In keeping with a previous study also based on the Danish population sample , we found that both the birth weight and birth length of Danish neonates displayed prominent seasonal fluctuations. The amplitude of these seasonal fluctuations changed over time. The features of these fluctuations are consistent with models based on J-shaped ERR and left-shifting the PED over time (i.e. the population distribution of the exposure is shifted down to low values).
In general, we note that J-shaped or U-shaped models of exposure-risk result in substantial changes in the nature of seasonally-generated time series. The findings have implications for those wishing to synthesise or understand diverse and apparently conflicting seasonal data. Time series with both 6 and 12 month periodicities may be the result of an exposure with a J-shaped exposure-risk relationship.
Furthermore, if researchers are not aware of the underlying U- or J-shaped models, different populations facing different doses of the exposure could result in conflicting results. For example, as shown in Figure 4, a population with a low exposure would be restrained within one side of the U-shaped curve. This results in a negative correlation between the variables of interest. Studies done in populations restrained to the opposite side of the U-shaped curve would find a significant positive correlation, and a time series with peak 6 months shifted from the previous example. Populations in the base of the U-shape would find a relatively small amplitude, with an even, 6 month periodicity. Altering the distribution of the population around this curve results in summed signals with unexpected properties.
The results of this study highlight that interpreting overall seasonal patterns should be done with care. The simplest interpretation requires the assumption that exposure is constant across the population. Many researchers examining seasonal patterns may not realise that they are making this assumption, or the implications if it is violated. This could lead to them wrongly concluding that two seasonal exposures are at work (e.g. a 6 and 12 month exposure).
Our analyses of Danish birth anthropometric data clearly show that the amplitudes did not remain constant over time (Figure 2). This means that the time series of birth weights and lengths are non-stationary. Hence the assumptions of most standard time series methods–such as those based on the periodogram, or those that average results over months–are violated, and would have given incorrect estimates of the seasonal pattern. The method used here was able to model the non-stationarity, and hence gave a more accurate fit to the data.
Depending on the cancellation of seasonal components (as shown in Figure 4), a study may not have enough statistical power to detect an overall seasonal change in risk. Power depends on the amplitude of the seasonal change and the amount of data (number of subjects and length of time). This could lead researchers to wrongly conclude that a health outcome is not seasonal, when in fact its seasonality is strong but depends on group.
We note with interest that our new model provides a parsimonious explanation of the interaction between latitude and seasonal fluctuations in birth weight noted in both Japan  and Australia . Both of these studies reported bimodality (6 and 12 month periodicity). Both studies reported a prominent peak in spring and a secondary peak in autumn. Both studies reported that the amplitude of the spring peak was larger in sites closer to the equator. Researchers interested in synthesising these results may wish to examine candidate exposures that have both (a) non-linear ERRs, and (b) latitude gradients (i.e. the PED of the exposure will vary across latitudes).
With respect to seasonally-fluctuating exposures that may underlie these findings, we speculate that low prenatal vitamin D is a candidate that warrants closer inspection . Firstly, there is robust evidence that vitamin D levels fluctuate over the seasons and also across latitudes . Secondly, a nonlinear exposure-risk relationship between maternal vitamin D levels and neonatal anthropometry has been found . The high prevalence of hypovitaminosis D in pregnant woman and the potential adverse health outcomes for the fetus has been the focus of recent editorial comments [27, 28]. For example, low maternal vitamin D levels have been associated with altered birth anthropometry, with some studies finding an association between vitamin D deficiency and larger birth weight and length , while others have shown an association between lower maternal vitamin D levels and shorter gestation and smaller knee-heel length (a component of birth length) . Future studies would ideally examine the relationship between maternal vitamin D status and neonatal anthropometry across time (e.g. in different seasons) and in a range of sites (e.g. high and low latitude).
Apart from neonatal anthropometry, J-shaped ERRs have also been observed between temperature and cardiovascular disease [30, 31], with large increases in risk associated with cold-temperatures and smaller increases with hot temperatures. A J-shaped ERR may provide a parsimonious explanation for the observed 6 and 12 month bi-modal peaks in seasonal cardiovascular disease risk [13, 32].
With respect to the Danish data, our findings are in keeping with other studies [11, 33]. Mean birth weight rose in a steady linear fashion over the period of observation (1973–2002). Curiously, mean birth length appeared to decline from 1973 to approximately 1985, and then rose steadily in a parallel fashion to birth weight. Assuming that the population mix remained stable over this period (i.e. that the broad genetic 'mix' of the Danish population was essentially stable, which is a fair assumption since both the child and its parents were born in Denmark), the change in secular trend in birth length suggests that factors that impact on fetal growth changed over this time. The mechanisms underlying this change remain unclear. There is evidence showing that the estimation of gestational age improved over recent decades , however this can not readily explain the nature of the current findings.
The time series in the modelling exercises shown above were generated without any noise and thus show ideal shapes. In real-life situations measurement and random error will lead to observed time series that are noisier than those shown here. Methods to extract the seasonal component after removing noise and trend should be used . We have limited our modelling by only investigating models based on biologically plausible assumptions – many exposures have J- and U-shaped ERRs. Furthermore, the PED changes we used in the models are relatively modest and within the range of many latitude-associated exposures (e.g. vitamin D, temperature). Other combinations of exposure-risk relationships (e.g. S- or V-shaped) and population exposure distributions may also give good fits to the observed seasonal pattern. The range of possible ERRs and PEDs is very large, and it would require a lot of computer power to search through all possible models. Such a search would also be data-driven rather than hypothesis-driven. Nevertheless, it is still important to note that the models of seasonal change shown here are just one possible fit to the data. Modelling can not replace a thorough understanding of the biological properties of candidate exposures. As with all epidemiology, analytic methods such as the choice of linear versus non-linear ERRs should be built on the best-available biology and experimental data. Conversely, signals that emerge from observational epidemiology can also help guide more focussed biological research (e.g. dose-response studies in animal models in order to refine the properties of the ERR).