How we really can study divorce using just five questions and a giant sample

It would be great to know more about everything, but if you ask just these five questions of enough people, you can learn an awful lot about marriage and divorce.

Questions

First the questions, then some data. These are the question wordings from the 2013 American Community Survey (ACS).

1. What is Person X’s age?

We’ll just take the people who are ages 15 to 59, but that’s optional.

2. What is this person’s marital status?

Surprisingly, we don’t want to know if they’re divorced, just if they’re currently married (I include people are are separated and those who live apart from their spouses for other reasons). This is the denominator in your basic “refined divorce rate,” or divorces per 1000 married people.

3. In the past 12 months, did this person get divorced?

The number of people who got divorced in the last year is the numerator in your refined divorce rate. According to the ACS in 2013 (using population weights to scale the estimates up to the whole population), there were 127,571,069 married people, and 2,268,373 of them got divorced, so the refined divorce rate was 17.8 per 1,000 married people. When I analyze who got divorced, I’m going to mix all the currently-married and just-divorced people together, and then treat the divorces as an event, asking, who just got divorced?

4. In what year did this person last get married?

This is crucial for estimating divorce rates according to marriage duration. When you subtract this from the current year, that’s how long they are (or were) married. When you subtract the marriage duration from age, you get the age at marriage. (For example, a person who is 40 years old in 2013, who last got married in 2003, has a marriage duration of 10 years, and an age at marriage of 30.)

5. How many times has this person been married?

I use this to narrow our analysis down to women in their first marriages, which is a conventional way of simplifying the analysis, but that’s optional.

Data

I restrict the analysis below to women, which is just a sexist convention for simplifying things (since men and women do things at different ages).*

So here are the 375,249 women in the 2013 ACS public use file, ages 16-59, who were in their first marriages, or just divorced from their first marriages, by their age at marriage and marriage duration. Add the two numbers together and you get their current age. The colors let you see the basic distribution (click to enlarge):

2011-2013 agemar figures.xlsx

The most populous cell on the table is 28-year-olds who got married three years ago, at age 25, with 1068 people. The least populous is 19-year-olds who got married at 15 (just 14 of them). The diagonal edge reflects my arbitrary cutoff at age 59.

Divorce results

Now, in each of these cells there are married people, and (in most of them) people who just got divorced. The ratio between those two frequencies is a divorce rate — one specific to the age at marriage and marriage duration. To make the next figure I used three years of ACS data (2011-2013) so the results would be smoother. (And then I smoothed it more by replacing each cell with an average of itself and the adjoining cells.) These are the divorce rates by age at marriage and years married (click to enlarge):

2011-2013 agemar figures.xlsx

The overall pattern here is more green, or lower divorce rates, to the right (longer duration of marriage) and down (older age at marriage). So the big red patch is the first 12 years for marriages begun before the woman was age 25. And after about 25 years of marriage it’s pretty much green, for low divorce rates. The high contrast at the bottom left implies an interesting high risk but steep decline in the first few years after marriage for these late marriages. This matrix adds nuance to the pattern I reported the other day, which featured a little bump up in divorce odds for people who married in their late thirties. From this figure it looks like marriages that start after the woman is about 35 might have less of a honeymoon period than those beginning about age 24-33.

To learn more, I go beyond those five great questions, and use a regression model (same as the other day), with a (collapsed) marriage-age–by–marriage-duration matrix. So these are predicted divorce rates per 1000, holding education, race/ethnicity, and nativity constant (click to enlarge)**:

2011-2013 agemar figures.xlsx

The controls cut down the late-thirties bump and isolate it mostly to the first year. This also shows that the punishing first year is an issue for all ages over 35. The late thirties just showed the bump because that group doesn’t have the big drop in divorce after the first year that the later years do. Interesting!

Sigh

Here’s where the awesome data let us down. This data is very powerful. It’s the best contemporary big data set we have for analyzing divorce. It has taken us this far, but it can’t explain a pattern like this.

We can control for education, but that’s just the education level at the time of the most recent survey. We can’t know when she got her education relative to the dates of her marriage. Further, from the ACS we can’t tell how many children a person has had, with whom, and when — we only know about children who happen to be living in the household in 2013, so a 50-year-old could be childfree or have raised and released four kids already. And about couples, although we can say things about the other spouse from looking around in the household (such as his age, race, and income), if someone has divorced the spouse is gone and there is no information about that person (even their sex). So we can’t use that information to build a model of divorce predictors.

