Family rituals in postdivorce families: The role of family structure and relationship quality for parents’ and stepparents’ attendance at children’s birthdays

Objective: To investigate if divorced parents celebrate their children’s birthdays with their respective ex-partner and current partner, and whether they do so “jointly” with both. Background: Family rituals like birthday celebrations are important and meaningful events in people’s lives, but little is known about who partakes in these in contemporary postdivorce families. Method: We assessed whether divorced parents celebrated their child’s birthday together with their ex-partner (i.e., the child’s other biological parent), current partner (i.e., the child’s stepparent), and jointly with both. Dutch Data (N=2,451) was analyzed using linear probability models. Results: Most parents celebrated the child’s birthday without the ex-partner, but with the current partner. One quarter celebrated with both. The ex-partners’ presence was more likely when parents’ and their current partners’ relationship with the ex-partner was good; and less likely when parents had repartnered and when the ex-partners had sole custody or additional biological or stepchildren. The presence of the current partner was more likely in case of coresidence with the biological parent and when the ex-partner had a new partner; and less likely when the ex-partners had sole custody and when parents’ relationship with the ex-partners was good. Conclusion: Child-related family rituals mostly involve the “new” stepparent rather than both biological parents. The effects of relationship quality, co-residence, repartnering, and having additional biological or stepchildren highlight the importance of (step)parents’ willingness to interact with each other, structural opportunities for parent-child interactions, and parents’ shifting loyalties from their ex-partner to their new family.

It is presumed that all families practice family rituals (Fiese et al., 1993), but who partakes in them differs between family types (Bakker et al., 2015;Berg-Cross et al., 1993;Costa, 2014). In Western first-married families with biological children family rituals usually involve at least the child and both biological parents (Braithwaite et al., 1998). Who partakes in family rituals in postdivorce families is more ambiguous (Braithwaite et al., 1998;Costa, 2014). Little societal norms exist promulgating with whom family rituals are to be celebrated (Cherlin, 1978), leading to potentially great variation within the group of divorced parents.
These predominantly qualitative studies outline with whom parents may practice family rituals (e.g., with the ex-partner) and usually focus on parents' and children's perceptions about practicing family rituals in different configurations. Whereas some reasons for why parents practice family rituals with, e.g., the ex-partner, have been stated, the mechanisms proposed have not been explicitly tested. A better understanding of what determines who partakes in family rituals after divorce would benefit our overall understanding of family interactions after divorce. Additionally, contemporary postdivorce families may not only include multiple parental figures after parents repartner (i.e., biological and stepparents), but also multiple types of parental figures -such as cohabiting or living-apart-together (LAT) stepparents (Raley & Sweeney, 2020). Moreover, more children grow up in shared residence arrangements (i.e., joint physical custody) instead of mother-residence (Poortman & van Gaalen, 2017). Existing studies offer little leads about how these more recent and emergent types of postdivorce families practice family rituals.
In this large-scale, quantitative, study we investigate with whom divorced parents practice family rituals, focusing on the child's parental figures. We focus on children's birthdays as these are special, meaningful, family rituals (De Carlo & Widmer, 2011, p. 230), and norms and children's expectations about celebrating them with both biological parents may be stronger and less negotiable than for other family rituals (Costa, 2014). We consider whether divorced parents (i.e., the so-called focal parents) celebrated their (focal) child's birthday (1) with their ex-partner, (2) their current partner, and (3) whether they celebrate jointly with both. Investigating who attends children's birthdays and what factors shape who attends birthdays matters, first, because of the potential ramifications for children and the parents involved. First, children usually want both of their biological parents present at their birthdays and other important family rituals (Zartler, 2014), so the absence of parent's ex-partner might impact children's well-being. Divorced parents might wish their current partners to be present, for example, to promote their role as a stepparent and to show to them that they count as family members. If they are absent from the child's birthday, this could have implications for the strength of family relationships in the new stepfamily. A joint celebration with both the ex-partner and current partner might imply greater availability of social capital to parents and children (Widmer, 2006) and greater child well-being (King, 2006;Spagnola & Fiese, 2007), but potentially also loyalty conflicts (see Fiese et al., 2002). Second, a closer understanding of the forces that shape who celebrates family rituals together after divorce adds to existing theoretical explanations of the forces that shape postdivorce family interactions and how family is done after divorce (Fiese et al., 2002;Morgan, 2011a).
To show how family rituals in postdivorce families reify the old or new family and when that is the case, we focus on the family structural and qualitative determinants of which parental figures celebrate children's birthdays together in postdivorce families. First, we consider both the child's biological parents' new family context (i.e., their (type of) repartnering and having (step)children with the new partner). Having a new partner may lead parents to practice family rituals without the ex-partner (Braithwaite et al., 1998). Second, we focus on the relationship quality between the different parent-parent dyads, which affects biological parents' and current partners' willingness to celebrate the child's birthday with each other (Costa, 2014). Lastly, we consider the child's residence arrangement, which regulates parents' opportunities for access and bonding with the child (Bakker et al., 2015). In our analyses, we, furthermore, control for other factors that might influence who attends the child's birthday, such as the geographical distance between both biological parents and the age of the child. We, where possible, take the perspectives of all relevant actors into account (i.e., the focal parent, child, ex-partner, and current partner). This is in keeping with extant studies on family rituals, which point to the intricate processes that determine who attends family rituals and why (e.g., Bakker et al., 2015;Braithwaite et al., 1998;Costa, 2014). While the actual reasons are impossible to test using survey data, taking all relevant perspectives into account enriches and nuances our theoretical understanding of the forces that shape who attends children's birthdays, particularly in more recent types of postdivorce families not considered in previous studies (e.g., those with LAT stepparents) and provides a more global assessment of the overall effect of the different actors' respective actions and considerations.
We used the second wave of the survey New Families in the Netherlands (NFN) -a recent and especially suitable dataset for this study (Poortman & van Gaalen, 2019). NFN includes rich and unique data about a large sample of diverse, more recent, and emerging postdivorce families in the Netherlands -for example, those with shared residence arrangements and LAT stepparents. Compared to other countries, shared residence arrangements are common in the Netherlands, with about 20% of parents following such arrangements, though sole (mother) residence remains the most common arrangement (about two thirds; Poortman & van Gaalen, 2017). This makes the Netherlands an excellent setting for investigating the effect of this particular residence arrangement on the attendance at family rituals. NFN is also, to our knowledge, one of the few surveys that include information on the attendance of a child's birthday (but see e.g., Dykstra et al., 2005). By using this dataset, we were able to consider how a broad range of contemporary postdivorce families celebrates children's birthdays.

