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Running head: Nonlinearity in demographic models
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Nonlinear relationships between vital rates and state variables in demographic models
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JOHAN P. DAHLGREN1,3, MARÍA B. GARCÍA2 & JOHAN EHRLÉN1
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Department of Botany, Stockholm University, SE 10691 Stockholm, Sweden.
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Instituto Pirenaico de Ecología (CSIC), Apdo. 13034, Zaragoza, Spain.
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E-mail: johan.dahlgren@botan.su.se
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Manuscript type: Note
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Abstract. To accurately estimate population dynamics and viability, structured population models
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account for among-individual differences in demographic parameters that are related to individual state.
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In the widely used matrix models, such differences are incorporated in terms of discrete state categories
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whereas integral projection models (IPMs) use continuous state variables to avoid artificial classes. In
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IPMs, and sometimes also in matrix models, parameterization is based on regressions that do not
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always model nonlinear relationships between demographic parameters and state variables. We stress
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the importance of testing for nonlinearity and propose using restricted cubic splines in order to allow
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for a wide variety of relationships in regressions and demographic models. For the plant Borderea
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pyrenaica, we found that vital rate relationships with size and age were nonlinear and that the
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parameterization method had large effects on predicted growth rates (linear IPM: 0.95, nonlinear IPMs:
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1.00, matrix model: 0.96). Our results suggest that restricted cubic spline models are more reliable than
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linear or polynomial models. Because even weak nonlinearity in relationships between vital rates and
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state variables can have large effects on model predictions, we suggest restricted cubic regression
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splines to be considered for parameterizing models of population dynamics whenever linearity cannot
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be assumed.
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Key words: demography, integral projection models, linearity assumption, matrix models, natural
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splines, nonlinear regression, plant population dynamics, restricted cubic splines
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INTRODUCTION
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Individuals of different age or size often differ in survival probability and fecundity, and thus in their
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contribution to population growth. Due to such differences it is important to include a state or stage
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structure in models of population dynamics. In matrix models, populations are divided into discrete
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states and the transition rates between states over a time step are modeled using a transition matrix
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(Leslie 1945, Lefkovitch 1965, Caswell 2001). The Integral Projection Model (IPM; Easterling et al.
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2000) is an extension of the matrix model, allowing state variables to be continuous. In both matrix
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models and IPMs, effects of state variables on vital rates can be incorporated as regression models
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(Morris and Doak 2002, Ellner and Rees 2006). Parameterization using continuous functions is more
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statistically efficient than the traditional method of parameterizing classes separately in matrix models,
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and allows more reliable estimates for sparsely sampled groups, such as old individuals. However, one
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potential problem with using regression analyses for model parameterization is that relationships
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between vital rates and state variables may not be adequately described by simple or polynomial
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generalized linear models.
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Nonlinear relationships are sometimes accounted for in demographic models parameterized using
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regression analyses. For example, exponential relationships between state variables and vital rates are
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accounted for by log-transformations of state variables, or by the use of Poisson regression. In addition,
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probabilities of survival and flowering are typically modeled with logistic regressions, implicitly
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accounting for asymptotic relationships. Also incorporation of polynomial terms in regression models
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is a common practice to account for nonlinear relationships. However, using polynomial regressions to
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describe vital rates in demographic models may be risky since the shape of one tail of the resulting
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curve may be unduly influenced by observations at the other tail, in particular if the observations are
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not evenly spaced over the range of the predictor variable (Harrell 2001). Furthermore, neither
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linearizations nor the inclusion of polynomial terms may adequately describe nonlinear state variable –
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vital rate relationships if functional relationships differ between different regions of state variable
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values. Changes in functional relationships between state variables and vital rates may not be
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uncommon in organisms with marked transitions between life cycle stages. For example, the
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relationship between age and survival in birds has been shown to sometimes be “saw-toothed” (Low
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and Pärt 2009).
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Incorporating nonlinear regression models into demographic models is straightforward in many
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situations, and this possibility was highlighted in the paper introducing the IPM (Easterling et al. 2000).
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However, more complex nonlinear vital rate – state variable relationships have rarely been included in
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demographic models. Here we illustrate a practical and reliable method of flexibly incorporating
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nonlinear relationships into IPMs using restricted cubic regression splines, and present results
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suggesting that model output may be very sensitive to assumptions of linearity and that allowing
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nonlinear relationships in demographic models should be considered whenever there is no a priori
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reason to assume either linear or other simple function forms.
