5. Studies including the use of Hellin's law

Fellman and Eriksson [1, 2, 32] presented the temporal trends in TWR, the square root of TRR and the cubic root of QUR obtained from the Veit data [16]. Note that their figure shows stronger fluctuations in TRR than in TWR. However, the confidence bands included indicate that the TWR and the transformed TRR show good agreement for the whole period. The transformed QUR is too high for almost the whole period [32]. In Figure 1, we present a new version of the TWR, the transformed TRR and QUR per 103 for the Prussian data presented in [16]. In this figure and later, the transformed TRR and QUR per 103 are denoted by the initial untransformed names TRR and QUR. Note that the transformed QUR shows a marked excess compared with the TWR and the transformed TRR. This excess can be connected to the comparisons presented above between the rates from conceptions to deliveries. Furthermore, one can observe that all rates show slightly decreasing trends.

In this study, we investigate the temporal trends in TWR, TRR and QUR. The TRRs and QURs are in all figures transformed according to Hellin's law in order to show the association between TWR, TRR and QUR. In the figures, the transformed variables are still denoted TRR and QUR. The trends show variations during different periods and for different countries, but for different countries, one can observe similar patterns. During the eighteenth and nineteenth centuries, the rates are rather similar, but during the first half of the nineteenth century, there is a deficit in the TRR. During the second half of the twentieth century, the TRR shows an excess, and this finding is mainly caused by the influence of the artificial reproduction technologies, particularly the use of fertility-enhancing drugs. Below, we present graphs for different countries, and similar patterns can be noted.

The temporal trends in the twinning and triplet trends in Sweden (1751–2000) are presented in Figure 3. During 1750–1890 the rates are rather similar, but during the period 1900–1970, there

Figure 3. Temporal trends in twinning and triplet rates in Sweden (1751–2000). During 1750–1890 the rates are rather

Historical Studies of Hellin's Law

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similar, but during the period 1900–1970, there is a deficit in the TRR. After 1970, the TRR shows an excess.

Following [37] we present in Figure 4 the temporal trends in the twinning and triplet trends in Portugal (1930–2011). One changing point can be found in 1950. After 1950, TRR shows an

Fellman [3] presented the temporal trends in the twinning and triplet trends in the Netherlands (1950–2003). The findings are given in Figure 5. An excess among TRR can be observed

Figure 4. Temporal trends in twinning and triplet trends in Portugal (1930–2011). One changing point can be found at

after 1970. At the end of the twentieth century, there is a marked deficit in the TRR.

is a deficit in the TRR. After 1970, the TRR shows an excess.

1950. After 1950, the TRR shows an excess [37].

excess.

The temporal trends in the TWR and the transformed TRR in Finland 1751–2000 show variations during different periods. During 1750–1900 the rates are rather similar, but during the period 1900–1970, there is a deficit in the TRR. After 1970, the TRR shows an excess, and this finding is mainly caused by the influence of the artificial reproduction technologies, particularly the use of fertility-enhancing drugs (Figure 2).

Figure 2. Temporal trends in TWR and transformed TRR in Finland (1751–2000).

stronger fluctuations in TRR than in TWR. However, the confidence bands included indicate that the TWR and the transformed TRR show good agreement for the whole period. The transformed QUR is too high for almost the whole period [32]. In Figure 1, we present a new version of the TWR, the transformed TRR and QUR per 103 for the Prussian data presented in [16]. In this figure and later, the transformed TRR and QUR per 103 are denoted by the initial untransformed names TRR and QUR. Note that the transformed QUR shows a marked excess compared with the TWR and the transformed TRR. This excess can be connected to the comparisons presented above between the rates from conceptions to deliveries. Furthermore,

