Vol. 132,No. 2
The American
Naturalist
August1988
THE FALLACY OF AVERAGES
A. H. WELSH,* A. TOWNSENDPETERSON,tAND STUARTA. ALTMANNt
Department
ofStatistics,
University
ofChicago,Chicago,Illinois60637;tCommittee
on
Evolutionary
Biology,University
ofChicago,Chicago,Illinois60637
Submitted
February9, 1987;AcceptedOctober21, 1987
ofaveragevaluesoftraits
Manyquestionsaboutorganisms
requireestimation
that,forvariouspracticalreasons,aredifficult
to measuredirectly
butthatcan be
decomposedintotwoor moremultiplicative
each ofwhichcan be
components,
measured.(Such characteristics
are hereafter
referred
to as producttraits.)For
instance,althoughan animal'sdailyintakeoffoodenergyis difficult
to measure
itcan be estimated
oftimespentfeeding,
food
directly,
indirectly
usingestimates
intakeperunitofforaging
time,and energycontentperunitoffoodmass.
THE PROBLEM
twooftheauthorsofthispaper)haveestimated
the
Manybiologists(including
meansofproducttraitsbymeasuring
variables
population
eachofthecomponent
in separatesubsamplesof the populationand thenmultiplying
the component
meansto producethedesiredmeanof theproducttraits.For example,Janson
(1985,table 1) and Stacey (1986,eq. 1) bothestimatedthe meandailyenergy
intakeby primatesfroma givenfoodas theproductofthemeandailyintakeof
thatfood,themeanweightof an itemofthatfood,and thefood'smeanenergy
= ,u(J/g),u
content,in joules per gram.This can be represented
by ,u(J/day)
(g/item),u(items/day).
(Here and in whatfollows,,u(x)indicatesthemeanofx.)
Furtherexamplesof thisprocedurehave been presentedby Shea and Ricklefs
(1985,table1),Shank(1986,p. 645),EkmanandAskenmo(1986,table1) andcan,
indeed,be foundin nearlyeveryvolumeofmanybiologicaljournals.
A generalpattern
appearsinthesecalculations.For an individual
producttrait
withmultiplicative
variablesX and Y, thepopulationmeanof their
component
product,,u(XY),is commonly
assumedto equal ,u(X),u(Y).Alas,thiswidespread
The
is notgenerally
itis ifX and Y areuncorrelated.
assumption
correct,
although
between,u(XY) and ,u(X),u(Y)is givenbelow. The correexact relationship
is madeaboutdivisionofmeans,namely,
that,u(XIY)equals
sponding
assumption
ofmeans.
,u(X)/I,(Y),and aboutotherfunctions
* Present
address:Department
ofStatistics,
TheFaculties,Australian
NationalUniversity,
G.P.O.
Box 4, CanberraA.C.T. 2601,Australia.
Am.Nat. 1988.Vol. 132,pp. 277-288.
? 1988byThe University
reserved.
Allrights
ofChicago.0003-0147/88/3202-0002$02.00.
278
THE AMERICANNATURALIST
The assumption that ,u(XY) = ,u(X),u(Y) is a special case of what Wagnerreferred
to as thefallacyofaverages:"Givenan arbitrary
nonlinear
function
f(x, ... , xJ)of randomvariables x1, . .. , x, it is usually erroneousto assume
= f[E(xl)
, E(x,)]," where E indicatesan expected value
E[f(xl, . . . ,
(1969,p. 658).
The fallacyof averagesis perhapsthe mostwidespreadstatisticalerrorin
biology.Templeton
andLawlor(1981)gaveseveralexamplesofthisfallacyinthe
literature
on optimization
modelsin ecology,thoughtheydidnotindicateanyof
inwhichtheexpectedvalueofa function
thesituations
is thesameas thefunction
oftheexpectedvalue; and in onlyone case, themeanofa reciprocalversusthe
reciprocalof a mean, did theyshow how the two are related.Althoughthe
particularexamplesselectedby Templetonand Lawlor maybe inappropriate
(Gilliamet al. 1982;Turelliet al. 1982),theprinciple
is stillvalid.
