The Sex Dimorphism Ratio in our lifespan

Abstract

The purpose of this study is to examine the Sex Dimorphism Ratio (SDR) for a wide range of age groups in two environments, the suburbs of Los Angeles and Taiwan, which differ in height distribution. Using statistics, we examine if SDR will alter due to the change in woman’s reproductive viability. Two-way Analysis of Variance (ANOVA) in the form of multiple regression analysis with dummy variables and traditional regression analysis are applied. The results show that SDR approximately follows the required normal distribution, there is no interaction between age and environment, and there is no difference in SDRs among the 36 age groups. The only factor that affects the mean of the SDR is the environmental difference. The results imply that SDR remains constant throughout our lifespan despite changes in our personal reproductive viability. The ideal SDR is not fixed because different height probability distributions in different environments may have different optimal SDRs. The minimum SDR stays fairly constant, though statistically different from two different environments in our study. The maximum SDR from one environment in this study is much higher than that from the other.

Keywords

List-wise deletion method multiple regression analysis with dummy variables reproductive fitness Sex Dimorphism Ratio two-way analysis of variance

1. Introduction

Many studies have been done relating to absolute and relative preferences for human heights. In particular, Salska et al. (2008) made observations regarding those topics in their research. It has been mentioned that tall stature of males may index heritable fitness and serve as a useful cue of access to resources and socioeconomic status during development (Cassidy, 1991; Judge & Cable, 2004). Height is considered as an important feature of male attractiveness (Pawlowski & Koziel, 2002; Pierce, 1996), and women express a greater preference for taller men during the fertile phase of their ovulatory cycle (Pawlowski & Jasienska, 2005). Taller males are reported to date more often than short or average males (Shepperd & Strathman, 1989). Taller males also have generally higher reproductive success (Pawlowski et al., 2000).

Pawlowski (2003) has proposed that rather than having some fixed absolute height selection rule, the preferences for height from individuals are tailored to his or her own stature. In Pawlowski (2003) study, human mate attractions will depend on the Sexual Dimorphism Ratio (SDR), which is defined to be the male’s height divided by the female’s height. This coincides with the study of Park (2003), which indicated that being outside the typical range for any given morphological trait in height is a cue to developmental hardships or risk for chronic health problem.

Boyson (1999) also shows that preferences for sexual dimorphism are regulated by the internalized social norm that women should not be taller than men as a sign of being more dominant. So, taller women will still have stronger preferences for dating men who are taller than them.

Salska et al. (2008) show that women prefer being in a relationship with males who are shorter than the shortest with a SDR of 1.06, or the tallest at a SDR of 1.17. They predict, in general, that women will prefer men taller than themselves at an SDR of 1.10. Salska et al. (2008) also show that taller than average men and shorter than average women would be more accepting of a larger sexual dimorphism in a relationship in order to maximize their dating pool – taller men and shorter women reported a willingness to accept larger SDRs between themselves and their partner. Also, shorter men and taller women preferred a smaller SDR, which ensures that they do not choose a partner outside the typical range of variation in height.

Tovee et al. (2006) made a survey of United Kingdom (UK) Caucasian and South African Zulu observers. They found that in environments where the optimal values for different features, including height and weight, differ, the attractiveness preferences will also be different.

All of the above have made general observations about height and relative heights. This paper’s focus is slightly different from that of Tovee et al. (2006). We compared the differences of SDRs across age groups and environment within themselves, instead of studying whether features in one environment are perceived differently in another.

The purpose of this study is to compare the differences of partner’s height selections across age and environment groups, by means of Sex Dimorphism Ratio (SDR). It shows the SDR remains constant throughout our lifespan despite changes in our personal reproductive viability but the ideal mean of SDRs does not stay constant across different environments. In an environment with taller men and shorter women, like our study in the suburbs of Los Angeles, the ideal SDR will be slightly higher than the ideal SDR from an Asian society at Taiwan with less height gaps. According to our literature survey, our study is one of the first in cross-sectional comparisons about SDR.

