Dr Stephen Baillie (fax 01842 750030; e-mail firstname.lastname@example.org). ‡Present address: Laboratoire de Paléontologie, Institut des Sciences de l'Évolution, Université Montpellier II, F-34095 Montpellier, Cédex 05, France. §Present address: Royal Society for the Protection of Birds, The Lodge, Sandy, Bedfordshire SG19 2 DL, UK.
1. Spatial variation in breeding performance is of critical importance in understanding the large-scale distribution and abundance of living species, and in understanding species conservation. We studied the large-scale spatial variation in reproductive output of two species of declining British bird, the song thrush Turdus philomelos and the blackbird Turdus merula.
2. We developed a method to predict spatial variation in reproductive output. Brood size and nest failure rates during the incubation and nestling periods were related to environmental factors using generalized linear models. Predicted values obtained from these models were combined to give values of number of fledglings produced per nesting attempt for 10-km squares throughout Britain.
3. We observed substantial spatial variation in reproductive output for both species; the component that varied most was nest failure rate during incubation. We were more successful in relating environmental factors to spatial variation in reproductive output for song thrush than for blackbird.
4. Reproductive output in both species was affected mainly by factors that vary on a small spatial scale. Nest failure rate during incubation increased significantly where corvids were more abundant, suggesting a role for avian nest predators in determining spatial variation in reproductive output.
5. Our approach can be extended readily to other species of birds, to other taxonomic groups and to finer spatial scales. Such models could be used to evaluate the implications of current and proposed wider countryside management for spatial variation in breeding performance. Evaluations based on breeding success as well as numbers are likely to be more robust than those based solely on abundance.
A number of studies have highlighted the importance of spatial variation in reproductive performance in maintaining local bird populations, from both theoretical (Blondel 1985; Blondel et al. 1990; Hatchwell, Chamberlain & Perrins 1996; Holmes, Marra & Sherry 1996) and applied perspectives (Galbraith 1988; Green 1988, 1995, 1999; Green et al. 1997). In an applied context, once those factors responsible for variation in breeding success can be identified, and prove to be tractable, steps can be taken to improve the conservation status of the species (Caughley 1994; Caughley & Gunn 1995). Applied studies have, understandably, often concentrated on rare and/or declining species and it is unclear whether their findings apply to common and widespread species. The more theoretical studies have focused on how reproductive performance varies with habitat among common species and how this interaction might influence population dynamics. Both types of study have tended to consider relatively small populations at small spatial scales. At a broader scale, spatial patterns of variation in breeding performance and the factors influencing them are not well understood (Lambrechts et al. 1999).
Individuals belonging to the same species usually live in a range of environmental conditions that are likely to affect their breeding performance. Such spatial variation has consequences for population dynamics (Pulliam 1988; Pulliam & Danielson 1991; Watkinson & Sutherland 1995; Dias 1996); mathematical models with different assumptions about spatial variation in breeding performance make different predictions about population dynamics (Diffendorfer 1998). In this paper, we use the term ‘breeding performance’ as a general concept linking reproduction and population dynamics, while we refer to ‘reproductive output’ as a measure of expected individual fecundity (per breeding attempt, or per breeding season). We avoid the term ‘reproductive success’, which is usually used in the context of a lifetime contribution (Grafen 1988).
Bird populations provide a good empirical model to address this spatial variation because breeding performance and its components (brood size and nest success) are relatively easily measured in natural conditions. In order to construct predictive models of large-scale bird population dynamics, it is essential to know not only how breeding performance varies in space but also what are the causes of this variation. Environmental factors are likely to change through time, and it is possible to predict corresponding changes in breeding performance through time if we quantify the influence of relevant environmental factors. It is important here to distinguish between the factors that affect breeding performance, and the mechanisms through which these factors operate (Peters 1991). Two main categories of factors can potentially affect breeding performance in birds: resources and predation. These factors operate through many different mechanisms (Lack 1954; Newton 1989; Martin 1995). Limiting resources are likely to affect breeding performance through intra- or interspecific competition (Gosler 1990). Spatial variation in the phenology of resources may result in differences in laying date (Blondel 1985). Spatial variation in the types of resources may affect breeding performance with respect to possible plastic or genetic variation (Via et al. 1995). The effects of predation on breeding performance are expected to vary in more or less the same way as resources. Unfortunately the quantity and quality of resources is difficult to quantify, especially for generalist and relatively wide-ranging species such as many birds.