A comparableproblemarisesrepeatedly
inthestudyofallometric
relationships
whenevertwoor moreallometric
regression
equationsare multiplied
together
or
dividedby one another,because theestimatesoftheregression
parameters
are
just weightedaverages.For example,in quadrupedalmammals,regressionof
distancemovedperday,D, againstbodymass,M (inkg),yieldsD = 0.875M022
km/day
(Garland1983),and regression
oftheincrement
in energyexpendedper
kilometer
walked,E, againstbodymassyieldsE = 10700M? 684J/km
(Tayloret
al. 1982);and on thisbasis, Altmann
(1987)calculatedaverageenergyexpended
on locomotion
perdayas ED, thatis, at 10700M06840.875M022. Thiscalculation
overlookspossiblecovariancebetweendistancetraveledper day and energy
expendedperunitofdistance.
Similarexamplescan be foundin recentsurveysof allometric
relationships
(Peters1983;Calder1984).Atthetransition
fromtrottogallop,mammalian
stride
frequency
regressedon body mass yields11.8M-014 strides/s
(Heglundet al.
1974),andtheregression
ofstridelengthon bodymassyields0.35M0O38
m/stride;
thus,Calder(1984)estimatedthevelocityas theproduct11.8M-0?14
0.35M? 38
m/s.This estimateignoresthe possiblecovariancebetweenstridelengthand
As a further
stridefrequency.
oftheenergycontentofa
example,theregression
fullmammaliangut on body mass yields757M1.02 kJ,and the regressionof
metabolicdemandon bodymassyields504M?176kJ/day;
therefore,
gutturnover
timewas calculatedfromtheratio757M1.02/504M076 days/gutfull
(Calder1984).
variablesis ignored.
Again,possiblecovariancebetweenthecomponent
In orderto demonstrate
thepotential
oftheseerrors,we used data
magnitude
the
fromDonaldson(1919,table 1) to calculatethreequotienttraitsdescribing
of bodymass,M. The meanmass of
mass of ratskeletons,K, as a percentage
skeleton(i.e., >i(K/M)) as a percentage
ofbodymassis 6.51%;however,>i(K) =
12.99 and >i(M) = 222.74; therefore,1i(M)/IA(K)= 5.83%. Thus, calculatingthis
foraxialtraitas a quotientofmeansyieldsan errorof10.5%.Similarcalculations
massas a
ofbodymassandappendicular-skeleton
skeletonmassas a percentage
percentageof bodymass yielderrorsof 10.2%and 10.8%,respectively.
Using
calculateproducttraitsbutpresentmeans
information
frompapersthatcorrectly
covariancesamongcomvariablesas well,we foundthatignoring
ofcomponent
THE FALLACY OF AVERAGES
279
sex
fromnearzero(as inestimating
ponentvariablescan introduce
errorsranging
aggregaratiosofhamsters;
Labov et al. 1986)to wellover50% (as in estimating
tionlevelsin hummingbirds;
Tamm1985).
of
the potentialmagnitude
For regressions,
two examplesserveto illustrate
covarerrorscaused by combiningregressionequationswithoutconsidering
the
(1962,table1,Coleoptera)to estimate
iances.We used datafromGreenewalt
frequency
(s- 1)to winglengthon bodymass
regression
oftheratioofwing-beat
usingnaturalloga(mg)in 28 beetlesof 12 species. All data weretransformed
ofwing-beat
frequency,
B, on bodymass,
rithms
beforeanalysis.The regression
M, is B = 4.942M-0?077;the regressionof winglength,L, on body mass is L =
1.054M? 321.The ratioofthese equationsis thenB/L = 4.942M-007711.054M? 321
= 4.690M0- 398.However,theactualregression
ofindividual
valuesofBIL on M
of
is BIL = 3.889 M- 398.Thus, althoughthe exponentis correct,the coefficient
M estimated
is offby 17.1%.
by theratiooftheregressions
In a secondexample,we useddatafromGreenewalt
(1962,table15,Corvidae)
to estimatethe regressionof the ratioof "large" pectoralmuscle(= M. pectoralis) mass (g) to winglength(cm) on body mass (g) in nine corvids of nine
usingnaturallogarithms
beforeanalysis.The
species.All dataweretransformed
the
regression
ofpectoralmusclemass,P, onbodymassM is P = - 2.576M1l096;
regression
ofwinglengthL on bodymassis L = 0.885M0404.The ratioofthese
equations is PIL = - 2.576 M1.096/0.885M0404 = - 2.910M0O692.However, the
actualregression
ofindividual
valuesofPIL on M is PIL = - 3.461M0692. The
ofM is 15.9%.
errorin theestimateofthecoefficient
Our susceptibility
to the fallacyof averagesstemsin partfromdimensional
andthenignore
algebraically
analysis,inwhichwe treatourunitsofmeasurement
ofbirds,the
thefactthatwe are usingmeans.So, forexample,in a population
meanlifetime
reproductive
success (numberof chicksfledged)wouldequal the
ofclutchesper
meannumberofchicksfledged
perclutchtimesthemeannumber
seems
wereuncorrelated-which
lifetime
ifand onlyifthesetwocharacteristics
the fact thatchicks/lifetime
equals (chicks/clutch)unlikely-notwithstanding
bird.In usingmeans,naivedimensional
foran individual
analy(clutches/lifetime)
sis is likelyto be misleading.
