4. Investigations of Hellin's law

Hellin's law has played a central role in the history of research on multiple maternities. The interest in Hellin's law is mainly the result of its being mathematically simple and approximately correct, but it shows discrepancies that are difficult to explain or eliminate. Statistical studies on empirical rates of multiple maternities can never confirm the law but serve only to identify errors too large to be characterized as random. It is of particular interest to ask why the rates of higher orders of multiple maternities are sometimes too high and sometimes too low when Hellin's law is used as a benchmark [32].

Fellman and Eriksson [34] proposed an alternative transformation arcsin ffiffi

ffiffiffiffi r \_ � � � � p

rð Þ 1 � r

Var arcsin

1 2 ffiffi <sup>r</sup> <sup>p</sup> ffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>r</sup> <sup>p</sup> � �<sup>2</sup>

> ffiffiffiffi r

<sup>r</sup> <sup>p</sup> <sup>¼</sup> ffiffi <sup>r</sup> <sup>p</sup> <sup>þ</sup> 1 2 r 3 2 3 þ

> ffiffi br <sup>p</sup> � <sup>k</sup>

TRR <sup>p</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>TWR [1].

does not depend on r. Consider the difference

<sup>r</sup> � � <sup>p</sup> � ffiffi

r p the corresponding transformed CI is

arcsin

ffiffiffiffi r <sup>p</sup>\_ � � is

The variance of arcsin

r p ð Þ� ffi r p ffi <sup>r</sup> <sup>p</sup> <sup>≈</sup> <sup>r</sup>

construct the CI for ffiffi

TRR <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>TWR2

Now, the model is ffiffiffiffiffiffiffiffiffi

relation TRR <sup>¼</sup> <sup>0</sup>:<sup>000013</sup> <sup>þ</sup> <sup>0</sup>:656TWR<sup>2</sup>

and arcsin ffi

sion 10�<sup>4</sup>

ffiffi

arcsin ffiffi

values of r, the difference between the two transformations is minute. The variance of

dr arcsin ffiffi <sup>r</sup> � � <sup>p</sup> � �<sup>2</sup>

4rð Þ 1 � r

<sup>5</sup> <sup>þ</sup> … � ffiffi

. The square root is a monotone-increasing function, and consequently, one can

<sup>r</sup> <sup>p</sup> <sup>¼</sup> <sup>1</sup> 2 r 3 2 3 þ

6. This relative difference between the transformed variables is of the dimen-

r p by a square root transformation of the limits of the CI for r. Hence, for

ffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> �b<sup>r</sup> 4n

. The intercept indicated that the line did not pass

� � <sup>r</sup>ð Þ <sup>1</sup> � <sup>r</sup>

Var r \_� � <sup>¼</sup>

<sup>n</sup> <sup>¼</sup> <sup>1</sup>

4n

ffiffiffiffi r

> r 5 2 <sup>5</sup> <sup>þ</sup> …<sup>≈</sup> <sup>1</sup> 6 r ffiffi <sup>r</sup> <sup>p</sup> (4)

<sup>¼</sup> <sup>d</sup>

<sup>n</sup> <sup>¼</sup> <sup>1</sup>

<sup>p</sup>\_ � � is slightly larger than the variance of

r 5 2

> ffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> �b<sup>r</sup> 4n

; ffiffi br <sup>p</sup> <sup>þ</sup> <sup>k</sup>

! r

Fellman and Eriksson [34] gave a mathematical proof that Hellin's law cannot hold in general. If one aggregates heterogeneous data, the fluctuations are smoothed out, but according to Hellin's law, the relation between the TWR and the TRR is not linear, and consequently, the

Jenkins [26, 35], Jenkins and Gwin [27], Bulmer [29] and later Fellman and Eriksson [30] have tried to modify the law in order to improve it. Using linear curves is the best method for identifying discrepancies from a presumptive model because graphs containing linear curves are easy to interpret. There are two possibilities for checking Hellin's law with linear curves. One is to use graphs with TWR<sup>2</sup> as abscissa and TRR as ordinate, that is, to use the model

