Why Evidence for the Fetal Origins of Adult Disease Might Be a Statistical Artifact: The “Reversal Paradox” for the Relation between Birth Weight and Blood Pressure in Later Life

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DOI: 10.1093/aje/kwi002 · Source: PubMed
Abstract
Some researchers have recently questioned the validity of associations between birth weight and health in later life. They argue that these associations might be due in part to inappropriate statistical adjustment for variables on the causal pathway (such as current body size), which creates an artifactual statistical effect known as the "reversal paradox." Computer simulations were conducted for three hypothetical relations between birth weight and adult blood pressure. The authors examined the effect of statistically adjusting for different correlations between current weight and birth weight and between current weight and adult blood pressure to assess their impact on associations between birth weight and blood pressure. When there was no genuine relation between birth weight and blood pressure, adjustment for current weight created an inverse association whose size depended on the magnitude of the positive correlations between current weight and birth weight and between current weight and blood pressure. When there was a genuine inverse relation between birth weight and blood pressure, the association was exaggerated following adjustment for current weight, whereas a positive relation between birth weight and blood pressure could be reversed after adjusting for current weight. Thus, researchers must consider the reversal paradox when adjusting for variables that lie within causal pathways.
27 Am J Epidemiol 2005;161:27–32
American Journal of Epidemiology
Copyright © 2005 by the Johns Hopkins Bloomberg School of Public Health
All rights reserved
Vol. 161, No. 1
Printed in U.S.A.
DOI: 10.1093/aje/kwi002
ORIGINAL CONTRIBUTIONS
Why Evidence for the Fetal Origins of Adult Disease Might Be a Statistical Artifact:
The “Reversal Paradox” for the Relation between Birth Weight and Blood Pressure in
Later Life
Yu-Kang Tu
1,2
, Robert West
1
, George T. H. Ellison
3
, and Mark S. Gilthorpe
1
1
Biostatistics Unit, Centre for Epidemiology and Biostatistics, University of Leeds, Leeds, United Kingdom.
2
Leeds Dental Institute, University of Leeds, Leeds, United Kingdom.
3
St. George’s Hospital Medical School, London, United Kingdom.
Received for publication March 11, 2004; accepted for publication June 3, 2004.
Some researchers have recently questioned the validity of associations between birth weight and health in later
life. They argue that these associations might be due in part to inappropriate statistical adjustment for variables
on the causal pathway (such as current body size), which creates an artifactual statistical effect known as the
“reversal paradox.” Computer simulations were conducted for three hypothetical relations between birth weight
and adult blood pressure. The authors examined the effect of statistically adjusting for different correlations
between current weight and birth weight and between current weight and adult blood pressure to assess their
impact on associations between birth weight and blood pressure. When there was no genuine relation between
birth weight and blood pressure, adjustment for current weight created an inverse association whose size
depended on the magnitude of the positive correlations between current weight and birth weight and between
current weight and blood pressure. When there was a genuine inverse relation between birth weight and blood
pressure, the association was exaggerated following adjustment for current weight, whereas a positive relation
between birth weight and blood pressure could be reversed after adjusting for current weight. Thus, researchers
must consider the reversal paradox when adjusting for variables that lie within causal pathways.
birth weight; blood pressure; computer simulation; confounding factors (epidemiology); statistics
Inverse associations observed between low birth weight
and markers of chronic disease in later life have generated
what is termed the “fetal origins of adult disease hypothesis”
(1, 2). The idea is that an unfavorable environment, or insults
during fetal life, might induce lifetime effects on the subse-
quent development of body systems and hence give rise to
major disease processes such as hypertension (2), diabetes
(3), arteriosclerosis (4), asthma (5), and obesity (6). Over the
last decade, many studies have been undertaken in many
parts of the world to examine these proposed relations (7).
Although some researchers have questioned the biologic
basis of the hypothesis as well as its clinical importance, the
concept that low birth weight is an independent risk factor
for a range of chronic diseases in later life is now widely
recognized as scientifically plausible and linked to poor fetal
nutrition (8). One consequence of this seemingly plausible
mechanism is that the fetal origins hypothesis is increasingly
viewed as an important issue for public health and preven-
tive medicine (9).