Theoretical background
Below, we outline our theoretical expectations regarding the effects of family structure (repartnering, having biological children and stepchildren, and postdivorce residence arrangements) and relationship quality on birthday attendance. Though postdivorce family structures, e.g. in terms of children's residence arrangements or parental repartnering, typically differ between men and women (e.g., Bakker & Mulder, 2013;Vanassche et al., 2015), we do not expect gender differences in birthday attendance as previous studies on family rituals in the broader sense do not mention gender differences as to who partakes in them (Bakker et al., 2015;Braithwaite et al., 1998;Costa, 2014). We also a priori do not expect clear differences in the effects of family structure and relationship quality on birthday attendance for men and women, i.e., interactions between these factors and parents' gender. The limited and indirect evidence (pertaining to parental involvement instead of family rituals) regarding interactions with gender is mixed: for example, some studies found different effects of repartnering for fathers and mothers (e.g., Meggiolaro & Ongaro, 2015), whereas others found the opposite or no conclusive gender differences (e.g., Koster et al., 2021). We thus discuss the impact of family structure and relationship quality on birthday attendance irrespective of gender.

Repartnering, having (step)children, and birthday attendance
Repartnering and having shared biological or stepchildren (in short: (step)children) with the new partner may indicate that parents have established a "new" family. Consequently, they might be less involved with their "old" family, i.e., the ex-partner (Smart & Neale, 1999, p. 72;Whiteside, 1989), and prefer practicing family rituals with the current partner (Bakker et al., 2015;Braithwaite et al., 1998;Costa, 2014;Whiteside, 1989). Practicing family rituals without the ex-partner, and with the current partner, might also serve to spend time with and show loyalty to the current partner, or be a result of the current partner opposing involvement with the ex-partner, e.g., out of jealousy (Ganong & Coleman, 2017, p. 11). We, therefore, theorize that the repartnering of either the focal parent or the ex-partner reduces the probability of the ex-partner, and increases the probability of the current partner being present at the child's birthday. We, furthermore, presume that joint presence is reduced by the repartnering of the ex-partner, as then both the focal parent and the ex-partner have a new family and may seek to integrate the child into their own -separate -new families by practicing family rituals separately. These effects may be stronger the more committed parents are to their new union, with marriage potentially embodying relatively more (formalized) commitment than cohabitation or LAT (Brines & Joyner, 1999). Note that, in the following, we use "formalization" as a shorthand to indicate the stronger interpersonal and legal commitment of marriage versus cohabitation and LAT, respectively.
Similarly, we argue that either the focal parent or ex-partner having (step)children with a new partner may imply even more commitment to their respective new family than repartnering alone. Furthermore, the focal child having half-or stepsiblings might cause the child or the focal parent to prefer celebrating the birthday with the new family members, for example, because of bonds between the focal child, its (step)siblings, and the current partner. Therefore, we expect a positive effect of having (step)children on the current partner's presence at the child's birthday, and, logically, a negative effect on joint presence. The effect of having biological children may be stronger than that of having stepchildren, as having biological children is typically a deliberate decision signifying commitment to the new partner, whereas stepchildren are frequently an involuntary "package deal" (Ganong & Coleman, 2017, p. 136). We hypothesize that: H1A: Repartnering of either biological parent reduces the probability of the ex-partner being present at the child's birthday, with this effect being stronger the more formalized the new union is. H1B: Repartnering of the ex-partner, and the extent to which the new union of either the focal parent or the ex-partner is formalized, increases the probability of the current partner being present at the child's birthday and/or reduces the probability of the expartner and current partner being jointly present at the child's birthday. H1C: Either biological parent having (step)children with their respective current partner reduces the probability of the ex-partner being present at the child's birthday, increases the probability of the current partner being present, and decreases the probability of their joint presence at the child's birthday.