In this study, we investigate the temporal trends in TWR, TRR and QUR. The TRRs and QURs are in all figures transformed according to Hellin's law in order to show the association between TWR, TRR and QUR. In the figures, the transformed variables are still denoted TRR and QUR. The trends show variations during different periods and for different countries, but for different countries, one can observe similar patterns. During the eighteenth and nineteenth centuries, the rates are rather similar, but during the first half of the nineteenth century, there is a deficit in the TRR. During the second half of the twentieth century, the TRR shows an excess, and this finding is mainly caused by the influence of the artificial reproduction technologies, particularly the use of fertility-enhancing drugs. Below, we present graphs for different coun-

The temporal trends in the TWR and the transformed TRR in Finland 1751–2000 show variations during different periods. During 1750–1900 the rates are rather similar, but during the period 1900–1970, there is a deficit in the TRR. After 1970, the TRR shows an excess, and this finding is mainly caused by the influence of the artificial reproduction technologies, particu-

one can observe that all rates show slightly decreasing trends.

tries, and similar patterns can be noted.

14 Multiple Pregnancy - New Challenges

larly the use of fertility-enhancing drugs (Figure 2).

Figure 2. Temporal trends in TWR and transformed TRR in Finland (1751–2000).

Figure 3. Temporal trends in twinning and triplet rates in Sweden (1751–2000). During 1750–1890 the rates are rather similar, but during the period 1900–1970, there is a deficit in the TRR. After 1970, the TRR shows an excess.

The temporal trends in the twinning and triplet trends in Sweden (1751–2000) are presented in Figure 3. During 1750–1890 the rates are rather similar, but during the period 1900–1970, there is a deficit in the TRR. After 1970, the TRR shows an excess.

Following [37] we present in Figure 4 the temporal trends in the twinning and triplet trends in Portugal (1930–2011). One changing point can be found in 1950. After 1950, TRR shows an excess.

Fellman [3] presented the temporal trends in the twinning and triplet trends in the Netherlands (1950–2003). The findings are given in Figure 5. An excess among TRR can be observed after 1970. At the end of the twentieth century, there is a marked deficit in the TRR.

Figure 4. Temporal trends in twinning and triplet trends in Portugal (1930–2011). One changing point can be found at 1950. After 1950, the TRR shows an excess [37].

Figure 5. Temporal trends in the twinning and triplet rates in the Netherlands (1950–2003). An excess among TRR can be observed after 1970.

Eriksson and Fellman [33] compared the rates of twin, triplet and quadruplet maternities in England and Wales for the period 1938–2003. In this study, we develop these findings. In Figure 6, one observes that before 1970 the graph lines are close to another, but after 1970 the lines rise and diverge. QUR shows the strongest increase and TWR the slightest. Furthermore, Figure 6 indicates that during the last years, TRR and QUR show a slight decline. Our opinion is that this change is caused by changes in fertilization policies, especially a reduction in the number of fertilized eggs implanted. To clarify the fluctuations, trend lines of sixth degree are included in the figure. Furthermore, Figure 6 indicates that for data sets after 1970, the TRRs and QURs are markedly too high. It is a remarkable finding that the rates are too high rather

than too low, but fertilization policies may result in the extreme sets of multiple maternities. The decreases at the end of the twentieth century are ascribed to changes in the treatment

Figure 7. Temporal trends in TWR and the Hellin-transformed rate of multiple maternities (MUR). The figure indicates

Otta et al. [38] analyzed the TWR and the MUR for Brazil data (2003–2014). They discussed the influence of artificial reproduction technologies, particularly the use of fertility-enhancing drugs. They included in their analyses the effect of maternal age. In this study, we look at their data from a different point of view. In Figure 7, we present the TWR and the Hellintransformed rate of multiple maternities (MUR). We assume that the number of multiple maternities is dominated by triplet maternities, and we use the square root transformation. Figure 7 indicates that the TWR is still increasing, but that the MUR decreases. The excesses coincided with the introduction of subfertility treatments, mainly ovulation inductions. Our opinion is that the difference in the changes between TWR and MUR is caused by changes in fertilization policies, especially a reduction in the number of fertilized eggs implanted. Finally, there is common agreement that discrepancies obtained during the era of fertility treatments are of less interest when Hellin's law is considered because no natural stochastic model is applicable.