Nonetheless, two recent articles outlined substantive chal-
lenges to the fetal origins hypothesis (7, 8). One article raised
concerns about the statistical methodology used and the
improper interpretation of epidemiologic analyses invoked
in support of the hypothesis (8). The second article suggested
that the inverse association between birth weight and adult
diseases might “chiefly reflect the impact of random error”
in the measurement of birth weights, as well as “selective
emphasis on particular results” (by which they meant publi-
cation bias in favor of analyses describing inverse relations
between birth weight and adult blood pressure), and “inap-
propriate adjustment for current weight and for [other]
confounding factors” (7, p. 659). A recent meta-analysis also
Correspondence to Dr. Yu-Kang Tu, Biostatistics Unit, Centre for Epidemiology and Biostatistics, University of Leeds, 30/32 Hyde Terrace,
Leeds, LS2 9LN, United Kingdom (e-mail: y.k.tu@leeds.ac.uk).
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Am J Epidemiol 2005;161:27–32
indicated that the relation between blood pressure and birth
weight might suffer publication bias because small studies
were more likely to report stronger inverse associations (10).
Indeed, although a number of retrospective studies have
found a direct relation between birth weight and adult health
outcomes, others have found that a significant relation
emerged only after adjusting for subsequent body size
(notably current adult weight or body mass index) in the
statistical analyses. It is the latter that forms the focus of this
article.
The inappropriate use of statistical adjustment(s) for so-
called confounders is an important source of potential bias
among observational studies of the fetal origins of adult
disease hypothesis. We sought to demonstrate how adjust-
ment for current body mass (or any similar measure of
current body size) is inappropriate because it is not a true
“confounder” but part of the causal pathway between birth
weight and adult blood pressure. The resulting phenomenon
has been given the generic name the “reversal paradox” (or
the “amalgamation paradox”) (11), although it is perhaps not
widely known as either. Whatever the name, the reversal
paradox makes it very challenging to correctly interpret the
findings of observational studies of the causal links between
fetal growth and adult disease where covariates are inappro-
priately treated as confounders.
Because the inverse association between birth weight and
blood pressure is considered the most statistically consistent
of the associations between birth weight and health in later
life (7), our article adopts this example for illustration.
Nevertheless, what follows is essentially applicable to all
health outcomes in later life and to any other descriptive
epidemiologic analyses in which similar statistical adjust-
ments are undertaken inappropriately.
MATERIALS AND METHODS
To avoid complicating the issues surrounding adjustment
for multiple confounders, such as age and gender, we
decided to consider only a single hypothetical sample
comprising adult males of equal age. Synthetic data for these
men were generated for three variables—birth weight,
current adult weight, and adult systolic blood pressure—and
for three scenarios in which there is 1) no direct relation
between birth weight and blood pressure (i.e., the Pearson
correlation between birth weight and blood pressure is 0); 2)
a modest inverse relation between birth weight and blood
pressure (i.e., the Pearson correlation between birth weight
and blood pressure is –0.05); and 3) a modest positive rela-
tion between birth weight and blood pressure (i.e., the
Pearson correlation between birth weight and blood pressure
is 0.05). Mean values for the three variables and their stan-
dard deviations were derived from the literature (12) and
from the results of surveys conducted by the United
Kingdom Department of Health (http://www.doh.gov.uk):
birth weight = 3.38 kg (standard deviation, 0.57 kg), current
weight = 82.60 kg (standard deviation, 14.75 kg), and
systolic blood pressure = 130.0 mmHg (standard deviation,
11.2 mmHg).
Since the effects of the reversal paradox depend on the
pairwise correlations of birth weight, current weight, and
blood pressure, each scenario was simulated for a range of
different assumptions. To illustrate the impact of the reversal
paradox on changes in the relation between birth weight and
blood pressure after statistically adjusting for current weight,
we assumed three different values for the birth weight–
current weight correlation (0.15, 0.25, and 0.35) and four
different values for the current weight–blood pressure corre-
lation (0.15, 0.25, 0.35, and 0.45). The range of correlation
values adopted was motivated by typical values encountered
in the literature. The theoretical basis of these simulations is
outlined in the Appendix.
With adult blood pressure as the outcome variable, we
estimated a median value for the partial regression coeffi-
cient with birth weight after adjusting for current adult
weight, using simulations based on a sample of 500
persons—the mid-range sample size of previous empirical
studies (7)—and with 1,000 iterations for each scenario. All
simulations and statistical evaluations were performed by
using the statistical package R, version 1.7.1 (13). For each
scenario, the function “mvrnorm” in the MASS package in R
was used to generate multivariate normal data using the
mean values, standard deviations, and given correlation
matrix for the three variables birth weight, current weight,
and blood pressure (14).