. An alternative is graphs with TWR as abscissa and ffiffiffiffiffiffiffiffiffi

Jenkins and Gwin [27] considered US data for the periods 1923–1924 and 1927–1936. They used TWR2 as abscissa and TRR as ordinate. From their figure, they obtained the linear

through the origin and the parameter estimate was markedly below the value one, indicating a deficit in triplet sets. When Fellman and Eriksson [32] applied a regression model to the same

r

aggregated and disaggregated data cannot simultaneously satisfy Hellin's law [1].

r p

http://dx.doi.org/10.5772/intechopen.79583

Historical Studies of Hellin's Law

� � (3)

<sup>p</sup>\_, but it is simpler and

(5)

TRR <sup>p</sup> as ordinate.

ð Þ. For small

11

Usually, the arguments for Hellin's law are based on stochastic models for multiple fertilizations and fissions of fertilized eggs. The influence of both multiple fertilizations and fissions of fertilized eggs has inspired scientists to associate the rates of higher multiple maternities with both monozygotic (MZ) and dizygotic (DZ) TWRs (e.g., [25–30]). The contributions by Zeleny [25] have resulted in the law also being known as the Hellin-Zeleny law.

Peller [31] was the first, at least indirectly, to connect Hellin's law to interindividual variation in mothers' chances for multiple maternities. Later, Eriksson [12] considered recurrent twin maternities in families on the Åland Islands (Finland) and presented a modified model (in the paper, the law was called Fellman's law). When Eriksson applied this law to his Åland data, he obtained better congruence with Hellin's law than if Peller's version had been applied. Fellman and Eriksson [1] reviewed papers where the genesis of Hellin's law was traced and where the strengths and weaknesses of the law were analyzed and improvements suggested [32].

Hellin's law presupposes strong correlations between TWR and TRR, but even strong correlations do not prove Hellin's law, establishing only a linear relationship. Fellman and Eriksson [30] considered the correlation between the TWR and the square root of the TRR in Sweden. After elimination of influential temporal factors, they found that the correlation was positive, but not very strong. This finding indicates that, in general, Hellin's law cannot be exact. One application of Hellin's law is to compare TWR and the square root of TRR, the cubic root of QUR and so on [14, 32, 33].

In the following, we consider formulae applicable in the statistical analysis of Hellin's law. Let the theoretical TRR be r. One has different possibilities to study the random errors of the TRR and particularly of the square root of the TRR. The first one is to estimate the standard deviations (SDs) of the TRR and construct confidence intervals (CIs) for r [32].

Let the observed TRR be <sup>b</sup>r, then SD <sup>b</sup><sup>r</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffi rð Þ 1�r n q , and the observed standard CI of r is

$$\left(\widehat{r} - \mathbf{k}\sqrt{\frac{\widehat{r}(1-\widehat{r})}{n}}, \ \widehat{r} + \mathbf{k}\sqrt{\frac{\widehat{r}(1-\widehat{r})}{n}}\right) \tag{1}$$

where the factor k defines the confidence level. The Hellin-transformed TRR is ffiffiffiffi r <sup>p</sup>\_. The variance of ffiffiffiffi r <sup>p</sup>\_ is

$$\operatorname{Var}\left(\sqrt{\hat{r}}\right) = \left(\frac{d}{dr}(\sqrt{r})\right)^2 \operatorname{Var}\left(\hat{r}\right) = \left(\frac{1}{2\sqrt{r}}\right)^2 \frac{r(1-r)}{n} = \left(\frac{1}{4r}\right) \frac{r(1-r)}{n} = \left(\frac{1-r}{4n}\right) \tag{2}$$

