Binary outcomes—which have two distinct levels (e.g., disease
yes/no)—are commonly collected in global health research. The relative
association of an exposure (e.g., a treatment) and such an outcome can be
quantified using a ratio measure such as a risk ratio or an odds ratio.
Although the odds ratio is more frequently reported than the risk ratio,
many researchers, policymakers, and the general public frequently interpret
it as a risk ratio. This is particularly problematic when the outcome is
common because the magnitude of association is larger on the odds ratio
scale than the risk ratio scale. Some recently published global health
studies included misinterpretation of the odds ratio, which we hypothesize
is because statistical methods for risk ratio estimation are not well known
in the global health research community.
Objectives:
To compare and contrast available statistical methods to estimate relative
measures of association for binary outcomes and to provide recommendations
regarding their use.
Methods:
Logistic regression for odds ratios and four approaches for risk ratios: two
direct regression approaches (modified log-Poisson and log-binomial) and two
indirect methods (standardization and substitution) based on logistic
regression.
Findings:
Illustrative examples demonstrate that misinterpretation of the odds ratio
remains a common issue in global health research. Among the four methods
presented for estimation of risk ratios, the modified log-Poisson approach
is generally preferred because it has the best numerical performance and it
is as easy to implement as is logistic regression for odds ratio
estimation.
Conclusions:
We conclude that, when study design allows, studies with binary outcomes
should preferably report risk ratios to measure relative association.
Publisher's note: A correction article relating to this article has been published and can be found at https://annalsofglobalhealth.org/articles/10.5334/aogh.2998/.
Introduction
Binary outcomes—which have two distinct levels (e.g., disease yes/no)—are
commonly measured in global health research. Examples include depression status
[1], disease status [2], and mortality [3],
among others. These binary outcomes may either be true “yes or no”
variables (e.g., mortality) or be created from an underlying continuous variable
(e.g., when depression status is determined by dichotomizing a psychological scale).
For example, one recent article created a binary HIV-related knowledge variable by
dichotomizing a total HIV-related knowledge score at the median [4].
The relative association of an exposure and binary outcome can be quantified through
the use of a ratio measure such as a risk ratio or odds ratio. The risk ratio is
defined as the risk of the outcome in the exposed group over the risk of outcome in
the unexposed group, where an exposure could be a treatment (intervention)
assignment or some other binary predictor (e.g., obesity yes/no). For example, using
data from a randomized controlled trial (RCT) of an intervention to increase the
proportion of febrile individuals testing for malaria [5], the estimated “risk” of testing for malaria is
higher in treatment than control (Table 1;
risk ratio = 1.45). The odds ratio is defined as the odds of the outcome in the
exposed group over the odds of the outcome in the unexposed group, where the odds of
the outcome in a group is the proportion with the outcome over the proportion
without the outcome. The odds ratio for this example is 2.7, which is larger than
the risk ratio.
Example of two relative measures of association, adapted from results of a
randomized controlled trial in febrile individuals published in BMJ
Global Health.^{a}
^{a} RCT by O’Meara et al. (2016) [5] was a 2 × 2 factorial design of two
interventions for febrile individuals. Here we have adapted the example
to focus on one of those interventions, namely a subsidy for a rapid
diagnostic test, where “intervention” denotes the group that
received the subsidy and “control” denotes the group that
did not receive the subsidy. Specifically, we have extracted outcome
data from Table 2 of
O’Meara et al. (2016) [5]
for the two groups which did not receive the second intervention.
^{b} Row counts correspond to the number of participants with
each level of the outcome within each exposure group.
^{c} For intervention group vs. control group, where we note that
O’Meara et al. (2016) [5]
reported neither of these results in their Table 2 because they instead reported absolute measures of
effect.
The odds ratio (OR) is the only valid measure of relative association in traditional
case-control studies, namely cumulative case-control studies, because the sampling
of controls (e.g., survivor sampling) does not provide a valid estimate of the risk
of exposure in the source population [6]. But
for studies that use sampling that is dependent on the exposure of
interest—including cohort and cross-sectional studies, and randomized
controlled trials—the risk ratio (RR) is a valid alternative measure of
relative association. Yet, in many of these studies, the OR is the only relative
measure of associated reported [7]. This
popularity is likely because it is straightforward to implement the logistic
regression approach that is typically used to estimate ORs.
