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Statistical Ideas: Celebrations, & annihilations

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Posted August 12, 2015
This news or article is intended for readers with certain scientific or professional knowledge in the field.

What are the chances that a child will predecease a parent?

  1. <4% (~1 in 50 children)
  2. 4%-8% (~1 in 20 children)
  3. 8%-16% (~1 in 8 children)
  4. 16%-24% (~1 in 5 children)
  5. >24% (~1 in 2 children)

This is of course a dispiriting question, since many of us are parents, and all of us are by definition children who have parents. None of us want to fantasize the loss of anyone in our family.  Psychology teaches us that the only natural drive is to see comfort and life for everyone we love.  We write this article because these are still important dynamics (life and death) that we are all -at some point- subject to.  And this has been the case since well before the dawn of mathematics.  Intriguing looks into probability models would allow us to better think through and appreciate how these transitions occur and make any life adjustments accordingly.

The answer to the question, of course, is that it could narrowly depend, on some key assumptions one might make about the problem and broad hereditary characteristics.  But otherwise the correct answer is simply B (4%-8%).  One could justify, basedonly on the specific assumptions we note later, that some of the other answer choices are feasible.  But if we take away the highly-tailored accompanying arguments, none of these other choices would ever apply.

The intensely agonizing concept of a parent losing a beloved child has been around since well before Joe Biden introduced this part of his biography in the 2007 primaries and ultimately stated to military families in 2012:

No parents should be pre-deceased by their sons or daughters … I, unfortunately, have that experience, too.

Earlier this summer Vice President Biden became the first sitting President or Vice President, in 50 years, to again experience the loss of a child.  But the odds from this statistic misguides you in a couple directions, since it only reflects sitting officials (commonly older than age 30), and also reflects the death of any child of a parent.  The probability of the latter, of course, increases with multiple children (or when considering both parents instead of the mid-parent).  This makes it doubly unfortunate this year for the Vice President.

But what happened with 46 year-old Beau Biden has been sadly copied through all time, for many people and in many stations of life.  I personally know nearly 50 people in this situation. 16th century English poet William Shakespeare wrote in King John:

Grief fills the room up of my absent child,

Lies in his bed, walks up and down with me,

Puts on his pretty look, repeats his words,

Remembers me of his gracious parts,

Stuffs out his vacant garments with his form.

18th century polymath and one of the Founding Fathers of the U.S., Benjamin Franklin had a more uplifting take in a condolence letter to a family member:

… it is the will of God and Nature that these mortal bodies be laid aside, when the soul is to enter into real life … a man is not completely born until he be dead: Why then should we grieve that a new child is born among the immortals?

Similar to our working with the U.S. government actuary tables in previous articles, we can not probabilistically model this in a theoretical closed-form, but must instead rely on an open-form computer model.  We prove the solution here through myriad of 10,000 simulations per assumption and per life table.  We also introduce the 1980 life table, and repurpose the 1950 life table that we’ve learned from prior articles (here, here, here).  The 1950 table allows for the mid-parent to be roughly 30 years old (1980-1950), when the child is born ~1980.  These assumptions cut through the middle of the population distribution.

Below we show these 2 death distributions, superimposed on top of one another, for the mid-parent and the child.  For the more modern actuary table (see the yellow distribution below), we can freely censor beyond the 2nd highest life increment of 85 years. Since even with a number of humans ever to live, well exceeds 100 billion (so a remarkable ~95% have expired, mostly with heartbreak), there has still been zero cases where a child lives to even 80 and still predecease their parent.  So the entire probability math here remains correct even with this difference in approximation at the high-end of the two government life tables.

1

To get accustomed to this death chart above, we can appreciate that 94% of mid-parents born in 1950, are alive at age 30.  Babies born in 1950 had a 3.5% mortality rate before they reached age 5.  By 1980 we see this statistic improve to 1.5%.  But things balance out a little later, as both cohorts have an equal 6% mortality (through age 30).  And this bias is pertinent, since we only use the mortality for those born in 1980 in our analysis.  The mortality of children born in 1950 could never later become parents!

Additionally, ~45% of children born about 1980 will be alive beyond their 80th birthday; a segment we noted has always outlived their parents in absolute time (assuming similar age difference tables).  The popular media should never quantitatively andgrammatically baffle medical research in headline language, as has been the case recently (here, here, here), that children currently are expected to die before their parents.  A quick look at the chart above shows that is absurd (used twice) to say this is the typical case, in absolute time.  What the news media confusingly meant is that relative to their own birth, the life expectancies of children are briefer (incidentally due to obesity for which Coca-Cola foolishly purports to have no responsibility).

