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DemoKin matrix kinship models tutorial for Dutch Demography Day (2022)

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Getting started with matrix kinship models in R using DemoKin

Instructor: Diego Alburez-Gutierrez (MPIDR); The Formal Demography of Kinship: Theory and Application - Dutch Demography Day; Nov 16 2022

1. Installation

Install the development version from GitHub (could take ~1 minute). We made changes to the DemoKin package ahead of this workshop If you had already installed the package, please uninstall it and and install it again.

# remove.packages("DemoKin")
# install.packages("devtools")
devtools::install_github("IvanWilli/DemoKin", build_vignettes = TRUE)

Load packages:

library(DemoKin)
library(dplyr)
library(tidyr)
library(ggplot2)
library(fields)

2. Built-in data

The DemoKin package includes data from Sweden as an example. The data comes from the Human Mortality Database and Human Fertility Database. These datasets were loaded using theDemoKin::get_HMDHFD function.

2.1. swe_px matrix; survival probabilities by age (DemoKin’s U argument)

This is what the data looks like:

data("swe_px", package="DemoKin")

swe_px[1:4, 1:4]
##      1900    1901    1902    1903
## 0 0.91060 0.90673 0.92298 0.91890
## 1 0.97225 0.97293 0.97528 0.97549
## 2 0.98525 0.98579 0.98630 0.98835
## 3 0.98998 0.98947 0.99079 0.99125

And plotted over time and age:

image.plot(
  x = as.numeric(colnames(swe_px))
  , y = 0:nrow(swe_px)
  , z = t(as.matrix(swe_px))
  , xlab = "Year"
  , ylab = "Survival probability"
  )

2.2. swe_asfr matrix; age specific fertility rate (DemoKin’s f argument)

This is what the data looks like:

data("swe_asfr", package="DemoKin")

swe_asfr[15:20, 1:4]
##       1900    1901    1902    1903
## 14 0.00013 0.00006 0.00008 0.00008
## 15 0.00053 0.00054 0.00057 0.00057
## 16 0.00275 0.00319 0.00322 0.00259
## 17 0.00932 0.00999 0.00965 0.00893
## 18 0.02328 0.02337 0.02347 0.02391
## 19 0.04409 0.04357 0.04742 0.04380

And plotted over time and age:

image.plot(
  x = as.numeric(colnames(swe_asfr))
  , y = 0:nrow(swe_asfr)
  , z = t(as.matrix(swe_asfr))
  , xlab = "Year"
  , ylab = "Age-specific fertility (f)"
  )

3. The function kin()

DemoKin can be used to compute the number and age distribution of Focal’s relatives under a range of assumptions, including living and deceased kin. The function DemoKin::kin() currently does most of the heavy lifting in terms of implementing matrix kinship models. This is what it looks like in action, in this case assuming time-invariant demographic rates:

# First, get vectors for a given year
swe_surv_2015 <- DemoKin::swe_px[,"2015"]
swe_asfr_2015 <- DemoKin::swe_asfr[,"2015"]
# Run kinship models
swe_2015 <- kin(U = swe_surv_2015, f = swe_asfr_2015, time_invariant = TRUE)

3.1. Arguments

  • U numeric. A vector (atomic) or matrix of survival probabilities with rows as ages (and columns as years in case of matrix).
  • f numeric. Same as U but for fertility rates.
  • time_invariant logical. Assume time-invariant rates. Default TRUE.
  • output_kin character. kin types to return: “m” for mother, “d” for daughter, …

3.2. Relative types

Relatives for the output_kin argument are identified by a unique code. Note that the relationship codes used in DemoKin differ from those in Caswell (2019). The equivalence between the two set of codes is given in the following table:

demokin_codes()
##    DemoKin Caswell                      Label
## 1      coa       t    Cousins from older aunt
## 2      cya       v  Cousins from younger aunt
## 3        d       a                   Daughter
## 4       gd       b             Grand-daughter
## 5      ggd       c       Great-grand-daughter
## 6      ggm       h          Great-grandmother
## 7       gm       g                Grandmother
## 8        m       d                     Mother
## 9      nos       p   Nieces from older sister
## 10     nys       q Nieces from younger sister
## 11      oa       r     Aunt older than mother
## 12      ya       s   Aunt younger than mother
## 13      os       m               Older sister
## 14      ys       n             Younger sister

3.4. Value

DemoKin::kin() returns a list containing two data frames: kin_full and kin_summary.