To understand large-scale ecological patterns it is necessary to evaluate how processes operate at different spatial scales (Caldow & Racey 2000; Ormerod & Watkinson 2000). At a small spatial scale, a particular environmental factor may have a predominant effect on variation in breeding performance, but this is not likely to be true at a large spatial scale where several environmental factors potentially affect breeding performance. Furthermore, these factors may have different patterns of spatial variation. As large-scale variations in breeding performance can be critical for population dynamics, it is important to assess the different effects of environmental factors on breeding performance. In this large-scale perspective, we have identified three questions that need to be addressed in order to develop reliable models of spatial population dynamics. (i) How do the components of reproductive output vary spatially and which vary most? (ii) What is the scale of variability of the factors that affect reproductive output? (iii) How important is predation for spatial variation in reproductive output?
In this paper, we report a study of large-scale spatial variation in reproductive output for two species of birds in Britain, the song thrush Turdus philomelos L. and the blackbird T. merula L. In spite of being very common in Britain, both species are of national conservation concern (Gibbons et al. 1996; Siriwardena et al. 1998) because of marked population declines. Their abundance on farmland has declined by 63% and 41%, respectively, between 1968 and 1995 (Baillie, Gregory & Siriwardena 1997). Our analyses are based on extensive national data collected by volunteer ornithologists through the Nest Record Scheme of the British Trust for Ornithology (Crick & Baillie 1996). Our specific objectives were to: (i) develop a statistical methodology to analyse the variations in the different components of reproductive output in birds; (ii) identify significant spatial correlates of reproductive output for song thrush and blackbird; and (iii) construct a model derived from the statistical analyses that predicts reproductive output with respect to the relevant environmental variables. In forming part of the NERC/SERAD special topic on large-scale ecology, this work complemented other ornithological contributions (Gaston et al. 2000; Pettifor et al. 2000), and added to other empirical model approaches (Collingham et al. 2000; Cowley et al. 2000).
The Nest Record Scheme started in 1939. Volunteers are asked to complete standard nest record cards for each nest they find in a given area, giving details of location, habitat, nest site, date and contents of the nest at each visit and evidence for success or failure. Single-visit nest record cards are discouraged because they cannot be used to estimate nesting success. We only used data from nests that were visited at least twice. For the present study, we restricted the time span of the nest record cards analysed to 1981–95 in order to avoid the confounding effect of substantial long-term changes in breeding performance (Crick et al. 1997; Siriwardena et al. 2000). Very large numbers of cards were available for the two species considered and it was not possible to computerize all of the available data. Samples of nest record cards were selected for computerization in a way that minimized biased representation from any one observer (i.e. location): cards for each year were selected randomly from those sent by each observer such that the number of cards from each observer was kept to the minimum value consistent with the sample size required. Many observers had contributed cards for song thrushes and blackbirds so any bias towards particular recording areas will have been very small. We considered the individual components of reproductive output separately (Grafen 1988). We estimated the mean number of fledglings per nesting attempt, which we called reproductive output per attempt. Given that nests were visited intermittently and that many were found after the start of laying, it was not possible to have a direct measure of the number of young fledged from each nest. Hence, we estimated reproductive output per attempt with the following equation:
ROA = BS(1 − DFI)IT(1 − DFN)NT(eqn 1)
where ROA is the reproductive output per attempt, BS is the brood size (maximum number of chicks known to have hatched), DFI and DFN are the daily nest failure rates during the incubation period and the nestling period, and IT and NT are the incubation period and the nestling period (both in days). Values for the last two parameters were taken from Cramp (1988). Although each nest observed during the nestling stage gave an estimate of brood size, a sample of nests was necessary to estimate daily failure rates. We used the Mayfield method, which takes into account the fact that nests failing at an early stage have a low probability of being found, resulting in an underestimated nest failure rate (Mayfield 1975). Each day during which a nest was observed is considered as a unit in the sample (= exposure day). Periods of observation were assigned to the egg or nestling periods as appropriate. Where a period of observation spanned hatching, the days were partitioned between the egg and nestling periods according to the estimated hatch date. Nest failure was assumed to occur half way between the penultimate observation and the date on which the failed nest was found. The daily nest failure rate was estimated as the ratio of total number of observed failures to the total number of exposure days (Mayfield 1975). This can be modelled as a binomial distribution for the outcomes of each exposure day (Johnson 1979; Hensler & Nichols 1981; Aebischer 1999). The probabilities of failure for each exposure day within the same nest are not independent and this may result in overdispersion. However, simulation studies have shown that the binomial approximation gives satisfactory results for a wide range of realistic situations (Johnson & Shaffer 1990).