Here’s an example of what we can only hint at. Remarriages are more likely to end in divorce, for a variety of reasons, which is why we simplify these things by only looking at first marriages. But what about the spouse? Some of these women are married to men who’ve been married before. I can’t how much that contributes to their likelihood of divorce, but it almost certainly does. Think about the bump up in the divorce rate for women who got married in their late thirties. On the way from high divorce rates for women who marry early to low rates for women who marry late, the overall downward slope reflects increasing maturity and independence for women, but it’s running against the pressure of their increasingly complicated relationship situations. That late-thirties bump may have to do with the likelihood that their husbands have been married before. Here’s the circumstantial evidence:

2011-2013 agemar figures.xlsx

See that big jump from early-thirties to late-thirties? All of a sudden 37.5% of women marrying in their late-thirties are marrying men who are remarrying. That’s a substantial risk factor for divorce, and one I can’t account for in my analysis (because we don’t have spouse information for divorced women).

On method

Divorce is complicated and inherently longitudinal. Marriages arise out of specific contexts and thrive or decay in many different ways. Yesterday’s crucial influence may disappear today. So how can we say anything about divorce using a single, cross-sectional survey sample? The unsatisfying answer is that all analysis is partial. But these five questions give us a lot to go on, because knowing when a person got married allows us to develop a multidimensional image of the events, as I’ve demonstrated here.

But, you ask, what can we learn from, say, the divorce propensity of today’s 40-year-olds when we know that just last year a whole bunch of 39-year-olds divorced, skewing today’s sample? This is a real issue. And demography provides an answer that is at once partial and powerful: Simple, we use today’s 39-year-olds, too. In the purest form, this approach gives us the life table, in which one year’s mortality rates — at every age — lead to a projection of life expectancy. Another common application is the total fertility rate (watch the video!), which sums birth rates by age to project total births for a generation. In this case I have not produced a complete divorce life table (which I promised a while ago — it’s coming). But the approach is similar.

These are all synthetic cohort approaches (described nicely in the Week 6 lecture slides from this excellent Steven Ruggles course). In this case, the cohorts are age-at-marriage groups. Look at the table above and follow the row for, say, marriages that started at age 28, to see that synthetic cohort’s divorce experience from marriage until age 59. It’s neither a perfect depiction of the past, nor a foolproof prediction of the future. Rather, it tells us what’s happening now in cohort terms that are readily interpretable.

Conclusion

The ACS is the best thing we have for understanding the basic contours of divorce trends and patterns. Those five questions are invaluable.


* For this I also tossed the people who were reported to have married in the current year, because I wasn’t sure about the timing of their marriages and divorces, but I put them back in for the regressions.

** The codebook for my IPUMS data extraction is here, my Stata code is here. The heat-map model here isn’t in that code file, but this these are the commands (and the margins command took a very long time, so please don’t tell me there’s something wrong with it):

logistic divorce i.agemarc#i.mardurc i.degree i.race i.hispan i.citizen
margins i.agemarc#i.mardurc

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That Sunday New York Times Style section trend piece

Just folks trying to survive their divorces.

Just folks trying to survive their divorces.

Does it matter which one?

You know it from the opening paragraphs:

The women are architects, film industry executives, skin care consultants, product managers at tech companies, psychologists. They have worked in finance, publishing and television, though some had scaled back or left the work force when their children were born.

Divorce is what they have in common. Their stories are varied: the breadwinner wife whose husband’s career hadn’t quite taken off and who found comfort in an affair; the husband who never really adapted to parenthood; the wife with Ivy League degrees who stayed home with her child but lost her way in the marriage while the husband thrived in his international career.

Really. Divorce is what they have in common? How hard would it be to include a single mention of how rich and privileged these women are compared to the typical woman getting divorced? Penelope Green’s story never mentions the possibility.

Here is what a five-minute effort would have looked like:

top10divoccs

These 10 occupations account for 25% of all women age 40+ who reported getting divorced in the previous year.

In addition, 34% of those just-divorced, 40+ women are not non-Hispanic Whites (14% Black, 13% Hispanic, 4% Asian/Pacific Islander).

