Introduction
 Top of page
 Summary
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 References
 Supporting Information
Prenatally, this process has been called foetal programming (Lucus 1991); it can influence birth weight, which in turn can affect adult reproductive success (Albon, CluttonBrock & Guinness 1987; Kruuk et al. 1999). Environmental factors that influence prenatal development are therefore of importance when considering both evolutionary processes and population dynamics, because they are likely to influence both an individual's lifetime fitness and the agespecific vital rates of individuals from a given birth cohort (Albon, CluttonBrock & Langvatn 1992). Lumma (2003) states that foetal programming occurs when a stimulus during a critical period of early development subsequently affects body structure, physiology or metabolism. This highlights the importance of the timing of a stimulus. It is this crucial temporal element that this paper aims to identify.
The situation we address is when a densityindependent environmental factor, such as a weather variable, has been measured in a sequence of time intervals; for example, a week or month in pregnancy. We wish to fit a linear model in which the response variable is the lifehistory trait and weather records at each time interval are the explanatory variables. As different stages of pregnancy are associated with different patterns of foetal development (Rhind, Rae & Brooks 2003), it is more likely that the effects of weather on the lifehistory trait are comparable during similar stages of pregnancy than at other times during pregnancy. Therefore, we can reasonably assume the relationship with weather changes smoothly across the pregnancy period. For example, the relationship between a lifehistory trait and weather may be strongest during midpregnancy and decline towards the beginning or end of pregnancy (Fig. 1). Standard techniques, such as an ordinary linear regression, could be used to investigate the timing of critical periods and we would expect the estimated regression coefficients to show a similar smooth pattern.
The estimated regression coefficients are interpretable when the explanatory variables are uncorrelated or nearly so. However, given the temporal structure of the data (seasonality) we expect weather conditions to be similar in neighbouring time intervals and the strength of this correlation to decline with increasing separation between intervals. Multicollinearity is encountered when there are a large number of correlated explanatory variables in a regression analysis. If this problem is ignored, uncertainty in the regression coefficient estimates will be inflated relative to if each covariate had been fitted alone and detection of time periods when weather is important is less likely. Multiple regression is therefore inappropriate and alternative procedures should be sought. One solution is to reduce the number of explanatory variables in the model perhaps by summarizing weather across a few broad temporal intervals. For example, some studies have found correlations between local weather conditions and lifehistory traits (see Albon et al. 1987; Gaillard, Delorme & Jullien 1993; FestaBianchet, Jorgenson & Reale 2000). Typically, these authors considered just one period of pregnancy, using intervals of 1 or 2 months, often within the last trimester when the foetus is growing most rapidly. However, recent analysis of the longterm study of red deer on the Isle of Rum, using a longer run of years, found birth weight was better correlated with temperature in a 3month interval from February to April (Coulson et al. 2003) than an April to May interval used previously (Albon, Guinness & CluttonBrock 1983; Albon et al. 1987; Kruuk et al. 1999). In practice, there has been little effort to develop methods to evaluate the influence of weather using finerscale temporal intervals. Yet such methods would highlight the periods when the relationship between foetal development and weather is strongest and hence suggest the possible mechanisms underlying observed cohort effects. However, this requires examination of many intervals and hence having a large number of explanatory variables in the regression model.
Regression methods have been developed that will both (1) lessen the effects of multicollinearity while keeping a large number of explanatory variables in the model, and (2) use the information about the structure of regression coefficients by smoothing the estimated coefficients towards a solution expected given their known interrelationship (Eilers 1991). The procedures are based on ridge regression (Hoerl & Kennard 1970), a standard method for dealing with multicollinearity. An unknown parameter controls the amount of smoothing and ‘optimal’ values are chosen using crossvalidation. Here, we adopt their general concepts and fit them into a linear mixed model framework. We show that using our method the value of the smoothing parameter is automatically selected within the mixed model itself, thus regularizing the procedure without requiring the use of classical selection methods such as crossvalidation.
