A Multistate Life Table Analysis of Union Regimes in the United States: Trends and Racial Differentials, 1970–2002

Abstract

We estimate trends and racial differentials in marriage, cohabitation, union formation and dissolution (union regimes) for the period 1970–2002 in the United States. These estimates are based on an innovative application of multistate life table analysis to pooled survey data. Our analysis demonstrates (1) a dramatic increase in the lifetime proportions of transitions from never-married, divorced or widowed to cohabiting; (2) a substantial decrease in the stability of cohabiting unions; (3) a dramatic increase in mean ages at cohabiting after divorce and widowhood; (4) a substantial decrease in direct transition from never-married to married; (5) a significant decrease in the overall lifetime proportion of ever marrying and re-marrying in the 1970s to 1980s but a relatively stable pattern in the 1990s to 2000–2002; and (6) a substantial decrease in the lifetime proportion of transition from cohabiting to marriage. We also present, for the first time, comparable evidence on differentials in union regimes between four racial groups.

Appendices

Appendix 1: Definitions of the summary measures of the multistate union regimes life table

For the sake of clarity, we omit dimensions of race, sex and period in all variables, but we should keep in mind that all estimates are race-sex-period-specific in this study.

Lifetime proportions of marriage/union formation and dissolution

Let m _ij (x) denote the age-specific o/e rate of transition from marital/union status i to j; the codes of i, j represent: 1. never-married & not-cohabiting, 2. currently married, 3. widowed and not-cohabiting, 4. divorced and not cohabiting, 5. never-married and cohabiting, 6. widowed and cohabiting, 7. divorced and cohabiting; L _i (x), person-years lived in marital/union status i between age x and x + 1 in the life table population; α and ω, the lowest and highest age considered in constructing the multistate life tables. In our current application, we consider that α = 15 and ω = 99.

Let SC denote the lifetime proportion of transition from never-married to cohabitation.

$$ {\text{SC }} = \frac{{\sum\nolimits_{x = \partial }^{\omega } {[L_{1} (x)m_{15} (x)]} }}{{100,000 + \sum\nolimits_{x = \partial }^{\omega } {[L_{5} (x)m_{51} (x)]} }}; $$

Let WC denote the lifetime proportion of transition from widowed to cohabitation.

$$ {\text{WC }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{3} (x)m_{36} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{2} (x)m_{23} (x) + L_{6} (x)m_{63} (x)]} }} $$

Let DC denote the lifetime proportion of transition from divorced to cohabitation.

$$ {\text{DC }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{4} (x)m_{47} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{2} (x)m_{24} (x) + L_{7} (x)m_{74} (x)]} }} $$

Note that SC, WC, and DC cannot be interpreted as proportion of ever experiencing cohabitation before first marriage and after divorce or widowhood, because the numerators of SC, WC, and DC include the events of second and higher order of cohabitation before first marriage or after divorce or widowhood. In other words, SC, WC, and DC include multiple events for some people of entering cohabitation union and entering the risk populations of never-married, widowed and divorced.

Let CM denote the overall lifetime proportion of transition from cohabiting to marrying.

$$ {\text{CM }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{5} (x)m_{52} (x) + L_{6} (x)m_{62} (x) + L_{7} (x)m_{72} (x)]} }}{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{15} (x) + L_{3} (x)m_{36} (x) + L_{4} (x)m_{47} (x)]} }} $$

Let CS denote the lifetime proportion of cohabitation union dissolution.

$$ {\text{CS }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {\left[ {L_{5} (x)m_{51} (x) + L_{6} (x)m_{63} (x) + L_{7} (x)m_{74} (x)} \right]} }}{{\sum\nolimits_{x = \alpha }^{\omega } {\left[ {L_{1} (x)m_{15} (x) + L_{3} (x,t)m_{36} (x) + L_{4} (x)m_{47} (x)} \right]} }} $$

