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Keywords:

  • demography;
  • dispersal;
  • McMC;
  • population growth rates;
  • projection matrices;
  • Strix occidentalis;
  • survival

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References
  • 1
    Effective resource management ultimately influences vital rates of fecundity and survival for target species. Meta-analysis can be used to combine results from multiple demographic studies replicated in time and space to obtain estimates of vital rates as well as metrics of population growth.
  • 2
    Workshop formats were used to conduct meta-analyses of mark–recapture experiments on spotted owls Strix occidentalis in the western USA. The implied motivation for demographic studies of spotted owls has been that changes in vital rates and population growth, λ, reflect the success of conservation strategies, but how to interpret results may not be obvious. Demographic analysis is of little practical utility until vital rates can be linked to management. In the case of spotted owls, future meta-analyses must focus on co-variation between vital rates and habitat variables, and experiments will be necessary.
  • 3
    Sensitivity of population growth to variation in vital rates is central to demographic analysis, but results must be interpreted cautiously because these sensitivities are not likely to identify the vital rates most responsible for variation in population size, and cannot reveal which vital rates will be most responsive to conservation investments.
  • 4
    Difficulties in documenting dispersal seriously compromised estimates of juvenile survival and thereby biased estimates of λpm from a projection matrix, a problem that was resolved in later workshops by estimating λRJS directly using a reparameterized Jolly–Seber mark–recapture method.
  • 5
    Several sources of bias for estimates of vital rates and λ were reviewed. Bias exists in meta-analysis estimates of λ combined over spatial replicates because λ is a non-linear function of vital rates. Bias also exists in estimates of average population growth where λt varies over time. This problem can be reduced by calculating the geometric mean of λ. Research to measure biases associated with the estimation of vital rates and the selection of study areas will be necessary to validate meta-analyses of demography for spotted owls.
  • 6
    Synthesis and applications. Meta-analysis is ideally suited to studies of the demography of long-lived species because of the large areas involved, high costs for each individual study, and multiple jurisdictions within which the organisms occur. Mixed models selected using information–theoretic approaches provide a powerful way to combine research results from several studies in a meta-analysis.

Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

Ecological processes are highly variable in space and time. This variation has a major bearing on the design of research studies (Hurlbert 1984) and compromises the ability of ecologists to generalize from the results of their individual studies. Conducting a wide-scale investigation to address such variation may not be feasible for a variety of reasons, including funding restrictions, agency jurisdictions, logistics and motivation. A possible solution to these problems is to conduct a meta-analysis, which is a statistical method for combining the results of a number of studies (Arnqvist & Wooster 1995).

Meta-analysis was first defined by Glass (1976) as ‘the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings’, in the context of psychology and education research. Since then, meta-analysis has seen extensive use and development, especially in medical sciences (Gates 2002), as an objective way to summarize the results of a number of experiments (Hedges & Olkin 1985). Various statistical procedures have been evaluated for objective synthesis of a number of independent studies (Adams, Gurevitch & Rosenberg 1997; Lajeunesse & Forbes 2003). Applications of meta-analysis in ecology first appeared in the early 1990s, but since then nearly 200 papers have been published describing meta-analysis in ecology (see reviews by Arnqvist & Wooster 1995; Gurevitch, Curtis & Jones 2001; Gates 2002), including several in this Journal of Applied Ecology (Clergeau, Jokimaki & Savard 2001; Kleijn & Sutherland 2003; Pywell et al. 2003; Vesk, Leishman & Westoby 2004).

The advantages of meta-analysis of ecological studies are clear. The approach allows us to learn of effects and trends that are common across related studies, and by combining results from a number of studies patterns can be documented with considerably greater statistical power (Cohen 1977; Burnham, Anderson & White 1996). By synthesizing studies over a large area or over time, meta-analysis may permit inferences at larger scales. Also, a meta-analysis can be used to identify patterns amongst studies conducted in different places and at different times.

Although there have been several published demographic investigations on spotted owls Strix occidentalis Xantus de Vesey 1860 (Lande 1988; Noon & Biles 1990; Forsman 1993; Franklin et al. 2000; Blakesley, Noon & Shaw 2001; Seamans et al. 2001; Forsman et al. 2002), a number of long-term studies exist that have not been published. Many of these are monitoring programmes by federal agencies, the timber industry, aboriginal tribal organizations and state governments, for example to support habitat conservation plans (Bingham & Noon 1997). To facilitate a synthesis of several mark–recapture data sets on spotted owls, the Colorado Cooperative Wildlife Research Unit organized a series of four meta-analysis workshops (Anderson & Burnham 1992; Forsman et al. 1996; Franklin et al. 1999; Franklin et al. 2004). Workshop protocol required adherence to strict guidelines to ensure a co-operative venture (Anderson et al. 1999). Such a workshop format to bring together multiple data sets has been used on numerous occasions for natural resource management, including adaptive management workshops (Holling 1978) and for population viability analysis (Ellis & Seal 1995).

