How architectural principles can help conceptualize and analyze breakups among intergenerational households

Guided by two architectural design principles, we investigated whether differences in the ways intergenerational households were structured could predict the odds of intergenerational households breaking apart. The two architectural design principles guiding our study were: (a) to classify structures, such as intergenerational households, according to a hierarchical ordering, in our case, a generational hierarchy, first, second, third, and so forth); and (b) to identify a central entity or focal point responsible for maintaining the structure, again for us, the focal generation responsible for a household. Applying both principles to a rich source of data that contained a large sample of intergenerational households, we found striking differences in odds of breakups by generational ordering, focal generation, and race. Whereas white three-generation households headed by grandparents were the most likely to break up, black skipped-generation households headed by grandparents were the least likely to break up.


Introduction
T oday's children grow up in many different living arrangements (Casper and Bianchi, 2002;Kreider and Ellis, 2011), some of which change quickly and others very little. The stability of children's diverse living arrangements has been the subject of debate (Acs, 2007;Brown, 2010;Kennedy and Bumpass, 2008;Kreider and Ellis, 2011;Smock and Greenland, 2010) because research documents the negative effects on children's well-being and development when arrangements end suddenly, or there is a succession of uncertain, short-term arrangements (Cavanagh and Huston, 2006;Krohn et al., 2009;Magnuson and Berger, 2009;Sun and Li, 2011).
Among the diverse living arrangements in which today's children can grow up, two of the more prevalent are three-and skipped-generation households. Between 2001 and 2012, there was a 30% increase in the proportion of children living in threegeneration households (Dunifon et al., 2014). Research estimates that nearly 30% of US children will live with grandparents at some point during their childhood (Amorim et al., 2017). In 2012, 4.2 million households had both grandchildren under 18 years of age and grandparents-i.e., about 3% of all households and 10% of all children. Sixty-seven percent of these 4.2 million households were maintained by a grandparent while another 31% were maintained by a parent, and the remaining 2% was maintained by someone else (Ellis and Simmons, 2014). Since the 1990s many grandparents have raised grandchildren on their own due to the absence of both parents (Kreider and Ellis, 2011;Lugailia, 1998;Simmons and Dye, 2003). In 2012, among the grandchildren in the homes of grandparents, about 20% had no parents present (Ellis and Simmons, 2014). Overall, with 12% of children living with a grandparent in 2015 (Pilkauskas and Dunifon, 2016, as cited in Amorim et al., 2017), the trend of grandchildren living with a grandparent shows no signs of abating.
Though inroads have been made in our understanding of three-and skipped generation households (Dunifon et al., 2014;Fuller-Thomson et al., 1997;Goodman and Silverstein, 2002;Pilkauskas, 2012), there is more to learn about the permanency of these households, especially since breakups can affect grandchildren's well-being (Aquilino, 1996;Dunifon and Kowaleski-Jones, 2007). Some studies offer insights into the permanency of three-and skipped-generation households, but questions remain, such as, whether characteristics that can distinguish among threegeneration households and can also distinguish between threeand skipped-generation households could help explain differences in the odds of breakups (Blustein et al., 2004;Dunifon et al., 2014;Dunifon and Kowaleski-Jones, 2007;Oberlander et al., 2009;Pilkauskas, 2012;Pilkauskas and Martinson, 2014).
Essentially, the challenge is to employ a conceptual framework that applies to a diverse set of intergenerational households that will allow us to investigate the odds of breakups. This challenge is much more difficult than one might first imagine since there are many demographic characteristics within intergenerational households based upon generation: first, second, third, even fourth! Another complication is that because only one generation within a generational hierarchy usually has responsibility for overseeing a household's affairs and maintenance that generation must first be identified and then its characteristics evaluated as to mattering to the permanence or impermanence of the household.
We believed that the challenge was surmountable, however, by framing the question in terms of two architectural design principles: (1) characterize the hierarchy of a structure that orders its elements into hierarchical positions by size, shape, or placement; and (2) identify a central entity or focal point of a structure that orders "form and space" and seeks to maintain symmetry and balance (Ching, 2014;Arnheim, 2009;Mallgrave and Goodman, 2011;Ching and Eckler, 2012). Intergenerational households are shaped by ordering of parentage and functions within these households usually coordinated by or accredited to a specific focal generation.
