vol. 176, no. 6

the american naturalist

december 2010

Matrix Dimensions Bias Demographic Inferences:
Implications for Comparative Plant Demography
Roberto Salguero-Gómez* and Joshua B. Plotkin
Biology Department, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Submitted December 8, 2009; Accepted August 12, 2010; Electronically published October 22, 2010
Online enhancement: zip file.

abstract: While the wealth of projection matrices in plant demography permits comparative studies, variation in matrix dimensions complicates interspecific comparisons. Collapsing matrices to
a common dimension may facilitate such comparisons but may also
bias the inferred demographic parameters. Here we examine howmatrix dimension affects inferred demographic elasticities and how
different collapsing criteria perform. We analyzed 13 # 13 matrices
representing nine plant species, collapsing these matrices (i) into even7 # 7, 5 # 5, 4 # 4, and 3 # 3 matrices and (ii) into 5 # 5 matrices
using different criteria. Stasis and fecundity elasticities increased
when matrix dimension was reduced, whereas those of progression
and retrogression decreased. We suggest a collapsing criterion thatminimizes dissimilarities between the original- and collapsed-matrix
elasticities and apply it to 66 plant species to study how life spanand growth form influence the relationship between matrix dimension and elasticities. Our analysis demonstrates that (i) projection
matrix dimension has significant effects on inferred demographic
parameters, (ii) there are better-performing methods than previously
suggested for standardizing matrix dimension, and (iii) herbaceousperennial projection matrices are particularly sensitive to changes inmatrix dimensionality. For comparative demographic studies, we recommend normalizing matrices to a common dimension by collapsing higher classes and leaving the first few classes unaltered.
Keywords: collapsing, comparative plant demography, elasticity, matrix dimension, population growth rate (l), projection matrix
models.

Introduction
Although most ecologists carry out specific experimentswith a limited set of study species, their ultimate goal is
to discover general rules, often by compiling and comparing studies. Indeed, most general ecological rules (e.g.,
exponential growth potential and regulated population
growth, the principle of competitive exclusion, robust coexistence, life-history trade-offs) arose and have been con* Corresponding author; e-mail: salguero@sas.upenn.edu.
Am. Nat. 2010. Vol. 176, pp. 000–000. 䉷 2010 by The University of Chicago.
0003-0147/2010/17606-51781$15.00. All rights reserved.
DOI: 10.1086/657044

firmed through the extensive application of standardized,
comparative methodologies.
In the case of demography, projection matrix models
have become a widely used, powerful tool for studying
ecological and evolutionary questions. A matrix model
classifies individuals in a population according to a discrete
range of ages (Leslie 1945), sizes, or developmental stages(Lefkovitch 1965) or a mixture of these variables (e.g.,
Ehrlén et al. 2005; Jacquemyn et al. 2005). The projection
matrix quantifies three fundamental processes for each of
the classes that together determine the life cycle of thespecies: (i) the probability of survival of its individuals,(ii) their probability of staying in the same class (stasis)or transition to another class (progression to higher classes
or retrogression to lower classes), and (iii) their contributions with newborns to the population (fecundity). Matrix models have been used to project population growth(e.g., Lefkovitch 1965; Marrero-Gómez et al. 2007) and to
examine the relative contributions of the different demographic processes to the population growth rate (e.g.,
Bierzychudek 1999; van Mantgem and Stephenson 2005)
using elasticity analysis (de Kroon et al. 1986). Matrix
models have also been used to explore the importance ofalternative life-history strategies (e.g., Moloney 1988; Morris and Doak 1998), to identify the most vulnerable classesof a life cycle for the purposes of population management(Werner and Caswell 1977; Lubben et al. 2008), and toestablish exploitation regimes that are compatible withlong-term demographic viability (Nault and Gagnon 1993;
Freckleton et al. 2003).
A great potential for comparative demographic studies
has emerged from the large body of literature based on
population projection matrices. Such matrices have now
been estimated from data and published for more than
475 plant species (R. Salguero-Gómez, unpublished data),
whose life spans range across four orders of magnitude(e.g., Namkoong and Roberds 1974; Dostal 2007). Because
these demographic dynamics are compiled in the sameformat—a matrix—the data can, in principle, be used to

