ProFamy: The Extended Cohort-Component Method for Household and Living Arrangement Projections

Abstract

Chapter 1 ended with a description of the origins of the extended cohort-component method for projections of a population classified by household types and living arrangements, as well as the associated software known as ProFamy. This chapter presents and discusses the methodology in detail.

Appendices

Appendix 1: Procedure to Correct the Inaccurate Accounting of Household Size Distribution Due to the Lack of Capacity to Identify the Reference Person’s Co-residence Status with Other Relatives or Non-relatives

Based on the census data set, we can derive h(i,j,t), the proportion of households with i direct family members and j other relatives or non-relatives among the total number of households with i direct family members in year t. The term “direct family members” here refers to spouse (or cohabiting partner), children, and parents of the reference person. ${\displaystyle \sum_{j=0}^M\mathrm{h}\left(\mathrm{i},\mathrm{j},\mathrm{t}\right)=1.0}$, for all i. The maximum value of i in our model is 2 + 2 + P; i.e., the largest three-generation household has two grandparents, two parents, and P (highest parity distinguished) children. j = 0, 1, 2, 3, …, M, where M is the largest number of other relatives or non-relatives living in a household. We chose M as 5 in our current version of the ProFamy software since the number of single households with more than five other relatives or non-relatives in modern societies is negligible.

Denote by H(i,t) the number of households of size i accounted for by our model before the adjustment. Denote N(i,j,t) as the number of households with i direct family members and j other relatives or non-relatives in year t. N(i,j,t) = H(i,t)h(i,j,t). The actual household size of H(i,j,t) is i + j. Regrouping H(i,j,t) with the sum of i and j as z, we obtain the adjusted number of households with size z in year t, which is denoted as H(z,t), where z = 1, 2, 3, …, 2 + 2 + P + M (i.e., the largest household size is 2 + 2 + P + M).

The average number of other relatives or non-relatives among all households with i direct family members is $a\left(i,t\right)={\displaystyle \sum_{j=0}^M\mathrm{h}\left(\mathrm{i},\mathrm{j},\mathrm{t}\right)\ \mathrm{j}}$. We can allow a(i,t) to change over time during the projection period. We may assume that the relative changes in h(i,j,t) for all j > 0 in year t as compared with year t-1 is the same as the relative changes of a(i,t) as compared with a(i,t−1); more specifically, assuming h(i,j,t) = h(i, j, t − 1)a(i,t)/a(i, t − 1) for all j > 0. If the sum of h(i,j,t) (j > 0) over j is greater than one, which will usually never happen in the real world, we will have to standardize h(i,j,t) (j > 0) to ensure their sum is not greater than one. We then estimate h(i,0,t) as $h\left(i,0,t\right)= 1.0-{\displaystyle \sum_{j=1}^M\mathrm{h}\left(\mathrm{i},\mathrm{j},\mathrm{t}\right)}$

To help readers to understand how this procedure works, we present a numerical example as follows. Based on the U.S. 1990 census data set, we know that the proportion of American households with four direct family members and 0, 1, 2, 3, 4, 5 other relatives or non-relatives were 0.9320 0.0516 0.0102 0.0040 0.0012, 0.0011, respectively; the average number of other relatives or non-relatives among the households of four direct family members was 0.09 in 1990. If we assume that this average will become 0.11 in year 2000, we then estimate:

$$ h\left( 4, 1, 2000\right)=h\left( 4, 1, 1 990\right)\times 0.11/ 0.09= 0.0516\times 1.222= 0.0631; $$

$$ h\left( 4, 2, 2 000\right)=h\left( 4, 2, 1990\right)\times 0.11/ 0.09= 0.0102\times 1.222= 0.0125; $$

$$ h\left( 4, 3, 2000\right)=h\left( 4, 3, 1990\right)\times 0.11/ 0.09= 0.0040\times 1.222= 0.0049; $$

$$ h\left( 4, 4, 2000\right)=h\left( 4, 4, 1990\right)\times 0.11/ 0.09= 0.0012\times 1.222= 0.0015; $$

$$ h\left( 4, 5, 2000\right)=h\left( 4, 5, 1990\right)\times 0.11/ 0.09= 0.0011\times 1.222= 0.0013; $$

$$ h\left( 4, 0, 2000\right)= 1.0-\left( 0.0631+ 0.0125+ 0.0049+ 0.0015+ 0.0013\right)= 0.9167. $$