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We used four different methods of parameterizing structured population models of the long-lived plant
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Borderea pyrenaica, where vital rates may depend on both age and size. First, we used continuous
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classes parameterized using generalized linear models (GLMs) where linearity is assumed for
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continuous vital rate rates, logistic linear for proportions and log linear for counts (“linear IPMs”).
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Second, we used continuous classes parameterized using GLMs where the linearity assumption is
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relaxed by including also quadratic and cubic age and size effects on vital rates (“polynomial IPMs”).
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Third, we used continuous classes parameterized using GLMs where more complex effects of age and
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size on vital rates were allowed by using restricted cubic regression splines (“restricted cubic spline
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IPMs”). Fourth, we used a few discrete classes parameterized based on observed class transitions
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(“matrix models”), representing the most commonly used type of demographic model. We used models
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based on these four parameterization methods and asked (1) if relationships of vital rates with size and
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age were significantly nonlinear, (2) if incorporating statistically significant nonlinear relationships into
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demographic models influenced projections, (3) if modeling vital rates with restricted cubic splines
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yielded more plausible results than modeling vital rates using polynomial terms, (4) if predictions of
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the matrix model, where states are discrete but relationships with vital rates may be nonlinear, more
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closely resemble the nonlinear models with continuous state variables than the linear model, and finally
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(5) we simulated data based on linear and nonlinear relationships between state variables and vital rates
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to investigate consequences of modeling linear relationships as non-linear and vice versa.
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METHODS
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The dataset
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Borderea pyrenaica Miégeville (Dioscoreaceae) is a small dioecious herb occurring in the central
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Pyrenees. The plant has morphological features that enable easy and precise ageing by inspection of the
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underground storage organ (García & Antor 1995), making it possible to include age as a state variable
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in the population models, in addition to size. Individuals sometimes reach ages of above 300 years. The
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study system is described in detail in García and Antor (1995). To parameterize the structured
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population models, we used recordings on survival, size and fecundity over four years (1995–1998) on
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a total of 519 female individuals in permanent plots in one population (‘Pineta’). One more census in
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1999, unearthing individuals, was used to determine if missing plants were dead or dormant
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(individuals that were missing in two consecutive censuses were assumed to be dead), and to record the
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age of plants by inspecting buried tubers. When calculating survival probabilities, male plants were
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also included since no differences between sexes have been found in previous analyses and very few
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observations of death were available despite the sample size (M. B. García, J. P. Dahlgren and J.
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Ehrlén, unpublished manuscript). Size was calculated as ln(number of leaves × shoot length2) since this
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has been found to more closely correlate with total plant biomass than other measures. Age estimates
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for individuals of unknown age but known size were imputed based on restricted cubic spline
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regressions of age on size. In order to avoid having zero mortality for the oldest individuals in the
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population models, an additional death of an individual of the median age of individuals in the oldest
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age class in the matrix model (145 years, see below) was added to the data. This addition did not
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change whether relationships between vital rates and state variables were significantly nonlinear or not.
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Relaxing the linearity assumption in generalized linear models of vital rates
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There exist several methods for relaxing the linearity assumption in GLMs. Apart from simply adding
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polynomial terms, such methods include GAM (Generalized Additive Models; Hastie and Tibshirani
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1990), fractional polynomials (Royston and Altman 1994), and B-spline regression (e.g. Easterling et
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al. 2000). In this note we used restricted cubic spline regression (Stone and Koo 1985, Harrell 2001) to
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test nonlinearity and model vital rates due to the ease of incorporating complex nonlinearities in
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relationships between vital rates and one or several variables. Regression spline models are constituted
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by piecewise polynomial regressions that are fitted between join points (knots). Restricted cubic splines
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are cubic polynomial splines that are forced to have similar curvature at the join points, and with the
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tails of the curve restricted to be linear so as to make them more reliable for predictions than in other
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regression spline techniques. These restrictions mean that the number of parameters needed to describe
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the fitted curve equals the number of knots minus one. The exact location of knots has been found to be
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of little importance and 4 or 5-knot restricted cubic splines with evenly spaced knots satisfactorily
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model many kinds of nonlinear relationships, assuming a sufficient number of observations to avoid
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overfitting (Harrell 2001). More specifically, we used ‘5-knot restricted cubic splines’ fitted using the
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‘Design’ package for R 2.9.1 (Harrell 2001, R development core team 2009). The ‘rcs’ function in the
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Design package was used to automatically divide predictor variables with evenly spaced knots, and
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models were fitted using the generic ‘glm’ function. The function named ‘Function’ in the Design
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package was used obtain an expanded function describing the fitted regression model in standard S (R)
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language – for incorporation into the IPM code.