Fellman and Eriksson [34] proposed an alternative transformation arcsin ffiffi r p ð Þ. For small values of r, the difference between the two transformations is minute. The variance of arcsin ffiffiffiffi r <sup>p</sup>\_ � � is

correct, but it shows discrepancies that are difficult to explain or eliminate. Statistical studies on empirical rates of multiple maternities can never confirm the law but serve only to identify errors too large to be characterized as random. It is of particular interest to ask why the rates of higher orders of multiple maternities are sometimes too high and sometimes too low when Hellin's law

Usually, the arguments for Hellin's law are based on stochastic models for multiple fertilizations and fissions of fertilized eggs. The influence of both multiple fertilizations and fissions of fertilized eggs has inspired scientists to associate the rates of higher multiple maternities with both monozygotic (MZ) and dizygotic (DZ) TWRs (e.g., [25–30]). The contributions by Zeleny

Peller [31] was the first, at least indirectly, to connect Hellin's law to interindividual variation in mothers' chances for multiple maternities. Later, Eriksson [12] considered recurrent twin maternities in families on the Åland Islands (Finland) and presented a modified model (in the paper, the law was called Fellman's law). When Eriksson applied this law to his Åland data, he obtained better congruence with Hellin's law than if Peller's version had been applied. Fellman and Eriksson [1] reviewed papers where the genesis of Hellin's law was traced and where the strengths and weaknesses of the law were analyzed and improvements

Hellin's law presupposes strong correlations between TWR and TRR, but even strong correlations do not prove Hellin's law, establishing only a linear relationship. Fellman and Eriksson [30] considered the correlation between the TWR and the square root of the TRR in Sweden. After elimination of influential temporal factors, they found that the correlation was positive, but not very strong. This finding indicates that, in general, Hellin's law cannot be exact. One application of Hellin's law is to compare TWR and the square root of TRR, the cubic root of

In the following, we consider formulae applicable in the statistical analysis of Hellin's law. Let the theoretical TRR be r. One has different possibilities to study the random errors of the TRR and particularly of the square root of the TRR. The first one is to estimate the standard

> ffiffiffiffiffiffiffiffiffiffi rð Þ 1�r n q

> > ; <sup>b</sup><sup>r</sup> <sup>þ</sup> <sup>k</sup>

! r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>b</sup>rð Þ <sup>1</sup> �b<sup>r</sup> n

where the factor k defines the confidence level. The Hellin-transformed TRR is

<sup>¼</sup> <sup>1</sup> 2 ffiffi r p � �<sup>2</sup>

, and the observed standard CI of r is

(1)

(2)

ffiffiffiffi r <sup>p</sup>\_. The

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>b</sup>rð Þ <sup>1</sup> �b<sup>r</sup> n

rð Þ 1 � r

<sup>n</sup> <sup>¼</sup> <sup>1</sup>

4r

� � <sup>r</sup>ð Þ <sup>1</sup> � <sup>r</sup>

<sup>n</sup> <sup>¼</sup> <sup>1</sup> � <sup>r</sup> 4n � �

deviations (SDs) of the TRR and construct confidence intervals (CIs) for r [32].

r

<sup>b</sup><sup>r</sup> � <sup>k</sup>

Var r \_� �

[25] have resulted in the law also being known as the Hellin-Zeleny law.

is used as a benchmark [32].

10 Multiple Pregnancy - New Challenges

suggested [32].

variance of

Var

QUR and so on [14, 32, 33].

Let the observed TRR be <sup>b</sup>r, then SD <sup>b</sup><sup>r</sup> <sup>¼</sup>

ffiffiffiffi r <sup>p</sup>\_ is

ffiffiffiffi r \_ � � p <sup>¼</sup> <sup>d</sup> dr

ffiffi <sup>r</sup> � � <sup>p</sup> � �<sup>2</sup>

$$\text{Var}\left(\arcsin\left(\sqrt{\hat{r}}\right)\right) = \left(\frac{\text{d}}{\text{d}\mathbf{r}}\arcsin\left(\sqrt{\hat{r}}\right)\right)^2 \text{Var}\left(\hat{r}\right) = $$

$$\left(\frac{1}{2\sqrt{r}\sqrt{1-r}}\right)^2 \frac{r(1-r)}{n} = \left(\frac{1}{4r(1-r)}\right) \frac{r(1-r)}{n} = \left(\frac{1}{4n}\right) \tag{3}$$

