, we can rewrite Equation 1
in Definition 1 as

$\gamma_{i,\alpha}^{-1}=\frac{1}{\mathbb{E}_{i,\alpha}[D]}+\frac{n\mathbb{E}_{i,\alpha}[Y]}{\mathbb{E}_{i,\alpha}[D]},\quad
i=0,1.$ (2)

In the first term on the right hand side (RHS) of Equation 2, we have
$\mathbb{E}_{i,\alpha}[D]=n\alpha(1-\beta_{i,\alpha})$. Theorem 3.2 showed
that $\lim_{\alpha\to 0^{+}}\beta_{1,\alpha}\leq\lim_{\alpha\to
0^{+}}\beta_{0,\alpha}$. Therefore, to examine the number of positive cases
identified per PCR test, it suffices to focus on comparing the second term of
the RHS of Equation 2. We formally define this quantity.

###### Definition 3.5

Let $\eta_{0,\alpha}$ and $\eta_{1,\alpha}$ denote the expected number of
followup individual tests consumed per positive case identified under
prevalence $\alpha$, in the naive and correlated pools, respectively. That is,
$\eta_{i,\alpha}=\dfrac{n\mathbb{E}_{i,\alpha}[Y]}{\mathbb{E}_{i,\alpha}[D]},\
i=0,1.$

In Theorem 3.6, we show that in the low prevalence setting, the two-stage
Dorfman procedure using correlated pooling consumes no more followup tests per
positive case identified than using naive pooling by a constant fraction. This
fraction is determined by the viral load distribution among infected
individuals, the PCR test mechanism, and the pooling strategy. By Equation 2,
this bound also applies to $\gamma_{i,\alpha}^{-1}$.

###### Theorem 3.6

$\lim_{\alpha\to 0^{+}}\dfrac{\eta_{1,\alpha}}{\eta_{0,\alpha}}\leq 1+\delta$
where $\delta=\dfrac{\mathbb{P}_{1,\alpha}(Y=1,S_{D}=0\mid
S>0)}{\mathbb{P}_{1,\alpha}(Y=1,S_{D}>0\mid S>0)}$ and
$S_{D}=\sum_{j=1}^{n}W_{j}$.

In a relatively simple case where a PCR test result deterministically reports
whether the sample viral load exceeds a threshold value, correlated pooling
consumes no more followup tests per positive case identified than naive
pooling. This is formulated in Corollary 3.7.

###### Corollary 3.7

Suppose the sensitivity function is $p(v)=\mathbbm{1}\\{v\geq u_{0}\\}$ for
some non-negative constant $u_{0}$. Then, $\lim_{\alpha\to
0^{+}}\dfrac{\eta_{1,\alpha}}{\eta_{0,\alpha}}\leq 1$.

In reality, the sensitivity of a PCR test, albeit not exactly a step function
of the sample viral load $v$, closely resembles one in that it increases
rapidly from zero to one within a narrow range of $v$. (See, e.g., Figure 1 in
Section 5.3.) Section 5.3 further shows that, under a realistic sensitivity
function, viral load distribution and pool size, the bound in Theorem 3.6 is
almost equal to one.

###  3.4 Revisiting Efficiency

Existing literature claims that within-pool correlation leads to better
efficiency (Comess et al. 2021, Augenblick et al. 2020, Lendle et al. 2012,
Deckert et al. 2020, Lin et al. 2020, Basso et al. 2021). While the claim is
true under simplified assumptions such as noise-free tests, we show in this
section that it does not hold in general.

We first relate efficiency to metrics investigated in previous sections. For
any prevalence $\alpha$, efficiency can be expressed in terms of
$\beta_{i,\alpha}$ and $\eta_{i,\alpha}$ as follows:

$\displaystyle\text{efficiency}_{i,\alpha}$
$\displaystyle=\left(\frac{1+n\mathbb{E}_{i,\alpha}[Y]}{n}\right)^{-1}=\left(\frac{1}{n}+\frac{n\mathbb{E}_{i,\alpha}[Y]}{\mathbb{E}_{i,\alpha}[D]}\cdot\frac{\mathbb{E}_{i,\alpha}[D]}{\mathbb{E}_{i,\alpha}[S]}\cdot\frac{\mathbb{E}_{i\alpha}[S]}{n}\right)^{-1}=\left(\frac{1}{n}+\alpha\eta_{i,\alpha}(1-\beta_{i,\alpha})\right)^{-1}.$
(3)

We identify scenarios where correlated pooling could have lower efficiency.
First, Theorem 3.6 showed that $\eta_{i,\alpha}$ may be higher under
correlated pooling, so by Equation 3, it is certainly possible that the
efficiency is also lower under correlated pooling. Second, even in settings
where correlated pooling has lower $\eta_{i,\alpha}$ and $\beta_{i,\alpha}$,
the product $\eta_{i,\alpha}(1-\beta_{i,\alpha})$ may still be higher under
correlated pooling, which leads to lower efficiency. Indeed, in Appendix 9, we
construct a stylized example where both of the above scenarios can occur,
resulting in correlated pooling having lower efficiency than naive pooling.
The example also shows that $\lim_{\alpha\to 0^{+}}\eta_{1,\alpha}$ can be
strictly larger than $\lim_{\alpha\to 0^{+}}\eta_{0,\alpha}$, which
necessitates the $(1+\delta)$ bound in Theorem 3.6.

##  4 Modeling at the Population Level

The goal of this section is to show that our pool-level model in Section 3 is
well-justified in a population-wide screening context. To achieve this, we
first describe a model at the population level and use it to justify
Assumptions 3.1, 3.1 and Condition 1 made in Section 3.

###  4.1 Forming Pools from the Population

We consider a population of $N$ individuals, where each individual is
associated with a unique index in $\\{1,\ldots,N\\}$. We call this the
“population index”. We slightly abuse the notation and let
$\\{V_{i}\\}_{i=1}^{N}$ denote the viral loads of the population. If
individual $i$ is infected, then $V_{i}>0$; otherwise $V_{i}=0$. The viral
loads $\\{V_{i}\\}_{i=1}^{N}$ are correlated random variables and follow some
joint distribution. This can model, for example, the spread of disease in a
population based on geographic locations and demographics. We let $\alpha$
denote the overall prevalence, the probability that a person chosen uniformly
at random from the population has a positive viral load. In Section 3 we used
$\alpha$ to denote the prevalence in an individual pool. Later, we will see
that the prevalence in an individual pool constructed as described in this
section is equal to the overall prevalence.

Section 3 defined correlated pooling and naive pooling at the pool level. Here
we describe how these two pooling strategies are implemented at the population
scale. For simplicity, we assume $N$ is a multiple of $n$ so that we can
divide this population into $\frac{N}{n}$ groups of size $n$444If $N$ is not a
multiple of $n$, for modeling purpose we fill the empty spaces in the last
pool with artificial negative samples. Then all of our subsequent analyses
still apply, with the prevalence among the pools changed to
$\hat{\alpha}=\frac{\alpha N}{\lceil N/n\rceil\cdot
n}\in(\alpha(1-O(\frac{1}{N})),\alpha)$ which converges to $\alpha$ when
$N\to\infty$.. Individuals assigned to the same group will participate in the
same pool in the two-stage Dorfman procedure.

Let any pooling assignment be represented by
$\mathcal{A}:=\\{A_{j}:j=1,\ldots,N/n\\}$, a random partition defined below of
$\\{1,\cdots,N\\}$ into $N/n$ groups of size $n$. Pool $j$ contains samples of
viral loads $\\{V_{i}:i\in A_{j}\\}$. Naive pooling and correlated pooling
form the pools in the following manner:

##### Naive Pooling:

In naive pooling, each pool is formed by picking $n$ individuals uniformly at
random from the population without replacement.

##### Correlated Pooling:

In correlated pooling, pools are formed in ways that preserve correlation
among samples in a pool. The within-pool correlation either occurs naturally
or can be enhanced by explicit measures. For example, at testing centers
established on college campuses and in neighborhoods, samples are most likely
from groups of people that live, study, and work in proximity to each other,
preserving correlation. Moreover, test kits can be mailed to households for
self-collection (Stanford Medicine 2020). Samples from the same household can
then be transported to laboratories together and tested in the same PCR test,
preserving correlation.

For both correlated and naive pooling, once the pools are formed, we reorder
the samples in each pool by applying independent random permutations of 1
through $n$.

We think of the social structure of the population (which has not yet been
specified in our model) as influencing both the pooling assignments and the
identities of infected individuals in the population. Below we will model one
aspect of this social structure, namely the set of close contacts associated
with each individual, as deterministic. We assume the pooling assignments are
being chosen independently from the viral loads in the population. Indeed,
dependence on the social structure does not necessarily break this
independence in our formal model.

We define the probability measures $\mathbb{P}_{0,\alpha}^{(N)}$ and
$\mathbb{P}_{1,\alpha}^{(N)}$ to correspond to the distributions of the random
variables defined above ($\mathcal{A}$, $V_{1},\ldots,V_{N}$) under naive and
correlated pooling respectively555From a measure theoretic perspective, the
random quantities $\mathcal{A}$, $V_{i}$ are mappings from the event space to
the (measurable) state space. The mappings themselves do not depend on $N$ but
the distributions of these random quantities under
$\mathbb{P}_{i,\alpha}^{(N)}$ depends on $N$ because the measure
$\mathbb{P}_{i,\alpha}^{(N)}$ itself depends on $N$.. We define
$\mathbb{E}_{0,\alpha}^{(N)}[\cdot]$ and $\mathbb{E}_{1,\alpha}^{(N)}[\cdot]$
to be the expectations taken under $\mathbb{P}_{0,\alpha}^{(N)}$ and
$\mathbb{P}_{1,\alpha}^{(N)}$ respectively. Here, we use a superscript $(N)$
to index quantities computed for a size-$N$ population; as in Section 3, we
use subscripts $i,\alpha$ to index quantities computed under pooling method
$i$ (where $i=0$ is naive pooling, and $i=1$ is correlated pooling) and
prevalence level $\alpha$. The subscript $i$ (or $\alpha$) is dropped when the
quantity does not depend on the pooling method (or prevalence level).

###  4.2 Embedding the Pool-Level Model in the Population-Level Model

To study properties of the pools constructed, it is sufficient to choose a
pool uniformly at random and analyze its properties. Formally, let $J$ be
chosen uniformly at random from $\\{1,\ldots,N/n\\}$. That is, $A_{J}$ is a
pool chosen uniformly at random. Mirroring notation used in Section 3, we let
$S=\sum_{i\in A_{J}}\mathbbm{1}\\{V_{i}>0\\}$ be the number of positives in
the randomly chosen pool.