Let SM denote the overall lifetime proportion of first marriage regardless of cohabiting status before first marriage. S_ncM denote the proportion of first marriage while not-cohabiting before marrying among all first marriages; S_cM, the proportion of first marriage while cohabiting before marrying among all first marriages;

$$ {\text{SM }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{12} (x)]} + \sum\nolimits_{x = \alpha }^{\omega } {[L_{5} (x)m_{52} (x)]} }}{100,000}; $$

$$ {\text{S}}_{\text{nc}} {\text{M }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{12} (x)]} }}{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{12} (x)]} + \sum\nolimits_{x = \alpha }^{\omega } {[L_{5} (x)m_{52} (x)]} }}; $$

$$ {\text{S}}_{\text{c}} {\text{M }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{5} (x)m_{52} (x)]} }}{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{12} (x)]} + \sum\nolimits_{x = \alpha }^{\omega } {[L_{5} (x)m_{52} (x)]} }}; $$

$$ {\text{S}}_{\text{nc}} {\text{M }} + {\text{ S}}_{\text{c}} {\text{M }} = { 1}.0 $$

Let MD denote the lifetime proportion of divorce.

$$ {\text{MD }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{2} (x)m_{24} (x)]} }}{{\sum\nolimits_{i} {\sum\nolimits_{x = \alpha }^{\omega } {L_{i} (x,t)m_{i2} (x,t)} } }}, \, \quad {\text{i}} = 1,{ 3},{ 4},{ 5},{ 6},{ 7} $$

Let WM denote overall lifetime proportion of remarriage among those who are widowed regardless of cohabiting status before remarriage; W_ncM, the proportion of remarriage while not-cohabiting before remarrying among all remarriages of widowed; W_cM, the proportion of remarriage while cohabiting before remarrying among all remarriages of widowed.

$$ {\text{WM }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{3} (x)m_{32} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{6} (x)m_{62} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{2} (x)m_{23} (x)} }} $$

$$ {\text{W}}_{\text{nc}} {\text{M}} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{3} (x)m_{32} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{3} (x)m_{32} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{6} (x)m_{62} (x)} }}; $$

$$ {\text{W}}_{\text{c}} {\text{M}} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{6} (x)m_{62} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{3} (x)m_{32} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{6} (x)m_{62} (x)} }}; $$

$$ {\text{W}}_{\text{nc}} {\text{M }} + {\text{ W}}_{\text{c}} {\text{M }} = { 1}.0 $$

Let DM denote overall lifetime proportion of remarriage among those who are divorced regardless of cohabiting status before remarriage; D_ncM, the proportion of remarriage while not-cohabiting before remarrying among all remarriages of divorcees; D_cM, the proportion of remarriage while cohabiting before remarrying among all remarriages of divorcees.

$$ {\text{DM }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{4} (x)m_{42} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{7} (x)m_{72} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{2} (x)m_{24} (x)} }} $$

$$ {\text{D}}_{\text{nc}} {\text{M }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{4} (x)m_{42} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{4} (x)m_{42} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{7} (x)m_{72} (x)} }}; $$

$$ {\text{D}}_{\text{c}} {\text{M }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{7} (x)m_{72} (x)} }}{{\sum\nolimits_{x = \alpha }^{\omega } {L_{4} (x)m_{42} (x)} + \sum\nolimits_{x = \alpha }^{\omega } {L_{7} (x)m_{72} (x)} }}; $$

$$ {\text{D}}_{\text{nc}} {\text{M }} + {\text{ D}}_{\text{c}} {\text{M }} = { 1}.0 $$

Average life-time numbers of cohabitations, marriages, divorces, and widowhoods per person

AC—the average number of cohabitation unions per person in the lifetime;

$$ {\text{AC}} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{1} (x)m_{15} (x) + L_{3} (x)m_{36} (x) + L_{4} (x)m_{47} (x)]} }}{100,000} $$

