1. The relative importance of density-dependent and -independent factors on interannual variation in over-winter survival was investigated in the fluctuating population of Soay sheep on St Kilda, Scotland, over the period 1985–96.
2. Population density had a negative effect on survival in lambs and adult males while adult female survival showed no evidence of density dependence over the observed range of population densities.
3. Climatic fluctuations associated with the winter North Atlantic oscillation index (NAO) also affected survival, which decreased in winters that were relatively warm, wet and windy. The effect was most pronounced in lambs.
4. Survival was modelled using logistic regression analysis with and without year fitted as a random effect. The former incorporated stochastic year to year variation in survival. Results from the two modelling approaches were similar in terms of the regression coefficients estimated. However, the standard errors of the year-dependent covariates, population size and NAO, were underestimated when the random year effect was ignored, leading to incorrect inferences about the relative significance of terms being made.
5. Using both modelling approaches, density dependence was found to have a greater influence on survival than the effect of NAO in lambs and adult males, whereas in adult females NAO was the more important.
6. Once random between-year effects were taken into account, the individually varying terms such as body weight and faecal egg count were the most significant factors explaining differences in survival.
There has been a recent increase in interest in the influence of density-independent factors and environmental stochasticity on survival and population dynamics (Leirs et al. 1997; Sæther 1997; Gaillard, Festa-Bianchet & Yoccoz 1998; Grenfell et al. 1998). A growing body of evidence shows that ecological processes are affected by climatic fluctuations (Grant & Grant 1989; Forchhammer, Post & Stenseth 1998; Post & Stenseth 1998). This has implications for the debate concerning the relative importance of intrinsic (density-dependent) and extrinsic (density-independent) factors on population changes. Sæther (1997) suggested that, in the absence of predation, the population dynamics of ungulates were determined by a combination of both density-dependent and stochastic environmental effects, operating through changes in survival and fecundity rates. An example is the density-independent cohort effects of spring temperatures on birth weight of red deer (Cervus elaphus L.) which were found to significantly influence over-winter survival rates (Albon, Clutton-Brock & Guinness 1987; Rose, Clutton-Brock & Guinness 1998), while the effects of birth weight on survival were intensified at high densities (Clutton-Brock et al. 1987). Post & Stenseth (1998) have shown that growth in the moose (Alces alces L.) population on Isle Royale, USA, and increases in white-tailed deer (Odocoileus virginianus Zimmerman) abundance in Superior National Forest, USA, were both influenced by delayed density-dependent feedback, as well as global climatic fluctuations and predation by wolves (Canis lupus L.).
Measures of age- or sex-specific survival rates have, until recently, been based on transversal life-table methods, with little knowledge of their reliability (Gaillard et al. 1993). However, several long-term studies of ungulate populations with individually known animals (reviewed by Sæther 1997; Gaillard et al. 1998) have enabled accurate estimates of vital rates to be made, partly due to advances in the modelling of capture–mark–recapture data (Lebreton, Pradel & Clobert 1993). Jorgenson et al. (1997) remarked that the degree to which survival rates varied between years remained largely unknown. Since then, Gaillard et al. (1998) have demonstrated that across 16 species of large herbivore the coefficients of variation in survival between-years varied little in prime-aged females (from 2 to 15%) but were very variable in juveniles (from 12 to 88%). Furthermore, it has been shown that juvenile survival is more sensitive to both density dependence and, in particular, to seasonal food availability and therefore stochastic variation, than adult survival (Sinclair 1977; Gaillard et al. 1998). We conducted our analyses separately on lambs, adult females and adult males to investigate the different susceptibilities of these components of the population.
It has previously been demonstrated that over-winter survival of the Soay sheep (Ovis aries L.) on St Kilda was strongly density-dependent (Clutton-Brock et al. 1991; Grenfell et al. 1992), but in recent high population years anticipated crashes have not occurred, especially within the adult population. It has become apparent that at high population sizes the system is particularly sensitive to a combination of density-dependent and -independent factors. Consequently populations above a certain threshold can increase, decrease or remain constant in size depending on the extrinsic environmental conditions (Grenfell et al. 1998). It would generally be expected that the effects of weather on population dynamics should become more evident as a system approaches the ecological carrying capacity (Sinclair 1989). In support of this, it has been shown that population growth rates or survival were more variable at high density, when density-independent effects were stronger, in populations of red deer on Rum (Benton, Grant & Clutton-Brock 1995), bighorn sheep (Ovis canadensis Shaw) in the Canadian Rocky mountains (Portier et al. 1998) and in Dall's sheep (Ovis dalli Nelson) in Alaska (Bowyer, Leslie & Rachlow, in press).