In Section 4.3, we assume the joint distributions on $\\{V_{i}:i\in A_{J}\\}$
and $S$ induced by $\mathbb{P}^{(N)}_{0,\alpha}$ and
$\mathbb{P}^{(N)}_{1,\alpha}$ (for naive and correlated pooling respectively)
have a limit as $N$ goes to infinity. We show that these limiting joint
distributions satisfy the assumptions and conditions assumed for
$\mathbb{P}_{i,\alpha}$ in Section 3 for $i=0,1$. Thus, we consider
$\mathbb{P}_{i,\alpha}$ to be equal to the limit of
$\mathbb{P}^{(N)}_{i,\alpha}$ as $N\to\infty$. Hence, we can view the analysis
of a single pool in Section 3 as being an analysis of the randomly chosen pool
$A_{J}$ for large $N$, and thus producing quantities equal to population-level
averages.

###  4.3 Justification of Pool-Level Analysis

Having established the population-level model and justified the use of a
randomly chosen single pool for analyzing a pooling strategy, we examine the
distribution of the sample viral loads in the chosen naive and correlated
pool.

First, we show that sample viral loads in the two pools are identically
distributed for each $N$, which justifies Assumption 3.1 in Section 3. It
follows that the pool prevalence in Section 3 is equal to the population-level
prevalence $\alpha$ here in Section 4.

###### Proposition 4.8

For each $N$, the viral loads of individuals in a randomly chosen naive pool
are identically distributed. The viral load of any individual in a randomly
chosen correlated pool follows the same distribution.

Then, we take $N$ to the asymptotic regime to justify Assumption 3.1 in the
cases where screening is implemented at a large scale, e.g., in a city or
town. We argue that samples in a naive pool are asymptotically independent as
$N\to\infty$. To achieve this, we first define a measure of association
between the viral loads of one individual and a group of individuals. See
Appendix 10.2 for details. We then assume that for any subset of a pool, the
number of individuals in the remaining population with association stronger
than some threshold scales sublinearly in population size (Assumption 10.2).
This enables us to derive the following asymptotic independence property of
naive pooling, which justifies Assumption 3.1 in Section 3.

###### Proposition 4.9

Under Assumption 10.2, the viral loads of individuals in a randomly chosen
naive pool are asymptotically independent as $N\rightarrow\infty$.

We now characterize the correlation between sample viral loads in a correlated
pool based on the notion of “close contacts”. Infected individuals and their
close contacts are assumed to be correlated in infection status and are likely
to be placed into the same pool under correlated pooling. We formulate these
assumptions mathematically.  For each individual $i$ in the population, let
$C_{i}$ denote the set of his/her close contacts. We model $C_{i}$ as
deterministic. The following hold:

  1. 1.

(Bounded infection risk) For any $\alpha$,
$\mathbb{P}_{\alpha}^{(N)}(V_{i}>0)\in\\{0\\}\cup[\epsilon_{0}\alpha,\Pi_{0}\alpha]$
where $0<\epsilon_{0}\leq 1\leq\Pi_{0}$.

  2. 2.

(Existence of close contacts for non-isolated individuals)
$C_{i}\neq\emptyset$ if $\mathbb{P}_{\alpha}^{(N)}(V_{i}>0)>0$.

  3. 3.

(Correlation in infection status) There exists $c_{1}>0$ such that
$\mathbb{P}^{(N)}_{\alpha}(V_{j}>0\mid V_{i}>0)\geq c_{1}$ $\forall j\in
C_{i}$. This holds for any $\alpha$ and any $N$.

  4. 4.

(Correlated pooling) There exists $c_{2}>0$ such that, under correlated
pooling, $\mathbb{P}^{(N)}_{1,\alpha}(j\text{ is in the same pool as }i)\geq
c_{2}$ $\forall j\in C_{i}$. This holds for any $\alpha$ and any $N$.

Assumption 4.3 captures important features of the spread of infectious
diseases and the correlated pooling strategy. The first sub-assumption
prescribes that each individual in the population either (i) could never be
infected due to social isolation; or (ii) could be infected but both the lower
and upper bounds of infection risk are on the same order as the population-
level prevalence. The second sub-assumption is well-justified since for an
individual with non-zero infection risk, he/she must have at least some human-
to-human contact. The third sub-assumption is supported by ample evidence in
the literature for transmission between infected individuals and their close
contacts (World Health Organization 2020, Madewell et al. 2020). The fourth
sub-assumption describes the key feature assumed for correlated pools, namely
that individuals that are close contacts of each other are placed into the
same pool with a non-vanishing probability even as $N$ goes to infinity. This
is justified because in large-scale screening using group testing, correlation
either arises naturally or can be enhanced through explicit measures, as
discussed in Section 4.1.

Assumption 4.3 allows us to derive the following property of correlated pools,
which justifies Condition 1 in Section 3.

###### Proposition 4.10

Under Assumption 4.3, a correlated pool selected uniformly at random satisfies
$\lim_{\alpha\to 0}\mathbb{P}_{1,\alpha}^{(N)}(S>1\mid S>0)>0$ for any $N$,
where $S$ denotes the number of positives in the pool.

Finally, we argue in Appendix 10.5 that the metrics studied in Section 3
($\beta_{i,\alpha}$, $\gamma_{i,\alpha}$ and $\eta_{i,\alpha}$) are
appropriate for evaluating a group testing protocol in the population-wide
screening context. Specifically, we show that quantities computed at the
population level (e.g., the fraction of positive samples in the population
missed by testing) converge in probability to their corresponding pool-level
metrics in Section 3.

Our arguments rely on two mild assumptions (not used elsewhere in this paper),
one on the number of positives identified in each pool given the number of
positives in the pool (Assumption 10.5), the other on the distribution of the
number of positive samples in a pool across all the pools as $N\to\infty$
(Assumption 10.5). For details of the arguments, see Appendix 10.5.

##  5 A Biological Perspective

In this section, we present a realistic model for viral load among infected
individuals. We then describe a realistic sensitivity function for the PCR
test that captures the randomness in the subsampling and pooling processes, an
aspect overlooked by most existing literature studying group testing
protocols. We find that, under these two models, correlated pooling consumes
no more followup tests per positive identified than naive pooling. In Section
6 we will use these models to perform a realistic case study of correlated
pooling implemented based on household membership.

###  5.1 Viral Load Distribution

We first specify a probability distribution governing viral loads across
infected individuals. One way to quantify the viral load in a sample is with
the so-called Ct value. A PCR test amplifies the viral RNA copies in a sample
by approximately doubling them in each cycle of the reaction. The minimum
number of cycles required for the RNA copies to reach a detectable threshold
is called the cycle threshold, denoted Ct (Heid et al. 1996). The lower the
initial viral load in the sample, the more duplicating cycles it requires to
become detectable, and the larger its Ct value is.

Jones et al. (2020) obtains an empirical distribution of Ct values based on
large-scale asymptomatic screening for SARS-CoV-2 conducted in Germany. Brault
et al. (2021) fits a censored Gaussian mixture model (GMM) to this data, where
the censoring accounts for the detection limit of PCR assays (high Ct values
above a certain threshold are not observed in data). It argues that the
fitted, uncensored GMM represents the distribution of Ct values among the
entire infected population. Using an assay-specific formula from Jones et al.
(2020) for converting Ct values to viral load, we transform the GMM on Ct
values into a GMM on the $\log_{10}$ viral load. We include the details of the
GMMs on Ct and $\log_{10}$ viral load in Appendix 11.1.

In our simulation, we assume the viral load of any individual is independent
from the viral loads of all other individuals given his/her infection status.
This assumption is mild given the heterogeneity in the individual biological
response to the virus, which we consider independent. Hence, for each infected
individual, we can sample his/her viral load from the distribution specified
in Table 11.1.

###  5.2 PCR Sensitivity

Whether an individual is infected can be detected by a PCR test. To
investigate the performance of a pooling strategy, we now specify how the
sensitivity of PCR tests depends on viral load.

Most existing mathematical models of group testing treat false negatives of
PCR tests in an oversimplified way, either assuming a fixed false negative
rate or one that is a simple deterministic function of the sample viral load.
(See Section 2.1.) In reality, before entering the PCR machine, a sample
undergoes multiple steps of processing (e.g., subsampling and extraction),
each of which introduces stochasticity into the amount of viral RNA that
remains. Based on the sample handling methodology described in Wyllie et al.
(2020) and the mathematical modeling for liquid partitioning in Basu (2017),
we lay out the steps in a size-$n$ pooled test. We discuss the randomness
associated with each step and how it impacts the final test outcome in
Appendix 11.2. Our modeling of the PCR test is one instantiation of the
general sensitivity function $p(v)$ discussed in Section 3.1 with one
exception that in this realistic model $p(v)=0$ for very small $v$.

###  5.3 Implications of Theorem 3.6 for Test Consumption in Practice

Recall that Theorem 3.6 derived a bound $1+\delta$, where
$\delta=\frac{\mathbb{P}_{1,\alpha}(Y=1,S_{D}=0\mid
S>0)}{\mathbb{P}_{1,\alpha}(Y=1,S_{D}>0\mid S>0)}$, for the ratio of test
consumption of correlated pooling to naive pooling. We now examine this bound
in a realistic setting, given the PCR model and the viral load distribution in
Sections 5.1 and 5.2. We show that in this setting, correlated pooling
consumes no more followup tests per positive identified than naive pooling for
a wide range of pool sizes and PCR test sensitivities ($80\%-97.5\%$).

Since the distribution of $S$ in the pool is not specified in our model, we
give an upper bound $\delta^{\prime}$ for $\delta$ which can then be estimated
directly using Monte Carlo simulation:

$\delta^{\prime}=\frac{\mathbb{P}(Y=1\mid S_{D}=0,S=n)}{\mathbb{P}(Y=1\mid
S_{D}=S=1)}\cdot\frac{\bar{\beta}}{1-\bar{\beta}},$

where $\bar{\beta}$ is the population-average individual test FNR, i.e.,
$\bar{\beta}=\mathbb{E}[1-p(V)\mid V>0]$666 Note that $\bar{\beta}$ is not to
be confused with $\beta_{i,\alpha}$ ($i=0,1)$ introduced in Section 3 which
represents the overall FNR of a specific group testing protocol.. The detailed
derivation of $\delta^{\prime}$ is given in Appendix 12.

Using Monte Carlo simulation with $10^{6}$ replications, we find that across a
wide range of $\bar{\beta}$ and pool sizes, $\delta^{\prime}$ is consistently
close to zero. The maximum value of $\delta^{\prime}$ is $8.96\times 10^{-5}$
(95% CI: $(8.90\times 10^{-4},9.02\times 10^{-5})$), obtained when $n=2$ and
$\bar{\beta}=2.5\%$. As $n$ increases, the relaxed bound converges to 1,
suggesting that in this realistic setting correlated pooling consumes no more
followup tests per positive identified than naive pooling. For detailed
methodology and results of the simulation, see Appendix 12.3.