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No, you should get married in your late 40s (just kidding)

Please don’t give (or take) stupid advice from analyses like this.

Since yesterday, Nick Wolfinger and Brad Wilcox have gotten their marriage age analysis into the Washington Post Wonkblog (“The best age to get married if you don’t want to get divorced”) and Slate (“The Goldilocks Theory of Marriage”). The marriage-promotion point of this is: don’t delay marriage. The credulous blogosphere can’t resist the clickbait, but the basis for this is very weak.

Yesterday I complained about Wolfinger pumping up the figure he first posted (left) into the one on the right:

wolfbothToday I spent a few minutes analyzing the American Community Survey (ACS) to check this out. Wolfinger has not shared his code, data, models, or tables, so it’s hard to know what he really did. However, he lists a number of variables he says he controlled for using the National Survey of Family Growth: “sex, race, family structure of origin, age at the time of the survey, education, religious tradition, religious attendance, and sexual history, as well as the size of the metropolitan area.”

The ACS seems better for this. It’s very big, so I can analyze just the one-year incidence of divorce (did you get divorced in the last year?), according to the age at which people married. I don’t have family structure of origin, religion, or sexual history, but he says those don’t influence the age-at-marriage effect much. He did not control for duration of marriage, which is messed up in his data anyway because of the age limits in the NSFG.

So, in my model I used women in their first marriages only, and controlled for marriage duration, education, race, Hispanic ethnicity, and nativity/citizenship. This is similar to models I used in this (shock) peer-reviewed paper. Here are the predicted probabilities of divorce, in one year, holding those control variables constant.

agemar-divorce

Yes, there is a little bump up for the late 30s compared with the early 30s, but it’s very small.

Closer analysis (added to the post 7/19), generated from a model with age-at-marriage–x–marital duration interactions, shows that the late-30s bump is concentrated in the first five years of marriage:

newheatmap

This doesn’t much undermine the “conventional wisdom” that early marriage increases the risk of divorce. Of course, this should not be the basis for advice to people who are, say, dating a person they’re thinking of marrying and hoping to minimize chance of divorce.

If you want to give advice to, say, a 15-year-old woman, however, the bottom line is still: Get a bachelor’s degree. You’ll likely earn more, marry later, and have fewer kids. If you or your spouse decide to get divorced after all that, it won’t hurt that you’re more independent. For what it’s worth, here are the education effects from this same model:

educ-div

(The codebook for my IPUMS data extraction is here, my Stata code is here.)

Anyway, it’s disappointing to see this in the Wonkblog piece:

But the important thing, for Wolfinger, is that “we do know beyond a shadow of a doubt that people who marry in their thirties are now at greater risk of divorce than are people who wed in their late twenties. This is a new development.”

That’s just not true. I wouldn’t swear by this quick model I did today. But I would swear that it’s too early to change the “conventional wisdom” based only on a blog post on a Brad-Wilcox-branded site.

Aside

One interesting issue is the problem of age at marriage and education. They are clearly endogenous — that is, they influence each other. Women delay marriage to get more education, they stop their education when they have kids, they go back to school when they get divorced — or think they might get divorced. And so on. And, for the regression models, there are no highly-educated people getting married at really young ages, because they haven’t finished school yet. On the other hand, though, there are lots of less-educated people getting married for the first time at older ages. Using the same ACS data, here are two looks at the women who just married for the first time, by age and education.

First, the total number per year:

age-ed-mar-count

Then, the percent distribution of that same data:

age-ed-mar-distInteresting thing here is that college graduates are only the majority of women getting married for the first time in the age range 27-33. Before and after that most women have less than a BA when they marry for the first time. This is also complicated because the things that select people into early marriage are sometimes but not always different from those that select people into higher education. Whew.

It really may not be reasonable to try to isolate the age-at-marriage effect after all.

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The latest get-married-young thing tells you all you need to know

Just a quick note for people wondering about this new thing by Nicholas Wolfinger on Brad Wilcox’s blog. He says it used to be (before 1995) that getting married young increased the odds of divorce. Since then, however, he says getting married either before or after age 32 raises the odds of divorce.

Why is that? His explanation — in his very own words, from his very own post: “my money is on a selection effect.” In other words, do not follow the advice in the headline, which is: “Want to Avoid Divorce? Wait to Get Married, But Not Too Long.” Because if the mechanism is selection, then changing your behavior to ride that curve will not work.