RESULTS
Scenario 1: the correlation between blood pressure and
birth weight is 0
When the Pearson correlation between blood pressure and
birth weight was 0, the simple (bivariate) regression coeffi-
cient of birth weight, for blood pressure regressed on birth
weight, was unsurprisingly close to 0. In contrast, when
current weight was included in the model, the regression
coefficient for birth weight became negative, and the magni-
tude of this coefficient increased as the birth weight–current
weight and blood pressure–current weight correlations
increased (table 1). When both of the latter correlations were
set to 0.15, the estimated effect of a 1-kg increase in birth
weight was a 0.42-mmHg reduction in adult blood pressure.
With the birth weight–current weight correlation increased
to 0.35 and the blood pressure–current weight correlation
increased to 0.45, the estimated effect of a 1-kg increase in
birth weight was a 3.51-mmHg reduction in adult blood
pressure.
Scenario 2: the correlation between blood pressure and
birth weight is –0.05
When a modest inverse relation between birth weight and
blood pressure was adopted, the theoretical value of the
simple (bivariate) regression coefficient of birth weight, for
blood pressure regressed on birth weight, was –0.98 mmHg/
kg. This value can be derived by multiplying the correlation
coefficient by the ratio of standard deviation for blood
pressure to that of the standard deviation for birth weight
(–0.05 × [11.2 ÷ 0.57]), and the results of the simulations
were very close to this (ranging from –0.95 to –1.01; table 2).
When the birth weight–current weight and blood pressure–
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Reversal Paradox for Fetal Origins Hypothesis 29
Am J Epidemiol 2005;161:27–32
current weight correlations were both 0.15, the estimated
effect of a 1-kg increase in birth weight was a 1.48-mmHg
reduction in adult blood pressure, similar to that estimated by
a previous meta-analysis (7). With the birth weight–current
weight correlation increased to 0.35 and the blood pressure–
current weight correlation increased to 0.45, increasing birth
weight by 1 kg yielded a 4.47-mmHg reduction in adult
blood pressure.
Scenario 3: the correlation between blood pressure and
birth weight is 0.05
When a modest positive relation between birth weight and
blood pressure was adopted, the theoretical value of the coef-
ficient of birth weight, for blood pressure regressed on birth
weight, was 0.98 mmHg/kg (0.05 × [11.2 ÷ 0.57]), as antici-
pated from the symmetry of scenario 3 with scenario 2; refer
to table 3. When the birth weight–current weight and blood
pressure–current weight correlations were both 0.15, a 1-kg
increase in birth weight was associated with a 0.55-mmHg
increase in adult blood pressure. When the birth weight–
current weight and blood pressure–current weight correla-
tions were both 0.25, the regression coefficient was reversed,
and increasing birth weight by 1 kg became associated with
a 0.26-mmHg reduction in adult blood pressure. With the
birth weight–current weight and blood pressure–current
weight correlations increased to 0.35 and 0.45, respectively,
the estimated effect of a 1-kg increase in birth weight was a
2.41-mmHg reduction in adult blood pressure.
DISCUSSION
For categorical variables within the field of probability
and statistical science, the reversal paradox is best known as
“Yule’s paradox” or “Simpson’s paradox.” George U. Yule
noticed this phenomenon as early as 1903 (15), when he
referred to a paper published by Karl Pearson et al. dating
from 1899 (16). The issue was later mentioned in a 1951
paper by Edward H. Simpson on the way in which the rela-
tion between two variables changed after a third variable was
factored into a two-by-two contingency table (17). When
such data are analyzed by regression methods, the reversal
paradox is more often referred to as “Lord’s paradox,”
particularly within the behavioral sciences (18), ever since
Frederic M. Lord published his 1967 paper on this phenom-
enon with respect to the use of analysis of covariance (19).
Within any generalized linear modeling framework, this
phenomenon is more generally known in the statistical liter-
ature as the “suppression effect,” with the third variable
termed a “suppressor” (20, 21). Thus, in whatever form and
under whatever name, the reversal paradox has been recog-
nized ever since the statistical methods of correlation and
regression became established. Indeed, the paradox was
discussed in 1910 by Karl Pearson and Arthur C. Pigou,
Professor of Political Economics at Cambridge University in
the United Kingdom, when they debated the role of parental
alcoholism and its impact on the performance of children
(11). However labeled, the paradox has been extensively
explored in the statistical literature, especially in the behav-
TABLE 1. Association between birth weight and adult blood pressure in simple regression models and multiple regression models
after adjustment for adult body weight in scenario 1*
* The bivariate correlation between birth weight and blood pressure is 0.