Yet anotherproblemarisesfortraitshavingadditiveorsubtractive
components
(e.g., body mass, lifespan, skeletalelements).Here, althoughthe meanof an
additivetraitdoes equal thesumofthecomponent
means,thatis, ,u(X+ Y) =
bythesumor averageof
,u(X) + ,u(Y),thevarianceofthetraitis notestimated
thecomponentvariances(Lande 1977;Soule 1982).For example,althoughthe
meanage ofhumanfemalesat menopauseequalstheirmeanage at menarche
plus
and menopause,
themeanduration
phasebetweenmenarche
ofthereproductive
the varianceof the age at menopausedoes not equal the varianceof age at
ofthereproductive
phaseif,say,females
menarche
plusthevarianceofthelength
Lande (1977)
whoreachmenarchelatetendto reachmenopauseearly.Although
this
with
wingmeasurepointed out
regardto a biologicalproblem,chiropteran
on the
to base conclusions
ments(BaderandHall 1960),somebiologists
continue
of
additive
variances.
(unstated)assumption
THE AMERICANNATURALIST
280
THE SOLUTION
Products
producttraitcan be decomposedintotwo
thatan individual
Supposeinitially
componentvariablesX and Y. Then,providedthemomentsare
multiplicative
itfollowsfromthedefinition
ofcovariancethat
finite,
p(XY) =
M(X)1(Y)+ cov(X, Y),
where cov(X, Y) denotes the covariance of X and Y. Thus, ,u(XY) = ,u(X),u(Y)if
,u(XY)=
thus,ifX and Y are independent,
andonlyifX and Y are uncorrelated;
butthat
theyareuncorrelated,
,u(X),u(Y).(RecallthatifX and Y areindependent,
theconverseneed notbe true.)
Now, suppose thatthe componentvariablescan be measuredon the same
the data can thenbe
are made on n individuals,
individual;if measurements
writtenas (Xl,Y,), (X2Y1), . . ., (Xn,Yj. It is usual to assume thateach pair of
oftheotherpairs.Considertheestimators
is notindependent
measurements
n
,uxy= n-1 XjYj
j=1
and
IXIy=
n
n
j=l
j=l
(n ZXj)(n 1Yj)
for
for,u(XY) and p4( xy) = ,u(XY),but ,ux,uyis consistent
Now, jixyis consistent
,u(X),u(Y) and
11Ax>r= ji(X)ji(Y) + cov(X, Y)In = pL(XY) - cov(X, Y)(n - 1)/n;
traitif
themeanoftheproduct
meansthusestimates
theproductofthecomponent
Goodman(1960)derivedtheexactsampling
andonlyifX and Y areuncorrelated.
ifX and Y are independent,
variancesof Ai yand Ai Aiy. In particular,
=
or(AxY)2
[gX(X)2o( y)2 + i4 y)2or(X)2 +
r(X)2o( Y)2]In
and
=
(AA>Y)2
[11(X)2(F y)2 + gy(Y)2oF(X)2 + (r(X)2or(y)2ln]ln,
withfinite
whereU(X)2 denotesthe varianceofX. If X and Y are independent
variancesand at least one of ,u(X),,u(Y)is nonzero,itfollowsfromthecentral
forlargen,
limittheoremthatapproximately,
AY~
_N{I(XY),
[g(X)2o(y)2 + g(Y)2o(X)2 + r(X)2o(y)2]In}
and
A
A
~
N{p(XY),
[g(X)2o(y)2
+ gy)2r(X)2]ln}
withmeana and varianceb2."
where"- N(a,b2)" means "is normallydistributed
than uixyand hence is a
smaller
variance
has
a
If X and Y are independent,
,ux,iy
is
estimator.
