Differential models of twin correlations in skew for body-mass index (BMI)
Differential models of twin correlations in skew for body-mass index (BMI)
Siny Tsang 0 1 2 3
Glen E. Duncan 0 1 3
Diana Dinescu 0 1 3
Eric Turkheimer 0 1 3
0 Funding: Siny Tsang is supported by the research training grant 5-T32-MH 13043 from the National Institute of Mental Health. Glen E. Duncan is supported by R56AG042176 (PI: G Duncan) from the National Institute on Aging. Eric Turkheimer is supported by the John Templeton Foundation, "Genetics and Human Agency."
1 Data Availability Statement: Data are obtained from the Washington State University Twin Registry at https://wstwinregistry.org , USA
2 Department of Epidemiology, Columbia University , New York, NY , United States of America, 2 Department of Nutrition & Exercise Physiology, Washington State University±Health Sciences , Spokane, WA , United States of America, 3 Kennedy Krieger Institute and Johns Hopkins School of Medicine , Baltimore, MD , United States of America, 4 Department of Psychology, University of Virginia , Charlottesville, VA , United States of America
3 Editor: Calogero Caruso, University of Palermo , ITALY
Body Mass Index (BMI), like most human phenotypes, is substantially heritable. However, BMI is not normally distributed; the skew appears to be structural, and increases as a function of age. Moreover, twin correlations for BMI commonly violate the assumptions of the most common variety of the classical twin model, with the MZ twin correlation greater than twice the DZ correlation. This study aimed to decompose twin correlations for BMI using more general skew-t distributions.
Same sex MZ and DZ twin pairs (N = 7,086) from the community-based Washington State
Twin Registry were included. We used latent profile analysis (LPA) to decompose twin
correlations for BMI into multiple mixture distributions. LPA was performed using the default
normal mixture distribution and the skew-t mixture distribution. Similar analyses were
performed for height as a comparison. Our analyses are then replicated in an independent
A two-class solution under the skew-t mixture distribution fits the BMI distribution for both
genders. The first class consists of a relatively normally distributed, highly heritable BMI with
a mean in the normal range. The second class is a positively skewed BMI in the overweight
and obese range, with lower twin correlations. In contrast, height is normally distributed,
highly heritable, and is well-fit by a single latent class. Results in the replication dataset were
Our findings suggest that two distinct processes underlie the skew of the BMI distribution.
The contrast between height and weight is in accord with subjective psychological
experience: both are under obvious genetic influence, but BMI is also subject to behavioral
control, whereas height is not.
Body Mass Index (BMI) is one of the more intensively studied phenotypes in the genetic
epidemiology literature. A 2012 meta-analysis identified 88 individual effect sizes from twin studies
] and estimated a mean heritability of 0.73 in males and 0.76 in females, with only a small
percentage originating in the shared environment and the remainder from the non-shared
environment. Several characteristics of BMI suggest that there may be more to its heritability
than meets the eye, however [
]. First, the distribution of BMI has a strong positive skew ,
which appears to shift towards the upper end of the distribution as a function of age [
Second, twin correlations for BMI commonly violate the assumptions of the standard version of
the classical twin model, called the ACE model, which partitions the variability of the
phenotype into components attributable to the additive effect of genes (A) and shared and
nonshared environments (C and E respectively). An expectation of this simple model is that the
correlation between DZ twins should be at least half the correlation between MZ twins. In an
earlier meta-analysis that listed the individual MZ and DZ twin correlations , the MZ twin
correlation was greater than twice the DZ correlation for 24 out of 64 reported effects (38%).
These findings suggest that something other than the additive effects of individual genetic loci
plus the independent effects of environmental factors is contributing to twin resemblance.
The primary goal of this study was to use latent profile analysis (LPA) to decompose the
twin correlations and corresponding biometric components for BMI into multiple
distributions using a large sample of twins. LPA estimates statistical models in multiple classes
determined by the structure of the data. Our analysis differs from typical applications of LPA to
twin studies in several ways. First, it has recently become possible to loosen the classical
method of decomposing variables into normally distributed latent profiles (or classes); recent
developments using the software Mplus allow dependent variables to be decomposed into
more general skew-t distributions [
], making them much more flexible in the analysis of
skewed outcomes. Although BMI is often log-transformed in statistical analyses [
] to create
a normal distributed phenotype, we contend that in transforming BMI important information
about the scale and skewness of the distribution of body mass is lost.
