Alaska governor Sarah Palin is the Republican candidate for Vice President of the United States of America.

Whether you like her or you don’t like her, you must admit that she is the subject of intense fascination by the media and by the public.

She has been described as “Sarah Barracuda.” She has created for herself the description “pit bull with lipstick.” She can field-dress a moose.

PowerPoint prepared by Gary Simon, September 2008.

The youngest child of Vice-Presidential candidate Sarah Palin was born April 18, 2008. Trig Palin is Sarah Palin’s fifth child, and he has Down’s Syndrome.

Source: Anchorage Daily News archive, 22 APR 2008, at Web site http://www.adn.com/626/story/382864.html

Shortly after Sarah Palin was selected to run with John McCain, rumors claimed that Trig was not really her child. These rumors put forward the idea that Trig’s mother was in fact Bristol Palin, Sarah’s teenage daughter.

We will ask, from a statistical perspective, whether these rumors can be believed.

… but what allows such a rumor to get started in the first place?

It’s not hard to imagine that a family might obscure a teenager’s pregnancy by just incorporating the baby into the family as a new “sibling.” Moreover, Bristol Palin had been kept out of view with a five-month case of mononucleosis.

The rumors were helped along by the fact that Sarah Palin seemed to manage her pregnancy in a strange way. While eight months pregnant, she travelled to Texas to give a speech at an energy conference. While in Texas, her amniotic fluid leaked (or broke open?), but she persevered with the speech. She then travelled back to Alaska. She did not appear visibly pregnant, and airline personnel were unaware of her condition.

She did not deliver the baby at a hospital near the airport in Anchorage, but instead was driven back home to Mat-Su Medical Center in Wasilla.

This is odd strategy for a woman with four children. In her defense, we have this story:

Sarah, 44, had her water unexpectedly break (just a slow leak of amniotic fluid, she shared) while attending an energy conference in Texas, but checked in with her doctor, was able to stay, deliver a 30 minute speech, and then get on a plane to return to her home state in time for the delivery (which ended up needing to be induced) early this morning.

“I am not a glutton for pain and punishment. I would have never wanted to travel had I been fully engaged in labor. After four kids I knew what labor felt like, and I wasn't in labor.”

This dates back to April 2008. It’s from the Web site http://www.celebrity-babies.com/2008/04/alaskan-governo.html.

But let’s get to the statistical question.

The problem has to be framed very carefully, as it is always tricky to apply a probability calculation to the situation that caused the question to be asked in the first place. That is, it’s very hard to ask directly about Trig, Sarah, and Bristol.

So let’s ask first about babies in general. What are the ages of their mothers?

This chart shows the percentage by age of mother:

Source: http://www.guttmacher.org/pubs/journals/2926897.html (downloaded 5 SEP 2008)These are for white women, New Jersey 1990. (Yes, there may be other, better sources.)

These values are not fertility numbers of the form .

Instead, they are ratios of the form , done for babies born in the target year and counting the women by age groups. We will use these as P(Age).

babieswoman

womenbaby

For our purposes, it helps to present these in single years of mothers’ ages. This can be done through smoothing and interpolation. The numbers at the outer ends (young mothers, old mothers) are less well-determined.

Here is a smoothed/interpolated version of P(Age) :

We need also information about the rate of Down’s Syndrome. These are conditional probabilities in the form

P(DS | Mother’s age = x).

The data on the chart given next were obtained from http://www.mothers35plus.co.uk/down.htm (downloaded on 5 SEP 08). These were smoothed and extrapolated to cover the range 14 years to 49 years.

Let’s convert this to posterior, or ex-post, information through Bayes’ formula:

DS

xxDSDSxP

AgePAge|P|AgeP

The denominator will be computed as

P P | Age P Agey

DS DS y y

The counter y will run from 14 to 49. The denominator works out to 0.001637.

As a particular example of this,

P | Age 32 P Age 32P Age 32 |

PDS

DSDS

= 0.0695230.001714 0.0664

0.001637

These can be assembled (by Excel, say) together on one detailed chart.

The prior P(Age) and posterior P(Age | DS ) are here shown together.

The bumpiness is related to data quality issues.

Sarah Palin was 44 at the time Trig was born.

P(Age = 44 | DS ) = 0.011492 ≈ 1.15%

Bristol Palin was 16 at the time Trig was born.

P(Age = 16 | DS ) = 0.002126 ≈ 0.21%

This is strong, but not overwhelming, evidence that Trig really is Sarah’s child, as opposed to Bristol’s.

The statistical calculations are about randomly selected babies. It is misleading to apply these probabilities to any particular baby.

Certainly Trig Palin is not a randomly selected baby.

This has compared P( Age = 44 | DS ) to P( Age = 16 | DS ) in a general context.

This can not be used as a direct commentary on Trig’s parentage. Still, these conditional probabilities help push aside the rumor.

We should believe that Trig really is Sarah’s baby.

The Down’s Syndrome diagnosis makes this case. If Trig was not a DS baby, would we compare

P( Age = 16 ) = 0.0060to

P( Age = 44 ) = 0.0008 ?

The question of parentage can be answered, but it is not likely that the Palin family will permit the collection of decisive data.

All babies (boys and girls) have mitochondrial DNA (mt-DNA) that are direct copies of their mother’s mitochondrial DNA . Since Sarah and her daughter Bristol have the same mt-DNA, this particular line of inquiry will not decide the question at hand.

More detailed genetic tests would resolve the question.

The “virtually” here refers to the possibility that there will a few mutations. These mutations are of such low probability that there can be only a small number of them. The issue of paternity will be decided correctly.

Paternity can also be determined for girl babies, but the procedure is more statistically complicated.

Trig is a boy, and therefore his male sex chromosome (Y-chromosome) should be virtually identical to that of Sarah’s husband Todd.

If Trig’s Y-chromosome is not a match to Todd’s Y-chromosome, then Todd is not the father.

So what would we conclude?

It could mean that the father is Bristol’s boyfriend and that Bristol is the mother.

It could also mean that Sarah is still the mother, but the father is someone other than her husband Todd. This is a pure scientific statement, but it would be scurrilous gossip to even suggest it.

If Trig’s Y-chromosome is a match to Todd’s Y-chromosome, then Todd is the father.

Then what would we conclude?

This strongly supports the notion that Todd is the father and Sarah is the mother.

It could also be possible that a brother of Todd (who would have the same Y-chromosome) could be the father. This is also in the category of scurrilous gossip.

Yes, it could be that Todd is the father and his own daughter Bristol is the mother. This is scientifically possible, but this gets beyond scurrilous. Let’s not go there.

Let’s make a clean honest summary. The numeric facts support the claim that Trig is Sarah’s son. Trig’s Down’s Syndrome makes the case rather strong.

There are many, many relevant things that we know about Sarah Palin. This election can really be decided on issues.

…. but it probably won’t be…..