str(swe_2015)
## List of 2
##  $ kin_full   : tibble[,7] [142,814 x 7] (S3: tbl_df/tbl/data.frame)
##   ..$ year     : logi [1:142814] NA NA NA NA NA NA ...
##   ..$ cohort   : logi [1:142814] NA NA NA NA NA NA ...
##   ..$ age_focal: int [1:142814] 0 1 2 3 4 5 6 7 8 9 ...
##   ..$ kin      : chr [1:142814] "d" "d" "d" "d" ...
##   ..$ age_kin  : int [1:142814] 0 0 0 0 0 0 0 0 0 0 ...
##   ..$ living   : num [1:142814] 0 0 0 0 0 0 0 0 0 0 ...
##   ..$ dead     : num [1:142814] 0 0 0 0 0 0 0 0 0 0 ...
##  $ kin_summary: tibble[,10] [1,414 x 10] (S3: tbl_df/tbl/data.frame)
##   ..$ age_focal     : int [1:1414] 0 0 0 0 0 0 0 0 0 0 ...
##   ..$ kin           : chr [1:1414] "coa" "cya" "d" "gd" ...
##   ..$ year          : logi [1:1414] NA NA NA NA NA NA ...
##   ..$ cohort        : logi [1:1414] NA NA NA NA NA NA ...
##   ..$ count_living  : num [1:1414] 0.2752 0.0898 0 0 0 ...
##   ..$ mean_age      : num [1:1414] 8.32 4.05 NaN NaN NaN ...
##   ..$ sd_age        : num [1:1414] 6.14 3.68 NaN NaN NaN ...
##   ..$ count_dead    : num [1:1414] 0 0 0 0 0 0 0 0 0 0 ...
##   ..$ count_cum_dead: num [1:1414] 0 0 0 0 0 0 0 0 0 0 ...
##   ..$ mean_age_lost : num [1:1414] NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN ...

kin_full

This data frame contains expected kin counts by year (or cohort), age of Focal, and age of kin.

head(swe_2015$kin_full)
## # A tibble: 6 x 7
##   year  cohort age_focal kin   age_kin living  dead
##   <lgl> <lgl>      <int> <chr>   <int>  <dbl> <dbl>
## 1 NA    NA             0 d           0      0     0
## 2 NA    NA             1 d           0      0     0
## 3 NA    NA             2 d           0      0     0
## 4 NA    NA             3 d           0      0     0
## 5 NA    NA             4 d           0      0     0
## 6 NA    NA             5 d           0      0     0

kin_summary

This is a ‘summary’ data frame derived from kin_full. To produce it, we sum over all ages of kin to produce a data frame of expected kin counts by year or cohort and age of Focal (but not by age of kin). This is how the kin_summary object is derived:

kin_by_age_focal <- 
  swe_2015$kin_full %>% 
  group_by(cohort, kin, age_focal) %>% 
  summarise(count = sum(living)) %>% 
  ungroup()

# Check that they are identical (for living kin only here)

kin_by_age_focal %>% 
  select(cohort, kin, age_focal, count) %>% 
  identical(
    swe_2015$kin_summary %>% 
      select(cohort, kin, age_focal, count = count_living) %>% 
      arrange(cohort, kin, age_focal)
  )
## [1] TRUE

4. Example: kin counts in time-invariant populations

Following Caswell (2019), we assume a female closed population in which everyone experiences the Swedish 1950 mortality and fertility rates at each age throughout their life. We then ask:

How can we characterize the kinship network of an average member of the population (call her ‘Focal’)?

For this exercise, we’ll use the pre-loaded Swedish data.

# First, get vectors for a given year
swe_surv_2015 <- DemoKin::swe_px[,"2015"]
swe_asfr_2015 <- DemoKin::swe_asfr[,"2015"]
# Run kinship models
swe_2015 <- kin(U = swe_surv_2015, f = swe_asfr_2015, time_invariant = TRUE)

4.1. ‘Keyfitz’ kinship diagram

We can visualize the implied kin counts for a Focal woman aged 35 yo in a time-invariant population using a network or ‘Keyfitz’ kinship diagram (Keyfitz and Caswell 2005) with the plot_diagram function:

swe_2015$kin_summary %>% 
  filter(age_focal == 35) %>% 
  select(kin, count = count_living) %>% 
  plot_diagram(rounding = 2)

4.2. Living kin

Now, let’s visualize how the expected number of daughters, siblings, cousins, etc., changes over the lifecourse of Focal (now, with full names to identify each relative type using the function DemoKin::rename_kin()).

swe_2015$kin_summary %>%
  rename_kin() %>% 
  ggplot() +
  geom_line(aes(age_focal, count_living))  +
  geom_vline(xintercept = 35, color=2)+
  theme_bw() +
  labs(x = "Focal's age") +
  facet_wrap(~kin)

Note that we are working in a time invariant framework. You can think of the results as analogous to life expectancy (i.e., expected years of life for a synthetic cohort experiencing a given set of period mortality rates).