000 The American Naturalist
examine broad ecological and evolutionary questions
about demographic dynamics.
Studies using large numbers of projection matrices haveallowed for linkage between specific demographic processes and stages along ecological succession gradients (Silvertown et al. 1992); for the establishment of methodologies to study the responses of populations to ecotones(Angert 2006), herbivory (Maron and Crone 2006), or
habitat fragmentation (Bruna et al. 2009); and for comparing demographic dynamics of native and invasive plant
species (Ramula et al. 2008), phylogenetic relationships oflife-history strategies (Burns et al. 2010), or relationships
between short-term (transient) and long-term (asymptotic) population dynamics (Stott et al. 2010). The numberof demographic studies based on projection matrices is
growing rapidly (fig. A1, in the online edition of the American Naturalist). Consequently, matrix models will likely
serve an important role in future studies of broad ecological and evolutionary questions.
Using published projection matrices for comparative
studies is attractive, but it is complicated by the fact thatthese matrices vary in the number of classes employed, or
matrix dimension (fig. 1). The dimension of projection
matrices influences the apparent demographic processesof a specific class, such as per capita fecundities (de Matosand Silva Matos 1998), as well as other parameters derived
from the matrix, such as population growth rates (Lamarand McGraw 2005; Ramula and Lehtilä 2005); transient
dynamics (Tenhumberg et al. 2009; Stott et al. 2010); elasticities of matrix elements (Silvertown et al. 1993; Enrightet al. 1995) and vital rates (Zuidema 2000; SalgueroGómez and Casper 2010); elasticities of demographic pathways (Salguero-Gómez and Casper 2010), as analyzed byloop analysis (van Groenendael et al. 1994); and demographic relationships based on phylogeny (Burns et al.
2010; Stott et al. 2010). This is a nontrivial issue because
matrix dimension varies a great deal, from two (Sohn and
Policansky 1977) to 24 (Meagher 1982). Although several
algorithms have been proposed to determine optimum
criteria to establish cutoffs for classes in constructing theoriginal matrices (Vandermeer 1978; Moloney 1986; Shimatani et al. 2007), thus offering some standardization,
most plant demographers still classify systems on the basisof the specific biology of the species and the question being
asked (Caswell 2001).
Concerns about matrix dimension were first addressedby Enright et al. (1995), who suggested ways to overcome
the problem of variation in dimension among projection
matrices. First, they proposed keeping the number of classes proportional to the life span of each species. However,
populations of many short-lived herbaceous species areusually classified using many life stages (e.g., 15 classes in
Allium tricoccum [Nault and Gagnon 1993] and 17 in Ari-

Figure 1: Life span (mean Ⳳ SE, gray bars) and matrix dimension (meanⳲ SE, white bars) by growth form, collated from 475 published projection
matrices of plant species. Letters indicate significant differences amonggroups (Tukey-Kramer HSD test).

saema serratum [Kakehashi and Kinoshita 1990]; fig. 1).
It would be laborious and probably meaningless from abiological perspective to follow the suggestion of Enrightet al. (1995) and enforce a correspondingly higher numberof life stages for longer-lived herbaceous, shrub, palm, andtree species (Ebert 1999; Caswell 2001). On the other hand,decreasing the number of classes for shorter-lived species
matrices would blur the study of their demographic dynamics. Alternatively, Enright et al. (1995) recommended
collapsing matrices to the same dimension for all speciesof a comparative study. However, we know of no comparative plant demographic study that has adopted thissuggestion (but see Salguero-Gómez and Casper 2010),
perhaps because this would require raw census data foreach species, which are frequently unavailable.

Matrices in Plant Comparative Demography 000
Currently, comparative demographic studies either do
not control for varying matrix dimensions (Silvertown etal. 2001; Franco and Silvertown 2004; Bruna et al. 2009)or treat dimension as a covariate in the analysis (Ramula
et al. 2008; Burns et al. 2010; Stott et al. 2010), which isinappropriate unless the residuals are normally distributedwith linear effects on inferred parameters. Integral projection models (IPMs; Easterling et al. 2000), on the other
hand, offer an appealing alternative for demographic comparisons because they are more robust to life-cycle complexity than classical projection matrices (Ramula et al.
2008; Zuidema et al. 2010). However, we know of only adozen plant species that have been studied with IPMs. For
classical projection matrices, Enright et al. (1995) suggested specific collapsing criteria to minimize alterations
to demographic inferences. These criteria include forming
collapsed classes of similar residence times (the averagetime spent by an individual in a given class) and not
merging nonreproductive and reproductive classes, because they are biologically different. However, no study
has explicitly evaluated whether these collapsing criteria
minimize associated changes to demographic inferences
drawn from the collapsed matrix.
In this article, we explore the inferred demographic
parameters associated with matrices of different dimensions with the goal of providing rational guidelines forinterspecific comparative studies. Specifically, we evaluate
how the relative importance of each demographic parameter (elasticity) varies as a function of matrix dimension. We pay special attention to the population
growth rate and elasticities because they are of primaryinterest in comparative studies (Silvertown et al. 2001).
We examine an overlooked algorithm that allows the reduction of matrix dimension without using raw data andwithout altering the inferred population growth rate, reproductive output, or stable class distribution (Hooley2000). We apply this algorithm to the projection matricesof nine plant species with different growth forms andlongevities in order to (i) study the dissimilarities in theinferred elasticities of demographic processes in collapsed
matrices and (ii) evaluate Enright et al.’s (1995) collapsing criteria, as well as (iii) propose alternative collapsing criteria that minimize the differences in elasticity
values between the original and the collapsed matrices.
We demonstrate that matrix dimension affects the estimation of all parameters of interest, particularly in herbaceous perennial species, and that such relationships are
not always monotonic. We discuss the limitations of
Hooley’s (2000) algorithm regarding stable stage distribution of the population. Finally, we conclude with somesuggestions for implementing meaningful comparative
demographic studies based on projection matrices.

