A central problem in research on twins is the estimation of the rates of monozygotic and dizygotic twin maternities. The estimation usually follows Weinberg's differential rule. According to this rule, the rate of dizygotic twinning is twice the rate of twin maternities in which the twins are of opposite sex. The monozygotic twinning rate is the difference between the rates of same-sex and opposite-sex twin sets. Weinberg's differential rule is implicitly based on the assumptions that the probability of a male equals the probability of a female and that the sexes in a dizygotic twin set are independent. Although Weinberg's differential rule has been the target of continuing discussions, the reliability of these assumptions has never been conclusively verified or rejected. In this study we present new variance formulas for the monozygotic and dizygotic twinning rates obtained using Weinberg's differential rule and stress that these new formulas have to be used. We analyze the accuracy of Weinberg's differential rule by considering alternative attempts. Especially, we build a general likelihood function and show that the maximum-likelihood estimates differ only slightly from the rates obtained using Weinberg's differential rule. In addition, our methods are applied to twinning data from Finland and Sweden. We compare our results with findings in the literature. In conclusion, our findings indicate that Weinberg's differential rule is rather robust and that despite its simplicity, it gives reliable results when official birth registers are analyzed.

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ABSTRACT.Purpose: To investigate exfoliation syndrome (ES) in order to elucidate gender distribution and the roles of genetic and climatic factors in its manifestation. Material and Methods:We studied the gender distribution of ES and the association between ES and the appearance of certain climatically induced disorders (pterygium, climatic droplet keratopathy) and the size of pingueculae in populations living in the Arctic region, in temperate regions and in tropical regions. This involved a total of 11 samples taken in eight different countries, comprising 2206 persons of both genders over the age of 50 years. Results: A total of 1051 males were investigated for ES and 147 of them (14.0%) were found to have it. The corresponding figures for females were 1093 and 177 (16.2%). After standardization according to age, no systematic difference between the genders was found in the study. The frequency of ES varied greatly. It was not observed at all in the Inuit (Eskimos) but was found most frequently (about 30%) among the Saami (Lapps), Åland Islanders, Finns, Icelanders and Russians over 70 years of age. After the age of 50, the frequency of ES increases rapidly with age in all populations. However, the curves for ES in people living in the tropics show a delay of about 10 years. In contrast, the frequency of climatically caused changes (pterygium, climatic keratopathy and pronounced pinguecula) mostly peaks at the age of 50 years and is highest in the tropics and in the Arctic. Males in these regions tended to be more affected by climatically caused changes than females. Likewise, in tropical climates, where radiation from the sun is strong, and in Lapland and Novosibirsk, where there is radiation from snow, males showed more evidence of ES than females. However, examination of 506 patients from a private practice in South Finland, who were under observation for glaucoma or for risk of glaucoma, showed females to be in the majority of those with ES. Conclusions: As a rule, climate does not appear to influence the occurrence of ES. However, in tropical countries, where radiation from the sun is strong, ES was more common among males than among females. In the light of its peculiar population distribution, even when climatic factors are considered, there would appear to be an important genetic factor involved in the manifestation of ES.

Theorems, proofs, laws and rules are commonly named according to the presumed investigator, but often earlier investigators have contributed substantially to the findings. One example of this is Hellin's law, which was named after Hellin, although he was not the first to discover it. In research on twinning and higher multiple maternities, the law has played a central role because it is approximately correct, despite showing discrepancies that are difficult to explain or eliminate. Several improvements to this law have been proposed. In this study, we re-examine some old papers to provide an overview of the scientists who have contributed to the genesis and the improvements of this law. In addition, we consider more recent contributions in which Hellin's law has been discussed and evaluated. It has been mathematically proven that Hellin's law does not hold as a general rule. However, most studies are based on empirical rates of multiple maternities, ignoring random errors. Such studies can never confirm the law, but only serve to identify errors too large to be characterized as random.

During the history of research on multiple maternities, Hellin's law has played a central role as a rule of thumb. It is mathematically simple and approximately correct, but shows discrepancies that are difficult to explain or to eliminate. It has been mathematically proven that Hellin's law does not hold as a general rule. Varying improvements to this law have been proposed. In this paper, we consider how Hellin's law can be used and tested in statistical analyses of the rates of multiple maternities. Such studies can never confirm the law, but only identify errors too large to be characterized as random. It is of particular interest to determine why the rates of higher multiple maternities are sometimes too high or too low when Hellin's law is used as a benchmark. Excesses of triplet and quadruplet maternities are particularly unexpected and challenging. Our analyses of triplet and quadruplet rates indicated that triplet rates are closer to Hellin's law than quadruplet rates. According to our analyses of the twinning rate and the transformed triplet rate and quadruplet rate for Sweden , both triplet and quadruplet rates showed excesses after the 1960s. This is mainly caused by artificial fertility-enhancing reproduction technologies. Regression analyses of twinning and triplet rates yield rather good fits, but deficiencies in the triplet rates are commonly present. We introduced measures of concordance between triplet rates with Hellin's law. According to these measures, historic data showed deficiencies in triplet rates, but recent data revealed excesses, especially found among older mothers. The excesses obtained are in good agreement with other studies of recent data.