Our objective was to highlight the use of meta-analysis for the quantitative synthesis of demographic data with a particular focus on conservation applications. We believe that important lessons have been learnt from the application of meta-analysis procedures in the study of spotted owl demography, so we have highlighted this case study.

We begin by summarizing briefly the key elements of a meta-analysis as identified by Gates (2002), and we then use this outline to structure our review of demographic meta-analysis.

Components of a meta-analysis

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

sources of data

Ideally data would come only from randomized experiments with proper control, or observational studies that used random sampling. Rarely can this ideal be met and so most meta-analyses will depend heavily on model assumptions. In this way they can be viewed as a special case of observational studies, where the assumed model plays a central role in summarizing complex data.

metrics

Choice of summary parameter (metrics) is an important part of meta-analysis (Osenberg, Sarnelle & Cooper 1997). In the case of spotted owls, focus has been on the estimation of λ, and so E[λ] across the population from which the contributing studies have been drawn would appear to be a natural choice. In studies where different parameters have been estimated by the contributing studies, the choice is less clear (Osenberg et al. 1999; Gurevitch, Curtis & Jones 2001). Gurevitch, Curtis & Jones (2001: table 2) present a summary of common effect metrics and associated variance formulae. These include Hedges’d, the response ratio, Fisher's z-transform of the correlation coefficient, and the odds ratio. A recommended way to present results is to show point estimates and confidence interval bars so that the error is readily apparent, and to reveal variation among studies (Gates 2002).

Table 2.  Summary of juvenile survival, φ̂J, SE(φ̂J), apparent adult survival φ̂A and SE(φ̂A) estimated using the program mark for respective study areas in three meta-analyses of the demography of the northern spotted owl (Anderson & Burnham 1992; Burnham, Anderson & White 1996; Franklin et al. 1999). Covariates in models varied as detailed in original reports. Study area acronyms are defined in Table 1
Study area19921996199919921996 1999
φ̂JSE(φ̂J)φ̂JSE(φ̂J)φ̂JSE(φ̂J)φ̂ASE(φ̂A)φ̂ASE(φ̂A)φ̂ASE(φ̂A)
CAL0·1950·0510·330·0430·2950·0310·8510·0220·8680·012 0·860·009
HJA0·3110·1030·2880·0520·3050·0630·8370·0310·8210·016 0·8710·01
SCS0·20·0510·320·0380·7850·0260·8240·009 
RSB0·2830·0370·4180·0420·8580·0130·8430·01 
KLA0·3640·036 0·8330·016
TYE0·4460·036 0·8690·014
OLY0·0710·0280·2450·0640·2520·060·8830·0280·8620·017 0·8260·012
SAL0·4020·1050·8510·022 
CLE0·140·0260·1950·0280·850·031 0·8390·015
EUG0·2320·0780·8530·026 
COO0·2180·0450·8620·019 
SIU0·2430·0920·8220·027 
SIS00·830·045 
AST0·3780·348 0·8420·041
CAS0·2840·198 0·8160·017
COA0·3660·042 0·8870·01
ELC0·3940·04 0·8820·009
EEU0·1590·149 0·8230·02
HUP0·3660·131 0·820·025
RAI 0·8950·031
SIM0·3650·029 0·8590·008
WEN0·1430·03 0·8360·012
WSR0·0640·029 0·8430·022
Means0·2120·0540·25780·0590·29170·08330·84280·0240·84420·021 0·850·0169

statistics

Meta-analysis is simply statistical modelling: the use of probability models to summarize data with estimates of parameters from the summarizing model. In this way it is no different from ordinary data analysis. The issues are the same and one must have a model in mind, implicitly or explicitly. For example, weighting effect sizes by the inverse of variances (Gurevitch, Curtis & Jones 2001) to summarize across studies is a simple approach but one that depends heavily on the validity of the simple linear model theory that underlies it. Mixed models seem to be the natural way to approach this problem, where the two sources of variation, within- and between-studies, are explicitly modelled (Bennington & Thayne 1994; Gurevitch & Hedges 2001). In these, the contributing studies are linked by a distribution that is assumed to have generated the individual parameters that have been estimated in each study. This distribution provides the formal means for summarizing the results from the different studies. An advantage of formally specifying the summarizing model is that it allows the adoption of a modelling approach where the emphasis is on parameter estimation and perhaps model selection, as advocated by Burnham & Anderson (2002).

investigating bias

Meta-analysis often involves the synthesis of published reviews. This can lead to a number of biases resulting from the publication process. For example, studies failing to find a significant effect of treatment are less likely to be published, a problem that is sometimes called the ‘file-drawer problem’ (Rosenthal 1991; Arnqvist & Wooster 1995). As a consequence, a summary of only published results would be expected to show more of an effect than really exists (Palmer 1999, 2000).