Implementation of the Collapsing Algorithm inSize- and Developmental Stage–Based Models
The algorithm of Hooley (2000) describes a method to
reduce the dimension of a projection matrix, once it has
been decided which classes to collapse. In its original form,
Hooley’s algorithm was intended for age-based models.
These models provide poor descriptions of plant demographic patterns because plants are often difficult to age
and because age is usually not the best predictor of plant
demographics (Harper 1977; Werner and Caswell 1977;
Ebert 1999). Here, we expand Hooley’s algorithm to sizeand developmental stage–based matrix models. Such models allow individuals not only to advance several classes
per unit time but also to retrogress (e.g., Bruna and Oli
2005; Lucas et al. 2008).
Briefly, three steps must be taken to reduce a projection
matrix of n dimensions, A n#n, to one of m dimensions,Cm#m: (1) choose classes to collapse according to somespecified “collapsing criteria,” (2) collapse rows of thoseclasses in the matrix, and (3) collapse the respective columns. Choosing the appropriate classes to collapse (i.e.,
specifying the collapsing criteria) is a nontrivial decision,
and we discuss it below. Collapsing rows is performed bya simple element-by-element addition. Collapsing the columns is the most complex step of the process because,
without the raw data, we do not know the proportion ofindividuals of the original classes that will contribute toforming the matrix elements aij of the newly collapsedclass.
In the simplest case possible, that of collapsing the twoclasses k and k ⫹ 1 of an n-dimensional matrix A n#n into
a new class k  in the collapsed matrix C(n⫺1)#(n⫺1), we first
add the matrix elements ak, j ⫹ ak⫹1, jGj along the k andk ⫹ 1 rows. The resulting intermediate matrix B(n⫺1)#n isthen collapsed for columns k and k ⫹ 1. For age-basedmodels, determining the entries in column k  of the newcollapsed matrix is rather intuitive: simply calculate theproportion of individuals in class k ⫺ 1 who survive, andconsequently age, entering class k after one transition period, as well as the proportion of individuals who survive
from class k and will automatically age into class k ⫹ 1.
Surviving individuals from class k ⫹ 1 do not represent achallenge for this calculation because they will transition
into another class that is not collapsed in this example.
The calculation of the entries of column k  for size- anddevelopmental stage–based models is somewhat challenging, because it involves the calculation of survival for theindividuals in the two classes to be collapsed (k and k ⫹1), transition probabilities with which the individuals ineither class will not leave the collapsed class, and transition
probabilities with which the individuals from all other classes (e.g., k ⫺ 1, k ⫹ 2) may enter the collapsed class k ;000 The American Naturalist
abrupt changes in size and development, typically expressedby some plant species, may lead individuals to leap over
several classes over one time interval. For age-, size-, ordevelopmental stage–based models, the collapsed vector
that can be obtained from the intermediate matrix
B(n⫺1)#n will report only the total number of individuals
in the collapsed class and not the breakdown in the number of individuals coming from each class involved. Thus,in order to collapse the columns, we must assume thatthe population was close to equilibrium during the field
censuses. In this case, it is possible to take the elementsof the stable class distribution (w) as the weighting factors
to construct the new collapsed class. Consequently, wedefine the weighting factors rk p wk /(wk ⫹ wk⫹1) andrk⫹1 p wk⫹1/(wk ⫹ wk⫹1), which are the proportions of themerged classes that we expect to form from the originalclass k and class k ⫹ 1, respectively. Column k  in the collapsed matrix C is made of entries that are the weighted
averages, using weighting factors rk and rk⫹1, respectively,of columns k and k ⫹ 1 of the partially collapsed matrix
B. The stable class distribution is an intrinsic parameterof the projection matrix that can be obtained with a twoline command in Matlab: the right eigenvector associatedwith the dominant eigenvalue. This knowledge regarding
the stable class distribution is critical because it can free
us from needing the raw field data to be able to quantify
matrix entries in collapsed matrices (app. D, in the online
edition of the American Naturalist, describes whether our
techniques are robust when the assumption of stable equilibrium is violated).

Matrix Algorithm Implementation
Mechanistically, the collapsing algorithm can be applied
to almost any (see “Discussion”) projection matrix following the calculation:
Pm#n 7 A n#n 7 Qn#m p Cm#m ,(1)

where P is a matrix of m # n dimensions containing
dummy holders (“1”) on the elements that are to be collapsed (ak , k and ak , k⫹1), with the rest of the matrix being
formed by identity submatrices I,
0 
Ik⫺1 0 0
Pp 0 1 10 ,
 0 0 0 I n⫺k⫺1m#n(2)

and Q is a matrix of n # m dimensions that contains theweighting factors for the elements in the columns k andk ⫹ 1 that are to be collapsed, with the rest of the matrix
elements characterized by identity submatrices,0 
Ik⫺1 00 rk
0Qp.
0 rk⫹100 I n⫺k⫺1n#m
 0





(3)