Next, we show the typical random match-ups in the family death years.  See the sample distributions below (sorted by death of mid-parent).

2

It is imperative to also assume that due to genetic factors, those children’s deaths tend to deviate from the norm somewhat similar to how their parent’s deaths deviate from the norm.  This is basis for modern regression analysis, invented by 19th century English scientist Francis Galton (and cousins to Charles Darwin).  One can see (here) my Statistics Topics book or recent Georgetown lecture notes, for the original crude sketches Francis Galton used to think about his tangled concepts, which we currently use in a simpler version.  Such as confidence intervals, and distributions of dependent and independent variables.  Studying the stature of hundreds of children and their mid-parents, Francis Galton authored in Anthropological Miscellanea that extreme heights in parents were only “partly” transferred to children:

The experiments showed further that the mean filial regression towards mediocrity was directly proportional to the parental deviation from it.

And therein lay the birthplace of the prevailing commonplace expression, “regression towards the mean” and the general popularity of the term regress.  Similar to our death data we see a similar “regression” dynamic on the extremes; else we’d currently be living in a universe exclusive to freakish extremes.  Infant deaths and centenarians (all of who are all either midgets or giants!)  While Francis Galton didn’t invent the concept of correlation (ρr), we’ll use it to encompass our dynamic here.  And so we’ll also redo this illustration with a correlation of ½ between the child and the mid-parent.  This is up from the 0% correlation we had earlier (implying perfect independence).  We’ll instead sort this time by the death of the child, so that you can see the illustration both ways.

3

One can notice that unlike with r=0, here whenever the yellow marker is on an early date, there are far more red markers belowthat time in the case of r=½.  In other words, with some correlation, fewer parents die at a later time (after their children).  We are still at (but on the low end of) the 4%-8% probability range for a child dying before the parent.  So answer B above is still valid.

Note that if we stretched ourselves to absurdly assume a r=-1, between the child and the parent, then we’d fall into answer C. Additionally, only if we assume the probability analysis begins at a later stage of a child’s life (say a 60 year old “child”, as opposed to a child at birth), then clearly only then could we justify a lower probability solution (i.e., answer A).

The math cleanly works so that -at some point- a child would grow up to be a parent himself or herself.  And this child is less likely to die ahead of his or her own parent, then to die ahead of his or her own child.  The opposite is senseless (though only rarely possible), where a grandparent sees his or her own grandchildren dying followed by the death of his or her children.

We can see an additional illustration to help show these death distributions.  In the two, dual illustrations below (one for r=0, and another for r=½), we first show -in the top row- the upper and lower halves of the parent’s death distribution, in addition to their paired distribution for the child’s death.  Second we flip this around and show –in the bottom row- the upper and lower halves of the child’s death distribution, in addition to the paired distribution for the parent’s death.  Again this one is for r=½.

4

Finally, we give a scatter point of the bi-variate death distributions, so that we can see the frequency of children dying in relation to the parent.  For ease, we show only one plot, at r=¼.

5

We can clearly see that roughly 5% (still within answer B) of the deaths fall beneath the dark red colored equality line.  And the distribution beneath this dashed line represents where the child predeceases the mid-parent.

We note that more than half of the U.S. Presidents have had children predecease them!  This high fraction is mostly due to anychild qualifying.  For the 6 most recent cases (covering 15 Presidents in a span of >85 years), we disclose those ages on the chart above for illustration only.  Albeit the life tables for most of these don’t fully correspond to the tables we have used above.  For George H. W. Bush, we simply show him at his current age since he is not deceased.  With these data all hugging the bottom of the chart, we see the bias referred to earlier.

Now some may have tried to answer different questions, including what is the probability that any parent will pass away after anyof their children.  Instead of focusing on a single child and a single parent, we open up to more complex probability mathematics. Clearly there could not be a specific upper-bound on how high this probability could go, particularly once we include the possibility of a various death scatterings among parents and children.  So only in this case either answer D or E could apply.  Again, if you were not thinking about specifically incorporating the trivial probability example of having multiple children and parents’ combinations, then answers D and E would never apply.

We wrap-up with a small sample of recent celebrities who have either been a child predeceasing the (adoptive) parents, or a parent being predeceased by a child.

Child Parent
1. River Phoenix 7. Charlie Chaplin
2. Gary Coleman 8. Johnny Carson
3. Cory Monteith 9. John Travolta
4. Steve Jobs 10. Marlin Brando
5. Amy Winehouse 11. Paul Newman
6. Michael Jackson 12. Mike Tyson

Source: Statistical Ideas

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