American Journal of Epidemiology
Copyright © 1997 by The Johns Hopkins University School of Hygiene and Public Health
All rights reserved
Vol. 146, No. 4
Printed in U.S.A.
Hypothesis Concerning the U-shaped Relation between Body Mass Index
and Mortality
David B. Allison,1 Myles S. Faith,1 Moonseong Heo,1 and Donald P. Kotler2
Numerous studies have documented a U- or J-shaped association between body mass index (BMI) (kg/m2)
and mortality, such that increased mortality rate is associated with relatively low and high BMI values. It has
been argued elsewhere that the elevated mortality rate observed at lower BMI values actually results from the
effects of unmeasured confounding variables, in particular smoking status and preexisting disease. In this
paper, the authors present an additional explanation for the phenomenon, i.e., nonspecific measurement. They
propose that differential health consequences of fat mass and fat-free mass can be masked by the use of BMI
when studied in relation to mortality. To illustrate this point, they use body composition data from 1,137 healthy
adults and specify a hypothetical underlying BMI-mortality model in which the logit of death increased linearly
with fat mass and decreased linearly with fat-free mass, and % fat increased monotonically with BMI. The
results indicate that, even under these specifications, the authors can recover a U-shaped association
between BMI and mortality. Consistent with previous suggestions in the literature, future epidemiologic studies
that examine the association between adiposity and mortality should prioritize the use of body composition
measures. Am J Epidemiol 1997;146:339-49.
body composition; body mass index; confounding factors (epidemiology); mortality; obesity; occult disease
The relation between body mass index (BMI) (kg/
m2) and longevity remains an area of interest, importance, and controversy (1-8). One of the main points
of controversy concerns the elevation in mortality risk
at the low end of the BMI continuum. That is, the
majority of studies observe a U- or J-shaped relation
between BMI and the risk of mortality within a defined period of time (e.g., references 9-15). In other
words, subjects with the highest and lowest relative
weights die earlier than subjects with more intermediate levels of relative weight. The robustness of this
association is most strongly suggested by Troiano et
al. (16) in their comprehensive meta-analysis of the
BMI-mortality association. Among a multinational
sample of studies, these investigators found clear evidence of a U- or J-shaped relation.
The increased risk of mortality at the low end of the
continuum seems quite counterintuitive to many in-
vestigators (2, 3). It is inconsistent with the generally
monotonic relation between BMI and indicators of
morbidity (17), including diabetes mellitus, hypertension, stroke, dyslipidemia, and cardiovascular disease
among others (18-20). Moreover, it stands in contrast
to the fairly consistent observation that weight loss
reduces risk factors for a variety of illnesses (21-24).
In response to this counterintuitive finding, several
authors have suggested that the elevated risk of mortality at the low end of the BMI continuum is an
artifact due to confounding. The two confounding
variables most frequently cited are smoking and preexisting "occult" disease (25, 26). In regard to occult
disease, several studies suggest that elimination of
subjects who die during the first several years of
follow-up results in a dampening of the left tail of the
BMI-mortality curve (27-29). For example, in an analysis of the Nurses' Health Study, Manson et al. (27)
suggested that excluding subjects who died during the
first 4 years of follow-up had such an effect on 16-year
mortality. The authors concluded that "the apparent
excess risks of leanness were found to be artifactual
and were eliminated after we accounted for .. . subclinical disease" (27, p. 683).
On the other hand, the majority of studies which
attempt to control for these potentially confounding
variables continue to show the U-shaped relation previously described (4-6, 30-33). In fact, the meta-
Received for publication November 25, 1996 and accepted for
publication May 7, 1997
Abbreviations: BMI, body mass index; FFM, fat-free mass; FM,
fat mass; OR, odds ratio; PF, %fat.
1
Obesity Research Center, St. Luke's/Roosevelt Hospital Center, Columbia University College of Physicians and Surgeons, New
York, NY.
2
St. Luke's/Roosevelt Hospital Center, Columbia University College of Physicians & Surgeons, New York, NY.
Reprint requests to Dr. David B. Allison, Obesity Research Center, St. Luke's/Roosevelt Hospital, 1090 Amsterdam Ave., 14th
floor, New York, NY 10025.