Synthesizing a number of studies can result in a variety of sources of bias. How studies are selected for inclusion, variation in methods, sample sizes, scale of study units (Steen & Haydon 2000), missing data, non-independence of observations, poor reporting and variable quality of data can all compromise the results of a meta-analysis (Gurevitch, Curtis & Jones 2001). Also, bias in an individual study might carry over across studies, for example if it is characteristic of the particular species under investigation.

Overview of meta-analyses on spotted owls

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

The northern spotted owl Strix occidentalis caurina was listed as a threatened species in the USA in 1990 and is currently managed under the North-West Forest Plan launched by President Bill Clinton (FEMAT 1993). This plan includes a large system of conservation reserves interconnected by riparian buffer strips. In total, 7·9 million of the 9·7 million ha included in the Clinton Plan are being managed to protect habitats for spotted owls and other wildlife species. In addition, on private, state and tribal lands, a number of habitat conservation plans have been developed that afford habitat protection for spotted owls (Bingham & Noon 1997). The California spotted owl Strix occidentalis occidentalis has not been listed under the Endangered Species Act but is protected by an interim conservation strategy developed by the US Forest Service (Franklin et al. 2004).

Five meta-analyses of spotted owl demography have been conducted. The first of these included only five study sites (Anderson & Burnham 1992), followed by a workshop in 1993 that included 11 reported studies (Forsman et al. 1996). The next demography workshop on the northern spotted owl was in 1998, synthesizing demographic data from 16 mark–recapture experiments (Franklin et al. 1999); another workshop was scheduled in 2004. Finally, a synthesis of demographic studies for five study areas within the range of the California spotted owl was conducted in 2001 (Franklin et al. 2004).

sources of data

Field sampling entails annual visits to owl nests or territories and recording the presence of pairs, uniquely marked with coloured leg bands (Franklin et al. 1996). New pairs of owls can be added to the study in the year following their discovery. Mark–recapture methods are used to estimate apparent survival, i.e. the probability that an owl survives and remains on the study area the next year. Emigration is not measured, so actual survival will be slightly higher than estimates of apparent survival.

Data sets included in the meta-analysis were subjected to detailed protocols for data screening, refined by the time of the 1998 workshop and involving four steps: (i) 10 records were drawn from each data set and field data forms were obtained for these records to verify records; (ii) each participant in the workshop was required to certify that the records had been verified and that no change would be made during the analysis; and (iii) in advance of the workshop, assumptions associated with the analysis were discussed and analysis protocols were agreed upon by the biologists who collected the data.

Demographic study areas were initially established in areas with extensive old forests that are now largely protected from timber harvest, with subsequent additions to the study areas on selected state, tribal and private lands. Areas selected do not reflect a representative sample of land-management practices on the larger landscape, nor do they comprise a random sample of spotted owl habitats. Demographic trends for owls in those areas ought to reflect how effectively habitat protection is ensuring persistence, because many of the study areas are located in areas largely withdrawn from timber harvest. In other areas, for example northern California, extensive timber harvesting has occurred within spotted owl study areas. So careful thought must be given to the population for which any inference is intended to apply.

metrics

Gates (2002) excluded the spotted owl demography meta-analysis (Burnham, Anderson & White 1996) from his review of meta-analyses because the studies were a ‘combination of data to estimate survival parameters, not estimation of combined effect sizes’. Gates (2002) overlooked the fact that a primary objective in the spotted owl meta-analysis was not simply to estimate survival but to estimate change in adult female survival over time and to estimate lambda (Franklin et al. 1999), either of which is as appropriate as metrics in a meta-analysis, as is effect size. Indeed, this was Osenberg, Sarnelle & Cooper's (1997) point, that selection of a metric for meta-analysis should not be restricted to effect size alone and that ecologically more relevant metrics should be the target of ecological meta-analyses.

Demographic analyses of spotted owl data are based on mark–recapture methods to estimate survival, φi, and field observations of ≥ 2-year-old females to estimate fecundity, bi (Franklin et al. 1996). These vital rates of survival and fecundity are then combined into a demographic projection matrix:

  • image

for which the dominant eigenvalue, λpm (subscript for projection matrix), can be estimated characterizing the asymptotic trajectory of population size (Caswell 2001).

Methods used in the first three workshops for the northern spotted owl were largely the same. Study areas differed among meta-analyses (Table 1) and estimates of juvenile and adult female survival, and λpm varied among years and among sites (Tables 2 and 3). In addition to the key vital rates summarized in Tables 2 and 3, estimates were made of fecundity for yearlings and 2-year-olds, and survival for younger classes of owls.

Table 1.  Summary of study areas for demographic studies of the northern spotted owl included in the respective meta-analysis (MA) workshops of 1992, 1996 and 1996 (Anderson & Burnham 1992; Burnham, Anderson & White 1996; Franklin et al. 1999)
Study areaAcronymArea (km2)Caps (year)
1992199619991992 MA1996 MA1999 MA
  1. Caps, number of spotted owls captured; (year), number of years over which data were compiled for the workshop; MA, meta-analysis of 1992, 1996 or 1999.