Now we provide intuition for why $\delta^{\prime}$ is small. We first examine
a representative curve of PCR test sensitivity versus sample viral load under
$\bar{\beta}=5\%$. Based on the viral load distribution among infected
individuals given in Table 11.1, when $\bar{\beta}=5\%$, the PCR test
sensitivity grows rapidly from 0 to 1 over a narrow range of log viral load in
the sample (as shown in Figure 1).

Figure 1: PCR test sensitivity $p(v)$ under $\bar{\beta}=5\%$.

Specifically, a $\log_{10}$ viral load of 3.45 gives a PCR test sensitivity of
0.3%, while a $\log_{10}$ viral load of 3.65 gives a PCR test sensitivity of
99.8%. The fraction of infected individuals that have $\log_{10}$ viral load
between 3.45 and 3.65 is only 2.8%, indicating that the majority of positive
samples either test positive with high probability (if the $\log_{10}$ viral
load is above 3.65) or test positive with low probability (if the $\log_{10}$
viral load is below 3.45). Though not depicted here, the $p(v)$ curves
corresponding to different $\bar{\beta}$ follow the same pattern.

Based on the above observations, we argue that correlated pooling’s test
consumption per positive identified nearly meets or exceeds that of naive
pooling in practice. We first observe that $\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}=0,S=n)$, which is in the numerator of $\delta^{\prime}$, is small. If a
pool contains only $n$ positives that would all test negative individually,
i.e., $S_{D}=0$, then they likely all have viral loads below the narrow region
where an individual test’s sensitivity rises. Thus, the viral load in the
pool, which is the average of the viral loads of these positive samples, is
likely also below the narrow region, making it likely to test negative, i.e.,
$Y=0$.

On the other hand, we argue that $\mathbb{P}(Y=1\mid S_{D}=S=1)$, which is in
the denominator of $\delta^{\prime}$, is reasonably large. In other words, if
a pool contains only one positive sample and it would test positive
individually, then the pool is likely to test positive. With its viral load
drawn from the distribution described in Table 11.1, a positive sample that
would test positive individually has its viral load way above the narrow
region with a reasonably large probability. Hence, even when such a sample is
diluted by a factor equal to the pool size, the pooled sample likely still has
its viral load above the narrow region and is likely to test positive, i.e.,
$Y=1$. We support this argument with the numerical results presented in
Appendix 12.3, Table 12.3.

##  6 Case Study

To illustrate our analysis of correlated pooling above, we simulate a setting
where within-pool correlation is induced by household transmission. We
demonstrate that correlated pooling consistently outperforms naive pooling in
terms of both sensitivity and efficiency. More importantly, we show that
correlated pooling implemented at a large scale enables more effective
epidemic control.

The improvement in sensitivity and efficiency achieved by correlated pooling
relative to naive pooling depends on the joint distribution of viral loads
across samples in the pool. To model this effect realistically, we propose a
model where individuals belong to households, infections are transmitted
between household members, and correlated pooling is implemented based on
household membership. This is a representative and pragmatic instance of
correlated pooling in large-scale screening, as it is logistically practicable
to collect samples by households and subsequently pool households together. We
first describe the simulation setup in detail, separating our discussion into
assumptions about within-household transmission and pooling assignment. Then,
we demonstrate that correlated pooling consistently outperforms naive pooling
in both sensitivity and efficiency.

We further analyze how the demonstrated advantages of correlated pooling
translate to real-world policy-making. In particular, we focus on population-
wide screening as a mitigation measure for an epidemic (where infected
individuals are isolated once identified in screening)777In practice, large-
scale screening can be complementary to other mitigation measures, such as
contact tracing. Positives missed in contact tracing can be found in
screening.. We consider an epidemic to be under control if the number of
active infections in a population stabilizes or declines. Based on an SIR
model (Kermack and McKendrick 1927) that incorporates repeated large-scale
screening, we show that correlated pooling enables more effective epidemic
control.

###  6.1 Experiment Setup

####  6.1.1 Correlated Infections in Households

We model the population as consisting of households with size $H$ ranging from
one to six (since households of size larger than six are rare). We gather the
household size distributions of four countries from census data and assume
that all probability mass on $H>6$ is allocated to $H=6$ (Table 13.1). We also
explore variants of the U.S. census data, in which we either add to or
subtract from the weight on household size of one and adjust the weights on
other household sizes accordingly (Table 13.1).

A household is said to be infected if one person is infected as the index case
in the household. We assume different households are infected independently
with probability $p_{h}$, i.e., correlation through other social groups is
considered negligible. Within each infected household, we assume transmissions
occur independently with secondary attack rate (SAR) $q$. That is, given a
positive index case in a size-$h$ household, the remaining $h-1$ members
become infected independently with probability $q$. We consider the following
possible values for $q$: [0.166, 0.140, 0.193, 0.005, 0.446]. These are the
estimated mean, 95% CI lower and upper bounds, minimum and maximum values of
household SAR from 40 studies, respectively, reported by a meta-analysis
(Madewell et al. 2020).

The distribution of household size $H$, probability that a household is
infected $p_{h}$, and secondary attack rate $q$ together yield an expected
prevalence in the population, which matches the overall population-level
prevalence $\alpha$:

$p_{h}\cdot\mathbb{E}_{H}[(1+(H-1)q)]=\alpha.$ (4)

We now describe the steps for simulating correlated infections within
households, given a fixed population-level prevalence, SAR, and household size
distribution:

  1. 1.

Compute the household infection probability $p_{h}$ using Equation 4.

  2. 2.

Generate households with sizes drawn from the household size distribution.

  3. 3.

Let each household be infected independently with probability $p_{h}$, with
one member selected uniformly at random as the index case.

  4. 4.

In each infected household, generate secondary infections.

  5. 5.

Assign to each infection a viral load sampled from the distribution described
in Table 11.1.

####  6.1.2 Pooling Assignment

Having developed a model for correlated infections in households, we now
describe how we allocate samples into pools when using naive pooling and
correlated pooling, under the Dorfman procedure:

  * •

Naive pooling: we perform an independent random permutation on all the
individual samples from the population and place them sequentially into pools
regardless of household membership.

  * •

Correlated pooling: we aim to place samples of individuals from the same
household in the same pool. A collection of partially full pools are
maintained and households are added sequentially. To add a household, we look
for the first unfinished, capacity-permitting pool and place all samples of
the household into this pool. If this is infeasible, we split the household
across two or more pools.

As is in the Dorfman procedure, samples in the same pool undergo one pooled
test. All individuals in the pools testing positive take followup tests. We
assume the amount of sample collected from each individual is enough so that
no re-sampling is required if the followup test is necessary. This implies
that the viral loads in the subsamples used for the pooled test and followup
test are equal. As is assumed in Section 5.2, the subsample for the pooled
test is smaller than that for an individual test by a factor of the pool size,
which results in dilution in the pooled sample.

###  6.2 Simulation Results

We compare the performance of naive pooling and correlated pooling by
conducting multiple numerical experiments under different sets of parameters.
We investigate the robustness of correlated pooling’s advantage over naive
pooling. First, we pick a set of parameters as the baseline setting, shown in
Table 6.2. We consider this to be a representative setting for a medium-sized
town in the early stage of an epidemic. The choice of pool size is informed by
empirical implementations of group testing for COVID-19 (Fan 2020, Lefkowitz
2020, Barak et al. 2021).

Baseline set of parameters in the sensitivity analysis.  Parameter Value
Population level prevalence 1% Pool size 6 SAR 16.6% Household distribution US
Population-average individual test FNR 5% Population size 12000

We focus on two metrics to evaluate the performance of a group testing
protocol, namely sensitivity (i.e., $1-\mathrm{FNR}$) and efficiency. Both are
important for epidemic mitigation, as high sensitivity helps identify the
positives accurately, while high efficiency permits more frequent screening
under limited resources. Here we present efficiency as the metric for test
consumption because it is most widely used. The performance in the metric
$\gamma_{i,\alpha}$ proposed in Section 3.3, the number of positive cases
identified per PCR test, can be inferred by taking the product of sensitivity
and efficiency.

The performance of naive pooling and correlated pooling in the Dorfman
procedure under the baseline setting over 2000 iterations is shown in Table
6.2. As a reference, only using individual testing has a sensitivity of 95%
and an efficiency of 1. Correlated pooling has better performance in terms of
both sensitivity and efficiency than naive pooling. This is because correlated
pooling in general has more positive cases in a positive-containing pool (due
to correlation among samples from the same household). As a result, a sample
with low viral load, which might otherwise be missed in naive pooling, is more
likely to be “rescued” by other positive samples in the same pool in
correlated pooling, leading to higher sensitivity. (This is referred to as the
“hitchhiker effect” in Barak et al. (2021).) Meanwhile, the clustering of more
positive cases in the same pool also implies a smaller number of pools that
contain positive samples and require followup tests, resulting in a higher
efficiency of correlated pooling. We demonstrate that the advantage of
correlated pooling is robust against deviation in parameter values from the
baseline setting in Appendix 13.2.

Performance of naive and correlated pooling in the Dorfman procedure under the
baseline parameter setting, averaging over 2000 iterations.  Pooling strategy
Sensitivity Efficiency Naive pooling (NP) 81.9% 4.67 Correlated pooling (CP)
86.0% 4.83 Percent advantage of CP over NP 5.02% 3.51% Note: The standard
errors for the sensitivity and efficiency are within 0.1% and 0.01,
respectively.

While such improvement may seem small, it can have a significant impact on
real world policy making. We will show in Section 6.3 that, when pool size is
optimized for both pooling strategies separately, correlated pooling enables
more effective epidemic control than naive pooling.

####  6.2.1 Sensitivity Versus Efficiency Across Pool Sizes

Under the same population-level prevalence, we anticipate test accuracy and
efficiency will vary when we choose different pool sizes. Figure 2 reveals the
tradeoff between sensitivity and efficiency using the two pooling strategies
under different prevalence levels. All parameters other than the prevalence
level and the pool size take the values given in Table 6.2. In most scenarios
(except when under high prevalence and large pool size), correlated pooling
outperforms naive pooling in both sensitivity and efficiency.

(a) $\alpha=0.1\%$

(b) $\alpha=0.5\%$

(c) $\alpha=1\%$

(d) $\alpha=5\%$

(e) $\alpha=10\%$

Figure 2: Tradeoff between sensitivity and efficiency for correlated pooling
and naive pooling under different prevalence levels. As we prefer both higher
sensitivity and higher efficiency, a point in the upper right corner of the
plot is more preferable. Each point is obtained by taking the average outcome
over 2000 simulation runs using a pool size equal to the integer in grey next
to the point.