This algorithm also allows us to collapse multiple classes
into a single class or into fewer, separated classes in just
one step, which saves a significant amount of time when
dealing with projection matrices of high dimension in large
databases. The steps when dealing with multiple collapses
are the same: normalize the elements on the merging columns by calculating the weighted average of the appropriate stable class distributions.
Methods: Illustration of the Matrix Dimension–
Collapsing Algorithm in Nine Plant Species
We applied Hooley’s algorithm to population projection
matrices of nine different plant species, in order to examine the dissimilarities between the elasticities of demographic processes in matrices of different dimensions
and to establish the optimum collapsing criteria. The elasticity is a widely used measurement of the importance ofa demographic process in a population; elasticity reflects
the proportional change in population growth rate as afunction of a proportional change in a part of the projection matrix (de Kroon et al. 1986). Specifically, we explored (i) how the inferred elasticities are influenced bymatrix dimension when a fixed collapsing criterion is used(“even collapsing”; see below) and (ii) how collapsing alarge matrix into a fixed, smaller dimension in different
ways (i.e., using different collapsing criteria; fig. 2) influences the associated elasticities. We refer to the first ofthese studies as the “matrix-dimension test” and the second as the “collapsing-criterion test.” We then investigated
how the elasticities of the collapsed matrices had changed
relative to the elasticities of the original, uncollapsed matrices. We used these comparisons to determine guidelines
for optimal collapsing (i.e., guidelines for both the size ofthe collapsed matrices and for the collapsing criteria thatminimize the dissimilarities between the inferred
elasticities).
We chose nine plant species of different growth forms
and longevities for which size- or developmental stage–
based projection matrices with relatively high dimension
were available in the literature (table 1). These species’
matrices differ because the demographic dynamics arestrongly influenced by the growth forms and life historiesof the species (Silvertown and Franco 1993). For instance,
herbaceous species display a higher fluctuation in sizes anddevelopmental stages because they may skip over morethan one class, in both positive (progression) and negative(retrogression) directions, after one annual transition. By

Matrices in Plant Comparative Demography 000

Figure 2: Collapsing approaches used to reduce the dimensions of
13 # 13 projection matrices. Adjacent classes represented with the samecolor were collapsed. The first class was kept unaltered because it contains
reproduction probabilities that are not to be merged with transition
probabilities. A, The original 13 # 13 matrix was collapsed evenly into
7 # 7, 5 # 5, 4 # 4 and 3 # 3 matrices to study the effect of decreasing
matrix dimension on its elasticity structure. B, The original 13 # 13
matrix was collapsed into 5 # 5 matrices in different ways to evaluate
the effect of several collapsing criteria on the matrix elasticity structure.
Criteria are (I) collapsing an even number of classes, resulting in the
structure {1, 2–4, 5–7, 8–10, 11–13}; (II) leaving unaltered the first three{1, 2, 3, 4, 5–13}, (III) the middle two {1, 2–6, 7, 8, 9–13}, or (IV ) thelast three classes {1, 2–10, 11, 12, 13}; (V) collapsing every two classes
for the first six classes and collapsing the remaining classes into a large
class {1, 2–3, 4–5, 6–7, 8–13}, (VI) leaving unaltered the middle and lastclass and collapsing the remaining classes {1, 2–6, 7, 8–12, 13}, (VII)collapsing the central classes an leaving unaltered the first and last twoclasses {1, 2, 3–11, 12, 13}; and collapsing classes so that the newly formed
classes will keep their respective stasis probabilities (VIII), residence times(IX), and mortality probabilities (X) similar for each new class. Collapsing
criteria VIII–X are species specific (the example here presented is for
Cryptantha flava; fig. A2, in the online edition of the American Naturalist).

contrast, the dynamics of longer-lived shrubs and trees are“slower” because biomass decreases or dramatic increases
occur rarely. We did not include annuals because theirmatrix dimensions are typically too small for us to implement any reduction with a meaningful outcome (fig.
1).
We chose an original, uncollapsed dimension of 13 # 13
in order to study the dissimilarities of elasticities in collapsed matrices, starting from a matrix of a dimension that

falls within the range typically reported in the literature.
For collapsing purposes, we treated the original 13 # 13
matrices as (1 ⫹ 12) # (1 ⫹ 12) matrices in which the firstclass was never merged with other classes so as to avoid
mixing the transition probabilities pij 苸 (0, 1) with the percapita contributions of fecundity a 1i 苸 (0, ⬁) (fig. 2). Notethat in some projection matrix models, per capita contributions such as seed bank, seedling recruitment, and clonal
reproduction can expand beyond the first row (e.g., Eriksson 1988; Cipollini et al. 1994; but this is not the casefor any of the species in table 1) and that, likewise, theseclasses should not be merged with those that contain transition probabilities.
Because the matrix dimensions of the chosen species
were not identical (table 1), we slightly modified them toproduce matrices of initial dimension 13 # 13. For thosespecies whose original dimensions were higher than
13 # 13 (Ulex minor and Araucaria cunninghamii), we
used Hooley’s algorithm to collapse the higher classes toform a life cycle of 13 classes. For one species (Cryptantha
flava), we recalculated its projection matrix on the basisof a 13-class life cycle, using the available raw data (B.
Casper, personal communication). The matrix of anotherspecies (Rourea induta) was already of a 13 # 13 dimension. All other species had matrix dimensions below
13 # 13 (Calathea ovandensis, Viola frimbriatula, Pterocereus gaumerii, Neobuxbaumia macrocephala, and Thrinax
radiata), and here we expanded the number of classes by
replicating the matrix elements of the last classes and readjusting the transition probabilities and per capita reproductive contributions to resemble the dynamics described by the original matrices. In all cases, we calculated
the primary demographic parameters (life span, population growth rate, stable stage distribution, and elasticities)
before and after the readjustment to make sure that themodification did not alter their demographic traits (app.
B, in the online edition of the American Naturalist). Nevertheless, we emphasize that our goal is not to study theprecise details of these particular species but rather toexplore the effects of matrix dimension on the demographic dynamics of realistic exemplars of a range of plant
species. Life spans were calculated as described in Forbis
and Doak (2004). We calculated the demographic parameters using MatLab, version 7.1 (MathWorks 2001), and
PopTools, version 3.0 (Hood 2003).
For every projection matrix, either 13 # 13 or of reduced
size, we calculated the population growth rate l, the stableclass distribution w, the reproductive output v, and theelasticities of the demographic processes involved. We calculated two types of elasticities: matrix-element elasticities
and vital-rate elasticities. Matrix-element elasticity measures
the proportional change in the population growth rate (l)due to a proportional change in a group of matrix cells