339
340
Allison et al.
analysis by Troiano et al. (16) yielded comparable
findings, which led the authors to conclude, "Failure to
exclude persons with disease was not associated with
a rise in mortality at low BMI with either short or long
follow-up. Moreover, the increased mortality observed
in the lowest BMI groups could not be fully explained
by pre-existing illness .. . because the relationship was
still present after controlling for these factors. .." (16,
p. 71). Similarly, the investigation by Waaler (13) of
over 1 million Norwegians found no significant effect
of early death exclusion specifically on the left tail of
the BMI-mortality curve. Waaler concluded, "The
hypothesis that the U-shape is a product of already
existing fatal diseases can therefore be dismissed" (13,
p. 22). Thus, whether or not the presence of occult
disease causes the elevated mortality seen among individuals with the lowest BMIs remains open to question and an area for further research.
In this paper, we suggest an additional possible
explanation for the elevated mortality risk at the low
end of the BMI continuum. Specifically, we conjecture
that this elevation may be due to a measurement problem, i.e., the use of BMI as a measure of "adiposity"
may create an artifactual relation. We suggest that,
while other factors may also underlie the observed Uor J-shaped association between BMI and mortality,
the measurement issue described herein may itself be
an important factor that deserves greater attention.
To begin, it is well known that BMI is highly
correlated and monotonically associated with % body
fat (34, 35). However, consider the following reexpression:
BMI-
mass
fat mass + fat-free mass
stature
stature
fat mass
fat-free mass
stature
stature
Expressed in this manner, it is obvious that BMI
includes (at least) two distinct components, fat mass
(FM) and fat-free mass (FFM), relative to stature. The
index (FM)/(stature2) has been suggested as a measure
of adiposity and referred to as the body fat mass index
(BFMI) (36). Similarly, the use of (FFM)/(stature2)
has been suggested as an indicator of relative fat-free
mass (36).
It is generally thought that increasing amounts of
FM pose a threat to health and longevity (7). Conversely, it may well be that, all other things being
equal, increasing amounts of FFM enhance health and
longevity (37). Indeed, past research has suggested
that body composition rather than body mass is a
major determinant of cardiovascular disease risk factors (18, 38-40).
Given the above, it is possible that the logit of the
probability of death within some finite period of time
could be expressed with the following model:
FM
m
Logit(7r) = /30 + )3i —22
FFM
,2 '
m
(1)
where Logit(-) = log(-/(l - •)), ^ = P{D = 1IFM,
FFM), D = 1 if "dead," and D = 0 otherwise. The
coefficients are restricted as follows: /3, > 0 and /32 <
0. This reflects a positive effect of FM and a negative
effect of FFM on mortality. Data published by Keys
(41) further support the assignment of positive and
negative signs to /3j and j32, respectively. Investigating
all-cause mortality at 35-year follow-up in the Twin
Cities Prospective Study, Keys found a positive association between "body density" and mortality (controlling for other fatness indexes), but a negative association between BMI and mortality.
Were this the true model underlying risk of death,
then the risk of death at any BMI relative to some
standard would be a composite of the relative difference in FM and FFM at that BMI level compared to
the standard BMI level. Under these circumstances, it
is possible that the risk of mortality within a defined
period of time can be non-monotonic over the range of
BMI even when the following conditions are met:
1. The logit of death increases linearly with increasing FM.
2. The logit of death decreases linearly with increasing FFM.
3. Percent fat increases monotonically with BMI.
We illustrate this point with a hypothetical example.
EXAMPLE
Sample
For this example, measurements of height, weight,
FM, and FFM on individual subjects were originally
pooled from several data sets of apparently healthy
adults. The data are retrieved from several references
(42-48). Subjects with BMIs <16 or >40 were then
excluded from this pooled sample for the present
study. The resulting pooled sample (with extreme BMI
excluded) includes 1,137 apparently healthy adults
aged between 18 and 65 years. Detailed descriptive
statistics for this pooled sample are described in
table 1. In this table, the method of the body composition measurement used in each source of the data sets
is also included. Correlations between measurements
are described in table 2.