However,ifX and Y havenonzerocovariance,[3x,uy
moreefficient
estimator.
biasedand A3 yis theappropriate
decomposedintoseveralcompoAnalogousresultscan be derivedforproducts
THE FALLACY OF AVERAGES
281
variablesX, Y, andZ,
nentvariables.For example,ifthereare threecomponent
ii(XYZ)
+
where C111(X,Y,Z)
+
1(X)1i(Y)(Z)
=
+ C 1 (X,Y,Z),
pL(X)cov(Y,Z)
- ji(X)][ Y - i( Y)][Z - ,u(Z)]}. Again,independence
,[X
=
+ ji(Y)cov(X,Z)
4(Z)cov(X,Y)
ensuresthatthe productof the meansequals the meanof the products.The
estimators
can be analyzedas before,using,forexample,resultsfromGoodman
(1962),and thesame qualitativeconclusionsobtained.
Ratios
The resultsformultiplicative
can be usedto deriveresultsforthe
components
ratiooftwocomponents.
SupposethatY > 0 andthatbotho(y)2 ando(XIY)2 are
finite.
Then,
>(X)
= 1i(XY/Y)
= 1(Y)1(X/Y)
+ cov(X/Y,Y),
so that
=
1i(X/Y)
-
11(X)/L(Y)
cov(X/Y,Y)/I(Y).
Thus, ,u(XIY) = ,u(X)/I,(Y) if and only if Y and XIY are uncorrelated,which is
constraints
extremely
unlikelybecause of mathematical
(Atchleyet al. 1976).
Sinceitis usuallydifficult
to thinkaboutthedependencebetweenY andXIY, the
is complicated.However,providedthatUr(1IY)2
is finite,
situation
,u(X/Y)= ,u(X),(uI/Y)+ cov(X,1/Y),
so that ,u(XIY) = ,u(X),u(1IY) if and only ifX and IY are uncorrelated.
variablescan be measuredon thesameindividual,
suchthat
Ifthecomponent
the data consistof n pairsof observations,(X1,Y1),(X2,Y2),...
n
Lx/y=
,
(Xn, Yn),consider
n-Z(XiIYi)
j=l
Xj (n
(nA
AX>/Y=
I/Yj)
and
n
XIIY
1=
(n lxi)
n
(n
ZY)
is ofinterest
itis
The lastestimator
onlyifXIY and Y areindependent;
therefore,
Now ,y is consistent
for,u(XIY) and pX(,A,Y)= ,u(XIY),
notdiscussedfurther.
forpL(X)ji(1IY)and
but AIi1/yis consistent
A(A
A
/Y)
=
j(X)ji(1IY) + cov(X,1IY)In
= ,u(X/Y) - cov(X,1IY)(n - 1)/n.
does notestimate,u(XIY)ifX and IIY are correlated.
Therefore,
,uxuL2Iy
By the
THE AMERICANNATURALIST
282
resultsof Goodman(1960),ifX and Y are independent,
ar(, xy)2
=
[1(X)2o(ljY)2
+
pj(1/Y)2o7(X)2
+ (r(X)2o(1IY)2]In
and
oQiX Ii/Y)2 = [g(X)2o(lI/y)2
+ >j(l/y)2a(X)2
+
r(X)2r(lI/y)2In] In
In largesamples,providedthatX and Y are independent,
xiy - N[i(XIY)a('IxY)2]
and
i(1Y/)2ez(X)2]/n},
>1/Y~N{p1(XIy),[p1(X)2o(1 Iy)2 +
as longas X and Y areindepenis thepreferred
estimate
so that,as before, yXi11Y
dent.
AllometricRelationships
is to takethenatural
decomposition
approachto a multiplicative
An alternative
Of course,if
of the variablesto obtainan additivedecomposition.
logarithm
X > 0, Y > 0, and themoments
are finite,
,u(lnXY) = ,u(lnX) + ,u(lnY)
and
u(lnXY)2 = o(lnX)2 +
-(lny)2 + 2 cov(lnX,lnY),
betweenlnXandlnY entersonlyinthevariancecalculasuchthattherelationship
tion. It is usual on an additivescale to assume thatthe variableslnX and
lnY are normally
distributed,
distributed.
If lnX and lnY are normally
,u(X) = exp[,u(lnX) + o(lnX)2/2],
,u Y) = exp[,u(ln Y) + c(ln Y)2/2],
and
,u(XY) = exp[,u(lnXY) + o(lnXY)2/2]
= ,u(X),u(Y)exp[cov(lnX,lnY)].
so thatifX
Thus,,u(XY) = ,u(X),u(Y)ifandonlyiflnXandlnY areuncorrelated,
and Y are independent,,u(XY) = ,u(X),u(Y).