North-west CaliforniaCAL 4 000 1 784 1 790 400 (7) 548 (9) 795 (14)
H.J. Andrews (Oregon)HJA   116 1 075 1 526 358 (5) 515 (7) 751 (12)
Medford (SW Oregon)SCS 4 05015 216 703 (7)1920 (9)
Roseburg (SW Oregon)RSB 1 700 6 044 589 (7)1022 (9)
Roseburg (Klamath)KLA 1 377 752 (14)
Roseburg (Tyee)TYE 1 741 737 (14)
Olympic PeninsulaOLY   965 8 145 8 152 302 (5) 548 (7) 869 (12)
Salem BLM (Oregon)SAL 3 249 260 (8)
Cle Elum (Washington)CLE 1 763 1 784 332 (5) 589 (10)
Eugene BLM (Oregon)EUG 2 082 176 (5)
Coos Bay (Oregon)COO 2 477 377 (4)
Siuslaw (Oregon)SIU 2 749 251 (4)
Siskiyou (Oregon)SIS 1 262 110 (4)
Astoria Forest (Oregon)AST   358  47 (8)
South Cascades (Oregon)CAS 2 590 446 (8)
Oregon Coast RangeCOA 3 918 772 (9)
Elliott State ForestELC 4 295 853 (9)
East EugeneEEU 2 537 179 (9)
Hoopla Tribal (California)HUP   356 188 (7)
Ranier (Washington)RAI 2 133 143 (7)
Simpson (California)SIM 1 2651011 (9)
Wenatchee (Washington)WEN22 048 957 (9)
Warm Springs (Oregon)WSR 1 001 318 (7)
Totals 10 83145 84656 871235260599407
Table 3.  Summary of adult fecundity, φ̂A, and λpm, with standard errors (SE) of these estimates, for females from each study area used in three meta-analyses of the northern spotted owl (Anderson & Burnham 1992; Burnham, Anderson & White 1996; Franklin et al. 1999). See Table 1 for study area definitions
Study area199219961999199219961999
φ̂ASE(φ̂A)φ̂ASE(φ̂A)φ̂ASE(φ̂A)λpmSE(λ)λpmSE(λ)λpmSE(λ)
CAL0·3580·02450·3330·0290·3490·0320·9150·04330·9660·0170·9510·015
HJA0·3270·050·3480·0340·2890·0410·9280·04370·9110·0210·9250·014
SCS0·3230·4880·3130·016– pAn0·8440·03040·9110·012
RSB0·330·0390·3210·0220·9410·01820·9570·015
KLA0·3940·030·9590·021
TYE0·2990·0310·9840·02
OLY0·3330·0780·380·0360·3440·0530·8830·0280·9470·0260·8760·016
SAL0·3810·0511·0190·073
CLE0·5650·0610·5680·0640·9240·0320·9410·023
EUG0·2720·0490·9130·031
COO0·3230·0440·9270·022
SIU0·2310·0430·8740·031
SIS0·2820·0720·83
AST0·2240·0840·920·077
CAS0·3210·0330·8460·021
COA0·2580·0240·970·014
ELC0·2630·0210·9720·014
EEU0·1030·0230·8280·021
HUP0·1720·030·8790·034
RAI0·2910·0330·940·031
SIM0·3490·0340·9690·015
WENWithdrawn0·4920·0460·880·015
WSR0·420·0720·8690·025
Means0·3340·13590·3410·04150·3210·04070·9020·03270·9250·0280·91930·0235

Survival estimation for all four meta-analyses used open population capture–recapture methods for each study area (program mark; White & Burnham 1999). For the 1998 analysis, a meta-analysis of adult female survival was conducted using two methods, one including all 15 study areas and one using just eight areas selected as long-term monitoring areas. The general approach can be summarized in five steps: (i) goodness-of-fit statistics were evaluated and an estimate was made of an overdispersion parameter, c; (ii) a set of alternative models was suggested prior to analysis; (iii) program mark was used to analyse the data; (iv) the overdispersion parameter was used to adjust the covariance matrices and an adjusted Akaike's information criterion (AIC) metric was calculated; and (v) the model with the lowest AIC value was selected for each study area.

A well-recognized difficulty in the estimation of λpm is that the dispersal of birds outside the study area that survive but do not return is difficult to document (Anderson & Burnham 1992; Bart 1995; Burnham, Anderson & White 1996; Franklin et al. 1999, 2004). Even though recent studies of dispersal have been reported in the literature (LaHaye, Gutiérrez & Dunk 2001; Forsman et al. 2002), these studies focus on dispersal distances and do not provide estimates of survival to compare with estimates used in the meta-analyses. Additional research is needed to test the assumption that dispersers survive as well as those that do not disperse (Burnham, Anderson & White 1996; Franklin et al. 1999). Certainly this assumption is violated in many species because dispersers suffer higher mortality because of predation and other risks (Tinkle, Dunham & Songdon 1993) (indeed, this would be a suitable topic for a meta-analysis).