The primary source of data from Kotler et al. (42)
consists of a group of 1,083 healthy volunteers (31
percent white, 69 percent black), which alone constiAm J Epidemiol
Vol. 146, No. 4, 1997
miol
<
vJo. 4. 1997
50.2(10.5)
26.9-84.0
24.7(11.1)
2.1-55.1
FFM* (kg)
% Fat
6
0
26.3 (6.8)
19.0-37.0
43.1 (1.9)
41.5-46.0
15.8 (5.5)
10.8-24.4
22.0 (2.5)
18.3-25.2
163.8(6.7)
152-172
58.9 (5.3)
53.5-65.9
Method used for body Bio-impedance Dual energy x-ray
absorptiometry
composition
measurements
* SD, standard deviation; FM, fat mass; FFM, fat-free mass.
t Range.
1,083
17.2(9.8)
1.0-62.3
FM* (kg)
Sample size
24.1 (4.2)
16.1-39.7
BMI
48
166.9(8.4)
125-200
Height (cm)
Sex (% male)
67.4(13.7)
38-115
Weight (kg)
54.5 (2.4)
52-58
Mean (SD)
Mean (SD»)
41.9(10.7)
18-65t
Hoover et al. (43)
Kotler et al. (42)
Age (years)
Variable
activation analysis
Bio-impedance
14
11
In vitro neutron
0
44.8 (4.3)
38.6-53.4
50.2 (4.9)
41.1-58.4
41.0(6.6)
34.1-55.2
34.0 (2.9)
29.1-39.1
163.9(6.4)
155-174
91.2(8.8)
79.6-110
39.7 (5.9)
24-46
Mean (SD)
Racette et al. (45)
0
37.1 (7.6)
22.9-50.0
54.2 (4.4)
49.6-63.6
33.1 (10.7)
14.7-51.6
30.9 (4.5)
22.3-36.8
168.4(6.9)
155-180
87.4(12.2)
42.5-64.4
47.3 (7.5)
31.5-60.3
Mean (SD)
Kreitzman et al. (44)
Data source
Densitometry
7
100
19.7(9.0)
7.0-31.0
63.0 (6.4)
53.4-70.2
16.7(10.1)
4.3-31.6
26.1 (5.3)
19.2-33.2
175.1 (5.9)
166-181
79.7(14.8)
61.4-102
28.0 (4.1)
20.5-32.0
Mean (SD)
Ravussin et al. (46)
8
Densitometry
Isotope dilution
0
16.5(11.3)
0.74-36.1
50.7 (7.7)
36.2-60.0
10.1 (6.8)
0.38-20.4
21.4(1.5)
19.8-23.2
168.5(5.7)
161-178
60.9 (5.9)
51.4-66.9
8
100
14.4 (3.7)
10.9-19.0
55.7(12.0)
42.0-72.3
9.1 (1.7)
6.2-11.0
21.5(2.0)
17.9-23.2
172.9(8.9)
164-188
64.8(12.0)
48.2-81.5
21.1 (1.5)
19-23
Mean (SD)
Mean (SD)
25.5 (1.9)
23-28
Veitl et al. (48)
SjSdin et al. (47)
TABLE 1. Descriptive statistics for the measurements in the example sample with extreme body mass index (BMI) excluded
1,137
46
24.9(11.3)
0.74-55.1
50.3(10.4)
26.9-84.0
17.5(10.2)
0.38-62.3
24.3 (4.4)
16.1-39.7
167.0(8.4)
125-200
67.8(14.0)
38.0-115
41.6(10.9)
18-65
Mean (SD)
Total (pooled)
Bod}1 Mass Ir dex and Morta
342
Allison et al.
