Calculating Divorce

Several days ago, Creative Minority Report posted a video interview with comedian Steven Crowder on the state of marriage in our country.  Before I get on with my own comments, I should say that Crowder makes several good points, and overall his spiel is very pro-marriage.  Give it a watch if you haven’t already seen it.

The “myth” that caught my attention is the one about a 50% divorce rate.  If it is indeed a myth, then I have certainly been taken in by it.  For, not only have I believed it for several decades, but I have found myself irresponsibly quoting it without having an actual source.  (Such is the case with myths, yes?)  I suppose the purpose of this post is not much better, because still don’t have a source.  However, the mathematician in me go to thinking about how one might go about “measuring” the rate of success in marriage at a given point in time.  Rarely do numbers lie, but people (and people’s lack of basic statistical understanding) often lie with numbers.  I made a similar point a while back with the the myth of the “99% effectiveness” of Natural Family Planning.

In other words, studies are often perfectly clear on their methodology, but most people have no idea what the studies actually measure, and they misapply the end results.

Let’s think about two different methods one might use to measure the current “divorce” rate.

The first method is the obvious one.  It is entirely accurate, but altogether impractical.  If we want to know the divorce rate for marriage that occurred in the year 2011, we take all those who were married and wait until one of two things happen: the couple divorces or one of the spouses passes away.  The marriage in which a couple passes away are deemed “successful”, whereas the ones that divorce are not.  With a simple division, we have our divorce rate.  Unfortunately, this means we have to wait until at least a half a decade in order to report on the success of marriage in any one given year.  For, although it is unlikely that a couple who is married past fifty years will end up divorcing, we cannot be sure – so we must wait it out.  (Of course, at any given moment, we could count the number of divorces and say, “The divorce rate for 2011 is at least x%.”)  This method seems to assume that divorce is a product of cultural attitude at the time of marriage.  In other words, we blame the failure of marriage on the year in which the marriage occurred.

The second method is the flip side of the first method.  It is quite easy to do, but perhaps not all that accurate.  We count the number of marriages that occurred in 2011, and we count the number of divorces that occurred in 2011, and we divide.  The upside is that all the information is available at the close of the year.  The down side is that we are comparing apples to oranges.  (Additionally, in theory very strange results could occurs, such as divorce rates above 100% .. unlikely, of course, but in this scheme, theoretically possible).  This method assumes that marriages fall apart based on current cultural attitudes, not on the attitudes in the year in which the couple was married.  Perhaps that is better, yet there still seems something wrong with counting divorces and marriages with an entirely different set of couples and then attributing the result to that particular year.

To illustrate how these calculations might differ, let’s come up with some hypothetical data.  I admit that I am over-simplifying the situation, but the goal is to point out the difference that results between the two calculations, not to give an accurate description of divorce in our country.  Because it is easier to begin with method one, we will assume that we have a 40% divorce rate that never changes.  Further, we will assume that 10% of the marriages end within the first year, 10% in the second year, 10% in the third year, and then 5% per year in years 4 and 5.  After year 7, no more divorces occur for that cohort.  (We attempt here to model the phenomenon that marriages that last tend to last!)  We will also assume for the sake of simplicity, that the number of marriages climbs by 10% every year.  Finally, we have a hypothetical starting data for the year 2000.  In order to compare results, we will need to wait through at least one cohort length, but we will extend it to two cohorts, or ten years.  Thus, our data looks like this

(My apologies for the small image.  Open it in a new window to see the full calculations and results.)

I have only totaled the years after 2004 because this is the first year we have all the divorce information (due to our assumption that no divorce takes place after five years of successful marriage).

Let’s look at the year 2005.  We know from our assumption that Method One yields a 40% divorce rate.  What does Method Two yield?  Method two suggests that we divide the number of divorces by the number of marriage in that year.  This gives us 505,510/1,610,510 = 31.4%.  There is quite a difference, yes?  (An 8.6% difference to be precise.)

Let’s see what happens as we progress through 2010.  Remember, we decided to keep a constant “Method One” divorce rate of 40%.  It turns out, and I’ll leave the reader to check this, that the 31.39% rate continues into the subsequent years.   (As a challenge, can you prove that a constant “Method One” rate yields a constant “Method Two” rate?)  Why is Method Two lower?  Because it is counting divorces with a higher cohort than might be appropriate – a number that ends up in the demoninator.  Of course, this is because the number of marriages is increasing throughout the years.  (Again, as a challenge, can you prove that if the number of marriages stays constant, there is no difference between the Method One rate and the Method Two rate?)  If the number of marriages decreases, then the Method One rate is less than the Method Two rate.  As an example, suppose that the number of marriages decreases by 10% rather than increases.  The Method One rate is still 40%, but the Method Two rate comes out to be 53.2%.

If you are savvy with a spreadsheet or a programming language, you can play around with the Method One rate and the way in which it is broken down (I broke 40% into 10%, 10%, 10%, 5%, and 5%) to see just how far apart the two method can get.  For instance, when I broke down the 40% into 10%, 10%, 5%, 5%, 5%, 1%, 1%, 1%, 1%, and 1%, the Method One 40% rate came out to a Method 2 rate of 30.1%.  The farther into a marriage that divorce is allowed to go in our model, the farther apart the two calculations get.  (Incidentally, that was with a 10% growth in marriages every year.  With a 10% decline, the 40% rate led to a 57.4% Method Two calculation.)

There are, of course, all sorts of auxiliary points.  For instance, the comedian seemed to suggest that people were afraid to get into marriage at all, in which case the rate we are really interested in is the divorce rate for first time marriages.  This will clearly be different than when we take into account all marriages.  Further, while it might be true that divorce numbers (in any calculation) might be dropping, let us not conclude that this means that marriage itself is becoming more successful.  It could mean that the number of marriages itself it dropping (or at least not growing as much as it once was).  With an increase in cohabitation, I would have to imagine that we are experiencing less marriage than perhaps would have been predicted given the rate of growth of population.  More to the point, those who chose not to get married are also those that would have been more susceptible to divorce.  (This is my intuition, not the result of actual data.)

Completely tangental, perhaps a more interesting number, especially as an educator, would be to look at the percent of the population who are the children of either a divorce or an out of wedlock relationship.  Conversely, this would mean looking at the percent of the population whose parents are either still together or have suffered the loss of a spouse.  If we are talking about the impact of divorce on future society, this seems like a valuable number to know, and the calculation is much more straightforward the the divorce rate.

I can’t say that I have read the research in front of me that proposes a near 50% divorce rate.  Likewise, I haven’t seen the research that backs up the numbers quoted by Steven Crowder.  What I can say is that it is not altogether unthinkable that both numbers were arrived at in scientific papers, each calculating the rate of divorce differently.  What this means for our casual conversation is this: try to understand what a statistic means before quoting it, and I include myself in this docile chastisement.

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