000 The American Naturalist
Table 1: Description of species used in this study, ranked by life spanSpecies
Cryptantha flava
Calathea ovandensis
Ulex minor
Viola frimbriatula
Pterocereus gaumerii
Neobuxbaumia
macrocephala
Rourea induta
Thrinax radiata
Araucaria
cunninghamiia

FamilyArchitectural
type

Life span(years)Population
growthrate (l)MatrixdimensionaSourceBoraginaceae
Marantaceae
Fabaceae
Violaceae
Cactaceae

Herbaceous
Herbaceous
Shrub
Herbaceous
Succulent

14
20
21
23
31

.86
.99
1.31
1.22
1.00

13
8
16
11
10

Lucas et al. 2008
Horvitz and Schemske 1995
Stokes et al. 2004
Solbrig et al. 1988
Méndez et al. 2004

Cactaceae
Connaraceae
Arecaceae

Succulent
Shrub
Palm

41
52
136

1.03
.99
1.13

10
13
9

Esparza-Olguı́n et al. 2005
Hoffmann 1999
Olmsted and Alvarez-Buylla 1995

Araucariaceae

Tree

256

1.02

17

Enright and Ogden 1979

Original matrix dimensions in the publication were 7 # 7, but we used the raw data to recalculate the matrix on the basis of a life cycle of 13 stages.

corresponding to a demographic process, while vital-rateelasticity measures proportional changes in l based on proportional changes of the underlying demographic processes
that correspond to each projection matrix element, such as
survival (Franco and Silvertown 2004).
To examine the effect of matrix dimension on the inferred elasticities (hereafter “matrix-dimension test”), we
collapsed the 13 # 13-class life cycle into new classes ofeven size, with the exception of the first class, which was
not collapsed. This resulted in matrices of dimensions
7 # 7, 5 # 5, 4 # 4, and 3 # 3. We then calculated thedissimilarities between the elasticity structure of the original and the sequentially collapsed matrices. In order toexplore how different ways of collapsing the matrices affects differently the elasticities and to establish an optimum
criterion (“collapsing-criterion test”), we collapsed the
13 # 13 matrices into 5 # 5 matrices (the median dimension in demographic studies; fig. A3), using various
collapsing criteria. We then calculated the dissimilarity between the elasticities of the original and its respectively
collapsed matrices; the collapsing criterion that resulted in
the lowest dissimilarity was then established as the optimum criterion. We performed the matrix-dimension test
and the collapsing-criterion test for both matrix-element
elasticities and vital-rate elasticities. We carried out a principal-component analysis (PCA) on the elasticities of survival, stasis, retrogression, progression, and fecundity to
calculate elasticity dissimilarities. The PCA reveals relationships and measures of proximity based on Euclidean
distances (Davidson 1975). The comparisons of elasticities
based on Euclidean distances were carried out for bothmatrix-element elasticities and their underlying vital-rateelasticities.
In the collapsing-criterion test, we applied 10 different
collapsing criteria to produce collapsed matrices of dimension 5 # 5. Briefly, these criteria involved collapsingclasses evenly (I), collapsing a large number of classes
together but in different positions of the life cycle (II–IV),
collapsing irregularly (V–VII), and collapsing so that themean stasis probabilities (VIII), residence times (IX), andmortality rates (X) were as similar as possible for eachnew class (see fig. 2 for a detailed description of thesecriteria). For criterion IX, the class-specific residence times
were calculated according to Caswell (2001, p. 112). The
last three collapsing criteria (VIII–X) are species specificbecause of the species’ different demographic life histories(app. B). Similarly, because each species has to passthrough a different number of classes before attaining maturity, the criteria that separate nonreproductive and reproductive classes (an approach suggested by Enright etal. [1995]), are also species specific, although there wasfor each species at least one criterion that fulfilled such a
condition (fig. A2).
Finally, we also measured the dissimilarity of the elasticity structures before and after collapsing the population
projection matrices of a large database as a function of
their growth form and life span. We applied the optimum
collapsing algorithm that was identified from the collapsing-criterion test to a set of 66 species’ projection matrices.
We calculated the Euclidean distance between the elasticities before and after collapsing each original matrix to a
5 # 5 matrix. We then performed linear regressions tostudy how life span and growth form correlate with these
Euclidean distances, using R (R Development Core Team
2009).
Results
The collapsed matrices for the matrix-dimension andcollapsing-criterion tests of the nine species had the sameor extremely similar asymptotic growth rates and life spansas their respective 13 # 13 matrices (app. B), as guaran-

Matrices in Plant Comparative Demography 000
teed by the Hooley algorithm. Their stable class distributions and reproductive outputs were also the same, withthe exception of the collapsed classes, whose class distribution and reproductive outputs were equal to the sumof the classes that were merged (not shown).