Applying these two architectural design principles, we conceptualized intergenerational households as structures that contain two or more related generations that are sequentially ordered by lineage, thereby reflecting a hierarchical structure, 1 (design principle 1), and a focal generation, one of those sequentially ordered generations, that is responsible for the maintenance and functioning of the household, (design principle 2). 2 For the second design principle, the focal generation has been labeled in studies of households and by governmental agencies as the head of a household or the householder (U.S. Census Ruggles and Brower, 2003;Smith, 1992).
Guided by these architectural design principles, we then tied the focal generation with the generational hierarchy for three types of intergenerational households: (1) a three-generation household headed by a second generation, (3G2G); (2) a threegeneration household headed by a first-generation (3G1G); and (3) a skipped-generation household headed by a first-generation (SG1G). 3 For brevity, the acronyms 3G2G, 3G1G, and SG1G are used. The conceptualization of these three intergenerational households using these architectural principles enabled us to distill and refine a set of demographic factors to help explain breakups among them.

Background
The amount of time that grandchildren live with grandparents varies greatly and depends upon whether households are three-or skipped-generation households. Several studies find that threegeneration households are often transitory as young parents with infants wish to move out of their parents' households to form their households (Oberlander et al., 2009;Pilkauskas, 2012). In one study, 27% of teenage mothers in three-generation households had moved out during their first 24 months postpartum (Oberlander et al., 2009). Among the remaining mothers in the study, over 80% of them wanted to leave their parents' households for other types of living arrangements (Oberlander et al., 2009).
Findings from Oberlander et al.'s (2009) study are consistent with a broader literature suggesting that the longevity of threegeneration households is short. Pilkauskas and Martinson (2014) found that among grandchildren living in three-generation households, only one-fifth remained in those households by their sixth birthday. In another study using the Fragile Families and Child Wellbeing Study, though about 43% of mothers lived in three-generation households for at least one of five consecutive waves, only 1.8% of them lived in a three-generation household for all five waves (Pilkauskas, 2012). Lastly, the Health and Retirement Survey used by Blustein et al. (2004) indicated that only about 5% of grandparents heading three-generation households had coresident grandchildren for all 8 years surveyed.
Most studies of three-generation households have found that teenage childbearing, race, and family structure are key predictors of the longevity of three-generation households. Pilkauskas (2012) found that single mothers with children were more likely to stay in three-generation households than married mothers. The significance of family structure and race to three-generation household dissolutions was highlighted by Dunifon and Kowaleski-Jones (2007). These scholars found that Black grandchildren spent almost two years on average in three-generation households whereas White grandchildren spent less than half a year. ARTICLE HUMANITIES AND SOCIAL SCIENCES COMMUNICATIONS | https://doi.org/10.1057/s41599-022-01107-6 Not all grandparent-maintained households are threegeneration households, however. About a third are skippedgeneration households (Ellis and Simmons, 2014), meaning both parents of grandchildren are absent from households. Skippedgeneration households have the potential to break up also, especially since these households are the most economically fragile of all grandparent-maintained households (Baker and Mutchler, 2010;Dunifon et al., 2014;Goodman and Silverstein, 2002). Though theory would suggest that more economic adversity should put skipped-generation households at higher odds of dissolution, to the contrary skipped-generation households remain intact longer and are less likely to break up than three-generation households. Musil et al.'s study (2011) found that about 75% of grandmothers in skipped-generation households who raised grandchildren had cared for them continuously for at least 2 years. Another study found that grandparents in skipped-generation households cared for grandchildren for 4 years on average, with 19% of those grandparents having cared for them for 6-10 years (Landy-Meyer and Newman, 2004). In a 2009 study, Dunifon et al. (2014) reported that grandchildren lived in skipped-generation households for about 10 years. Hence, evidence suggests, though comparisons are scarce, that despite greater economic hardship skipped-generation households last longer than three-generation households.
Overall, the literature suggests that the "lifespans" of skippedor three-generation households are related to their organizational design. Yet, notably for three-generation households, it is still unclear if their "lifespans" depend on which generation is the focal one. In other words, does the longevity of a three-generation household depend on whether the supporting focal generation is grandparents or the adult children of grandparents? Smaller studies offer some insights into whether breakups of threegeneration households are related to the focal generation that maintains households, (Beck and Beck, 1989;Musil et al., 2011;Oberlander et al., 2009), but a broader study that compares the odds of dissolutions based upon two alternative focal generations that could head households is absent. Our approach permits juxtaposing results for two alternative focal generations within three-generation households and then comparing those results with those for skipped-generation households in which the focal generation is unambiguous. Thus, this study's wider scope, encompassing skipped-and two forms of three-generation households, permits analyzing the role of the focal generation heading the household on the dissolution of intergenerational households while also exploring the impact of other demographic factors. Our study expands the literature on grandchildren's living arrangements since we regard their time in these arrangements as partly determined by a generational hierarchy and focal generation, i.e., adults responsible for the daily workings and maintenance of the household.