Grenfell et al. (1998) found that the effects of March gales and April temperatures had a greater influence on Soay sheep survival above a threshold density than at low density. Here, we have investigated the influence of a stochastic environmental variable, climatic fluctuation, on over-winter survival. In northern Europe, fluctuations in winter climate are strongly correlated with interannual variations in the atmospheric circulation over the North Atlantic (Wilby, O’Hare & Barnsley 1997). An annual index of this North Atlantic oscillation (NAO) can be measured by the difference in normalized sea level pressures between Lisbon, Portugal and Stykkisholmur, Iceland, between December and March (Hurrell 1995). In the British Isles, high positive values are associated with warm, wet winters with strong westerly winds, whereas low negative values indicate cold, dry winters (Wilby et al. 1997). Variations in the abundance of zooplankton species have already been linked with fluctuations in NAO, with the implication that the NAO may play a comparable role to the El Niño southern oscillation in pelagic ecosystems (Fromentin & Planque 1996). In terrestrial systems, breeding phenologies of a number of species of birds and amphibians have been shown to be well correlated with fluctuations in NAO (Forchhammer et al. 1998a). Furthermore, direct and delayed effects of the NAO on population dynamics of red deer have been found in Norwegian populations (Post et al. 1997; Forchhammer et al. 1998b), and in moose and white-tailed deer populations in the United States (Post & Stenseth 1998).
Most of the previous analyses of survival of Soay sheep on St Kilda have used logistic regression analysis (Clutton-Brock et al. 1992; Bancroft et al. 1995; Illius et al. 1995; Clutton-Brock et al. 1996; Moorcroft et al. 1996). This has advantages over other techniques such as the analysis of one-way contingency tables in that several factors can be controlled for simultaneously. However, conventional logistic regression, as applied through generalized linear modelling, does not allow a distinction to be drawn between fixed and random effects and cannot take account of more than one source of variation in the data. Here we investigate the effects of density-dependent and stochastic year-to-year variation in the survival of the Soay sheep, using an extension of logistic regression from the more statistically sophisticated class of generalized linear mixed models (GLMMs). This allowed random year effects to be fitted within the framework of conventional logistic regression. We assessed the appropriateness of conventional logistic regression, without random effects, for analysing survival data across years, by comparing the results with those obtained using mixed modelling.
Materials and methods
The population dynamics of the Soay sheep on Hirta, St Kilda, have been monitored intensively from 1959 to 1968 (Jewell, Milner & Boyd 1974) and from 1985 to the present (Clutton-Brock et al. 1991; unpublished data). During this time the population has fluctuated between 600 and 2000 individuals, with dramatic crashes occurring in the winter of years when high population density was combined with extreme weather conditions, particularly March gales (Grenfell et al. 1998). In crash years 50–70% of animals died of starvation (Grubb 1974; Clutton-Brock et al. 1991; Grenfell et al. 1992), exacerbated by high gastrointestinal parasite burdens (Gulland 1992), particularly in lambs and yearlings. In post-crash years of low population density, over-winter survival was high (greater than 90%) and this, coupled with high fecundity (Clutton-Brock et al. 1992), led to a rapid increase in population size.
Life-histories of tagged individuals in the Village Bay area (175 ha) of the main island, Hirta, have been monitored since 1985 by regular censusing at three times of year and daily mortality searches from February to April (Clutton-Brock et al. 1991). An individual was considered to have survived a winter if it was known to be alive on 15 May the following year. Animals that disappeared and were not resighted within 3 years were assumed to have died in the winter after which they were last sighted. Animals that have disappeared since 1994, and have not been resighted have been excluded from the analysis. Years ran from spring to spring such that the winter of 1985 covered the period from autumn 1985 until spring 1986.
The Village Bay population size was taken as the number of sheep using the study area at the start of winter and was estimated from census data (Fig. 1). This number correlated well with the total island sheep population, representing approximately one-third of it (Clutton-Brock et al. 1991).
