In this lab we will look at patterns of intermarriage among traditional opposite-sex couples and more recent same-sex couples in California. Our data comes from the American Community Survey from 2015. The Census recently changed its coding so that same-sex married couples would not have their responses edited to “non-married.” (See http://www.pewresearch.org/fact-tank/2014/05/29/census-says-it-will-count-same-sex-marriages-but-with-caveats/)

# Do not edit this chunk, but *do* press the green button to the answer key for the quiz info (the unreadable string below)
tot = 0
answer.key = "eJytUrtuwzAM3PMVhJe0gCHUj7VD0MeYAGn6AYpD20RtKRHlpP77Sorj1vWaRQJJ8Hh3pDiprt2jSUQCz+DehSi0MVhYIRVfboXoJVqImpQdwg9EeHUpUmxNV1jSiofSFo+NLBBsjbBewYVsDb3uDFzx4AFFJWJYrpZACk6dtsiPDgq/XZ+SHmpA2jkErZoe1psd7LafbzcIYvB8RuppoJ7OqftCtBqppzfqpTZt10j/e55WgtzrM4qZovRuitJR0eHAYHzW68gykcWu7UCFS6kKjprJ0hlBMuuCQrf4KzYLYrO52Gy6pxC+k2EbQ6P1F0gbFFi5bxDIMjYlWA3s/LjUQ9FgEyZyTUdHb25IdjdDAtJmNIOh0lAa3cKTZzXaQKokRbaPr2MSYKoUlb3zaiGUnjCe+JQHn/K5T/n0KK6h+ruXyqC06E9DKjdx2A7yLyvWxm9rZk9+N3sCUuJ/dyVtV9T/WInoB0H2Mdw="
library(quizify)
source.coded.txt(answer.key)

Reading in data

df <- read.csv("/data175/ca_marriage_out.csv", as.is = T)
nrow(df)
[1] 34214
## we see there are about 34,000 couples in our sample

Let’s look at the variables

head(df)
ABCDEFGHIJ0123456789
 
 
sex
<chr>
sex_sp
<chr>
x
<int>
x_sp
<int>
college
<chr>
college_sp
<chr>
racere
<chr>
racere_sp
<chr>
incwage
<int>
1malefemale4132collegenocollegeww50000
2femalemale3434collegecollegeww0
3malefemale5849nocollegenocollegeww29000
4femalemale4546nocollegenocollegeww67000
5femalemale4347nocollegenocollegeww95000
6femalemale5856nocollegenocollegeww30000

Each line is a couple. Variables ending in “_sp" are for the spouse of the person answering the questionnaire. Only one person per household filled out the questionnaire.

To make our analysis easier, we’ve restricted the sample to - California only - respondents and spouses aged 19-59 - races: Asian, White and Black

Let’s just check to see if there are same sex marriages.

head(df[df$sex == df$sex_sp,])
ABCDEFGHIJ0123456789
 
 
sex
<chr>
sex_sp
<chr>
x
<int>
x_sp
<int>
college
<chr>
college_sp
<chr>
racere
<chr>
racere_sp
<chr>
incwage
<int>
38malemale4646collegecollegeww73000
105malemale4547collegenocollegeww50000
248femalefemale5455nocollegenocollegewa75000
296femalefemale3919nocollegenocollegeww16000
387malemale2422collegecollegeww0
395malemale5453collegecollegebb0

Let’s save the variables in a form that’s easier to type

sex = df$sex
sex_sp = df$sex_sp # sex of spouse
age = df$x
age_sp = df$x_sp
college = df$college
college_sp = df$college_sp
incwage = df$incwage
incwage_sp = df$incwage_sp
race = df$racere
race_sp = df$racere_sp

Measuring positive and negative assortative mating

Correlation: for a continuous variable taking numeric vales

The correlation between two vectors of numbers x and y tell you how the x & y are “co-related”. A value of 1 means that x and y are perfectly aligned, so that like is with like. A value of -1 means that they are negatively related, so that like and unlike pair together. A value of 0 means that they are unrelated to each other.

An example of zero correlation would be two independent sets of randomly generated numbers

set.seed(1)
x = rnorm(100)
y = rnorm(100)
plot(x,y)
title(paste("correlation is:", round(cor(x,y),3)))

An example of positive correlation

 x = rnorm(100, mean = 100, sd = 30)
 epsilon = rnorm(length(x), mean = 0, sd = 25)
 y = x + epsilon
 plot(x,y)
 title(paste("correlation is:", round(cor(x,y),3)))

An example of negative correlation

 x = rnorm(100, mean = 100, sd = 30)
 epsilon = rnorm(length(x), mean = 0, sd = 25)
 y = 300 -x + epsilon
 plot(x,y)
 title(paste("correlation is:", round(cor(x,y),3)))

Q1.1 What is NOT true about the correlation between married partners?