The variance of arcsin ffiffiffiffi r <sup>p</sup>\_ � � is slightly larger than the variance of ffiffiffiffi r <sup>p</sup>\_, but it is simpler and does not depend on r. Consider the difference

$$\arcsin\left(\sqrt{r}\right) - \sqrt{r} = \sqrt{r} + \frac{1}{2}\frac{r^{\frac{3}{2}}}{3} + \frac{1 \times 2}{3 \times 4}\frac{r^{\frac{5}{2}}}{5} + \dots \\ - \sqrt{r} = \frac{1}{2}\frac{r^{\frac{3}{2}}}{3} + \frac{1 \times 2}{3 \times 4}\frac{r^{\frac{5}{2}}}{5} + \dots \\ \approx \frac{1}{6}r\sqrt{r} \tag{4}$$

and arcsin ffi r p ð Þ� ffi r p ffi <sup>r</sup> <sup>p</sup> <sup>≈</sup> <sup>r</sup> 6. This relative difference between the transformed variables is of the dimension 10�<sup>4</sup> . The square root is a monotone-increasing function, and consequently, one can construct the CI for ffiffi r p by a square root transformation of the limits of the CI for r. Hence, for ffiffi r p the corresponding transformed CI is

$$\left(\sqrt{\hat{r}} - \mathbf{k}\sqrt{\frac{1-\hat{r}}{4n}}, \ \sqrt{\hat{r}} + \mathbf{k}\sqrt{\frac{1-\hat{r}}{4n}}\right) \tag{5}$$

Fellman and Eriksson [34] gave a mathematical proof that Hellin's law cannot hold in general. If one aggregates heterogeneous data, the fluctuations are smoothed out, but according to Hellin's law, the relation between the TWR and the TRR is not linear, and consequently, the aggregated and disaggregated data cannot simultaneously satisfy Hellin's law [1].

Jenkins [26, 35], Jenkins and Gwin [27], Bulmer [29] and later Fellman and Eriksson [30] have tried to modify the law in order to improve it. Using linear curves is the best method for identifying discrepancies from a presumptive model because graphs containing linear curves are easy to interpret. There are two possibilities for checking Hellin's law with linear curves. One is to use graphs with TWR<sup>2</sup> as abscissa and TRR as ordinate, that is, to use the model TRR <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>TWR2 . An alternative is graphs with TWR as abscissa and ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> as ordinate. Now, the model is ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>TWR [1].

Jenkins and Gwin [27] considered US data for the periods 1923–1924 and 1927–1936. They used TWR2 as abscissa and TRR as ordinate. From their figure, they obtained the linear relation TRR <sup>¼</sup> <sup>0</sup>:<sup>000013</sup> <sup>þ</sup> <sup>0</sup>:656TWR<sup>2</sup> . The intercept indicated that the line did not pass through the origin and the parameter estimate was markedly below the value one, indicating a deficit in triplet sets. When Fellman and Eriksson [32] applied a regression model to the same data set, they obtained the slightly different result: TRR <sup>¼</sup> <sup>0</sup>:<sup>000039</sup> <sup>þ</sup> <sup>0</sup>:584TWR2 . The coefficient of determination is R<sup>2</sup> <sup>¼</sup> <sup>0</sup>:842, indicating a rather good fit. They obtained a deficit in the TRR when they tested the parameter estimate against one with a one-sided t test. The SE β � �\_ ¼ 0:113 yielded t ¼ �3:7, and the estimate was significantly below one [1]. As an alternative model, Fellman and Eriksson used TWR as abscissa and ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> as ordinate. The estimated model was ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> <sup>¼</sup> <sup>0</sup>:<sup>0029</sup> <sup>þ</sup> <sup>0</sup>:679TWR and R2 <sup>¼</sup> <sup>0</sup>:844, SE <sup>β</sup> � �\_ ¼ 0:130 and t ¼ �2:5, and the obtained estimate is significantly below one. Both alternatives indicate deficits in the TRR. The parameter estimates are slightly higher for the first model, but the goodness of fit for both models is comparable. Their analyses confirm the results given in [27].

