OIKOS 88: 335–340. Copenhagen 2000
Confidence intervals for population growth rate of organisms with two-stage life histories Richard M. Sibly, Flemming T. Hansen and Valery E. Forbes
Sibly, R. M., Hansen, F. T. and Forbes, V. E. 2000. Confidence intervals for population growth rate of organisms with two-stage life histories. – Oikos 88: 335–340. Although life histories can be modelled with great generality using projection matrices, for organisms with life histories that can be accurately described by a simplified set of parameters, e.g. when adult fecundity and mortality are independent of age, more accurate estimates of life table parameters and of population growth rate and its standard error can be readily obtained. Here an analytic method for calculating approximate confidence intervals for population growth rate is given for two-stage life histories that can be described by four variables representing age at first breeding, fecundity per unit time, and juvenile and adult survivorships per unit time. The method is applied to experimental data on Capitella sp. I obtained by Hansen et al., and quite good agreement is found between the analytic and bootstrap estimates of the standard error of l. The analytic estimates were a little conservative, probably because of the way the action of mortality was modelled. Alternative life-history models are briefly discussed, and the desirability of formulating life-history models so that the variables involved are independent of each other is stressed. Analytic estimates of l may be biassed if an inappropriate model is chosen or if variables are not independent and the correlations between them are not measured. To allow for these possibilities, where necessary a conservative approach should be taken to significance testing using the analytic method. R. M. Sibly, Di6. of Zoology, School of Animal & Microbial Sciences, Uni6. of Reading, PO Box 228, Reading, UK RG6 6AJ (
[email protected]). – F. T. Hansen and V. E. Forbes, Dept of Life Sciences and Chemistry, Roskilde Uni6., PO Box 260, DK-4000 Roskilde, Denmark.
Life tables are generally constructed using age classes of equal span. Thus life tables of birds and large mammals generally present survivorships and birth rates year by year, and life table parameters of small invertebrates are generally presented day by day or week by week. Statistical estimation of parameter values is improved, however, if it can reasonably be assumed that birth rates and adult mortality rates do not change with age. Life histories that can be so described will here be referred to as two-stage life histories. The assumptions that birth rates and adult mortality rates do not change with age are reasonable for many iteroparous organisms that do not grow after the age of first breeding,
e.g. many avian species (Deevey 1947, Lande 1988; but see Newton et al. 1997) and mammals, at least until the age of senescence (Sibly et al. 1997). The assumptions may also be reasonable for some organisms that do grow after the age of first breeding, as in the case of Capitella sp. I analysed here. Where the assumptions are applicable, their use in statistical modelling should increase accuracy and precision in the estimation of birth and mortality rates, with consequent improvement in the estimation of population growth rate, l, and its confidence interval. Despite these advantages only a small number of studies have used the two-stage lifehistory assumptions in estimating confidence intervals
Accepted 17 May 1999 Copyright © OIKOS 2000 ISSN 0030-1299 Printed in Ireland – all rights reserved OIKOS 88:2 (2000)
335
for population growth rate. These include Lande (1988) and Levin et al. (1996). Lande (1988) was the first to use the statistical theory of standard errors of functions of random variables to calculate approximate confidence intervals for population growth rate l, as part of an analysis of the life table of the northern spotted owl (Strix occidentalis caurina). Methods using the statistical theory of standard errors will here be referred to as analytic methods. Lande used a two-stage five-variable life-history model. The general analytic method he introduced has subsequently been applied to life histories described by population projection matrices (Caswell 1989, Alvarez-Buylla and Slatkin 1991, 1993, 1994). The entries in these matrices – the so-called ‘vital rates’ – are the random variables to which the statistical theory applies. Population growth rate l is an implicit function of these variables, being the solution of the matrix’s characteristic equation, or equivalently for age-dependent life histories, of the Euler-Lotka equation. Alvarez-Buylla and Slatkin (1993) used Monte Carlo methods to evaluate the performance of the analytic method applied to a number of life-history types, but not two-stage life histories. They concluded that the analytic method is reliable if correlations between vital rates have been measured and if coefficients of variation of estimates of vital rates are not very high ( B50%). When dealing with vital rates with high coefficients of variation, they concluded that the analytic method may still be used to assign rough confidence intervals. Further support for these conclusions was provided by the application of both methods to three real biological cases (Alvarez-Buylla and Slatkin 1994). Some differences between the analytic and the Monte Carlo estimates were found in cases in which l was very sensitive to vital rates whose estimates had high coefficients of variation ( \50%). It seems, then, that there are advantages to using the two-stage life-history assumptions where they are applicable, and that analytic approximations of confidence intervals for l can then be readily calculated. Here we apply the analytic method to real examples of two-stage life histories that have recently been analysed by bootstrapping. The results of the two methods of analysis are then compared.