How does overall family size (and family composition) vary over life for an average woman who survives to each age?

counts <- 
  swe_2015$kin_summary %>%
  group_by(age_focal) %>% 
  summarise(count = sum(count_living)) %>% 
  ungroup()

swe_2015$kin_summary %>%
  select(age_focal, kin, count_living) %>% 
  rename_kin(., consolidate_column = "count_living") %>%
  ggplot(aes(x = age_focal, y = count)) +
  geom_area(aes(fill = kin), colour = "black") +
  geom_line(data = counts, size = 2) +
  labs(x = "Focal's age", y = "Number of living female relatives") +
  coord_cartesian(ylim = c(0, 6)) +
  theme_bw() +
  theme(legend.position = "bottom")

4.3. Age distribution of living kin

How old are Focal’s relatives? Using the kin_full data frame, we can visualize the age distribution of Focal’s relatives throughout Focal’s life. For example when Focal is 35, what are the ages of her relatives:

swe_2015$kin_full %>%
DemoKin::rename_kin() %>%
filter(age_focal == 35) %>%
ggplot() +
geom_line(aes(age_kin, living)) +
geom_vline(xintercept = 35, color=2) +
labs(y = "Expected number of living relatives") +
theme_bw() +
facet_wrap(~kin)

4.4. Deceased kin

We have focused on living kin, but what about relatives who have died during her life? The output of kin also includes information of kin deaths experienced by Focal.

We start by considering the number of kin deaths that can expect to experience at each age. In other words, the non-cumulative number of deaths in the family that Focal experiences at a given age.

loss1 <- 
  swe_2015$kin_summary %>%
  filter(age_focal>0) %>%
  group_by(age_focal) %>% 
  summarise(count = sum(count_dead)) %>% 
  ungroup()

swe_2015$kin_summary %>%
  filter(age_focal>0) %>%
  group_by(age_focal, kin) %>% 
  summarise(count = sum(count_dead)) %>% 
  ungroup() %>% 
  rename_kin(., consolidate_column = "count") %>% 
  ggplot(aes(x = age_focal, y = count)) +
  geom_area(aes(fill = kin), colour = "black") +
  geom_line(data = loss1, size = 2) +
  labs(x = "Focal's age", y = "Number of kin deaths experienced at each age") +
  coord_cartesian(ylim = c(0, 0.086)) +
  theme_bw() +
  theme(legend.position = "bottom")
## `summarise()` has grouped output by 'age_focal'. You can override using the
## `.groups` argument.

Now, we combine all kin types to show the cumulative burden of kin death for an average member of the population surviving to each age:

loss2 <- 
  swe_2015$kin_summary %>%
  group_by(age_focal) %>% 
  summarise(count = sum(count_cum_dead)) %>% 
  ungroup()


swe_2015$kin_summary %>%
  group_by(age_focal, kin) %>% 
  summarise(count = sum(count_cum_dead)) %>% 
  ungroup() %>% 
  rename_kin(., consolidate = "count") %>% 
  ggplot(aes(x = age_focal, y = count)) +
  geom_area(aes(fill = kin), colour = "black") +
  geom_line(data = loss2, size = 2) +
  labs(x = "Focal's age", y = "Number of kin deaths experienced (cumulative)") +
  theme_bw() +
  theme(legend.position = "bottom")
## `summarise()` has grouped output by 'age_focal'. You can override using the
## `.groups` argument.

A member of the population aged 15, 50, and 65yo will have experienced, on average, the death of 0.5, 1.9, 2.9 relatives, respectively.

5. Exercises

For all exercises, assume time-invariant rates at the 2010 levels in Sweden and a female-only population. All exercises can be completed using datasets included in DemoKin.

Exercise 1. Offspring availability and loss

Use DemoKin (assuming time-invariant rates at the 2010 levels in Sweden and a female-only population) to explore offspring survival and loss for mothers.

Answer: What is the expected number of surviving offspring for an average woman aged 65?

# Write your code here

Answer: What is the cumulative number of offspring deaths experienced by an average woman who survives to age 65?