When prevalence is low (e.g., 0.1%, Figure 2(a)), as pool size increases,
sensitivity decreases and efficiency increases. Under low prevalence, most of
the pools have either zero or one positive sample even when the pool size is
large. A larger pool size causes a stronger dilution effect which lowers the
pooled test sensitivity. Meanwhile, efficiency increases with pool size
because fewer pools are needed, and under low prevalence not many pools
require followup tests even if they are large.

When prevalence is intermediate (e.g., 0.5% or 1%, Figures 2(b) or 2(c)), as
pool size increases, sensitivity decreases because of the dilution effect.
Efficiency, however, reaches a peak first before declining. This is because
large pool size under intermediate prevalence results in many positive pools.
The heightened demand for followup tests offsets the savings in the number of
pooled tests.

When prevalence is high (e.g., 5% or 10%, Figures 2(d) or 2(e)), as pool size
increases, sensitivity first decreases and then increases. This is because a
larger pool size under high prevalence leads to multiple positive samples in
the same pool, offsetting the dilution effect. Efficiency drops dramatically
as pool size increases, since a majority of pools test positive and most
samples require followup tests. The efficiency of large pools under 10%
prevalence, for example, is close to 1, indicating little reduction in test
consumption compared to individual tests. In this scenario, one should
consider using individual testing instead of group testing, as is also
suggested in Eberhardt et al. (2020).

Figure 3 visualizes the advantage of correlated pooling over naive pooling
under different prevalence levels and pool sizes. Except when prevalence
$\alpha=10\%$, pool size $n=40$, correlated pooling is more advantageous. The
advantages in sensitivity and efficiency are both more significant under low
prevalence and when the pool size is large.

Figure 3: The advantage of correlated pooling in (left) sensitivity and
(right) efficiency, over naive pooling. In both heatmaps, the value in the
cell is the metric value of correlated pooling minus that of naive pooling; a
positive value implies that correlated pooling is more advantageous.

####  6.2.2 Test Specificity

As discussed in Section 1, false positives pose challenges to large-scale
screening, including waste of public health and economic resources, disruption
of personal lives, and increased exposure risk during unnecessary treatment.
Though false positives are not explicitly included in our modeling, here we
argue that they are not a significant concern if pooling is used. In
particular, we demonstrate that group testing has substantially lower false
positive rate (FPR) than individual testing, and, moreover, correlated pooling
achieves a lower FPR than naive pooling.

For our discussion, we start by assuming that false positives originate mainly
from lab contamination that occurs independently across tests. We assume any
PCR test on a negative sample has a small constant FPR (e.g., 0.01% as
reported in Public Health Ontario (2020)), which is much smaller than the
probability that a typical positive-containing pool tests positive. Under
these assumptions, the probability that a negative sample in an all-negative
pool is declared positive is negligible (e.g., $10^{-8}$) compared to when it
is in a positive-containing pool. Hence, we estimate the FPR of a testing
protocol by the fraction of negative samples that receive individual tests,
assuming they are all in positive-containing pools. This can be directly
inferred from our simulation results.

First, we compute the fraction of samples in the population receiving
individual tests using
$\texttt{frac}_{\text{indiv}}=\text{efficiency}^{-1}-\frac{1}{n}$. Second, we
estimate $\texttt{frac}_{\text{pos, indiv}}$, the fraction of samples that are
positive and receive individual tests, using
${\alpha\cdot\text{sensitivity}}.$888Note that not all positives receiving
individual tests test positive. Hence, this estimate is an underestimate,
which eventually leads to an upper bound on FPR. We take the difference of the
above two quantities to estimate $\texttt{frac}_{\text{neg, indiv}}$, the
fraction of samples that are negative and receive individual tests.
Multiplying this difference by 0.01% then gives $\texttt{frac}_{\text{neg,
indiv pos}}$, the fraction of samples that are negative and test positive in
individual tests. Finally, we divide the $\texttt{frac}_{\text{neg, indiv
pos}}$ by $1-\alpha$, the fraction of samples that are negative, to obtain the
estimate for FPR.

We summarize the above calculations for correlated pooling and naive pooling
in Table 6.2.2 based on the simulation results for the baseline setting in
Table 6.2. We see that both pooling strategies achieve an FPR on the order of
$10^{-6}$, with correlated pooling slightly outperforming naive pooling. In
our regime of discussion, the FPR roughly scales linearly with pool size and
prevalence. Hence, for a prevalence of up to 1% and a pool size of up to 20,
we expect the FPR of either pooling strategy to be at least as good as
$10^{-5}$. This is a ten-fold reduction from the FPR of individual testing.
Such specificity is sufficiently high in many uses of repeated screening for
infection control.

FPR estimates for naive and correlated pooling under the baseline setting.
Quantity Correlated pooling Naive pooling $\texttt{frac}_{\text{indiv}}$ 4.03%
4.75% $\texttt{frac}_{\text{pos, indiv}}$ 0.86% 0.82%
$\texttt{frac}_{\text{neg, indiv}}$ 3.17% 3.93% $\texttt{frac}_{\text{neg,
indiv pos}}$ 3.17E-6 3.93E-6 FPR estimate 3.20E-6 3.97E-6

We also argue that false positives from PCR tests have little impact on
efficiency, i.e., they incur only a small number of extra tests. In the pooled
stage, 0.01% of the all-negative pools are expected to test positive and
require follow-up tests for their samples. As the number of samples in all-
negative pools is upper bounded by $N$, the extra tests due to PCR false
positives translate to a less than $10^{-4}$ increment in the number of tests
per person. Besides, sensitivity is not affected by false positives of PCR
tests.

###  6.3 Practical Implications for Epidemic Control

In this section, we study how the improvement in sensitivity and efficiency
due to correlated pooling translates to more effective epidemic control. In
specific, we show that, when used for repeated large-scale screening,
correlated pooling requires 12.9% fewer tests per day than naive pooling to
stabilize or reduce the number of active infections in a population with 1%
prevalence.

We consider a setting where policy makers of a city wish to choose a pool size
and screening frequency in using group testing for population-wide screening.
We represent the epidemiological dynamics with a deterministic SIR model
(Kermack and McKendrick 1927) that incorporates screening. We let $S$ and $I$
denote the fraction of susceptible individuals and active infections in the
population, and let $R$ denote the fraction of population “removed” due to
either natural recovery or being detected and isolated in screening followed
by recovery. We assume, for simplicity, that an infected individual is
infectious and a recovered individual does not become susceptible again. We
also assume a constant fraction of the non-isolated population is screened
every day.

We use a set of three discrete-time equations to represent the disease
dynamics, where a time step corresponds to a day:

$\displaystyle S(t+1)-S(t)$ $\displaystyle=\ -b_{I}\cdot S(t)I(t)$
$\displaystyle I(t+1)-I(t)$ $\displaystyle=\ b_{I}\cdot
S(t)I(t)-(b_{R}+f\cdot\text{sensitivity})\cdot I(t)$ (5) $\displaystyle
R(t+1)-R(t)$ $\displaystyle=\ (b_{R}+f\cdot\text{sensitivity})\cdot I(t),$

where $b_{I}$ is the rate of transmission given an interaction between a
susceptible and an infected person; $b_{R}$ is the rate at which an infected
individual recovers on any day (we assume $b_{I}>b_{R}$, since the epidemic
dies out naturally even without intervention if $b_{I}\leq b_{R}$); $f$ is the
frequency of screening for non-isolated individuals, i.e, those in the $S$ and
$I$ groups.

We first derive the critical screening frequency required to control the
epidemic, i.e., stabilize or reduce the number of active infections. To
quantify the epidemic growth, we define the “growth factor” $\lambda$ at time
$t$ as the ratio of the number of new cases at time $t$ to the number of cases
removed at time $t$: $\lambda(t)=\dfrac{b_{I}\cdot
S(t)I(t)}{(b_{R}+f\cdot\text{sensitivity})\cdot I(t)}$.

According to Equation 6.3, the number of infected individuals grows when
$\lambda(t)>1$ and declines when $\lambda(t)<1$. We further construct a time-
invariant upper bound on $\lambda(t)$ by setting $S(t)=1$:
$\lambda^{\prime}=\dfrac{b_{I}}{b_{R}+f\cdot\text{sensitivity}}$.
Alternatively, $\lambda^{\prime}$ can be interpreted as the growth factor in
the early stage of the epidemic, where the majority of the population is
susceptible, i.e., $S(t)\approx 1$.

Since $\lambda(t)\leq\lambda^{\prime}$ for all $t$, any screening frequency
$f$ that results in a $\lambda^{\prime}$ less than 1 also implies
$\lambda(t)<1$ for all $t$. Therefore, we use $\lambda^{\prime}=1$ as a
threshold that characterizes whether the epidemic is brought under control. At
this threshold, the screening frequency has a critical value $f^{*}$
satisfying $f^{*}\times\text{sensitivity}=b_{I}-b_{R}$, which implies that

$f^{*}\propto\text{sensitivity}^{-1}.$ (6)

A larger value of $f$ would reduce $\lambda^{\prime}$ even further, but it
would increase test consumption, a key quantity of practical concern. Hence,
we next use $f^{*}$ to derive the minimum test consumption required for
epidemic control. For a screening frequency $f$, test consumption per day
satisfies:

test consumption per day $\displaystyle\propto\ \text{screening frequency}\
\times\ \text{\\# tests consumed per person}$ $\displaystyle=\ f\ \times\
\text{efficiency}^{-1}.$ (7)

By Equations 6 and 7,

minimum test consumption per day $\displaystyle\propto\ f^{*}\ \times\
\text{efficiency}^{-1}$ $\displaystyle\propto\ \text{sensitivity}^{-1}\
\times\ \text{efficiency}^{-1}\propto\ \gamma_{i,\alpha}^{-1},$ (8)

where we recall that both sensitivity and efficiency depend on the pool size,
prevalence level, and pooling choice (whether correlated or naive pooling is
adopted). As discussed in Section 3.3, $\gamma_{i,\alpha}$, the expected
number of positives identified per PCR test, is directly proportional to
$\text{sensitivity}\times\text{efficiency}$. Hence, for a certain pool size,
$\gamma_{i,\alpha}^{-1}$ provides a proxy for the minimum test consumption
that enables epidemic control. Therefore, one should maximize
$\gamma_{i,\alpha}$ (or, equivalently, sensitivity $\times$ efficiency) when
optimizing the pool size for a group testing protocol in real-world decision
making.