Theoretical perspectives
The process leading to the dissolution of intergenerational households is undoubtedly complex. Theoretically, a compound set of biological, psychological, familial, and ecological factors could interact or align to change a household from a skipped-or three-generation one to an alternative form. Though these sorts of theoretical possibilities and complexities are beyond the scope of this study, we nevertheless posit that amid the constellation of elements a subset of socio-demographic and economic factors can prolong or shorten the longevity of skipped-and three-generation households. Yet, understanding whether the generational hierarchy of an intergenerational household and whether the focal generation heading a household is associated with the odds of intergeneration household dissolutions presently remains unclear.
We conjecture that the odds of dissolution among intergenerational households partly depend upon which focal generation, within the generational hierarchy, heads a household. Since grandchildren are dependents in our study, the head of the household is either a second-generation individual, i.e., an adult child, or a first-generation individual, i.e., a grandparent. 4 As to which of these two generations will have higher odds of dissolutions: we hypothesized that it will be three-generation households headed by grandparents, i.e., the first generation rather than three-generation households headed by parents, the second generation. This hypothesis is based on the literature documenting that in three-generation households, adult parents and their children are highly likely to stay for only a short time in their own parents' households (Pilkauskas and Martinson, 2014).
Conversely, again drawing on the literature we can infer that skipped-generation households, headed by grandparents, will outlast three-generation households headed by either grandparent or by adult parents (Musil et al., 2011;Landy-Meyer and Newman, 2004;Dunifon et al., 2014;Beck and Beck, 1989;Blustein et al., 2004). The absence of a second-generation, i.e., parents of grandchildren, means the responsibility for raising grandchildren rests solely with grandparents who, by definition, also head these households. In this type of household, the generational hierarchy and focal generation have merged. This rarer structure, and the circumstances that led to both parents' absences in the first instance, (e.g., drug addiction, incarceration, mental illness), compelled grandparents to assume the parenting role, thereby reinforcing the permanency of these households. And, if the only alternative to their care was foster care, this alternative could also give grandparents an incentive to maintain the intergenerational nature of the household.
Finally, we argue that if the focal generation in a threegeneration household is the second generation, the odds of dissolution for these households should fall between odds for skipped-generation households and three-generation households headed by the third generation. Parents raising children while also sharing their households with their own parents are often referred to as the "sandwich generation" (Schwartz, 1979). We believe that households headed by this second "sandwich" generation are less likely to dissolve than three-generation, grandparent-headed households because adult parents raise their children while caring for their parents in their homes. And we suppose that odds of dissolution among three-generation households headed by the second generation are higher than those of skipped-generation households because the latter has a much larger imperative, as discussed above, to stay intact.
Apart from establishing the focal generation heading a household within the generational hierarchy, we hypothesize that certain demographic characteristics of the focal generation will also lower the odds of dissolution. Increasing age of household heads should lower the odds of dissolutions. Headship age acts as an indirect measure of the ongoing intergenerational relationships among household members-the older the household head, the more likely that relationships among household members will remain unchanged. As well, studies indicate that minority intergenerational households last longer (Beck and Beck, 1989;Dunifon and Kowaleski-Jones, 2007;Luo et al., 2012). We believe that our study will confirm that households headed by Non-Hispanic Blacks will have lower odds of dissolution (Stack, 1975).
Several other household head characteristics of the focal generation should increase the odds of dissolutions. Stresses experienced by unmarried heads of households and a higher likelihood of poverty should lead to households with unmarried household heads having higher odds of dissolution. Also, if higher levels of education are proxies for heads of households having greater command over resources through earnings and accumulated wealth, then odds of dissolution should rise among households headed by those who have higher levels of education. These household heads, regardless of whether they are a second or third generation, should have greater personal and material resources to help other family members establish themselves elsewhere. Finally, given the demands of caring for family members and managing a household, we predict that work-preventing disabilities among household heads curtail economic resources sufficiently to increase the odds of household dissolution.