Analytic method The statistical theory of standard errors of functions of random variables (e.g. Kendall and Stuart 1969) states that if x1, x2, …, xk are random variables with variances and covariances of order o, and if l(x1, x2, …, xk ) is a function of x1, x2, …, xk then 336
k
variance l = %
! "
i=1
#l #xi
k
+ %%
i"j=1
2
variance xi
#l #l covariance(xi, xj ) +o(o) #xi #xj (1)
where all partial derivatives are evaluated at the mean values of x1, x2, …, xk. The partial derivative #l/#xi indicates the sensitivity of l to xi. The variances in eq. (1) are the squares of standard errors. Eq. (1) can therefore be used to assign approximate confidence intervals for l if the coefficients of variation (SE/ mean) of x1, x2, …, xk are small. The equation can be understood as stating that the variance of l is the weighted sum of the variances and covariances of the life-history variables x1, x2, …, xk. The weights are the squares and products of the sensitivities. In most applications of this theory to population growth rates to date the random variables have represented the entries in the population projection matrix. Here, however, we apply the theory to other ways of representing the life history, in particular to the two-stage life-history model previously analysed by Sibly and Calow (1986a), Lande (1988), Levin et al. (1996), Calow et al. (1997), and others. In this model individuals are classified only as juveniles and adults. Lifehistory variables relating to juveniles and adults will here be denoted by subscripts j and a respectively. We use the notation that it takes juveniles tj time units to reach the age of first reproduction. They then produce on average F female offspring each time unit, potentially for ever, but are subject to mortality. Adult survivorship between time units is pa and juvenile survivorship between time units is pj. Thus the model has four variables, tj, F, pj and pa. It simplifies the mathematics to define an additional variable Sj = p tj j, representing juvenile survivorship from birth to age tj. If deaths are independent of each other then the deaths in a given time interval can be considered the outcome of a binomial process with binomial parameter pi, where i= a or j, and so standard error of the estimate of pi =
'
pi (1− pi ) n (2)
where n is the number of individuals in the sample (Lande 1988). l is calculated from the relevant form of the Euler-Lotka equation, i.e. 1= FSj l − tj + pa l − 1
(3)
(Calow and Sibly 1990). Writing T= FSj tj + pa ltj − 1, the sensitivities of l to the life-history variables are calculated by implicit differentiation as OIKOS 88:2 (2000)
#l = #tj
FSj l ln
pj l
(4)
T
#l Sj l = T #F
(5)
#l FSj tj l = #pj Tpj
(6)
#l ltj = #pa T
(7)
The difference between these equations and those given in Levin et al. (1996) arises because of small differences in model formulation.