† CI, confidence interval.
‡ The expected regression coefficient should be 0 (0 × [11.2 ÷ 0.57]).
Simple regression
coefficient (mmHg/kg)
Birth weight–
current weight
correlation
Multiple regression coefficient (mmHg/kg): blood pressure–current weight correlation
0.15 0.25 0.35 0.45
Median 95% CI† Median 95% CI Median 95% CI Median 95% CI Median 95% CI
0.02‡ –1.69, 1.77 0.15 –0.42 –2.13, 1.31 –0.75 –2.40, 0.97 –1.05 –2.66, 0.60 –1.35 –2.87, 0.22
0.07‡ –1.79, 1.74 0.25 –0.77 –2.49, 1.04 –1.31 –2.99, 0.47 –1.79 –3.43, –0.19 –2.36 –3.91, –0.73
0.01‡ –1.76, 1.82 0.35 –1.17 –2.94, 0.71 –1.95 –3.69, –0.11 –2.74 –4.44, –0.96 –3.51 –5.14, –1.86
TABLE 2. Association between birth weight and adult blood pressure in simple regression models and multiple regression models
after adjustment for adult body weight in scenario 2*
* The bivariate correlation between birth weight and blood pressure is –0.05.
† CI, confidence interval.
‡ The expected regression coefficient should be –0.98 (–0.05 × [11.2 ÷ 0.57]).
Simple regression
coefficient (mmHg/kg)
Birth weight–
current weight
correlation
Multiple regression coefficient (mmHg/kg): blood pressure–current weight correlation
0.15 0.25 0.35 0.45
Median 95% CI† Median 95% CI Median 95% CI Median 95% CI Median 95% CI
–1.01‡ –2.75, 0.70 0.15 –1.48 –3.20, 0.25 –1.79 –3.48, –0.10 –2.08 –3.70, –0.45 –2.39 –3.93, –0.83
–0.95‡ –2.87, 0.69 0.25 –1.86 –3.62, –0.10 –2.38 –4.11, –0.67 –2.87 –4.67, –1.29 –3.41 –5.00, –1.84
–0.96‡ –2.58, 0.66 0.35 –2.32 –4.13, –0.51 –3.10 –4.86, –1.36 –3.87 –5.59, –2.17 –4.47 –6.26, –3.01
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ioral sciences (22, 23), yet comparatively few of these anal-
yses acknowledge that they are in fact different
manifestations of the same phenomenon; that is, they are all
just one paradox.
Moreover, while the original definition and naming of the
reversal paradox drew on the notion that the direction of any
relation between two variables is reversed after a third vari-
able is introduced, it may nevertheless be generalized to
scenarios in which the relation between two such variables is
enhanced, not reversed nor reduced, after a third variable is
introduced. This difference is not widely appreciated, which
is perhaps why the impact of the reversal paradox is not
always recognized in empirical, rather than theoretical,
statistical analyses. However, the potential problem of statis-
tical adjustment in the fetal origins hypothesis has been
known for some time. Paneth et al. (24, 25) specifically
addressed the issue of “overcontrolling” for current body
mass index when analyzing the relation between birth weight
and disease risk in later life. To illustrate the reversal
paradox and explore this point further, we used simulations
of hypothetical data to mimic three scenarios pertaining to
the fetal origins hypothesis for adult blood pressure. For
those interested in the statistical theory underpinning these
ideas, refer to the Appendix.
The reversal paradox invokes bias due to the inappropriate
“controlling” of alleged confounders that are not in fact “true
confounders” (26). The concept of what constitutes a
confounder has been revised in recent years, with greater
emphasis given to the definition of “causality” in the associ-
ations among outcomes, exposures, and confounders.
Detailed expositions on this issue have emerged only
recently (27–29), and these stricter, revised definitions of
what constitutes a confounder may not have been dissemi-
nated, or universally accepted, throughout the discipline of
epidemiology. Consequently, the reason why the reversal
paradox is a problem in some instances and not others may
be more of a philosophical issue than a statistical one.