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Demographic models
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One linear, one polynomial and one restricted cubic spline IPM, as well as a matrix model, were
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specified and population growth rates obtained from the four models were compared. For
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parameterization of the IPMs, generalized linear models were used to fit relationships between the state
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variables (age and size) and the vital rates survival, mean and variance in growth, flowering, and seed
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number. The data were pooled over years. Random individual effects were not significant and omitted
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in final models (M. B. García, J. P. Dahlgren and J. Ehrlén, unpublished manuscript). The integral
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projection models were specified as:
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n(y, a + 1, t + 1) = ∫ K(y, x, a) n(x, a, t) dx,
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where a is age, x is size year t and y is size year t + 1 and integration is performed over all possible
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sizes. Age was assumed to be continuous in the regressions determining relationships with vital rates,
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but we did not integrate over age in the population model since age of an individual is increased by one
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for each time step (modeled year). n is a density function describing the distribution of individuals
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among ages and sizes, and K is the projection kernel given by functions of survival (s), growth (g),
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flowering probability (fp), seed number (fn) and the size distribution of establishing seedlings (pd):
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K(y, x, a) = s(x, a) g(y, x, a) + fp(x, a) fn(x, a) pd(y).
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Growth was modeled as a normal distribution with the mean from a linear regression of y on x and a
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and the variance as the variance around this regression line. Survival was modeled as a logistic
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regression of presence or absence on x and a, flowering as a logistic regression on flowering or not
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flowering on x and a, and seed number as a Poisson regression on x and a. In the implementation of the
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IPM, the integration over size was performed using the midpoint rule and a 100 × 100 sized matrix (cf.
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Ellner and Rees 2006) for each of the 300 possible ages. Changes in survival probability were assumed
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to cease when individuals reached age 200, as further projections depended on few data. At age 300,
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individuals were entered into an absorbing “300+” class (few individuals reached this class, which is in
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accordance with the field data).
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For each vital rate in the IPMs, a linear, a polynomial and a restricted cubic spline regression model on
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age and size were fitted separately. Non-significant effects (P > 0.05), based on likelihood ratio tests,
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were removed. Significance tests of nonlinear spline components were based on the null-hypothesis
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that all nonlinear spline components equaled zero (Harrell 2001).
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In order to compare estimated population growth rates between the IPMs and a matrix model, we also
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specified a model with three size classes and five age classes. This model was parameterized based on
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observed transitions between classes (cf. Morris and Doak 2002). Size classes were distributed
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approximately evenly over the observed size ranges and were separated at sizes 6.5 and 8.5. Age
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classes were wider for more advanced ages and separated at ages 20, 40, 80 and 150.
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Population growth rates (λ) were estimated by iterating the models until stable distributions were
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reached. Elasticity values showing the proportional consequences of small changes in either
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reproduction or in state transition probabilities for population growth rate were calculated based on the
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stable distributions and reproductive value distributions as in Ellner and Rees (2006). Confidence
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intervals of λ were calculated for each model from 1000 bootstrap replicates using the package ‘boot’
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in R. Observations (individuals) in the same number as recorded were resampled with replacement
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from the data matrix and the models were re-parameterized and λ calculated for each replicate.
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Simulations
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To investigate the consequences of choosing the “wrong” parameterization method and to decide which
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regression modeling approach may be the most reliable, we performed simulations where true vital rate
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curves were known. We modeled three different scenarios corresponding to the true relationships being
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the linear, polynomial and spline models presented in Table 1, respectively. For each scenario, 1500
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individuals of random age and size were drawn based on the observed distributions, taking the
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covariance into account by using Cholesky decomposition in the “mvtnorm” R package. In a second
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step, vital rates were assigned to individuals using the parameters from the “true” regression models. In
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accordance with the regression analyses, random error distributions were assumed to be binomial for
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survival and flowering, normal for growth, and Poisson for seed number. To the simulated data, we
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fitted linear, polynomial and restricted cubic spline models, thus resulting in nine combinations of true
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and modeled relationships. Quadratic and cubic age and size terms were included in all polynomial
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models, and 5 knots were used in the restricted cubic spline models. As we used a simulated dataset
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with many observations we included all terms regardless of their statistical significance, because full
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models have been found to produce more reliable predictions than reduced models in other studies
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(Harrell 2001). The vital rate functions fitted to the simulated data were used as components of IPMs
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and population growth rate was calculated as above. All steps were repeated 200 times for each of the
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three scenarios. In addition, we used 500 observations per vital rate to examine if relative differences
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between methods remained the same.