TABLE 2. Correlations between measurements in the
example sample
BMI*
BMI
FM
FFM
% Fat
FM*
1
0.85
0.33
0.67
1
-0.07
0.93
FFM*
TABLE 3. Hypothesized parameter sets and estimated
coefficients of the body mass index (BMI) quadratic model
(model 4)
% Fat
Hypothesized
parameters
1
-0.37
1
* BMI, body mass index; FM, fat mass; FFM, fat-free mass.
tutes 95 percent of the total sample size. This data set
was split roughly evenly between males (48 percent)
and females (52 percent) (table 1), as well as between
Africans (52 percent) and Americans (48 percent). All
of the African subjects were human immunodeficiency
virus (HlV)-seronegative employees of the National
Bank of Zaire. The Americans were volunteers from
the New York City metropolitan area. The body composition was measured by means of an RJL 101A
bio-impedance analyzer with standard tetrapolar lead
placement (RJL Equipment Co., Detroit, Michigan).
ft
ft-
0.05
0.05
0.05
0.05
0.04
0.04
0.04
0.04
0.03
0.03
0.03
0.03
0.02
0.02
0.02
0.02
-0.20
-0.15
-0.10
-0.05
-0.16
-0.12
-0.08
-0.04
-0.12
-0.09
-0.06
-0.03
-0.08
-0.06
-0.04
-0.02
Estimated coefficients
Nadir t
4.41
3.47
2.88
2.29
3.92
3.22
2.51
1.94
3.31
2.72
2.13
1.64
2.59
2.16
1.74
1.39
«i*
a2 X 1 0 3 *
-0.253
-0.191
-0.161
-0.122
-0.214
-0.171
-0.128
-0.090
-0.168
-0.131
-0.094
-0.062
-0.114
-0.087
-0.061
-0.037
4.82
3.84
3.48
3.00
4.05
3.41
2.74
2.21
3.15
2.58
2.00
1.52
2.12
1.70
1.29
0.94
26.25
24.92
23.13
20.42
26.47
25.18
23.33
20.30
26.69
25.44
23.51
20.17
26.91
25.67
23.67
20.01
* p values are all <0.001.
t Estimated BMI level associated with minimum mortality, that is,
Hypothetical model
Using the actual data collected as described, we
estimated what the shape of the BMI-mortality distribution would be assuming a hypothetical underlying
model. Specifically, we assumed that the logit of the
probability of death within some defined period of
time could be characterized by model 1, i.e.,
FFM
FM
Logit(Tr) = /30 + /3i —Y
m
m
Assuming /30 = 0 (we are not interested in the intercept but in the effects of FM and FFM), the probability
7T of death within the defined period of time can then
be calculated for each subject as follows:
TT =
P(D= 1IFM, FFM)
FM
FFM
exp j3, — T + Pi
m
m
FM
FFMV
1 + expl /3i —2"
(2)
At each BMI level, the % fat (PF) was averaged over
all subjects in that interval as well as the calculated
probability of death for each set of /3j and /32.
The average % fat was fitted by a quadratic regression onto BMI and BMI2, weighted by the sample size
of each bin, as follows:
PF = 70 +
(3)
This model was postulated to see the statistical significance of the estimated coefficients for BMI2, and to
verify the monotonic relation of % fat with BMI.
Based on the average death probability in each bin
of BMI level, the odds ratio (OR) between each BMI
level and the BMI level associated with minimum
death probability was obtained and similarly fitted by
the following weighted quadratic regression in terms
of BMI level:
OR = OQ + a,BMI + a2BMI2
—r
i
It is also assumed that J3j > 0 and /32 < 0m2(condition
1 and 2). To calculate the death probabilities, we set a
range of the parameter sets of 0, and /32 as listed in the
first two columns of table 3.
+ 72BMI2
(4)
This model was tried to see if the odds ratios (generated by means of FM and FFM) can also be fitted by
a (usually practiced) BMI-quadratic model.
Results
Methods
For each set of /3, and /32 individual probabilities of
death were calculated according to equation 2. Subjects were then placed into bins of unit BMI length
(i.e., 16 to <17, 17 to <18, 18 to < 19 . . . 39 to 40).
The resulting weighted quadratic regression (model
3) fit of PF is depicted in figure 1. There clearly
appears to be a strong (monotonic increasing) linear
relation between PF and BMI (/?2 = 0.977); 7, was
estimated as 1.37 (p = 0.011) and 7 2 as 0.007 (p =
Am J Epidemiol
Vol. 146, No. 4, 1997
Body Mass Index and Mortality
S
343
i
16
20
25
30
35
39
BMI
FIGURE 1. A weighted quadratic regression fit of % fat onto body mass index (BMI) and its square. Each data point represents the average
% fat for all scores falling within the given BMI bin.
