Tanya Khovanova's Math Blog

Once I was at a party and a woman was complaining that her car insurance bills were enormous. Her expensive car was hit three times while it was parked. She was whining about how unfair it was for her to be paying increased insurance premiums when it hadn’t been her fault. I didn’t tell her my opinion then, but I’m going to write about it now.

Though such things can happen, it is possible to reduce the probability of your parked car being hit.

In my personal experience the most frequent parking accident happens when someone backs out of a driveway and there is a car parked in a space which is usually empty. People often back out of their driveways on autopilot. If you park on a narrow street with no other cars — a sign that people don’t usually park there, do not park across from a driveway or close to a driveway.

There are many other common sense ideas. Don’t park at a corner. Choose the better lit areas. Don’t park next to a truck or a van, because they might not see you very well and if they hit you, they’ll do more damage. Don’t park next to an old, battered car because they have less to lose than you do. New cars are the best neighbors. Not only are owners of new cars usually more careful, but new cars are also often leased. And people who lease a car are even more careful, because they have to return it in good order.

When you are choosing a perpendicular parking spot, here’s a cute idea. Pick cars with four doors as your neighbors. Cars with two doors have bigger doors and if you are too close, they might scratch you.

Here’s what I would have told that woman: If your car has been hit so many times while parked, you should rethink your parking strategy.

When you talk over the phone with an adult stranger, you can generally determine if this person is male or female. From this, I conclude that the voice characteristics are often noticeably different for males and females. There are many other characteristics that have a different distribution by gender — for example, height.

My question is: “Can we find a trait such that the distributions are the same for both genders?”

Trying to find the answer, I remembered what we learned in high school about the genetics of eye color. I checked the Internet on the subject and discovered that the story is somewhat more complicated than what I studied 30 years ago, but still we can say that eye color is defined by several genes, which are located on non-sex chromosomes. That means, your eye color depends on the genes your parents have and doesn’t depend on your sex. A boy and a girl from the same parents have the same chances for any particular eye color.

Since eye coloring has nothing to do with gender, women and men are equal in the eyes of eye colors.

Does that mean that if we check the distribution of eye color for the world population, the distribution histogram will be the same for men and women? That sounds like a logical conclusion, right? I would argue that this is not necessarily the case.

Let me remind you that the distribution of eye color depends on the country. China has an unprecedented gender imbalance, with 6% more men than women in its population. As the eye color of Chinese people is mostly dark brown, this creates an extra pool for a randomly chosen man in the world to have a darker eye color than for a randomly chosen woman. If we exclude China from consideration, we can still have different distributions. For example, in Russia the life expectancy for women is 15 years longer than the life expectancy for men. Consequently, Russia has 14% less men than women, while globally the male/female sex ratio is 1.01. Therefore, eye colors common in Russia will contribute to female eye colors more than those of male.

What if we consider only one country? Let us look at the US. Immigrants to the US are mostly males. If the distribution of eye color for immigrants is different than the distribution for non-immigrants, then male immigrants contribute more to the eye color distribution than female immigrants.

There are so many factors impacting eye color distribution, that it isn’t clear whether it’s possible to find a group of people other than siblings in which the distribution of eye color would be the same for women and men.

We see that eye color distribution, which theoretically doesn’t depend on gender, when measured in a large population can produce different distributions for men and women.

Recently I wrote a theoretical essay titled “Math Career Predictor”, where I assumed that the distribution of math ability is different for men and women. In reality, there is no good way to measure math ability, hence we do not have enough data to draw a complete picture. For the purposes of this discussion let us assume that we can measure the math ability and that Nature is fair and gave girls and boys the same math ability. My example with eye color shows that if we start measuring we might still see different distributions in math ability in boys and girls.

My conclusion is that if we measure some ability and the distribution is different for boys and girls, or for any other groups for that matter, we can’t just conclude that boys and girls are different in that ability. For some distributions, like voice, we probably can prove that the difference is significant, but for other characteristics, different distribution graphs are not enough; we need to understand the bigger picture before drawing conclusions.

Suppose N mothers live in a city. Half of them have one child and half of them have two children. That means that an average mother has 1.5 children.

Suppose we pick the sexual orientation of every child by rolling dice. Let’s assume that a child has a 10% probability of being homosexual.

The number of mothers with one child who is homosexual is 0.05N. The number of mothers with two children both of them homosexual is 0.005N. The number of mothers with two children with only the first child homosexual is 0.045N, which is the same as the number of mothers of two children with only the second child homosexual. The total number of mothers who have two children with at least one of them homosexual is 0.095N.