Appendix 2: A procedure to Meet the Requirement that Other Relatives and Non-relatives Cannot Be Reference Persons of the Household

The k status is equal to 3 (not living with parents) for those who are relatives other than parents and children or non-relatives of the reference person, since they do not live with parents. The c status of these persons is equal to 0, since they do not live with children. If no adjustment is made, these people would be counted as a one-person household if they are not married and not cohabiting (most likely), or counted as a one-couple household if they are married or cohabiting (less likely). Therefore, adjustment of the number of persons whose k = 3 and c = 0 must be done in order to derive a correct account of one-person and one-couple households.

Based on the census sample data set of the starting year, we calculate the 5-year-age-specific and marital-status-specific proportions of those, who are relatives (other than parents and children) or non-relatives in reference to the household reference person, among all persons with the same age and marital status not living with parents and children. We may assume that these proportions either remain constant or change over time. Multiplying these projected (or assumed) proportions by the corresponding number of persons whose k status is equal to 3 and c status is equal to 0, we estimate the number of other relatives and non relatives in the future years. Subtracting them from the number of persons whose k status is equal to 3 and c status is equal to 0, we have met the requirement that other relatives and non-relatives cannot be reference persons of the households.

Appendix 3: Parity Transition Probabilities in the 1st and 2nd Half of the Year

The age-specific probabilities of parity status change occurring throughout the first and second halves of the interval are “gross” probabilities, in the absence of the mother’s mortality since it had already been taken into account in the middle of the single-year age interval.

Let f _p (x, m) denote the occurrence/exposure rates of the pth birth by age x and marital status m of the mother, which is defined as the number of pth births by women aged x to x + 1, with marital status m divided by the person-years lived in parity p-1 and marital status m of women aged x to x + 1. The probability that a woman of parity p-1 and marital status m at exact age x will be in parity p at exact age x + 1 in the absence of mortality and marital status change, b _p (x, m), can be estimated in a familiar manner with the assumption of a uniform distribution of births between ages x and x + 1 (analogous to the estimation of death probabilities from death rates):

$$ {b}_p\left(x,m\right)=\frac{f_p\left(x,m\right)}{1+{\scriptscriptstyle \frac{1}{2}}{f}_p\left(x,m\right)}\left(p=1,2,3....,N\right). $$

(2.22)

As stated earlier, we will calculate the parity status change in the first half and in the second half of the age interval, respectively, so the corresponding formulas are needed. It should be stated that the following derivation is based on the assumption that no multiple parity transitions take place within a single age interval. There are at least two reasons for making this assumption. First, the multiple parity transitions are very rare. Second, birth rates are usually defined as the number of births divided by the number of women at risk. Multiple births and multiple deliveries in a single year have already been counted in the number of births, which is the numerator of the birth rates to be used.

Define _1/2 b _p (x,m) and _1/2 b _p (x + 0.5, m) as the probabilities of giving a pth birth between exact ages x and $x+{\scriptscriptstyle \frac{1}{2}}$ and between exact ages $x+{\scriptscriptstyle \frac{1}{2}}$ and x + 1, respectively, in the absence of mortality. Define W as the number of women of parity p-1 at exact age x. Assuming the uniform distribution of births in a year, the number of pth births to these W women in the first half of the year is equal to those occur in the second half of the year; both are ${\scriptscriptstyle \frac{1}{2}}W{b}_p\left(x,m\right)$ . Therefore, the probability of giving a pth birth in the first half of the year is:

$$ {}_{1/2}b_p\left(x,m\right)=\frac{1}{2}W{b}_p\left(x,m\right)/W={b}_p\left(x,m\right)/2. $$

