In a long series of papers, scientists have studied the distribution of birth weight. The interest in birth weight arises from its central role in the future health condition of a child. Recorded birth weight data showed skewing from the normal distribution. Erkkola et al. (Reference Erkkola, Kero, Seppälä, Grönroos and Rauramo1982) compared perinatal and neonatal mortality rates in different birth weight groups. They stated that while the neonatal mortality rates are indicators of general obstetrical and neonatal care, rates in different weight groups are extremely important for obstetricians. Wilcox and Russell (Reference Wilcox and Russell1983) discussed the frequency distribution of birth weight and identified a predominant Gaussian distribution and a residual distribution, with the complete distribution characterized by three parameters: the mean and standard deviation of the Gaussian component and the proportions of births in the residual distribution. Umbach and Wilcox (Reference Umbach and Wilcox1996) assumed that the distribution of birth weight is a Gaussian distribution contaminated within the tails by an unspecified ‘residual’ distribution. They proposed a technique for measuring certain features for birth weight distributions useful for epidemiologists: the mean and variance of the predominant distribution, the proportion of births in the high- and low-birth weight residual distributions, and the boundary support for these residual distributions.
Yokoyama et al. (Reference Yokoyama, Sugimoto, Pitkäniemi, Kaprio and Silventoinen2011) recently analyzed size at birth and growth trends among Japanese triplets. They noted that triplets are small at birth and their height deficit relative to the general population in Japan remained between 2% and 5% until 12 years of age.
In our birth weight data set from the Åland Islands (1885–1998), the maternity types, that is, singletons, twins, and triplets, were registered and these data are used here. In fact, in this data set, additional variables were also registered, but their effect is discussed elsewhere (Eriksson & Fellman, Reference Eriksson and Fellman2013). Fellman and Eriksson (Reference Fellman and Eriksson2013) compare the birth weight among Finnish triplets with their future survival.
Materials and Methods
Material
Our birth data were collected from official birth certificates from the Åland Islands for more than one century (1885–1998). From 1921 onward, Åland has been a county of its own and the number of births has been officially registered. Earlier, Åland was a part of the county of Turku and Pori. For the period 1885–1920, we estimated the total number of births from Eriksson (Reference Eriksson1973). The total registered number of births during 1885–1998 was about 46,940. Our birth data with known maternity type (single, twin, or triplet) consist of 18,821 births and, consequently, our data comprise about 40% of all births on Åland for this period. On the birth certificates, much information could be registered, but the long period and the large number of midwives working in different parishes resulted in missing values, and the quality of the completed certificates varied. For some births (about 200), the weights were missing. During the 19th century, some birth weights were registered in Russian pounds. We have transformed these weights to grams using the conversion of one pound to 409.5 gram. A detailed presentation of the data set is presented in Eriksson and Fellman (Reference Eriksson and Fellman2013).
Statistical Methods
Recently, Lindsay and Liu (Reference Lindsay and Liu2009) discussed inter alia how to test the normal distribution by quantile–quantile (QQ) plots. In a two-dimensional coordinate system, the quantiles (QW) of the observed variable (birth weight in this study) are distributed over the horizontal axis, and the quantiles (QN) of the normal variable, when the parameters of the normal distribution are estimated from the sample, are distributed over the vertical axis. If the scatter points (QW, QN) are linearly distributed, then the observed variable can be assumed to be normal. Lindsay and Liu used the Kolmogorov–Smirnov goodness-of-fit test to test the normality assumption. The test statistic is the greatest standardized absolute vertical distance 
$K = \sqrt n \mathop {sup}\limits_x \left| {F_n (x) - F(x)} \right|$ where Fn(x) is the observed distribution and F(x) is the hypothetical normal distribution whose parameters are estimated from the sample. The critical values for K are K 0.05=1.358, K 0.01=1.628, and K 0.001=1.949 for p=.05, .01, and .001, respectively. Lindsay and Liu (Reference Lindsay and Liu2009) stressed that for large samples the normality is rejected, although the QQ plot looks quite linear in the center. This finding is attributed to the Kolmogorov–Smirnov test measuring absolute deviations and therefore being more sensitive to discrepancies in the center than in the tails.