Table 1 compares the optimal naive pooling and correlated pooling policies (by
choosing a pool size that maximizes sensitivity $\times$ efficiency) under
different prevalence levels. The last column of Table 1 illustrates the
reduction in minimum test consumption required for epidemic control using the
optimal correlated pooling policy relative to the optimal naive pooling
policy.

Table 1: Comparison of optimal correlated pooling and naive pooling policies in terms of sensitivity $\times$ efficiency under different prevalence levels.  Prevalence | Optimal naive pooling | Optimal correlated pooling |  Reduction in test consumption when using correlated pooling
---|---|---|---
Pool size | Sensitivity $\times$ Efficiency | Pool size | Sensitivity $\times$ Efficiency
0.1% | 40 | 13.52 | 40 | 15.86 | 14.8%
0.5% | 15 | 6.29 | 20 | 7.26 | 13.4%
1% | 12 | 4.56 | 12 | 5.23 | 12.9%
5% | 6 | 2.17 | 6 | 2.44 | 10.9%
10% | 4 | 1.59 | 4 | 1.72 | 7.4%

For example, when prevalence is 1% and we only consider correlation among
samples from the same household, a pool size of 12 is optimal for both naive
pooling and correlated pooling in terms of maximizing sensitivity $\times$
efficiency. Using Equation 8, we derive that compared to the optimal naive
pooling policy, the optimal correlated pooling policy uses
$\frac{1/4.56-1/5.23}{1/4.56}=12.9\%$ fewer tests.

We argue that such difference has a substantial impact on policy-making in
real-world scenarios. As illustrated earlier, correlation in infection
statuses exists due to interaction within social groups such as households,
schools and offices, and is preserved to some extent in pools. Hence, policies
informed by analyses ignoring correlation tend to be too pessimistic because
the predicted test consumption overestimates the reality. We describe two
possible resulting scenarios below:

  * •

The available testing capacity of the city meets the minimum test consumption
required by the optimal correlated pooling strategy but not the optimal naive
pooling strategy. Assuming naive pooling, the policy maker decides that no
screening policy can permit safe reopening and thus issues a lockdown.
However, had the policy maker taken into account the existence of within-pool
correlation, he/she could have safely reopened the economy with a feasible
screening policy.

  * •

The available testing capacity of the city meets the minimum test consumption
required by the optimal naive pooling strategy, so the policy maker decides to
reopen. However, since naive pooling underestimates the actual efficiency, the
policy maker chooses a lower screening frequency than allowed by the available
testing capacity. Had the policy maker accounted for correlation, he/she could
have picked a higher screening frequency and achieved better epidemic
mitigation.

Furthermore, if the naturally-induced within-pool correlation is weak,
explicit measures can be taken to facilitate correlated pooling. For example,
one can mandate that individuals from the same household get tested together
so that their samples can be placed in the same pool without many logistical
difficulties. For a city with limited resources, such measures could enable a
safe reopen with population-wide screening, while it may not be feasible
otherwise.

##  7 Conclusion

In this paper, we proved that under a general correlation structure in the
population and other mild assumptions, for the same pool size, correlated
pooling achieves higher sensitivity than naive pooling and consumes comparable
or fewer tests per positive identified compared to naive pooling. We used
numerical experiments to quantify the advantage of correlated pooling over
naive pooling in both sensitivity and efficiency and substantiated its real
world implications for epidemic control.

Our work can be extended in several directions in future research. First, we
focused our study on correlated pooling in the standard two-stage Dorfman
procedure. However, in other group testing protocols, such as hierarchical and
combinatorial group testing, each sample is assigned to multiple pools.
Within-pool correlation structures are more convoluted in these settings.
Second, study has shown that a simple SIR model and a similarly configured
stochastic compartmental model can behave differently (Koopman et al. 2002).
Hence, it would be interesting to explore the effect of correlation in a
stochastic compartmental model that incorporates social dynamics and
transmission within the network. One might also consider studying the
heterogeneity in infection time, infection duration, and viral load of
different individuals, which imposes a more complicated correlation structure.
An agent-based simulation model may be a good approach to achieve this. Third,
it would be meaningful to incorporate sampling noise, where the sample viral
load could be zero for an infected individual. The additional transmission due
to undetected individuals may counteract the benefits offered by correlated
pooling, and such consideration is of practical interest for large-scale
epidemic control. This could be addressed using latent variable models.

The authors are grateful to Diego Diel and Jeff Pleiss for conversations on
the implementation details of PCR tests. The authors also thank Stephen Chick,
Saskia Comess, Claire Donnat, and Susan Holmes for providing valuable
feedback. This work was conducted under the support from Cornell University
when the authors served in the Cornell COVID mathematical modeling team.
Additional support was provided by Air Force Office of Scientific Research
FA9550-19-1-0283. J.W. and Y.Z. contributed equally to this paper.

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Online Appendix: Correlation Improves Group Testing

##  8 Proofs of Theorems 3.2, 3.6 and Corollary 3.7

###  8.1 Proof of Theorem 3.2

###### Proof 8.11

Proof of Theorem 3.2. For $i=0,1$, we have that the overall false negative
rate is given by

$\displaystyle\beta_{i,\alpha}$
$\displaystyle=1-\frac{\mathbb{E}_{i,\alpha}[\text{\\# positives identified in
a pool}]}{\mathbb{E}_{i,\alpha}[\text{\\# positives in a pool}]}$
$\displaystyle=1-\frac{\mathbb{E}_{i,\alpha}[D]}{n\alpha}$
$\displaystyle=1-\frac{\mathbb{E}_{i,\alpha}[\sum_{j=1}^{n}YW_{j}]}{n\alpha}$
$\displaystyle=1-\frac{1}{n\alpha}\cdot\sum_{j=1}^{n}\mathbb{E}_{i,\alpha}[YW_{j}]$
$\displaystyle=1-\frac{1}{n\alpha}\cdot\sum_{j=1}^{n}\mathbb{E}_{i,\alpha}[\mathbb{E}_{i,\alpha}[YW_{j}\mid
V_{1:n}]]$
$\displaystyle=1-\frac{1}{n\alpha}\cdot\sum_{j=1}^{n}\mathbb{E}_{i,\alpha}[\mathbb{E}_{i,\alpha}[Y\mid
V_{1:n}]\mathbb{E}_{i,\alpha}[W_{j}\mid V_{j}]].$

In both correlated pooling and naive pooling, all $V_{j}$’s are identically
distributed by Proposition 4.8, which follows that
$(\mathbb{E}_{i,\alpha}[Y\mid V_{1:n}],\mathbb{E}_{i,\alpha}[W_{j}|V_{j}])$
are also identically distributed. Hence, we obtain that

$\displaystyle\beta_{i,\alpha}$ $\displaystyle=1-\frac{1}{n\alpha}\cdot
n\cdot\mathbb{E}_{i,\alpha}[\mathbb{E}_{i,\alpha}[Y\mid
V_{1:n}]\mathbb{E}_{i,\alpha}[W_{1}\mid V_{1}]]$
$\displaystyle=1-\frac{1}{\alpha}\cdot\mathbb{E}_{i,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\right]\quad\text{where
$\bar{V}_{n}=\frac{1}{n}\sum_{j=1}^{n}V_{j}$}$
$\displaystyle=1-\frac{1}{\alpha}\mathbb{E}_{i,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0\right]\mathbb{P}_{\alpha}(V_{1}>0)$
$\displaystyle=1-\mathbb{E}_{i,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0\right].$ (9)

For naive pooling, the $V_{i}$’s are i.i.d. Hence,

$\displaystyle\beta_{0,\alpha}$
$\displaystyle=1-\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]\mathbb{P}_{0,\alpha}\left(S=\ell\mid
V_{1}>0\right)\quad\text{recall that
$S=\sum_{j=1}^{n}\mathbbm{1}\\{V_{i}>0\\}$}$
$\displaystyle=1-\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]{n-1\choose\ell-1}\alpha^{\ell-1}(1-\alpha)^{n-\ell}.$

Taking $\alpha\to 0^{+}$ gives

$\displaystyle\lim_{\alpha\to 0^{+}}\beta_{0,\alpha}$
$\displaystyle=\lim_{\alpha\to
0^{+}}\Bigg{(}1-\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=1\right]{n-1\choose 1-1}\alpha^{1-1}(1-\alpha)^{n-1}\Bigg{)}$
$\displaystyle=1-\mathbb{E}\left[p\left(\frac{1}{n}V_{1}\right)p(V_{1})\mid
V_{1}>0\right].$

Similarly, we derive $\beta_{1}$ for correlated pooling. Following Equation 9
we have that

$\displaystyle\beta_{1,\alpha}$
$\displaystyle=1-\sum_{\ell=1}^{n}\mathbb{E}_{1,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]\mathbb{P}_{1,\alpha}\left(S=\ell\mid V_{1}>0\right)$
$\displaystyle\overset{\Delta}{=}1-\sum_{\ell=1}^{n}A_{\ell}\cdot\mathbb{P}_{1,\alpha}(S=\ell\mid
S>0)$

where
$A_{\ell}\overset{\Delta}{=}\mathbb{E}_{1,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]$.

When $\ell=1$, $A_{1}=\mathbb{E}[p(\frac{1}{n}V_{1})p(V_{1})\mid V_{1}>0]$.
When $\ell\geq 2$, we have $\bar{V}_{n}>\frac{1}{n}V_{1}$ because there exists
at least one $j\neq 1$ such that $V_{j}>0$. Assuming $p(v)$ is a monotone
increasing function in $v$, we obtain $p(\bar{V}_{n})\geq
p(\frac{1}{n}V_{1})$, which, combined with $p(V_{1})>0$ given $V_{1}>0$,
implies that $A_{\ell}\geq A_{1}$.

Therefore, taking $\alpha\to 0^{+}$ gives

$\displaystyle\lim_{\alpha\to 0^{+}}\beta_{1,\alpha}$
$\displaystyle=1-\lim_{\alpha\to
0^{+}}\sum_{\ell=1}^{n}A_{\ell}\cdot\mathbb{P}_{1,\alpha}(S=\ell\mid S>0)$
$\displaystyle=1-\sum_{\ell=1}^{n}A_{\ell}\cdot\lim_{\alpha\to
0^{+}}\mathbb{P}_{1,\alpha}(S=\ell\mid S>0)\quad\text{$A_{\ell}$'s do not
involve $\alpha$ because they condition on $S=\ell$}$ $\displaystyle\leq
1-\sum_{\ell=1}^{n}A_{1}\cdot\lim_{\alpha\to
0^{+}}\mathbb{P}_{1,\alpha}(S=\ell\mid S>0)$ $\displaystyle=1-A_{1}$
$\displaystyle=\lim_{\alpha\to 0^{+}}\beta_{0,\alpha}.$

The inequality is strict if $p(v)$ is strictly increasing in $v$ and Condition
1 holds. Condition 1 implies that there exists $\ell\geq 2$ such that
$\lim_{\alpha\to 0^{+}}\mathbb{P}_{1,\alpha}(S=\ell\mid S>0)>0$. If $p(v)$ is
strictly increasing in $v$, $A_{\ell}>A_{1}$ for $l>1$, which implies that
$\lim_{\alpha\to 0^{+}}\beta_{1,\alpha}<\lim_{\alpha\to
0^{+}}\beta_{0,\alpha}$.