AM—the average number of marriages (including the first marriage and remarriages) per person in the lifetime;

$$ {\text{AM }} = \frac{{\sum\nolimits_{i} {\sum\nolimits_{x = \alpha }^{\omega } {L_{\text{i}} (x)m_{i2} (x)} } }}{100,000},\quad {\text{i}} = 1,{ 3},{ 4},{ 5},{ 6},{ 7} $$

AD—the average number of divorces per person in the lifetime;

$$ {\text{AD }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{2} (x)m_{24} (x)} }}{100,000} $$

The proportion of life span after age $\partial$ (the lowest marriageable or cohabiting age) spent in different marital/union statuses

Let P _i denote the proportion of life span after age $\partial$ spent in marital/union status i.

$$ P_{i} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{i} (x)} }}{{\sum\nolimits_{i} {\sum\nolimits_{x = \alpha }^{\omega } {L_{i} (x)} } }},\quad {\text{i }} = { 1},{ 2},{ 3},{ 4},{ 5},{ 6},{ 7};\quad \sum\nolimits_{i} {P_{i} = 1.0} $$

Appendix 2: A procedure to adjust the o/e rates of marital/cohabiting union status transitions based on the NSFH and NSFG data for consistency with the o/e rates of marital status transitions based on the CPS, SIPP, NSFH, and NSFG data

We perform the adjustments for each of the race groups, men, women, and the periods (1970s, 1980s, 1990s, and 2000–2002), respectively, while we omit the dimension indices of race, sex and period in the formulas for simplicity of the presentation.

Let m4_ij(x) denote the o/e rate of transition from marital status i to marital status j between age x and x + 1 based on the CPS, SIPP, NSFH, and NSFG data, using a classic 4 marital statuses model (i,j = 1,2,3,4, represent never-married, married, widowed and divorced, respectively, excluding cohabitation).

m*_ij(x), observed and unadjusted age-specific o/e rates of transitions from marital/union status i to j (i,j = 1,2,3,4,5,6,7, including cohabitation, see the definitions in Appendix 1), based on NSFH and NSFG data;

m_ij(x), the final adjusted age-specific o/e rates of transitions from marital/union status i to j (i,j = 1,2,3,4,5,6,7, including cohabitation) based on pooled survey data, and adjusted to be consistent with m4_ij(x); m_ij(x) can be analytically transferred into P_ij(x), age-specific probabilities of transitions from marital/union status i to j using the standard formula in multistate demography (see, e.g., Willekens et al. 1982; Schoen 1988; Preston et al. 2001).

$$ {\text{l}}_{\text{i}} \left( {{\text{x}} + 1} \right) \, = {\text{l}}_{\text{k}} \left( {\text{x}} \right){\text{P}}_{\text{ki}} \left( {\text{x}} \right),\quad {\text{k}} = 1,{ 2}, \ldots , 7 $$

(1)

$$ {\text{L}}_{\text{i}} \left( {\text{x}} \right) = 0. 5[{\text{l}}_{\text{i}} \left( {\text{x}} \right) + {\text{l}}_{\text{i}} \left( {{\text{x}} + 1} \right)] $$

(2)

The goal of the adjustment is to make the average number of marriages including first and re-marriages (AM7) and average number of divorces (AD7) in the life time in the 7 marital/union status life table (including cohabitation) based on NSFH and NSFG data equal to the corresponding average numbers (AM4 and AD4) in the life table of 4 marital statuses excluding cohabitation based on all of the data from CPS, SIPP, NSFH, and NSFG.