Let’s calculate the average fertility of a mother with at least one homosexual child. It is (1*0.05N + 2*0.095N)/(0.05N + 0.095N) = 0.24/0.145 = 1.66. The resulting number — 1.66 — is much bigger than 1.5, the average number of children for a mother.

This means there is a correlation between homosexuality and the fertility of mothers. This suggests that there is a gay gene which at the same time is responsible for female fertility.

But the model is completely random — there can’t be any correlation.

Where is the mistake?

Obviously, you can substitute homosexuality with having blue eyes or math ability or whatever, but I invented this paradox while I was working on my “Fraternal Birth Order Threatens Research into the Genetics of Homosexuality” post. Besides, there is some research on correlation between homosexuality and fertility.

I look forward to your solution to this puzzle.

According to a famous study: ” [E]ach older brother increases a man’s odds of developing a homosexual sexual orientation by 28%-48%.”

This means that sexual orientation is not uniquely defined by genes in the way that our blood type is defined. Indeed, if homosexuality was similar to blood type, the probability of giving birth to a homosexual would be the same for the same parents, independently of birth order. Furthermore, the fact that identical twin brothers quite often have different sexual orientation also supports my claim that it isn’t like blood type. On the other hand, there is some research that shows that there is a genetic component to sexual orientation. This might mean that there are genes that increase a predisposition in a man to become gay or it could be that there is a gay gene which determines their homosexuality for some part of the gay population — or both.

The research into a genetic component of gayness shows that there might be some genes in the X chromosome that influence male homosexuality. It also shows that the same genes might be responsible for increased fertility in females.

By the way, this fascinating research provides an explanation of why “gay genes” — if they exist — do not die out, as evolutionary laws might lead us to expect.

The theory that the probability that someone will be gay is dependent on the fraternal birth order impacts in several interesting ways on the whole field of research into the genetics of homosexuality.

Let us create a theoretical model where homosexuals would be born randomly with a fixed probability if they are the first sons of a woman, and increasing probability for subsequent sons. In this model there would be two interesting consequences:

• Mothers of homosexual men would be more fertile on average than mothers of men in general. Remember, this random consequence can be misinterpreted as a genetic correlation between homosexuality and fertility.
• If a homosexual person has a brother, then the probability that this brother is homosexual might be very different from the probability that a random person is homosexual. Again, this might create the suggestion that there is a genetic component, when it is not there.

My point is that all genetics research on homosexuality should take into account these two consequences and adjust for them. That means that it is not enough to show that brothers are more likely to be homosexual in order to prove that there is a genetic component. It needs to show that the correlation between relatives is much stronger than the correlation resulting from the birth order.

For people who are not mathematicians, I will build simple models that illustrate my points. Let us consider some extreme theoretical examples first.

First model. Suppose mothers only give birth to sons and only to one or two sons. Suppose the first son has a zero probability of being gay (which means that first sons are never gay) and the second son has a probability of one of being gay (which means that second sons are always gay). Then all mothers of gay men will have two sons, while mothers of random boys will have somewhere between one and two sons. Another result will be that all gay boys in this model will never have a gay brother.

Second model. Suppose mothers only give birth to one or three sons. Suppose the first son has a zero probability of being gay and the second and the third sons have a probability of one of being gay. Then all mothers of gay men will have three sons, while mothers of random boys will have somewhere between one and three sons. Another result will be that all gay boys in this model will always have a gay brother.

Third model. Let us take some more realistic numbers. Let us consider only the case of women who have one or two boys. Let a be a probability of a woman to have one boy, and correspondingly, 1-a to have two boys. Suppose N is the total number of women in consideration. Suppose x is the probability of the first boy to be homosexual and y is the probability of the second boy to be homosexual. We assume that these probabilities are independent of each other.

Let us first estimate the total number of boys born. It will be aN + 2(1-a)N. The number of homosexuals that are born is expected to be xN + y(1-a)N. The probability of a random boy to be a homosexual is (x + y(1-a))/(2-a).

Let us see what happens with fertility. For a random boy (including both homosexual and non-homosexual boys), what is the average number of sons his mother has? It would be one son for aN boys who are from one-boy families and 2 sons for 2(1-a)N boys who are from two-brother families. Hence a mother of a random boy has on average 2 – a/(2-a) children.