A. A negative value means opposites attract.

B. A positive value means like prefers like.

C. A value of 0.5 means that there is no tendency for married partners to be similar.

D. A value of 0 means there is no tendency for married partners to be similar.

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
answer1.1 = 'C'
quiz.check(answer1.1)
Your  answer1.1 : C
Correct.
Explanation:  The only NOT TRUE answer is C

Measuring association in a 2x2 table

Correlation is useful for continuous numeric variables. Sometimes we have variables that take on only two values, which we call binary. The relationship between two binary variables can be summarized with the odds ratio.

Say we have a 2x2 table of the sexes of spouses in California

table(sex, sex_sp)
        sex_sp
sex      female  male
  female    215 13402
  male    20366   231

To measure the association we can calculate the odds ratio, which is defined as

θ=(F[1,1]/F[1,2])(F[2,1]/F[2,2])
In our case this would be

theta = (215/13402) / (20366/231)
print(theta) 
[1] 0.0001819596

The odds-ratio can take values from 0 to positive infinity. A value of 1 indicates a neutral relationship, with no positive or negative sorting. A value close to 0 indicates very negative sorting; high values (10, 100, or even 1000) indicate very positive sorting.

The reason that this is called the odds-ratio is because the numerator is the odds of being in cell [1,1] compared to [1,2], and the denominator is odds of being in cell [2,1] compared to [2,2]. If there were no tendency to marry one sex rather than the other, the odds of marrying a “male” would be the same for females and males and the ratio would be 1.

Here’s an example of a table with an odds ratio of 1.

3 5 6 10

A value of θ>1 tells us that there is positive association, as with a correlation greater than 0. An example would be

10 3 2 20

Q1.2: The odds ratio in this table is: A. 200/ 6 B. 5 / 200 C. 20/ 60 D. 30 / 40

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
answer1.2 = 'A'
quiz.check(answer1.2)
Your  answer1.2 : A
Correct.
Explanation:  The odds ratio is 33.3, indicating positive association.

A value of θ<1 tells us that there is negative association. An example is our above table of spouses by sex in California. We expect a number that is much less than 1, because we know that most marriages in California are between people of opposite sexes.

Here’s a function to calculate the odds ratio

get.odds.ratio <- function(my.table)
{
  if (!all(dim(my.table) == c(2,2)))
    stop("table needs to be 2x2")
  theta = (my.table[1,1]/ my.table[1,2]) / (my.table[2,1] / my.table[2,2])
  return(theta)
}

Let’s check our calculation of the odds ratio of our table by sex

sex.tab <- table(sex, sex_sp)
print(sex.tab)
        sex_sp
sex      female  male
  female    215 13402
  male    20366   231
sex.theta <- get.odds.ratio(sex.tab)
print(paste("theta = ", round(sex.theta, 5)))
[1] "theta =  0.00018"

Q1.3: Which of the following is true for the odds ratio found for the table of marriages by combination of sex. A. There’s no tendency for marriages to be same- or opposite- sex. We know this because the odds ratio is close to 0. B. There’s some tendency for marriages to be same-sex. We know this because the odds ratio is positive. C. There’s a strong tendency for marriages to be opposite-sex. We know this because the odds ratio is much less than 1.

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
answer1.3 = 'C'
quiz.check(answer1.3)
Your  answer1.3 : C
Correct.
Explanation:  Odds ratios go from 0 to positive infinity, with 1 signifying
.no relationship.

Sorting for opposite-sex married couples

Here we will look at assortative pairing for opposite-sex couples. At the end of the lab, you will repeat this exercise for same-sex couples to see if same-sex spouses appear to be more similar to each other (or more different) than opposite-sex couples.

1. Education

Our variable “college” tells us if the person has a 4-year college degree. The predictions from Becker’s theory are ambiguous. On the one hand, we would expect education to be associated with market wages, which Becker thinks should be negatively correlated across spouses. On the other hand, there are many reasons to think that education should positively sort, including complementarity in investments in children and complementary consumption (of books, films, and conversation). And then there is the issue of search costs: since future spouses often meet in an educational or work setting, and since these are highly sorted by education, we might expect likes to marry likes simply out of convenience, even if there were no preference for assortative marriage in education.

In order to compare same-sex to opposite sex marriages, we create an index variable.

ss <- sex == sex_sp ## index for "same sex" ("ss")  couples
os <- !ss ## index for "opposite sex" ("os") couples

We can now look at the association of education for opposite sex couples

Now we can construct two tables, one for the college status of same-sex couples and one for the college status of opposite sex couples

college.tab.os <- table(college[os], college_sp[os])
print(college.tab.os)
           
            college nocollege
  college     11342      4890
  nocollege    2973     14563
print(get.odds.ratio(college.tab.os))
[1] 11.36153

Q1.4: Would you agree or disagree with the following statement

We see an odds ratio of 11.4, which is much bigger than one, showing strong positive sorting between opposite-sex spouses.