Climatic conditions, in particular the severity of March gales, April temperatures (Grenfell et al. 1998) and March rainfall (E.A. Catchpole et al., in press), and variations in plant productivity (Forchhammer et al. 1998b; M.J. Crawley, unpublished data) were important extrinsic factors that could influence the scale of over-winter survival. The North Atlantic oscillation index (Climate Analysis Section, http://http://www.cgd.ucar.edu:80/cas/climind/) provided a single variable to encapsulate between-year differences in a number of weather variables such as temperature, wind speed and direction and precipitation at the time of year when most mortality occurred. Winter NAO indices covered the period from December to March so were used with data collected during the preceding summer and the prewinter population size. For example, the NAO index for 1985/86 was matched with the sheep population size and individual variables from 1985. Other weather data came from the Meteorological Office's station on the island of Benbecula (Outer Hebrides) but unfortunately this station was closed at the end of 1995 so subsequent data were unavailable. Correlations between these variables are shown in Table 1. Once NAO was fitted in the models, the effects of March gale days and rainfall were not significant. Despite a visual impression of association between population size and NAO (Fig. 1) these two variables were not significantly correlated (r = 0·106, P > 0·6), allowing the estimates of their effects on survival to be nearly independent.
Table 1. Correlations between weather variables and population size on St Kilda. Population and NAO data were available over the period 1985/86–1996/97, but correlations involving the other weather variables were restricted to the period 1985/86–1994/95. Population size was the number of sheep in the Village Bay area entering the winter and NAO indices covered the same winter period. The weather variables were collected during the period of winter mortality and were correlated with the prewinter population size, i.e. in the previous calendar year
Population size and NAO were common to all sheep but varied from year to year. Together with catch date (see below), which varied more between years than within years, these variables will be considered as year-dependent covariates. For all other variables the main source of variation was considered to be between individuals within years.
A number of morphometric measurements including body weight, hindleg length and incisor arcade breadth have been recorded for all animals caught in August roundups. Body weight, the live mass measured to the nearest 0·1 kg, was the measure of body size found to have the best explanatory power for survival and therefore was used in the analysis presented here. Gutfill and wetness of fleece could not be controlled for, thus contributed to the error of this measure (Illius et al. 1995). In addition, the date of capture and measurement influenced body weight because of daily weight gain throughout the summer (Milner et al. 1999) and this was controlled for, to some extent, by including catch date as an explanatory variable.
The number of strongyle eggs/g of faeces, recorded since 1988, was used as an index of gastrointestinal parasite burden (Gulland & Fox 1992). Models for lamb and adult female survival, which included faecal egg count, were restricted to the period 1988–96, whereas adult male models without faecal egg count ran from 1985 to 1996 to maximize the use of available data. In the years 1988–91 and 1995 some of the individuals caught were dosed with anthelminthic treatments to combat intestinal parasites. Subsequent records for these sheep have been excluded from all analyses as their survival was influenced by the treatment (Gulland et al. 1993).
Survival of lambs (individuals less than 1 year old) over their first winter was analysed separately from that of yearlings (individuals aged 12–23 months) and adults (over 2 years old) because of the particular susceptibility to mortality of juvenile animals (Clutton-Brock et al. 1992). Differences in survival between yearlings and adults were less marked and it was found that the use of two age classes in adult females, prime (yearlings and 2–6-year-olds) and old (7 years and over) individuals was the most parsimonious way of explaining variation due to age. Adult males rarely lived to 7 years so insufficient data were available to test for a decline in survivorship with old age. Although survival of yearlings was lower than that of adults over 2 years, differences were not significant. Consequently all adult and yearling males were grouped in a single age category. Data from adult males and females were analysed separately because of the differential survival of the sexes, brought about by male rutting activity (Stevenson & Bancroft 1995). There were a number of marked animals of unknown birth year, and hence age, in the study population. These individuals have been excluded from the analysis.
Logistic regression models
Over-winter survival data were analysed using logistic regression models (Cox & Snell 1989) to ascertain the relationship between survival, density, climatic fluctuations and individual attributes. Two modelling approaches were contrasted, conventional logistic regression with fixed effects only and logistic regression with both fixed and random effects. Both approaches are types of generalized linear models but the latter belongs to the specific class of generalized linear mixed models (GLMMs). The additional random effects in mixed models allow for the analysis of stratified data with more than one error term. By including a random error term, for example a year effect, one automatically introduces a dependency between individuals in each year. This parallels the way in which a random plot factor accounts for the dependency of subplots within a plot in classical split-plot anova analysis, allowing valid inference when treatments are applied at both the plot and subplot level.