Let’s see what happens with homosexual boys. We have axN homosexual boys from one-brother families and (1-a)(x+y)N gay boys from two-brother families. Hence the average number of sons for their mothers is (axN + 2(1-a)(x+y)N)/(axN+(1-a)(x+y)N) = (ax + 2(1-a)(x+y))/(ax + (1-a)(x+y)) = 2 – ax/(x+y-ay). If we denote by r the ratio y/x, then the result will be 2 – a/(1 + (1-a)r). If r is more than 1, then mothers of gays are more fertile then average mothers. If we assume that y = 1.5x and a = .5, then the average number of sons for a mother of a random boy is 1.667 and for a homosexual boy, it is 1.714, a 3% increase. The impact will be stronger if we take into consideration three-boy families.

Now let us look at brothers in two-son families. There is a total of (1-a)N such families and they have a total of (x+y)(1-a)N gay boys. We have 2xy(1-a)N gay boys who live in families in which both brothers are gay. Hence, the probability of a gay boy who has a brother to have a gay brother is 2xy/(x+y). We saw earlier that the probability of a random person to be gay is: (x + y(1-a))/(2-a).

Let’s look at these numbers more closely. You can easily see that if x = y these two probabilities are the same — as they should be. If x = 0 and y > 0, then a gay person never has a gay brother, suggesting negative correlations with a genetic component. Suppose y = rx, where r is a constant. Then the first number is 2rx/(1+r) and the second number is: x(1+r(1-a))/(2-a). We see that the ratio of these two numbers doesn’t depend on x and is equal to 2r(2-a)/(1+r)(1+r-ar). Suppose r = 1.5, then the ratio is 2.4(2-a)/(5-3a). If a = .5, this ratio is 1.03. So, in this model, for a gay boy who has a brother, the probability that this brother will be gay is 3% higher than the probability that a random man will be gay.

I am so fascinated with the fact that a property that depends on the birth order can create an illusion of a genetic component. I am not discounting the possibility of a genetic component for male homosexuality, but I urge researchers to recalculate their proofs, adjusting for the impact of the fraternal birth order.

To be fair, the female fecundity correlation with male homosexuality was shown not only in mothers of gay men, but also in maternal aunts. Also, the increase in the probability of being gay for a brother of a gay man is very much higher than 3% in my model. That means these researches might survive my critique. Still, they ought to look at their numbers again.

Finally, this discussion is not really about homosexuality, but about any property that depends on birth order. In this case, such a property might imply a genetic component that doesn’t really exist.

I read an article published in US News and World Report: Alcohol in Teens Leads to Adult Woes. This article describes the discovery that teenagers who drink heavily are much more likely to become alcoholics and have mental disorders and depression when they become adults, and that they are much less likely to finish college or be satisfied with their jobs.

This correlation is not surprising. Have you ever seen a depressed alcoholic satisfied with his/her job?

For me, the interesting question is what the word “leads” in the title “Alcohol in Teens Leads to Adult Woes” means. One might interpret “leads” as indicating that alcohol in teens causes the adult woes. If we persuade our teenagers to abstain from alcohol, will they have fewer problems in their adult lives? Will it help if you install pictures of a cirrhotic liver as a screen saver for your child’s computer?

In the middle of the article, there is a sentence that correctly states:

“What these data don’t tell us is whether those kids were already predisposed to have problems or whether drinking helped cause the trouble.”

Who is the genius who came up with a title that contradicts the article? Did they even read the article? Flashy titles sell better, but such contradictions show disrespect to the reader.

The truth is that correlations are usually insufficient to prove causality; a different type of research is needed. It appears that some of it was actually done. An interesting article, “A longitudinal study of alcohol use and antisocial behaviour in young people,” describes the study that investigated the causality between alcohol and woes. In this study, they started with three hypotheses about the long-term causality:

• Alcohol use causes antisocial behavior
• Antisocial behavior causes alcohol use
• Both above statements are true: alcohol use causes antisocial behavior and the reverse

I didn’t check this study, but the fact that they are trying to compare different hypotheses is encouraging. The result of this study is that the data supports the second hypothesis — in the long run, antisocial behavior causes alcohol use. That means the correct title for the article in the US News and World Report should have been: “Teen Woes Lead to Adult Alcohol.”

So what can you do to stop your teen’s antisocial behavior? There are many studies on that subject too. I do not know if they are correct, but you might consider a fish diet for your teen or sign up your child to train dogs.