Despite its widespread use, the OR is frequently misinterpreted as an RR by
researchers, journalists, policymakers, and the general public [8]. As shown by the example above, interpreting
the OR of 2.7 as an RR would considerably overstate the impact of the intervention
evaluated in this RCT. Such a large difference in magnitude between the two relative
measures of associations arises here because the reference (control) arm risk (51%)
shows that malaria testing is a common outcome in the study setting. In contrast, in
situations where the reference risk is not large because the outcome is not common
(e.g., <10%), the odds ratio would approximate the risk ratio and therefore the
potential for misinterpretation is greatly reduced.
The purpose of the current paper is to provide researchers with the tools to be able
to obtain appropriate and interpretable measures of relative association in studies
with binary outcomes. To do so, we compare and contrast the risk ratio and odds
ratio, provide examples of the misinterpretation of odds ratios from the recent
global health literature, describe methods for obtaining risk ratios in analyses of
binary outcomes, and make recommendations for selecting the most appropriate
analysis. In addition, we briefly discuss the merits of including an absolute
measure of association along with a relative measure. Our goal is to assist global
health researchers in making informed decisions about when to report the odds ratio
or the risk ratio to measure relative association.
Relative Measures of AssociationMotivating Example
In the introduction, we presented an example from O’Meara et al. who
reported the results of a 2 × 2 factorial RCT, examining the independent
and combined effects of two different subsidy interventions (subsidies for rapid
diagnostic tests and subsidies for malaria treatment) on the proportion of
febrile individuals testing for malaria, a binary outcome [5]. For simplicity, we considered only one of the
interventions, namely subsidies for rapid diagnostic tests (RDTs), ignoring the
fact that a second intervention was evaluated in the study, and reproduced the
reported outcome data from Table 2 of
O’Meara et al. [5] (Table 1). The probability (“risk”) of
testing for malaria in the RDT subsidy arm is 73.8%, whereas this probability is
51% in the no subsidy (control) arm. Thus, as noted above, the estimated RR is
1.45, and the estimated OR is 2.7. That is, the RDT subsidy is associated with
1.45 times the “risk”, or 2.7 times the odds of malaria testing when
compared with no subsidy.
Unadjusted measures of relative association from three articles in the
global health literature.
Exposure
Outcome
Unexposed group outcome proportion
Risk Ratio^{a}
Odds Ratio^{b}
Magnitude of odds ratio relative to risk
ratio^{c}
1
Surviving Ebola virus [32]
Safe sexual behavior
14%
2.71
3.67
35%
2
Point-of-care testing [33]
Antibiotic use
78%
0.82
0.50
178%
3
Drinking [34]
Feelings of aggression
20%
3.1
6.7
116%
^{a} Risk ratio (for “exposed” vs.
“unexposed”) computed directly from outcome proportions
reported in the article as none of the three articles used the risk
ratio as a measure of relative association.
^{b} Odds ratio is obtained from unadjusted logistic
regression [32] or directly
from outcome proportions reported [3334].
^{c} In these examples where the outcome is relatively common
(i.e., >10%), if the odds ratio were to be incorrectly
interpreted as a risk ratio, this is the magnitude of overstatement
of relative association.
At this stage, it is valuable to make a note on the terminology of
“risk” and “risk ratio”. Although in the strictest
sense, a risk is defined in epidemiology as the “probability of an event
during a specified period of time [9]”, in common usage, “risk” refers more generally to
a probability, and the term risk ratio or relative risk is commonly used in
research to describe a relative association, even when the probability does not
involve an element of time (e.g., cross-sectional prevalence of an outcome).
Hence, we use the term “risk ratio” (RR) throughout to refer to a
ratio of probabilities or prevalences.
Uses and (Mis)interpretations
Although the authors did not misinterpret the degree of association of the
subsidy intervention and the proportion who tested for malaria, the
O’Meara example can be used to demonstrate the potential for
misinterpretation of the OR. We have stated that the RR of 1.45 can be
interpreted as “receiving RDT subsidy is associated with being 1.45 times
more likely to test for malaria compared to those not receiving a
subsidy”. But, as Schwartz et al. [8] point out in critiquing a prominent study and the media’s
interpretation of the results, this is also how the OR is
commonly interpreted. It is natural to want to interpret the OR of 2.7 as
“2.7 times more likely to test for malaria if receiving RDT
subsidy.” However, this is not the correct
interpretation. In the O’Meara et al. example, if the OR is incorrectly
interpreted as an RR, it overstates the intervention effect by almost double
(1.45*2 = 2.9).