Suppose thatn independentpairs of measurements(X1,YI), .
beenmade,and put
. .,
(X,, Yj) have
n
AIxy =
exp(n-
IZlnXjYj
+ c2I2)
and
n
n
FLX>L
exp(n
IEnXj
+X/
cx2exp
n
IEnyYj
(rY/2)
THE FALLACY OF AVERAGES
283
where
n
2
2~~~~~~~~~~~~~~~In
- (n
1)-l,Xj
I
'2
n
Yj - n-
n
= (n
ax
I)1(nXj
j=l
'I,nXiYi)
n
- n
2
I
EnXij
2
i=1l
and
(y=
(n
-
1)11(ln
J=
Yj - n 11nYi)2.
i=1
If lnX and InY are normally distributed,,ixy is consistent for ,u(XY) but
exp(n- 17= lnXjYj)is consistentonlyif lnX and lnY have degeneratedistriand
butionsand otherwiseunderestimates
,u(XY).If X and Y are independent
it
and
,u[(ln
Y)4]
are
finite,
follows
that
approximately,
for
large
n,
u[(lnX)4]
vixy
N(m(XY),m(XY)2{
(InX)2 + o-(lnY)2 + K3(lnX) + K3(ln Y)
+ [K4(lnX) - u(lnX)4 + K4(ln Y) -
u(lnY)4] /4 + u(lnX)2u(InY)2}/n)
and
xVlt
- N(m(XY),m(XY)2{
(InX)2 + o-(lnY)2 + K3(lnX) + K3(ln Y)
+ [K4(lnX) - u(lnX)4 + K4(lnY) - c(lnY)4]/4}/n),
wherem(XY) = exp[,u(lnXY) + o-(lnXY)2/2]and K(lfnX) = V{[lnX - (InX)]t}.
(IflnXand lnY are normally
distributed,
m(XY) = ,u(XY).)Again,ifX and Y are
independent,,iLx2iyis more efficient
than ,ixy,but iflnX and ln Y are correlated,
is
not
consistent
for
V2xVi
,u(XY).
The above resultscan be extendedto the decompositionof allometricrelation-
ships.If a producttraitcan be decomposedintotwo multiplicative
component
variablesX and Y, each ofwhichis relatedto bodymassM via Huxley's(1932)
allometricequation X = oaxMIxand Y = otyM13Y,
thenon the additivelogarithmic
scale it is convenientto suppose thatlnX givenM is normalwithmean ,u(lnX) =
otx+ f3xlnM andvarianceo-(lnX)2
and,similarly,
thatlnYgivenM is normalwith
mean,u(lnY) = oty + f3ylnM and variancecr(ln
y)2. Ofcourse,on thelog scale,
themeansare additive,thoughthevariancesneednotbe. On theoriginalscale,
givenM,
,u(X)= exp[ax + I3xlnM + or(lnX)2/2],
,u(Y)=
exp[oty+
j3y lnM
+
-(lnY)2/2],
and
,u(XY) = exp[ox + oCy+ (Px + Py)lnM + o-(lnX)2/2
+ c(ln Y)2/2+ cov(lnX,lnY)].
THE AMERICANNATURALIST
284
If n independenttriplesof measurements(X1,Y1,M1),(X2,Y2,M2),
Mn)have been made,let
exp{& + ,lnM +
Ax y=
,
(Xn,Y,
/212}
and
ALX>r=
exP{&x
+
+ IBxlnM + c2/2}exp{&y + ,ylnM
2y12},
where
A
=
+
Ax
Ay,
Oy
Ii
n
oXx= n-1
lnXj
-
j=1
Ii
n
iBxn 1lnMj,
j=1
n
n
y - Pyn-1lEnMj,
oty= n- In1
j=1
j=l
A
=
x
[_j(lnj
n
n
- n 1
lnMi)lnX
j] Z(lnMj
-
n
=
= [Z(lnMj - n j1,
n
Z (lnMj
lnMi)lnY]
-
2
1iZlnnM;)
n
n
n
y
Y,
x+
n
n
=
A
A
2
n1 Z, lnM;)
n
6.2 =
(n - 2) -1
(lnXYj
j=1
Z (lnXj
-
a
-
lnM1)2,
n
ax = (n - 2)1
-
ox
-
dy -_ylnMj)2.