Matrix-Dimension Test: How Does Matrix Dimension
Affect the Estimation of Elasticities?
Matrix dimension affected the matrix-element elasticities
consistently across all species in our study for matrices
collapsed to dimensions of more than 3 # 3. When evencollapsing to produce 7 # 7, 5 # 5, or 4 # 4 matrices,
the matrix-element elasticities of stasis and fecundity increased, whereas the elasticities of retrogression and progression decreased. The relationships between matrix dimension and inferred elasticities were not always linear(fig. 3). A similar pattern was detected for the vital-rateelasticities, except for Calathea ovandensis (fig. 3A), Pterocereus gaumerii, and Neobuxbaumia macrocephala (app. C,
in the online edition of the American Naturalist), where
the matrix-element elasticities of stasis, after having
reached a maximum value in 5 # 5 matrices, decreased
for the 4 # 4 matrices.
When the matrices were further collapsed to a dimension of 3 # 3, an unexpected behavior was detected. In
the case of the matrix-element elasticities of the herbaceous
C. ovandensis and the palm Thrinax radiata (fig. 3), theelasticity of stasis decreased drastically in the 3 # 3 matrices. Nonetheless, this behavior was not consistent acrossall species. In the case of the vital-rate elasticities, survival
decreased from 4 # 4 matrices to 3 # 3 matrices, not only
in C. ovandensis and T. radiata but also in Viola fimbriatula,
N. macrocephala, and Araucaria cunninghamii (app. C). In
these five species, the decreases in the vital-rate elasticity
of survival (C. ovandensis: ⫺17%; T. radiata: ⫺3%; allother species experienced a less drastic decrease of survival,
between 0% and ⫺1%), and the increases in fecundity (C.
ovandensis: ⫹435% and ⫹115%; T. radiata: ⫹317% and⫹197%, for element matrix and vital-rate elasticities, respectively) were rather steep.
The effects of collapsing matrix dimension on inferred
elasticities were typically smaller for vital-rate elasticities
than for matrix-element elasticities. In particular, the Euclidean distances between the collapsed and original elasticities were smaller for vital-rate elasticities than for matrixelement elasticities in 29 out of 36 cases (four matrix
dimensions [7 # 7, 5 # 5, 4 # 4, 3 # 3] # nine species;
P ! .001, binomial test; fig. 4).

Collapsing-Criterion Test: What Is the Best Criterion forSpecifying Which Classes to Collapse?
The collapsing criterion that did not modify the first fourclasses and collapsed classes k4–k13 into class k 5 (fig. 2B,
criterion II) resulted in the smallest dissimilarity between
the associated elasticities of the 13 # 13 original and5 # 5 collapsed matrices (fig. 5). Such dissimilarity wasparticularly small for the comparison based on vital-rateelasticities (x̄criterion II p 0.07).
The criteria that formed 5 # 5 matrices such that thestasis probabilities (VIII), residence times (IX), or mortality probabilities (X) of each collapsed class were similar
did not rank among the most optimal collapsing criteria.
The Euclidean distances ranked the collapsing criteria as
follows when based on matrix-element elasticities: II(Euclidean distance p 0.16) ! V (0.17) ! IX (0.20) ! III(0.20) ! VIII (0.20) ! VI (0.20) ! I (0.21) ! VII (0.23) !
X (0.23) ! IV (0.27); the criteria were ranked as follows
when based on vital-rate elasticities: II (0.07) ! V(0.13) ! VIII (0.16) ! X (0.16) ! IX (0.16) ! I (0.17) ! III(0.17) ! VI (0.18) ! VII (0.19) ! IV (0.27); see figure 2B.
The collapsing criteria that separated nonreproductive
and reproductive classes (fig. A2) did not systematicallyresult in lower dissimilarities between the elasticity structures of the original and the collapsed matrices. For
instance, for Cryptantha flava and C. ovandensis, all collapsing criteria separated nonreproductive and reproductive classes, but the criterion that collapsed classes k5–k13(fig. 2B, criterion II) performed the best, in that the dissimilarity between the elasticities of the original and collapsed matrices was lowest. A cursory evaluation of theresults in figure 5, contrasted with figure A2, would suggestthat this criterion (criterion II; fig. 2B) is the optimum
one because it splits nonreproductive and reproductiveclasses in five of the nine species in our study. However,the same is true of the criterion where classes k3–k11 werecollapsed (criterion VII; fig. 2B), and yet it performed
relatively poorly. Furthermore, the criterion that collapsed
only the last classes into a new class (criterion II; fig. 2B)
was the best approach even for those species in which thiscriterion did not separate reproductive and nonreproductive classes (e.g., P. gaumerii, N. macrocephala, T. radiata,
and A. cunninghamii).
We found that collapsing matrices had a larger effecton inferred elasticities for shorter-lived species. While thelinear regression between life-span and dissimilarity measurements of the nine study species was not significant for
matrix-element elasticities (t 1, 8 p 6.14, P p .104) and wasborderline significant for vital-rate elasticities (t 1, 8 p
5.15, P p .057), this relationship was highly significant
when a more extensive study of 66 species was used, inwhich their projection matrices had been collapsed to a

Figure 3: Effects of matrix dimension on the elasticity of vital rate– and matrix element–based demographic processes for three of the nine species
our study: Calathea ovandensis (A), Ulex minor (B), and Thrinax radiata (C). In each case, the original 13 # 13 projection matrix was collapsed by
combining two, three, four, or six adjacent classes, to produced 7 # 7 , 5 # 5 , 4 # 4 , and 3 # 3 matrices, respectively. Retrogression vital-rateelasticities are represented as absolute values to fit into the positive axis, but they are negative.