We modelled the survival of whole nests rather than that of individual eggs and chicks because eggs and chicks within nests are not statistically independent, which would make some of the statistical procedures used here inappropriate. Partial losses are less important than whole nest losses in these species (O'Connor & Shrubb 1986). Partial losses of eggs and small chicks were incorporated into our estimate of reproductive output per attempt via our measure of brood size.
We aggregated the nest record cards into 100-km squares and computed the mean brood size and daily failure rates during the egg and nestling periods for each square with at least 25 nest record cards. These components were then used to estimate reproductive output per attempt. This gave 24 100-km squares for song thrush (mean cards per square 177, range 27–532) and 21 100-km squares for blackbird (mean cards per square 113, range 27–217). We computed the among-squares variances and coefficients of variation (CV) of brood size, and daily failure rates during the incubation and nestling periods, in order to investigate the spatial variation in components of reproductive output per attempt throughout Britain.
We used logistic regressions to investigate the effects of environmental variables on the components of reproductive output per attempt. Thus our approach is similar to the methods that are often used to model spatial variation in occurrence (Collingham et al. 2000; Cowley et al. 2000). We selected a set of variables (referred to as the predictors; Table 1) that we considered to have a potential effect on breeding performance in our study species. These predictors included: (i) habitat, as recorded by the observer and classified into five categories, woodland (referred to as habitat a), open (habitat b), farmland (habitat c), wetland (habitat d) and urban (habitat e); (ii) frequency of occurrence of individuals of the study species during the breeding season (which is closely correlated with actual abundance; Gibbons, Reid & Chapman 1993); (iii) combined frequency of occurrence of potential avian nest predators (corvids: jay Garrulus glandarius L. and magpie Pica pica L.); (iv) frequency of occurrence of sparrowhawk Accipiter nisus L., the most important potential predator of breeding adults; (v) mean spring temperature and rainfall; (vi) the percentage of agricultural land in the 10-km square where the nest was found; and (vii) landscape type classified in four categories (arable, pasture, upland and marginal upland). The last two predictors are measures of human activities that are likely to impact on the success of bird nests. Our approach to the selection of these predictors was to consider only those that were likely to affect reproductive output per attempt and its components, hence avoiding possible indirect or spurious relationships. We did not include carrion crows Corvus corone L. as potential nest predators as recent studies suggest that most predation is from smaller corvids, particularly magpies (Groom 1993; Hatchwell, Chamberlain & Perrins 1996). However, carrion crows were the main nest predators of blackbirds in the Oxford botanic gardens during the 1950s (Snow 1958).
Table 1. Predictors used in the logistic regressions
Geometric distance to all occupied 10-km squares (1988–91)
We were limited in our selection of predictors by the availability of suitable data. For example, we had no suitable data on the abundance of wild or domestic mammalian predators. We cannot exclude the possibility that other predictors, not considered here, may have a significant effect on spatial variation in breeding performance. Thus, we also included some geographical predictors: (i) latitude; (ii) longitude; (iii) altitude; (iv) large-scale region (region 1, north; region 2, south-west; region 3, south-east; Fig. 1); and (vi) the distance to the edge of range, defined as the geometric mean distance to all occupied 10-km squares (Blackburn et al. 1999); because we considered only the range in Britain this measure of distance to edge of range tends to be larger in the centre of Britain than near the coasts. Finally, we also included two temporal parameters: (i) year, to control for long-term trends in productivity (Crick et al. 1997); and (ii) laying date, to control for seasonal variations in clutch size (Crick, Gibbons & Magrath 1993). Before analysis, continuously varying predictors were log-transformed, or logit-transformed if they varied between 0 and 1 (logit(x) ≡ ln[x/(1 − x)]; Table 1). The categorical predictors were transformed into dummy variables as described by Agresti (1996). We did not include polynomial or interaction terms in these models because this would have given rise to increased problems of model selection and interpretation.