We will demonstrate the usefulness of our method in two examples investigating the timing of the influence of weather conditions prevailing during gestation on the birth weight of red deer Cervus elaphus (L.) and North Country Cheviot sheep Ovis aries (L.) in Scotland. As described above, birth weight in red deer appears to be associated with daily air temperature sometime between February and May. In contrast, rainfall appears to be the important variable, both earlier in pregnancy (8–14 weeks) and later in pregnancy (15–21 weeks), influencing the birth weight of sheep (Larkham, personal observations). Thus, previous studies identified weather effects within broad temporal intervals. The application of our method will enable examination of multiple finescale temporal intervals throughout the whole of the pregnancy period thus expanding our understanding of when the relationship with weather is strongest.
We explore the performance of our approach against two commonplace statistical methods for fitting regression models with many explanatory variables. This involves comparing how useful the methods are at obtaining good estimates of the regression coefficients in the presence of multicollinearity. However, a comparison is only possible when we know the true values; therefore, simulated data are used, rather than field data, as the true regressions are then known. We begin by introducing these different statistical methods for model fitting. The aim here is to illustrate the development of our approach in order to fully appreciate the methodology described in this paper. We show from the simulation study that our method, which we shall call difference penalty regression (DPR), provides a robust technique for estimating the effects of intercorrelated explanatory variables. In particular, we are able to get a better understanding of the role weather plays in the early development of mammals.
Although we develop the method in the context of traits affecting population demography, the applicability of the methods is very general. Another application in which the covariates would correspond to time intervals is phenology, with attention focusing on the effect of weather (often temperature) on the timing of an event. Alternatively, the covariates may correspond to spatial intervals. Examples in this case might be in studies of the effects of surrounding habitats on breeding success at a sample of nest sites, or the effect of surrounding water pollution on the growth of a marine sedentary organism. The paper concludes with a discussion of the observed weather–birth weight relationship and a general evaluation of the developed method.
Results
 Top of page
 Summary
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 References
 Supporting Information
Difference penalty regression (DPR) with a penalty on first differences revealed a significant negative effect of rainfall during midpregnancy on the birth weight of lambs (Fig. 3). Females showed a negative response of –0·03 kg mm^{−1} at approximately 13 weeks before birth, which is on average between late January and early February. The negative effect is most distinct when the penalty is on second differences (Fig. S1 in Supplementary material). The estimated effects of rainfall were not significantly different from zero during early and late pregnancy (Fig. 3 shows that the confidence intervals of the estimates include zero). Despite attempts to smooth the regression coefficients, we still see an irregular estimate at 5 weeks before birth compared with its neighbouring values.
Fitting a penalty on the first differences of the regression coefficients reveals a positive relationship between temperature and the birth weight of red deer (Fig. 4). The individual relationships were not significantly different from zero until about 14 weeks before birth, then rise from about 0·03 kg °C^{−1} at 14 weeks before birth to 0·045 kg °C^{−1} at 10 weeks before birth but then declined again, with the regression coefficients in the last 6 weeks of gestation again being not significantly different from zero. Thus, on average it was temperature during March and April, particularly the last 2 weeks of March and the first 2 weeks of April, when the relationship was strongest. The estimated coefficients for the two broad time intervals (February–April and April–May) were similar to each other and to the estimate for the peak (early April) identified by DPR. An even smoother result is obtained with a second differences penalty on the regression coefficients (Fig. S2 in Supplementary material).
The addition of a temperature effect in the linear mixed model for birthweight of red deer decreased the magnitude of the variance component for calf's year of birth (Table 1). In the models with an April–May or February–April temperature covariate, the size of the variance component was reduced by 41% and 45%, respectively. The DPR model revealed the greatest reduction of 52%, i.e. temperature was explaining more of the betweenyear variation in birthweight. There was little difference in the estimated variance for the mother and residual effect between the models.