As pointed out by many authors, the interpretation of an OR is not intuitive, and
ORs are easily misinterpreted [71011]. And, importantly, even when authors are careful in their
interpretation, for instance by using language such as “receiving
treatment is associated with 2.7 times the odds of outcome compared with
control”, it is still natural for the news media and other readers of the
research to interpret it as an RR—that is, a ratio of probabilities rather
than a ratio of odds [89101112].
In the example above, the OR and RR were so different because the proportion of
the study sample testing for malaria (the outcome) was so high (51.0% in
control). In the case of a high outcome proportion, the OR is pulled away from
the null value (i.e. an OR and an RR of 1.0) more than the RR. In Figure 1, we display this relationship by graphing
the OR and RR for various levels of the reference probability (e.g., probability
of testing for malaria in the no subsidy arm), including for settings where the
intervention is associated with a reduction in the probability of the outcome
(i.e. with the RR and OR both <1). As can be seen, the higher the reference
probability of the event (e.g., tested for malaria) the more the OR overstates
the RR (i.e., OR < RR if both are <1 and OR > RR if both are >1), if
the OR is incorrectly interpreted as an RR. Similarly, the figure shows that
when the outcome is not common (e.g., <10%), the OR closely approximates the
RR, and therefore, in such cases the OR may be interpreted as an RR.
Relationship between the odds ratio and risk ratio at various levels of
the reference risk.
Why does this matter?
Although there is some literature on methods to compute RRs for binary outcomes,
most of it has appeared in epidemiological [13141516171819], medical [112021222324252627], or statistics journals [28293031]. This may partly explain why ORs are still commonly used and
misinterpreted across a wide array of papers and throughout the media. The
following three examples from the global health literature, and which are
summarized in Table 2, present further
evidence to the global health research community about why this issue is
important.
In the first example, it was reported that Ebola virus disease survivors
“were more than five times as likely to engage in safe sexual behavior
compared with the comparison group”, based on an adjusted OR of 5.59,
obtained from a logistic regression model with adjustment for relevant
covariates selected by the authors [32].
Given that the corresponding unadjusted OR is 35% larger than the unadjusted RR
(3.67 vs. 2.71), the adjusted RR is expected to be approximately 4.14 (vs.
5.59), which would indicate that survivors were “more than four times as
likely to engage in safe sexual behavior”. This example shows that even in
studies with a relatively low reference probability of the outcome of interest,
interpreting the OR as an RR would lead to an overstatement of the
association.
In a second example from the global health literature, an adjusted OR of 0.49
comparing antibiotic use in the intervention (64%) and control (78%) groups was
estimated in a RCT of a point-of-care testing intervention to reduce antibiotic
use [33] The unadjusted OR of 0.50
corresponds to an unadjusted RR of 0.82. If this OR were interpreted as an RR,
the association of the intervention and outcome would be overstated almost
three-fold (i.e., 50% vs. 18% reduction). This article can be used to further
emphasize why the distinction between the two relative measures of association
is important. Suppose that the intervention is associated with an increase in
adverse events. Reporting the OR instead of the RR may make it more challenging
to balance the pros of the intervention in terms of reduction in antibiotic
overuse vs. the cons due to adverse events. The more high-impact the study, the
more likely the conclusions will affect policy and, hence, will affect
people’s lives.
The third example from the global health literature is an example of
misinterpretation of the OR in the news media. The article reports on the
results of a study examining the emotions associated with alcohol consumption
[34], in which an adjusted OR of 6.41
comparing heavy drinkers with light drinkers for the outcome of “feelings
of aggression” was interpreted as heavy drinkers were “just over six
times more likely to report feelings of aggression” than light drinkers
[34]. In reporting on the study the
news article picked up on this interpretation and stated that “those who
showed signs of alcohol dependence were six times more likely to say they felt
aggression while drinking [12]”.
However, using numbers provided in Table 2 of the paper we find an unadjusted OR for this association of 6.7
while the unadjusted RR is 3.1. Thus, the OR overstates the RR more than twofold
and, instead, a more appropriate interpretation would be that “heavy
drinkers were just over three times more likely to report feelings of aggression
than light drinkers”.