Matrices in Plant Comparative Demography 000

Figure 4: Euclidean distance (x̄dimension ) between the elasticities of theoriginal 13 # 13 matrix and the elasticities of the 7 # 7, 5 # 5, 4 # 4,
and 3 # 3 matrices for each species (using criteria described in fig. 2A).
Species are ranked from top to bottom by increasing life span.

difference in elasticities between the original and collapsed
matrices, contrary to the suggestion of Enright et al.
(1995). Specifically, we found that for all species in thisstudy, keeping the first life-cycle classes unaltered (criterion II; fig. 2B) resulted in the lowest effect on the inferred
elasticities. This likely occurs because individuals of younger/smaller/less developed classes are usually more susceptible to changes in the environment and thus typicallyexhibit distinct, class-specific vital rates (Harper 1977).
These conclusions are further supported by the fact thatwhile the criterion that grouped the first classes and left
the last four classes unaltered (IV, the opposite of II; fig.
2B) was the worst performer, the criterion where the firstclasses were collapsed only two at a time, in a form very
similar to criterion II, always ranked in the second position. Our results are also supported by a study that suggests
ways to save time, resources, and field effort while constructing projection matrices from small populations of
threatened species (Ramula and Lehtilä 2005).
Our findings suggest that fecundity or age at first
reproduction are not the most important biological pro-5 # 5 dimension via the previously established optimum
criterion (criterion II; fig. 2B; t 1, 64 p 18.49, P ! .001,
R 2 p 22.4%; fig. 6). Furthermore, the correlation between
dissimilarity and life span was growth form specific(F5, 60 p 7.45, P ! .001; growth form as covariate: F4, 60 p
5.02, P ! .001), and it was primarily driven by the significant life span–dissimilarity correlation of herbaceous perennials (t 1, 36 p 17.95, P ! .001); this correlation was nonsignificant for succulent, shrub, palm, and tree species.
Discussion
Our study offers insights into how and why the dimensionof projection matrices influences the estimation of fundamental demographic parameters. Using an overlooked
algorithm (Hooley 2000) for collapsing matrix dimension
without relying on raw field data, we have (i) establishedcollapsing criteria better than those previously suggested(Enright et al. 1995); (ii) found that the effect of matrix
dimension on the associated elasticity structure is often
nonmonotonic and not always lower on vital-rate elasticities than on matrix-element elasticities, as previously reported (Zuidema 2000); (iii) observed unexpected behavior for very small matrices (e.g., 3 # 3) that allows us toestablish a dimension threshold for comparative plant demographic studies; and (iv) found a negative relationshipbetween life span and the robustness of elasticities to matrix dimension in herbaceous perennials.
Our results indicate that collapsing a population projection matrix by similar residence times leads to a large

Figure 5: Euclidean distance between the elasticities of the original
13 # 13 matrix and the elasticities of the 5 # 5 collapsed matrices obtained with the various collapsing criteria described in figure 2B. Overall
Euclidean distances are averaged per species (x̄species ) and per criterion(x̄criteria).

000 The American Naturalist
licated so that they comprise all biotic and abiotic phenomena that are inherent to the species’ life cycle. Doingso would approximate the mean observed class distribution to the stable class distribution. We suggest that class
distributions be calculated and compared to the stable class
distribution as a prerequisite to deciding whether to incorporate a given annual transition for later interspecificcomparison. Applying methods to do so (Keyfitz’s D, in
Caswell 2001, p. 101), we have found evidence that
Hooley’s algorithm provides reliable collapsed projection
matrices for at least one population subjected to extreme
weather events (app. D). We also recommend averaging
the control transition matrices for as many sites and years
as there are data available to reduce the effect of abnormal
biotic and abiotic fluctuations.
Figure 6: Euclidean distance between the elasticities of the original andthe 5 # 5 collapsed matrices (using collapsing criterion II in fig. 2B) asa function of life span for 66 plant species: 38 herbaceous perennial, 6
succulent, 6 shrub, 7 palm, and 9 tree species.