A. Agree B. Disagree

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
answer1.4 = 'A'
quiz.check(answer1.4)
Your  answer1.4 : A
Correct.
Explanation:  11.4 is much greater than 1.

2. Age

We expect ages to be similar between spouses both because of the relative ease of meeting people of one’s own age group and also because age will be correlated with a number of other traits including cultural taste, political outlook, energy level, and other characteristics that we would expect to positively sort.

plot(head(age[os], 1000), head(age_sp[os], 1000)) # we only plot first 1000 points
title(paste("ages of spouses, correlation:", round(cor(age[os], age_sp[os]),3)))

3. Wage income

plot(head(incwage[os], 1000), head(incwage_sp[os], 1000))
title(paste("wage income of spouses, correlation:",
            round(cor(incwage[os], incwage_sp[os]),3)))

Note: The data is top-coded, so that all values above a certain level are assigned the same value, in this case $483,000, which is the average of the top-coded values.

Here we see a more complicated story. We see a large number of points in which one of the spouses has zero income and the other a positive income, so this is consistent with the specialization argument of Becker. But overall there is no strong positive or negative relationship.

We could plot only couples where both partners had income greater than zero.

both_nonzero = incwage > 0 & incwage_sp > 0
plot(head(incwage[os & both_nonzero], 1000),
     head(incwage_sp[os & both_nonzero], 1000))
title(paste("wage income of spouses, correlation:",
            round(cor(incwage[os & both_nonzero],
                      incwage_sp[os & both_nonzero]),3)))

But Becker would probably say this is uninformative, since where there are big differences in potential earnings, one spouse will choose to specialize in home production and have zero earnings. So if we leave out the zero earners, we’re missing the important part of the story.

Perhaps a better test would be to see if zero earnings tend to be only in one spouse.

zerowage = incwage == 0 
zerowage_sp = incwage_sp == 0
zerowage.tab.os <- table(zerowage, zerowage_sp)
print(zerowage.tab.os)
        zerowage_sp
zerowage FALSE  TRUE
   FALSE 19704  7991
   TRUE   4625  1894
print(get.odds.ratio(zerowage.tab.os))
[1] 1.009768

Surprisingly, we get an odds ratio close to 1. So even here, we don’t see negative assortative marriage. We could further refine the analysis, say by considering people of a particular age range. But the bigger point is probably that potential wages before marriage, not observed wages after marriage, are probably what people are considering when they choose a spouse. Education is a great measure of potential wages, because it is observed both for those in and out of the labor force.

3. Race

Our data set has been restricted to people who identify as Asian (“a”), Black (“b”), or White (“w”). In order to calculate the association, we will consider three possible 2x2 combinations (“a” & “w”), (“b” & “w”), and (“a” & “b”)

First we make our tables of the race of spouses of each 2x2 combination:

os.wb <- os & race %in% c("w", "b") & race_sp %in% c("w", "b")
os.wa <- os & race %in% c("w", "a") & race_sp %in% c("w", "a")
os.ab <- os & race %in% c("b", "a") & race_sp %in% c("b", "a")
race.wb.tab.os <- table(race[os.wb], race_sp[os.wb])
print(race.wb.tab.os)
   
        b     w
  b   839   155
  w   245 24023
race.wa.tab.os <- table(race[os.wa], race_sp[os.wa])
print(race.wa.tab.os)
   
        a     w
  a  6636   655
  w  1117 24023
race.ab.tab.os <- table(race[os.ab], race_sp[os.ab])
print(race.ab.tab.os)
   
       a    b
  a 6636   43
  b   55  839

We see that most marriages in all three tables are “same race”. But it is hard to tell by looking at the numbers which pairs of races have the highest tendency to in-marry and inter-marry. For this we use the odds-ratio.

print(get.odds.ratio(race.wb.tab.os))
[1] 530.7517
print(get.odds.ratio(race.wa.tab.os))
[1] 217.8909
print(get.odds.ratio(race.ab.tab.os))
[1] 2354.167

So it appears that crossing the White-Asian racial divide is easier than crossing the White-Black racial divide, and both are easier than crossing the Asian-Black racial divide.

Conclusions for opposite-sex spouses

Our findings are consistent with Becker’s statements about positive sorting by race, age, and education, but not particularly supportive of the negative assortative pattern he predicted (in the 1970s) for market wages.

Same-sex couples

Becker’s main interest is in explaining why opposite-sex couples form. But his theory has implications for same-sex couples. He writes

Households with only men or only women are less efficient because they are unable to profit from the sexual difference in comparative advantage.