Methods of Obtaining Risk Ratios For Binary Outcomes
Given the difficulty in interpreting the odds ratio, several methods of obtaining
risk ratios have been proposed in the literature and have been implemented in a
range of studies. For simplicity and brevity, here we describe some advantages and
disadvantages of four commonly used methods [29]. Summaries and examples of use of these four methods in the global
health literature are given in Table 3. Two
of the methods are regression-based approaches that directly estimate the RR
(log-binomial [35] and modified log-Poisson
[16]). In this case, the estimated
coefficient of the association between exposure and outcome, when exponentiated, is
directly interpreted as a risk ratio. The other two methods indirectly obtain the RR
(substitution [25] and standardization [18]) by estimating an OR from a logistic
regression model, then computing the RR as a function of the OR through some form of
transformation. Although commonly used in many settings, these methods are only
necessary when adjusting for covariates. This is because when there is no need to
adjust for covariates, simple formulas can be used instead of modeling.
Nevertheless, in many cases, researchers will use model-based methods even when not
adjusting for covariates as they are similarly valid in unadjusted models and are
straightforward to implement.
Brief summary of four methods of obtaining risk ratios for binary
outcomes.
Name of method
Type of method
Background literature
Some advantages
Some disadvantages
Example of use in the global health
literature
Exposure
Binary Outcome
Log-binomial
Direct
Wacholder (1986) [35]
Easy to implement.
May not converge; may estimate
individual-level probabilities (and/or the upper bound of their 95%
confidence intervals) above 1.
Gibson et al. (2017) [37]
Mobile phone based intervention to improve
immunization rates, in a cluster-randomized trial
Full immunization by 12 months of age.
Modified log-Poisson
Direct
Zou (2004) [16]
Easy to implement; almost always
converges.
May estimate individual-level probabilities
(and/or the upper bound of their 95% confidence intervals) above 1.
Chan et al. (2017) [38]
AIDS-related stigma
Probable depression (PHQ-9 score ≥10 or
recent suicidal thoughts).
Substitution
Indirect
Zhang and Yu (1998) [25]
Easy to implement. Uses output from logistic
regression.
Generally produces biased estimates and 95%
confidence intervals are expected to be too narrow, on average [18].
Agweyu et al. (2018) [39]
Various demographics and health-related
exposures
Mortality.
Marginal or Conditional Standardization
Indirect
Localio et al. (2007) [18]
Uses output from logistic regression.
May be more difficult to implement and
interpret than other methods, especially in certain software
packages.
Weobong et al. (2017) [40]
Psychological intervention for depression, in
a randomized trial
Remission from depression as measured by the
PHQ-9.
Abbreviation: PHQ-9 – Patient Health Questionnaire 9-item [36], a screening tool for
depression.
The first direct estimation method, the log-binomial approach, is a generalized
linear model like logistic regression which also uses a binomial outcome
distribution but uses a log link rather than a logit link function [35]. While this is an attractive option because
it is simple to implement in standard statistical software, the model may fail to
converge to a solution, especially when the outcome is common [1841].
The second direct estimation method is the modified log-Poisson model [16]. In this approach, a Poisson model with
log-link is fitted to the binary data, which is “modified” by using
robust standard errors to obtain valid statistical inference. This approach is
simple to implement in many statistical software packages, and generally does not
suffer from the same convergence issues as the log-binomial.
While the log-binomial and modified log-Poisson regression approaches are appealing,
both may estimate individual-level outcome probabilities and/or the upper bound of
their 95% confidence intervals above one for binary outcome data. If the intention
is inference about associations, this is generally not a major issue. However, if
the goal is estimation of individual-level risk, then these two methods will
sometimes be inappropriate and estimate individual-level risk above one, especially
when the outcome is very common and the variance of adjustment variables is high
[1142].
Two logistic-regression based approaches that indirectly obtain the RR through
transformation are substitution and standardization. Zhang and Yu [25] proposed a substitution method, in which a
simple formula—which includes the odds ratio and the prevalence of the outcome
in the unexposed group—is used to convert the OR (and its 95% confidence
interval [CI]) obtained using a standard logistic regression model to an RR (with
95% CI). Although this method is often cited and used in practice, simulations
suggest that 95% CIs obtained using this method suffer from poor coverage, such that
they are too narrow and type I error is inflated (too high) [18]. Additionally, several authors have pointed out that such
simple substitution methods produce biased RRs [13141820]. Thus, this method
is not recommended and there is currently no simple formula with desirable
statistical properties to convert an OR to an RR.