For the whole period 2003–2014, the TWR <sup>¼</sup> <sup>11</sup>:96 per 1000 and MUR <sup>¼</sup> <sup>357</sup>:97 per 106

A problem that complicates the discussion of Hellin's law is that the law is a mathematical rule concerning theoretical rates, but all checks of the law must be based on empirically obtained rates. In fact, one can only check whether the discrepancies are too large and cannot be explained by random errors. Although the discrepancies are small, Hellin's law cannot be accepted as a

theoretical one. In this way, no exact proof to support the law can be obtained [32].

HR ¼ 2:50 indicates a marked excess of multiple maternities.

that the TWR is still increasing, but the transformed MUR is decreasing [38].

. Hence,

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policies discussed above.

6. Discussion

Figure 6. Temporal trends in the twinning, triplet and quadruplet rates in the UK (1938–2003). During 1938–1970 the rates are rather similar, but after 1970 the rates increase. QUR shows the strongest increase and TWR the slightest. In order to clarify the fluctuations, trend lines of sixth degree are included in the figure.

Figure 7. Temporal trends in TWR and the Hellin-transformed rate of multiple maternities (MUR). The figure indicates that the TWR is still increasing, but the transformed MUR is decreasing [38].

than too low, but fertilization policies may result in the extreme sets of multiple maternities. The decreases at the end of the twentieth century are ascribed to changes in the treatment policies discussed above.

Otta et al. [38] analyzed the TWR and the MUR for Brazil data (2003–2014). They discussed the influence of artificial reproduction technologies, particularly the use of fertility-enhancing drugs. They included in their analyses the effect of maternal age. In this study, we look at their data from a different point of view. In Figure 7, we present the TWR and the Hellintransformed rate of multiple maternities (MUR). We assume that the number of multiple maternities is dominated by triplet maternities, and we use the square root transformation. Figure 7 indicates that the TWR is still increasing, but that the MUR decreases. The excesses coincided with the introduction of subfertility treatments, mainly ovulation inductions. Our opinion is that the difference in the changes between TWR and MUR is caused by changes in fertilization policies, especially a reduction in the number of fertilized eggs implanted. Finally, there is common agreement that discrepancies obtained during the era of fertility treatments are of less interest when Hellin's law is considered because no natural stochastic model is applicable. For the whole period 2003–2014, the TWR <sup>¼</sup> <sup>11</sup>:96 per 1000 and MUR <sup>¼</sup> <sup>357</sup>:97 per 106 . Hence, HR ¼ 2:50 indicates a marked excess of multiple maternities.

#### 6. Discussion

Eriksson and Fellman [33] compared the rates of twin, triplet and quadruplet maternities in England and Wales for the period 1938–2003. In this study, we develop these findings. In Figure 6, one observes that before 1970 the graph lines are close to another, but after 1970 the lines rise and diverge. QUR shows the strongest increase and TWR the slightest. Furthermore, Figure 6 indicates that during the last years, TRR and QUR show a slight decline. Our opinion is that this change is caused by changes in fertilization policies, especially a reduction in the number of fertilized eggs implanted. To clarify the fluctuations, trend lines of sixth degree are included in the figure. Furthermore, Figure 6 indicates that for data sets after 1970, the TRRs and QURs are markedly too high. It is a remarkable finding that the rates are too high rather

Figure 6. Temporal trends in the twinning, triplet and quadruplet rates in the UK (1938–2003). During 1938–1970 the rates are rather similar, but after 1970 the rates increase. QUR shows the strongest increase and TWR the slightest. In order

to clarify the fluctuations, trend lines of sixth degree are included in the figure.

Figure 5. Temporal trends in the twinning and triplet rates in the Netherlands (1950–2003). An excess among TRR can be

observed after 1970.

