Body size as a latent variable in a structural equation model: thermal acclimation and energetics of the leaf-eared mouse
Centro de estudios avanzados en Ecología y Biodiversidad, Departamento de Ecología, Facultad Ciencias Biológicas, Pontificia Universidad Católica de Chile, PO Box 6513677, Santiago, Chile
* Author for correspondence at present address: Instituto de Ecología y Evolución, Universidad Austral de Chile, Casilla 567, Valdivia, Chile (e-mail: robertonespolo{at}uach.cl)
Accepted 24 March 2003
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Summary |
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We measured these variables in Phyllotis darwini, a murid rodent from central Chile, under conditions of warm and cold acclimation. In addition to standard statistical analyses to determine the effect of thermal acclimation on each variable and the body-mass-controlled correlation among them, we performed a Structural Equation Modeling analysis to evaluate the effects of three different measurements of body size (body mass, mb; body length, Lb and foot length, Lf) on energy metabolism and thermal conductance. We found that thermal acclimation changed the correlation among physiological variables. Only cold-acclimated animals supported our a priori path models, and mb appeared to be the best descriptor of body size (compared with Lb and Lf) when dealing with energy metabolism and thermal conductance. However, while mb appeared to be the strongest determinant of energy metabolism, there was an important and significant contribution of Lb (but not Lf) to thermal conductance. This study demonstrates how additional information can be drawn from physiological ecology and general organismal studies by applying Structural Equation Modeling when multiple variables are measured in the same individuals.
Key words: basal metabolic rate, maximum metabolic rate, thermal acclimation, structural equation modeling, body size, leaf-eared mouse, Phyllotis darwini
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Introduction |
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The roles of BMR, NST and MMR (known collectively as `energy metabolism')
in maintaining the thermal homeostasis of endotherms depends on the efficiency
of heat conservation in the body, which is, in turn, a function of body mass
(mb) and thermal conductance (C). Thermal conductance,
among other things, is a function of the insulating properties of fur, the
thermal gradient, evaporative water loss, the composition of the atmosphere
surrounding the body, and environmental factors such as radiant temperature
and wind velocity (McNab,
1980; Wooden and Walsberg,
2002
). A reliable estimation of `wet' thermal conductance (i.e.
including evaporative water loss; McNab,
1980
) is obtained from the equation
C=VO2/(TbTa),
where Tb is body temperature and Ta is
ambient temperature (McNab,
1980
). This simplified calculation of C is useful as long as
O2 is recorded
below thermoneutrality (Anderson et al.,
1997
). It is known that C, as well as energy metabolism, can
change seasonally and in response to thermal acclimation
(Maddocks and Geiser, 2000
;
Merritt et al., 2001
). The
mechanisms and physiological processes responsible for these changes and their
ecological significance are well understood (for reviews, see
Jansky, 1973
;
Heldmaier et al., 1985
;
Wunder and Gettinger, 1996
;
McNab, 2002
).
Body mass, as a proxy for body size, is the main determinant of energy
metabolism and thermal conductance in animals
(Schmidt-Nielsen, 1995;
Schleucher and Withers, 2001
;
McNab, 2002
). This dependence
is not simple, and complicates statistical analyses, especially when defining
how to deal with body mass when analyzing mass-independent physiological data
(Hayes, 1996
,
2001
;
Christians, 1999
). Several
analytical methods exist for solving these problems, most of them related to
common statistical procedures such as analysis of covariance (ANCOVA),
multiple regression and residual analysis
(Christians, 1999
). Body size,
however, is an abstraction, whereas mb is a correlated
variable that can be measured. It has been shown that other correlates of body
size could yield different strengths of association with physiological
variables (Gosler, 2000
;
Tracy and Walsberg, 2002
), and
hence different results from the analyses
(Gosler, 2000
;
Milner et al., 2000
;
Tracy and Walsberg, 2002
). The
problem is that most statistical methods are good at treating each
physiological variable separately, but performing multi-variable analyses is
not easy, especially when physiological variables differ in their dependence
on mb.
Structural equation modeling (SEM) is a powerful set of procedures which,
combined with an adequate design, could solve these problems. With SEM it is
possible to test complete path diagrams of causation and correlation among
variables, and to include latent variables (variables that cannot be measured
without error) or theoretical constructions associated with observed variables
(Everitt, 1984;
Cox and Wermuth, 1996
). In
addition, comparisons among standardized path coefficients are straightforward
because they are weighted indices that represent the proportional contribution
of each causation path to an observed, manifest variable (i.e. they can be
compared because they are scale-independent). SEM is widely used as a research
tool in psychology, sociology and medicine (e.g.
