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ß&
tions in the Red GrouseLagopus
lagopusscoticus.
J. Anim. Ecol. 36: 97-122.
MOSS,R. 1987. Demography of Capercaillie Tetrao
urogallusin north-east Scotland. III. Production
and recruitment of young. Ornis Scandinavica
18: 141-145.
[Auk, Vol. 105
1987.
The mechanics of annual
changesin ptarmigan numbers:a reply to Bergerud, Mossop,and Myrberget. Can. J. Zool. 65:
1043-1047.
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P. ROT•ER¾, & R. P^R•.
1984.
De-
mographic causesand predictive models of pop-
ß A. WATSONß•: J. OLLASON. 1982. Animal
populationdynamics.LondonßChapman& Hall.
WATSON,
A. 1965. A population study of ptarmigan
(Lagopus
taurus)in Scotland.J. Anita. Ecol. 34:
ulation fluctuations in Red Grouse. J. Anita. Ecol.
53: 639-662.
Received
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9 September
1987.
135-172.
ß & R. Moss. 1979. Population cyclesin the
Tetraonidae.
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Fennica
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Growth-curve Analysis: A Critical Reevaluation
RETO ZACH 1
Growth-curveanalysis(Ricklefs1967)haslong enjoyed an ardent following. RecentlyßBrisbin et al.
(1987)predicteda bright future for this methodology,
but failed to addresscritical shortcomingsof growthcurve analysisand of the Richards(1959) model in
particular.They alsocompletelyignoredalternatives.
Growth-curveanalysisis a tool onlyßand not an end
in itself. The main question is not which model to
use,but whether to use growth-curveanalysisat all.
For many studies growth-curve analysis is inappropriateß and alternativesßsuch as simple observedgrowth statistics, may be more effective growth
indicators.Although the Richardsmodel with its variable sigmoidshapeis intuitively attractiveßother simpler modelscan be superiorfor studying growth.
There are many reasonsgrowth-curveanalysismay
not be an ideal methodology.Growth-curveanalysis
is labor intensive becausenumerous data are required
to satisfymodelswith severalfitted parameters.For
passerines,with their short nestling timeßthis usually
meansdaily measurementon a rigid scheduleso that
each nestling is always measuredat about the same
time. Alternatively, variable time must be taken into
accountduring curve fitting. Growth-curve analysis
requiresnumerousprecisedata.Smallchangesin only
and Mayoh 1986a).Subtle stressfrom disturbances
and handling is difficultto detectand confoundsinterpretation of treatment effects.
A consistentfinding is large variation in growth
amongbroodsß
commonlyexceeding50%of the total
variation (Ricklefsand Peters1979,Zach and Mayoh
1982).Consequentlyßstatisticalpower to test for treatment effectsis low unless a large number of broods
is studied. Unfortunatelyßgrowth-curve analysisrequires much effort per broodßmaking it difficult to
study many broodssimultaneously.
For many passerines,growth-curveanalysisis dif-
ficult to apply becausefledging occurswell in advance of growth completion. Meaningful fitting of
growth curvesrequirespreciseestimatesof final size.
Body-massrecessionmay also occurbefore fledging
(Ricklefs 1968a).There seemsto be little agreement
on how to deal with this phenomenon. We have used
data up to, and including, the highestbody massbefore fledging, but this may not be entirely appropriate. Actually, adult body massis difficult to define
becausethe mass of most passerinesfluctuates seasonally.
To summarize,growth-curveanalysisis unsuitable
when resourcelimitations prevent collection of sufa single measurementcan profoundly affectparam- ficient and precise dataßwhen disturbanceslead to
brood abandonmentor growth depressionß
and when
eter estimates. Sensitivity depends on the type of
adult sizes cannot be unequivocally measured.Formodel employed, however.
High data requirementsof growth-curve analysis tunately, there are viable alternativesto growth-curve
analysis.
can stressboth parents and nestlings. Frequent disturbances can cause nest abandonment
and conseOur studieshave consistentlyindicatedthat several
quent lossof data.During cold,wet weather it is often simpleobservedgrowth statisticsare equalßor betterß
difficult to keepsmallnestlingswarm and dry; henceß growth indicatorsthan fitted modelparameters(Zach
handling alone can significantlyaffectgrowth (Zach and Mayoh 1986a,b). These statisticsare body-mass
asymptote,and body massßprimary-featherlength,
and foot length at or near fledging.They are consis•Atomic Energy of Canada Limited, Whiteshell tent with each other and with fitted model parameNuclear Research Establishment, Pinawa, Manitoba
ters. Further, they are more readily determined than
ROE 1L0, Canada.
fitted parametervaluesbecausefewer measurements
January1988]
Commentaries
are required, precisedata can be securedmore easily
from large nestlings, disturbancesare reduced, and
statisticscan be readily defined and measured.Fewer
measurementsper brood allow study of more broods
to quantify properly variation among broods and to
compensatefor unforeseenlosses.When measuring
observedgrowth statistics,suchlossesdo not represent much wasted effort. Finally, observed growth
statistics
arebiologicallymeaningfulandcanbe readily
interpreted.