I’m not getting into the methods, which are not revealed, despite a link for “more information” — there is no paper, no tables, no code or data. However, something is off, and the post is off-gassing a discernible essence of Wilcox’s influence. In the new blog post, they show this graph:

wolfinger1Wow, that’s a pretty big boomerang effect. If it weren’t a selection effect, it might really be relevant for personal decision-making. But when you follow the link for “more information” you see this graph:

wolfinger2

The upward swing here is hardly enough to get your marriage promotion lather up. Clearly, something had to be improved from Wolfinger’s post from April and his post for Wilcox’s site in July. That’s the kind of data leadership we expect from this site. (Also, get rid of those dots, which show you the all those people with really low divorce odds at higher ages.)

Related:

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Goffman dissertation followup

I previously reviewed Alice Goffman’s book On The Run, and wrote a critique of the survey that was part of that project (including a formal comment sent to American Sociological Review). Then I complained that her dissertation was not made public, despite being awarded the American Sociological Association’s dissertation prize. I proposed a rule change for the association, requiring that the winning dissertation be “publicly available through a suitable academic repository by the time of the ASA meeting at which the award is granted.”

Here’s a quick followup.

shelf_of_dissertations_500

I was interested in Goffman’s 2010 dissertation because I thought it might have more information about the survey she conducted than the 2014 book did. When I inquired about the dissertation on June 4 of this year, Princeton’s director of media relations, Martin Mbugua, told me she “was granted an exemption from submitting her dissertation to the University Archives, so we do not have a copy of her dissertation in our collection.”

Jesse Singal at New York magazine reported yesterday that they now have the dissertation, and he’s read it. Not only does it not have more methodological information than the book, Singal reports, it actually has less, as the methodological appendix that’s in the book is not in the dissertation. In a saved-you-a-trip-to-Princeton email to me, Singal says the dissertation’s description of her survey is “basically identical” to what is in ASR. That speaks to my critique of her survey, which seems unaffected by the release of the dissertation. (I’m not in charge of dissertations at Princeton, so I’m not critiquing the dissertation anyway.)

With regard to the open-science-inspired rule change for ASA dissertation awards, Singal’s article just reinforces my desire to see the rule adopted. Mbugua told Singal that Princeton now allows up to two two-year embargo periods for PhD students who don’t want their dissertations publicly released. But why embargo it? I think most people do this because they don’t want to undermine their book deals. The need for this may be overstated, but it’s a thing. (Eric Schwartz who acquires sociology books for Columbia University Press, tweeted: “No problem. Book and dissertation are for different audiences.”)

Anyway, Singal quotes Goffman giving a quite different reason:

The dissertation contained very sensitive material about people who were vulnerable to arrest and incarceration. … I wanted to think through the ethical and human subjects issues of making it available beyond the committee members and I wanted some time to go by between the actual events and a public reading. That felt safer for the people who had granted me permission to write about their lives, and for me, than publishing right away.

Apart from the fact that this concern did not prevent Goffman from submitting her book to a reading by an awards committee — “beyond the [dissertation] committee members” — I do not find this very credible, and I don’t like that rationale. If it was wrong to release it in 2010 because it would endanger her subjects, then it was wrong to publish a book in 2014 with the same — actually, more — incriminating information. In fact, as we now know, identifying the individuals mentioned in the book was trivial using Google, and of course the police knew who they were anyway. By this rationale, I cannot understand why the dissertation would not be given to the library until 14 months after the book was published — or until three months after the commercial paperback edition was published. Oh, wait.

Look, if people want to embargo their dissertations for financial gain, and their elite private universities allow it, then so be it. But that doesn’t have to be ASA’s problem. We can add one small piece to that calculation: giving up the ASA Dissertation Award.

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The U.S. government asked 2 million Americans one simple question, and their answers will shock you

What is your age?

[SKIP TO THE END for a mystery-partly-solved addendum]

Normally when we teach demography we use population pyramids, which show how much of a population is found at each age. They’re great tools for visualizing population distributions and discussing projections of growth and decline. For example, consider this contrast between Niger and Japan, about as different as we get on earth these days (from this cool site):

japan-niger-pyramids

It’s pretty easy to see the potential for population growth versus decline in these patterns. Finding good pyramids these days is easy, but it’s still good to make some yourself to get a feel for how they work.