Brood size usually follows a strongly skewed distribution that makes it difficult to model with traditional regression approaches. Preliminary analyses showed that a normal regression of the log-transformed data or a Poisson regression with the raw data were not effective in selecting predictors. Instead, we used a cumulative logit model for ordinal response (Agresti 1996). The parameterization used in this modelling approach results in an inverted sign for the slope: a positive effect of a given predictor on brood size will result in a negative corresponding parameter (see the Appendix).
We used a stepwise forward selection procedure to select the relevant predictors. A null model (intercepts only) was first fitted to the data, then each predictor was included separately in the analysis: the null model was compared with each model with one predictor using a score test (which follows a χ2 distribution with d.f. = 1). The predictor with the most significant score test (i.e. the smallest P-value) was included in the model. This process was reiterated (comparing the one-predictor model with a two-predictor model, and so on) until no significant score test at the 0·05 level was observed.
The analyses of nest failure rates followed the same procedure as for brood size except that a simple logistic regression was used. Each exposure day was used as an independent observation, with success or failure as the possible outcomes. As noted above, the exposure days for the same nest were not independent. Such non-independence is likely to cause overdispersion in the data. Under these circumstances the estimators of daily failure rates during the egg and nestling periods will be unbiased but the estimates of their variances will be too small. If overdispersion is not taken into account, this may lead to spurious significance tests. Furthermore, other forms of non-independence, such as spatial autocorrelation (Legendre 1993), are likely to result in overdispersion. Hence, we corrected the score tests for overdispersion using a scale parameter (McCullagh & Nelder 1989). The selection procedure was the same as the one used for the analysis of brood size.
The parameter estimates from the logistic regressions were then used to compute the predicted reproductive output per attempt for each 10-km square in Britain where the species was present (using equation 1). The values of the coefficient of determination from the logistic regressions (R2) are not informative because they depend critically on the spread of the predictors and the magnitude of the effect(s) on the modelled probabilities (Cox & Wermuth 1992). Therefore, we assessed the predictive value of the present approach by computing the correlation between the values of predicted reproductive output per attempt (averaged on a 100-km basis) and the corresponding values of observed reproductive output per attempt. The model parameters and the observed values of reproductive output per attempt were estimated from the same data sets. Predicted values were calculated for all 10-km squares and not just for those where nests were recorded. All analyses were performed in SAS (1997).
Spatial variation in observed reproductive output
The components of reproductive output per attempt were estimated for 24 100-km squares for song thrush, and for 21 100-km squares for blackbird. In both species, the components of reproductive output per attempt showed considerable spatial variation (Table 2). For song thrush, brood size varied less than the other components of reproductive output per attempt: the CV of daily failure rate during the incubation period was about seven times that of brood size, and the CV of daily failure rate during the nestling period was about six times that of brood size (Table 2). A similar result was obtained for blackbird. There were some similarities in the values of CV of the components of reproductive output per attempt between the two species. This contrasted with the mean values of these components, which differed between the species, particularly failure rate during incubation. This resulted in a slight difference in overall reproductive output per attempt between the two species (Table 2). The general pattern of spatial variation in reproductive output per attempt looked very similar for both species: a general latitudinal gradient was apparent with greater output per attempt in northern Britain (Fig. 2a,b). Spatial variation in reproductive output per attempt did not follow this gradient tightly, however, and smaller scale patterns were also apparent.
Table 2. Mean and spatial variation in the observed (based on 100-km squares) and predicted (based on 10-km squares) components of reproductive output per attempt (ROA) and its components
BS, brood size; DFI, daily nest failure rate during the incubation period; DFN, daily nest failure rate during the nestling period; n = 24 for song thrush, n = 21 for blackbird; n = 2507 for song thrush, n = 2519 for blackbird, for observed and predicted, respectively.