(2.23)

There are $W-{\scriptscriptstyle \frac{1}{2}}W{b}_p\left(x,m\right)$ women of parity p-1 in the middle of the year at risk of giving a pth birth. Since we assume that no multiple births occur in a single age interval, we must assume that the women who were of parity p-2 at the beginning of the age interval but who give a (p-1)^th birth in the first half of the interval are not at risk of giving a pth birth in the second half of the interval. The probability of giving a pth birth in the second half of the year is

$$ \begin{array}{l}{}_{1/2}b_p\left(x+0.5,m\right)=\frac{1}{2}W{b}_p\left(x,m\right)/\left[W-0.5W{b}_p\left(x,m\right)\right]\\ {}={b}_p\left(x,m\right)/\left[2-{b}_p\left(x,m\right)\right].\end{array} $$

(2.24)

Note that the data f _p (x,m) are for an 1-year age interval, but the calculation of parity transitions between exact age x and x + 1 is divided into two steps by formulas (2.23) and (2.24). Fortunately, however, the parity distribution at the end of the age interval calculated by _1/2 b _p (x,m) and _1/2 b _p (x + 0.5, m) with two steps is the same as the parity distribution calculated by one step only, using b _p (x,m) estimated by formula (2.22). This equivalence can be demonstrated as follows: first, combining two steps, the probability of parity progression is

$$ {}_{1/2}b_p\left(x,m\right)+\left[1-{}_{1/2}b_p\left(x,m\right)\right]{}_{1/2}b_p\left(x+0.5,m\right)={}_{1/2}b_p\left(x,m\right)+\frac{\left[1-0.5{b}_p\left(x,m\right)\right]{b}_p\left(x,m\right)}{2-{b}_p\left(x,m\right)}={b}_p\left(x,m\right) $$

Second, the probability of no parity progression is

$$ \left[1-{}_{1/2}b_p\left(x,m\right)\right]\left[1-{}_{1/2}b_p\left(x+0.5,m\right)\right]=\left[1-\frac{b_p\left(x,m\right)}{2}\right]\left[1-\frac{b_p\left(x,m\right)}{2-{b}_p\left(x,m\right)}\right]=1-{b}_p\left(x,m\right) $$

This supports our two-step approach for calculating parity transitions (Zeng 1991a: 61–63).

Appendix 4: Procedures for Estimating Transition Probabilities of Status of Co-residence with Parents

Let w _ij (x,t,s,m) denote the probability of transition from co-residence status i at age x in year t to j at age x + 1 in year t + 1 for persons of sex s and marital status m, where i (=1,2,3) and j (=1,2,3);

q _m (x,t) and q _f (x,t), probabilities of death of an x-year-old person’s mother and father;

d _m (x,t) and d _f (x,t), probabilities of divorce of an x-year-old person’s mother and father in year t;

q ₁ (x,t) and q ₂ (x,t), female and male death probabilities in year t;

d ₁ (x,t) and d ₂ (x,t), female and male divorce probabilities in year t;

z, the average age difference between the male and female partners;

$$ {q}_m\left(x,t\right)={\displaystyle \sum_{i=15}^{49}{q}_1}\left(x+i,t\right){f}_1(i); {q}_f\left(x,t\right)={\displaystyle \sum_{i=15}^{49}{q}_2}\left(x+z+i,t\right){f}_2(i); $$

$$ {d}_m\left(x,t\right)={\displaystyle \sum_{i=15}^{49}{d}_1}\left(x+i,t\right){f}_1(i); {d}_f\left(x,t\right)={\displaystyle \sum_{i=15}^{49}{d}_2}\left(x+z+i,t\right){f}_2(i); $$

Note that f ₁ (i) and f ₂ (i) are the frequency distributions of a product of age-specific fertility rates and conditional survival probability.