We use the m4_ij(x) to compute P4_ij(x), age-specific probabilities of marital status transitions based on CPS, SIPP, NSFH, and NSFG data, using the standard formula. Based on P4_ij(x), we construct a multi-state life table to get L4_i(x), using formulas (1) and (2) presented above. We then use m4_ij(x) and L4_i(x) to compute the AM4 and AD4 in the 4 marital statuses model based on CPS, SIPP, NSFH, and NSFG data.

$$ {\text{AM4 }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L4_{\text{i}} (x)m4_{i2} (x)} }}{100,000},\quad {\text{i}} = 1,{ 3},{ 4} $$

$$ {\text{AD4 }} = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {L_{2} (x)m_{24} (x)} }}{100,000} $$

We then employ the following two-step procedure to adjust the observed o/e rates of 1^st marriage, divorce, and remarriages (m*₁₂(x), m*₅₂(x), m*₂₄(x), m*₃₂(x), m*₆₂(x), m*₄₂(x), and m*₇₂(x)), but do not need to adjust the observed o/e rates of cohabitation union formation and dissolution (m*₁₅(x), m*₃₆(x), m*₄₇(x), m*₅₁(x), m*₆₃(x), m*₇₄(x)) based on NSFH and NSFG.

Step 1: Adjustment for the o/e rates of first marriage, remarriage, and divorce

We use the unadjusted survey-based m*_ij(x) to compute P*_ij(x), and we then use P*_ij(x) to construct an initial multi-state life table and get the initial L*_i(x) using formulas (1) and (2); we then use m*_ij(x) and L*_i(x) to compute the initial AM7* and AD7* in the 7 marital/union statuses model based on the NSFH and NSFG data.

$$ {\text{AM7}}^* \, = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{\text{i}^*} (x)m_{i2}^* (x)} }}{100,000},\quad {\text{i}} = 1,{ 3},{ 4},{ 5},{ 6},{ 7} $$

(3)

$$ {\text{AD7}}^* \, = \frac{{\sum\nolimits_{x = \alpha }^{\omega } {[L_{2}^* (x)m_{24}^* (x)]} }}{100,000}, $$

(4)

We use AM4/AM7*, AD4/AD7* as adjustment factors (not age-specific) to adjust the corresponding age-specific o/e rates of first marriage, remarriage, and divorce for not-cohabiting and cohabiting persons at ages x (x = α to ω).

$$ {\text{m}}'_{ 1 2} \left( {\text{x}} \right) \, = {\text{ m}}_{ 1 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(5)

$$ {\text{m'}}_{ 5 2} \left( {\text{x}} \right) = {\text{m}}_{ 5 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(6)

$$ {\text{m'}}_{ 3 2} \left( {\text{x}} \right) = {\text{m}}_{ 3 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(7)

$$ {\text{m'}}_{ 6 2} \left( {\text{x}} \right) = {\text{m}}_{ 6 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(8)

$$ {\text{m'}}_{ 4 2} \left( {\text{x}} \right) = {\text{m}}_{ 4 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(9)

$$ {\text{m'}}_{ 7 2} \left( {\text{x}} \right) = {\text{m}}_{ 7 2}^* \left( {\text{x}} \right){\text{ AM4}}/{\text{AM7}}^* $$

(10)

$$ {\text{m'}}_{ 2 4} \left( {\text{x}} \right) \, = {\text{ m}}_{ 2 4}^* \left( {\text{x}} \right){\text{ AD4}}/{\text{AD7}}^* $$

(11)

Step 2: Check whether the goal of the adjustment is achieved

We use the first adjusted m’_ij(x) to compute the first adjusted P′_ij(x), and use m′_ij(x) to replace m*_ij(x) in the formulas (3) and (4) to get the first adjusted AM7′ and AD7′. If the absolute values of the relative difference between AM7′ and AM4 and between AD7′ and AD4 are all less than a selected criterion (e.g., 0.5%), we have completed Step 2 and have the final estimates of the o/e rates (m_ij(x)). Otherwise, we will have to use the first adjusted AM7′ and AD7′ to replace AM7* and AD7* in formulas (5–11) to repeat the iterative procedures described in Step 1 and Step 2 until the selected criterion is achieved.

Appendix 3

See Table 4.

Table 4 Comparisons of summary measures of the marital status life tables (excluding cohabitation) between our estimates based on the pooled survey data and Schoen’s estimates based on vital statistics, all races combined