2. Present research

In this study, we would like to find out if the ideal SDR stays constant across different age groups. Will younger women tend to have different SDR choices than older women? Will women grant more allowances to their ideal SDR when they cross certain biological turning points? In other words, will the ideal ratio for SDR stay the same when concerns about health and/or re-productivity change?

We also ask if the ideal SDR will stay the same in the western social environment, where there are more tall males available in contrast to its counterpart Asian society where there are fewer tall males, but also fewer tall females. Or, possibly, since females in the western social environment are taller than females in the eastern social environment the ideal SDR will remain constant.

It is possible that factors like environment and gender may have some interaction with the SDRs. A two-way ANOVA on Age and Environment about the SDRs from females (i.e., expected male height/female height) is presented in this paper. Similar two-way ANOVA’s on Age and Environment about SDRs from males (i.e., $=$ male height/expected female height), on Gender and Age, on Environment and Gender, or a three-way ANOVA on Age, Environment, and Gender about SDRs can be studied. However, to reduce the complexity of the analysis and the length of the paper, they are not included in this particular study. In summary, the following hypotheses from the two-way ANOVA on Age and Environment about the SDRs from females are tested:

(1) (1)
H ${}_{0,1}$ : The means of SDRs will stay the same across the age groups,H ${}_{a,1}$ : The means of SDRs will not stay the same across the age groups,
(2)
H ${}_{0,2}$ : The means of SDRs will stay the same across environment groups,H ${}_{a,2}$ : The means of SDRs will not stay the same across environment groups,
(3)
H ${}_{0,3}$ : There is no interaction between factors Age and Environment,H ${}_{a,3}$ : There is interaction between factors Age and Environment.

Tests for (1) and (2) are about the main effects from independent variables Age and Environment, respectively. They involve the independent variables one at a time. Any potential interaction is ignored in these tests. Therefore, the tests are similar to the one-way analysis of variance. The test for (3) is about the effect of interaction. It is the effect that one factor, such as Environment, has on the other, Age, regarding the SDR.

Finally, we apply a traditional regression analysis to confirm our findings. The number of observations is 3,600, and it will be very clear if there is any dependence or not from Age and Environment to SDR.
3. Data analysis

Random samples from suburban Los Angeles, similar to what Salska et al. (2008) have studied, and those from an Asian country at the age groups of 20 to 55 were obtained from Yahoo Dating sites. That is, a total of 3,600 random individuals are used in this study, coming from fifty random individuals taken from each of the thirty-six age groups ranging from the ages of 20 to 55, and two different height environments.

3.1 Data collection

Instead of using the pair-wise deletion method as in the paper by Salska et al. (2008), data were collected by the list-wise deletion method for this paper.

For the pair-wise deletion approach, if one participant reports her own height and the expectation of her date’s minimum height, but not his maximum height, the information is included in one data set. Vice versa, if the expectation of her date’s maximum height is reported, but not his minimum height, then the information is also included in another data set. In this approach, each element of a study is estimated using all available data from a survey. The problem with this approach is that the estimate of the smallest acceptable and the tallest acceptable population SDRs as well as their correlations to their own heights will be based on different sets of data with different sample sizes.

In practice, the most common approach is to use the list-wise deletion. This simply omits those cases with missing (default) data and runs the analysis on what remains. Under the assumption that data are missing completely at random, this leads to unbiased parameter estimates for the SDR. Unfortunately, when the data are not missing completely at random, a loss in power of the test statistics and/or bias may results. For example, if individuals with no photos posted in their online advertisement may be more likely not to specify a height restriction on who should contact them, the resulting SDR in this study will be biased in favor of those participants with photos posted there. In our study, since we can easily obtain a large amount of random samples from the dating website, the traditional list-wise deletion approach is adopted.

In this paper, Sampling with Replacement generated from Microsoft EXCEL under “Random Number Generation” of “Data Analysis” was used. When a random number was duplicated, it was deleted and replaced with the next available random number. If a participant opted to use the website’s default data range with Age marked 10–99, Expected Height marked 100–200 cm (or 3’0”–7’11”), and Expected Weight marked 30–150 kg (a feature only in the Asian dating site), then that data was list-wise deleted and replaced with the next one available without default inputs. Independent simple random samples of 50 observations are selected from each age group of ages from 20 to 55. Every data is carefully checked to avoid the possibility of replication.