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RESULTS AND DISCUSSION
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Relationships between states and vital rates in Borderea pyrenaica were nonlinear in several cases
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(Table 1), most markedly so for survival (logit transformed in the logistic regressions; Fig. 1). The
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nonlinear relationship between age and survival entailed that no effect of age was found when
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assuming linearity. Age also had a significant nonlinear effect on seed number in both nonlinear models
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(log transformed in the Poisson regressions; Fig. 2). Size effects on growth, flowering probability and
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seed number were significantly but only slightly curved after transformations. Nonlinear relationships
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between state variables and vital rates have been recorded also in other systems (e.g. Zuidema et al.
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2010, but see e.g. Easterling et al. 2000). Indeed, it seems unlikely that perfectly linear relationships are
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common even after various statistical transformations as it would imply that the total effect of all traits
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affecting vital rates is linearly related to the state variable used.
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Long-term population growth rate (λ) was estimated to be 1.00 in both the polynomial and restricted
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cubic spline nonlinear IPMs and 0.95 in the linear IPM (Fig 3). The true population growth rate is not
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known, but given that Borderea pyrenaica is very long-lived and population size has remained stable
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for many years, a mean population growth rate of 1.00 seems more reasonable than a growth rate of
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0.95. Fecundity elasticity constituted 0.08% of total elasticity in the polynomial model, and 0.01% in
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the spline model. Both differed markedly from the linear model where fecundity was found to be much
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more important for population growth rate and constituted 4.59% of total elasticity. This shows that
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assumptions of functional relationships can be important in demographic models (cf. Pfister and
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Stevens 2003). The substantial difference in λ was mainly the result of that positive age effects on
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survival were nonlinear, and no effects were found when assuming linearity (results not shown).
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However, also nonlinearities in size effects changed predicted growth rates by several percent.
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Polynomial and restricted cubic spline vital rate models explained a similar amount of variation (Table
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1), and population growth rates in the polynomial and restricted cubic spline IPMs were similar (Fig.
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3). However, there were important differences in both the vital rate and population models. Most
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notably, seed number and flowering probability were predicted to increase sharply for plants of very
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high ages in the polynomial models but not in restricted cubic spline models (Fig. 2). As a result,
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fecundity elasticity increased with age for very old ages (250-300 years) in the polynomial model,
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suggesting that changes in the fecundity of old individuals would affect population growth rate more
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than changes in middle aged individuals. This elasticity pattern appears highly unlikely and illustrates
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the problems with using polynomial regression for parameterization of demographic models.
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In the (matrix) population model with a few discrete classes, λ was estimated to be 0.96. Contrary to
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our expectations, predicted λ differed significantly from predictions of the nonlinear IPMs but not of
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the linear IPM. The substantial difference in λ compared to the nonlinear IPM may possibly have been
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caused by the fact that a matrix model with few classes was not as flexible in capturing nonlinear
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relationships in this case.
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Fitting linear and nonlinear models to simulated data where the true functional relationships were
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known showed that restricted cubic spline models were clearly more reliable than linear models, and
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slightly more reliable than using polynomial models. Simulation results suggested that modeling vital
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rates using restricted cubic splines when they are in fact linear would result in a slight overestimation
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of population growth rate (Δλ = 0.010) and that using polynomial regression would lead to a little
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larger overestimation (Δλ = 0.013). Variation in predicted λ was similar when using restricted cubic
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splines (s.d. = 0.0267) and polynomials (s.d. = 0.0263), and only a little larger than when using linear
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regression (s.d. = 0.0173). On the other hand, modeling relationships as linear when they were
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nonlinear (either based on polynomials or restricted cubic splines) would lead to a larger deviation
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from the true value (Δλ = 0.045), than applying nonlinear models to linear relationships. When the
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underlying relationship was nonlinear, standard deviations were similar using all three modeling
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techniques (0.0087 – 0.0102). Thus, taken together the simulation results suggest that spline models are
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stable as well as more accurate on average. Sample size did not seem to affect these relationships (i.e.