# Write your code here

Exercise 2. Mean age of kin

The output of DemoKin::kin includes information on the average age of Focal’s relatives (in the columns kin_summary$mean_age and kin_summary$$sd_age). For example, this allows us to determine the mean age of Focal’s sisters over the lifecourse of Focal:

swe_2015$kin_summary %>%  
  filter(kin %in% c("os", "ys")) %>% 
  rename_kin() %>% 
  select(kin, age_focal, mean_age, sd_age) %>% 
  pivot_longer(mean_age:sd_age) %>% 
  ggplot(aes(x = age_focal, y = value, colour = kin)) +
  geom_line() +
  facet_wrap(~name, scales = "free") +
  labs(y = "Mean age of sister(s)") +
  theme_bw()
## Warning: Removed 1 row(s) containing missing values (geom_path).

Instructions

Compute the the mean and SD of the age of Focal’s sisters over the ages of Focal by hand (i.e., using the raw output in kin_full). Plot separately (1) for younger and older sisters, (2) and for all sisters.

First, get mean and SD of ages of sisters distinguishing between younger and older sisters:

# Write your code here

Second, get ages of all sisters, irrespective of whether they are older or younger:

# Write your code here

Exercise 3. Living mother

What is the probability that Focal (an average Swedish woman) has a living mother over Focal’s live?

Instructions

Use DemoKin to obtain M_1(a), the probability of having a living mother at age a in a stable population. Conditional on ego’s survival, M_1{(a)} can be thought of as a survival probability in a life table: it has to be equal to one when a is equal to zero (the mother is alive when she gives birth), and goes monotonically to zero.

Answer: What is the probability that Focal has a living mother when Focal turns 70 years old?

# Write your code here

Exercise 4. Sandwich Generation

The ‘Sandwich Generation’ refers to persons who are squeezed between frail older parents and young dependent children and are assumed to have simultaneous care responsibilities for multiple generations, potentially limiting their ability to provide care. In demography, ‘sandwichness’ is a generational process that depends on the genealogical position of an individual vis-a-vis their ascendants and descendants. For this exercise, we consider an individual to be ‘sandwiched’ if they have at least one child aged 15 or younger and a parent or parent within 5 years of death. Alburez‐Gutierrez, Mason, and Zagheni (2021) defined the probability that an average woman aged a is ‘sandwiched’ in a stable female population as:

S(a) = \underbrace{\left(1 - \prod_{x=1}^{15} [1 - m_{a-x})] \right)}_{\substack{\text{fertility risk in the}\\ \text{$15$ years preceding age 'a'}}} \times \underbrace{M_1(a)}_{\substack{\text{Prob. that mother of ego}\\ \text{is alive when ego is 'a' years old}}} \times \underbrace{\left(1- \frac{M_1(a+5)}{M_1(a)}\right)}_{\substack{\text{Prob. that mother of ego}\\ \text{would die within $5$ years}}}

where

  • m_{a-x} is the fertility of women at age a-x, and
  • M_1(a) is the probability of having a living mother at age a in a stable population.

These estimates refer to an average woman in a female population, ignoring the role of offspring mortality.

Instructions

Use DemoKin to compute the probability that Focal is sandwiched, S(a), between ages 15 and 75. Assume time-invariant rates at the 2010 levels and a female-only population.

Answer: At which age is Focal at a highest risk of finding herself sandwiched between young dependent children and fragile older parents?

# Write your code here

6. Vignette and extensions

For more details on DemoKin, including an extension to time varying-populations rates, deceased kin, and multi-state models, see vignette("Reference", package = "DemoKin"). If the vignette does not load, you may need to install the package as devtools::install_github("IvanWilli/DemoKin", build_vignettes = TRUE).

For a detailed description of extensions of the matrix kinship model, see:

  • time-invariant rates (Caswell 2019),
  • multistate models (Caswell 2020),
  • time-varying rates (Caswell and Song 2021), and
  • two-sex models (Caswell 2022).

7. Appendix

7.1. Mean age of living kin

The output of the DemoKin::kin() function can also be used to easily determine the mean age Focal’s relatives by kin type. For simplicity, let’s focus on a Focal aged 35 yo and get the mean age (and standard deviation) of her relatives in our time-invariant population.

ages_df <- 
  swe_2015$kin_summary %>% 
  filter(age_focal == 35) %>% 
  select(kin, mean_age, sd_age)

ma <- 
  ages_df %>% 
  filter(kin=="m") %>% 
  pull(mean_age) %>% 
  round(1)

sda <- 
  ages_df %>% 
  filter(kin=="m") %>% 
  pull(sd_age) %>% 
  round(1)

print(paste0("The mother of a 35-yo Focal woman in our time-invariant population is, on average, ", ma," years old, with a standard deviation of ", sda," years."))
## [1] "The mother of a 35-yo Focal woman in our time-invariant population is, on average, 65.4 years old, with a standard deviation of 5.1 years."