Results Here we apply the analytic method to a study for which bootstrap estimates of the confidence interval of l calculated from an age-classified life table are available (Hansen et al. 1999) The study animal was the infaunal deposit-feeding polychaete Capitella sp. I. Bootstrapping followed the recommendations of Meyer et al. (1986). Estimates were bias-adjusted, normal-based, calculated according to Meyer et al.’s eq. 3. The life histories of 18–20 worms were recorded in each of four treatments (concentrations of 4-nnonylphenol). For the purposes of analytic estimation the four parameters tj, F, pj and pa were recorded. Age at first reproduction, tj, was measured as the time from an egg being hatched until the first eggs of that individual hatched. Mean fecundity, F, was defined as the average number of eggs (Capitella sp. I is hermaphroditic) per individual per week during the 11-week experiment. Survivorships and fecundity were not found to vary with age so there are good grounds for considering that the two-stage four-variable model is appropriate. The analytic approximation of the standard error of l was calculated from eq. (1) using variance (xi ) =(SE xi )2 and SE(l)=ãvariance(l). Estimates of covariances were not available and these were assigned values of zero. The values of the four life-history traits in control
conditions without 4-n-nonylphenol, together with the other parameter values used in calculating the standard error of l, are given in Table 1. The within-treatment standard errors of tj and F were calculated in the usual way and of survivorships using eq. (2). l was calculated from eq. (3) and the sensitivities of l to the life-history variables, #l/#xi, were calculated from eqs (4) – (7). The last column of Table 1 gives the components of the variance of l. Summing the last column gives the variance of l as 0.0105. The estimated standard error of l is therefore 0.102. This is the analytic approximation of the standard error. By comparison, the standard error obtained by bootstrapping as described in Hansen et al. (1999) was 0.073. The components of the variance of l that were attributable to each life-history parameter are shown for each of the four treatments in Table 2. It will be seen that F contributed relatively little, and pa contributed nothing because there was no adult mortality in the experiment. Most of the variance was attributable to pj, and the variability in the values of SEanalytic to some extent reflects the variability of pj. The values of l and its standard error obtained by the analytic method and by bootstrapping are shown in the bottom two rows of Table 2 and plotted against each other in Fig. 1. The values of l are well correlated but those obtained by the analytic method are 6 – 9% lower than those obtained by bootstrapping. Estimates of standard error obtained by bootstrapping are 5 – 35% lower than those obtained by the analytic method.
Discussion Overall there was quite good agreement between the analytic and bootstrap estimates of l and its standard error (Fig. 1, Table 1), although the bootstrapping estimates of standard error are a little lower, and of l a little higher than those obtained by the analytic method. We consider first the discrepancies in the estimates of standard error. In an extensive investigation of the performance of bootstrapping methods of estimating r (= loge l) in age-classified models, Meyer et al. (1986) found close agreement between full-sample and bootstrap estimates suggesting that bootstrap estimation of SE can be accurate to about 1%. Since the
Table 1. Estimates and standard errors of estimates of the life-history parameters of Capitella sp. I in control conditions, together with the coefficients of variation, CoV = SE/estimate, and the sensitivity of l to each life-history parameter, #l/#xi. The last column gives the contribution of each life-history parameter to the total variance of l. n =19. Data from experiments described in Hansen et al. (1999).
tj, wk F, wk−1 pj, wk−1 pa, wk−1
OIKOS 88:2 (2000)
Estimate
SE of estimate
CoV, %
#l #xi
7.1 459.9 0.976 1
0.19 33.9 0.035 0
2.7 7.4 3.6 0
−0.300 0.000696 2.330 0.216
! " #l #xi
2
variance xi
0.0032 0.0006 0.0067 0