Jenkins and Gwin [27] also considered data from Finland (1878–1916). They used the data given by Dahlberg [36]. However, Fellman and Eriksson [32] performed a check based on Finnish official registers and confirmed their suspicion that Dahlberg's data contained a misprint for the maternal age group 35 to 40 years. In the analyses, they used the corrected data and present the results in Figure 7 in [32]. When they applied the linear model to the Finnish data, they obtained the results TRR <sup>¼</sup> <sup>0</sup>:<sup>00003</sup> <sup>þ</sup> <sup>0</sup>:742TWR2 and R<sup>2</sup> <sup>¼</sup> <sup>0</sup>:930. The SE β � �\_ ¼ 0:091, t ¼ �2:8, and the obtained estimate is significantly below one. The linear relation between ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> and TWR is ffiffiffiffiffiffiffiffiffi TRR <sup>p</sup> <sup>¼</sup> <sup>0</sup>:<sup>0026</sup> <sup>þ</sup> <sup>0</sup>:768TWR with R2 <sup>¼</sup> <sup>0</sup>:906. The SE <sup>β</sup> � �\_ ¼ 0:111 and t ¼ �2:1, and the obtained estimate is significantly below one. All of these results indicate good fit but deficits in triplet maternities [32].

The discrepancies between the results concerning Finnish data given by Fellman and Eriksson and Jenkins and Gwin were mainly caused by two facts; Jenkins and Gwin did not use regression models, but a geometric attempt, and they excluded in their analyses the extreme TRR for the age group 45+ years. In addition, they did not perform any statistical tests. Fellman and Eriksson [32] introduced measures to check both Hellin's law and Jenkins' [35] model in formula (6). They introduced the ratio HR <sup>¼</sup> TRR=TWR<sup>2</sup> named Hellin's ratio and assumed that it is a measure of the agreement with respect to Hellin's law. If HR > 1, there is an excess, but if HR < 1, there is a deficit in the TRR. An alternative measure is based on Jenkins' model [32]:

$$\mathbf{J} = \mathbf{T} \mathbf{R} \mathbf{R} = \frac{1}{n} \sum\_{\mathbf{i}} \mathbf{T} \mathbf{W} \mathbf{R}\_{\mathbf{i}}^2 n\_{\mathbf{i}} \tag{6}$$

Equality is obtained if and only if

At birth, the observed rates are.

<sup>w</sup> <sup>¼</sup> <sup>w</sup>0ð Þ <sup>1</sup> � <sup>c</sup><sup>w</sup> and the transformed rates ffiffi

<sup>w</sup> ≈ 2cw.

An excess for the quadruplet rate is obtained if

<sup>w</sup> <sup>þ</sup> <sup>c</sup><sup>3</sup>

<sup>w</sup> ≈ 3cw.

5. Studies including the use of Hellin's law

comparisons between the levels of twinning, triplet and quadruplet rates.

An excess for the triplet rate is obtained if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>w</sup> <sup>¼</sup> <sup>w</sup>0ð Þ <sup>1</sup> � <sup>c</sup><sup>w</sup> , <sup>r</sup> <sup>¼</sup> <sup>w</sup><sup>2</sup>

that is, <sup>c</sup><sup>r</sup> <sup>&</sup>lt; <sup>2</sup>c<sup>w</sup> � <sup>c</sup><sup>2</sup>

that is, <sup>c</sup><sup>q</sup> <sup>&</sup>lt; <sup>3</sup>c<sup>w</sup> � <sup>3</sup>c<sup>2</sup>

for all i. Consequently,

TWRi

HR <sup>¼</sup> TRR

assume that Hellin's law holds for these rates [32]. Consequently, <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>w</sup><sup>2</sup>