Like growth-modelparameters,observedgrowth
statisticsare powerful stressindicatorsbecausethey
integrate growth performancethroughoutthe nestling period. Growth in passetinesseemsto follow a
rigid and rapid schedule,leaving little opportunity
for compensation.Thus, stressat any time in, or
throughout, the nestling period reducesthe bodymassasymptoteand is reflectedin the size measurementsat or near fledging.
Growth usually follows a sigmoid pattern in birds
(Ricklefs1968b,1973).Of the sigmoidgrowth models,
the von Bertalanffy,Gompertz,logistic,and Richards
models are commonly used. The first three have a
fixed shapeand involve three parameters(A = asymptote, K = growth-rate constant,c = constantof
integration);the Richardsmodel has a flexible shape
and four parameters(A, K, c, n = shapeparameter),
and in essence includes
therefore,lessdemandingin termsof data precision.
Further, their parametershave clearbiologicalmeanings and canbe readily interpreted (Pruitt et al. 1979).
In our studiesthe Richardsmodel usually failed to
explain a significantlylarger proportion of the variationin the datathan the best-fittingthree-parameter
model (Zachet al. 1984).This is not surprisingbecause
the best-fittingthree-parametermodel commonlyexplainedmorethan 99%of the variation.Clearly,threeparametermodels can have several distinct advantagesover the Richardsmodel.
Future studiesinvolving growth should selectthe
besttool for evaluatinggrowth performance.In many
casesthis meansthe use of simple observedgrowth
statisticsrather than growth-curveanalysis,or simple
growth models rather than the complex Richards
model.
LITERATURE CITED
BRADLEY,D. W., R. E. LANDRY, & C. T. COLLINS. 1984.
The use of jackknife confidenceintervals with
the Richards curve for describing avian growth
patterns. Bull. South. California Acad. Sci. 83:
133-147.
BRISBIN,][. L., JR., C. T. COLLINS, G. C. WHITE, & D.
A. MCCALLUM.1987. A new paradigm for the
analysisand interpretation of growth data: the
shape of things to come. Auk 104: 552-554.
the other three models. Given
sufficientdata to justify a four-parametermodel, the
Richards model seems to be most desirable; however,
there are several difficulties
Perturbation
with
studies have shown
DAVIS,O. L., & J. Y. Ku. 1977. Re-examination of the
this model.
fitting of the Richardsgrowth function. Biomet-
that the Richards
modelis particularlysensitiveto small changesin the
input data.Thus, thesedata mustbe preciseto obtain
good parameterestimates.In fact, some fitting pro-
rics 33: 546-547.
PRUITT,K. M., R. E. DEMUTH,& M. E. TURNERJR. 1979.
Practical application of generic growth theory
and the significanceof the growth curveparam-
cedures
(WhiteandBrisbin1980)requirethatd•tabe
measuredwithout error. The high sensitivitymay be
relatedto the fact that K and n are not independent.
Theseparametersseemto be consistentlycorrelated,
with a coefficient,r, exceeding0.9 (Bradleyet al. 1984,
Zach et al. 1984).This is a well-recognizedproblem
(Davis and Ku 1977, Ricklefs 1983). Thus, although
the Richardsmodelhasfour parameters,it effectively
has only three, like the modelswith a fixed shape,
because
of the close correlation
between
K and
209
eters. Growth
43: 19-35.
RICHARDS,
F.J. 1959. A flexible growth function for
empirical use. J. Exp. Bot. 10: 290-300.
RICKLEFS,
R. E. 1967. A graphicalmethod of fitting
equationsto growth curves.Ecology48: 978-983.
1968a. Weight recessionin nestling birds.
Auk 85: 30-35.
1968b. Patternsof growth in birds. Ibis 110:
419-451.
n.
1973. Patternsof growth in birds. II. Growth
Anotherproblemwith the Richardsmodelis param-
rateand modeof development.Ibis 115:177-205.
eter interpretation.It is difficult to attributebiological
meaningto the shapeparameter,n, and a priori state
whether,or how, it will be affectedby a growthstress- --,
Biol. 7: 1-83.
or. Unfortunately, becauseof the correlationbetween
K and n, and the variableshapeof curves,the growthrate constant,K, is also difficult to interpret and unsuitablefor strictcomparisons(Ricklefs1983,Bradley
et al. 1984).Fortunately,there are useful,proven alternatives
to the Richards
model.
With only three parameters,the von Bertalanffy,
Gompertz,and logisticmodelsrequire fewer data for
meaningful fitting than the Richardsmodel. These
modelsare alsolesssensitiveto changesin data and,
1983. Avian postnatal development. Avian
& $. PETERS.1979. Intraspecificvariation in
the growth rate of nestling EuropeanStarlings.
Bird-Banding 50: 338-348.
WHITE, G. C., & I. L. BRISBINJR. 1980. Estimation and
comparisonof parametersin stochasticgrowth
models
for Barn Owls.
Growth
44: 97-111.
ZACH, R., Y. LINER,G. L. RIGBY,8•:K. R. MAYOH. 1984.
Growth curve analysis of birds: the Richards
model and proceduralproblems.Can. J.ZooI. 62:
2429-2435.
--,
& K. R. MAYOH. 1982. Weight and feather
210
Commentaries
growth of nestling Tree Swallows.Can. J. Zool.
&
& --..
1986b. Gamma-radiation effects
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Received
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1987,accepted
9 October1987.