0.475). Therefore, it is verified that the pooled sample
meets condition 3.
For each set of /3t and j32 model 4 was fitted and the
resulting estimates of a 0 , a x , and a 2 are listed in table
3, along with the estimated BMI levels associated with
minimum death probabilities (estimated from model
4). All of the coefficients were statistically significant
with all p values <0.0001. This indicates that the
BMI-quadratic model fits the odds ratio very well even
though these are generated by means of FM and FFM.
The resulting fits are depicted in figures 2a-5d. As
can be seen, a U- or J-shaped relation clearly emerges
between BMI and mortality depending on the sets of
parameters through the absolute ratio I^1//32I of the
two coefficients. Specifically, when the absolute ratio
is small (a and b in figures 2-5), which implies that the
negative effect of FM is relatively smaller than the
positive effect of FFM, the relation becomes
U-shaped. On the other hand, when the absolute ratio
is large (c and d in figures 2-5), which implies that the
negative effect of FM is relatively larger than the
positive effect of FFM, the relation becomes J-shaped.
This is quite intuitive in this hypothetical setting,
because BMI is highly correlated with FM (correlation = 0.85, table 2); the large effect of FM might
have been manifested directly through the highly correlated BMI. In addition, closer inspection of the BMI
nadirs (the BMI associated with the minimum mortality) in table 3 reveals that they are also a function of
the absolute ratio in this hypothetical example. That is,
the combination of the effects of FM and FFM could
be a factor in estimating the nadir in terms of BMI.
The overall pattern of the shapes of the BMImortality relations over the given range of the paramAm J Epidemiol
Vol. 146, No. 4, 1997
eter sets is quite similar to that seen in actual studies of
BMI and mortality. Again, we emphasize that the Uor J-shaped relation between BMI and mortality in this
hypothetical example emerges despite the fact that the
"true" (hypothetical) relation between fatness and
mortality is monotonic increasing and the true (actual)
relation between BMI and percent body fat is also
monotonic increasing (figure 1).
DISCUSSION
We have suggested that the U-shaped relation frequently observed between BMI and mortality may
result at least in part from the fact that BMI is composed of separate components, mainly FM and FFM,
which have opposite effects on health and longevity.
We have shown that even in the circumstances where
adiposity expressed either as % fat or absolute FM
increases monotonically with BMI and where risk of
death increases monotonically with adiposity, this Ushaped relation between BMI and mortality can still
emerge. This finding shows that the "clinical wisdom"
that "one can never be too lean" is not inconsistent
with the epidemiologic observation that "one can be
too thin."
However, we must emphasize that we have only
demonstrated that this is a plausible explanation for
the frequently observed U-shaped relation. We cannot
say with any degree of certainty that this is the actual
explanation for the observed phenomenon. Nevertheless, such an explanation seems plausible. For example, Keys (41) found that, compared with individuals
who died at 35-year follow-up, those who survived
had greater body density but a somewhat smaller BMI
344
Allison et al.
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FIGURE 2. Weighted quadratic regression fits of odds ratios of death probability onto body mass index (BMI) and its square. Each data
point represents the odds ratio obtained from the average death probability, when /3, = 0.05, for all scores falling within the given BMI bin
with varying ft,: (a) ft, = -0.20; (b) ft, = -0.15; (c) ft, = -0.10; (d) ft, = -0.05.
on average. Decreased FFM has also been shown to
coincide with the increased mortality of individuals
exposed to traumatic life experiences involving starvation (49).
As stated above, research exists (18, 38-40) that
indicates that body composition more than BMI is a
primary determinant of health. Consistent with this
hypothesis, some research (31, 50) has shown a somewhat more monotonic relation between adiposity as
measured by skinfolds or circumferences and mortality than between BMI and mortality. In addition,
Baumgartner et al. (51) have argued that the measureAm J Epidemiol
Vol. 146, No. 4, 1997
Body Mass Index and Mortality
345
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FIGURE 3. Weighted quadratic regression fits of odds ratios of death probability onto body mass index (BMI) and its square. Each data
point represents the odds ratio obtained from the average death probability, when /3, = 0.04, for all scores falling within the given BMI bin
with varying ft,: (a) ft, = -0.16; (b) ft, = -0.12; (c) ft, = -0.08; (d) ft, = -0.04.
ment of body composition, rather than BMI, will ultimately shed the most light on this issue of mortality.