An important development occurred in the 1998 workshop with the introduction of λRJS, using the reparameterized Jolly–Seber method to estimate growth rate directly from the mark–recapture data (Pradel 1996; Hines & Nichols 2002; Nichols & Hines 2002). The primary advantage of this method is that population growth can be estimated without reliance on estimates of emigration. Also, annual estimates of λt are obtained. Although emigration was estimated on three study areas using radiotelemetry, these data were not available for each demographic study area (Burnham, Anderson & White 1996; Franklin et al. 1999). So use of λRJS was a distinct improvement.

The investigation of the demography of five studies of the California spotted owl involved a synthesis of information on each of the five areas, and meta-analysis of survival and λRJS was performed. However, in contrast with the meta-analyses for the northern spotted owl, no meta-analysis was attempted on the matrix-based λpm. The rationale for this was because of differences among study areas in the methods used to estimate fecundity (Franklin et al. 2004), i.e. the ‘apples and oranges’ problem (Rosenthal 1991).

statistics

Meta-analyses were conducted using mixed models (SAS proc mixed) for fecundity and adult survival estimates as well as for estimates of λpm for the northern spotted owl (SAS Institute 1997). Data were clearly nested by study area, and included repeated measures with a different number of years of observation collected at some sites. More standard statistical procedures would discard data not meeting model assumptions but mixed models permit all of the data to be used in an optimal fashion (Bennington & Thayne 1994).

For the 1993 analysis (Burnham, Anderson & White 1996), a declining trend in adult female survival was observed, causing some alarm, but this trend did not persist as additional data were accumulated for with the 1998 meta-analysis. The λpm estimates appeared to be biased low probably because of low estimates of juvenile survival, a problem that is rectified by using the Pradel (1996) modification of the Jolly–Seber model that allows estimation of λRJS and by year, tt), directly from the mark–recapture data. Although the interpretation is somewhat different (Franklin et al. 2004), the Pradel method more directly estimates population growth based on changes in the number of breeding birds in study areas, in contrast to λpm, which is a more abstract value that emerges asymptotically.

The two meta-analyses that present estimates of λt provide weighted means of these values for each study site (Franklin et al. 1999: table 21; Franklin et al. 2004: fig. 5). Averaging population growth, λt, when λt is time varying, can yield highly misleading measures of population growth (Lewontin & Cohen 1969) simply because of the geometric nature of population growth. It is much better to use the geometric mean of λt, which will always be < inline image if λt varies.

Overdispersion occurs when variance in the data is greater than predicted by the assumed model. This can occur because recaptures are not independent or because parameters assumed constant are actually varying. The simplest approach for coping with overdispersion is to assume that the underlying model is approximately correct and to increase variances using a variance inflation factor or overdispersion parameter, c. This is the approach used in the spotted owl meta-analyses (Franklin et al. 2000). Although the theory behind this adjustment seems reasonably straight forward, Schmutz et al. (1995) found that estimates of c did not achieve correct variance inflation in their ringing study of brant Branta bernicla L. in Alaska. Ultimately we wish to understand what factors are causing overdispersion, and using variance inflation is post hoc and does not give insight into reasons why the models do not fit.

AIC and related information–theory statistics were used instead of the traditional hypothesis-testing statistical inference that has been used in virtually all meta-analyses published to date. These information–theoretic approaches are more suitable in this instance because a number of covariates are possible, and AIC assists in the identification of the model that best explains the data with the fewest parameters (Gibson et al. 2004; Rushton, Ormerod & Kerby 2004). Models used for parameter estimation varied among study areas and varied depending on the vital rates that were being estimated (Franklin 2002). Covariates included in survival models included study area, time, ecological province, land ownership and selected interactions between these effects (Franklin et al. 1999). Simulation studies have confirmed that the information–theoretic protocol usually identifies the ‘correct’ model, or at least a similar one (Manly, McDonald & McDonald 1999). Once the best model is selected, coefficients for covariates can be examined to assess their influence on the model. Tests of hypothesis may be relevant at this stage.

investigating bias

Because the spotted owl meta-analyses were not a synthesis of published reports, as is common in the medical literature, these analyses cannot suffer from publication bias (Palmer 1999, 2000; Gates 2002). Nevertheless, there is great potential for bias in estimates of vital rates and λ associated with the selection of study areas and which studies were included in each analysis. Although inclusion of study areas was considered carefully during workshops, questions linger about whether the study areas adequately represent the spotted owl population (Raphael et al. 1996), especially on non-federal lands. Although the study areas were distributed broadly throughout the range of the northern spotted owl, no demographic studies have occurred in the western Cascades of Washington, the eastern slope of the Oregon Cascades, or the California Cascades.