7.2. Visualizing living kin

Finally, let´s visualize the living kin by type and mean age during Ego’s life course:

swe_2015$kin_full %>%
  filter(kin %in% c("m", "gm", "d", "gd")) %>%
  rename_kin() %>% 
  group_by(age_focal, kin) %>% 
  summarise(
    count = sum(living)
    , mean_age = sum(living * age_kin, na.rm=T)/sum(living)
    ) %>% 
  ggplot(aes(age_focal,mean_age)) + 
  geom_point(aes(size=count,color=kin)) +
  geom_line(aes(color=kin)) +
  scale_x_continuous(name = "Age Focal", breaks = seq(0,100,10), labels = seq(0,100,10))+
  scale_y_continuous(name = "Age kin", breaks = seq(0,100,10), labels = seq(0,100,10))+
  geom_segment(x = 0, y = 0, xend = 100, yend = 100, color = 1, linetype=2)+
  labs(color="Relative",size="Living")+
  coord_fixed() +
  theme_bw()
## `summarise()` has grouped output by 'age_focal'. You can override using the
## `.groups` argument.

## Warning: Removed 87 rows containing missing values (geom_point).

## Warning: Removed 87 row(s) containing missing values (geom_path).

8. Session info

sessionInfo()
## R version 4.0.2 (2020-06-22)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 19044)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_United Kingdom.1252 
## [2] LC_CTYPE=English_United Kingdom.1252   
## [3] LC_MONETARY=English_United Kingdom.1252
## [4] LC_NUMERIC=C                           
## [5] LC_TIME=English_United Kingdom.1252    
## 
## attached base packages:
## [1] grid      stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
## [1] fields_11.6     spam_2.6-0      dotCall64_1.0-1 ggplot2_3.3.3  
## [5] tidyr_1.1.3     dplyr_1.0.5     DemoKin_1.0.0  
## 
## loaded via a namespace (and not attached):
##  [1] highr_0.8        pillar_1.5.1     compiler_4.0.2   tools_4.0.2     
##  [5] digest_0.6.28    evaluate_0.17    lifecycle_1.0.0  tibble_3.1.0    
##  [9] gtable_0.3.0     pkgconfig_2.0.3  rlang_1.0.2      igraph_1.2.6    
## [13] cli_3.2.0        DBI_1.1.1        rstudioapi_0.13  yaml_2.2.1      
## [17] xfun_0.21        fastmap_1.1.0    withr_2.5.0      stringr_1.4.0   
## [21] knitr_1.31       maps_3.3.0       generics_0.1.0   vctrs_0.4.1     
## [25] tidyselect_1.1.0 glue_1.6.2       R6_2.5.0         fansi_0.4.2     
## [29] rmarkdown_2.7    farver_2.1.0     purrr_0.3.4      magrittr_2.0.1  
## [33] scales_1.1.1     ellipsis_0.3.2   htmltools_0.5.2  assertthat_0.2.1
## [37] colorspace_2.0-0 labeling_0.4.2   utf8_1.2.1       stringi_1.5.3   
## [41] munsell_0.5.0    crayon_1.4.1

References

Alburez‐Gutierrez, Diego, Carl Mason, and Emilio Zagheni. 2021. “The ‘Sandwich Generation’ Revisited: Global Demographic Drivers of Care Time Demands.” Population and Development Review 47 (4): 997–1023. https://doi.org/10.1111/padr.12436.

Caswell, Hal. 2019. “The Formal Demography of Kinship: A Matrix Formulation.” Demographic Research 41 (September): 679–712. https://doi.org/10.4054/DemRes.2019.41.24.

———. 2020. “The Formal Demography of Kinship II: Multistate Models, Parity, and Sibship.” Demographic Research 42 (June): 1097–1146. https://doi.org/10.4054/DemRes.2020.42.38.

———. 2022. “The Formal Demography of Kinship IV: Two-Sex Models and Their Approximations.” Demographic Research 47 (September): 359–96. https://doi.org/10.4054/DemRes.2022.47.13.

Caswell, Hal, and Xi Song. 2021. “The Formal Demography of Kinship. III. Kinship Dynamics with Time-Varying Demographic Rates.” Demographic Research 45: 517–46.

Keyfitz, Nathan, and Hal Caswell. 2005. Applied Mathematical Demography. New York: Springer.

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DemoKin matrix kinship models tutorial for Dutch Demography Day (2022)

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