Thinking about preparing for a course this Fall, I had a look at the current literature on assortative mating and found a pair of excellent papers on assortment by education [1,2]. Using census data they generated tables by decade from 1940 to 2000 with counts of marriages by level of education of husband and wife. There are two families of tables in these papers, one based on recent marriages (“newlyweds”) and the other based on marriages of five or more years duration (“established”). Since the established series is more up to date, I have looked at that series.

Here is the raw data table from a sample from the 1940 US census of established marriages. The data are given as percentages of the total number of marriages but the raw numbers are simple to generate from the published tables. Here for example is the data table from 1940:

1940 Marriages Male Education <10 10-11 12 13-15 16+ <10 69729 7053 4945 1236 634 10-11 11618 6150 4137 1093 570 Female 12 10382 5706 12887 3629 3027 Education 13-15 2092 1062 2330 2504 2567 16+ 507 253 745 855 2773

The education categories are dated from today’s perspective: fewer than ten years of schooling, ten to eleven years, high school graduate, some college, and college graduate. Of course the significance of these has changed since 1940: today’s associate degree is equivalent to finishing high school in 1940 some would say. Nevertheless we can look at what we have.

While the above table is complete, how do we make sense of it? Some things are easy, for example the largest number of marriages, 69729, were those in which both partners had fewer than ten years of education. In 2000 the largest number of established marriages is between two college graduates. The smallest entry in 1940, 507, is for college graduate women married to men with fewer than ten years of schooling. Interesting numbers, I suppose, but what is the big picture? Is there a little or a lot of assortative mating? For a full analysis using log-linear models and covariates see the above papers. The second, by Schwartz and Mare, is well written, easy to understand, but it is a full course meal. What I want is a snack, a tasty cracker, rather than a full meal.

One attractive approach to generating a light snack is called “raking” the table [3]. This technique was a hot topic way back when I was in graduate school [4]. Iterating, one adusts the marginals, alternating rows and columns, to some desired values. In more detail: first compute the column sums, then divide each column by its sum, then sum the rows, then divide each row but its sum, repeating these steps until the table stops changing. This adjusts the marginals, i.e. the row and column sums, to 1.

This method, applied to the 1940 marriages, gives this:

1940 Marriages 0.59 0.23 0.12 0.04 0.02 0.23 0.42 0.25 0.08 0.02 0.11 0.23 0.36 0.22 0.08 0.04 0.08 0.19 0.41 0.28 0.03 0.03 0.09 0.25 0.60

The row and column labels are the same as in the table above. What do these numbers mean? They are (supposed to be) estimates of what we might call “attraction.” (I want to say “preference” but sociologists use that for something else.) Since we have iterated to unit marginals (the rows and columns each sum to unity), the numbers tell us how many marriages of each mating-type (pair of education levels) would occur if the numbers of potential mates from each level were the same. For example, the rows correspond to female partners. The second entry in the first row, 0.23, tells us that if the mating pool numbers were uniform across levels, 23% of the women with fewer than ten years of schooling would marry men with ten or eleven years of schooling. We could have adjusted the marginals to anything we wished, of course, but the reduction to unit marginals in my opinion gives the simplest picture possible of assortative mating by education.

Here is the same table from the 2000 census:

2000 Marriages 0.64 0.19 0.11 0.05 0.02 0.19 0.48 0.21 0.09 0.02 0.11 0.21 0.39 0.22 0.08 0.04 0.1 0.22 0.42 0.21 0.01 0.02 0.08 0.22 0.67

Now things are starting to get strange. The diagonals in the 2000 table are slightly greater, indicating more within-group marriage attraction, but not much. Other than that is is essentially the same table. Hardly anything has changed in 60 years!

Here is another (to me) surprise. Female hypergamy (marrying up) is often thought to be common. A simple indicator is the average number of marriages in the upper triangle (male education>female), on the diagonal (both partners have same education level), and in the lower triangle(male education<female). There is scarcely any indication of preferential hypergyny, neither in 1940 nor in 2000 nor in the intervals between. There might be some if I had reported more signficant digits but it would not be large enough to be of any interest.

Average of upper triangle, diagonal, and lower triangle values Year Female<Male Same Female<Male 1940 0.37 0.47 0.37 1960 0.37 0.48 0.37 1980 0.38 0.50 0.37 2000 0.38 0.52 0.38

So this is the cracker I have gotten ready for my class this fall, perhaps 15 minutes worth. If you are interested in digging deeper I recommend a careful read of Schwartz and Mare [2]: good luck.

If you are interested in the numbers, here are the raw data straight from their paper, in a format such that you can copy one of the blocks and read it directly as a floating point array with numeric python’s loadtxt() function.

# year 1940 counts from schwartz and mare 2005 demography # men in columns, women in rows, schooling increases across and down #N=158512 28.83 4.37 3.53 0.62 0.42 7.23 4.96 4.67 0.72 0.22 7.01 5.80 14.29 3.92 2.30 0.74 0.52 1.97 1.97 2.05 0.25 0.12 0.52 0.64 2.32 #1960 #N=203117 9.32 2.94 2.85 0.47 0.19 4.70 5.04 6.94 1.42 0.37 5.84 6.41 22.61 6.93 3.01 0.63 0.76 3.35 5.15 3.81 0.10 0.08 0.84 1.54 4.69 #1980 #N=239980 2.68 1.51 1.92 0.44 0.09 1.51 3.16 5.00 0.94 0.16 2.23 4.33 25.51 8.26 3.03 0.42 1.03 7.08 9.48 5.51 0.09 0.14 1.66 3.37 10.46 #2000 #N=220209 3.47 0.60 1.42 0.52 0.16 0.68 1.01 1.79 0.65 0.13 1.80 2.02 15.54 7.33 2.41 0.76 1.06 9.26 14.91 6.98 0.17 0.18 2.80 6.33 18.02

REFERENCES

[1] R.D. Mare, Five decades of educational assortative mating, American Sociological Review. (1991) 15–32.

[2] C.R. Schwartz, R.D. Mare, Trends in educational assortative marriage from 1940 to 2003, Demography. 42 (2005) 621–646.

[3] A. Agresti, Categorical data analysis, Wiley, Hoboken, NJ, 2013.

[4] F. Mosteller, Association and estimation in contingency tables, Journal of the American Statistical Association. 63 (1968) 1–28.