Earlier today I reported on a study by M. V. Lee Badgett and Christy Mallory, who analyzed dissolution rates from same-sex marriages in New Hampshire and Vermont and reported a same-sex divorce rate of approximately 1% per year, which is quite a bit below the divorce rate of conventional marriages.

Here’s Badgett and Mallory describing what they did:

The table . . . shows that a total of 5.4% of New Hampshire couples and 3.6% of same-sex couples in Vermont divorced in the first four years or so of marriage equality. That corresponds to an average rate of 1.1% annually for the two states. This is slightly lower than the annual rate of divorce among different-sex couples, which is about 2% annually.

Take the observed four-year divorce rate of approximately 4%, divide by 4 because we have 4 years of data, and you get an annualized rate of about 1%.

Right? Wrong.

Badgett and Mallory were wrong in their calculation (I think), and so was I in reporting it.

The error is subtle, and I learned of it via an email from a demographer, who wrote:

Looking at the way they did things, it seems to me that they understate divorce rates by roughly a factor of two in their calculation. What they want is an occurence-exposure rate, which is obtained by dividing the number events by the person-years of exposure. They have the events (e.g. 132 divorces in Vermont). They then need to estimate the exposure. They do this by dividing by the total number of marriages (about 3,700 for VT) and dividing again by the years that same-sex marriage has been allowed (about 4.33 VT).

A moment’s reflection (or a bit longer in my case) makes it clear that this overstates the person-years of exposure. Since not all of the couples married 4.33 years ago, they should not all be counted as contributing 4.33 years. The average couple married half way through the interval, and so contributed only about 2 years.

So, this means we should double their “Average annual dissolution rate” to get something that is comparable to the divorce rate they are calculating for the general population.

A factor of 2—that’s a lot! In particular, it completely destroys the finding that same-sex marriage dissolution rates are lower than traditional-marriage divorce rates. Once you correct for that factor of 2, you get a rate of 2% per year, same as for traditional marriages.

OK, here’s the math

Calculating a rate sounds easy: it’s just numerator divided by denominator. For example, if you want to calculate the divorce rate for same-sex marriages, you just divide the number of divorces divided by the number of marriages. And if you want to compute the divorce rate by year, you need to divide by the number of years of each marriage, up to the point of divorce.

For example, suppose 4000 couples get married during a 4 year period–1000 in year 1, 1000 in year 2, 1000 in year 3, and 1000 in year 4–and suppose that, each year, there’s a 2% chance that a marriage will end in divorce. Then what will happen:

Year 1: 1000 couples get married, 10 get divorced. (If they all got married on January 1, it would be 20 getting divorced, but if they’re getting married uniformly throughout the year, 10 will get divorced.)

Year 2: 1000 more couples get married, 10 of them get divorced, and, of the remaining 990 from year 1, 19.8 get divorced (in expectation); call that 20. So 30 couples get divorced in year 2.

Year 3: 1000 more marriages, 10 of them get divorced, and of the remaining 1960, another 40 (approx) get divorced. So 50 get divorced in year 3.

Year 4: 1000 more marriages, 10 of them get divorced, and of the remaining 2910, another 60 (approx) get divorced. So 70 get divorced in year 4.

In total out of the 4 years: 4000 marriages, 160 divorces.

A quick calculation would show a 4-year divorce rate of 160/4000 = .04. And then if you divide by 4 to get a one-year divorce rate, you get a rate of .01, or 1%.

But that quick calculation would be wrong! As demonstrated in the above calculation, the divorce rate in this example is 2% per year, not 1%.

In summary

Unless I made a mistake (which is always possible!) or Badgett and Mallory’s calculations are not as they seem, they made a mistake and underestimated the per-year dissolution rate of same-sex marriages by a factor of 2.

Given the evidence available, the dissolution rate of same-sex marriages seems comparable to, not lower than, the divorce rate of traditional marriages.

In any case, as I noted in my earlier post, the statistics on same-sex marriages are likely to change rapidly over the next decade, as the population of people getting married changes.

For now, though, let me retract the earlier headline (“Same-sex couples less likely to get divorced than straight couples . . .”) and replace it with “Same-sex couples about as likely to get divorced than straight couples . . .”).