Blackbird reproductive output per attempt had a narrower range of variation than that for song thrush, but the CVs for the two species were similar (Table 2). This suggested that reproductive output per attempt varied in a more patchy way for blackbird than for song thrush, although there was less absolute variation in the former species. Observed values of reproductive output per attempt were clumped around the centre of the range for song thrush, while for blackbird they were spread more evenly across the entire range (Fig. 2; see also Fig. 4 below).
Effects of environmental variables on reproductive output
There were great differences in the variables selected for the six sets of logistic regressions (2 species × 3 components). For all three components of reproductive output per attempt, more predictors were selected for song thrush than for blackbird (Table 3). In order to check if this could be related to the smaller sample sizes for the latter species, we re-ran the logistic regressions for song thrush with randomly selected subsamples of sizes equal to the corresponding samples for blackbird. The logistic regressions consistently selected more predictors for song thrush, and the results were similar to those presented in Table 3. Furthermore, we conducted analyses similar to those reported here for some species with less data than for blackbird (e.g. dunnock Prunella modularis L., magpie), and more predictors were selected for those species than for blackbird (E. Paradis, S. R. Baillie & W. J. Sutherland, unpublished data). This implied that the smaller numbers of selected predictors for blackbird were due to a lack of effect on the components of reproductive output per attempt.
Table 3. Results of the logistic regressions of the components of reproductive output
In all regressions but one (daily failure rate during the nestling period for blackbird), the first predictors selected had a strongly significant effect on the modelled component (P ≤ 0·0001). In both species, brood size was influenced mainly by predictors with large-scale variations such as latitude, longitude, altitude, distance to edge of range, or large-scale region (Table 3). Corvid frequency of occurrence had a weakly significant effect on brood size; for both species, mean brood size was higher where corvids are more abundant.
Predictors with a simple pattern of large-scale variation (e.g. latitude or longitude) were less important for failure rates than for brood size (Table 3). For song thrush, laying date was important for both the incubation and nestling periods: the later nests in the course of the breeding season were the most successful. Corvid frequency of occurrence was important for daily failure rate during incubation in both species: nest failure rate was higher where corvids were more abundant (Table 3). This effect was strongly significant for blackbird.
Predicting large-scale variations in reproductive output
The parameter estimates reported in Table 3, together with the values of the relevant predictors, were used to predict reproductive output of song thrush and blackbird for each 10-km square where each species was present (Fig. 3). The mean, SD, range and CV for these predicted values of reproductive output per attempt were in good agreement with those observed on a 100-km basis (Table 2). In order to assess the quality of these predicted values with respect to the corresponding observations, we computed for each 100-km square the mean of the predicted values of reproductive output per attempt. The squared correlation between these averaged predicted values and the observed values gives a measure of the spatial variation in reproductive output per attempt explained by our logistic regressions. The results were r2 = 0·401 and r2 = 0·293 for song thrush and blackbird, respectively. A plot of observed against predicted values revealed that, in the case of song thrush, prediction was best for high values of reproductive output per attempt, while for blackbird prediction was best for low values (Fig. 4). However, the value of the r2 for the blackbird was greatly influenced by a single point that corresponded to an area in south-west England (square SX of the Ordnance Survey). In this area, reproductive output per attempt was estimated to be 2·26 while the predicted value was 1·43. Deleting this point would have increased r2 to the value of 0·500.
Spatial variation in breeding performance is critical for population dynamics as different assumptions on how breeding performance varies in space lead to different predictions with respect to the dynamics of populations (Hanski 1997; Diffendorfer 1998). As long as spatial variation in breeding performance is small and space can be divided into suitable and unsuitable habitat, then metapopulation models can predict the spatial distribution of species (Hanski 1997). Birds, and certainly most vertebrates, do not conform to this restriction, and more general spatial population models are necessary for these organisms (Conroy et al. 1995; Dunning et al. 1995; Baillie et al. 2000; Rushton et al. 2000). The two main competing models for the spatial population dynamics of continuously distributed organisms such as birds, namely the source–sink and balanced dispersal models (Diffendorfer 1998), have different assumptions about spatial variation in breeding performance (Pulliam & Danielson 1991; Doncaster et al. 1997). It is important to know how breeding performance varies in space, but also to know what environmental factors affect breeding performance in order to predict the dynamics of a spatial population in a changing environment.