Combining estimates of λpm across study areas can result in biased estimates because λpm is a non-linear function of age-specific vital rates (Boyce 1977; Daley 1979; Meyer & Boyce 1994), but this was not recognized in the spotted owl workshop reports. In spotted owls, λpm is a concave function of fecundity and juvenile survival but a convex function of adult survival (Meyer & Boyce 1994: fig. 4). According to Jensen's inequality, because there is always variation in vital rates, when λpm(·) is concave E(λpm) will always be less than that calculated using average vital rates, whereas if λpm(·) is convex, as it is for adult survival, E(λpm) will be greater than the λpm calculated from average vital rates (Meyer & Boyce 1994). Generally, juvenile survival and fecundity vary markedly more than adult survival, therefore the consequences of this bias will be an overestimation of λpm.

Likewise, variation in vital rates among years can result in bias, with the general pattern being that estimated long-term growth rates will be lower when there is variation in projection matrices among years (Tuljapurkar 1989). Again, this bias will tend to reduce estimates of long-term growth rates, and for a long-lived species like the spotted owl, the magnitude of this bias can be substantial.

A number of assumptions in application of mark–recapture methods to spotted owls have received attention, both in workshop reports and other publications (Van Deusen, Irwin & Fleming 1998; Manly, McDonald & McDonald 1999; Hines & Nichols 2002). The size of the study area can vary over time because of inclusion of owls near the boundary of a designated study area, and this could potentially bias estimates of λt simply because the area sampled has changed over time (Franklin et al. 1999; Hines & Nichols 2002). Indeed, we note that the size of the study areas varied substantially from one workshop to the next (Table 1). Some of this was due to expansion of study areas, but also some study areas were subdivided as additional data were collected and more was learned about the study areas. This was a fairly easy matter to resolve in the screening of data to ensure that the boundaries of the study area did not change during the period over which λt was being estimated.

One concern in the estimation of growth rates is that the bias created by emigration would be worse in small study areas. Indeed, Steen & Haydon (2000) recommend that study areas have a diameter at least twice the maximum range of juvenile dispersal. Given that maximum natal dispersal can exceed 100 km in the spotted owl (Forsman et al. 2002) this would eliminate many of the study areas from consideration. To evaluate whether study area size itself was an important determinant of estimates of population growth, we conducted a regression analysis of estimates of both λpm and λRJS as a function of the square root of study area. Neither metric of population growth was correlated with study area size (r = −0·096, P= 0·6; r=−0·14, P= 0·45).

sensitivity analysis

Although not an essential component of a meta-analysis (contra Gates 2002), standard methods for estimating sensitivity of λ to variation in vital rates are well known (Caswell 2001) and have been applied in spotted owl meta-analyses. Adult survival is the most sensitive component of a life table for a long-lived species, and Lande (1988) and Blakesley, Noon & Shaw (2001) claim that this should therefore be the focus of management attention. Yet adult survival exhibits relatively little variance and does not contribute much to year-to-year variation in population size (Franklin et al. 2000). Among the demographic study areas for the California spotted owl, only the Sequoia-Kings Canyon population had a higher survival rate, estimated adult survival for the other study areas being indistinguishable.

Serious problems can occur in the interpretation of demographic sensitivity (Meyer & Boyce 1994). This is helped somewhat by the use of elasticities instead of sensitivities (Caswell 2001), but the fundamental question is which vital rates can be influenced best by management (Nichols & Hines 2002). Even if adult survival is the most sensitive vital rate, if nothing can be done to increase an already high rate of survival there may be little consequence (Noon et al. 1999).

Are we asking the right questions?

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

The objectives stated for the demographic meta-analyses were to (i) estimate survival and fecundity and variances for each study areas; (ii) calculate range-wide trends in survival and fecundity; and (iii) estimate population growth rates. Clearly a primary focus was on evaluating whether populations were declining, i.e. if λ < 1. One might reasonably ask why there has been so much focus on vital rates and population trajectories when the analyses have not been tied to habitats (Raphael et al. 1996). All management focus for spotted owls has been on habitat protection, yet none of the objectives listed for the demographic meta-analyses have been placed in a context of management significance. Given the availability of modern remote-sensing methods, there should be no excuse for not associating vital rates and population trajectories for each study population to same-period changes in habitats.

The more appropriate question relates to which vital rates can be influenced by management. Franklin et al. (2000) found substantial variation in reproduction and recruitment amongst years and found these vital rates to be correlated with climate. In comparison, as is often the case for long-lived species (Gaillard, Festa-Bianchet & Yoccoz 1998), adult survival showed relatively little variation from year to year, although adult survival varied among study areas, suggesting habitat effects (Franklin et al. 2000). Program mark has the ability to accommodate covariates, such as habitat and climate effects (Franklin et al. 2000; Franklin 2002). Such models would permit a demographic analysis that would relate directly to the management issues at hand. Several studies are now attempting to link owl demography to climate and large-scale maps of vegetation conditions.