Koch et al., 2001
;
MacLullich et al., 2002
), and
has been applied by evolutionary biologists
(Crespi and Bookstein, 1989
),
geneticists (Dohm, 2002
)
community (Wootton, 1993
) and
ecosystem ecologists (Ferguson,
2002
) and, to a lesser extent, by physiological ecologists
(Hayes and Schonkwiler,
1996
).
This study had two aims: (1) to determine the mass-controlled effects of thermal acclimation on energy metabolism (BMR, NST and MMR) and thermal conductance (C) in a murid rodent Phyllotis darwini, and (2) to explore and compare the effect of different measurements of body size (mb, Lt and Lf) on energy metabolism and C, and their reliability as good approximations for body size using SEM. We hypothetized that cold thermal acclimation has a profound effect over physiological variables, modifying the effect of body size on them. Accordingly, we predict that SEM models will adjust differently in warm- and cold-acclimated individuals.
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Materials and methods |
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Basal metabolic rate and non-shivering thermogenesis
Upon completing each thermal acclimation, and prior to measurements of BMR
and NST, animals were fasted for 6h
(Nespolo et al., 2002). BMR
and NST were measured according to the following protocol. Oxygen consumption
(VO2) was measured in a computerized (Datacan
V) open-flow respirometry system (Sable Systems, Henderson, Nevada, USA).
Animals were kept in steel metabolic chambers of 1000ml volume, at
Ta of 30.0±0.5°C, which is within the
thermoneutral zone for this species
(Bozinovic and Rosenmann,
1988
). The metabolic chamber received dried air at a rate of
505±3mlmin-1 from mass flow controllers (Sierra Instruments,
Monterey, CA, USA), which was enough to ensure adequate mixing in the chamber.
Air passed through CO2-absorbent granules of
Baralyme® and Drierite® before and after passing
through the chamber, and was monitored every 5s by an Applied Electrochemistry
O2-analyzer, model S-3A/I (Ametek, Pittsburgh, PA, USA). Oxygen
consumption values were calculated using equation4a of Withers
(1977
). All metabolic trials
were completed between 08:00 and 16:00h. Body mass was measured prior to the
metabolic measurements using an electronic balance (to ±0.1g), and
rectal body temperature (Tb) was recorded at the end of
each measurement with a Cu/copper-constant thermocouple (ColeParmer,
Illinois, USA).
The experimental protocol was: (1)
O2 recorded for
a 1.5h period at rest, (2) intramuscular injection of norepinephrine (NE),
followed by (3) a final 30min period of
O2 recording.
Recording ended when
O2 reached a
sustained maximum value during a 10min period. Doses of NE were calculated
according to Wunder and Gettinger's equation
(Wunder and Gettinger, 1996
:
p. 133). Basal metabolic rate was estimated as the lowest mean value recorded
over a 3min interval after the first period of
O2 recording. We
previously measured BMR to determine the optimal time of recording needed to
reach minimum metabolism and we found that this species reaches a steady state
after 1520min, with no changes of
O2 >15% in
the following 3 h (see also Nespolo et
al., 2002
). To calculate NST from the recording, we used the
highest sample above BMR, following the standard procedure (i.e. maximum
O2 after NE
injection minus BMR; e.g. Klaus et al.,
1988
; Wunder and Gettinger,
1996
).
Minimum thermal conductance
In addition to NST and BMR measurements, we measured
O2 using the
same procedure as above but at a Ta of 6±2°C,
in order to determine `wet' minimum thermal conductance (C). To monitor
Ta inside the chamber, we used a copper-constant
thermocouple set on its geometric center and approximately 4cm above the
animal. Temperature was recorded every 4s with a Data Logger (Digi-Sense,
Illinois, USA). To avoid heat loss due to contact of the animal with the steel
floor of the chamber, or with urine and feces, we covered the chamber floor
with 0.5cm of sawdust, which was renewed before each new measurement. Rectal
Tb was measured within 30s of the last recording. The
lowest value of
O2 within the
last 10min period of
O2 measurements
was taken to determine C. For each individual we calculated minimum C using
the equation
C=VO2/(TbTa)
(McNab, 1980
).