Results
In our Åland series, the births are grouped according to maternity type and presented in Table 1. The data set consisted of 18,570 maternities including 18,323 singleton, 243 twin, and 4 triplet maternities. The twinning rate was 13.09 per 1,000, the triplet rate 21.54 per 100,000, and the Hellin-transformed triplet rate was 14.68 per 1,000.
TABLE 1 Overview of the Åland Data Set
The rates are the twinning rate per 1,000 maternities and the triplet rate per 100,000 maternities. The Hellin-transformed rate is the squared root of the triplet rate, that is, 14.68 per 1,000.
Comparisons between the mean birth weight of both sexes for singletons, twins, and triplets in the Åland data are presented in Table 2. The ANOVA test yields significant differences, F(5, 18, 604)=322.7; p<.001. When we compared the mean birth weight between singleton males and females, we also obtained significant differences, F(1, 18, 121)=299.20; p<.001, but when we compared the mean birth weight between twin males and females no significant differences were found. The mean birth weights for singletons and twins are presented in Figure 1. In the Åland data, there are only four triplet sets. Consequently, we did not apply an ANOVA test for the triplets, nor did we include them in Figure 1.
TABLE 2 Birth Weight Data Grouped According to Sex and Type of Births on Åland, 1885–1998
FIGURE 1 Mean birth weight for males and females among singletons and twins. For the twins, 95% confidence intervals are included.
The distribution of birth weight has been a central topic in all birth weight studies. In the following, we study the distribution for single males and females and for twin males and females in the Åland data. The distributions of the birth weight among male (n = 9,304) and female (n = 8,819) singletons on Åland are presented in Figure 2. The Kolmogorov–Smirnov test yields K=5.327, p<.001 for males and K=5.435, p<.001 for females. Both tests indicate significant deviations from the normal distribution. These deviations can also be identified in the corresponding QQ plots, which are included in Figure 2.
FIGURE 2 Birth weight among male (n = 9,304) and female (n = 8,819) singletons on Åland (1885–1998). The Kolmogorov–Smirnov tests yield significant deviations from the normal distribution (K=5.327,p<.001 for males and K=5.435, p<.001 among females). These deviations are apparent in the QQ plots (for details, see text).
For comparison, in Figure 3 we present the distribution of birth weight among male (n = 251) and female (n = 224) twins on Åland. The Kolmogorov–Smirnov test yields K=1.328, p>.05 for males and K=1.134, p>.05 for females, and the distributions can be assumed to be normal.
FIGURE 3 Birth weight among male (n = 251) and female (n = 224) twins on Åland (1885–1998). The Kolmogorov–Smirnov tests yield K = 1.328, p > .05 for males and K = 1.134, p > .05 for females, and the distributions can be assumed to be normal.
Discussion
Preliminary studies of the Åland birth data (Eriksson & Fellman, Reference Eriksson and Fellman2013) have indicated that several factors influence the birth weight distribution, but even after the elimination of these factors marked discrepancies remained when the residual distribution was compared with the Gaussian one. Among singletons, the grouping according to sex did not result in normal distributions. For twins, grouping according to sex yielded acceptable normal distributions.
Box (Reference Box1976) stated that since all models are wrong, the scientist cannot obtain a ‘correct’ one by excessive elaboration. Furthermore, he stressed that in nature while normal distributions and straight lines do not exist, with normal and linear assumptions known to be false, one can often derive results that provide useful approximations of the real world. Consequently, following Box (Reference Box1976) and Lindsay and Liu (Reference Lindsay and Liu2009), the distributions of birth weight found in our studies can be considered acceptable approximations of normal distributions.
Acknowledgments
This study was in part supported by grants from the Finnish Society of Sciences and Letters, Finska Läkaresällskapet, and the foundations Magnus Ehrnrooths Stiftelse, Ålands Kulturstiftelse, and Liv och Hälsa.