16 Multiple Pregnancy - New Challenges

A problem that complicates the discussion of Hellin's law is that the law is a mathematical rule concerning theoretical rates, but all checks of the law must be based on empirically obtained rates. In fact, one can only check whether the discrepancies are too large and cannot be explained by random errors. Although the discrepancies are small, Hellin's law cannot be accepted as a theoretical one. In this way, no exact proof to support the law can be obtained [32].

Jenkins ([35], Figure 8) and later Fellman and Eriksson [1] presented the association between the TWRs and the TRRs in the Prussian data [16]. They observed marked fluctuations for the different annual data but a good agreement between the TRR and the TWR for the total data set [32].

and Eriksson [32] discuss this deficiency in more detail. There is almost constantly an excess in the QUR for the whole period. After 1970, both the TRR and QUR show excesses, but this is mainly caused by the influence of the artificial reproduction technologies, particularly the use

TRR can show remarkable variations. Eriksson [12] studied the TWR and the TRR in the southwestern part of Finland. On the Åland islands, the TWR was continuously high. For the

good agreement with Hellin's law. In the Åboland (Turunmaa in Finnish) archipelago, close to the Åland islands, the TWR was also high. For the period 1655–1949, the TWR was 20.90 per

Lam and Ho [44] noted an increase in the number of multiple maternities in Hong Kong in 1981–1995. They also stressed the marked discrepancy between the observed data and Hellin's law. Zhang et al. [45] have observed similar increases in the rates of multiple maternities among older mothers in the USA in 1995–1997, and they also attributed this finding to the increased use of assisted reproductive technology. Simmons et al. [46] noted a dramatic

It is generally agreed that the main argument for Hellin's law is that the probabilities of additional ovulations and the fissions of fertilized eggs can be explained by stochastic models. Consequently, in large data sets, the averages could be stable and formulated by a mathematical relation (Hellin's law). A common argument for the discrepancies is that after the conceptions, there is a long process influenced by disturbing factors (intrauterine deaths, spontaneous abortions, etc., of one or more fetuses) [32]. The discussion of Hellin's law is complicated by the fact that the law is a mathematical rule concerning theoretical rates, but all checks have to be based on empirically obtained rates. In fact, one can only check if the discrepancies are so large that they cannot be explained by random errors. If the discrepancies are small, an exact Hellin's law

cannot be accepted. In this way, no exact proof to support the law can be obtained [32].

This work was supported in part by grants from the Magnus Ehrnrooth Foundation. I am very grateful to the personnel of the National Library of Health Sciences, University of Helsinki (Terkko), for providing copies of old publications concerning twin studies in the nineteenth

, yielding HR ¼ 0:58, and consequently, the Åboland archipelago data

period 1653–1949, the TWR was 19.21 per 1000, and the TRR was 375 per 106

. Agreements between TWR and transformed

. For the Åland data, HR ¼ 1:02, showing a

. According to Hellin's law, the expected

. According to

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of fertility-enhancing drugs. For references, see [14, 32].

Above the HR is defined as HR <sup>¼</sup> TRR=TWR2

Hellin's law, the expected TRR was 369 per 106

TRR was 437 per 10<sup>6</sup>

7. Conclusion

Acknowledgements

century, otherwise difficult to obtain.

1000. For the same period, the TRR was 252 per 10<sup>6</sup>

showed a marked deficit in TRR with respect to Hellin's law [32].

decrease in the proportion of triplet and higher-order births since 1998 [32].

Our impression is that the finding already noted by Strassmann [20] was the birth of Hellin's law. Furthermore, Jenkins [35] stressed that Hellin's law is a first approximation. It is generally agreed that the main argument for Hellin's law is that the probabilities of additional ovulations and the fissions of fertilized eggs can be explained by stochastic models. Consequently, in large data sets, the averages could be stable and formulated by a mathematical relation (Hellin's law). A common argument for the discrepancies is that after the conceptions, there is a long process influenced by disturbing factors (intrauterine deaths, spontaneous abortions, etc., of one or more fetuses). Jenkins [35] and Komai and Fukuoka [39], for instance, assumed that differential mortality in utero of twins and triplets could be one such factor. Consequently, the final result often shows only a weak resemblance to the outcome of a simple stochastic process associated with the initial conceptions. Excesses of higher multiple maternities in old birth registers must be considered paradoxical. One explanation can be the results of the comparisons between the changes in the rates of singletons, twins and triplets during the time from conception to confinement discussed above. Another probable explanation is that systematic errors in the registers may cause biases in the data. This explanation is less plausible if the data are collected in different countries, as is the case in data in ([32, 35], Table 1 and Figure 1).