Table 1. Variance components of linear mixed models fitted to the birth weights of red deer. Models were fitted with no temperature effect (none), with a fixed effect included for average April–May temperature (April–May) or average February–April temperature (Feb–April), and with temperature smoothed using DPR with a penalty on first differences (smoothing) Random effect  Model for temperature effect 

none  April–May  Feb–April  smoothing 

Mother  0·637  0·630  0·633  0·623 
Calf's year of birth  0·082  0·048  0·045  0·039 
Residual  0·856  0·859  0·856  0·853 
The simulation results show that the DPR performs better at obtaining estimates of regression coefficients in the presence of intercorrelated explanatory variables (Table 2; see Appendix S5 in Supplementary material for complete results). The LSR estimator appears to be unbiased but the variance is very large. In contrast, RR and DPR show an increase in bias but a substantial reduction in variance. Moreover, the mean square error is much reduced using DPR. Using DPR, we would expect to be an approximation of var(β) as is true for the LSR estimates. However, is consistently larger for all parameter estimates using either RR or DPR, the gap being more noticeable for the latter, arising because the variance components are estimated. This consequently increases the coverage of the 95% confidence intervals.
Table 2. Simulation results comparing least squares regression (LSR), ridge regression (RR) and difference penalty regression (DPR)  Bias  Variance  Mean square error 

Parameter  LSR  RR  DPR  LSR  RR  DPR  LSR  RR  DPR 

β_{1}  0·003  –0·002  0·044  0·204  0·178  0·075  0·204  0·201  0·077 
β_{5}  0·004  0·021  0·162  0·366  0·306  0·063  0·366  0·126  0·089 
β_{10}  0·007  –0·004  –0·053  0·390  0·316  0·057  0·390  0·200  0·060 
Parameter   Coverage 

LSR  RR  DPR  LSR  RR  DPR 


β_{1}  0·209  0·192  0·106  0·950  0·956  0·978 
β_{5}  0·365  0·329  0·118  0·948  0·956  0·980 
β_{10}  0·395  0·350  0·116  0·950  0·958  0·994 
Discussion
 Top of page
 Summary
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 References
 Supporting Information
In this paper we have demonstrated that a form of penalized multiple regression, with a penalty on the differences of neighbouring regression coefficients, identifies the pattern of the timing of influence of weather experienced during pregnancy on the birth weight of two species of mammals. Our approach is an improvement on previous analyses as it allows for the effect of weather at multiple, narrow time intervals to be estimated simultaneously. Hence, it provides a flexible method for identifying periods when a stimulus may have profound effects on early development and may help illuminate the potential mechanisms underlying weather–biological trait relationships. As birth weight is a major factor influencing lifetime fitness (CluttonBrock et al. 1996; Kruuk et al. 1999) and because whole cohorts experience similar weather, the consequences of weather conditions experienced during pregnancy may have lagged effects on the dynamics of populations (Albon et al. 1992).
The method uses the additional information regarding the structure of the regression coefficients to reduce the negative consequences of multicollinearity. By fitting the penalty as a random effect in a linear mixed model, the amount of smoothing is regulated within the model itself. Shrinkage of the regression coefficients towards neighbouring coefficients increases when estimates from the data are less reliable. This may occur when the relative variability between the random effect and the residual error, , decreases, as the size of the random effect may be partly due to the large residual error. Likewise shrinkage increases when the amount of information on each random effect decreases.
We assumed the explanatory variables are an ordered sequence at evenly spaced intervals. However, we can relax these assumptions to include variables that form a looped sequence, such as calendar month (Elston & Proe 1995) or wind direction (Eilers 1991), or are positioned at uneven intervals by modifying the differencing matrix D. Application of DPR depends on a suitable choice for the order of differencing. In some cases we might have prior knowledge about the pattern of the coefficients that will determine what level of differencing to penalize.
The simulation study shows that the DPR estimator is preferable to the LSR or RR when the explanatory variables form an ordered sequence; there was a noticeable decline in mean square error using DPR. However, the width of the confidence intervals calculated using DPR is overestimated resulting in higher coverage probabilities than expected. As the error is small we feel this should not hinder its applicability but rather recommend caution when interpreting results.