My English teacher and editor Sue Katz wrote a funny blog entry about masturbation: “Sex and the Single Hand: Stroke Your Way to Health”

I followed the link of one of the studies she mentions to the BBC article “Masturbation ‘cuts cancer risk'”, where ” … They found those who had ejaculated the most between the ages of 20 and 50 were the least likely to develop the [prostate] cancer.”

When I hear such results, my first question is, “How was the study conducted?” It appears that “Australian researchers questioned over 1,000 men who had developed prostate cancer and 1,250 who had not about their [past] sexual habits.” The problem with asking people about their sexual habits 30 years ago is that there are a large number of dead people you can’t ask. What if the most active masturbators have died from fatigue?

Should you masturbate more to reduce your cancer risk as the BBC suggests?

Prostate cancer might not be related to masturbation at all, but rather to something else that correlates with masturbation.

• It could be that men who have a higher libido have less prostate cancer.
• Or that men who have more free time have less prostate cancer.
• Or that men who are not depressed have less prostate cancer.
• Or that men who have higher speed Internet connections have less prostate cancer.

In case you are wondering how one’s Internet connection is related to all this, let me remind you of a joke about a conversation between two geeks.

— “When you look at a girl, what do you notice first?”
— “Her hair, then her eyes, then her nose, then her lips — I have dial-up.”

One thing I know for sure: women who masturbate have even less prostate cancer than men who masturbate. Hooray for masturbation!

I stumbled upon an article in the Boston Globe (February 29, 2008) titled “Study: Spanking children affects their sex lives as adults.” Here are some quotes:

New research by a University of New Hampshire domestic abuse expert says spanking children affects their sex lives as adults. …[C]hildren who are spanked are more likely as adults to coerce partners to have sex, to have unprotected sex and to have masochistic sex.

The classical method to prove scientifically that spanking affects something is to find many parents of newborn identical twins and persuade them to treat their children the same way, with the exception that they spank one and do not spank the other. The researchers should then compare these twins in their adulthood. Such a project is impossible, as parents are likely to start feeling guilty towards the child they spank, for you can’t separate spanking from the whole package of how parents treat their children.

I decided to study the study. I found a more detailed description of the study in the Concord Monitor. As I suspected, the study was a survey. The survey found a correlation between spanking and “undesirable” sexual behavior. As every statistician knows, correlation doesn’t prove causality.

Here’s another quote from the study: “The best-kept secret in child psychology is that children who were never spanked are among the best behaved.” Did it occur to anyone that the best behaved children do not need spanking?

Could it be that parents who spank their children are tired, impatient and less loving? Could it be that not being loved as a child affects your sexual behaviour as an adult much more than spanking?

Could it be that parents who spank their children are more aggressive in general? Could it be that they pass their aggressive genes to kids and their kids’ aggressive behavior is related not only to upbringing, but to genetics?

Do not get me wrong. I do think that spanking is bad. I am saying that the study doesn’t prove that spanking is affecting anyone’s future sex life.

I am surprised that so many magazines republished the article without thinking. Now all the country is fooled into believing that the easy way to improve their kids’ future sex life is to stop spanking.

Go ahead! Stop spanking. Love your children too.

I just read the following in Women’s Health Magazine (March, 2008; page 54): Visa conducted a study of 100,000 fast food restaurant transactions. They found that people who pay with credit cards spend 30% more on food than people who pay cash.

The article concludes with the suggestion to pay cash, so you spend less and lose weight.

My question is: Who is more incompetent, Visa or Women’s Health Magazine?

Perhaps people who do not have credit cards are poorer and more price-conscious; hence, they spend less on food. This might explain the correlation. Here’s another possible explanation: people who are ordering for large groups might prefer to pay with a credit card. Or, maybe stores do not like using credit cards for small transactions, so they encourage people to pay cash for modest orders.

The main rule of statistics is that correlation doesn’t mean causality.

There are several possible answers to my question about incompetence:

• The study wasn’t described correctly in the WH magazine. In this case we can’t say much about the competence of Visa, but WH looks bad.
• The study was described correctly, but the conclusion belongs to WH. In this case Visa is innocent and WH is incompetent.
• The study was described correctly and Visa suggested the conclusions. In this case both are incompetent  — Visa for its conclusions and WH for printing them.

It could well be that paying cash makes you stingier, or at least more price-conscious, but I can’t trust Women’s Health Magazine any more. One thing I know for sure is that math can help you lose weight. Math allows you to differentiate a good study from a dumb study.