In both types of logistic model, a probability curve was fitted through the binomially distributed survival data (0 died, 1 survived) using the logit link function.
where P(yij = 1) was the probability that an individual i would survive winter j, a was a constant and b1…bm were coefficients of the independent variables x1…xm. The random effects for individual and year, Ai and Tj, respectively, were only included in the mixed models. They were assumed to be uncorrelated with each other and drawn from normal distributions with zero means and variances and . If Ai and Tj were zero, or equally and were zero, the model would be equivalent to conventional logistic regression. GLMMs can be fitted in several statistical packages by combining algorithms that fit generalized linear models with algorithms for fitting linear mixed models (Schall 1991). We used Genstat 5, release 3·2 (Genstat 5 Committee 1993; Welham 1995) for all analyses.
Conventional logistic regression was used to select the best-fitting fixed effects models. All continuous variables were centred about the mean values. Model parameters were estimated by maximum likelihood (McCullagh & Nelder 1989) and significance was measured by the change in deviance that occurred when a term was dropped from the maximal model. Terms were dropped sequentially and refitted if significant, until the model included only significant terms. Change in deviance was assumed to follow a χ2 distribution, with degrees of freedom equal to the change in the residual degrees of freedom.
The fitting procedure for models including random effects proved to be slow and there were some problems with convergence, so for the mixed models random terms were simply added to the best-fitting fixed effects models. The significance of terms in GLMMs was assessed by the Wald statistic at the final iteration of the algorithm, for each term, when fitted last in the model. The significance of terms involved in interactions was determined from a model of the main effects. Wald statistics were assumed to be distributed as χ2, except for year-dependent covariates for which Wald statistics were tested against F1,r where r was the residual degrees of freedom of years. A comparison was drawn between parameter estimates, and their corresponding standard errors, made using the two modelling approaches.
In all mixed models, year was fitted as a random effect to account for stochastic between-year variation in survival. When modelling lamb survival each individual appeared once only in the data set. However, when modelling adult survival, there were multiple appearances in the data set of a small proportion of individuals that had both survived more than one winter as adults and been caught on more than one occasion and consequently had repeated records. The influence of repeated records was therefore investigated by fitting the identity of the individual as a second random effect in adult survival models. However, the variance component for individual was small and not significant in either sex, so this was discarded. Whether this arose due to particular features of our data set or was a general result was not clear and will be discussed later.
To avoid the potential problems of non-independence due to repeated records, previous logistic regression analyses of the Soay sheep survival data have (i) been restricted to animals that have never experienced a crash (Bancroft 1993), (ii) conducted age- or crash-specific analyses (Clutton-Brock et al. 1992; Stevenson 1994; Illius et al. 1995) or (iii) excluded multiple data records (Stevenson 1994). In the analysis conducted here all repeated records were used since no significant individual effect had been found using the mixed model approach.
Random between-year variation
When assessing the significance of year-dependent covariates, conventional logistic regression analysis was unable to make a direct allowance for stochastic variation in survival between years because no random effects could be included. A correction for this was therefore made by multiplying the standard error of the regression coefficients for these covariates (population size, NAO and catch date) by the square root of the mean deviance due to the year effect (McCullagh & Nelder 1989).
The relative significance of the year-dependent covariates was investigated by comparing the variation in survival that they explained with that due to the residual between-year variation. The comparison was made using F-ratios of the mean deviance due to the year-dependent covariate when fitted second-last with the mean deviance due to the year effect when fitted last.
Factors affecting survival
Over-winter survival probability declined sharply with increasing population size and index of the North Atlantic oscillation (Table 2a). The NAO effect indicated that survival was reduced in warm, wet winters with strong westerly winds. The relationship between predicted survival and population size fitted the observed points for each year well (Fig. 2a). The relationship between survival and NAO fitted the observed points less well, particularly in 1989 when the observed survival was very high and much greater than would have been predicted from the NAO index in that year (Fig. 2b). However, that year followed the crash of 1988/89 and the population density was sufficiently low that a high survival rate was possible regardless of NAO.