$$ {f}_1(i)=\left(b(i){l}_1\left(x+i\right)/{l}_1(i)\right)/{\displaystyle \sum_{i=15}^{49}\Big(b(i){l}_1}\left(x+i\right)/{l}_1(i) $$

$$ {f}_2(i)=\left(b(i){l}_2\left(x+i\right)/{l}_2(i)\right)/{\displaystyle \sum_{i=15}^{49}\Big(b(i){l}_2}\left(x+i\right)/{l}_2(i) $$

where b(i) are age-specific fertility rates, l ₁ (x) and l ₂ (x) are female and male survival probabilities from age 0 to x. It is ideal that b(i),l ₁ (x) and l ₂ (x) are cohort data, but it would be a good approximation if one employs the period data since the frequency distribution rather than the fertility and mortality level is used.

The events that cause transitions of the co-residence status from 1 to 2 are death of one of the parents or divorces of the parents. If the death of one parent occurs first, divorce cannot occur. Divorce, however, may precede death. Therefore,

$$ \begin{array}{ll}{w}_{12}\left(x,t,s,m\right)\hfill & ={q}_m\left(x,t\right)+{q}_f\left(x,t\right)+d\left(x,t\right)-{q}_m\left(x,t\right)\ {q}_f\left(x,t\right)\hfill \\ {}\hfill & -{q}_m\left(x,t\right)d\left(x,t\right)/ 2-{q}_f\left(x,t\right)d\left(x,t\right)/ 2\hfill \end{array} $$

(2.25)

where d(x,t) = (d _m (x,t) + d _f (x,t))/2.

The events that cause transitions of the co-residence status from 1 to 3 are an x-year-old person leaving the parental home or numbers of death of both parents. If the deaths of both parents occur first, the event of leaving parental home cannot occur. A person can leave home, however, before either parent dies or after one of them dies. Therefore,

$$ {w}_{13}\left(x,t,s,m\right)=l\left(x,t,s,m\right)+{q}_m\left(x,t\right)\ {q}_f\left(x,t\right)-{q}_m\left(x,t\right)\ {q}_f\left(x,t\right)\ l\left(x,t,s,m\right)\left( 2/ 3\right) $$

(2.26)

where l(x,t,s,m) is the probability of leaving the parental home at age x in year t for persons of sex s and marital status m.

The events that cause transitions of the co-residence status from 2 to 3 are death of the non-married parent or numbers of an x-year-old person leaving the parental home. If the death of the lone parent occurs first, the event of leaving the parental home cannot occur. Therefore,

$$ {w}_{23}\left(x,t,s,m\right)=l\left(x,t,s,m\right)+q\left(x,t\right)-\left(l\left(x,t,s,m\right)\ q\left(x,t\right)\right)/ 2 $$

(2.27)

where q(x,t) = (q _m (x,t) + q _f (x,t))/2.

The events that cause transitions of the co-residence status from 2 to 1 are remarriage of the non-married parent, who may be widowed or divorced. Denote r _d1 (x,t) and r _d2 (x,t) as divorced female and male remarriage probabilities in year t; r _w1 (x,t) and r _w2 (x,t) as widowed female and male remarriage probabilities in year t.

$$ \begin{array}{l}{w}_{21}\left(x,t,s,m\right)=\left({\displaystyle \sum_{i=15}^{49}{r}_{d1}\left(x+i,t\right){f}_1(i)\Big){g}_{d1}(x)}+\right({\displaystyle \sum_{i=15}^{49}{r}_{d2}\left(x+z+i,t\right){f}_2(i)\Big){g}_{d2}\left(x+z\right)}\\ {}+\left({\displaystyle \sum_{i=15}^{49}{r}_{w1}\left(x+i,t\right){f}_1(i)\Big){g}_{w1}(x)}+\right({\displaystyle \sum_{i=15}^{49}{r}_{w2}\left(x+z+i,t\right){f}_2(i)\Big){g}_{w2}\left(x+z\right)}\end{array} $$