Two other hypotheses relate to the economic resources available to households. Those resources usually derive from the household itself in the form of income or income from the government in the form of public assistance transfers, either cash or in-kind. The first hypothesis is that the level of household income should affect the odds of intergenerational breakups. Most households, including intergenerational ones, need income to offset the costs of necessities and raising children. Higher levels of household income should decrease the pressure these costs entail, thus lowering the odds of breakups. Second, not just household income but in-kind assistance received from public welfare programs reduce a household's costs. We hypothesize that participation in the Supplemental Nutritional Assistance Program, SNAP, a program aiming to alleviate food insecurity, should decrease the odds of intergenerational household dissolution since SNAP offsets the costs of providing sufficient food for a household, a key expenditure.
Our two economic conjectures reflect potential adaptions intergenerational households can make (Conger and Elder, 1994;Yeung and Hofferth, 1998;Swanson et al., 2008;Brandon 2000;Craine et al., 1992) as well as mirroring family resilience process as conceptualized by the Family Adjustment and Adaption Response (FAAR) model (McCubbin and Patterson, 1983). For example, the odds of dissolutions among intergenerational families are predicted to be lower if a household head-the focal generation-adapts to economic strains by participating in the SNAP as that participation lowers food costs. Or a household head demonstrates resilience to an unexpected loss of household income by working more or returning to the workforce. In both examples, the generational hierarchy has better odds of remaining intact if the focal generation acts. Of course, other household members can adapt as well to preserve the household's structure, but we argue that at some point a household head must at least coordinate the resources available to ensure intergenerational structure continues.
We recognize that our emphasis on generational dimensions of complex households operates within broader systems: family customs, traditions and history, cultural contexts, macroeconomic conditions, neighborhood, and sociohistorical circumstances. However, we cannot incorporate these valid layered systems because we lack indicators of family distinctiveness and ecological context that could affect the odds of household breakups (Bronfenbrenner, 1979;Mikiyasu and MaloneBeach, 2013;Dym, 2017). Even so, we include three measures of ecological context that can alter odds of breakups: (a) the state in which a household is located; (b) state unemployment rates; (c) and seasonable changes, e.g., summers when school vacations happen.
Our goal was to assess whether the conceptual idea of a generational hierarchy and focal generation borrowed from architectural principles would help us better understand the longevity of intergenerational households. Those principles, though unconventional, when put to the test using a longitudinal source of household panel data were indeed found to be helpful.
Data description and statistical approach Data description. We used data from the Survey of Income and Program Participation, the SIPP, a longitudinal survey of a random sample of the US population. The SIPP panels used were the 2004, 2008, and 2014 panels 5 . The SIPP provides monthly data on family structure, household headship and composition, labor market behavior, and income sources (U.S. Bureau of the Census, 2001, 2019.
The SIPP was well-suited for this study for three reasons. First, the survey could precisely identify and categorize households by generational hierarchy and by focal generation because of the rich information gathered on household composition. Census Bureau field representatives enumerated every person in a household, determined relationships within that household, and asked how each household resident was related to the head of the household, defined as the person who owns or rents the dwelling and permanently lives at the sampled address (U.S. Bureau of the Census, 2001, 2019. Having identified the household head enabled us to confirm whether that person was a grandparent who lived with grandchildren, with or without the presence of the grandchildren's parents, or whether that person was a parent who lived with dependent children and their parents. From this elaborate household data, we constructed a monthly generational hierarchy and established within that hierarchy the focal generation to which the head of the household belonged. A second reason why the SIPP is appropriate is that its longitudinal design can show changes in living arrangements. Having monthly household composition measures can improve the estimation of the timing and likelihoods of three-and skipped-generation households breaking apart. Equipped with a monthly matrix of generational relationships, we could track household exits by month for an adult, including parents, grandparents, and grandchildren. Thirdly, there are sufficiently large samples of three-and skipped-generation households to conduct analyses by race which were precluded in smaller past studies (Landry-Meyer and Newman, 2004;Oberlander et al., 2009;Ruiz, 2008). The SIPP panels yielded a sample of 5831 Non-Hispanic White, Non-Hispanic Black, and Hispanic grandchildren who at the start of the panels were 12 years of age or younger and living with grandparents. Most grandchildren, 61.5% (N = 3587), lived in 3G1G households; 19.8% (N = 1152), lived in 3G2G households, and 18.7% (N = 1092), lived in SG1G households.
Our sample of grandchildren would have increased if we had included grandchildren older than 12 years of age. But we did not want our analyses confounded by teenage grandchildren who might leave a household for reasons of their own. Nevertheless, we had a large sample of preadolescent grandchildren from which to make comparisons across three types of intergenerational households. That panel design generated 116,796 "grandchildmonths" from the sample of 5831 preadolescent grandchildren.