Maximum metabolic rate
We measured MMR in a HeO2 atmosphere according to the
procedure of Rosenmann and Morrison
(1974), using an open circuit
respirometer, as described by Chappel and Bachman (1995). In brief, a mixture
of He (80%) and O2 (20%) was passed through a volumetric flowmeter
before entering the chamber (i.e. a positive pressure system), which was
maintained at 1002±10 mlmin-1. This flow rate prevented the
partial oxygen pressure from falling below 20kPa, a value far above those
considered hypoxic (Rosenmann and Morrison, 1975). As in the case of BMR
measurements, the mixture passed through CO2-absorbent granules of
Baralyme® and Drierite® before and after passing
through the chamber, which was tightly sealed with Teflon® and
Vaseline®. Chamber temperature (5.0±0.5°C)
and Tb were measured. The highest steady-state 3min period
of recordings were taken as MMR.
Statistics and structural equation modeling
All statistical analyses were performed with Statistica version 6.0
(StatSoft, Tulsa, OK, USA). Differences in measured variables between
acclimation groups (BMR, NST, MMR and C) were tested by repeated-measures (RM)
ANCOVA with mb as a changing covariate (StatSoft). Partial
productmoment correlations were used to test associations between
variables within each acclimation temperature, controlled by
mb.
The Structural Equation Modeling (SEPATH) module of Statistica (StatSoft)
was used to compute standardized path coefficients among measured variables,
and to test the overall path diagram as a likely cause of observed data. We
used maximum likelihood (ML) to estimate parameters, with their respective
standard errors. Since this procedure is based on asymptotic statistics (large
sample sizes), we performed Monte Carlo simulations to assess the behavior of
our sample statistic and the iteration procedure at different sample sizes.
Our three measurements of body size were assumed as measured consequences of a
latent variable, BODY SIZE, that we included in the model (see Appendix).
Computationally, SEM makes a linear combination of these variables to build
the latent variable (Shipley,
2000). We hypothesized that energy metabolism and thermal
conductance were primarily a function of the amount of tissue in the body,
which is better described by mb, and not by linear
measurements (Lb and Lf). For this
reason, we included the paths from body mass to physiological variables. The
first model (the `indirect' model,
Fig.1) consisted of this causal
structure (i.e. mbBN
BMR,
mbBN
NST; mbM
MMR;
mbC
C) (see
Fig.1), which supposes that the
effects of body size on physiological variables are expressed indirectly
through body mass. The second model considered an additional path structure
from `BODY SIZE' to each metabolic variable (the `directindirect'
model, Fig.2). Hence, in this
model the effect of body size over physiological variables was decomposed into
indirect effects via mb, as in the first model, and direct
effects from `BODY SIZE' to physiological variables
(Fig.2). In the context of the
physiological variables studied here, if the indirect models were accepted, we
would conclude that mb is the variable that best explains
body size. In contrast, the acceptance of the directindirect model
would mean that both Lf and Lb are
important variables explaining body size, in addition to
mb.
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Results |
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The adjustments of path models are presented in
Table 3. Based on nominal
2 P-values, only three of eight path models presented
non-significant differences from the expected covariance structure (i.e. are
supported by the data). However, from the Monte Carlo simulated distribution,
two additional models are supported by the data
(Table 3). In general, for the
large dataset (i.e. using only mb as a measure of body
size; see Materials and methods), cold-acclimated individuals presented better
adjustment than warm-acclimated ones (Table
3; models 1 and 2), and for the small dataset (i.e. body mass plus
Lf and Lb, models 36, only in
cold-acclimated animals; see Materials and methods), the single path
Lf
C improved the adjustment significantly
(Table 4). This effect is also
observed in the path diagram that includes only indirect effects of body size
to physiological variables (model 3 versus 4,
Table 4), and in the path
diagram including direct and indirect effects of body size to physiological
variables (model 5 versus 6, Table
4). Comparisons between nested models (indirect model
versus directindirect model, see
Table 3) for warm- and
cold-acclimated individuals were both significant (warm-acclimated animals:
2[4]=15.44; P=0.004; cold-acclimated
animals:
2[4]=9.99; P=0.041).
|
Considering only the best-adjusted models (i.e. non-significant from Monte
Carlo P-values, Table
3), path diagrams that included direct and indirect effects of
body size over physiological variables were less explanatory than those that
included only indirect paths through mb (Figs
5,
6,
7,
8,
9). This is supported by the
fact that in all cases all direct paths from body size are non-significant,
and several paths mb`physiological variable' have
values above 1.0 (Figs 6,
7,
9), which suggest a poor
adjustment. This allows us to discard all directindirect models (models
2, 5, 6 in Table 3, and Figs
6,
7 and
9), and leaves us only with
model 1 for cold-acclimated animals (Fig.