Second, the usual strategy in LPA-based models of twin studies has been to estimate the
classes on the individual phenotypes, and subsequently estimate the twin correlations for the
estimated classes [
]. Here, we estimate the classes at the pair level, thus including the
twin correlations among the parameters being optimized by the LPA. Throughout, we
compare our BMI results to those for height, a normally distributed trait that does not produce
results at odds with the classical twin model. We hypothesize that the twin correlations for
BMI can be decomposed into a relatively normally distributed component within the normal
range of BMI, and a positively skewed component in the overweight and obese range of BMI.
In contrast, we hypothesize that the twin correlations for height will consist of one normally
The study was approved by the IRB at the University of Virginia (SBS #2014036900).
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This study included a sample of 7,086 (4,753 MZ; 2,300 DZ) twin pairs from the
communitybased Washington State Twin Registry within a cross-sectional study design. Twins include
same-sex male and female twin pairs aged 18±97 years, reared together. Participants were
recruited from Washington State driver's license and identification card applications [
twins completed an enrollment survey with questions related to childhood similarity to
evaluate twin zygosity (MZ vs. DZ), a common twin registry practice with an accuracy of 95±98%
compared to biological indicators [
]. Twins were mailed an invitation letter and
enrollment survey including questions related to height and weight. Data collected from completed
questionnaires received between 2009 and 2015 were analyzed.
Body mass index
The main outcome was BMI calculated from self-reported height and weight and expressed as
kg/m2. These measures were collected from responses to the survey questions ªWhat is your
current height?º in feet and inches and ªWhat is your current weight?º in pounds. In a sample
of twins (n = 144 pairs) participating in an ongoing funded study [
], there was excellent
agreement between mean self-reported and measured BMI (26.7 vs. 27.5 kg/m2, respectively;
r = 0.97), suggesting our use of self-reported height and weight for BMI is an acceptable
Latent profile analyses (LPA) of BMI were estimated for the one, two, and three class models
using the default normal distributions. As the distribution of BMI is positively skewed [
non-normal mixture modeling with the skew-t distribution was also estimated for the one,
two, and three class models. The skew-t distribution takes into account excessive skewness
and kurtosis of the BMI distribution by including parameters for skew and degrees of freedom
We conducted the profile analysis on a simple model of the between and within pair
variances of BMI in the twins. The models were fitted at the twin-pair level, with the latent classes
estimated based on the within-pair and between-pair means and variances of the twin pairs.
The between-pair means were constrained to be equal between MZ and DZ twin pairs but
varied across the classes, whereas the within-pair means were fixed at zero, as is typical of
twolevel models. The between- and within-pair variances were allowed to vary between MZ and
DZ twin pairs and across classes. The sum of the between- and within-pair variances (equal to
the total phenotypic variance) were constrained to be equal between MZ and DZ twin pairs,
but were allowed to vary across classes. Skew parameters for the between- and within-pair
variances, and degrees of freedom for each class were also estimated in the skew-t mixture models.
The ratio of the between pair variance to the phenotypic variance estimates the intraclass
correlation for the twins. We refer to these models as intraclass correlation models.
As a comparison, similar mixture models were performed for height. Considering the
differences in average BMI and height between men and women, the mixture models were fitted
separately for each sex.
All mixture models were estimated using 100 random starting values and 20 final stage
optimizations in order to replicate the best log-likelihood. The log-likelihood (LL), Bayesian
information index (BIC; [
]) and entropy were reported for each latent class mixture model. The
BIC imposes a penalty term to the LL for the number of model parameters, with a lower BIC
value indicative of better model fit [
]. Entropy is a measure of classification accuracy; a
model with entropy closer to 1 suggests greater classification accuracy . Three criteria were
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used to determine the best latent class solution: the lowest BIC value relative to the other
models, a substantively meaningful model, and adequate group membership per latent class (at
least 10% of the sample).
The probability that each twin pair belonged to each of the latent classes was estimated
based on the data and the maximum likelihood parameter estimates associated with the
mixture model. Twin pairs were assigned to membership in the latent class to which they had the
highest likelihood of being a member. All mixture modeling and twin analyses were performed
using Mplus version 7.4 [
Descriptive statistics and the distribution of BMI by gender and zygosity are presented in
Table 1 and Fig 1. As expected, BMI is positively skewed in both men and women. We also
compared BMI to the distribution of weight, with the linear and quadratic effects of height
partialed out. This distribution was identical to the BMI distribution (r = 1 for both gender, Figs 2
and 3), suggesting that the skew of the BMI distribution is not a peculiarity of the way BMI is
calculated, but is in fact, a structural property of human body size.