Though the SIPP has major assets, it had drawbacks. The timevarying monthly data in the three panels are limited to 48 or 60 months. This shorter time frame prevents studying dissolutions among intergenerational households occurring over even longer periods. Also, the grandchildren lived in three-and skipped-generation households when first observed, but the survey does not reveal if this coresidence has been a short-or long-term arrangement.
Statistical approach. The dissolution of a skipped-or threegeneration household in any given month is a rare event, (about 1.5% per grandchild-month), but one with potentially large implications for grandchildren, parents, and grandparents alike. To model this rare event, we implement a logistic regression method, which is specifically designed to address the "rare events" problem Zeng, 2001a, 2001b;Allison, 2012;Firth, 1993). Often researchers want to model rare events, but when there are few occurrences of those rare events, the estimates generated will be biased. Biased parameter estimates generated from logistic regression models can be corrected, however, as Firth (1993), King and Zeng (2001a, 2001b, 2001c, and Allison (2012) have demonstrated. In advanced logistic regression models, like the one implemented here, biased parameter estimates due to the rare events problem, (small-sample bias), are corrected. For more on logistic modeling of rare events see Heinze and Schemper (2002), Heinze and Puhr (2010), Gao and Shen (2007), Lee et al. (2006), Maalouf and Trafalis (2011), Qiu et al. (2013).
Besides estimating the effects of hierarchical structures on intergenerational household dissolution, we estimated the effects of other key socio-demographic and economic factors on the odds of dissolution, as well. Measures for grandchildren include age and gender while measures of a household head include age, gender, race, disability, nativity, marital status, labor market attachment, and educational attainment. Measures of households and context include household income, SNAP participation, household poverty level, living in a metropolitan area, state of residence, and state unemployment rates. Table 1's descriptive statistics for the three types of intergenerational households show notable differences. Bolstering past results on the economic insecurity that skipped-generation households face (Baker and Mutchler, 2010;Dunifon et al., 2014;Mutchler and Baker, 2004), we find that SG1G households, (45%), compared to 3G1G and 3G2G households, 33% and 30%, respectively, are more likely to be poor, receive more welfare income, have less household income, and less likely to own a home. More SG1G and 3G1G households, 39% and 40%, respectively, receive SNAP benefits to combat food insecurity compared to 3G2G households, 32%. Like other studies (Brandon, 2005;Fuller-Thomson and Minkler, 2001;Fuller-Thomson et al., 1997), we find that grandparents heading skipped-generation households are the oldest (57.4 years), more likely disabled and unmarried, and out of the labor force (55%). Even among the grandparents heading skipped-generation households who still work, they work fewer hours (13.1), compared with heads of 3G1G (19.1), and 3G2G households (26.2).

Findings
Reflecting recent immigration trends to the United States (Trevelyan et al., 2016), these older heads of SG1G households are less likely to be foreign-born (10%), especially compared to 3G2G households (32%). Curiously, 3G2G and 3G1G households receive similar levels of welfare income, but apart from that similarity, demographic differences between 3G2G and 3G1G households, e.g., SNAP receipt, education levels, and labor force participation, are apparent. When the focal generation heading households are grandparents, those households are more likely in the South and non-metropolitan areas compared with households headed by parents as well (Simmons and Dye, 2003).
Turning to differences among preadolescent grandchildren, more of them living in SG1G households are non-Hispanic Black and non-Hispanic White, but the opposite is found for Hispanic grandchildren. Most Hispanic grandchildren live in 3G2G households, again likely reflecting immigration trends. Also, grandchildren in SG1G households are older than their counterparts in 3G1G and 3G2G households.
Further, Table 2's results supported our conjectures that the odds of intergenerational households breaking apart over the course of the survey would differ. Event history, (survival), analyses indicated that the highest incidence rate of intergenerational household breakups over the survey period occurred among 3G1G households. The latter households' incidence rate compared to the incidence rate for 3G2G households suggested that grandparents left the households of their adult children-the focal generation-at a slower rate than that at which grandchildren and their parents left the households headed by grandparents-the other focal generation. By contrast to 3G1G and 3G2G households, SG1G households had the lowest incidence rate of breakups. Comparisons of all three incidence rates credibly implied that, though breakups were rare, differences in the odds of intergenerational households breaking apart were associated with the specific focal generation that headed the household.