The dating websites are free to register. For Yahoo in the Asian country, participants can opt to pay a fee to be registered as a VIP member. Approximately 5% of participants chose to become a VIP member. A VIP member can contact both VIP and non-VIP members, but non-VIP members can only contact VIP members and lose the opportunity to contact other non-VIP members, which constitute 95% of the participant pool.

We are unable to verify whether users of online dating services are truthful when indicating their own physical traits and preferences for mate choice. For instance, we will not be able to verify if some participants change from the default maximum height of 8’11” (272 cm) to 8’ (244 cm) or 7’11” (241 cm) intentionally or for convenience of data input. Due to the nature of online data collection, this study is only about the “expectation” of a mate’s SDR.

3.2 Test statistics

In this paper, we will apply two-way Analysis of Variance (ANOVA) with replication model to our data analysis.

In the two-way ANOVA, there are two independent variables, called factors, affecting the dependent variable. In our case, the two factors are Age and Environment, and the dependent variable is the SDRs. Each factor will have two or more levels within it. In our study, we are considering age groups from age 20 to age 55 (36 levels), and two different environment, western vs eastern (two levels). We will have 50 replications (observations) per each combination of age level and environment level. A typical two-way ANOVA table is as follows:

Source of variation	SS	df	MS $=$ SS/df	F-Ratio
Rows (age)	SSR	r $-$ 1	MSR $=$ SSR/(r $-$ 1)	F ${}_{1}=$ MSR/MSE
Columns (environment)	SSC	c $-$ 1	MSC $=$ SSC/(c $-$ 1)	F ${}_{2}=$ MSC/MSE
Interaction	SSI	(r $-$ 1)(c $-$ 1)	MSC $=$ SSC/(r $-$ 1)(c $-$ 1)	F ${}_{3}=$ MSC/MSE
Errors	SSE	rc(n $-$ 1)	MSE $=$ SSE/rc(n $-$ 1)
Total	SST	rcn $-$ 1

where SS $=$ the Sum of Squares, df $=$ the Degrees of Freedom, MS $=$ the Mean of Squares (MS) $=$ SS/df, r $=$ the number of rows (blocks), in our study, r $=$ 2 (Eastern and Western), c $=$ the number of column (treatments), in this study, c $=$ 36 (Age 20 to Age 55), n $=$ the number of observations (replicates) in each cell, in our study, n $=$ 50, SSR $=$ the Sum of Squares due to Rows, SSC $=$ the Sum of Squares due to Columns, SSI $=$ the Sum of Squares due to Interaction, SST $=$ the Total Sum of Squares.

F-ratios, F ${}_{1}$ , F ${}_{2}$ and F ${}_{3}$ above follow F-distributions with degrees of freedom (r $-$ 1, rc(n $-$ 1)), (c $-$ 1, rc(n $-$ 1)), and ((r $-$ 1)(c $-$ 1), rc(n $-$ 1)), respectively. They are used to test the hypotheses H ${}_{01}$ vs H ${}_{a1}$ , H ${}_{02}$ vs H ${}_{a2}$ , and H ${}_{03}$ vs H ${}_{a3}$ , respectively, as stated in the last session.

When the null hypothesis H ${}_{01}$ or H ${}_{02}$ is rejected, the T-tests for independent two samples can be used to test if there are any differences between two individual mean SDRs from different age group or environment.

3.3 Assumptions for test statistics

The assumptions for a two-way ANOVA model are as follows:

(1) (1)
The populations from which the samples were obtained must be normally or approximately normally distributed;
(2)
Samples must be independent;
(3)
The variances of the populations must be equal;
(4)
Each cell must have the same sample size.

Since the sample size is large at 50 in each cell, the first assumption requesting normality can be released for the ANOVA. Nevertheless, we performed a normality test on the populations in this study via converting the two-way ANOVA model to a multiple regression model with dummy variables.