Table 2. Comparison of parameter estimates for the best fitting over-winter survival models for (a) lambs (b) adult female and (c) adult male Soay sheep, calculated using logistic regression analysis with (GLMM) and without (GLM) random effects. The significance of terms was assessed by Wald statistics for GLMMs (distributed as χ2 for individual covariates and as F1,r for year-dependent covariates, where r was the residual degrees of freedom of years) and change in deviance for GLMs (distributed as χ2), when each term was fitted last in the model. Body weight and faecal egg count were transformed by natural logarithms and all continuous variables were centred about their mean . Interactions are denoted by ‘.’ between terms. Adjusted standard errors (SE) allowed for a between-year effect of year-dependent variables and F ratios compared the mean deviance of year-dependent variables with the mean deviance due to the residual between-year effect (see text). Degrees of freedom were 1 throughout. (a) Lamb over-winter survival model for the period 1988–96 (n = 442). Estimated variance component for year effect = 0·550 (SE = 0·480) in GLMM.
(b) Survival models for adult females between 1988 and 1996 (n = 676). Estimated variance component for year effect = 0·894 (SE = 0·786) in GLMM. Age was added as a two-level factor of prime (1–6 years) and old (≥7 years) individuals.
In addition, the probability of survival in lambs increased with body weight while parasite burden had a significant negative effect. Female lambs had higher survival rates than males (Table 2a). These terms had the most significant effect on lamb over-winter survival when random between-year variation was taken into consideration.
There were no significant interactions between NAO and any of the individual variables. However, there was a significant positive interaction between population size and body weight, resulting in an increased probability of survival of heavy individuals at high population density and a decline in survival of heavy lambs relative to light ones at low densities. This interaction could be illustrated by a survival surface predicted from the conventional logistic regression model (Fig. 3). The most important trend to note was the very much steeper decline in survival probability of light individuals relative to heavy ones as population size increased. Observed survival patterns from within the population generally supported the trends of the survival surface, including the tendency for large lambs to have lower survival than small lambs at low population densities (Table 3). However, this particular result should be interpreted cautiously because sample sizes from low population years were small, especially of heavy lambs born in years following severe winter mortality. Below a population size of 300, the model was based on only 37 data points, collected during 1989 and 1990.
Table 3. Observed number of lambs surviving the winter, expressed as a fraction of the total number of individuals of that weight class entering the winter at different population sizes. The proportion of survivors is given in parentheses. Body weights have been corrected for catch date but no allowance made for other factors
In adult females, the effect of NAO on survival was stronger than that of population size but neither had as pronounced an effect on survival as in lambs (Fig. 2c). The effect of increasing NAO on survival was again negative, indicating that warm, wet winters with strong westerly winds reduced survival but this was only significant at the 10% level. The weak and non-significant effect of density on adult female survival arose because there has been no major mortality event in the adult female component of the population since 1988/89. As a result there has been a considerable increase in the adult female population size throughout the 1990s.
The fit of the observed survival against the NAO in each year was good except for 1988, when survival was much lower than was predicted from its NAO index (Fig. 2d). There was a high population density in that year and a particularly large crash occurred. However, there was less disparity between the observed survival in 1988 and that predicted for its population size (Fig. 2c).
Body weight and age class were the variables with the greatest effect on over-winter survival in adult females (Table 2b). Again, survival chances improved with increasing body weight but there was no significant interaction between body weight and population size. Old age (≥ 7 years) was associated with a decrease in survival compared with individuals in their prime. A negative relationship between survival and faecal egg count also existed in adult females.
Adult male survival over the period from 1985 to 1996 was influenced by a strong negative effect of population size (Table 2c; Fig. 2e). It appeared to be more strongly density-dependent than adult female survival, despite both sexes having similar regression coefficients. While the difference between the sexes was not statistically significant, an apparent difference in density dependence arose because each sex was sampled from a different part of the range of the non-linear relationship between survival and population size.
Surprisingly, there was no significant effect of NAO on adult male survival. This was due to the inclusion of the winter of 1985/86 when mortality was high despite a moderate NAO (Fig. 2f). Body weight was the only other significant variable (Table 2c). Faecal egg count was not significant and there was no apparent age effect, presumably because high mortality rates meant very few males lived to old age. Again the interaction between population size and body weight observed in lambs was absent from the adult model.