Further underscoring the differences, the average number of months until a breakup, (18.6), was less for 3G1G households than for 3G2G and SG1G households, 20.4 and 20.6 months, respectfully. The relatively quicker turnover of 3G1G households, which is consistent with the literature and Table 2's incidence rates (Oberlander et al., 2009;Pilkauskas, 2012), may suggest that greater economic strains lead to fundamental changes in the composition of these households; alternatively, breakups of 3G1G households might reflect adult parents with children having stronger preferences to move on from their own parents' households. Tables 1 and 2 show differences among intergenerational households by generational hierarchy and focal generation. Those tables' results augur well for using the rare events logistic regression model described earlier to estimate the effects of generational hierarchy and focal generation on the odds of dissolutions. Table 3 presents results from the regression model. Table 3's results confirmed our hypothesis that variation in odds of dissolutions among intergenerational households is tied to generational hierarchy. Controlling for socioeconomic and demographic measures, the estimated coefficient for 3G1G households suggests that the odds of breakups are higher than for 3G2G households, our comparison group. By contrast, the estimated odds of dissolution for SG1G households indicate that breakups of SG1G households are less likely compared to 3G2G households. Opposite signs for these estimates of dissolution for 3G1G and SG1G households and further statistical tests 6 verified these households differed from each other, not only from 3G2G households. In the "Discussion and conclusions" section, we tie these significant findings to our hypotheses.
Other findings support our conjectures that characteristics of the focal generation are related to odds of household dissolutions. The model confirmed that the odds of dissolution are lower among intergenerational households headed by Non-Hispanic Blacks and Hispanics compared to households headed by Non-Hispanic Whites. And, a one-year increase in the household head's age was associated with a small but significant decrease in the odds of a household splitting apart. Education levels influenced the odds of dissolutions, as well. Compared to household heads without a high school diploma, those heading intergenerational households with a high school diploma had a lower likelihood of dissolving. By contrast, though statistically insignificant, the estimated coefficients for households headed by a generation with some college or a college degree suggested increased the odds of dissolutions. Lastly, estimates for our other two demographic measures on focal generation household heads, unmarried marital status, and having a work-preventing disability, failed to support our hypotheses that lacking a partner or an inability to work would increase the odds of dissolutions.
The SIPP gathered limited data on children, but the regression confirmed that the odds of intergenerational household dissolutions were not associated with grandchildren's gender. We expected that elementary-school-aged and high-school-aged grandchildren would have lower odds of dissolution than preschool-aged grandchildren. Household dissolution for the former two groups of school-aged children could mean changing schools which is more disruptive for them as compared to preschool-aged children. Though the estimated coefficient was as anticipated for high-school-aged children, only for elementary school-aged children was the estimated coefficient significant.
Findings supported our hypotheses about the effects of our two household economic measures. Increases in the log of monthly household income were significantly associated with decreases in intergenerational household dissolutions. Our second measure, SNAP use, which reflects a household's reliance on an in-kind public transfer to relieve food insecurity, supported our conjecture that public aid keeps needy intergenerational households intact, in other words, SNAP participation reduced the odds of breakups substantially.
Lastly, our measures of ecological context were associated with the odds of dissolution among intergenerational households. As predicted, intergenerational dissolutions varied seasonally: in the summer months, the odds were higher while in the winter months the odds were lower than the odds in the spring and fall 3G1G three-generation household focal generation heading household = grandparent, 3G2G three-generation household focal generation heading household = parent, SG1G skipped-generation household focal generation heading household = grandparent, with both parents of grandchild absent from household; summary statistics from survival analyses. a Measured in months. b Incidence rates = dissolutions per grandchild-month and 95% confidence intervals for incidence rates indicated each rate was significantly different from each other. Sample is grandchild-person-months generated from 3363 grandchildren 12 years of age or younger at start of SIPP panels; model controls for month in panel and region. 3G2G comparison group, three-generation household, household head is parent, SG1G skippedgeneration household with both parents of grandchild absent, grandparent is head of household; 3G1G three-generation household, head of household is the grandparent with at least one adult child parent present. *p < 0.10, **p < 0.05, ***p < 0.01. a 2001 dollars.
months. Several factors may account for seasonal variation in intergenerational coresidence. Perhaps during summer, grandchildren who are out of school lose social services provided by schools, e.g., school lunches and after-school care. Or summer may make searching for alternative housing and moving easier. Also, in states with higher rates of unemployment, which imply fewer job possibilities to support a household, intergenerational households are at greater likelihood of dissolution.