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results were similar when using 500 in place of 1500 observations in the simulations), and polynomial
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and restricted cubic regression spline models both predicted true relationships of the other model type
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well (Δλ < 0.003).
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In many cases of applied regression, assuming linearity, or strict adherence to for example exponential
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or logistic curves, may be unwarranted (Harrell 2001, Sauerbrei et al. 2007). Even though assuming
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linear relationships may be motivated in demographic models for many species (e.g. Morris and Doak
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2002), and taking into account the fact that nonlinear models may not provide better fits in all species
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(e.g. Easterling et al. 2000), the diverging predictions of the continuous models presented here show
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that this assumption is critical and that routine-like use of generalized linear models may hamper our
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understanding of population dynamics and of population viability. Restricted cubic regression splines
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are both relatively simple and sufficient to model continuous nonlinear relationships (Harrell 2001),
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making the method ideal for modeling vital rates when there is no mechanistic knowledge of effects of
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state variables. If there is knowledge about mechanisms, models such as the growth model used by
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Zuidema and colleagues (2010) in IPMs of tropical trees are important options. In the present study,
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join points between splines (knots) were evenly spaced among observations. If prior information
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regarding “class divisions” exists, users may also choose to specify knot locations in spline models.
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In conclusion, our results suggest that even weak nonlinearities may have important effects in IPMs.
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For our dataset polynomial and restricted cubic spline regressions yielded similar population growth
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rates, but both based on statistical theory and as shown by the unrealistic vital rate relationships and the
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simulations, restricted cubic spline regression is likely to be a more reliable method. We thus
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recommend careful tests for nonlinear relationships and the use of a method such as restricted cubic
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splines to parameterize models of population dynamics in cases where linearity can not be assumed.
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Inclusion of nonlinear relationships is likely to make demographic models more realistic without large
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increases in complexity. Such models should therefore in many cases be an optimal option for
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modeling population dynamics and examining population viability.
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ACKNOWLEDGMENTS
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This study was financially supported by the Strategic Research Programme EkoKlim at Stockholm
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University (J.P.D.), the National Parks project (ref 018/2008) (M.B.G), and the Swedish Research
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Council for Environment, Agricultural Sciences and Spatial Planning (FORMAS) (J.E.).
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TABLE 1. Significant relationships in generalized linear models of Borderea pyrenaica vital rates over linear and nonlinear (using either
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polynomial regression or restricted cubic splines) age and size predictors.
Linear model
Polynomial modela
Spline modelb
(deviance explained, AIC)
(deviance explained, AIC)
(deviance explained, AIC)
Survival
Size*** (11%, 707.6)
Age2** + Size2** (14%, 692.4)
Age*** + Size* (15%, 692.3)
1287
Growth
Size*** (88%, 1115)
Size2*** (88%, 1104)
Size*** (88%, 1103)
719
Flowering
Size*** (32%, 574.0)
Age3* + Size2* (33%, 567.7)
Size* (33%, 572.4)
659
No. seeds
Age** + Size*** (27%, 4401)
Age3*** + Size2*** (32%, 4206)
Age** + Size* (32%, 4213)
368
Vital rate
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Notes: AIC = Akaike Information Criterion, D.f. = degrees of freedom in the null model
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* P < 0.05, ** P < 0.01, *** P < 0.001, from likelihood ratio tests.
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a
Superscripts refer to whether models are quadratic or cubic, quadratic terms are also included in cubic models.
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b
All significant parameters were also significantly nonlinear and 5-knot restricted cubic spline functions were used for each included
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variable.
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D.f.
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Figure legends
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FIG. 1. Survival probability and growth as functions of size (at mean age) and age (at mean size) based
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on restricted cubic spline (solid light-gray lines), polynomial (dashed dark-gray lines) and linear (solid
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black lines) GLMs. Relationships with survival probability were modeled with logistic regression and
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growth with linear regression. Values are back-transformed in the figure. Age did not significantly
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affect growth in any of the models.
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FIG. 2. Flowering probability and seed number as functions of size (at mean age) and age (at mean size)
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based on restricted cubic spline (solid light-gray lines), polynomial (dashed dark-gray lines) and linear
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(solid black lines) GLMs. Relationships with flowering probability were modeled with logistic
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regression and seed number with Poisson regression. Values are back-transformed in the figure. Age
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did not significantly affect flowering probability in the linear and restricted cubic spline models.
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FIG. 3. Deterministic population growth rates (λ, with bootstrapped 95% confidence intervals) predicted
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from four structured population models.
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