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Table 2. The contributions of each life-history parameter (‘components of variance’, first four rows) to the total variance of l (row 5) in each of the four treatments (columns) in the experiments of Hansen et al. (1999). The bottom two rows give the values of l calculated from eq. (3) and by bootstrapping. The standard error of lanalytic was obtained as the square root of ‘total variance’ (row 5). The standard error of lbootstrap was obtained by bootstrapping assuming a normal distribution of l. Treatments (mg 4-n-nonylphenol/g dry weight of sediment)
Components of variance due to tj F pj pa Total variance lanalytic 9 SE, wk−1 lbootstrap 9 SE, wk−1
0 (control)
14
52
174
0.0032 0.0006 0.0067 0 0.0105 2.4990.102 2.679 0.073
0.0031 0.0013 0.0023 0 0.0066 2.59 9 0.081 2.83 90.078
0.0022 0.0003 0.0103 0 0.0128 2.44 9 0.113 2.67 90.073
0.0009 0.0005 0.0016 0 0.0031 1.94 9 0.055 2.05 9 0.052
bootstrap estimates here used Meyer et al.’s methods, they hopefully achieved similar accuracy. How then can we explain the discrepancies with the analytic estimates? The Monte Carlo investigations of Alvarez-Buylla and Slatkin (1993), though using different life-history models, suggest that analytic estimates of SE(l) should be accurate to B1% if coefficients of variation (CoV) are B5%, and to B4% if CoV are B25%. Small CoV are needed because analytic estimation using eq. (1) is based on a Taylor series expansion. In our data CoV were B5% for all variables and treatments except that for F which was in the range 5.7–10.8%. It seems unlikely therefore that the discrepancies are due to the magnitude of the coefficients of variation. Another possible cause of discrepancies in SE estimation is lack of information about correlations between life-history variables. This could bias analytic estimates based on eq. (1). By contrast the bootstrapping methods of Meyer et al. (1986) are unbiassed. To assess the severity of this possible problem with analytic estimation, Alvarez-Buylla and Slatkin’s (1993) Fig. 2b gives a ‘typical trend’ in which, with lognormal errors, SE rises more or less linearly with the size of the correlations, assumed positive. In the example given by AlvarezBuylla and Slatkin, ignoring correlations would produce a discrepancy of some 25% if correlations between vital rates were 0.45. It is possible therefore that the discrepancies between our analytic and bootstrapping estimates of SE result in part because covariances between life-history variables were not included in our analytic estimate. In our data there was no variation in pa and therefore no covariance between pa and other characters. Correlations involving pj were not measured (it is worth noting that large samples are needed to measure correlations involving survivorship). The correlations between F and tj are shown in Fig. 2; that at treatment 14 was significant (r16 = −0.50, p =0.035) and corresponded to a covariance of −4.44. Inserting this value in eq. (2) increases the value of SEanalytic from 0.081 to 0.087. The other possible correlation, at treatment 52, would also increase SEanalytic. In neither case, 338
therefore, is the fit with SEbootstrap improved by including the covariance term. If correlations between life-history variables have not or cannot be measured, then the life-history model should be formulated so that as far as possible its variables are independent of each other. The covariance terms are then expected to be zero. Unfortunately, however, it is not always easy to identify life-history models with the property that model variables are independent of each other. The problem of identifying independent life-history variables is particularly acute in modelling mortality/survivorship. In the model used here, pj and tj have been assumed to be independent variables. Thus if tj is extended Sj necessarily declines according to the equation Sj = p tj j. An alternative approach, adopted by Lande (1988), would be to use Sj and tj as independent variables. This assumes that Sj does not decline if tj is extended. The pj model is to be preferred if mortality rate per unit time is independent
Fig. 1. lbootstrap 9SE plotted against lanalytic 9SE, data of Table 2. Units are weeks − 1. OIKOS 88:2 (2000)
Fig. 2. Correlations between F (eggs per female per week) and tj (weeks) in each treatment. Correlation coefficients were r15 = 0.00, n.s., r16 = −0.50, p=0.035, r14 = − 0.33, n.s. and r18 = − 0.17, n.s. for, respectively, treatments 0, 14, 52 and 174 mg 4-n-nonylphenol/g dry weight of sediment.