###  8.2 Proof of Theorem 3.6

###### Proof 8.12

Proof of Theorem 3.6.

We first derive $\eta_{0,\alpha}$ for naive pooling. By similar arguments in
the Proof of Theorem 3.2, the denominator of $\eta_{0,\alpha}$ is given by

$\displaystyle\mathbb{E}_{0,\alpha}[D]$
$\displaystyle=\mathbb{E}_{0,\alpha}\left[\sum_{j=1}^{n}YW_{j}\right]$
$\displaystyle=n\alpha\cdot\mathbb{E}_{0,\alpha}[p(\bar{V}_{n})p(V_{1})\mid
V_{1}>0]$
$\displaystyle=n\alpha\cdot\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]{n-1\choose\ell-1}\alpha^{\ell-1}(1-\alpha)^{n-\ell}.$
(10)

The numerator of $\eta_{0,\alpha}$ is given by

$\displaystyle n\mathbb{E}_{0,\alpha}[Y]$
$\displaystyle=n\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\right]$
$\displaystyle=n\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\mid
S=\ell\right]\mathbb{P}_{0,\alpha}\left(S=\ell\right)$
$\displaystyle=n\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\mid
S=\ell\right]\binom{n}{\ell}\alpha^{\ell}(1-\alpha)^{n-\ell}$
$\displaystyle=n\alpha\cdot\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\mid
S=\ell\right]\binom{n}{\ell}\alpha^{\ell-1}(1-\alpha)^{n-\ell}.$ (11)

By definition of $\eta_{0,\alpha}$ and Equations 10 and 11, taking $\alpha\to
0^{+}$ gives

$\displaystyle\lim_{\alpha\to 0^{+}}\eta_{0,\alpha}$
$\displaystyle=\lim_{\alpha\to
0^{+}}\frac{n\mathbb{E}_{0,\alpha}[Y]}{\mathbb{E}_{0,\alpha}[D]}$
$\displaystyle=\lim_{\alpha\to
0^{+}}\frac{\cancel{n\alpha}\cdot\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\mid
S=\ell\right]\binom{n}{\ell}\alpha^{\ell-1}(1-\alpha)^{n-\ell}}{\cancel{n\alpha}\cdot\sum_{\ell=1}^{n}\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=\ell\right]{n-1\choose\ell-1}\alpha^{\ell-1}(1-\alpha)^{n-\ell}}$
$\displaystyle=\frac{\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)\mid
S=1\right]\cdot{n\choose
1}}{\mathbb{E}_{0,\alpha}\left[p\left(\bar{V}_{n}\right)p(V_{1})\mid
V_{1}>0,S=1\right]}$
$\displaystyle=\frac{n\cdot\mathbb{E}\left[p\left(\frac{1}{n}V_{1}\right)\mid
V_{1}>0\right]}{\mathbb{E}\left[p\left(\frac{1}{n}V_{1}\right)p(V_{1})\mid
V_{1}>0\right]}\quad\text{because $V_{i}$'s are iid}$ (12) (the denominator is
nonzero because $p(v)>0\ \forall v>0$)
$\displaystyle=\frac{n\cdot\mathbb{E}[p(\frac{1}{n}V_{1})\mid
V_{1}>0]}{\mathbb{E}[p(\frac{1}{n}V_{1})W\mid V_{1}>0]}$
$\displaystyle=n\cdot\frac{\sum_{j=0,1}\mathbb{E}[p(\frac{1}{n}V_{1})\mid
V_{1}>0,W_{1}=j]\cdot\mathbb{P}(W_{1}=j\mid
V_{1}>0)}{\mathbb{E}[p(\frac{1}{n}V_{1})\mid
V_{1}>0,W_{1}=1]\cdot\mathbb{P}(W_{1}=1\mid V_{1}>0)}$
$\displaystyle=n\cdot\left(1+\frac{\mathbb{E}[p(\frac{1}{n}V_{1})\mid
V_{1}>0,W_{1}=0]}{\mathbb{E}[p(\frac{1}{n}V_{1})\mid
V_{1}>0,W_{1}=1]}\cdot\frac{\mathbb{P}(W_{1}=0\mid
V_{1}>0)}{\mathbb{P}(W_{1}=1\mid V_{1}>0)}\right).$ (13)

Then, we derive $\eta_{1,\alpha}$ for correlated pooling.

$\displaystyle\eta_{1,\alpha}$
$\displaystyle=\frac{n\mathbb{E}_{1,\alpha}[Y]}{\mathbb{E}_{1,\alpha}[D]}$
$\displaystyle=\frac{n\mathbb{E}_{1,\alpha}[Y]}{\mathbb{E}_{1,\alpha}[\sum_{j=1}^{n}YW_{j}]}$
$\displaystyle=n\cdot\frac{\mathbb{E}_{1,\alpha}[Y\mid
S>0]\cancel{\mathbb{P}_{1,\alpha}(S>0)}}{\mathbb{E}_{1,\alpha}[YS_{D}\mid
S>0]\cancel{\mathbb{P}_{1,\alpha}(S>0)}}$ (14)
$\displaystyle=n\cdot\frac{\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}>0)\mathbb{P}_{1,\alpha}(S_{D}>0\mid S>0)+\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}=0,S>0)\mathbb{P}_{1,\alpha}(S_{D}=0\mid
S>0)}{\mathbb{E}_{1,\alpha}[YS_{D}\mid
S_{D}>0]\mathbb{P}_{1,\alpha}(S_{D}>0\mid S>0)}$ $\displaystyle\leq
n\cdot\frac{\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}>0)\mathbb{P}_{1,\alpha}(S_{D}>0\mid S>0)+\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}=0,S>0)\mathbb{P}_{1,\alpha}(S_{D}=0\mid
S>0)}{\mathbb{P}_{1,\alpha}(Y=1\mid S_{D}>0)\mathbb{P}_{1,\alpha}(S_{D}>0\mid
S>0)}$ because $\mathbb{E}_{1,\alpha}[YS_{D}\mid
S_{D}>0]\geq\mathbb{E}_{1,\alpha}[Y\mid S_{D}>0]=\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}>0)$ (15) (both terms in the denominator are nonzero because $p(v)>0\
\forall v>0$) $\displaystyle=n\left(1+\frac{\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}=0,S>0)}{\mathbb{P}_{1,\alpha}(Y=1\mid
S_{D}>0)}\cdot\frac{\mathbb{P}_{1,\alpha}(S_{D}=0\mid
S>0)}{\mathbb{P}_{1,\alpha}(S_{D}>0\mid S>0)}\right)$
$\displaystyle=n\left(1+\frac{\mathbb{P}_{1,\alpha}(Y=1,S_{D}=0\mid
S>0)}{\mathbb{P}_{1,\alpha}(Y=1,S_{D}>0\mid S>0)}\right).$ (16)

Lower-bounding Equation 13 by $n$ and using Equation 16 gives the desired
result.

###  8.3 Proof of Corollary 3.7

###### Proof 8.13

Proof of Corollary 3.7.

We apply the threshold sensitivity function to the calculation of
$\lim_{\alpha\to 0^{+}}\eta_{0}$ and $\eta_{1}$. In Equation 13, the first
term on the numerator in the parenthesis is given by

$\displaystyle\mathbb{E}\left[p\left(\frac{1}{n}V_{1}\right)\mid
V_{1}>0,W_{1}=0\right]$
$\displaystyle=\mathbb{E}\left[\mathbbm{1}\left\\{\frac{1}{n}V_{1}\geq
u_{0}\right\\}\mid V_{1}>0,V_{1}<u_{0}\right]=0,$

which implies $\lim_{\alpha\to 0^{+}}\eta_{0,\alpha}=n$.

In Equation 16, the numerator of the last term is given by

$\displaystyle\mathbb{P}_{1,\alpha}(Y=1,S_{D}=0\mid S>0)$
$\displaystyle=\mathbb{P}_{1,\alpha}(\bar{V}_{n}\geq u_{0},V_{j}<u_{0}\
\forall j\text{ s.t. }V_{j}>0\mid S>0)=0,$

which implies $\eta_{1,\alpha}\leq n$.

##  9 Example Where Correlated Pooling Has Lower Efficiency

We give an example of sensitivity and viral load distribution under which
correlated pooling has lower test efficiency, contrary to the claims in the
literature. Through this example, we also show the necessity of the bound in
Theorem 3.6.

Consider a piecewise constant sensitivity function $p$, where
$\theta_{1},\theta_{2}\in(0,1)$ and $\theta_{1}<\theta_{2}$:

$\displaystyle p(v)$ $\displaystyle=\begin{cases}0&v=0\\\
\theta_{1}&v\in(0,2)\\\ \theta_{2}&v\in[2,3)\\\ 1&v\in[3,+\infty)\end{cases}.$

Consider a correlated pool consisting of two samples with the joint viral load
distribution given in Table 9. By Assumptions 3.1 and 3.1, the corresponding
naive pool contains two samples whose viral loads are independent with the
same marginal distribution as that in Table 9. We set $\alpha=1\%$.