5) and model 4 (Fig.
8).
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Path coefficients of accepted models (Figs 5 and 8) are similar: large values in paths from body size to body mass and smaller paths relating body mass to physiological variables. A difference between both models is that in the more complex one (Fig.8) the path relating mbC to C is considerably larger, and error paths for mbC and C are smaller (see Figs 5 and 8).
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Discussion |
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We found significant partial correlations between BMR and NST for both
acclimations, and between BMR and MMR in warm-acclimated animals only.
Previous reports of (residual) intraspecific correlations between basal and
maximum metabolic capacities show that, in general, they are small but
significant (Hayes and Garland,
1995). Since in endotherms BMR measures maintenance costs (i.e.
operation of metabolically active organs), the coupling of BMR to maximum
capacities (in this case either MMR or NST) reflects the increase in systemic
performance when maximum capacities are higher. In cold-acclimated animals,
however, the association BMR to MMR was absent (see also
Bozinovic et al., 1990
), which
suggests that cold induced a disproportionate increment of maximum
performance, in agreement with systemic adjustments.
Thermal acclimation
Thermal acclimation had an effect over all energetic variables, which is
well known for MMR and NST in mammals
(Lynch, 1973;
Wickler, 1980
;
Hayes and Chappell, 1986
;
Merritt et al., 2001
).
Nevertheless, it is rather paradoxical that C increased following cold
acclimation since published studies seem to suggest either no change under
these conditions, or a decrease in C
(Maddocks and Geiser, 2000
;
Sharbaugh, 2001
). This means
that insulation is reduced in cold-acclimated individuals, which suggests a
poor performance under cold conditions. However, our measure of C (`wet'
thermal conductance; McNab,
1980
) is computed from
O2. This
measurement has several useful properties; for example in small bodies it has
been shown to be a better predictor of heat loss than dry C (i.e. heat loss
rate measured in dead animals or carcasses) (Klaassen et al., 2002). Actually,
carcasses increase the surface-area-to-volume ratio (Klaassen et al., 2002)
and dead animals lack piloerection and other physiological mechanisms that
reduce heat loss in live animals
(Turnpenny et al., 2000
). So,
wet C accounts for all the ways that heat loss occurs in live animals,
including those that stem from increases in metabolic rate (higher respiratory
gas exchange), and enhanced peripheral circulation due to increase in
interscapular brown adipose tissue (Lynch,
1973
). It is known that after cold acclimation, murids can
increase blood flow to brown adipose tissue by a factor of ten
(Puchalski et al., 1987
). This
change is enough to increase heat loss in peripheral areas of the thorax,
which is demonstrated by infrared thermography
(Jackson et al., 2001
). Since
our cold-acclimated animals showed an increase in C (which reflect the rate of
heat loss), all other things being equal, it would be reasonable to observe a
small increase in C, which was the case (less than 10%; see Results).
There still remains the question of why other studies have not detected
such an effect of thermal acclimation on C. We believe this is because our
analysis is more powerful, due to the control for body mass. In fact, when
expressed as mass-specific units, both BMR and C show non-significant effects
of thermal acclimation (Table
5). This illustrates the confounding effect of
mb when analyzing mass-specific data. When this ratio is
used, the residual variance is increased because mb is
measured with an error that is not controlled, because it is included as a
denominator of another variable (VO2) which, in
turn, is also measured with an error. It follows that using this procedure,
the residual variance of the linear model must be enlarged (in the case of an
ANOVA), with the consequent loss of power in the overall analysis
(Packard and Boardman, 1999;
Hayes, 2001
). The use of
mass-specific units has probably obscured several subtle effects of
acclimation and many other factors on the physiological variables published so
far (e.g. Holloway and Geiser,
2001
; Tracy and Walsberg,
2002
). In agreement with previous authors (Hayes,
1996
,
2001
;
Packard and Boardman, 1999
;
Christians, 1999
;
Lleonart et al., 2000
), this
evidence suggests that mass-specific variables in the literature should be
treated with caution.