Comparison between modelling approaches
There was very close agreement between parameter estimates determined by the two types of analysis in all age–sex classes (Table 2). There was a slight disagreement in the constants of the adult female survival models, with the GLMM estimate being lower. The discrepancy arose because, in calculating the overall mean survival rate, conventional logistic regression gave equal weight to each observation, so weighting the years according to the number of records. Conversely, the mixed model weighted the years in a more equal manner.
More seriously, a discrepancy also arose in the standard errors of the year-dependent covariates, population size, NAO and catch date, due to the random year effect being ignored in the conventional logistic regression model. However, once the standard errors of these terms were adjusted, agreement was better for population size and NAO, although worse for catch date which varied within years as well as between years. Without adjustment, the under-estimation of standard errors of the year-dependent covariates by conventional logistic regression has important consequences for the significance of terms and could lead to incorrect inferences being drawn from the data. This was illustrated very clearly by the differences in significance as assessed by the two modelling approaches (Table 2).
In lambs, conventional logistic regression identified population size and NAO as having the greatest effect on over-winter survival yet neither appeared among the most important terms when the mixed model approach was used. The individually varying parameters, body weight, faecal egg count and sex were in fact the most significant terms. Furthermore, conventional logistic regression indicated that catch date and a population–sex interaction were significant whereas the former was only significant at the 10% level and the interaction was not significant at all when modelled by GLMM. Similarly in adults catch date, which was significant in the logistic regression models without random effects, was not significant in the GLMM when year was fitted as a random effect (Table 2). The significance of NAO and population size were also lower in the GLMMs. In the adult female model population size, which was significant in the conventional logistic regression model, was not significant once the random year effect was taken into consideration (Table 2b). The apparently significant effect of density on adult female survival, revealed by conventional logistic regression, was therefore an artefact of the analysis technique, which failed to take account of random year-to-year variation.
The differences in the relative importance of terms between the two modelling approaches could be explained by the inclusion of some random between-year variation within the year-dependent covariates. This was demonstrated by examining the contribution of these covariates towards the total between-year variation using F-ratios (Table 2). The significance of these ratios were more in keeping with the significance of year-dependent covariates in the GLMMs. In lambs, both population size and NAO contributed significantly towards the total between-year variation. By contrast, catch date was only significant at the 10% level when compared with the residual between-year variation (Table 2a). In adult females, although NAO made a significant contribution towards the total between-year variation, stochastic between-year effects were more important in explaining variation than the effect of population size or catch date (Table 2b). In adult males, the effect of population size and catch date were only significant at the 10% level when compared with the residual between-year variation (Table 2c). This emphasized the considerable amount of year-to-year variation in over-winter survival, of which density dependence, NAO and catch date were only three components.
Our analyses have shown that both density-dependent and density-independent factors influenced survival of different components of the population. Lambs were strongly affected by both factors, with density dependence being the stronger of the two. In adults, survival of males was also density-dependent but to a lesser extent than in lambs, whereas over the range of population densities observed, female survival was not significantly affected by density dependence. Adult survival was also less strongly affected by density-independent factors than lamb survival. Juvenile survival is generally lower and more variable than adult survival and tends to be more sensitive to resource availability and changes in weather (Sinclair 1977; Fowler 1987; Owen-Smith 1990; Gaillard et al. 1998). Our results therefore corroborate these earlier findings.
Previous analyses of survival on St Kilda have found that both adult male and female survival were density-dependent (e.g. Clutton-Brock et al. 1991; Clutton-Brock et al. 1997). While there was no statistical evidence for a difference in density dependence between the sexes, there was also no significant effect of density on adult female survival over the range of population densities encountered during the course of this study, once stochastic between-year variation was accounted for. The difference between this and previous results could be explained in terms of the different statistical techniques used. However, the result has probably also been exaggerated by the longer run of data now available, in particular including the winter of 1996/97 when, despite the largest ever population density, adult female survival was high. This therefore provides some new evidence that on St Kilda adult female survival may, after all, be in keeping with the general principle of adult survival being buffered against density effects. Of nine long-term ungulate population studies reviewed by Sæther (1997) and a further four reviewed by Gaillard et al. (1998), the Soay sheep and red deer on Rum were the only populations to show a density-dependent decline in adult survival. Both are island populations, living close to their carrying capacities and are frequently resource-limited.