Indeed, the principal issue of statistical adjustment pertinent
to the fetal origins hypothesis is the one surrounding the
causal pathway and the position within it of current adult
body weight as an alleged confounder. If one defines a
causal pathway as the chain of events or factors leading in
sequence to an outcome, it only makes sense to examine an
outcome in relation to any one point along the causal
pathway (27, 30).
We differentiated between two different, but complemen-
tary causal pathways: 1) low birth weight affecting blood
pressure directly (e.g., poor nutrition in utero having an irre-
versible impact on the subsequent development of cardio-
vascular systems); and 2) low birth weight affecting blood
pressure via birth weight’s impact on current weight (e.g.,
through a genetic link between size at birth and current adult
body size), which in turn is causally related to high blood
pressure (i.e., birth weight current weight blood pres-
sure). In the latter model, it is sensible to examine either the
relation between birth weight and adult blood pressure or the
relation between current weight and adult blood pressure, in
isolation. It is not sensible to examine the relation between
birth weight and adult blood pressure while controlling for
current weight, because adult weight lies on the causal
pathway between the outcome (blood pressure) and the
exposure (birth weight). To statistically “adjust” for current
weight while exploring the impact of birth weight on adult
blood pressure invokes the reversal paradox.
In a slightly different formulation of the fetal origins
hypothesis, some researchers have argued that persons of a
relatively low birth weight and a relatively high adult body
size (be it weight, height, or body mass index) ought to be
considered a “high-risk” group for cardiovascular disease in
later life (31). Indeed, this is another possible interpretation
of the statistical models commonly used in analyses
exploring the fetal origins hypothesis. However, as
discussed previously, it is not appropriate to include current
weight as a confounder in these analyses if the relation
between birth weight and adult blood pressure is the primary
interest because weight gain, like current adult weight, is in
the causal pathway from birth weight to adult blood pressure.
Nonetheless, the same statistical model is useful and appro-
priate if the relation between adult blood pressure and
current weight is the primary interest; the precision of esti-
mating the association between adult blood pressure and
current weight can be enhanced by including birth weight in
the analyses. This asymmetric utility of adjusting for
“confounding” occurs because the adult body weight of
some adults is likely to be greater because they were born
heavier. For these adults, their greater body weight may not
be associated with the factors that
give rise to higher adult
TABLE 3. Association between birth weight and adult blood pressure in simple regression models and multiple regression models
after adjustment for adult body weight in scenario 3*
* The bivariate correlation between birth weight and blood pressure is 0.05.
† CI, confidence interval.
‡ The expected regression coefficient should be 0.98 (0.05 × [11.2 ÷ 0.57]).
Simple regression coefficient
(mmHg/kg)
Birth weight–
current weight
correlation
Multiple regression coefficient (mmHg/kg): blood pressure–current weight correlation
0.15 0.25 0.35 0.45
Median 95% CI† Median 95% CI Median 95% CI Median 95% CI Median 95% CI
0.97‡ –0.76, 2.63 0.15
0.55
–1.17, 2.18 0.24 –1.43, 1.84 –0.05 –1.68, 1.49 –0.36 –1.93, 1.13
1.00‡ –0.89, 2.77 0.25
0.26
–1.50, 1.94 –0.26 –2.02, 1.38 –0.74 –2.51, –0.93 –1.32 –2.95, 0.22
0.95‡ –0.72, 2.60 0.35
–0.06
–1.85, 1.69 –0.85 –2.65, 0.90 –1.64 –3.40, 0.04 –2.41 –4.08, –0.79
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Am J Epidemiol 2005;161:27–32
blood pressure but with their greater size at birth (which
might, for example, be genetically determined). Under these
circumstances, the association between adult blood pressure
and current weight will be attenuated, and adjusting for birth
weight deals with the way in which the relation between
adult blood pressure and current weight would otherwise be
diluted. Nevertheless, it should be stressed that, in this
analytic model, the relation between adult blood pressure
and birth weight does not have any empirical utility.
In the epidemiologic literature, body size measurements
(be they body weight, body height, or body mass index) are
frequently considered confounders for health-related
outcomes, presumably because there are well-known allo-
metric relations between body size and function (32).