of period survived, as will be true of mortality caused by e.g. predators or parasites that search at a constant rate. The Sj model is to be preferred if juvenile survivorship is independent of tj, the period survived, as might happen if all juvenile mortality occurred in a period experienced by all juveniles, e.g. during hatching. Where both types of mortality agent are known to be important a composite model should be used that incorporates both types of variable. The effects on l’s estimated confidence interval of choosing the pj as opposed to the Sj model are readily evaluated. The contribution to variance (l) of the tj term is larger under the pj model by a factor of −(loge pj −loge l)/loge l. In the present study pj was very close to 1, so the contribution of the tj term was similar under both models. The contribution to variance (l) of a pj term is larger than that of an Sj term by a factor of [p tj j − 1(1−pj )t 2j ]/(1−p tj j ). For the parameters of Table 1 this is a factor of 6.6. Since the contribution of the pj term is relatively large in this case, it is clear that choice of model has a major effect on the estimated confidence interval. SEanalytic would indeed be OIKOS 88:2 (2000)
closer to SEbootstrap if a composite model was used rather than the pj model presented. The reason why the analytic method gives a lower estimate of l than the bootstrap in Table 2 is not entirely clear, but we think it arises from the way between-individual variation in the age of first reproduction is handled. We think the discrepancy is not due to the type of method (analytic vs bootstrap) but to the underlying models here used by the two methods. In calculating l the two-stage life-history model, here used by the analytic method, assumes that there is no between-individual variation in tj. By contrast the ageclassified model, here used by the bootstrap method, allows such variation. In effect in the age-classified model tj is taken to be the youngest age at which any individual in the treatment reproduced. This assumption was not adopted in the version of the two-stage model implemented here, but it could be. The adult stage would then begin at the youngest age at which any individual in the treatment reproduces, and F and pa would be estimated from all individuals older than this age. For the model to be a good representation of reality it would be necessary that there were no varia339
tion with age, in either adult survivorship or the average fecundity of each age class, from the earliest age at which any individual in the treatment reproduces. This approach was taken by Levin et al. (1996) and Hansen et al. (1999) and gave good agreement with estimates based on the age-classified model. Effects of variation in tj could also be allowed for in the two-stage model using Taylor series methods (see e.g. Kendall and Stuart 1969: Exercise 10.17). In summary, in Table 2 the bootstrap estimates of l are probably more accurate than the analytic estimates. However if the variation in tj does not vary between treatments, then the relative values of ls should be approximately the same whichever method is used, as they are in Fig. 1. For some applications it may be desirable to modify the two-stage model. Semelparity is dealt with by setting pa =0 in eqs (2)–(6), and eq. (7) is not then needed. The possibility that individuals only reproduce for a limited period has been ignored so far. This is reasonable for the purposes of calculating l and its confidence interval, if as here the population is increasing quickly and the reproductive period is a large fraction of lifespan, since estimates of l and its confidence interval are then relatively insensitive to age at last reproduction. In these cases, pa should be estimated from the data obtained during the reproductive period. In other cases it may be necessary to include age at last reproduction as a fifth variable in the model (Sibly and Calow 1986b, Sibly 1996; see also Sibly et al. 1997). If it is desired to use mortality rate per unit time, mi = −loge pi, instead of pi, then pi should be replaced by e − mi and #/#pi by − 1/pi #/#mi in eqs (3)–(7). Population growth rate has here been measured by l, but loge l (=r) has the advantage of being symmetrical in its representation of population increases and decreases; l lies in the range (0, ) and constant population size gives l= 1, while r lies in the range (− , ) and constant population size gives r=0. The asymmetry of l about 1 would present a problem for analytic estimation if l’s confidence interval were large, since it must then be asymmetrical. The problem is avoided by using r instead of l. The formulae for r-sensitivities can be obtained from those for l by dividing by l, thus #r/#xi =(1/l)(#l/#xi ). In consequence, analytic estimates of standard errors for r can be obtained from those for l by dividing by l. It should be borne in mind, however, that the distribution of r is not necessarily normal (see e.g. Rago and Dorazio (1984) and Meyer et al. (1986) for discussions of skewed distributions). In conclusion, for organisms with life histories that can be accurately described by a simplified parameter set, e.g. those in which adult fecundity and mortality are independent of age, use of these parameters allows accurate analytic estimation of population growth rate and its standard error. Where there is doubt about how best to describe the life history, bootstrapping using an age-classified model and following the protocol of 340
Meyer et al. (1986) is the safest approach. The needed data collection and computer implementation of the bootstrapping protocol are, however, demanding, and where they cannot be achieved the analytic method, using a two-stage model if appropriate, is a realistic alternative. Analytic estimates of l may be biassed if an inappropriate model is chosen or if variables are not independent and the correlations between them are not measured. If it is necessary to allow for these possibilities a conservative approach should be taken to significance testing. Acknowledgements – We are very grateful to R. N. Curnow and H. Caswell for useful discussions and to RNC for comments on the manuscript.
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