Since samples are randomly chosen from the websites, the requirement for being independent is satisfied. However, through the residual analysis for a multiple regression model, independency of samples can also be checked.

We can check if all the variances are equal or not for the two-way ANOVA model in experimental design by transferring the two-way ANOVA to a multiple regression with dummy variables, especially given that no interaction between Age and Environment was discovered later in our study. The multiple regression model is suggested as follows:

$\displaystyle y_{ijk}=\beta_{0}+\beta_{1,1}D_{1,1}+\beta_{1,2}D_{1,2}+\ldots+% \beta_{1,35}D_{1,35}+\beta_{2,1}D_{2,1}+\beta_{2,2}D_{2,2}+\ldots+\beta_{2,36}% D_{2,36}+\varepsilon_{ijk},$ (1)

where $D_{ij}$ are dummy variables, and error terms $\varepsilon_{ijk}$ are independent, and follow a normal distribution with a mean of 0 and a constant variance.

Since we have thirty-six age groups (from the age of 20 to 55), and two cultural backgrounds (western and eastern environment), there will be $36*2=72$ possible categorical effects. In this situation, $72-1=71$ dummy variables will be occurred in the multiple regression model.

$D_{1,1}=$ 0, if observations are 21 years old from the eastern environment; $D_{1,1}=$ 1, otherwise;

$D_{1,2}=$ 0, if observations are 22 years old from the eastern environment; $D_{1,2}=$ 1, otherwise;

……….

……….

$D_{1,35}=$ 0, if observations are 55 years old from the eastern environment; $D_{1,35}=$ 1, otherwise;

$D_{2,1}=$ 0, if observations are 20 years old from the western environment; $D_{2,1}=$ 1, otherwise;

$D_{2,2}=$ 0, if observations are 21 years old from the western environment; $D_{2,2}=$ 1, otherwise;

……….

……….

$D_{2,36}=$ 0, if observations are 55 years old from the western environment; $D_{2,36}=$ 1, otherwise;

Thus, $\beta_{0}$ is the mean or the expected value of outcomes for the age group of 20 and from eastern environment; $\beta_{1,1}$ is the difference between the mean values of outcomes from the age group of 20 and the age group of 21 from the eastern environment; $\beta_{1,2}$ is the difference between the mean values of outcomes for the age group of 20 and the age group of 22 from the eastern environment; and $\beta_{2,36}$ is the difference between the mean values of outcomes for the age group of 20 and from eastern environment, and the age group of 55 from the western environment.

A F-test for the hypothesis H ${}_{0}$ : $\beta_{1,1}=\beta_{1,2}=\ldots=\beta_{1,35}=$ 0 indicates that there is no difference between the mean values of outcomes from the age groups of 20 to 55 in Eastern environment. Similarly, a F-test for the hypothesis H ${}_{0}$ : $\beta_{2,1}=\beta_{2,2}=\ldots=\beta_{2,36}=$ 0 indicates there is no difference between the mean values of outcomes from the age groups of 20 to 55 in Western environment.

Individual H ${}_{0}$ : $\beta_{i,j}=$ 0 indicate there is no difference between the mean values of outcomes from different age group versus that from the age group of 20 in Eastern, respectively.

Table 1
ANOVA table for SDR_Mini $=$ the expected lowest male height/female height

Source of variation SS Df MS F P-value F crit

Environment 0.048 1 0.048 27.026 0.000 3.844

Age 0.086 35 0.002 1.403 0.059 1.426

Interaction 0.064 35 0.002 1.048 0.392 1.426

Within 6.205 3528 0.002

Total 6.403 3599

Table 2
ANOVA table for SDR_Maxi $=$ the expected highest male height/female height

Source of variation SS df MS F P-value F crit

Environment 2.328 1 2.328 439.581 0.000 3.844

Age 0.249 35 0.007 1.341 0.087 1.426

Interaction 0.167 35 0.005 0.902 0.633 1.426

Within 18.686 3528 0.005

Total 21.430 3599

In our study, we will not show those testing hypotheses since the F-test for a two-way ANOVA analysis from SPSS as stated in Session 3.2 has already given us the result. We will only check the assumptions about normality, independency, and equal variance for the two-way ANOVA model via the residual analysis for the multiple regression.