The second indirect logistic regression-based approach is the standardization method
proposed by Localio et al. [18] and described
in further detail in Muller & MacLehose [43]. This method fits a logistic regression model and uses the estimated
regression coefficients to obtain an estimated RR by using marginal standardization
whereby the proportions with the outcome in the exposed and unexposed groups are
estimated and, from these, the corresponding RRs are estimated. We obtain these
proportions as the estimated probability of the outcome within each unexposed and
exposed group at specified values of the other covariates in the model (e.g., at the
mean of continuous variables). Under certain assumptions, this marginalized effect
at each level of the exposure is the prevalence of the outcome we would have
observed had everyone been assigned to that level of the exposure and to the
specified values of the other covariates in the model. That is, although termed a
“marginalized effect” when adjusted for covariates in this way, it is
also conditional on the level of those covariates [43].
Cummings compares these methods with others and finds that, except for the often
biased substitution method, they generally produce similar estimates of the RR
[29]. Therefore, on balance, we find that
the most easily implementable approach with the fewest drawbacks is the modified
log-Poisson approach. Given that it is as easy to implement as logistic regression
in all major software, there should be no barriers to global health researchers
estimating and reporting the RR as a measure of relative association of an exposure
and binary outcome, when study design allows. In Table 4, we provide code for fitting both the log-binomial and
modified log-Poisson models in four commonly used statistical software (R, SAS,
Stata, and SPSS), as well as code to implement the marginal standardization approach
in both R and Stata. For the marginal standardization approach, SAS and SPSS code
are not provided as to our knowledge it is not easily implemented in these two
programs.
Code to fit the log-binomial and modified log-Poisson models in four commonly
used statistical software packages, and to use the marginal standardization
method in two of the packages.
Abbreviations: Ind = Independent (i.e., non-clustered); Clust =
Clustered.
Variables: binaryoutcome = the binary outcome; exposure = exposure (e.g.,
treatment group indicator), assumed to be categorical; participantID =
participant identifier; cluster = cluster identifier.
^{a} The log-binomial code for direct estimation of the risk
ratio in the clustered setting is only shown in the generalized
estimating equations (GEE) framework. A generalized linear mixed model
(GLMM) could also be used.
^{b} For the log-Poisson approach, a robust standard error is
needed to account for misspecification of the outcome distribution
(i.e., Poisson instead of binomial); GEE is the natural approach to
obtain this robust standard error, in both the non-clustered and
clustered setting.
^{c} To our knowledge, the marginal standardization method is not
as straightforward to implement in SAS or SPSS, so no code is provided.
In addition, we are unaware of an easy-to-implement function in R to
perform marginal standardization in a clustered setting.
^{d} In the context of GEE to analyze clustered outcome data, we
have used an exchangeable working correlation matrix as an example. It
is natural to use such a working correlation matrix when the outcome
data are measured at a single point in time and the clustering arises
through some natural grouping of individuals (e.g., in schools or
hospitals). But, if the clustering arises from longitudinal data, other
working correlation structures may be preferred.
^{e} The standard errors from Stata may be slightly larger than
that obtained from the other programs. This is because Stata multiplies
the robust standard errors by K/(K–1), where K is the number of
clusters, whereas other programs do not do this.
^{f} The cbind R code illustrated here works only for a single
binary exposure variable. It will need to be modified for more complex
scenarios. Additionally, the gee function requires that the outcome be
set up as a numeric variable, rather than a factor variable, when
specifying the modified log-Poisson model.
Additionally, cluster randomized trials (CRTs) are common in global health research.
In such cases, statistical models must take into account the fact that the data
collected on participants within the same cluster are likely to be correlated. When
the outcome is binary, the generalized estimating equations (GEE) approach is an
appealing method to analyze CRT data because of its desirable statistical properties
(e.g., population-average interpretation; robustness to model misspecification;
ability to correct for small-sample bias in the case of fewer than 40 clusters
enrolled in the CRT) [444546]. Both the
log-binomial and modified log-Poisson [30]
models can be easily implemented in the GEE framework, and code to do so is provided
in Table 4. Similarly, in some software, the
marginal standardization procedure can also be easily adapted to clustered binary
outcome data. This code could also be used for non-randomized studies with
clustering of outcomes.
Absolute Measures of Association
Just as RRs can be directly estimated as a measure of relative association, direct
and indirect methods are available to estimate absolute measures of association such
as risk differences. Although a discussion of the advantages and disadvantages of
such methods is beyond the scope of the current article, it is important to note
that absolute measures of association are able to provide important and
complementary information about the public health impact of interventions. In fact,
both the Consolidated Standards of Reporting Trials (CONSORT) and the Strengthening
the Reporting of Observational Studies in Epidemiology (STROBE) statements recommend
that all RCTs and observational studies reporting on the association of an exposure
with binary outcomes provide both a relative measure and absolute measure [4748].