Joint viral load distribution in a correlated pool.  $V_{2}=0$ $V_{2}=2$
$V_{2}=3$ $V_{1}=0$ $1-3\alpha/2$ $0$ $\alpha/2$ $V_{1}=2$ $0$ $\alpha/2$ 0
$V_{1}=3$ $\alpha/2$ 0 0

Recall that, for $i=0,1$, referring to naive and correlated pooling
respectively, and prevalence $\alpha$, we have the following notation:

  * •

$\beta_{i,\alpha}$ is the overall FNR of the pooling strategy;

  * •

$\eta_{i,\alpha}$ is the number of followup tests consumed per positive
identified;

  * •

$\text{efficiency}_{i,\alpha}$ is the number of individuals screened per test
consumed. By Equation 3, it is also given by the following expression:

$\text{efficiency}_{i,\alpha}=\left(\frac{1}{n}+\alpha\eta_{i,\alpha}(1-\beta_{i,\alpha})\right)^{-1}\quad\text{for
}i=0,1.$ (17)

To derive efficiency, we first calculate $\beta_{i,\alpha}$ and
$\eta_{i,\alpha}$ for $i=0,1$. Using Equation 9, we derive the overall FNR for
naive and correlated pooling:

$\displaystyle\beta_{0,\alpha}$
$\displaystyle=1-\alpha\left(\frac{1}{2}\theta_{2}^{2}+\frac{1}{4}\theta_{2}+\frac{1}{4}\right)-(1-\alpha)\left(\frac{1}{2}\theta_{1}\theta_{2}+\frac{1}{2}\theta_{1}\right)$
$\displaystyle\beta_{1,\alpha}$
$\displaystyle=1-\left(\frac{1}{2}\theta_{2}^{2}+\frac{1}{2}\theta_{1}\right).$
(18)

Second, we derive $\eta_{i,\alpha}$ for $i=0,1$. By definition, we have

$\eta_{i,\alpha}=\frac{n\mathbb{E}_{i,\alpha}[Y]}{\mathbb{E}_{i,\alpha}[D]},$
(19)

where $\mathbb{E}_{i,\alpha}[D]=n\alpha(1-\beta_{i,\alpha})$ and
$\mathbb{E}_{i,\alpha}[Y]=\mathbb{E}_{i,\alpha}\left[p\left(\bar{V}\right)\right]$.
Hence, to compute $\eta_{i,\alpha}$ it suffices to compute
$\mathbb{E}_{i,\alpha}[Y]$ for $i=0,1$. We find that

$\displaystyle\mathbb{E}_{0,\alpha}[Y]$
$\displaystyle=2\alpha(1-\alpha)\cdot\theta_{1}+\frac{3}{4}\alpha^{2}\cdot\theta_{2}+\frac{1}{4}\alpha^{2}$
$\displaystyle\mathbb{E}_{1,\alpha}[Y]$
$\displaystyle=\alpha\cdot\theta_{1}+\frac{1}{2}\alpha\cdot\theta_{2}.$ (20)

Plugging Equations 18 and 20 into Equation 19 gives the expressions for
$\eta_{0,1}$, $i=0,1$. Plugging in the expressions for $\beta_{i,\alpha}$ and
$\eta_{i,\alpha}$ into Equation 17 gives the expressions for efficiency under
naive and correlated pooling. We plot the difference between
$\text{efficiency}_{1,\alpha}$ and $\text{efficiency}_{0,\alpha}$ in Figure
4(a). We also plot the ratio of $\eta_{1,\alpha}$ to $\eta_{0,\alpha}$, shown
in Figure 4(b).

Region ‘A’ in Figure 4(a) corresponds to $(\theta_{1},\theta_{2})$ pairs where
correlated pooling has lower efficiency than naive pooling. Within this
region, there are two scenarios:

  1. 1.

$\eta_{1,\alpha}>\eta_{0,\alpha}$. This occurs when $(\theta_{1},\theta_{2})$
also falls into region ‘B’ in Figure 4(b)999Though not depicted, a small area
at the top of region ‘B’ (where $\theta_{1}$ and $\theta_{2}$ are both close
zero) corresponds to $\beta_{1,\alpha}>\beta_{0,\alpha}$, both of which are
close to 1. This does not violate Theorem 3.2 which considers the asymptotic
scenario..

  2. 2.

$\beta_{1,\alpha}<\beta_{0,\alpha}$ and $\eta_{1,\alpha}<\eta_{0,\alpha}$.
This occurs when $(\theta_{1},\theta_{2})$ falls outside region ‘B’ in Figure
4(b)101010If correlated pooling has lower efficiency, i.e.,
$\eta_{1,\alpha}(1-\beta_{1,\alpha})>\eta_{0,\alpha}(0-\beta_{1,\alpha})$ in
Equation 17, it is not possible to have both $\eta_{1,\alpha}<\eta_{0,\alpha}$
and $\beta_{1,\alpha}\geq\beta_{0,\alpha}$..

Furthermore, we take $\alpha$ to the limit of zero and observe that the region
where $\lim_{\alpha\to 0^{+}}\frac{\eta_{1,\alpha}}{\eta_{0,\alpha}}>1$
resembles region ‘B’ in Figure4(b) where $\alpha$ takes 0.01. This
necessitates the $(1+\delta)$ bound in Theorem 3.6.

(a)

(b)

Figure 4: (a) The difference in test efficiency between correlated and naive
pooling for valid values of $(\theta_{1},\theta_{2})$. A negative value
(region enclosed by red lines, labeled ‘A’) means correlated pooling screens
fewer individuals per test than naive pooling. (b) The ratio of the number of
followup tests consumed per positive identified using correlated pooling to
that using naive pooling. A value larger than 1 (region enclosed by the red
curve, labeled ‘B’) indicates correlated pooling consumes more followup tests
per positive case identified.

##  10 Supplementary Analysis for the Population-Level Model

###  10.1 Proof of Proposition 4.8

###### Proof 10.14

Proof of Proposition 4.8.

Let $I$ denote the population index of an arbitrary individual from the naive
pool. Because naive pools are formed by picking individuals uniformly at
random from the population, $I\sim U(\\{1,\cdots,N\\})$. That is,
$\mathbb{P}_{0}^{(N)}(I=i)=\frac{1}{N}$ for all $i=1,\cdots,N$. The cdf of the
viral load of this sample is

$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(V_{I}\leq v)$
$\displaystyle=\sum_{i=1}^{N}\mathbb{P}_{0}^{(N)}(I=i)\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v)$
$\displaystyle=\sum_{i=1}^{N}\frac{1}{N}\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v).$

Suppose the correlated pool being studied is the $J$th of the $|\mathcal{A}|$
correlated pools. Because the correlated pool we are studying is chosen
randomly from the $|\mathcal{A}|$ pools,
$\mathbb{P}^{(N)}(J=j^{\prime})=\frac{1}{|\mathcal{A}|}$ for all
$j^{\prime}=1,\cdots,|\mathcal{A}|$. Now consider an arbitrary individual from
this pool, and suppose this individual is the $i$th of this pool. Recall that
we reordered samples in each pool by performing an independent random
permutation of 1 through $n$, denoted by $\pi$. Then, the index of $i$ before
the permutation is uniformly distributed over $\\{1,\cdots,N\\}$, that is,
$P^{(N)}(\pi(i^{\prime})=i)=\frac{1}{n}$ for all $i^{\prime}=1,\cdots,n$. Let
$V_{(j^{\prime},i^{\prime})}$ denote the viral load of the $i^{\prime}$th
sample in the $j^{\prime}$th pool before reordering, for each
$i^{\prime},j^{\prime}$. Then, the cdf of the sample viral load of this
arbitrary individual from the correlated pool is given by

$\displaystyle\mathbb{P}_{1,\alpha}^{(N)}(V\leq v)$
$\displaystyle=\sum_{j^{\prime}=1}^{|\mathcal{A}|}\sum_{i^{\prime}=1}^{n}\mathbb{P}^{(N)}(J=j^{\prime})\mathbb{P}^{(N)}(\pi(i^{\prime})=i)\mathbb{P}_{\alpha}^{(N)}(V_{(j^{\prime},i^{\prime})}\leq
v)$
$\displaystyle=\frac{1}{|\mathcal{A}|}\frac{1}{n}\sum_{j^{\prime}=1}^{|\mathcal{A}|}\sum_{i^{\prime}=1}^{n}\mathbb{P}_{\alpha}^{(N)}(V_{(j^{\prime},i^{\prime})}\leq
v)$
$\displaystyle=\frac{1}{N}\sum_{j^{\prime}=1}^{|\mathcal{A}|}\sum_{i^{\prime}=1}^{n}\mathbb{P}_{\alpha}^{(N)}(V_{(j^{\prime},i^{\prime})}\leq
v)$
$\displaystyle=\sum_{k=1}^{N}\frac{1}{N}\mathbb{P}_{\alpha}^{(N)}(V_{k}\leq
v),$

where the last equality follows from the observation that this double sum is
equivalent to summing over all individuals in $\\{1,\cdots,N\\}$. This is
identical to the cdf of the viral load of an individual chose uniformly at
random from the naive pool.

###  10.2 Association Model

We define a measure of association between the viral loads of one individual
and a group of individuals. Consider a collection of individuals $\BFj$, whose
population indices are denoted $\\{j_{1},\cdots,j_{|\BFj|}\\}$. For a
population of size $N$, we define the cumulative distribution function for the
viral load $v$ of individual $i\in[N]\backslash\BFj$ conditioning on the viral
loads $\BFz\in\mathbb{R}^{|\BFj|}$ of the individuals in $\BFj$:

$\displaystyle
F^{(N)}_{\alpha}(i,\BFj,v,\BFz)=\mathbb{P}^{(N)}_{\alpha}(V_{i}\leq v\mid
V_{j_{k}}=z_{k},k=1,\cdots,|\BFj|).$

Based on this, we define a measure of association between the viral loads of
$i$ and $\BFj$:

$\displaystyle\Delta_{\alpha}^{(N)}(i,\BFj)=\sup_{\begin{subarray}{c}v\in\mathbb{R}_{\geq
0}\\\ \BFz,\BFz^{\prime}\in\mathbb{R}_{\geq
0}^{|\BFj|}\end{subarray}}|F_{\alpha}^{(N)}(i,\BFj,v,\BFz)-F_{\alpha}^{(N)}(i,\BFj,v,\BFz^{\prime})|.$

This quantity is the maximum change in the cdf of $i$’s viral load that can be
created by varying the viral loads of $\BFj$. It reflects the degree to which
conditioning on the viral loads of $\BFj$ affects the viral load of $i$. A
larger $\Delta_{\alpha}^{(N)}(i,\BFj)$ indicates a stronger association
between $i$ and $\BFj$. The collection of individuals having association with
$\BFj$ stronger than $\epsilon$ is

$\displaystyle\\{i:\Delta_{\alpha}^{(N)}(i,\BFj)>\epsilon\\}.$

We denote by $m_{\alpha}^{(N)}(\epsilon)$ the maximum size of such sets,
across any collection $\BFj$ of at most $n-1$ individuals. When
$m_{\alpha}^{(N)}(\epsilon)$ is small relative to $N$, when we add an
individual $i$ to the pool who is chosen uniformly from the larger population,
they are unlikely to be in a set with high association
$\Delta_{\alpha}^{(N)}(i,\BFj)>\epsilon$ with the individuals already in the
pool. This makes the viral loads in the pool unlikely to be strongly
correlated. Recalling that the pool size is $n$, we have

$\displaystyle
m_{\alpha}^{(N)}(\epsilon)=\max_{\begin{subarray}{c}\BFj\subset\\{1,\cdots,N\\}\\\
|\BFj|<n\end{subarray}}\left|\\{i:\Delta_{\alpha}^{(N)}(i,\BFj)>\epsilon\\}\right|.$
(21)

Now we take $N$ to the asymptotic regime and make the following assumption.
There exists a sequence $\epsilon_{N}\downarrow 0$ such that
$\lim_{N\rightarrow\infty}\frac{1}{N}m_{\alpha}^{(N)}(\epsilon_{N})=0$.
Assumption 10.2 prescribes that as population size $N$ goes to infinity, for
any collection $\BFj$ of individuals of size less than $n$, the set of
individuals that have association stronger than $\epsilon_{N}$ with $\BFj$
grows slower than linearly in population size. In an epidemic like COVID-19,
transmission typically takes place between close contacts (World Health
Organization 2020). It is reasonable to assume that for two individuals to be
associated in infection statuses, they have to be within a few degrees of
contact with each other. Since the duration of the infectious period is
finite, and a person’s contact rate is typically bounded above by some
constant (Hu et al. 2013) even as population size grows large, the number of
people connected to an individual in $\BFj$ via within a few degrees of
contact grows slower than linearly in population size. Hence, this assumption
is well-justified.