Predicted odds of breakups. The logistic regression model yielded monthly predicted odds of dissolution. A graph of these predicted odds confirmed that our model was appropriate since dissolutions were rare events. We calculated average predicted odds over all months concatenated by generational hierarchy, focal generational headship, and race. 7 The averaged predicted odds provide insights into which intergenerational households faced the greater or lesser odds of dissolutions. Figure 1 suggests that Non-Hispanic White 3G2G and 3G1G households possess the greatest odds of dissolutions (2.2% and 2.3%, respectively), 8 followed by Non-Hispanic Black 3G2G households (1.83%). In contrast, Non-Hispanic Black SG1G households have the least odds of dissolution, (0.8%), followed by Non-Hispanic White SG1G households (1.3%). Non-Hispanic Black households, regardless of focal generational headship, have lower odds of breaking apart compared to Non-Hispanic White households. The odds of dissolution were much lower among skipped-generation households compared to three-generation households, which was consistent with Table 3's generational hierarchy estimates and the extant literature. 9

Discussion and conclusions
This study was motivated by research suggesting that the duration of an intergenerational household might be surprisingly short (Oberlander et al., 2009;Pilkauskas, 2012;Pilkauskas and Martinson, 2014). But which type of intergenerational household has the shortest duration relative to other types, and if that briefer period is associated with a household's generational and hierarchical features have remained open questions. Yet, answers to these questions are needed because intergenerational household dissolutions can have adverse impacts on members of these households, e.g., grandchildren's later outcomes in adulthood (Aquilino, 1996;Monserud and Elder, 2011;Pilkauskas and Dunifon, 2016).
Answers to these questions are elusive, however. For a start, selecting among the many demographic characteristics within intergenerational households that have an association with breakups is challenging to identify and couch within a structural framework. Difficulties arise when deciding which generation plays a bigger role in the continuance or end of the household organization; those analytical challenges persist when adjudicating among the numerous demographic measures that could hasten or hinder dissolutions.
Our adoption of two architectural principles enabled us to conceptualize intergenerational households as ordered, hierarchical structures. That conceptualization aided us in addressing questions about the odds of dissolutions across different sorts of intergenerational households. Our approach was unconventional, but it had leverage. The ideas of "generational hierarchy" and "focal generation" provided a scaffolding from which to build concatenations of focal generation and generational hierarchy and then apply measures of those concatenations to predicting household dissolutions.
The appeal of these architectural principles was clarifying which intergenerational household-distinguished by its generational hierarchy intertwined with its focal generation-was associated with longevity. The study verified ( Table 3) that skipped-generation households, though the most disadvantaged (Table 1), had greater durability than three-generational households. Supporting past studies that documented young parents' early departures from parents' households (Oberlander et al., 2009), this study showed that compared to three-generation households headed by the second generation, three-generation households headed by the first generation broke apart faster.
Findings reveal that what matters to determining the continuance of an intergenerational household is not only having classified it as a skipped-or three-generation household, but pinpointing which generation has responsibility for the household, or in our architectural parlance, identifying the focal point (or "focal generation").
We can also conclude that the continuance of these households is further mediated by race. Table 3's estimates for "Non-Hispanic Black" and the predicted odds of household dissolutions suggested what has been noted before: Black skipped-generation households last longer than other types of intergenerational households, (Luo et al., 2012). Black intergenerational households (Fig. 1), outlasted white intergenerational households. Overall, differences in breakups between whites and Non-Hispanic Blacks intergenerational households are pronounced, probably reflecting differences in kinship patterns (see Stack, 1975). Though there are some stark differences in dissolution odds across generational hierarchies and focal generations, ( Fig. 1 and Table 3), nevertheless, odds of breakups are small. Households stayed intact over 4-and 6-year time frames rather than moving towards breaking apart.
We conclude that the regression model tapped into the synergy of a generational hierarchy and a focal generation working in tandem to adapt, preserve, and promote household stability. Without reviewing each time-varying predictor in Table 3, two household predictors, "Log of monthly household income", and "Monthly household receipt of SNAP", and some focal generation predictors, "Focal generation's age", "Non-Hispanic Black", "High school only" support the argument that these interconnected generational features are associated with household longevity or dissolution. Not every predictor for a focal generation supported our theory. Coefficients for "Work-preventing disability" and "Focal generation head unmarried" were statistically nonsignificant. Furthermore, we recognize that an intergenerational household break up could result from forces like (a) vagaries in the economic cycle or seasonal changes as shown (see Table 3); (2) an unanticipated change in household or focal generation characteristics, e.g., loss of income, loss of public assistance benefits, the onset of a disability or health condition, divorce; or (3) unobserved inclinations of a non-focal generation to leave a focal generation's household.