(2.28)

where,

$$ {g}_{d1}(x)={\displaystyle \sum_{i=15}^{49}{N}_{d1}\left(x+i\right)/{\displaystyle \sum_{i=15}^{49}\left[{N}_{d1}\left(x+i\right)+{N}_{d2}\left(x+z+i\right)+{N}_{w1}\left(x+i\right)+{N}_{w2}\left(x+z+i\right)\right]}} $$

$$ {g}_{d2}(x)={\displaystyle \sum_{i=15}^{49}{N}_{d2}\left(x+i\right)/{\displaystyle \sum_{i=15}^{49}\left[{N}_{d1}\left(x+i\right)+{N}_{d2}\left(x+z+i\right)+{N}_{w1}\left(x+i\right)+{N}_{w2}\left(x+z+i\right)\right]}} $$

$$ {g}_{w1}(x)={\displaystyle \sum_{i=15}^{49}{N}_{w1}\left(x+i\right)/{\displaystyle \sum_{i=15}^{49}\left[{N}_{d1}\left(x+i\right)+{N}_{d2}\left(x+z+i\right)+{N}_{w1}\left(x+i\right)+{N}_{w2}\left(x+z+i\right)\right]}} $$

$$ {g}_{w2}(x)={\displaystyle \sum_{i=15}^{49}{N}_{w2}\left(x+i\right)/{\displaystyle \sum_{i=15}^{49}\left[{N}_{d1}\left(x+i\right)+{N}_{d2}\left(x+z+i\right)+{N}_{w1}\left(x+i\right)+{N}_{w2}\left(x+z+i\right)\right]}} $$

and N _d1 (x + i), N _d2 (x + z + i), N _w1 (x + i), and N _w2 (x + z + i) are the number of divorced females, divorced males, widowed females, widowed males, age x + i or x + z + i, all living with at least one child in year t.

The event that causes a transition of the co-residence status from 3 to 1 is an x-year-old person returning home to join her or his two parents; the event that causes a transition of the co-residence status from 3 to 2 is an x-year-old person returning home to join her or his one non-married parent, so that

$$ {w}_{31}\left(x,t,s,m\right)=h\left(x,t,s,m\right)\left\{{N}_{k 1}\left(x,t,s,m\right)/\left[{N}_{k 1}\left(x,t,s,m\right)+{N}_{k 2}\left(x,t,s,m\right)\right]\right\} $$

(2.29)

$$ {w}_{32}\left(x,t,s,m\right)=h\left(x,t,s,m\right)\left\{{N}_{k 2}\left(x,t,s,m\right)/\left[{N}_{k 1}\left(x,t,s,m\right)+{N}_{k 2}\left(x,t,s,m\right)\right]\right\} $$

(2.30)

where h(x,t,s,m) is the probability of returning home between age x and x + 1 to join the parental home in year t, for persons of sex s and marital status m. N _k1 (x,t,s,m) and N _k2 (x,t,s,m) are numbers of x-year-old persons of sex s and marital status m who are living with two parents and one parent, respectively.

In addition,

$$ {w}_{11}\left(x,t,s,m\right)= 1-{w}_{12}\left(x,t,s,m\right)-{w}_{13}\left(x,t,s,m\right) $$

(2.31)

$$ {w}_{22}\left(x,t,s,m\right)= 1-{w}_{21}\left(x,t,s,m\right)-{w}_{23}\left(x,t,s,m\right) $$

(2.32)

$$ {w}_{33}\left(x,t,s,m\right)= 1-{w}_{31}\left(x,t,s,m\right)-{w}_{32}\left(x,t,s,m\right) $$

(2.33)

Appendix 5: Procedures for Estimation of Probabilities of Change in Number of Children Living Together

Denote by q ₁ the average probability of dying for the children of an x-year-old mother or father;

q ₂ , average probability of leaving the parental home for the children of an x-year-old mother or father;

q(x−i), age-specific average death rates for male and female children;

h(x−i), age-specific average rates of leaving the parental home for male and female children;

f(i), the frequency distribution of fertility rates from age α to age x; ${\displaystyle \sum_{i=\alpha}^xf(i)=1.0}$, i as the age at birth of the mother or father; α, the lowest age at birth.

