Calculating Divorce

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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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  1. Forgive me please if I missed it, but why not simply take the total number of marriages of any given year that subsequently was terminated with a civil divorce? Statistically you could then average out the rate over the entire population by comparing ALL marriages with living partners with the total number of all divorces. It would give you a number, but it wouldn’t reflect reality with any degree of sophistication or accuracy.

    Either way, I believe your argument is sound. It is certainly evident to me as a clergyman with 22 years experience under my belt that the 50% figure was never representative of the situation, at least in this corner of Canada.

    Fr. Tim Moyle
    Mattawa, Ontario

  2. Fr. Moyle,

    Once you find “the total number of marriages of any given year that subsequently was terminated with a civil divorce”, what do you propose to do with it? If you props to divide it by the number of marriages in that year, then I believe you have described what I referred to as “Method One.” In terms of accuracy, it is wonderful. The difficultly is the amount of time that needs to pass in order to get an accurate count. How long do you wait until you think your number is stable? (Statisticians play this game, by the way. They use what they call “Life Tables,” and typically estimate how long it takes for, say, half the divorces that are “going” to happen. They then extrapolate based on this number.)



  3. Jake: Thank you for your kind response. That’s the point that I was trying to make. Any statistic that professes to be the ‘divorce rate’ is like a quantum measurement. It’s a number that doesn’t really mean much. (apologies to Einstein)

    Fr. Tim

  4. Here’s how I’d explain it. The notion of a 50% divorce rate is based on the fact that the number of divorces in any given year is about half the number of marriages that take place in the same year.

    If you applied the same assumption to birth and death rates, since the number of deaths in any given year tends to be roughly half the number of births in that same year, one would have to conclude that only 50 percent of those children will ever die and that the remainder are immortal.

  5. Elaine,

    While agree that the method of calculation is misleading at best, I am not sure I understand the analogy you offered. The conclusion for marriage/divorce is not that the marriages outside of the divorce rate will never end, but that they will end of “natural causes”. Perhaps a better analogy would involved murder rates and death by natural causes.

    The real issue is the comparison on entirely different cohorts. A different analogy might be the following. Suppose that I have been given 100 gold coins over the course of the last 50 years. It takes me a while, but I am interested in finding out which of these are counterfeit. I go through several of these every year. It turns out that this year I was able to discover that 10 of them are counterfeit. (By the way, this doesn’t mean that these are the only counterfeits, just that ten of them were discovered this year.) It also turns out that I was given 20 more coins this year. Because I was given 20 new coins and I discovered 10 counterfeit coins in my existing collection, I report this as 10/20 = 50% counterfeit rate. It simply makes no sense. First, the counterfeits are from an entirely different collection that the current collection. Second, the counterfeits discovered this year are not even constitutive of all counterfeits in the original 100 coins (some may have been discovered in previous years, some may yet to be discovered).

    Now, the (10,20) pair is not entirely useless. It does tell us the net gain of “good” coins (or rather, “not yet discovered as bad” coins, depending on whether you are a half-full or half-empty kind of person). It tells us that we gained 20 coins and tossed out 10, so there is a 20-10=10 net gain in “good” coins. The same can be said with the marriage-divorice numbers in Method One. It tells us how many marriage have been gained in society during the course of the year.

    I would like to emphasize, however, that if the number of marriages stays constant, this faulty method does actually give a correct result. Bad-reasoning, perhaps, but a correct rate nevertheless. This is true because the “incorrect” denominator being used just happens to be the right “number”, even though it was collected from the wrong data. It is only if the number of marriage begins to change radically (either decreasing or increasing) that the two Methods begin to diverge. The greater the change in marriages, the greater the discrepancy in the two Methods.

  6. Why not simply divide the total number of married couples (new and existing) by number of divorces each year? So instead of option 2 using only new marriages that year as the numerator, you’d use all (existing) marriages. It would be something like per capita income; call it per couple divorce rate. It could be viewed for each year going back to discern trends. This would not tell us anything about the attitudes of marriage and/or divorce based on the year of marriage, or falsely give an impression about how many married this year will ultimately divorce at some point in the future. But it would wash out any significant jumps due to an aberrantly large number of marriages or divorces in any given year. So if there is a general societal openness toward divorce that is increasing, this statistic alone would be useful over time. (Of course, it would be way below the 50% myth and its publication would risk headlines declaring “Concerns about divorce found to be highly exaggerated,” etc.)

    To perhaps have something more useful and get closer to a “true” divorce rate, we could use probability to make assumptions about when the ending marriages had taken place. So even without the married year datum tied to the divorced year datum, we could say that 60% (or whatever it is) of all divorced couples have been married between 0 and X years (based on some single good study or average findings of multiple studies). Then you could count back to the range when those marriages would have taken place, and use the new marriages for each of those years to get a total, which you would divide by X years and call it M. Take 60% (or whatever the factor is) of the current year’s divorces call it D. Whatever D/M is each year could be a kind of moving divorce rate, call it estimated rate for couples married this year to get a divorce in the next X years. This stat would, better than the per couple divorce rate, reflect any societal and generational trends toward increased marriage instability.