So, thinking I might make a video lesson to follow up my blockbuster total fertility rate performance, I gathered some data from the U.S., using the 2013 American Community Survey (ACS) from IPUMS.org. I started with 10-year bins and the total population (not broken out by sex), which looks like this:

totalbinned

There’s the late Baby Boom, still bulging out at ages 50-59 (born 1954-1963), and their kids, ages 20-29. So far so good. But why not use single years of age and show something more precise? Here’s the same data, but showing single years of age:

totalsingleyears

That’s more fine-grained. Not as much as if you had data by months or days of birth, but still. Except, wait: is that just sample noise causing that ragged edge between 20 and about 70? The ACS sample is a few million people, with tens of thousands of people at each age (up age 75, at least), so you wouldn’t expect too much of that. No, it’s definitely age heaping, the tendency of people to skew their age reporting according to some collective cognitive scheme. The most common form is piling up on the ages ending with 0 and 5, but it could be anything. For example, some people might want to be 18, a socially significant milestone in this country. Here’s the same data, with suspect ages highlighted — 0’s and 5’s from 20 to 80, and 18:

totalsingleyearsflagged

You might think age heaping results from some old people not remembering how old they are. In the old days rounding off was more common at older ages. In 1900, for example, the most implausible number of people was found at age 60 — 1.6-times as many as you’d get by averaging the number of people at ages 59 and 61. Is that still the case? Here it is again, but with the red/green highlights just showing the difference between the number of people reported and the number you’d get by averaging the numbers just above and below:

totalsingleyearsflaggedhighlightProportionately, the 70-year-olds are most suspicious, at 10.8% more than you’d expect. But 40 is next, at 9.2%. And that green line shows extra 18-year-olds at 8.6% more than expected.

Unfortunately, it’s pretty hard to correct. Interestingly, the American Community Survey apparently asks for both an age and a birth date:

acs-age

If you’re the kind of person who rounds off to 70, or promotes yourself to 18, it might not be worth the trouble to actually enter a fake birth date. I’m sure the Census Bureau does something with that, like correct obvious errors, but I don’t think they attempt to correct age-heaping in the ACS (the birth dates aren’t on the public use files). Anyway, we can see a little of the social process by looking at different groups of people.

Up till now I’ve been using the full public use data, with population weights, and including those people who left age blank or entered something implausible enough that the Census Bureau gave them an age (an “allocated” value, in survey parlance). For this I just used the unweighted counts of people whose answers were accepted “as written” (or typed, or spoken over the phone, depending on how it was administered to them). Here are the patterns for people who didn’t finish high school versus those with a bachelor’s degree or higher, highlighting the 5’s and 0’s (click to enlarge):

heapingbyeduc

Clearly, the age heaping is more common among those with less education. Whether it’s really people forgetting their age, rounding up or down for aspirational reasons, or having trouble with the survey administration, I don’t know.

Is this bad? As much as we all hate inaccuracy, this isn’t so bad. Fortunately, demographers have methods for assessing the damage caused by humans and their survey-taking foibles. In this case we can use Whipple’s index. This measure (defined in this handy United Nations slideshow) takes the number of people whose alleged ages end in 0 or 5 and multiplies that by 5, then compares it to the total population. Normally people use ages 23 to 62 (inclusive), for an even 40 years. The amount by which people reporting ages 25, 30, 35, 40, 45, 50, 55, and 60 are more than one-fifth of the population ages 23-62, that’s your Whipple’s index. A score of 100 is perfect, and a score of 500 means everyone’s heaped. The U.N. considers scores under 105 to be “very accurate data.” The 2013 ACS, using the public use file and the weights, gives me a score of 104.3. (Those unweighted distributions by education yield scores of 104.0 for high school dropouts and 101.7 for college graduates.) In contrast, the Decennial Census in 2010 had a score of just 101.5 by my calculation (using table QT-P2 from Summary File 1). With the size of the ACS, this difference shouldn’t have to do with sampling variation. Rather, it’s something about the administration of the survey.

Why don’t they just tell us how old they really are? There must be a reason.

Two asides:

  • The age 18 pattern is interesting — I don’t find any research on desirable young-adult ages skewing sample surveys.
  • This is all very different from birth timing issues, such as the Chinese affinity for births in dragon years (every twelfth year: 1976, 1988…). I don’t see anything in the U.S. pattern that fits fluctuations in birth rates.