Finally, we would like to mention that the SDRs are ratios of two possible (normal) random variables (regarding heights). Therefore, it may result in a non-central Cauchy distribution with no means so that Central Limit Theorem for parametric statistics will not apply. That is, none of the above statistical analyses under the constraints of normality may apply. Fortunately, the non-central Cauchy distribution situation did not happen to our study. The SDRs in our study do follow a normal probability distribution. We have also run the analysis for the differences in heights. The results are similar to what we have for the ratios, SDRs.
4. Results

Source of variation	SS	Df	MS	F	P-value	F crit
Environment	0.048	1	0.048	27.026	0.000	3.844
Age	0.086	35	0.002	1.403	0.059	1.426
Interaction	0.064	35	0.002	1.048	0.392	1.426
Within	6.205	3528	0.002
Total	6.403	3599

Source of variation	SS	df	MS	F	P-value	F crit
Environment	2.328	1	2.328	439.581	0.000	3.844
Age	0.249	35	0.007	1.341	0.087	1.426
Interaction	0.167	35	0.005	0.902	0.633	1.426
Within	18.686	3528	0.005
Total	21.430	3599

4.1 The variable age has no effect on the means of SDR’s

The F-ratios and the $p$ -values from the two-way ANOVA for the minimum and maximum SDRs are listed in Tables 1 and 2. If the $p$ -value is greater than 0.05 or the F-ratio is less than or equal to the F critical value in the last column, it implies there is no difference among mean SDR’s; otherwise, if the $p$ -value is less than 0.05 or the F-ratio is greater than the F critical value in the last column, we conclude that there is a difference among the means for SDRs. From the statistical results in Tables 1 and 2, we conclude that there is no interaction between age and environment on the SDRs. The variable Age has no effect on the means of the SDRs either. The main cause of the differences among the means of SDRs is the difference of environment. Therefore, we can apply the multiple regression model with dummy variables as stated in Eq. (1), Session 3.3 to check the validity of model assumptions for a two-way ANOVA model.

Figure 1.

P-P plot for the minimum SDRs.

Figure 2.

Residual plot for the minimum SDRs.

4.2 The assumptions for a two-way ANOVA analysis is valid

In regression analysis, the residual for the i-th observation is defined to be the difference between the observed value of the dependent variable (SDR) and the estimated value of it from Eq. (1) in Session 3.3. Residuals provide the best information about the assumptions for the error term $\varepsilon_{ijk}$ in Eq. (1). To check if observations are independent and follow a normal distribution with a mean of 0 and a constant variance, we can examine the graphical plots of residuals, or standardized residuals. Standard residuals are defined to be the residuals divided by their corresponding standard deviations. Figures 1 and 2 show the p-p plot, and the standardized residual plots for the minimum SDRs. When residuals or the standardized residuals cluster around the 45 degree line, it indicates that the error term follows a normal probability distribution. Similar results can be found for the maximum SDRs.

From the p-p plot for the standardized residuals, it is reasonable to conclude that the error terms, as well as the observations, have a normal probability distribution, even though it appears to deviate slightly from the normal distribution guidelines for the maximum SDRs. Since there is no specific pattern observed, for instance, a trend, a curve, or an increasing of the variations, in the residual plots for the minimum and maximum SDRs, it is reasonable to conclude that the residuals (as well as the observations) are independent with a constant variance.

4.3 Environment difference may cause the difference on the mean (or the optimal) SDR’s

Table 3 shows the statistical results from the follow-up T-test from two independent samples (East vs West). It shows that the average of the minimum SDR from the eastern environment is 1.05, which is close to that of 1.06 from the western environment, but they are statistically significantly different. The average of the maximum SDR from the eastern environment and the western environment are 1.15 and 1.20, respectively. They are statistically significantly different also. It implies that the variations of heights among men and women from different environment may cause the difference of ideal SDRs.

