552
Commentaries
[Auk,Vol. 104
A New Paradigmfor the Analysis and Interpretation of Growth Data:
The Shapeof Things to Come
I. LEHRBRISBIN,JR.,• CHARLEST. COLLINS,
2 GARYC. WHITE,3 AND
D. ARCHIBALD MCCALLUM 4
The processof growth is basicto all organisms,and
the path or trajectory taken by the growth process.
studiesof growthhavelongbeena subjectof concern The parametersof size,rate,and shapein the Richards
to ornithologists.At a workshop held at the recent sigmoidmodelare, at leasttheoretically,free to vary
meeting of the American Ornithologists'Union (1821 August 1986), a series of presentationsand subsequentdiscussionslaid the foundation for what we
feel to be a new paradigm for studying growth. The
strengthsof this new paradigmlie in its potential for
moredetailedquantitativecomparisons
of growth in
general and intraspecificcomparisonsin particular.
It is our purposeto summarizethe majoraspectsof
this paradigm and to provide information on its development.The paradigmwas derived mainly from
studiesof birds,but shouldhave broadapplicability
to studiesof a variety of organisms.
Growth was first studied qualitativelyby describing developmentalstagesin chronologicalseries.This
was followed by quantitative formulationsthat considered growth as the net result of simultaneousanabolic and catabolicprocesses.Many of these early
concepts(e.g.Huxley 1932,von Bertalanffy1957)still
influence current thinking. A third stagewas introducedby Ricklefs(1967),basedon a graphicalmethod
of fitting datato S-shapedor sigmoidgrowth models.
This methodology,applied to interspecificcomparisons,hascontributedsignificantlyto our understanding of the meaning of the patternsof variation in
aviangrowth--particularlyfroman evolutionarypoint
of view (Ricklefs 1983). The new paradigm is also
basedon the fitting of data to sigmoidmodelsbut is
independently of each other.
In the Richardsmodel,growth-curveshapeis quantified by the so-called"shape parameter," m. When
m has values of 0, 0.67, or 2.0, the Richards model is
identical to the monomolecular (also known as the
negativeexponential),von Bertalanffy,or logisticsigmold models, respectively,and as m approachesa
valueof 1.0the Richardsmodelapproachesthe Gompertz. In this sensethe Richardsmodel may be considereda generalized"parent" growth function, from
which almostall of the other commonlyusedgrowth
functionsmay be generatedby changesin the value
of the curve shapeparameter,m.
The quantificationof growth rate has been a particular problem in the past. In its original form the
Richardsmodel contained a growth-rate parameter,
K, which becameunstablestatisticallyas the value of
m approached1.0. Recentreparameterizationsof the
Richardsmodel,however,producedalternativemeans
of describinggrowth rate that do not have this problem and that are more meaningfulbiologically(e.g.
the parameterG of Bradley et al. 1984,or the parameter T of Brisbin et al. 1986b).
One important hypothesistested within the new
paradigm concernedthe suggestionthat growth-curve
shape,as quantifiedby the Richardsshapeparameter
m, is severaltimesmorelikely to changein response
to environmentalstressthan is either asymptoticsize
specificas well as interspecificcomparisons.
(A) or growth rate (T) (Brisbin et aL 1986a-c).These
The Richardssigmoid growth function (Richards experimentsdealt with a variety of speciesexposed
directed more toward questions that involve intra-
1959) and its expressionin terms of parameterswith
to various stressors,with comparisonsat the intra-
specificbiologicalmeaning(Bradleyet al. 1984,Bris-
specificlevel. The findings suggestedthat comparisonsbasedon the useof modelsin which curveshape
is setand not allowed to vary (e.g. Fendley and Brisbin 1977) may not be useful in understanding the
effectsof environmentalstresson growth. Moreover,
others (Pasternakand Shalev 1983) suggestedthat
changesin growth-curve shapealone can reflect an
alterationin the energeticefficiencieswith which birds
may grow to a specifiedsize within a specifiedtime.
Considerationsof changes of growth-curve shape,
within the approachesoutlined here, may thus be
bin et al. 1986b) have been of basicimportance to the
new growth paradigm.Theseapproachesnow allow
independent quantitative assessmentsof three bio-
logically meaningful aspectsof growth: (1) size,the
upper or asymptoticlimit; (2) rate, a measureof the
time requiredfor specifiedgrowth incrementsto take
place;and (3) shape,
a quantitativemeasuredescribing
• SavannahRiver EcologyLaboratory,P.O. Drawer
E, Aiken, South Carolina 29801 USA.
important to studies of basic bioenergetics, as well as
2Departmentof Biology,California State University, Long Beach,California 90840 USA.
3 Department of Fishery and Wildlife Biology,Colorado State University, Fort Collins, Colorado 80523
to commercialinterestsof the poultry industry.