As an example of the importance of reporting both types of measures, suppose that the
probability of malaria testing uptake in the RDT subsidy arm was 0.02%, while in the
no subsidy arm it was 0.01%. In this case, the risk ratio is 2, whereas the risk
difference is 0.01 percentage points. From a public health perspective, such a small
risk difference may be considered of insufficient magnitude to justify the increased
costs and potential adverse effects of providing subsidies. However, if only the
relative measure were reported, those involved in scaling up the intervention may be
misled to believe the intervention is more effective than it actually is. Thus, we
also strongly recommend reporting both a relative and absolute measure of
association when reporting on binary outcomes.
The statistical methods for obtaining an absolute measure of association
(specifically, the risk difference) are straightforward to implement. For both the
modified log-Poisson and log-binomial models, instead of using a log link, an
identity link can be used. In this case, the regression coefficients will have a
straightforward interpretation as the difference in risk between the levels of the
predictor (e.g., intervention and control). Additionally, for the marginal
standardization method, once we have obtained the estimated mean risks for the
levels of the predictor, we simply subtract the estimated mean risks to obtain the
risk difference. In sum, the same methods that can be used to obtain risk ratio can
also be used to obtain risk differences with only minor modifications.
Conclusion
We have shown that many methods exist to estimate risk ratios for binary outcome data
and that the global health researcher need not feel compelled to present odds ratios
for studies with sampling which depends upon the exposure, such as cohort and
cross-sectional observational studies and randomized controlled trials. Overall, the
modified log-Poisson regression approach to generate RRs is generally preferred to
alternative approaches due to its ease of implementation and desirable statistical
properties. While we have not provided an exhaustive review of methods for
estimating RRs, Cummings [20] provides an
excellent review of alternative methods, and also provides Stata code for
implementing various approaches [29]. In
addition, Muller & MacLehose discuss marginal standardization (as well as other
methods), and provide Stata code for implementing marginal standardization [43]. As noted earlier, sample code for fitting
the two direct regression approaches in four statistical programs, as well as
performing marginal standardization in two of those programs, is provided in Table
4. As with fitting any model, researchers
should be aware of and test assumptions underlying the model, and consider how
interpretation changes when adjusting for covariates.
Given concerns with interpretation, especially since results of research are commonly
used to implement and scale up interventions, we believe that estimation and
reporting of odds ratios should be reserved for use mainly when performing
case-control studies. In this case, the risk ratio will be directly estimated by the
OR if base population sampling (i.e., a case-cohort design) is used. It should be
noted, however, that for other forms of control sampling (i.e., risk-set sampling
and survivor sampling), the risk ratio would not be validly estimated by the odds
ratio [49].
We have provided researchers with the information needed to decide upon the most
appropriate and interpretable measures of relative association to present in studies
with binary outcomes, while also describing in detail the tools needed to obtain
these relative measures. Our hope is that this information will assist researchers
in providing the best evidence on the association between exposures and binary
outcomes in observational studies, as well as on the effectiveness of interventions
evaluated in RCTs in global health settings.
Acknowledgements
The authors wish to thank Joseph Egger, Kristie Kusibab, and Monica Harding of the
Duke Global Health Institute Research Design and Analysis Core for their helpful
comments on earlier versions of this manuscript. This paper was inspired by the
authors’ work on the Bachpan study (PI: Dr. Joanna Maselko) and the Innovative
public-private partnership to target subsidized antimalarials study (PI: Dr. Wendy
Prudhomme O’Meara). The paper received partial funding from National
Institutes of Health awards R01AI110478 and K01MH104310 (ELT), and R01HD075875 (both
JAG and ELT), and award OPP1171753 from the Bill and Melinda Gates Foundation (ELT).
Disclaimer: The content is solely the responsibility of the authors and does not
necessarily represent the official views of the National Institutes of Health or the
Bill and Melinda Gates Foundation.
Funding Information
Partial funding from National Institutes of Health awards R01AI110478 and K01MH104310
(ELT), and R01HD075875 (both JAG and ELT), and award OPP1171753 from the Bill and
Melinda Gates Foundation (ELT).
Competing Interests
The authors have no competing interests to declare.
Author Contribution
All authors had access to the data and a role in writing the manuscript.
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