Source: Becker (1991), Treatise_on_the_Family, p. 37-38, cited by Jepsen and Jepsen(2001).

This inability to “profit from the sexual difference in comparative advantage” could mean that same-sex couples would seek to profit more from non-sexual differences, perhaps in demographic characteristics like age or education. Or it might mean that same-sex couples may derive their utility less from comparative advantages in production and more from complementarities in consumption.

Another factor to think about is limited choice. If one is looking for a partner in a large pool, then it may be easier to find exactly what one wants. On the other hand, if there is a decisive constraint – like the person must be over 6 feet tall or the person must be interested in a same-sex marriage – then a less than ideal match on another characteristic might well be worth it.

Replicating our analysis for same-sex couples

We can use the “ss” index we created to look at assortative marriage patterns for same sex couples. For example, for education, we can calculate

college.tab.ss <- table(college[ss], college_sp[ss])
print(college.tab.ss) 
           
            college nocollege
  college       202        74
  nocollege      49       121
print(get.odds.ratio(college.tab.ss))
[1] 6.740761

We see that our odds ratio of 6.7 for same sex couples is much less than the odds ratio of 11.4 observed for opposite-sex couples. This means that there appears to be a greater tendency to cross educational divides for same-sex couples.

Graded Questions

  1. Do you expect same-sex marriage spouses to be more like each other or less like each other than opposite-sex spouses? Is your expectation the same for age, race, income and education? If not, specify further. Explain your reasoning. Make sure to write down somewhere your expectations BEFORE you do the analysis. (You won’t be graded on this. The goal is for you to get the most out of your data analysis and to prevent you from coming up with a purely post hoc explanation after you’ve seen the result.)

  2. Replicate our analysis of opposite-sex marriages for same-sex marriages. Complete the (5) cells marked “X” in the following table. Transfer your answers to bCourses.

Characteristic Opposite-sex Same-sex Opposite-sex Same-sex
Measure odds ratio odds ratio correlation correlation
————– ————— ————- ————– ———
Education 11.4 6.7
Age 0.86 X
Wage Income 0.03 X
Race (“w”,“b”) 531 X
Race (“w”,“a”) 218 X
Race (“a”,“b”) 2354 X
# same-sex race analysis
ss.wb <- ss & race %in% c("w", "b") & race_sp %in% c("w", "b")
ss.wa <- ss & race %in% c("w", "a") & race_sp %in% c("w", "a")
ss.ab <- ss & race %in% c("b", "a") & race_sp %in% c("b", "a")
race.wb.tab.ss <- table(race[ss.wb], race_sp[ss.wb])
print(race.wb.tab.ss)
   
      b   w
  b   4   7
  w  12 347
race.wa.tab.ss <- table(race[ss.wa], race_sp[ss.wa])
print(race.wa.tab.ss)
   
      a   w
  a  37  14
  w  24 347
race.ab.tab.ss <- table(race[ss.ab], race_sp[ss.ab])
print(race.ab.tab.ss)
   
     a  b
  a 37  1
  b  0  4
print(get.odds.ratio(race.wb.tab.ss))
[1] 16.52381
print(get.odds.ratio(race.wa.tab.ss))
[1] 38.21131
print(get.odds.ratio(race.ab.tab.ss))
[1] Inf
# same-sex age
plot(head(age[ss], 1000), head(age_sp[ss], 1000)) # we only plot first 1000 points
title(paste("ages of spouses, correlation:", round(cor(age[ss], age_sp[ss]),3)))

# same-sex wage
plot(head(incwage[ss], 1000), head(incwage_sp[ss], 1000))
title(paste("wage income of spouses, correlation:",
            round(cor(incwage[ss], incwage_sp[ss]),3)))

  1. How does positive sorting differ for same-sex spouses, compared to opposite-sex spouses? (For example, “The smaller odds ratios for education mean that same-sex marriages sort less positively on education than do opposite-sex marriages.”). [Describe the remaining 5 relationships in a sentence each.]

  2. Describe a potential explanation for the pattern of your results. Try to use the words “complementarity” and “comparative advantage” in your answer. If some characteristics sort more and some less, try to make sense of the pattern. (There is no wrong explanation here. Just make sure your explanation is consistent with your evidence.) [ < 75 words]

  3. We didn’t distinguish same-sex male couples from same-sex female couples. If we did, which would you expect to have a less positive correlation in wage income: same-sex female couples or same-sex male couples? Explain briefly why. [ < 25 words] (Hint: you don’t need to calculate anything, just explain your logic. There is no single right answer.)

  4. Let’s extend the example we did in class (on about slide 20 of the lecture on assortative pairing). Let the payoffs have the form