How do components of reproductive output vary spatially, and which vary most?
In order to compare the observed spatial variation in reproductive output with the corresponding temporal variation, we computed the mean reproductive output per attempt for each year (from 1981 to 1995) and each species for the whole of Britain (i.e. ignoring spatial variation); we then calculated the CV among years to assess temporal variation in reproductive output. The observed spatial variation in reproductive output (22·6% and 21·4% for song thrush and blackbird, respectively) is higher than the corresponding temporal variation (15·2% and 6·9%). The range of variation was greater for song thrush than for blackbird, but spatial heterogeneity of reproductive output within its range was greater for blackbird than for song thrush (see the x-axis on Fig. 4).
For both species the component of reproductive output that varied most through space was nest failure rate during the incubation period. Several recent studies have shown that increasing nest failure in some passerine populations may be enough to cause population declines in some areas (Robinson et al. 1995; Arcese, Smith & Hatch 1996; Siriwardena et al. 2000). There is further evidence that nest failure in some of these passerines may vary greatly spatially (Donovan et al. 1997). On the other hand, Beauchamp et al. (1996) showed that the decline in nest success of several species of ducks in North America occurs at a large spatial scale and does not vary between habitats. Martin & Clobert (1996) showed that nest success may vary at a continental scale, but they studied only interspecific variations, while the present study and those cited above show that nest success may vary greatly at the intraspecific level. Studies on other species of birds also suggest that nest success is the most important component of variation in breeding performance (Wiklund 1995).
What is the scale of variability of the factors that affect reproductive output?
Our models based on environmental predictors gave a better correlation between the predicted and observed values of reproductive output for song thrush than for blackbird. However, our predicted values are based on the mean of all 10-km squares where the species is present, while the observed nests are not distributed evenly within each 100-km square; as there is, within each 100-km square, some variation in habitat, altitude and land use, this may lead to discrepancies between the observed and predicted values. Thus our plots of predicted against observed reproductive output give a conservative assessment of the predictive power of the models.
The predictors that had the most significant effect on brood size vary on a large spatial scale (altitude and latitude for song thrush, region and distance to edge of range for blackbird). However, as mentioned previously, brood size varied only slightly overall. On the other hand, nest failure rates during the incubation and nestling periods were mostly affected by predictors that vary on a small spatial scale (e.g. frequency of occurrence, corvid frequency of occurrence, agricultural land). However, these predictors also show large-scale patterns of variation, for instance frequencies of occurrence of song thrush, blackbird and corvids follow a general latitudinal gradient in Britain (Gibbons, Reid & Chapman 1993). This results in similar large-scale patterns of variation in predicted reproductive output (Fig. 3). Overall, however, patchy variation in reproductive output is predicted. These large-scale patterns of reproductive output are thus determined by factors operating at different spatial scales, as is frequently found in large-scale ecological studies (Caldow & Racey 2000).
How important is predation for spatial variation in reproductive output?
Several recent studies of bird populations have shown that nest predation can greatly affect local dynamics (Groom 1993; Robinson et al. 1995; Kurki et al. 1997; Rogers et al. 1997). Andrén (1992) showed that corvid nest predators may have a significant impact on nest success of passerines in relation to landscape structure. Groom (1993) showed that nest predation attributable to magpies in urban parkland had a large impact upon the local blackbird population, and speculated that it could be sustained only by immigration. Our study shows that corvid nest predators may have a significant effect on song thrush and blackbird reproductive output more generally, and that this is due to an increased nest failure rate during incubation where corvids are abundant. Whether nest predation by corvids is sufficient to create ‘sink populations’ in some areas remains to be proved because this depends on the survival of juveniles and adults in these and other areas. The numbers of most British corvids have increased in recent decades (Gregory & Marchant 1996), but previous work has shown that the concomitant declines in songbird populations have not been driven by corvid predation (Gooch, Baillie & Birkhead 1991; Thomson et al. 1998). So while nest predators may have an impact upon particular stages in the life cycle in particular areas, they do not appear to influence the overall population level.