They state, "Efforts to recommend 'optimal' weights,
as well as cutoff values for underweight or overweight, deny the biological reality . . . that individuals
Am J Epidemiol
Vol. 146, No. 4, 1997
of the same weight, or even weight-for-stature, can
have widely differing amounts of fat and lean mass as
well as fat distribution" (51, p. 90). The authors offer
plausible theoretical relations among FM, FFM, and
mortality risk which might yield the observed U- or
346
Allison et al.
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FIGURE 4. Weighted quadratic regression fits of odds ratios of death probability onto body mass index (BMI) and its square. Each data
point represents the odds ratio obtained from the average death probability, when 0, = 0.03, for all scores falling within the given BMI bin
with varying ft,: (a) ft, = -0.12; (b) 0 = -0.09; (c) ft, = -0.06; (d) ft, = -0.03.
J-shaped association between BMI and mortality risk
(51, figure 3, p. 91); however, those models differ
from the one we propose here.
The most obvious implication of this conjecture is
that, were it correct, meaningful inferences about the
relation between adiposity and mortality would be
better achieved by conducting prospective cohort studies that take actual measurements of body composition
rather than by relying solely on BMI. In this regard, it
is noteworthy that the third National Health and Nutrition Examination Survey (NHANES-III) measured
body fat with bio-impedance analysis (52) and that
Am J Epidemiol Vol. 146, No. 4, 1997
Body Mass Index and Mortality
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35
39
I
16
20
25
BMI
BMI
C
d
30
35
39
30
35
39
m
CM
m
d
CM
d
o
g
Is
CM
O
d
I
CO
a
a
o
a
o
o
o
m
p d
S
o
d
o
d
d
16
20
30
25
35
39
BMI
16
20
25
BMI
FIGURE 5. Weighted quadratic regression fits of odds ratios of death probability onto body mass index (BMI) and its square. Each data
point represents the odds ratio obtained from the average death probability, when 0, = 0.02, for all scores falling within the given BMI bin
with varying /32: (a) ft, = -0.08; (b) ft, = -0.06; (c) ft, = -0.04; (d) ft, = -0.02.
proposals for NHANES-IV include the measurement
of fatness with dual energy x-ray absorptiometry
(Steven B. Heymsfield, Obesity Research Center, St.
Luke's/Roosevelt Hospital, New York, NY, personal
communication, 1996). Longitudinal follow-ups of
Am J Epidemiol
Vol. 146, No. 4, 1997
these studies would therefore be enlightening. From a
clinical point of view, such results would also suggest
that weight loss would only be beneficial if the ratio of
FM lost to FFM lost exceeds the absolute value of
the ratio of /V/3 2 . This result also holds if the model
348
Allison et al.
is expressed as a proportional hazards model (i.e.,
hj = h0ep>FMj+^2FFMj). However, both assume (for this
simple ratio statement to be true) that the logit or log
hazard of death increase and decrease linearly with
increasing FM and FFM, respectively. Were actual
estimates of /32 and J3, available, these could be used
to guide clinical practice by establishing under which
methods and conditions of weight loss the ratio of FM
to FFM lost exceeded the desired level. Treatment
components that affect these changes (e.g., dietary
composition, exercise regimen, etc.) could be investigated to examine the relative rate at which they produced changes in FM and FFM.
In conclusion, we have suggested an additional
hypothesis, different from that of occult disease, to
explain the U- or J-shaped relation frequently observed between BMI and mortality. We concur with
Baumgartner et al. (51) that future investigators should
strongly consider conducting prospective longitudinal
studies using actual measures of body composition
rather than simply BMI.
ACKNOWLEDGMENTS
Supported in part by National Institutes of Health grant
nos. ADDK353572, DK51716, DK26687, R29DK47256,
DK42618, and T32DK37352.
The authors thank Drs. Richard N. Pierson, Jr., Steven B.
Heymsfield, Angelo Pietrobelli, Donald Thea, Gerald T.
Keusch, and Michael E. St. Louis for the use of their body
composition data.
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