The main application of this paper was to identify the role weather plays during pregnancy in determining birth weight of mammals. In order to control for the effects of seasonality on birth weight, for example that individuals born later experience warmer weather and hence may be born heavier, we included birth date as a covariate in both models. Weather regression coefficients therefore represent the effects of departures from the seasonal average. The results from both examples have potentially important consequences because the variation in birth weight is a major determinant of lifetime reproductive success in large herbivores (Kruuk et al. 1999; Steinheim et al. 2002).
Our analyses confirm that temperature during late pregnancy was having a positive effect on birth weight in red deer, presumably because warm spring temperatures advanced the onset of grass growth and increased the mother's plane of nutrition (Albon et al. 1992). Previous studies have used April–May mean temperatures (Albon et al. 1983, 1987; Kruuk et al. 1999) or February–April mean temperatures (Coulson et al. 2003) in analyses of birthweight in the same population. Our results suggest these estimates encapsulated much of the time period when temperature most strongly influenced birthweight. However, we found from the smoothed regression coefficients that the time window when weather most strongly influenced birthweight was between midMarch and midApril. The main difference between these modelling approaches was the estimated pattern of the timing of influence of temperature during pregnancy. The smoothed set of regression coefficients revealed using our approach is what we would expect given the biological background of the problem rather than effectively a step function that was produced when temperature was summarized across a single broad time window of multiple months.
Rainfall was most strongly influencing birth weight of North Country Cheviot sheep during midpregnancy, between the end of January and early February. A possible explanation for the negative effect might be that harsh environmental conditions, such as high rainfall, increase maternal thermoregulatory costs and may result in repartitioning of nutrients from the foetus to the mother. This may reduce placental growth, which is at its maximum rate during midpregnancy (Ehrhardt & Bell 1995; Wallace et al. 2000). Reduced placental size can in turn reduce foetal growth rate (Mellor 1987), potentially resulting in reduced birth weight.
In addition to developing a better understanding of the timing of the impact of prevailing weather on lifehistory traits, penalized regression should increase the likelihood of detecting other factors contributing to the variation in fitness. For example, by explaining more of the variation due to densityindependent weather factors, DPR may permit a better estimate of both the additive effects of density dependence and the often more obscure interactions between densitydependent and stochastic, densityindependent factors (Milner, Elston & Albon 1999; Coulson et al. 2001; Hallett et al. 2004).
The linear mixed model approach provides an alternative method for fitting penalized regression, requires no additional smoothing parameterselection method and alternative forms of penalty terms can be implemented by a simple transformation of the intercorrelated explanatory variables. In this paper we use our method to produce estimates of the critical periods during pregnancy when weather is influencing lifehistory traits of mammals. More generally, the procedure presented here provides a valuable technique for ecologists from a broad range of fields who wish to study the effects of sequences of intercorrelated explanatory variables.
Acknowledgements
 Top of page
 Summary
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 References
 Supporting Information
The red deer data were collected from Rum, and we acknowledge the contribution of all involved in the collection process, including SNH for permission to work there, their staff for local support, Fiona Guinness, Ali Donald, Sean Morris, and many other field workers on the Kilmory deer project. We would also like to thank Tim CluttonBrock for permission to use the red deer data, Simon Wood for advice on running the analysis in R, as well as Tim Coulson, Chris Glasbey, Xavier Lambin and Josephine Pemberton for helpful comments on earlier drafts of this paper. The work was supported by funds from the Scottish Executive Environment and Rural Affairs Department.
Supporting Information
 Top of page
 Summary
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 References
 Supporting Information
Appendix S2. Example Genstat code to fit a DPR model with a penalty on first differences.
Appendix S3. Simulated data used to fit DPR in Fig. 2c.
Appendix S4. Example R code to fit a DPR model with a penalty on first differences.
Appendix S5. Simulation study results comparing least squares regression (LSR), ridge regression (RR) and difference penalty regression (DPR).
Fig. S1. The relationship between rainfall experienced during pregnancy and the birthweight of North Country Cheviot sheep.
Fig. S2. The relationship between temperature experienced during pregnancy and the birthweight of red deer.
Please note: Wiley Blackwell is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.