<sup>0</sup>ð Þ <sup>1</sup> � <sup>c</sup><sup>r</sup> and <sup>q</sup> <sup>¼</sup> <sup>w</sup><sup>3</sup>

ffiffiffiffi ni p ffiffiffiffi ni

ð Þ TWR <sup>2</sup> <sup>≥</sup>

The following step is a simple analysis of the data to show that the transformations may cause excesses in the transformed TRRs and QURs. Fellman and Eriksson [29] simplified their studies by ignoring any random effects. Assume that after the fertilization and any fissions of the fertilized egg, the twinning rate is w0, the triplet rate is r<sup>0</sup> and the quadruplet rate is q0, and

pregnancy the rates may decrease, and let the relative reductions be cw, c<sup>r</sup> and c<sup>q</sup> for the twinning, triplet and quadruplet rates, respectively. An obvious assumption is that c<sup>w</sup> ≤ c<sup>r</sup> ≤ cq.

> <sup>0</sup> 1 � c<sup>q</sup> � �,

and the variables w, r and q do not satisfy Hellin's law. A fundamental question is whether excesses in the transformed rates of triplets and quadruplets are possible. Compare

<sup>r</sup> <sup>p</sup> <sup>¼</sup> <sup>w</sup><sup>0</sup>

ð Þ 1 � c<sup>r</sup> <sup>p</sup> <sup>&</sup>gt; ð Þ <sup>1</sup> � <sup>c</sup><sup>w</sup> ,

These conditions are conceivable, and if the relative reductions in the triplet and quadruplet rates are not too strong, excesses are possible. If one speculates about these results, the extreme excesses observed for transformed quadruplet rates compared with triplet rates, would be explained by the fact that c<sup>q</sup> < 3c<sup>w</sup> is more likely than c<sup>r</sup> < 2c<sup>w</sup> [32]. Consequently, the transformations should be applied with caution and used only for descriptive purposes and not for

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 � c<sup>r</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � c<sup>q</sup> � � <sup>3</sup> q

<sup>p</sup> and ffiffi <sup>3</sup> <sup>q</sup> <sup>p</sup> <sup>¼</sup> <sup>w</sup><sup>0</sup>

> ð Þ 1 � c<sup>w</sup> ,

p ¼ TWRi

TRR <sup>J</sup> <sup>¼</sup> JR:

<sup>0</sup> and <sup>q</sup><sup>0</sup> <sup>¼</sup> <sup>w</sup><sup>3</sup>

Historical Studies of Hellin's Law

13

http://dx.doi.org/10.5772/intechopen.79583

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � c<sup>q</sup> � � <sup>3</sup> q

.

<sup>0</sup>. During

Fellman and Eriksson [29] defined Jenkins' ratio as JR ¼ TRR=J, where TRR is the total triplet rate. If JR > 1, there are excesses, and if JR < 1, there are deficits in the TRRs. Hellin's ratio can be defined for both age-specific and total rates, but Jenkins' ratio applies only to total rates. In addition, Eq. (6) indicates that JR can be calculated only for data grouped according to maternal age. Based on Schwarz's inequality, a comparison between HR for the total set of maternities and JR yields [32].

$$\left(\text{TWR}\right)^2 = \left(\frac{1}{n}\sum\_{i} \left(\text{TWR}\_i\right)n\_i\right)^2 \le \frac{1}{n}\sum\_{i} \left(\text{TWR}\_i^2 n\_i\right)\frac{1}{n}\sum\_{i} n\_i = \text{J.}$$

Equality is obtained if and only if

$$\frac{\text{TWR}\_{\text{i}}\sqrt{n\_{\text{i}}}}{\sqrt{n\_{\text{i}}}} = \text{TWR}\_{\text{i}}$$

for all i. Consequently,

data set, they obtained the slightly different result: TRR <sup>¼</sup> <sup>0</sup>:<sup>000039</sup> <sup>þ</sup> <sup>0</sup>:584TWR2

alternative model, Fellman and Eriksson used TWR as abscissa and ffiffiffiffiffiffiffiffiffi

TRR

SE β � �\_

SE β � �\_

between ffiffiffiffiffiffiffiffiffi

TRR

maternities and JR yields [32].