###  10.3 Proof of Proposition 4.9

###### Proof 10.15

Proof of Proposition 4.9.

Let random variables $[1],[2],\cdots,[n]$ be the population indices of the
individuals placed into this randomly chosen naive pool $J$. We use
superscript $(N)$ and subscript $\alpha$ to index the quantities computed for
a size $N$ population at prevalence $\alpha$. We assume
$\lim_{N\to\infty}\mathbb{P}_{i,\alpha}^{(N)}(\\{V_{i}:i\in A_{J}\\}\in\cdot)$
exists for $i=0,1$.

To prove the proposition, we want to show that the joint cdf of viral loads in
a naive pool factors into a product of cdf’s of individual viral loads as
$N\rightarrow\infty$. Let $v\in\mathbb{R}_{\geq 0}^{n}$. We first use the law
of conditional probability to expand the joint cdf:

$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(V_{[1]}\leq
v_{1},\cdots,V_{[n-1]}\leq v_{n-1},V_{[n]}\leq v_{n})$ $\displaystyle=~{}$
$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(V_{[1]}\leq
v_{1},\cdots,V_{[n-1]}\leq
v_{n-1})\cdot\mathbb{P}_{0,\alpha}^{(N)}(V_{[n]}\leq v_{n}\mid V_{[1]}\leq
v_{1},\cdots,V_{[n-1]}\leq v_{n-1}).$ (22)

To analyze the conditional probability in the second term, we first make the
following claim: For all $\BFj\subset\\{1,\cdots,N\\}$ with $|\BFj|=n-1$ and
$i\notin\BFj$,

$\displaystyle\left|\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid
V_{j_{1}}\leq v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v_{n})\right|\leq\Delta_{\alpha}^{(N)}(i,\BFj).$ (23)

To prove the claim, we first apply the law of iterated expectations to the
second term on the left hand side of Equation 23:

$\displaystyle\left|\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid
V_{j_{1}}\leq v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n})\right|$
$\displaystyle=~{}$ $\displaystyle\Big{|}\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v_{n}\mid V_{j_{1}}\leq v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-\mathbb{E}_{\BFz\in\mathbb{R}_{\geq
0}^{n-1}}[\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid
V_{j_{1}}=z_{1},\cdots,V_{j_{n-1}}=z_{n-1})]\Big{|}$ $\displaystyle=~{}$
$\displaystyle\Big{|}\mathbb{E}_{\BFz\in\mathbb{R}_{\geq
0}^{n-1}}\left[\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid V_{j_{1}}\leq
v_{1},\cdots,V_{j_{n-1}}\leq v_{n-1})-\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v_{n}\mid V_{j_{1}}=z_{1},\cdots,V_{j_{n-1}}=z_{n-1})\right]\Big{|}$
$\displaystyle=~{}$ $\displaystyle\Big{|}\mathbb{E}_{\BFz\in\mathbb{R}_{\geq
0}^{n-1}}\left[\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid V_{j_{1}}\leq
v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz)\right]\Big{|}$
$\displaystyle\leq~{}$ $\displaystyle\mathbb{E}_{\BFz\in\mathbb{R}_{\geq
0}^{n-1}}\left[\left|\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid
V_{j_{1}}\leq v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz)\right|\right].$ (24)

We expand and bound the first conditional probability in Inequality 24

$\displaystyle\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid V_{j_{1}}\leq
v_{1},\cdots,V_{j_{n-1}}\leq v_{n-1})$ $\displaystyle=~{}$
$\displaystyle\frac{\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n},V_{j_{1}}\leq
v_{1},\cdots,V_{j_{n-1}}\leq v_{n-1})}{\mathbb{P}_{\alpha}^{(N)}(V_{j_{1}}\leq
v_{1},\cdots,V_{j_{n-1}}\leq v_{n-1})}$ $\displaystyle=~{}$
$\displaystyle\frac{\int_{\BFz^{\prime}\in[0,v_{1}]\times\cdots\times[0,v_{n-1}]}\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq
v_{n}\mid
V_{j_{1}}=z_{1}^{\prime},\cdots,V_{j_{n-1}}=z^{\prime}_{n-1})f(V_{j_{1}}=z_{1}^{\prime},\cdots,V_{j_{n-1}}=z^{\prime}_{n-1})d\BFz^{\prime}}{\int_{\BFz^{\prime}\in[0,v_{1}]\times\cdots\times[0,v_{n-1}]}f(V_{j_{1}}=z_{1}^{\prime},\cdots,V_{j_{n-1}}=z^{\prime}_{n-1})d\BFz^{\prime}}$
$\displaystyle=~{}$
$\displaystyle\frac{\int_{\BFz^{\prime}\in[0,v_{1}]\times\cdots\times[0,v_{n-1}]}F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz^{\prime})f(V_{j_{1}}=z_{1}^{\prime},\cdots,V_{j_{n-1}}=z^{\prime}_{n-1})d\BFz^{\prime}}{\int_{\BFz^{\prime}\in[0,v_{1}]\times\cdots\times[0,v_{n-1}]}f(V_{j_{1}}=z_{1}^{\prime},\cdots,V_{j_{n-1}}=z^{\prime}_{n-1})d\BFz^{\prime}}$
$\displaystyle\in~{}$
$\displaystyle\left[\inf_{\BFz^{\prime}\in\mathbb{R}_{\geq
0}^{n-1}}F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz^{\prime}),\
\sup_{\BFz^{\prime}\in\mathbb{R}_{\geq
0}^{n-1}}F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz^{\prime})\right].$

Therefore,

$\displaystyle\left|\mathbb{P}_{\alpha}^{(N)}(V_{i}\leq v_{n}\mid
V_{j_{1}}\leq v_{1},\cdots,V_{j_{n-1}}\leq
v_{n-1})-F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz)\right|$
$\displaystyle\leq\sup_{\BFz,\BFz^{\prime}}\left|F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz^{\prime})-F_{\alpha}^{(N)}(i,\BFj,v_{n},\BFz)\right|$
$\displaystyle\leq\sup_{v,\BFz,\BFz^{\prime}}\left|F_{\alpha}^{(N)}(i,\BFj,v,\BFz^{\prime})-F_{\alpha}^{(N)}(i,\BFj,v,\BFz)\right|$
$\displaystyle=\Delta_{\alpha}^{(N)}(i,\BFj)\quad\text{by definition}.$

That is, the absolute difference is bounded by a constant, which does not
change when taken expectation over $\BFz$. This proves the claim.

Claim 23 enables a closer analysis of the conditional probability in Equation
22. Using the law of iterated expectations where we condition on
$[1],[2],\cdots,[n]$ (hereafter abbreviated as $[1:n]$), we have that

$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(V_{[n]}\leq v_{n}\mid V_{[1]}\leq
v_{1},\cdots,V_{[n-1]}\leq v_{n-1})$ $\displaystyle=~{}$
$\displaystyle\mathbb{E}_{0,\alpha}^{(N)}\left[\mathbb{P}_{\alpha}^{(N)}(V_{[n]}\leq
v_{n}\mid V_{[1]}\leq v_{1},\cdots,V_{[n-1]}\leq v_{n-1},[1:n])\right]$
$\displaystyle\leq~{}$
$\displaystyle\mathbb{E}_{0,\alpha}^{(N)}\left[\mathbb{P}_{\alpha}^{(N)}(V_{[n]}\leq
v_{n}\mid[n])+\Delta_{\alpha}^{(N)}([n],[1:n-1])\right]$ $\displaystyle=~{}$
$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(V_{[n]}\leq
v_{n})+\mathbb{E}_{0,\alpha}^{(N)}\left[\Delta_{\alpha}^{(N)}([n],[1:n-1])\right].$
(25)

We consider two cases for the expectation in the second term. For any
$\epsilon>0$, $\Delta_{\alpha}^{(N)}([n],[1:n-1])$ could either be less than
$\epsilon$, or greater than $\epsilon$ but upper bounded by 1. That is,

$\displaystyle\mathbb{E}_{0,\alpha}^{(N)}\left[\Delta_{\alpha}^{(N)}([n],[1:n-1])\right]$
$\displaystyle\leq
1\cdot\mathbb{P}_{0,\alpha}^{(N)}(\Delta_{\alpha}^{(N)}([n],[1:n-1])>\epsilon)+\epsilon\cdot\mathbb{P}_{0,\alpha}^{(N)}(\Delta_{\alpha}^{(N)}([n],[1:n-1])\leq\epsilon)$
$\displaystyle\leq\mathbb{P}_{0,\alpha}^{(N)}(\Delta_{\alpha}^{(N)}([n],[1:n-1])>\epsilon)+\epsilon.$
(26)

Now, let’s unpack the first term in this expression.

$\displaystyle\mathbb{P}_{0,\alpha}^{(N)}(\Delta_{\alpha}^{(N)}([n],[1:n-1])>\epsilon)$
$\displaystyle=\mathbb{E}_{0,\alpha}^{(N)}\left[\mathbb{P}_{0,\alpha}^{(N)}(\Delta_{\alpha}^{(N)}([n],[1:n-1])>\epsilon\mid[1:n-1])\right]$
$\displaystyle=\mathbb{E}_{0,\alpha}^{(N)}\left[\frac{|\\{i:\Delta_{\alpha}^{(N)}(i,[1:n-1])>\epsilon\\}|}{N-(n-1)}\mid[1:n-1]\right]$
because under $\mathbb{P}_{0,\alpha}^{(N)}$, $[n]$ takes values other than
$[1:n-1]$ with equal probability