Fitting growth curves requires nonlinear leastsquarescurve-fittingtechniques.Thesehave now been
adaptedto nearly all of the standardstatisticalpack-
USA.
ages, and some can now be used on personal micro-
4Department of Biology, University of New Mexico, Albuquerque,New Mexico 87131 USA.
computers.Thesetoolssimplify the fitting of the new
growth equations. Proceduressuch as PROC NLIN
July1987]
Commentaries
of SAS (SAS Inst. 1982) or program AR of BMDP
(BMDP 1983) permit the fitting of growth data (enteredasdatesor agespairedwith corresponding
body
massesor measurements)without writing complex
differentialequations.As suggested
by Ricklefs(1983),
these nonlinear curve-fitting techniques can replace
the graphical method of selectingand analyzing nonlinear growth models.
Along with theseadvanceshave comeconcomitant
developmentsin readily usablestatisticalpackages
for testing the parametersestimatedby the curvefitting programs.This easein detectingstatisticaldifferencesamong treatmentgroupsencouragesthe use
of growth data to rigorously test important hypothesesconcerningthe biology of birds and other organisms.In the caseof the Richards model, for example, the parametersquantifying growth size, rate,
and shapecan now be tested for differencesamong
such groups as males vs. females, birds maintained
on different diets or exposedto different levels of
some toxicant, or birds from different parts of the
breeding range.
The proceduresfor fitting growth data to a model
and then comparingstatisticallythe resultsfor size,
rate,and shapewill vary accordingto the type of data
available. As pointed out previously (Ricklefs 1983,
Bradley et al. 1984), several different experimental
designsexistby which growth data can be collected.
Each design imposescertain restrictionsfor fitting
data to models and then comparing the estimated
parameters.Three of the more important of thesedesigns are:
(1) Longitudinaldata sets,where individual organismsare weighed or measuredrepeatedlythroughout
the growing period, ideally until a sizeasymptoteis
reached.Becauseof problemsintroducedby autocorrelation and becausesuccessivedata points from the
same individual are not independent, such longitudinal growth data shouldbe analyzedwith a processerror modelasdescribedby White and Brisbin(1980)
for the Richardsfunction. In a process-errormodel,
growth rate is considereda function of increasing
body size or mass. Failure to use the process-error
model approachfor longitudinal data might produce
excessivelynarrow confidenceintervals and increase
the likelihood that significancemay be claimed erroneouslybetween parameterestimatesfor different
treatment groups (White and Brisbin 1980). Application of a multivariate analysisof variance (MANOVA) and then one-way analysesof variance (ANOVA) procedurescan then be utilized to identify the
parametersthat differ betweentreatmentgroup(Brisbin et al. 1986a-c).
(2) Cross-sectional
data sets,where each individual
is weighed or measuredonly once (as, for example,
in the caseof destructivepopulation sampling). In
such casesit is not possible to test for differences
among individuals. Differences between treatment
groupsare testedwith an F-statisticcomparing "com-
553
plete" vs. reducedmodels(White and Brisbin 1980).
Cross-sectional
data cannotbe analyzedwith the process-errormodel approach but must be fit directly
using the integrated form of the growth equation in
which size or massis consideredasa function of age.
Fitting the integrated form of the Richards model
requires the addition of a fourth parameter to the
three describedearlier. This fourth parameter is usually the size or massof the organismat birth (or
hatching)and either maybe enteredinto the equation
as a constant (the value of which is determined from
observationsof the sizes or massesof newly born or
hatchedindividuals) or may be estimatedby the curvefitting procedure.Further discussionof the analysis
of cross-sectional
data sets with
the Richards
model
is provided by Bradley et al. (1984).
(3) Mixed data sets, where some individuals are
weighed or measuredonly oncewhile othersmay be
weighed or measuredseveral times. These include
the interval data setsdescribedby Ricklefs(1983)and
Fabens(1965). Such data may be analyzed by curvefitting proceduresusing a jackknife techniqueas describedby Bradley et al. (1984).
One danger that accompaniestheseanalytical tools
is their use to analyze data setsthat do not warrant
their application.It is now possibleto use curve-fitting proceduresthat will createan estimateof a final
asymptotewhether or not one existsin the data. Frequently, for example,growth data setsare truncated
longbeforethe subjects
approachasymptote.One such
situationis the growth dataobtainedfor specieswhose
young leave the nest (and thus can no longer be
weighed or measured)before they reach adult size.
In somecases,asymptotesestimatedfrom such data
setsare obviously unrealistic and are readily recognized as artifacts.The danger lies in caseswhere a
realisticvalue is generatedfor the asymptotebut must
still be consideredunreliable in light of the limitations of the data used for the estimate.
Each researcher must consider which growth parameters are to be estimated and what specific hypothesesthe growth data will be used to test. The
burden is also placed on the investigatorto demonstratethat the dataare sufficientto warrant the degree
of model complexity used in their analysisand interpretation. Editors and referees should demand
nothing less in evaluating such studiesfor publication.