  7. Erich,

    I think you have presented two different takes on the two Methods I outline. I actually agree with your first method, which is an alternative to “Method Two.” It simply changes the denominator. There is nothing wrong with this, and in some ways is preferred. I think you hit the nail on the head in indicating that we would have to describe what the measure actually measures, which in this case is decidedly not “how likely a marriage is to end in divorce”, but rather “the per capita number of divorces” among married people. This certainly reflects a society’s attitude towards marriage and divorce, and in a way reminds me of my call to calculate the percent of the population that is a child of a divorce or out of wedlock birth. It tell us something about the current state of marriage attitudes and impacts.

    Your second suggestions is very close to what statisticians call “life tables.” Think of it this way. I said, in describing “Method One” that one could potentially have to wait upwards of fifty year in order to know the final outcome of marriages that occurred in any one given year. However, past (I am guessing here) twenty years or so, the number of marriages that end is very low, perhaps low enough to not effect the actual rate by any statistically significant measure. Thus, maybe we only have to wait twenty years. Of course, this is still not feasible, so statisticians like to predict how long we would have to wait in order for half of the divorces that eventually will occur to actually occur. They wait this long, and then they double the number of divorces. Because marriages that break up are far more likely to do so earlier, this turn out to work pretty well. However, it is all based on probabilities, which in turn are based on past data. Essentially, this is what you are suggesting, though in a modified version. The upside is, it is doable, even in the short term, and it actually reflects what people think of as a “divorce rate” (i.g., the likelihood that a marriage will last/fail). The down side is, the result is only as good as the working probabilities. I don’t know the research enough to comment on their confidence intervals. Do they reach the 95% that we usually look for? I’m not sure.

    Thanks for your comment.

  8. Thanks for your response, Jake. I agree the probability factor used in my second suggestion would have to be fairly certain. Also, I imagine you’d preferably want it updated every year, so you are not making assumptions based on the probability rates from marriages ten years ago. As people get married later and/or people chose not to get married, it could affect the probability for divorce within a certain number of years. I don’t really have a sense of any studies or annual measure for this, as I was only “thinking out loud.” It makes me want to look a little deeper though. And to see if anyone has been calculating the per capita divorce rates; if so, I haven’t run across them. Your article really does a good job of discussing the issues in not only calculating divorce rates, but also in being precise in telling people what exactly is being measured. Thanks again!

  9. Jake, it’s been a long time since I’ve studied demography, but it looks to me like your article is correct. You see the same problems in total fertility rates (your type 1) and crude birth rates (type 2). The best you can do is apply the current rates of divorce for each year into a marriage across the current marriages. It’s an approximation.

  10. Pinky,

    Yes. Actually, in the end, I would not disagree so much with the method as I do with how it is used (misused)? Using divorced versus marriage in a particular year does tell us something … it tells us the net gain of marriage in society. If we were to look at that in a per capita sort of way, as suggested by another commenter, it would tell us something about the change in attitudes towards marriage over time. What it doesn’t tell us is how likely a marriage in a particular year is to last.

    I made very similar points a while back in a post on NFP effectiveness ratings. I have no problem with the ratings themselves (which often boast upwards of 99%), so long as people understand what they mean. They are calculated over a period of only one year. The problem is that people misunderstand this and think that the 99% is a “life long” success rating. That number is decidedly lower than 99%.

    This is tangental, of course, but confirms the original point. Numbers don’t lie, but people can lie with numbers (either purposefully or inadvertently).

  11. Unfortunately, there is no way of gathering divorce statistics that are entirely accurate. Even if there were, they would still not reflect just how successful marriages are as quite literally thousands of couples separate but do not proceed with a divorce.

  12. J.,

    (I had a student named Jason Christian when I was teaching at the seminary.)

    Yes, this is the technical term for the process describe by Erich (okay, in fairness, his was a modified version of this), and the process with which I used the phrase “life tables” as a summary. It seemed to me not prudent to go into the Calculus of probability distributions for the average reader. The difficulty, as I am sure you know, with survival analysis is that it requires the survival function, or the lifetime distribution function (its complement). This is the “probability” to which Erich refers (again, in a modified way). It begs the question, how we we construct such a function? The answer: Life Tables. It is based on data from the past and being sued to predict data in the future (hence “right censored”). This all begs the same concerns, though. Just how accurate is it?

    Here is where I have to admit that I am not an expert. (I try …emphasize “try” … never to speak past the point where I know what I am talking about.) While I understand the method, I have not had the opportunity to see what confidence intervals come out of the calculations. Are they 95%? 99%? Perhaps you are more familiar with the literature.

    In the end, the main point stands. Numbers are fine – measurements are great – so long as we know what they are measuring and using them out of context.

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