An intriguing result was that, in both species, corvid frequency of occurrence had a positive effect on brood size. For both species, brood size was higher where corvids were more abundant. Since corvid frequency of occurrence increases the failure rate during the incubation period, a possibility is that song thrushes and blackbirds may lay more eggs in areas of higher nest failure (see McCleery et al. 1996 for a similar trade-off). In contrast to this result from the logistic regressions, the direct correlations between brood size and corvid frequency of occurrence were negative: r = −0·120, n = 1429, P = 0·0001, and r = −0·022, n = 958, P = 0·503, for song thrush and blackbird, respectively. An explanation may be that the effect of predation is masked in the raw correlation because brood size increases with latitude (because of the larger clutch sizes at higher latitudes), and corvid frequency of occurrence decreases with latitude (Gibbons, Reid & Chapman 1993). However, our explanation of the interrelationships between brood size, breeding performance and corvids is only tentative, and would require further investigation.
Conclusion and perspectives
This paper presents an approach to the analysis of large-scale data on reproductive output of birds in order to quantify spatial variation in reproductive output, relate these variations to environmental factors, and predict spatial variation in reproductive output using these relationships. We believe this approach can be extended to other species of birds, and even to other taxonomic groups where reproductive output can be measured in terms of success or failure and/or number of offspring (note that other forms of regression can be used instead of the ordinal logistic regression depending on the distribution of the number of offspring). Our approach has potential conservation applications, for instance to target and guide conservation effort for declining species, to predict changes in reproductive output in relation to hypothetical large-scale changes in environmental conditions, and to identify those factors that might influence reproductive performance. Thus large-scale spatially explicit approaches are required to address the conservation of widespread and migratory bird populations (Pettifor et al. 2000) in common with many other ecological issues (Ormerod & Watkinson 2000).
However, we emphasize that the kind of study presented here should be integrated with other elements, such as the number of broods per year and survival rates, in order to give a full picture of spatial variation in population dynamics. The development of such models is the subject of a separate paper (Baillie et al. 2000).
We are grateful to the many volunteers who contributed to the BTO Nest Record Scheme, and to the Institute of Terrestrial Ecology for providing the data on land cover and land characteristics. The maps were produced using dmap software written by Alan Morton. Special thanks to David Thomson for suggesting the use of the ordinal logistic regression, to Gavin Siriwardena for helpful advice with SAS, and to two anonymous referees for their comments on the manuscript. Financial support for the present study was provided by NERC grant GST/02/1197 as part of the NERC/SERAD Thematic Programme on Large-Scale Processes in Ecology and Hydrology Thematic Programme. The Nest Record Scheme is funded by a partnership of the British Trust for Ornithology and the Joint Nature Conservation Committee (on behalf of English Nature, Scottish Natural Heritage and the Countryside Council for Wales, and the Environment and Heritage Service in Northern Ireland).
Received 22 March 1999; revision received 21 April 2000
Interpreting the ordinal logistic regression
The ordinal logistic regression was originally proposed by McCullagh (1980) to analyse data where the response variable can take only a restricted number of values but there is a natural ordering of these values. This approach is appropriate for the analysis of clutch size or brood size where the number of eggs or chicks is limited, and there is a natural ordering related to the quantity. Specifically, the ordinal logistic regression models the cumulative probabilities that are defined by Pr(BS ≤ j), i.e. the probability to have a brood size (BS) of no more than j. If the maximum brood size is M, then M − 1 such probabilities are defined. These probabilities are then modelled simultaneously in a logistic framework with the only assumption that they are all affected in the same way by the predictors (but with different intercepts). This leads to the interpretation that a positive effect of a predictor in such a logistic regression means that this predictor has a negative effect on brood size because Pr(BS ≤ 1), the probability to have no more than one chick, will tend to unity (Fig. 5). The advantages of this approach are: (i) the natural ordering of the modelled variable (here brood size) is conserved in the analyses; (ii) no assumption are made on the variance structure in the data; and (iii) the number of categories is assumed to be limited to what is observed (which is not the case with a Poisson regression where the modelled variable is assumed to vary between 0 and +∝). See Thomson, Furness & Monaghan (1998) for a detailed comparison of different approaches.