Table 3
T-test results from two independent samples

	Minimum SDR		Maximum SDR
	East	West	East	West
Mean	1.054	1.061	1.147	1.198
Variance	0.001	0.002	0.002	0.008
Observations	1800	1800	1800	1800
Pooled variance	0.002		0.005
Hypothesized mean difference	0.000		0.000
df	3598		3598
t stat	$-$ 5.187		$-$ 20.941
P (T $\leqslant$ t) one-tail	0.000		0.000
t critical one-tail	1.645		1.645
P (T $\leqslant$ t) two-tail	0.000		0.000
t critical two-tail	1.961		1.961

Table 4

Regression analysis results for minimum SDR from two environments

	Coefficients	Standard error	T-stat	P-value
Intercept	1.050833417	0.002715698	386.9478	0.0000
Age	7.578E-05	6.74287E-05	1.123853	0.2611
Dummy	0.007267333	0.001400939	5.187474	0.0000

Table 5

Regression analysis results for maximum SDR from two environments

	Coefficients	Standard error	T-stat	P-value
Intercept	1.151942284	0.006421298	179.3940028	0.0000
Age	$-$ 0.000132615	0.00016502	$-$ 0.803633636	0.4216
Dummy	0.069456888	0.009081086	7.648522051	0.0000
Age*dummy	$-$ 0.000495864	0.000233373	$-$ 2.124767267	0.0336

Figure 3.

P-P plot for the maximum SDRs.

5. Traditional regression analysis

We can alternatively express the relationship between our variables through the following traditional regression model:

$\displaystyle Y=\beta_{0}+\beta_{1}(\text{Age})+\beta_{2}(D)+\beta_{3}(\text{% Age}*D)+\varepsilon$

where $D$ is the dummy variable for two different environments and $\text{Age}*D$ is the interaction variable of the Age and the Environment. The results shown in Tables 3 and 4 are consistent with our findings from our two-way ANOVA analysis, with $p$ -values for the Age being greater than 0.05 indicating no impact on the SDR, and the $p$ -values for the Environment (Dummy Variable) being less than 0.05 indicating the impact on the SDR. There are also no interactions between the Age and the Environment for SDR_Mini but a minor interaction between them for SDR_Maxi.

Normality tests in Fig. 3 also shows that SDR_Maxi is slightly off from the normal distribution, which may be due to SDR_Maxi outliers.

6. Conclusion

There appears to be an ideal SDR across all ages in different environments. In our study, the ideal average SDR from the sample from an Asian environment at Taiwan is around 1.10, and that from the counter environment in western in the suburbs of Los Angeles is slightly higher at about 1.13. Somewhat unexpectedly, there was no difference on SDR due to different ages, given previous assumptions about height preferences driven by mate selection for reproductive fitness. It is possible that SDR remains constant throughout our lifespan despite changes in our personal reproductive viability because preferences for height may have formed earlier in life and developed into an entrenched predisposition. The variations of the heights between women and men from two different regions may cause the difference of ideal SDR. Studies of valuating ideal SDRs from the male perspective can also similarly be done.

In summary, this article compares the differences of partner’s height selections across the age and environment groups by means of Sex Dimorphism Ratio (SDR). It shows the SDR remains constant throughout our lifetime span despite changes in our personal reproductive viability. The means of the ideal SDRs do not stay constant (as claimed to be 1.10 by Salska et al., 2008). By comparing different environments, when an environment with more taller men and shorter women, like our study in the suburban of Los Angeles, the ideal SDR will be statistically significantly higher than the ideal SDR from an Asian society with less height gaps.

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The Sex Dimorphism Ratio in our lifespan

Abstract

Keywords

1. Introduction

2. Present research

3.1 Data collection

3.2 Test statistics

3.3 Assumptions for test statistics

4.1 The variable age has no effect on the means of SDR’s

4.3 Environment difference may cause the difference on the mean (or the optimal) SDR’s

Table 3 T-test results from two independent samples

6. Conclusion

References

Table 3
T-test results from two independent samples