We began by estimating twin correlations and a classical twin (ACE) model for BMI in
men and women. In men, the MZ twin correlation (rMZ) was 0.71 (SE = 0.02), the DZ twin
correlation (rDZ) was 0.36 (SE = 0.04). In women, rMZ was 0.73 (SE = 0.01), rDZ was 0.44 (SE =
0.02). These correlations are consistent with the heritability coefficient [2(rMZ ± rDZ)] of 0.70
and 0.58 for men and women; a small proportion of the variance attributed to the shared
family environment (2rDZ ± rMZ = 0.01 and 0.15, for men and women) and non-shared
environment and measurement error (1 - rMZ = 0.29 and 0.27 for men and women). These findings
are consistent with previous reports of twin studies of BMI [
Latent profile analyses
BMI. The fit statistics of the mixture models under normal and skew-t distributions for
BMI are presented in Table 2. The one-class skew t distribution mixture models fit better than
the one-class normal distribution mixture models for both men and women (BICskewt = 30805
and BICnormal = 31570 for men; BICskewt = 58163 and BICnormal = 60924 for women), indicating
that excessive skewness and kurtosis need to be taken into account when modeling the
distribution of BMI. For both men and women, the two-class skew-t distribution mixture models
had lower BICs than the one- and three-class skew-t mixture models, suggesting that the
twoclass solutions were better fit for both gender. The entropy value probabilities were 0.723 and
0.783 for men and women, respectively, suggesting adequate latent class separation (18). For
men, the three-class skew-t distribution mixture model had zero individuals in one of the
Fig 1. Distribution of BMI by gender.
latent classes, suggesting that a third latent class was not needed. Similar results were found for
women: one of the latent classes had less than 2% of the participants, suggesting that the
threeclass skew-t distribution mixture model was not a good fit.
Table 3 shows the estimated parameters from the two-class skew-t mixture models. In both
men and women, the solution consisted of one relatively non-skewed distribution, with mean
BMI in the normal range (M ~ 22kg/m2 for both men and women), and a skewed distribution,
with mean BMI in the overweight range (M ~ 29kg/m2 for both men and women). We
subsequently labeled the latent profile with mean BMI in the normal range as the "normal" class,
and the other one the "overweight" class.
Table 3 presents the number and proportion of twin pairs in each latent profile by gender
and zygosity. Fig 4 illustrates the distributions of BMI in the normal (in grey) and overweight
(in tan) latent classes. Consistent across gender and zygosity, the normal profile shows a
relatively normal distribution, with a mean BMI of between 21 and 22 units, and a small variance.
All twins in the normal class have BMI 30. The overweight class shows a positively skewed
distribution, with a mean BMI of about 29 units and a much larger variance.
The number of individual twins in each latent class stratified by their measured BMI is
presented in Fig 5. For both men and women, the majority of twins in the normal class have BMIs
within the normal range (BMI < 25), and only a very small proportion have BMIs in the
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Fig 2. Distribution of weight, with the linear and quadratic effects of height partialed out, by gender.
overweight range (25 BMI < 30). In the overweight class, the proportion of twins with
normal BMI was relatively small, whereas most twins have BMIs in the overweight or obese range
The twin correlations in the normal class (rMZ = 0.79 and rDZ = 0.28 for men; rMZ = 0.71
and rDZ = 0.29 for women) are indicative of high heritability, no effect of family environment,
a relatively small effect of non-shared environment, and a modest violation of the ACE model.
In contrast, the smaller twin correlations in the overweight class (rMZ = 0.5 and rDZ = 0.01 for
men; rMZ = 0.5 and rDZ = 0.14 for women) suggest a large non-shared environmental effect
and a severe violation of the ACE model. These twin correlations are illustrated in Fig 6. MZ
twin correlations are higher than DZ twin correlations in both classes for men and women;
correlations are higher in the normal class compared to the overweight class for both men and
women, and for MZ and DZ twins.
To explore the extent to which the normal/overweight class distribution was associated
with age, we computed the Pearson correlations between age and the estimated class
probabilities of being in the overweight class. The estimated class probability was used to take into
account measurement error in the estimation of the most likely latent class membership.
Twins in the overweight class were, on average, older (mean age = 44 to 48 years old) than
those in the normal class (mean age = 34 to 40 years old).
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Fig 3. Association between BMI and weight, with the linear and quadratic effects of height partialed out, by gender.
df = degrees of freedom. LL = log-likelihood. BIC = Bayesian information index. Entropy is not available for one-class models as there was no class separation.
a The three-class solutions under the normal mixture distribution are omitted here as one latent class had zero observations for men, and two latent classes had less than
10% of the sample for women. Only the one-class solutions are presented for the skew-t mixture distribution as results showed only excessive kurtosis, but not excessive
skewness, needed to be accounted for.