In practice, the estimates of lambda and adult female survival received considerable attention by managers. An observation from the second meta-analysis, that adult female survival was declining at an increasing rate, was taken as an indication that the owl populations were in serious trouble (Harrison, Stahl & Doak 1993). The fact that estimates of λpm were still less than 1 by the time of the 1998 workshop (Franklin et al. 1999) was taken as an indication that the populations were continuing to decline, even though in the same report the new λt estimates, not burdened with unknown juvenile survival, were presented indicating that the populations were not declining. Again, in none of these analyses was it clear how the demographic estimates actually reflected management.

Is population growth rate the right metric? Sibly & Hone 2002) argue that population growth rate is the ‘key unifying variable linking the various facets of population ecology’. Although population growth rate allows a forecast of future population size, in a regulated population we would expect population growth rate, whether increasing or decreasing, to be transient. With persistence, long-term growth rates, lnλ, should converge to 0 (Royama 1992). More fundamental, and of greater management significance, is how the population is regulated (Franklin 1992; Boyce, Sinclair & White 1999).

Consider the implications of two alternatives. In the first instance, we can imagine that spotted owl populations are governed by the habitat heterogeneity hypothesis (HHH), where poor-quality territories are occupied only at high densities or, if a floater population exists, in particularly good years (Both 1998). Population-level changes in abundance would be greater than the within-territory responses in vital rates. Indeed, there is recent evidence indicating that spotted owls appear to follow an ideal despotic distribution (Zimmerman, LaHaye & Gutiérrez 2003). Core area populations would be expected to have long-term growth rates converging to 1. Yet, if habitats were being eroded outside of the study areas, for example, the population could be declining rapidly but this would not be detected by growth rates occurring in the study areas. In fact this is likely to be the situation, because significant portions of the demographic study areas were sites set-aside for protection of old growth by the North-West Forest Plan.

In the second instance, the individual adjustment hypothesis (IAH) is that population regulation, i.e. density dependence, occurs through changes within individual territories. Under this hypothesis, we expect the same variation within territories as at the population level. This would be the underlying model under which we might expect a demographic meta-analysis to reflect declining populations, as measured by vital rates on the demographic study areas. If habitats are deteriorating more broadly, instead of focused on areas currently being logged, reduced reproduction and survival would occur throughout. One possibility might be that owls displaced from logged territories would move into existing territories and competitive interactions could result in reduced reproduction and survival as a consequence. What is lacking from the demographic meta-analyses are clear statements of hypothesis that will allow an analysis of the data to help resolve some of these fundamental questions about how spotted owl populations work.

McMC and meta-analysis

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

Gelman et al. (1995) list three possibilities for studies that are being considered for meta-analysis: (i) different studies may be regarded as replications from the same population (i.e. parameters are identical in the different studies); (ii) the studies may be so different that any summary is meaningless; and (iii) various studies may be related but with some differences among studies that can be described using relatively simple statistical models. For spotted owls the different studies cannot be regarded as identical replications from the same study population. Although common parameters were measured on each study population, vital rates vary among studies. For example, differences in parameters might be due to among-site differences in habitats. Spotted owl meta-analyses fall into the third class listed by Gelman et al. (1995).

One approach to the meta-analysis is to combine data in a single mark–recapture analysis and assume that some parameters are the same in each study. This approach reflects a view that the studies represent a mixture of class (i) and class (ii) above: for parameters that are shared among studies it is assumed that either no variation exists among studies or any variation has been explicitly accounted for using study-site covariates. The parameters that are not shared are assumed to be distinct for each study and unrelated. An intermediate approach is to assume a common distribution for the mark–recapture model parameters. That is, an assumption that, although the parameters are distinct for each study, they can be regarded as having been sampled from a common probability distribution. In this approach between-site covariates may still be used to account for variation, but a random error term is included in the model to account for unexplained differences. Parameters in one experiment carried out on a local population become random variables in the description of the entire study population. Such models are known as random-effects, mixed or hierarchical models.

There are two advantages to the hierarchical modelling approach to meta-analysis. First, it offers a compromise between the view that the parameters are either all the same, or all different. The former invites oversimplified summary and the latter none. In reality, analyses in which the parameters are regarded as all different usually present tables of parameter estimates that invite summarizing (see, for example, Table 2). The assumption that parameters are related through a common probability distribution is much weaker than assuming that they are all the same, and at the same time it offers a valid means of summarizing parameters. Secondly, by modelling the relationship between parameters in different studies explicitly, the summarizing model can be brought into an information–theoretic model-selection process as has become standard in mark–recapture analysis.

Existing software for fitting mark–recapture models is poorly suited to hierarchical models. Analysis of deviance based on restricted models, as described by Lebreton et al. (1992), can be used to explore deterministic relationships between parameters, but methods for fitting random-effects models that allow for stochastic relationships between parameters are relatively crude. For example, the variance-components procedure incorporated in program mark uses method-of-moments, a procedure that can result in estimates of variances less than zero and for which sample sizes must be large for inference to be valid. Parameters that are close to boundaries (e.g. survival probabilities close to 1·0) also cause problems in the method-of-moments approach. An alternative approach is to adopt a Bayesian framework for analysis, in which the distribution of parameters of the mark–recapture model is governed by hyperparameters and prior distributions are specified for the hyperparameters. If little prior information is available then a flat or vague prior may be specified.