j=l
IlnMj)2,
and
n
ay = (n - 2)-1
j=l
(lnYj
distributed,
As before,noticethatiflnXandInY arenormally
,ixyis consistent
then,Jxyand
for,u(XY) butexp(cx+ lnM) is not.IfX and Y are independent,
it
and
be
shown
that
both
estimate
can
,uX,uy
,u(XY)
asymptotically
,uX2iyis more
for,u(XY),
not
consistent
efficient
than,uxy;iflnXandln Y arecorrelated,uX,u
yis
theproblemsin takingtheproductof
and Au'y shouldbe used. Consequently,
are thesameas thosein takingtheproductofmeans.
relationships
allometric
ofResults
Summary
ofmeansto meansoffunctions.
functions
We summarize
theresultsrelating
A. Functionsof a singlevariable
1. Linear function:
,u(aX + b) = a,u(X) + b.
THE FALLACYOF AVERAGES
,u(1/X)$ 1144X);
2. Reciprocalfunction:
>(1IX) _ 11/(X) +
285
and Lawlor1981).
r(X)24i(X)3 (Templeton
j,[exp(X)] $ exp[,u(X)];
3. Exponentialfunction:
exp[,u(X)][1+ u(X)2/2] or
,i[exp(X)]
,[exp(X)]
=
exp[,u(X) + u(X)2/2]
ifX is normal.
B. Functionsof twovariables
1. Productofrandomvariables: ,u(XY) = ,u(X),u(Y)+ cov(X,Y) or
,(XY) = ,[exp(lnX + ln Y)]
= exp[,u(lnX) + o(lnX)2/2
+ ,u(lnY) + o(lnY)2/2
+ cov(lnX,ln
Y)].
IfX and Y are independent,
,u(XY) = ,u(X),u(Y).
- cov(X/Y,
2. Ratioofrandomvariables: ,u(XIY) = pu(X)/IR(Y)
Y)/4(Y)
= ,u(X),(1I/Y) + cov(X,1I/Y)
:
(X)4L(Y) + R(X)U(y)24i(y)3.
If Y andXIY are independent,
,u(XIY)= ,u(X)I/,(Y).IfX and Y are independent,
u(X/Y) = R(X)pu(1Y).
3. Sums of randomvariables: ,u(X + Y) = ,u(X) + ,u(Y),
u(X + y)2 = u(X)2 + u((y)2 + 2cov(X,Y).
C. Allometric
relationships
1. Singlerelationship:
IfX = aMO exp(Z) withZ
-
N(O,u2), then
= ctM3 exp(u2/2).
,(X)
2. Productoftworelationships:
IfX = ct1M0exp(Z1)and Y =
withZi - N(O,4or),
wherei = 1,2, then
R(XY) = OlO2
+
02
exp(Z2)
22M12
exp[c2l/2 + u2I2 + cov(Z1,Z2)].
If Z1 and Z2 are independent,then ,u(XY) = ala203
+1 2 exp(r2/2 + U2/2).
For each of thefunctions
involving
tworandomvariables,iftheappropriate
means(on theright
independence
structure
is present,combining
thecomponent
side of each equation)yieldsan estimator
witha smallerstandarderrorin the
finalresultthanthatobtainedby combining
and thenaveraging
thecomponents
them.
286
THE AMERICANNATURALIST
DISCUSSION
itis
othersintheliterature,
Fromtheexamplesgivenhereandfromnumerous
arebasedon an assumption,
apparent
thatmanyresultsinthebiologicalliterature
components.
Yet, the
ofmultiplicative
one rarelymadeexplicit,ofindependence
ofthetraitsofindividuals
natureofbiologicalsystemsis suchthatindependence
is an exceptionratherthanthecommonsituation.
inmostofthegivenexamplesof
is a setofindividuals
Although
thepopulation
hereapplyequallyto "populations"
means,all oftheresultspresented
population
meanvaluesfor
inthegeneralstatistical
senseoftheterm.Thus,whenobtaining
arisesif the
thetraitsof an individual,the same problemof nonindependence
meanvaluesof traitsare to be multiplied
or otherwise
combinedin a nonlinear
walksper
themeandistancethatan individual
fashion.Forexample,inestimating
The
day,thepopulationis theset of distancestraveleddailyby thatindividual.