Structural Equation Modeling and body mass as a proxy of body
size
Our three measurements of body size were highly intercorrelated, reflecting
the fact that all were reliable estimators of body size. Also, path
coefficients relating latent body mass with mb,
Lf and Lb were large and significant
in all cases. It is interesting, however, that while mb
appears to be the strongest determinant of energy metabolism, there is an
important and significant contribution of Lb (and not
Lf) to C. This is not surprising since C is a function of
surface-area-to-volume, and Lb is the total length of the
body, and hence is geometrically related to surface area.
In spite of the generalized use of SEM in biology, in addition to the study
of Hayes and Shonkwiler
(1996), we were unable to find
other published works that have addressed the problem of causation of
physiological variables in small endotherms. Nevertheless, SEM has been
recognized as a powerful tool for correlational experiments by several
organismal biologists. For example, Pigliucci and Schlichting
(1998
) applied path analysis
to investigate the differences among populations and the plasticity of plant
architecture in fruits of Arabidopsis; Kause et al.
(1999
) used the full
capabilities of SEM to test the effects of leaf quality on larval traits in
six sawfly species; Miles et al.
(2000
) used SEM to predict
survivorship from life history variables in a lizard; and Ferguson
(2002
) used SEM to
discriminate between direct and indirect effects of demographic and
environmental variables on age at maturity in a moose. These examples,
together with our results, illustrate how SEM could be a powerful tool to
distinguish between contrasting causal models in organismal biology.
SEM proved to be useful for our data analysis in the sense that we could
test specific models and subtle effects, such as the inclusion of specific
paths, to explore the change in the overall goodness of fit. This enabled us
to conclude that our models were good explanations of the relationship between
body size and energetic variables. However, this was only true for
cold-acclimated animals; warm-acclimated animals did not adjust to our models,
possibly because NST and MMR only have physiological significance in
cold-acclimated individuals. This raises the importance of thermal acclimation
when measuring respirometric variables in endotherms. Likewise, the partial
correlation that was detected among physiological variables was not detected
by the SEM analysis (when we included such paths, the iteration did not
converge because of the appearance of singular matrix). This occurs because
SEM models can often be structurally identified, but numerically
underidentified (Shipley,
2000, p. 167). These are limitations of SEM that cannot be
ignored, and make it especially important to complement the SEM procedure with
standard statistical analyses.
An interesting outcome from the SEM analysis is that the single path
LbC significantly improved the goodness of fit. Our
best models were those that related body size with physiological variables
indirectly, through mb, and not by direct paths. Moreover,
by examining the accepted path models, and comparing error paths, we could
infer that the strength of association between body size and morphological
traits is considerably larger than the (indirect) association between body
size and physiological variables. Similarly, it is clear that C has a higher
(indirect) dependence from body size than energy metabolism. None of these
conclusions would have been possible using only standard statistical
procedures.
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Appendix |
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Nested models (i.e. models differing in path number but with same number of
variables) were compared with 2 ratio tests since the
difference in the maximum likelihood
2 values between nested
models is, itself, asymptotically distributed as a
2
distribution (Shipley, 2000
).
The degrees of freedom for the resultant
2 distribution are
equal to the number of parameters that have been freed in the nested model,
which is the same as the change in the degrees of freedom between the nested
models (Shipley, 2000
). This
test allowed us to evaluate whether different models that included the same
variables were statistically different, using the same set of data. To select
the best models, we proceeded as follows: first, we looked at the statistics
of the adjustment of the different models and selected (and presented) those
that yielded a non-significant
2. Second, we performed
2 ratio tests to analyze the specific contribution of single
paths to the overall model, between nested models. Third, path coefficients of
selected models (i.e. non-significant from the Monte Carlo P-value)
were examined, and those that presented more significant paths were judged to
be the best. Although the effects of thermal acclimation were evaluated
individually for each physiological variable, the condition of measurement
(cold- or warm-acclimated) is explicitly stated in each path diagram.
This work was funded by Fondecyt grant 2000002 to R.N. and FONDAP grant 1051-0001 (program 1) to F.B. M.A. acknowledges a DIPUC fellowship.
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