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Zyg = zygosity. M = mean. SD = standard deviation. Skew = skewness. Kurt = kurtosis. r = rMZ for MZ twin pairs; rDZ for DZ twin pairs. 95% CI = 95% confidence
Fig 4. BMI density of the two latent classes by gender and zygosity. Note. The densities of the single-class BMI are illustrated in dashed lines.
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Fig 5. Distribution of participants in normal and overweight class, stratified by BMI.
Height. The fit statistics of the mixture models under normal and skew-t distributions for
height are presented in Table 2. Among the one-class models, the skew-t distribution mixture
models had lower BICs than the normal mixture models. The degree of freedom parameters
(df = 5.80, SE = 0.51, p < .0001 for men; df = 6.27, SE = 0.47, p < .0001 for women) deviated
from that of the normal distribution (df 30), whereas the skew parameters were
non-significant (skew = -0.07, SE = 0.16, p = 0.687 for men; skew = 0.18, SE = 0.14, p = 0.188 for women).
These results suggested the need to account for excessive kurtosis but not excessive skewness
for height. Subsequent multi-class mixture models were only estimated using normal
For both men and women, although the two-class mixture models showed improvements
in BIC over the one-class mixture model, more than 90% of the participants were estimated to
belong to one class (94% for men and 95% for women), suggesting that the one-class solution
was a sufficient fit to the current data for both men and women. Descriptive statistics of height
by sex and zygosity are presented in Table 3. The distributions of height are almost identical
between MZ and DZ twin pairs, with men being slightly taller than women (Fig 7). It should
also be noted that the distribution of height is relatively normal for both men and women.
The twin correlations of height by sex and zygosity are also illustrated in Fig 8. The height
of MZ twin pairs, regardless of sex, are highly correlated with each other, whereas moderate
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Fig 6. Associations of BMI between twin pairs of the two latent classes by gender and zygosity.
correlations were observed between DZ twin pairs. These results suggest high heritability of
height, with very small family environment effects.
We applied the same statistical analyses to decompose the twin correlations and corresponding
biometric components for BMI into multiple distributions in another large independent
cohort of twins for a replication study. This cohort included 13,553 (5,965 MZ, 7,588 DZ) twin
pairs from the National Academy of Sciences-National Research Council Twin Registry
(NAS-NRC Twin Registry [
]). Twins were same-sex male twin pairs, average aged 45
(SD = 3.9, range = 40±56) years at the time height and weight data was collected.
The distribution of BMI is positively skewed for MZ (M = 21.9, variance = 6.0, skew = 0.98,
kurtosis = 2.21) and DZ (M = 22.1, variance = 6.3, skew = 1.09, kurtosis = 2.79) twins. With
rMZ = 0.80 and rDZ = 0.42, twin correlation is consistent with the heritability of 0.76 for BMI; a
small proportion of the variance attributed to the shared family environment (2rDZ − rMZ =
0.04) and non-shared environment and measurement error (1 - rMZ = 0.20).
Results showed that the two-class skew-t distribution mixture model was the best fit for the
intraclass correlation model (S1 Table). The two-class skew-t solution consisted of a normal
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Fig 7. Distribution of height (in feet) between twin pairs by gender and zygosity.
BMI class (M ~ 21kg/m2) that is relatively normally distributed, and an overweight class (M ~
24kg/m2) that is positively skewed (S2 Table and S1 Fig). The twin correlations in the normal
class (rMZ = 0.81 and rDZ = 0.24) are indicative of high heritability, whereas those in the
overweight class (rMZ = 0.56 and rDZ = -0.05) suggest a large non-shared environmental effect and
a severe violation of the classical twin model (S2 Fig). Twins in the overweight class were, on
average, older than those in the normal class (r = 0.12 and 0.16, for MZ and DZ twins,
respectively; all ps < .001).
The current study presents a unique investigation of the skewed nature of BMI using latent
profile analyses in a large sample of twin pairs. The BMI latent classes were estimated by
including the twin correlations in the optimization process; twin pairs were therefore assigned
to be in the same latent class. Our use of the skew-t distribution in the estimation accounts for
the excessive skewness and kurtosis of the skewed distribution of BMI (4, 13, 14). The findings
show that the skewness of the BMI distribution can be decomposed into a relatively normally
distributed component within the normal BMI range, and a positively skewed component in
the overweight and obese range of BMI, with the former more highly correlated in twin pairs
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Fig 8. Associations of height (in feet) between twin pairs by gender and zygosity.
than the latter. Twin correlations in the first class conformed to the assumptions of the classical
twin model; those in the second class did not. Similar results were obtained when the analyses
were replicated using another large sample of twin pairs.