cesses on which to base the classes chosen for matrix
construction and collapsing. Enright et al. (1995) recommended against merging nonreproductive and reproductive classes because trade-offs between reproduction,
growth, and survival would result in distinctive demographic dynamics in both life-cycle stages. However, weshow that collapsing criteria that violate this suggestion
can produce more robust elasticities than collapsing criteria that differentiate nonreproductive from reproductiveclasses.
We also found some inconsistencies with previous works
on the robustness of vital-rate versus matrix-element elasticities to varying matrix dimension. Zuidema and Zagt
(in Zuidema 2000) studied the effect of matrix dimension
on inferred elasticities and found that progression andretrogression matrix-element elasticities decreased with increasing dimension, consistent with our results. However,they also reported that the vital-rate elasticities were morerobust to varying matrix dimension than their matrixelement counterparts. Our results generally agree with
theirs, but in 20% of the cases (n p 9 species) in ourstudy, vital-rate elasticities were more robust than matrixelement elasticities to changes in matrix dimension. Nevertheless, vital-rate elasticities seem more relevant for the
study of ecological patterns because they reflect independent demographic processes and allow for the separationof the survival vital rates (e.g., Bruna et al. 2009).
Although the collapsing algorithm we have analyzed
here has the virtue of preserving population growth rates,
stable class distributions, and reproductive output, thisalgorithm assumes demographic stability. In an ideal scenario, field censuses would be extensively temporarily rep-

Intuition behind the Impact of Matrix Dimension
on Elasticities
Although we cannot provide a mechanistic explanation for
all the specific details of our results on the nine plant
species, there is a simple intuition for the general effectsof collapsing dimension on associated elasticities. The elasticities of fecundity tend to increase as matrix dimension
is reduced because the merging of nonreproductive andreproductive classes inevitably makes reproductive statusattainable in a shorter period of time by the faster-growing
individuals in the population (Enright et al. 1995). Faster
demographic processes have been shown to have greater
elasticities (Zuidema et al. 2009). Similarly, the elasticityof stasis also increases because, with fewer classes in acollapsed matrix, the amount of stasis is again augmented.
Since stasis elasticity increases while the overall population
growth rate l is kept constant, the elasticities of retrogression and progression must decrease. Although there issome variation across species and some aberrant behavior
at very small dimensions, this simple intuition explains
most of the general patterns we have observed. We attribute the unexpected increase in matrix-element progression elasticities in small matrices of some herbaceousperennials and succulent species, and the subsequent decrease in stasis elasticity, to the high degree of phenotypic
plasticity of these growth forms and their fast demographic
dynamics (Chien and Zuidema 2006). Furthermore, studies of herbaceous perennials often suffer from small sample
sizes in large size classes.
Guidelines for Comparative Demographic Studies
Our results on the relationship between matrix dimension
and inferred demographic properties have important implications for comparative studies. We have shown thatthe apparent importance of fecundity, stasis, progression,

Matrices in Plant Comparative Demography 000
retrogression, and survival is highly sensitive to the dimensions of the projection matrix. Consequently, demographic comparisons of species based on elasticities must
carefully account for the varying matrix dimensions.
For comparative work based on published projection
matrices, we suggest two alternatives: either account for
matrix dimension with a nonlinear analysis or collapse
matrices into an equal dimension, at least for the herbaceous perennials, whose elasticities are most sensitive tomatrix dimension (fig. 6). Surprisingly, most recent comparative studies using projection matrices have not usedmatrix dimension as a covariate (Franco and Silvertown
2004; Bruna et al. 2009). On the other hand, a trade-off
emerges between the matrix dimension cutoff and how
many studies to include for comparative purposes because(i) matrix collapsing will mean that species with fairly
dissimilar traits will be treated as equals from a demographic standpoint (e.g., Crouse et al. 1987; Cochran andEllner 1992); (ii) decreasing dimension, regardless of thecollapsing criteria, will tend to artificially promote the apparent importance of stasis and fecundity, compared toprogression and retrogression; and (iii) an extreme decrease in the number of classes will not allow for the studyof processes such as class-specific mortality rates, time tomaturation, senescence, clonal growth, or vegetative dormancy. In the extreme, a very small matrix will inevitably
merge transition probabilities for stasis, progression, and/or retrogression pij 苸 (0, 1) with sexual and/or clonal reproduction a i 苸 (0, ⬁), which makes little biological sense.
Based on the results of our matrix-dimension test, in which
the elasticities exhibited unexpected behaviors for dimensions smaller than 4, we suggest dimensions of at least 5
in complex and 4 in simple life cycles for comparative
studies, despite the fact that this rules out a large numberof published projection matrices with smaller dimensions(n p 107; R. Salguero-Gómez, unpublished data) for demographic comparisons. Finally, our results suggest an
optimum criterion for choosing which stages to collapse
in a projection matrix (criterion II; fig. 2B): keeping thefirst few life-cycle classes unaltered and collapsing all other
older/larger/more developed classes, regardless of whether
that means merging nonreproductive and reproductiveclasses or keeps the residence times constant in the newly
collapsed classes.
Acknowledgments
We thank B. Casper, M. Franco, B. Waring, and P. Zuidema
for valuable comments on the manuscript. Thanks are also
due to R. Lucas and B. Casper, who made available theraw demographic data for Cryptantha flava. The manuscript also greatly benefited from useful comments fromtwo reviewers. J.B.P. acknowledges support from the Bur-roughs Wellcome Fund, the David and Lucile Packard
Foundation, the James S. McDonnell Foundation, and theAlfred P. Sloan Foundation. R.S.-G. was supported by aDissertation Research Fellowship of the University ofPennsylvania.
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Associate Editor: Carol Horvitz
Editor: Donald L. DeAngelis

Cryptantha flava (Boraginaceae), in full bloom in Utah, frequently undergoes drastic changes in size between years. Photograph by Roberto SalgueroGómez.