The influence of the North Atlantic oscillation on the survival of Soay sheep was that warm, wet and windy winters were negatively associated with survival in lambs and adult females. The lack of influence of NAO on adult male survival was surprising. If, however, the analysis of adult male survival was restricted to the period 1986–96, the effect of NAO became highly significant (χ2 = 21·3, d.f. = 1, P < 0·001; Fig. 2f). Furthermore, as in the case of adult female survival, NAO was then more important than population size (χ2 = 19·6, d.f. = 1, P < 0·001). This switch appeared to be due to the exclusion of the high density-dependent winter mortality of 1985/86, despite a NAO index suitable for moderately good survival (Fig. 2f).
Although it might initially be surprising that lower survival was not associated with cold winters, the negative effect of mild, wet and windy weather could be accounted for by a decrease in time spent foraging while animals sheltered from gales and hailstorms (Stevenson 1994). In red deer populations in Norway, winters with high positive NAO indices were also associated with decreases in apparent abundance (Forchhammer et al. 1998b). However, as well as the direct negative effect on survival observed, Forchhammer et al. suggested that NAO also had a delayed positive effect on the population size 2 years later, operating through female fecundity which was enhanced by improved plant growth and female body condition. The influence of NAO on plant productivity on St Kilda has yet to be examined but it seems likely that the growing season may start earlier in high NAO winters and, while improved forage may come too late for individuals already at the end of their energy reserves, may benefit survivors. A fuller analysis of the interactions between climate, plant productivity and population dynamics is now being carried out.
We might have expected to find an interaction between population size and NAO, since environmental fluctuations tend to have greater effects at high population density (Bowyer et al., in press). However, no significant interaction was found. This was also the case for over-winter survival in lambs of bighorn sheep, which was affected both by density and weather the previous spring, but the effects of weather were not mediated by changes in population density (Portier et al. 1998). By contrast, in the same study, the effects of winter and spring temperature on neonatal survival were found to interact with density, being more important in high density years. The analysis conducted by Portier et al. used conventional logistic regression and took no account of stochastic year-to-year variation in survival. This raises the question of whether density and weather effects would remain significant if a random year effect was included.
The individually varying parameters, body weight and faecal egg count were found to be the most important factors affecting survival. This was perhaps not surprising, since there was a lot of individual variability in attributes between years. Body weight was the primary factor influencing survival in all three age–sex classes studied. Although its influence on survival has long been acknowledged (Peters 1983; Calder 1984), especially in juveniles (Clutton-Brock et al. 1987; Clutton-Brock et al. 1992; Sedinger, Flint, & Lindberg 1995), relatively little attention has been paid to the changing effects of body weight on survival of mature animals (Festa-Bianchet et al. 1997). In bighorn sheep the probability of survival was found to increase with body weight in lambs but among adults significant effects were found only in old females (Festa-Bianchet et al. 1997). However, it should be noted that mortality of bighorns was largely due to predation, accident or disease and unlike the Soay sheep population, no evidence was found of starvation. Consequently, we would not expect body weight to play such an important role in survival. By contrast, under the conditions of extreme mortality due to resource shortages in some years on St Kilda (Grubb 1974; Clutton-Brock et al. 1991), heavier individuals with larger energy reserves were at an advantage. We would expect this effect to be most pronounced at high densities when competition for resource is at its greatest (Lindstedt & Boyce 1985). As anticipated, a significant interaction between population size and body weight was found in lamb survival, although not in the adult population.
Our analyses have shown good agreement between parameter estimates made using logistic regression models of over-winter survival, with and without random effects. Since, from the population dynamics perspective, the magnitude of the coefficients and range of variation of the covariates are the important results, broadly similar conclusions would be reached by both modelling approaches. However, the standard errors of year-dependent covariates were substantially underestimated by logistic regression without random effects. This concurs with findings from other areas of statistics (Glasbey 1988) that standard errors are more sensitive than regression coefficients to any change in assumptions about correlation between observations. Conventional logistic regression, which treated the observations as independent, had considerably smaller standard errors for the covariates population size and NAO than GLMM, which treated the sample size for these covariates as the number of years and inflated their standard errors according to the unexplained between-year variation.
