Matthew A. Carlton

Human Sex Ratios and Sex Distribution in Sibships of Size 2

ABSTRACT We previously analyzed data from the U.S. National Health Interview Survey (NHIS, 1998 to 2002) on families with two biological children (10 years of age and younger) and found that the distribution of families with two boys, two girls, and one boy + one girl did not statistically conform to a binomial distribution regardless of the boy/girl sex ratio used. Using the best estimate of the sex ratio from the data, we found that there were significantly more families with opposite-sex siblings than families with same-sex siblings. No biological mechanism could explain these results at the time. In the present study we conducted an analysis of the first two children in sibships of size 3 from the same data source and found that there are significantly more same-sex sibships than unlike-sex sibships. Combining the two sets of data for the first two children produced observed numbers in close agreement with the expected numbers. A hypothesis of parental choice (family planning) appears to be strongly supported as an explanation for the discrepancies in the two sets of data individually. For example, parents who have a boy and a girl (either order) as their first two children are more likely to stop having children (“stopping rule”) than are parents whose first two children are of the same sex.

Making Babies by the Flip of a Coin

Many probability and genetics textbooks pose standard questions about eye color, birth defects, sexes of children, and so on. Solutions to these questions, specifically about sexes, generally make two assumptions: first, that a randomly selected embryo is equally likely to be male or female; second, that the sexes of successive children from the same parents are independent. In other words, probabilists (and some geneticists) treat sexes of children like flips of a fair coin: two possible outcomes, each equally likely, with outcomes independent from trial to trial. But are these assumptions realistic? Demographic data suggest that neither a balance of sexes nor true independence exist in nature. Yet most textbooks, both in genetics and probability theory, continue to use the binomial distribution as an acceptable approximation for solving genetics problems involving live-birth sex ratios in species where sex is determined by an XX versus XY chromosome mechanism. We look at a widely circulated article in Parade magazine regarding the gender distribution in human families with two children and analyze comparable data from federal sources to show that such families do not conform to any binomial distribution. The sequence of investigations we take here could be followed in an introductory or intermediate probability and statistics course.

Probability with applications in engineering, science, and technology

Probability.- Discrete Random Variables and Probability Distributions.- Continuous Random Variables and Probability Distributions.- Joint probability distributions and their applications.- The Basics of Statistical Inference.- Markov chains.- Random processes.- Introduction to signal processing.

The Most Widely Publicized Gender Problem in Human Genetics

Abstract In two-child families containing at least one boy, the expected probability that such a family has two boys is 1/3, provided that the boy/girl (B/G) ratio is 1.0 and the population to which they belong has a binomial distribution of BB, (BG + GB), and GG families. It is commonly known that in most human populations the sex ratio at birth (i.e., the ratio of the number of boys to the number of girls) is greater than 1.0. Teachers and textbook writers seldom discuss the more realistic expected distributions in populations where the sex ratio is greater than 1.0. We present data from two federal surveys with sex ratios greater than 1.0 and find that the observed proportions of two boys in families of size 2 with at least one boy range from 0.3335 to 0.3941. It has been reported in the literature that the probability (p) of a male birth is subject to both within-sibship variation (Poisson variation), for which our data are suggestive, and possibly also between-sibship variation (Lexis variation). These deviations (biases) from the assumptions of a simple binomial distribution are involved in the calculation of values of p and standard 95% confidence intervals, thereby foiling attempts to make reliable statistical inferences from the data. Analysis of the data is also complicated by family planning that falsifies the assumption of randomness in the binomial gender distribution model. Families of size 2 (and their sex composition) are often discussed in a wider context. Overpopulation in some parts of the world has caused mass starvation and threatens to do the same worldwide unless the birth rate drops to agriculturally sustainable levels. Even if every woman of fertile age has only two children on average from now on, the worlds population is predicted to continue growing toward 9 billion people by 2050. Other sociological problems are bound to follow. Although the birth rate in China has recently dropped, the average age of the population has risen, so that by 2035 it is projected that for each person over age 65 there will be just three working-age people. Furthermore, Chinas one-child policy has already led to a sex imbalance where there is a large excess of men for whom marriage and parentage is denied.

Bayesian Statistics for Biological Data: Pedigree Analysis

I n teaching biology, there may be a tendency to concentrate too much on the descriptive aspects of the subject. A well-rounded education in the biological sciences also requires experience in the gathering and statistical analysis (interpretation) of quantitative data from field or laboratory studies. There are numerous mathematical tools and computer programs to help us do this today. Introducing students to some of these tools and their practical applications should be part of every biology class. One of these tools is known as Bayesian analysis. The specific purposes of this report are to:

The Truth about Models: How Well Do Mechanical Models Mimic the Observed Gender Distributions in Two-Child Families?

ABSTRACT We question the use of mechanical models, such as coin flipping, to represent the probabilities of gender distributions in sibship families consisting of two children. Both the assumptions of the models and the reliability of the data should be evaluated. Using models without these critical evaluations may tend to perpetuate myths rather than elucidate biological realities.

Probability and Statistics for Computer Scientists

Devore, J. L. (2003), Probability and Statistics for Engineering and the Sciences, Pacific Grove, CA: Duxbury. Montgomery, D. C. and Runger, G. C. (2007), Applied Statistics and Probability for Engineers, Hoboken, NJ: Wiley. Ryan, T. P. (2007), Modern Experimental Design, Hoboken, NJ: Wiley. Vining, G. G., and Kowalski, S. (2005), Statistical Methods for Engineers, Pacific Grove, CA: Duxbury. Walpole, R. E., Myers, R. H., Myers, S. L., and Ye, K. (2007), Probability & Statistics for Engineers & Scientists, New York: Prentice Hall.

The Basics of Statistical Inference

The overarching objective of statistical inference is to draw conclusions (make inferences) based on available sample data. In this chapter we generally assume that data have been acquired by observing the values of a random sample X1, X2, …, X n ; recall from Sect. 4.6 that a random sample consists of rvs that are independent and have the same underlying probability distribution (what we also called iid). For example, highway fuel efficiency of a certain type of vehicle might have a normal distribution with mean μ and standard deviation σ. Then each observed fuel efficiency value would come from this normal distribution, with the various observed values obtained independently of one another—a normal random sample. Or the number of blemishes on a new type of DVD might have a Poisson distribution with mean value μ. If n of these disks were to be randomly selected and the number of blemishes on each one counted, the result would be data from a Poisson random sample. In either example, the values of the parameters would typically not be known to an investigator. The sample data would then be used to draw some type of conclusion about these values.

Joint Probability Distributions and Their Applications

In Chaps. 2 and 3, we studied probability models for a single random variable. Many problems in probability and statistics lead to models involving several random variables simultaneously. For example, we might consider randomly selecting a college student and defining X = the student’s high school GPA and Y = the student’s college GPA. In this chapter, we first discuss probability models for the joint behavior of several random variables, putting special emphasis on the case in which the variables are independent of each other. We then study expected values of functions of several random variables, including covariance and correlation as measures of the degree of association between two variables.