Recent developments in methods for fitting models, such as Markov chain Monte Carlo (McMC), have made it easier to fit such hierarchical models (Gilks, Richardson & Spiegelhalter 1996). In a conventional likelihood-based analysis we begin with the distribution of the data under the assumed model that depends on unknown parameters, θ. If we denote this distribution by [data|θ], then the likelihood function that we maximize is L(θ|data) ∝[data|θ]. Inference about the unknown parameter is made from the function L(θ|data). In a Bayesian analysis, instead of the likelihood function L(θ|data) we find the distribution [θ|data]∝[θ] ×[data|θ], where [θ|data] denotes the posterior distribution (i.e. the distribution of θ after the data have been observed) and [θ] denotes the prior distribution (i.e. the distribution of θ after the data have been observed). The key difference between the two approaches is that L(θ|data), although it may be proportional to a probability function, is not a probability function. In contrast, [θ|data] is a probability function and statements of inference about parameters can be expressed directly in terms of probability.

Historically, there have been two problems with Bayesian inference. First, there have been widespread objections to the Bayesian philosophy (Fisher 1932), and secondly, the calculation of [θ|data] requires (often) high-dimension numerical integration. However, McMC provides an effective solution to the second problem, and the widespread adoption of Bayesian methods since the advent of McMC suggests that the problem people have had with Bayesian inference has not been philosophical but technological.

McMC is a Monte Carlo method for generating correlated samples from distributions, which is particularly useful for the complicated distributions typical of Bayesian posterior distributions. The software WinBUGS (Spiegelhalter, Thomas & Best 2000) has made McMC accessible and relatively easy to use. To fit a hierarchical model in WinBUGS, the user specifies the distribution(s) that governs the data in terms of the hyperparameters, and then specifies the distribution that governs the hyperparameters in terms of the prior. Model specification is done using simple model syntax. Examples where WinBUGS has been to fit mark–recapture models are given by Brooks, Catchpole & Morgan (2000) and Johnson & Hoeting (2003).

Conclusions and recommendations

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

Meta-analysis can be a powerful tool for combining the results of demographic studies, and we expect this method to be especially important for species with large area requirements, where individual studies are expensive to conduct and multiple jurisdictions are involved. However, meta-analysis is not a panacea, and serious problems can arise because of selection of studies for inclusion, publication bias, and methods used to combine data. Indeed, Palmer (2000) warns of quasi-replication, i.e. the ‘replication’ of previous studies using different species or systems instead of proper replicative experimental research. We would argue, however, that the replication of demographic studies of the spotted owl provides an excellent example where the methods of data synthesis are very appropriate.

Furthermore, a meta-analysis cannot correct problems caused by defective sampling. For example, the bias in estimates of juvenile survival remains a problem in each spotted owl study area, and likewise in the meta-analysis. But meta-analysis brings additional assumptions related to the inclusion of studies that can further bias parameter estimates when the data are considered collectively. This becomes a serious issue for the spotted owl study areas because they are not a random sample of the landscape, being primarily on federal lands and often selected because the areas contained high populations of spotted owls.

The meta-analysis of owl demographic data has evolved through five workshops, using more sophisticated methods in successive iterations. Perhaps the most important development has been the recognition that estimates of juvenile survival were not obtainable with available data, and new mark–recapture methods could be used to estimate λRJS directly (Pradel 1996; Nichols & Hines 2002). We suggest that McMC offers a powerful tool more appropriate to the data.

A focus for future work should be to design management actions targeted to manipulate vital rates, for example habitat alteration (Irwin & Wigley 1993; Boyce 1998; Meyer, Irwin & Boyce 1998; Noon & Franklin 2002; Bennett & Adams 2004) or less-invasive management actions (Boyce 1993). Such active adaptive management was clearly outlined in the Thomas et al. (1990) plan for the northern spotted owl but, in a fashion typical for attempts at adaptive management (Walters 1997), administrative barriers have developed (Boyce 1998).

In addition, attention should be given to identifying sampling problems relative to management jurisdictions and to enlist more representative samples weighted by ownership patterns on the land base. One might ask if the objective of the meta-analysis is to evaluate the status of the species or simply the status on federal lands? Given the conservation mandate of the Endangered Species Act, clearly the target should be the former, calling for better sampling.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References

Thanks to J. Gurevitch, R. Mickey, J. Nichols and S. Rushton for comments on a draft manuscript, and to A. Franklin for providing unpublished reports. We received support from the Association of O & C Counties and NSERC.

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  4. Components of a meta-analysis
  5. Overview of meta-analyses on spotted owls
  6. Are we asking the right questions?
  7. McMC and meta-analysis
  8. Conclusions and recommendations
  9. Acknowledgements
  10. References
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