Mystery-partly-solved addendum

I focused one education above, but another explanation was staring me in the face. I said “it’s something about the administration of the survey,” but didn’t think to check for the form of survey people took. The public use files for ACS include an indicator of whether the household respondent took the survey through the mail (28%), on the web (39%), through a bureaucrat at the institution where they live (group quarters; 5%), or in an interview with a Census worker (28%). This last method, which is either a computer-assisted telephone interview (CATI) or computer-assisted personal interview (CAPI), is used when people don’t respond to the mailed survey.

It turns out that the entire Whipple problem in the 2013 ACS is due to the CATI/CAPI interviews. The age distributions for all of the other three methods have Whipple index scores below 100, while the CATI/CAPI folks clock in at a whopping 108.3. Here is that distribution, again using unweighted cases:

caticapiacs

There they are, your Whipple participants. Who are they, and why does this happen? Here is the Bureau’s description of the survey data collection:

The data collection operation for housing units (HUs) consists of four modes: Internet, mail, telephone, and personal visit. For most HUs, the first phase includes a mailed request to respond via Internet, followed later by an option to complete a paper questionnaire and return it by mail. If no response is received by mail or Internet, the Census Bureau follows up with computer assisted telephone interviewing (CATI) when a telephone number is available. If the Census Bureau is unable to reach an occupant using CATI, or if the household refuses to participate, the address may be selected for computer-assisted personal interviewing (CAPI).

So the CATI/CAPI people are those who were either difficult to reach or were uncooperative when contacted. This group, incidentally, has low average education, as 63% have high school education or less (compared with 55% of the total) — which may explain the association with education. Maybe they have less accurate recall, or maybe they are less cooperative, which makes sense if they didn’t want to do the survey in the first place (which they are legally mandated — i.e., coerced — to do). So when their date of birth and age conflict, and the Census worker tries to elicit a correction, maybe all hell breaks lose in the interview and they can’t work it out. Or maybe the CATI/CAPI households have more people who don’t know each other’s exact ages (one person answers for the household). I don’t know. But this narrows it down considerably.

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Colorado leads drop in teen birth rate, 2008-2013

Yesterday I tweeted a figure of teen birth rate changes based on the fertility question in the American Community Survey. It showed Colorado with an above-average drop in teem births from 2008 to 2013, but not the biggest drop in the country. I have a better chart on this below.

The reason for the attention was this story in the New York Times, which reported:

Over the past six years, Colorado has conducted one of the largest experiments with long-acting birth control. If teenagers and poor women were offered free intrauterine devices and implants that prevent pregnancy for years, state officials asked, would those women choose them?

They did in a big way, and the results were startling. The birthrate among teenagers across the state plunged by 40 percent from 2009 to 2013, while their rate of abortions fell by 42 percent, according to the Colorado Department of Public Health and Environment.

Since the article didn’t provide data for comparisons, and I knew teen births were declining all over, I wanted to see if Colorado’s experiment was really such a standout. The figure was republished by German Lopez at Vox.com in a post on the Colorado program. Although the figure showed Colorado with a big drop, it still cast doubt on the program because it showed four states and DC with bigger drops.

I’m retracting that figure today, because I realized — and I should have known this — that we have teen birth rates by state and year from the vital records data reported by the National Center for Health Statistics. In these reports we can see that Colorado did, in fact, have the largest decline in teen births from 2008 and 2013 (their program started in 2009). Here’s the new figure:

teenbirthratechangestates

The story isn’t that different between NCHS and ACS data, but Colorado is trying to raise money to continue the program, and it sure is nice for them to have this comparison. It’s great to have data right away — and share it — and it’s also great, even greater, to have better data. The vital records data is more complete and reliable, since it is not based on a sample, and teen births are rare enough now that sampling variation matters, even in a big sample like the ACS. So I regret that I published the earlier figure.

That said…

The teen birth rate is declining all over the country, even in places with terrible policies, so the Colorado program — valuable as it may be — is swimming with the tide.

The reason teen births are declining all over is because the teen birth rate is a myth — what’s really happening is women in the U.S. are having their children later, for economic and social reasons that go way beyond what’s happening with teens per se. I have written about this a few times:

See also:

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