In his study of the rates of multiple maternities for total, "white" and "colored" in US populations (1922–1936), Strandskov [40] evaluated how well his data satisfy Hellin's law. Applying χ<sup>2</sup> tests, he found that in none of the populations tested did the observed plural birth frequencies agree closely with Hellin's law [3].

Based on hospital data, Sarkar [41] studied the TWR in India and on Ceylon (Sri Lanka). His paper is interesting because he defined the TWR as 1 : n and the triplet rate as 1 : m2, that is, he indirectly used a modified Hellin's law without any reference to Hellin. One finds a deficit of triplet maternities (m > n). In addition, one observes that on Ceylon the TWR was low (1 : 161:1), yielding a TWR of 6.21 per 1000. On Ceylon, the TRR followed Hellin's law more exactly because it was 1 : 154:4<sup>2</sup> .

Das [42] formulated Hellin's law such that "the frequency of twin confinements bears to that of total confinements a ratio which is equal to the ratio borne by the frequency of the triplet confinements to that of the twin confinements". This modified definition is in congruence with Strassmann's version of the law. He reviewed earlier studies concerning Hellin's law and stressed the discrepancies presented in them [3]. Das concluded that Hellin's law has no sound basis and that exceptions to the rule have been the rule. In a later paper, Das [43] also considered the relation TRR <sup>¼</sup> ð Þ TWR <sup>2</sup> . He constructed an advanced model based on the zygosity of both twins and triplets. His mathematical analyses of these models did not support Hellin's law [1].

Fellman and Eriksson [2] compared in Figure 3 the TWR and the transformed TRR and QUR for Sweden (1751–2000). For the period 1871–1960, there is a deficiency in the TRR. Fellman and Eriksson [32] discuss this deficiency in more detail. There is almost constantly an excess in the QUR for the whole period. After 1970, both the TRR and QUR show excesses, but this is mainly caused by the influence of the artificial reproduction technologies, particularly the use of fertility-enhancing drugs. For references, see [14, 32].

Above the HR is defined as HR <sup>¼</sup> TRR=TWR2 . Agreements between TWR and transformed TRR can show remarkable variations. Eriksson [12] studied the TWR and the TRR in the southwestern part of Finland. On the Åland islands, the TWR was continuously high. For the period 1653–1949, the TWR was 19.21 per 1000, and the TRR was 375 per 106 . According to Hellin's law, the expected TRR was 369 per 106 . For the Åland data, HR ¼ 1:02, showing a good agreement with Hellin's law. In the Åboland (Turunmaa in Finnish) archipelago, close to the Åland islands, the TWR was also high. For the period 1655–1949, the TWR was 20.90 per 1000. For the same period, the TRR was 252 per 10<sup>6</sup> . According to Hellin's law, the expected TRR was 437 per 10<sup>6</sup> , yielding HR ¼ 0:58, and consequently, the Åboland archipelago data showed a marked deficit in TRR with respect to Hellin's law [32].

Lam and Ho [44] noted an increase in the number of multiple maternities in Hong Kong in 1981–1995. They also stressed the marked discrepancy between the observed data and Hellin's law. Zhang et al. [45] have observed similar increases in the rates of multiple maternities among older mothers in the USA in 1995–1997, and they also attributed this finding to the increased use of assisted reproductive technology. Simmons et al. [46] noted a dramatic decrease in the proportion of triplet and higher-order births since 1998 [32].