However, there appears to be no consistent practice in the
literature as to when and how such variables should (or
should not) be “controlled for,” and most studies do not offer
any justification for their choice of confounders. For
instance, a recent study of the fetal origins hypothesis that
adjusted for current body weight and body mass index simul-
taneously found a significant negative relation between birth
weight and adult systolic blood pressure, even though there
was no significant (simple, bivariate) correlation between
them prior to adjustment (33). Furthermore, because body
weight is the numerator of body mass index, including both
variables in the analyses simultaneously means that they will
have undergone mathematical coupling (34, 35). A recently
published meta-analysis, along with other studies, shows
that removing the adjustment for current weight (or a compa-
rable measure of current body size) reduces the association
between birth weight and adult blood pressure (7); in some
instances, the association is no longer statistically significant
(36). Our scenario 2 might therefore be closest to reality.
Note that the estimated effect size in this simulated study,
or in any empirical study, is affected by the sample ratio of
the blood-pressure standard deviation to that of the birth-
weight standard deviation. If this ratio is large in any partic-
ular study, for instance, when the adult age range is wide and
thereby yields a wider range of adult blood pressures, the
effect size (i.e., the extent of bias) caused by the reversal
paradox will be exaggerated. On the other hand, if the
sample ratio of blood-pressure standard deviation to birth-
weight standard deviation is small, as, for instance, among
studies of children or young adults in which blood-pressure
variation tends to be smaller than among older adults, the
effect size caused by the reversal paradox will be dimin-
ished. Nevertheless, testing the significance of an associa-
tion is affected by sample size, and it is well known that even
a small effect size will be statistically significant if the
sample size is large enough. Most studies examining the fetal
origins of adult blood pressure have sufficiently large
samples to yield statistical significance for relatively small
effect sizes. Thus, the exaggerating effect of the reversal
paradox tends to give the misleading impression that the
relation between blood pressure and birth weight, after
adjustment for current weight, is not only statistically signif-
icant (because of the power available to detect the biased
difference from 0) but also biologically and clinically signif-
icant (as a result of the biased effect size caused or enhanced
by the reversal paradox).
Frequently, statistical methodology in studies of the fetal
origins hypothesis is at risk of the reversal paradox, thereby
bringing into question the validity and reliability of some of
the purported evidence. It is thus difficult, if not impossible,
to compare results across studies where so many varied
attempts have been made to control for confounders, without
consistent reasoning concerning the choice of these
confounders. This difficulty does not invalidate the fetal
origins hy
pothesis per se; rather, it implies that any direct
interpretation of inverse relations between birth weight and
any adult condition, while adjusting for current adult body
measurements, cannot be taken to mean that birth weight has
a direct impact on the adult outcome. For our hypothetical
example, a more appropriate interpretation is that, if all
babies grew to the same size in adulthood, lower-birth-
weight babies would, on average, have higher blood pressure
in adulthood. Yet, this conclusion is counterfactual since
low-birth-weight babies will, on average, be smaller than
heavy-birth-weight babies in adulthood; there is a positive
correlation between birth weight and adult weight, as studies
supporting the fetal origins hypothesis have shown.
As an understated and poorly recognized issue, the
reversal paradox, in whatever form it takes, has the potential
to severely affect data analyses undertaken in empirical
research, which increasingly rely on the methods of general-
ized linear modeling of observational (i.e., nonrandomized)
data. Our simulations highlight this issue with respect to the
fetal origins of adult disease hypothesis, where the paradox
is perhaps instrumental in generating a great deal of the
evidence cited within the hypothesis’ burgeoning orthodoxy.
However, the aim of this article was not to refute the fetal
origins hypothesis but to remind epidemiologists that, to
arrive at the correct interpretation of evidence supporting or
refuting this hypothesis, one must fully understand the statis-
tical methods used and the implications of making statistical
adjustment for “confounders.”
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APPENDIX
Denoting adult blood pressure by BP, birth weight by BW,
and current weight by CW, and assuming that the three vari-
ables follow a trivariate normal distribution, we may write
BW = a
1
+ b
1
CW + e
1
and BP = a
2
+ b
2
CW + e
2
. The uncon-
ditional covariance between BW and BP is thus
b
1
b
2
var(CW) + cov(e
1
, e
2
). If both BW and BP are positively
correlated with CW, then b
1
and b
2
must both be positive,
and hence so is their product. Thus, the only way to obtain a
zero covariance between BW and BP—that is, a zero
bivariate correlation between them, as was imposed in
scenario 1—is for cov(e
1
, e
2
) to be negative. Now, if we look
at cov(BW, BP) while conditioning on CW (as in a regres-
sion model that includes both CW and BW as predictors), the
conditional covariance is equal to cov(e
1
, e
2
), which we have
just indicated must be negative. The sign of the coefficient of
BW as determined by cov(BW, BP) must therefore be nega-
tive in the model that includes CW, provided that both BW
and BP are positively correlated with CW and that BW and
BP are also uncorrelated.