Besidesthe need for data at and beyond the attainment of asymptote,considerationmust be given to
the total number of data points available and their
spacingthroughout the growing period. Data setsof
only a few points determined at widely spacedin-
tervalsthroughout
thegrowingperiodcannotbe expected to give an adequateestimateof the value of
the shapeparameterfrom any sigmoidmodel even
if someof the data pointsare taken well after asymptotic sizeis attained. Practicalconsiderationsmay thus
often prevent the collection of sufficient data for a
554
Commentaries
[Auk,Vol. 104
ß
, --,
& L. A. MAYACK.1986c. Sigsigmoidanalysis.In suchcases,
or wheneveradequate
mold growth analysesof Wood Ducks:the effects
post-asymptoticdata cannot be obtained,it may be
of sex, dietary protein and cadmium on paramnecessaryto choosea simpler model (e.g. linear or
eters of the Richards model. Growth 50: 41-50.
exponential) to test hypotheses.Such procedures,
A.J. 1965. Propertiesand fitting of the von
while failing to provideall the informationcontained FABENS,
Bertalanffygrowth curve. Growth 29: 265-289.
in a sigmoidanalysis,are neverthelesspreferableto
analytical proceduresthat are unwarrantedly com- FENDLEY,T. T., & I. L. BRISBIN,JR. 1977. Growth curve
analyses:investigationsof a new tool for studyplex with respectto the nature of the dataavailable.
ing the effectsof stressupon wildlife populaWe thank the many individuals who have contribtions. Proc. XIII Intern. Congr. Game Biol.: 337uted to our thinking on this subjectover the years,
350.
as well as at the recent Avian Growth Workshop.
Manuscriptpreparationand the participationof Brisbin and White in the Workshop were supportedby
HUXLEY,
J.S. 1932. Problemsof relative growth. Lon-
a contract (DE-AC09-76SROO-819) between the In-
PASTERNAK,H., & B. A. SHALEV. 1983. Genetic-eco-
stitute of Ecologyat the University of Georgiaand
the United StatesDepartmentof Energy.
nomic evaluationof traits in a broiler enterprise:
reductionof food intake due to increasedgrowth
rate. Brit. Poultry Sci. 24: 531-536.
RICHARDS,
F. 1959. A flexible growth function for
empirical use.J. Exper. Bot. 10: 290-300.
RICKLEFS,
R. E. 1967. A graphicalmethod of fitting
equationsto growth curves.Ecology48: 978-983.
1983. Avian postnatal development. Pp. 283 in Avian biology, vol. 7 (D. S. Farner, J. R.
King, and K. C. Parkes, Eds.). New York, Aca-
don, Methuen.
LITERATURE CITED
VON BERTALANFFY,L. 1957. Quantitative
laws in me-
tabolismand growth. Quart. Rev. Biol. 32: 217231.
BMDP. 1983. BMDP statisticalsoftware,1983printing. LosAngeles,Univ. California Press.
demic
BRADLEY,D. W., R. E. LANDRY,& C. T. COLLINS. 1984.
Press.
The useof jacknife confidenceintervalswith the
Richardscurve for describingavian growth pat-
SASINSTITUTE,
INC. 1982. SASuser'sguide:statistics,
1982 ed. Cary, North Carolina, SAS Inst.
terns. Bull. Southern California
WHITE, G. C., & I. L. BRISBIN,JR. 1980. Estimation
Acad. Sci. 83: 133-
and comparison of parameters in stochastic
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147.
BRISBIN,I. L., JR.,K. W. MCLEOD,& G. C. WHITE. 1986a.
Sigmoidgrowthandthe assessment
of hormesis:
111.
a casefor caution. Health Physicsin press.
--,
G. C. WHITE, & P. B. BUSH. 1986b. PCB intake
Received9 September
1986, accepted24 February1987.
and the growth of waterfowl:multivariateanalysesbasedon a reparameterizedRichardssigmold
model.
Growth
50: 1-11.
On Paradigmsvs. Methods in the Study of Growth
MICHAEL
Paradigmsprovidemodelsfor restructuringour investigations.New paradigmsallow usto answerquestionsthat formerlyeludedus.Because
growth isboth
a universal biologicproperty and a complex,usually
nonlinear phenomenon, new approachesto quanti-
fyinggrowtharehighly desirable.
Thesigmoidcurve,
long heraldedasthe growth curve,reappearsin many
mathematical, economic, and scientific analyses.Ac-
cordingly,access
to mathematicaltoolsfor analyzing
GOCHFELD •
and characterizingsigmoidcurvesaffordsthe opportunity to use parametersof the curve as discreteindependent variables.Brisbin et al. (1987) detail new
methods for approachingthe study of the sigmoid
growth function. With powerful statisticalpackages
available to crunch the numbers, the iterative proceduresfor solving nonlinear function problemsare
now readily available.
Apart from lauding the introductionof suchprocedures, I want to call attention to the fact that Brisbin
• Environmental and Community Medicine,
UMDNJ--Robert Wood JohnsonMedical School, Piscataway, New Jersey08854 USA.
et al. (1987)do indeedoffera paradigmand not merely a tool. I emphasizethat the availability of these
procedures,by themselves,allows us to restructure
our investigationsand to ask new questions.In fact,