ð Þ TWR <sup>2</sup> <sup>¼</sup> <sup>1</sup>

n X i

<sup>p</sup> and TWR is ffiffiffiffiffiffiffiffiffi

good fit but deficits in triplet maternities [32].

TRR

estimated model was ffiffiffiffiffiffiffiffiffi

12 Multiple Pregnancy - New Challenges

cient of determination is R<sup>2</sup> <sup>¼</sup> <sup>0</sup>:842, indicating a rather good fit. They obtained a deficit in the TRR when they tested the parameter estimate against one with a one-sided t test. The

t ¼ �2:5, and the obtained estimate is significantly below one. Both alternatives indicate deficits in the TRR. The parameter estimates are slightly higher for the first model, but the goodness of fit for both models is comparable. Their analyses confirm the results given in [27]. Jenkins and Gwin [27] also considered data from Finland (1878–1916). They used the data given by Dahlberg [36]. However, Fellman and Eriksson [32] performed a check based on Finnish official registers and confirmed their suspicion that Dahlberg's data contained a misprint for the maternal age group 35 to 40 years. In the analyses, they used the corrected data and present the results in Figure 7 in [32]. When they applied the linear model to the Finnish data, they obtained the results TRR <sup>¼</sup> <sup>0</sup>:<sup>00003</sup> <sup>þ</sup> <sup>0</sup>:742TWR2 and R<sup>2</sup> <sup>¼</sup> <sup>0</sup>:930. The

¼ 0:113 yielded t ¼ �3:7, and the estimate was significantly below one [1]. As an

<sup>p</sup> <sup>¼</sup> <sup>0</sup>:<sup>0029</sup> <sup>þ</sup> <sup>0</sup>:679TWR and R2 <sup>¼</sup> <sup>0</sup>:844, SE <sup>β</sup>

¼ 0:091, t ¼ �2:8, and the obtained estimate is significantly below one. The linear relation

and t ¼ �2:1, and the obtained estimate is significantly below one. All of these results indicate

The discrepancies between the results concerning Finnish data given by Fellman and Eriksson and Jenkins and Gwin were mainly caused by two facts; Jenkins and Gwin did not use regression models, but a geometric attempt, and they excluded in their analyses the extreme TRR for the age group 45+ years. In addition, they did not perform any statistical tests. Fellman and Eriksson [32] introduced measures to check both Hellin's law and Jenkins' [35] model in formula (6). They introduced the ratio HR <sup>¼</sup> TRR=TWR<sup>2</sup> named Hellin's ratio and assumed that it is a measure of the agreement with respect to Hellin's law. If HR > 1, there is an excess, but if HR < 1, there is a deficit in the TRR. An alternative measure is based on Jenkins' model [32]:

<sup>J</sup> <sup>¼</sup> TRR <sup>¼</sup> <sup>1</sup>

ð Þ TWRi n<sup>i</sup> !<sup>2</sup>

n X i

Fellman and Eriksson [29] defined Jenkins' ratio as JR ¼ TRR=J, where TRR is the total triplet rate. If JR > 1, there are excesses, and if JR < 1, there are deficits in the TRRs. Hellin's ratio can be defined for both age-specific and total rates, but Jenkins' ratio applies only to total rates. In addition, Eq. (6) indicates that JR can be calculated only for data grouped according to maternal age. Based on Schwarz's inequality, a comparison between HR for the total set of