Fitting year as a fixed effect after the year-dependent covariates, and in the absence of a random effect, allowed estimation of a mean deviance, attributable to the differences between years that were unexplained by the covariates. This was applied as a scaling parameter to adjust the standard errors for the year-dependent covariates. The adjustment worked reasonably well for population size and NAO. However, when the standard errors for catch date were treated in the same manner they became double those of the GLMM value. This indicated that much of the information about catch date came from within-year rather than between-year comparisons and highlighted the inability of conventional logistic regression to deal with such terms in a satisfactory manner.
Conventional logistic regression tended to find year-dependent covariates more highly significant than GLMM leading to incorrect inferences about the relative importance of year-dependent covariates and individual variables. This enhanced significance was due to the pseudo-replication effect of failing to treat year as a random effect (Diggle 1990). Logically it seems reasonable that mortality will vary substantially from year to year due to unrecorded factors, that the influences will be similar on all animals in the population and that, consequently, our year-dependent covariates should be measured against unexplained year-to-year variation.
We found little evidence in any of our modelling to suggest additional unexplained differences in survival between individuals once the fixed and random year effects had been accounted for. In part this may be a question of power, in that there was little information in the binary survival data and therefore a lack of heterogeneity between individuals. This was exacerbated by a relatively small proportion of individuals being recaught in subsequent years and having more than one data record (169 of 303 adult females and 65 of 262 adult males). Consequently, unexplained differences between individuals were probably quite small and the use of individual as a random effect within GLMM was unnecessary. The trends can therefore be interpreted as being across animals within years, rather than within animals across years (Diggle et al. 1994).
Of the two methods used, the mixed model approach was preferred to conventional logistic regression analysis, in that it correctly incorporated unexplained variation between years. However, logistic regression without random effects was much faster to perform so was used for model selection. The drawback of conventional logistic regression is that it can be non-conservative, but final fitting of models using GLMM overcame this problem. It also made good use of the robustness of the iteratively reweighted least squares algorithm for fitting logistic regression, in contrast to the Schall algorithm (Schall 1991) for fitting GLMMs, whose implementation in Genstat we found to suffer from convergence problems.
An alternative approach to analysing the Soay sheep data is to build a probability model for the sighting history of each individual, leading to the integrated mark–recapture–recovery analysis (mrra) method of Catchpole et al. (1998). One justification for using the mrra is that crude mortality rates estimated from animals found either dead or alive in each spring will be affected by the recovery and resighting probabilities (Catchpole et al., in press, a). However, our analyses are ‘conditional’: animals not seen alive or found dead are assumed to have died in the winter after the final sighting. This assumption is rather inelegant but provided the recapture rates are high, which they are, it should not introduce substantial errors into the analysis, even though there will be some wrong assignments in the year of death. The advantage of undertaking our GLMM analysis is that the data can then be analysed in standard statistical packages, with the inclusion of time-varying, individual-specific covariates, such as weight and year, as a random effect. It should come as no surprise that, within the constraints outlined above, the mrra gives broadly similar results to those presented in this paper (Catchpole et al. in press, b). Further methodological developments are now required to encompass the merits of both approaches.
This study has demonstrated that the considerable between-year variation in over-winter survival of Soay sheep arose from the effects of density dependence, density-independent climatic fluctuations and other unaccounted-for stochastic variation. While the influence of these factors on the population dynamics at the whole island level has already been explored (Grenfell et al. 1998), the implications for the dynamics of the population from the individual perspective should now be examined in greater depth. Our model provides a framework for estimating the magnitude of random year-to-year variation, which would be of interest for developing dynamic stochastic population models.
Our analysis has also emphasized the importance of including random annual effects in survival models and of taking into consideration the variation in demographic parameters expected between age and sex classes. In addition, our results add to the growing evidence of the influence of large-scale climatic fluctuations on the structure and demographic trends of ungulates at northern latitudes.
We are grateful to the National Trust for Scotland and Scottish Natural Heritage for permission to work on St Kilda and for their assistance in many aspects of the work. Much assistance and logistical support has been given by the Royal Artillery Range, Hebrides, its St Kilda Detachment and the Royal Corps of Transport. Special thanks go to David Green, Tony Robertson, Andrew MacColl and Jill Pilkington for their important contributions to the long-term data collection and to many volunteers for helping them. We thank Tim Clutton-Brock, Bryan Grenfell and Mick Crawley for advice and support. Peter Rothery, Andrew Illius and Xavier Lambin made many helpful comments on the manuscript. The Soay Sheep Project is supported by grants from NERC, BBSRC and the Wellcome Trust.