Despite the study's contributions to understanding dissolutions among intergenerational households, it has shortcomings. We cannot incorporate valid layered ecological systems and contexts that could affect dissolutions. Besides ecological factors, conflicts among generations or other stressors (Goodman, 2007;Musil et al., 2011), either new or longstanding, could result in dissolutions. In an ideal study, we would model when an intergenerational household began to preclude the influence of left censoring on our model estimates (see Miller and Halpern, 1982). Many studies of various demographic transitions confront this common methodological concern (Ferraro et al., 2016;Manning et al., 2016).
Notwithstanding caveats, the study's approach adds knowledge, with some of that knowledge offering useful insights for policymakers. The study helps policymakers appreciate that the lifespans of intergenerational households differ, and one set of income support policies may not fit equally across all intergenerational households. Our findings suggest that at least 3G1G and SG1G households can benefit from public assistance programs, like SNAP, but perhaps for different purposes. Since 3G1G households appear resilient in the face of economic adversity, public assistance support would be most effective at supplementing grandparent investments in grandchildren, not necessarily to ensure that households will remain intact. Alternatively, 3G1G households might benefit from public aid, but it is unclear whether income support should be aimed at keeping these households whole or used to support younger parents' maintenance of their households once they have moved out of their parents' households.
Maybe cash income support policies are not the correct policy instruments in the first place? For 3G1G, policies protecting the rights of custodial grandparents might matter more than policies subsidizing household stability (Letiecq et al., 2008;Meara, 2014). While for 3G2G households, (our comparison group), policies that encourage access to in-home nursing care (Markle-Reid et al., 2006) might have the most benefit.
In summary, the architectural principles we adopted facilitated a deeper understanding of dissolutions among intergenerational households. The approach provided novel analytical tools to compare across different intergenerational households. In future work, we will search for a source of data permitting us to continue this line of inquiry, incorporate new contextual factors, deal with noted methodological concerns, and develop policy recommendations. Overall, this study enhanced our understanding of how the different intergenerational configurations of households lead some of them to endure and others to end.

Data availability
The dataset analyzed during the current study is available in the Dataverse repository: https://doi.org/10.7910/DVN/UKMC3N. The dataset was derived from the following public domain resources: https://www.census.gov/programs-surveys/sipp/data/ datasets.2004.html; https://www.census.gov/programs-surveys/ sipp/data/datasets.2008.html; https://www.census.gov/programssurveys/sipp/data/datasets.2014.html. Notes 1 The relationship across the generations could be by blood or by adoption. 2 The Oxford English Dictionary defines architecture as, "The complex or carefully designed structure of something" (Architecture, n.d.). Further discussions introducing the discipline and practice of architecture are found in Ching (2014), Ching andEckler (2012), andArnheim (2009). In architecture, which many contend is either a science, or an art, or both (Moore, 1965;Hillier and Leaman, 1976), there are two principles which focus on notions of hierarchy and axis or focal point. For more on foundational principles, see GharPedia (n.d.), GharPedia (n.d.), and Learn Design Principles (n.d.). We argue, metaphorically, that these two principles are useful tools for conceptualizing complex social structures, such as, intergenerational households. 3 Other types of intergenerational households exist, but our focus is the main types of such households, which are three-and skipped-generation households (Kreider and Ellis, 2011;Pilkauskas and Cross, 2018). 4 Clearly at some earlier time one of the two generations moved into the other generation's household to create an intergenerational household. Perhaps when the change in household composition occurred that headship status might also have changed between the two generations, but we believe this change is uncommon. 5 The calendar periods covered by the three panels are Available from request from authors. 7 We present averaged odds because differences among monthly odds ratios by household hierarchical generation and racial categories are uniform over durations of the panels. The uniformities of differences in monthly odds ratios by household hierarchical generation and racial categories over the months makes sense since the SIPP panels are limited to either 48 (2004, 2014), or 60 (2008) months, respectively. 8 We expected a priori that the odds for 3G1G households would have been the highest, but we think that the monthly estimates probably slighted attenuated the weighted averages. 9 95% confidence intervals for each predicted odds of breaking up for each intergenerational household showed that predictions were significantly different from each other.