Our analyses suggest that two distinct processes underlie the skew of the BMI distribution
and the tendency for twin correlations for BMI to violate the assumptions of the classical twin
model. The first process is relatively normally distributed, in the normal range of BMI, and
highly heritable. The second process is positively skewed, in the overweight and obese range of
BMI, with lower twin correlations, especially for DZ twins. The near-zero DZ twin correlations
represent a severe violation of the assumptions of the classical ACE twin model, and for that
matter any genetically based twin model, none of which predict correlations of zero in DZ
twins. In contrast to BMI, height is normally distributed, highly heritable, does not violate the
classical twin model, and is well-fit by a single latent class. This contrast between height and
weight is in accord with human psychological and physiological experience of height and BMI:
both are under obvious genetic influence, but BMI is also subject to individual behavioral and
environmental control, whereas height is not [
Our finding that twins in the overweight class are more likely to be older than those in the
normal class is consistent with the positive shift in BMI with age in other samples [
]. It is
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possible that these changes in the BMI distribution over age may be largely due to the
positively skewed distribution of BMI, reflecting increased environmental contribution to the
change in BMI with age. That is, BMI in the normal range may be largely inherited, whereas
BMI above the normal range accrues during the lifespan according to individual level
processes that are less correlated in twin pairs. A meta-analysis of 40 twin cohorts showed that the
heritability of BMI decreases with age in both men and women , suggesting that the gain
in BMI, which is reflected in the skewness of the BMI distribution, may be less heritable than
the BMI distribution within the normal range. Nonetheless, this hypothesis with respect to the
two distinct processes of BMI needs to be tested with data in which repeated measures BMI
are collected across ages.
A few limitations of this study should be noted. First, height, weight and zygosity were all
self-reported. Second, participants were primarily Caucasians, which limits generalizability of
our findings to other race/ethnic groups. Future research should replicate our findings among
populations of different race/ethnic composition. Third, the cross-sectional nature of our data
limits our ability to examine potential differential causal relationships between BMI, genetic,
behavioral, and environmental factors. Future research should make use of longitudinal
studies to examine such associations, and investigate how components of BMI change or stabilize
across the lifespan.
Deeper understanding of the nature of these weight classes will require more information
at the genetic, environmental, and phenotypic level. It would be interesting to know, for
example, whether either candidate genes or polygenic risk scores are more correlated with one
distribution than the other. Similarly, one could investigate how dietary or activity behaviors, or
food and activity environments, differ between the classes, or how the classes change or
stabilize across the lifespan.
We have explored statistical models of twin development that emphasize reciprocal effects
of differences in phenotype and individual behavior, demonstrating that such processes have
the result of systematically depressing DZ twin correlations [
]. To date these models have
mostly been applied to cognitive development, but they would appear to be broadly applicable
to BMI as well. It should also be noted that positively skewed distributions are characteristic of
a wide range of human phenotypes, like alcohol consumption [
] and mood [
comprise both normal and disrupted behavior. Many of these same human phenotypes frequently
violate the classical twin model in the same way as BMI, with MZ twin similarity more than
twice that of DZ twins [
]. We hypothesize that many of these phenotypes consist of a
normally distributed portion in the normal range under strong genetic control, and a skewed
portion with a mean in the pathological range, representing environmental or reciprocal
processes at the individual level.
S1 Table. Fit statistics of the mixture models under normal and skew-t distributions for
BMI (NAS-NRC Twin Registry sample).
S2 Table. Descriptive statistics and twin correlations of BMI for the intraclass correlation
mixture models (NAS-NRC Twin Registry sample).
S1 Fig. BMI density of the two latent classes by zygosity (NAS-NRC Twin Registry
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S2 Fig. Associations of BMI between twin pairs of the two latent classes by zygosity
(NAS-NRC Twin Registry sample).
Conceptualization: Eric Turkheimer.
Data curation: Glen E. Duncan.
Formal analysis: Siny Tsang, Eric Turkheimer.
Investigation: Eric Turkheimer.
Methodology: Siny Tsang, Eric Turkheimer.
Visualization: Siny Tsang.
Writing ± original draft: Siny Tsang, Eric Turkheimer.
Writing ± review & editing: Siny Tsang, Glen E. Duncan, Diana Dinescu, Eric Turkheimer.
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