> ≤ 1 n X i

TWRi 2 ni � � 1

n X i

TWR2

<sup>p</sup> <sup>¼</sup> <sup>0</sup>:<sup>0026</sup> <sup>þ</sup> <sup>0</sup>:768TWR with R2 <sup>¼</sup> <sup>0</sup>:906. The SE <sup>β</sup>

. The coeffi-

¼ 0:130 and

TRR

� �\_

<sup>p</sup> as ordinate. The

� �\_

<sup>i</sup> n<sup>i</sup> (6)

n<sup>i</sup> ¼ J:

¼ 0:111

$$\text{JHR} = \frac{\text{TRR}}{\text{(TWR)}^2} \ge \frac{\text{TRR}}{\text{J}} = \text{JR}.$$

The following step is a simple analysis of the data to show that the transformations may cause excesses in the transformed TRRs and QURs. Fellman and Eriksson [29] simplified their studies by ignoring any random effects. Assume that after the fertilization and any fissions of the fertilized egg, the twinning rate is w0, the triplet rate is r<sup>0</sup> and the quadruplet rate is q0, and assume that Hellin's law holds for these rates [32]. Consequently, <sup>r</sup><sup>0</sup> <sup>¼</sup> <sup>w</sup><sup>2</sup> <sup>0</sup> and <sup>q</sup><sup>0</sup> <sup>¼</sup> <sup>w</sup><sup>3</sup> <sup>0</sup>. During pregnancy the rates may decrease, and let the relative reductions be cw, c<sup>r</sup> and c<sup>q</sup> for the twinning, triplet and quadruplet rates, respectively. An obvious assumption is that c<sup>w</sup> ≤ c<sup>r</sup> ≤ cq. At birth, the observed rates are.

$$w = w\_0(1 - c\_{\le}), \; r = w\_0^2(1 - c\_{\le}) \text{ and } q = w\_0^3(1 - c\_{\le}).$$

and the variables w, r and q do not satisfy Hellin's law. A fundamental question is whether excesses in the transformed rates of triplets and quadruplets are possible. Compare <sup>w</sup> <sup>¼</sup> <sup>w</sup>0ð Þ <sup>1</sup> � <sup>c</sup><sup>w</sup> and the transformed rates ffiffi <sup>r</sup> <sup>p</sup> <sup>¼</sup> <sup>w</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 � c<sup>r</sup> <sup>p</sup> and ffiffi <sup>3</sup> <sup>q</sup> <sup>p</sup> <sup>¼</sup> <sup>w</sup><sup>0</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � c<sup>q</sup> � � <sup>3</sup> q .

An excess for the triplet rate is obtained if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 � c<sup>r</sup> <sup>p</sup> <sup>&</sup>gt; ð Þ <sup>1</sup> � <sup>c</sup><sup>w</sup> ,

that is, <sup>c</sup><sup>r</sup> <sup>&</sup>lt; <sup>2</sup>c<sup>w</sup> � <sup>c</sup><sup>2</sup> <sup>w</sup> ≈ 2cw.

An excess for the quadruplet rate is obtained if ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � c<sup>q</sup> � � <sup>3</sup> q > ð Þ 1 � c<sup>w</sup> ,

that is, <sup>c</sup><sup>q</sup> <sup>&</sup>lt; <sup>3</sup>c<sup>w</sup> � <sup>3</sup>c<sup>2</sup> <sup>w</sup> <sup>þ</sup> <sup>c</sup><sup>3</sup> <sup>w</sup> ≈ 3cw.

These conditions are conceivable, and if the relative reductions in the triplet and quadruplet rates are not too strong, excesses are possible. If one speculates about these results, the extreme excesses observed for transformed quadruplet rates compared with triplet rates, would be explained by the fact that c<sup>q</sup> < 3c<sup>w</sup> is more likely than c<sup>r</sup> < 2c<sup>w</sup> [32]. Consequently, the transformations should be applied with caution and used only for descriptive purposes and not for comparisons between the levels of twinning, triplet and quadruplet rates.