From the model, we know an x-year-old person has c (c = 0, 1, 2, …) children living together, but the ages of these c children are not kept track of to make the model manageable. The chance that the x-year-old person gave a birth at age i and the child is x−i years old is f(i). The weighted average of the probability of dying of a child of an x-year-old person can be estimated as

$$ {q}_1={\displaystyle \sum_{i=\alpha}^xq\left(x-i\right)*f(i)} $$

(2.34)

The weighted average of the probability of leaving the parental home of a child of an x-year-old person can be estimated as

• 1. If three-generation households are considered,

  $$ {q}_2={\displaystyle \sum_{i=\alpha}^xh\left(x-i\right)*f(i)} $$

  (2.35)

• 2. If three-generation households are negligible, such as in the Western countries, we assume all children who have not left the parental home before marriage (or cohabitation) will do so in the same year of their marriage (or cohabitation). In other words, children who remain single until the end of the year have a leaving home probability of h(x−i); children who newly marry or enter a union in the year have a leaving home probability of 1.0.

  $$ {q}_2={\displaystyle \sum_{i=\alpha}^x\left[h\left(x-i\right)\left(1-m\left(x-i\right)\right)+1.0m\left(x-i\right)\left]*f\right(i\right)} $$

  (2.36)

where m(x−i) is the age-specific average probability of first marriage/union formation for male and female children.

The probability that a child will survive and continue to live at home is p = (1−q ₁ ) (1−q ₂ ); the probability that a child will either leave home or die is 1−p.

Assuming the events of death and leaving the parental home are locally independent, we could easily estimate the probability of changes in the c status of the number of co-residing children. To simplify the presentation, we assume that the highest parity is 3 here (the calculation method will be the same when the highest parity is larger than 3).

let s ₁ (t), s ₂ (t), and s ₃ (t) denote the probabilities that the one child, two children, or three children who were living at home at the beginning of the year will survive and live at home at the end of the year t;

d ₁ (t), d ₂ (t), and d ₃ (t), the probabilities that the one child, both of the two children, and all of the three children will die or leave home during the year t.

d ₁₂ (t), d ₁₃ (t), and d ₂₃ (t), the probabilities that one of the two children, one of the three children, and two of the three children will die or leave home at the end of the year t.

The estimators of s ₁ (t), s ₂ (t), s ₃ (t), d ₁ (t), d ₂ (t), d ₃ (t), d ₁₂ (t), d ₂₃ (t) and d ₁₃ (t) are as follows:

$$ {s}_1(t)=p $$

(2.37)

$$ {s}_2(t)=p\times p $$

(2.38)

$$ {s}_3(t)=p\times p\times p $$

(2.39)

$$ {d}_1(t)= 1-p $$

(2.40)

$$ {d}_2(t)=\left( 1-p\right)\times \left( 1-p\right) $$

(2.41)

$$ {d}_3(t)=\left( 1-p\right)\times \left( 1-p\right)\times \left( 1-p\right) $$

(2.42)

$$ {d}_{12}(t)= 2\times p\left( 1-p\right) $$

(2.43)

$$ {d}_{23}(t)= 3\times p\times \left( 1-p\right)\times \left( 1-p\right) $$

(2.44)

$$ {d}_{13}(t)= 3\times p\times p\times \left( 1-p\right) $$

(2.45)

In the case in which two children leave home or die in the year t among the three co-residing children (d ₂₃ (t)), for example, there are three combinations of one of the three children leaving home or dying while the other two survive and continue to stay at home. Therefore, we multiply ‘p × (1−p) × (1−p)’ by ‘3’.