mean daily distancefor an individualcan be estimatedaccuratelyfromthe
only
meanpace length
andmeandailypace frequency
productofthatindividual's
if the covariancebetweenthe two is negligible;the covariancewouldnot be
longerfordays
negligible
if,say,thelengthoftheanimal'space was appreciably
ifan animal'sfood
whichitwalkedmore(i.e., tookmorepaces). Similarly,
during
unitsbeingsampled,the
foodpatchesare thepopulation
intakeratesat different
morefood;thus,themean
animalmayfeedmorerapidlyin patchescontaining
intakeperpatchmaynotbe onlytheproductofthemeanintakerateand ofthe
meantimeperpatch.
providesnumerousexamplesof the false
Althoughthe empiricalliterature
thatthemeanoftheproductoftwoor morerandomvariablesequals
assumption
papersare not immune.For example,
the productof theirmeans,theoretical
inoptimalforaging
(KrebsandMcCleery1984;Lucas 1985,eq. 1) define
theorists
E fromforaging
fora givenamountof
theexpectedreward(e.g., energyreturn)
encounter
timeT as E = TIV, whereI is themeanprey-item
rate,and V is the
meanrewardfroma singlepreyitem.Thisformula,
however,ignorespotential
betweenencounter
rateandvalue(e.g.,foragers
highcorrelations
mayencounter
yieldpreyat a lowerratethanlow-yieldprey).In anotherveinof theorizing,
Studd and Robertson(1985) defineda male's expectedlifetimereproductive
tobreed,(2) the
ofsubadults
successas theproductof(1) theproportion
surviving
byadultmales,and (3) theexpected
production
averageannualrateofoffspring
lifespanofa breeding
anypatterns
ofcovariancebetween(1), (2),
male,ignoring
of these
and (3), whichare expectedvalues. An exampleof nonindependence
that
was givenbyPartridge
and Farquhar(1981),whodemonstrated
components
have shorter
lifespans.
fruitflymalesthatmatefrequently
For situationsin whichmeasurement
of componentvariableson the same
individuals
values shouldbe calculatedand
is possible,individualproduct-trait
ifthecovariancesare known,theformulas
thenaveraged.Alternatively,
given
valuesalmostinvaritheindividual
above can be used,thoughin suchsituations
ablyare knownand the meanof theproductscan be obtaineddirectly.When
is notpossibleor not
componentvariableson the same individuals
measuring
of
independence
feasible,one of severaloptionsis available.In somesituations,
THE FALLACY OF AVERAGES
287
componentvariablescan reasonablybe assumed,and thejustification
forthat
can be provided.If so, theproductofcomponent
meanscan be used
assumption
as an estimateof themeanoftheproductsofthecomponents.
For situations
in
betweenthevariablesis unknown,
whichtherelationship
theproductofcomponentmeanscan be presentedalongwiththecaveatthattheproductequals the
meanonlyifthevariablesareindependent.
between
Finally,whenthedifferences
meansof producttraitsin two different
populations(e.g., different
individuals,
groups,or species) are compared,if the covariancesbetweenthe component
variablesin thetwogroupscan be assumedequal, theycan be ignoredbecause
theywouldcancelout in calculating
thedifferences.
SUMMARY
In thebiologicalliterature,
the meanof theproductof two or morerandom
variablesis frequently
calculatedfromthe productof theirmeans.However,
unless the variablesare independent,
an exceptionaloccurrencein biological
thetwoare notequivalent.Corresponding
systems,
falseassumptions
commonly
are made about ratiosof meansand variousotherfunctions
of means.These
are examplesofperhapsthemostcommonstatistical
assumptions
fallacyin the
biologicalliterature,
thefallacyofaverages:thefalseassumption
thatthemeanof
a nonlinear
function
ofseveralvariablesequalsthefunction
ofthemeansofthose
variables.We providetherelationship
betweenfunctions
ofmeansandmeansof
functions
forcommonfunctions
ofone variable(linear,reciprocal,
andexponentialfunctions),
fortwoormorevariables(product,
ratio,sum),andfortheproduct
ofallometric
relationships.
ACKNOWLEDGMENTS
Thisresearchwas partially
supported
by NationalScienceFoundation(NSF)
grantDMS-8601732.Thismanuscript
was preparedusingcomputer
facilities
supto theDepartment
portedinpartbyNSF grantsDMS-8601732andDMS-8404941
of Statisticsat theUniversity
ofChicago.
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