If we denote growth weight by GW, as per Lucas et al. (8),
the following models could be rearranged:
BP = β
0
+ β
1
BW + β
2
CW, (1)
and
BP = β
0
+ β
1
BW + β
2
(BW + GW)
= β
0
+ (β
1
+ β
2
)BW + β
2
GW. (2)
In model 2, the regression coefficient for birth weight is
attenuated because β
2
is always positive. In other words,
whenever weight gain after birth (i.e., growth weight) is
adjusted for, the inverse association between birth weight
and blood pressure will be less than that obtained while
“controlling” for current weight. However, the two
approaches—controlling for current weight or growth
weight—are no different from a statistical viewpoint since
birth weight is contained in some aspect within current adult
weight. The difficult question then arises: which coefficient
is the more accurate estimate of the contribution of birth
weight to adult blood pressure?
by guest on November 6, 2015http://aje.oxfordjournals.org/Downloaded from
    • "Results of the BMI stratification analyses support the argument that BMI needs to be included in the model for adjustment in the analyses related to smoking-central obesity association, especially for normal-weight adults. The phenomenon that the direction of any relation between two variables is reversed after a third variable is introduced is statistically known as " reversal paradox " [36,39]. The reversal paradox makes it very challenging to correctly interpret the findings seen in observational studies. "
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    Full-text · Article · Apr 2015
    Jun LvJun LvWei ChenWei ChenDianjianyi SunDianjianyi Sun+1more author...[...]
    • "In our study the association between birth weight and SBP was seen without adjustment for current body size in women, but became apparent only after adjusting for current body size in men. Whether adjustment for current body size is appropriate has been the subject of much debate in the literature [7,363738. Weinberg notes that both the adjusted and unadjusted models are mathematically correct, therefore inference depends on the hypothesized causal mechanisms [38]. In the fetal origins paradigm birth weight serves as a proxy for fetal growth and is not itself a 'cause' of adverse adult outcomes; additionally higher birth weight is associated with higher adult body size suggesting that current body size is not the 'causal pathway' in the inverse birth weight BP relationship [36, 38]. "
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    Full-text · Article · Feb 2015
    • "The remarkable and complete reversal of association between the average tail lengths of eubacterial and eukaryotic common membrane lipids with their mutually exclusive subsets, shown in Figures 2E and 3E, presented a fascinating statistical anomaly to us. After a substantial literature review, we found that this statistical anomaly, called " Simpson's paradox " (Wagner, 1982), has been reported as a rare phenomenon in biological data, especially clinical data (Tu et al., 2005; Rucker and Schumacher, 2008). To our knowledge, our findings in this work are the first example of Simpson's paradox in biological data at a molecular level through the compositional analysis of chemical structures. "
    [Show abstract] [Hide abstract] ABSTRACT: Compositional analyses of nucleic acids and proteins have shed light on possible origins of living cells. In this work, rigorous compositional analyses of ∼5000 plasma membrane lipid-constituents of 273 species in the three life domains (archaea, eubacteria and eukaryotes) revealed a remarkable statistical paradox, indicating symbiotic origins of eukaryotic cells involving eubacteria. For lipids common to plasma membranes of the three domains, number of carbon atoms in eubacteria was found to be similar to those in eukaryotes. However, mutually exclusive subsets of same data show exactly the opposite - number of carbon atoms in lipids of eukaryotes was higher than eubacteria. This statistical paradox, called Simpson's paradox, was absent for lipids in archaea and for lipids not common to plasma membranes of the three domains. This indicates presence of interaction(s) and/or association(s) in lipids forming plasma membranes of eubacteria and eukaryotes, but not for those in archaea. Further inspection of membrane lipid structures, affecting physico-chemical properties of plasma membranes, provides the first evidence (to our knowledge) on symbiotic origins of eukaryotic cells based on the "third front" (i.e. lipids) in addition to the growing compositional data from nucleic acids and proteins. © 2015 by The American Society for Cell Biology.
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