Child's residence arrangements and birthday attendance
Residence arrangements define biological and stepparents' opportunities for access to the child and contact with each other. These include shared residence, where the child resides about equally with both biological parents or sole (mother/father) residence, where the child primarily resides with one of its biological parents. In the Netherlands, mother residence is the most frequent, followed by shared residence, and, lastly, father residence (Poortman & van Gaalen, 2017). Shared residence approximates the pre-divorce situation: both a child's biological parents make joint parenting decisions. Therefore, it has been argued that parents who practice shared residence might also practice family rituals with each other (Bakker et al., 2015;Smart & Neale, 1999). Additionally, low-conflict parents might select themselves into shared residence arrangements, which may increase the probability of them jointly celebrating the child's birthday (Poortman & van Gaalen, 2017). Sole-residence, conversely, is the classical approach to post-divorce parenting, where one parent (usually the mother) makes largely autonomous parenting decisions (Smart & Neale, 1999). The choice for sole-residence might, in itself, indicate that parents prefer minimal involvement with each other, and, therefore, prefer to celebrate children's birthdays without the ex-partner (Bakker et al., 2015). Sole-resident parents, furthermore, have the opportunity to celebrate the child's birthday "first", and without the nonresident ex-partner.
The probability of the current partner being present at the child's birthday is expected to be highest in case of sole residence of the child with the focal parent. First, it appears obvious that parents who live together would also celebrate their (step)child's birthday with each other. Possible reasons are that sharing a household with the child in question means that the current partner has more access to the child and more opportunities for interaction. Research on closeness with stepparents after divorce, for example, showed that coresiding with a child fosters the development of strong bonds between the child and the stepparent (King, 2006), potentially resulting in them wanting to celebrate the birthday with each other. Second, sole-resident parents might be especially keen on cementing their new family by practicing family rituals with their new partner, e.g., to solidify their role as the stepparent (Whiteside, 1989). In comparison, shared residence provides current partners with less access to the child, albeit likely more frequent and regular than if the child were residing with the ex-partner. In the latter arrangement the child, current partner, or even focal parent might not wish the current partner to be present at the child's birthday, as there might be few meaningful bonds between the child and current partner.
Consequently, joint presence of the ex-partner and current partner may be especially likely in case of shared residence vis-à-vis sole-residence, as sole-residence may lead family rituals to be practiced in the new family context (i.e., with the current and without the expartner). We hypothesize that: H2A: Compared to shared residence, sole-residence of the child with either biological parent reduces the probability of the ex-partner, and the ex-partner and current partner being jointly present at the child's birthday. H2B: Compared to shared residence, sole-residence of the child with the focal parent increases the current partner's probability of being present at the child's residence, while sole-residence with the ex-partner decreases it.

Parental relationship quality and birthday attendance
Researchers have argued that the better the relationship with someone, the greater the willingness to interact with that person (Lawler, 2001). Hence, the better the relationship between the focal parent and the ex-partner or the current partner, the more likely it is that they will, respectively, celebrate the child's birthday together (Costa, 2014;Whiteside, 1989). Low relationship quality may decrease their willingness to interact with each other and lead to biological parents celebrating the child's birthday separately (Bakker et al., 2015). Although parents might hide or put aside conflict for the sake of the child, many may avoid practicing family rituals together in such a situation to protect their own or the child's emotional well-being (Costa, 2014). These arguments imply that the better these relationships are, the more likely it is that the ex-partner and current partner are jointly present at the child's birthday. Similarly, it has been suggested the better the relationship between the ex-partner and the current partner, the more they might be willing to celebrate the child's birthday together (Costa, 2014;Johnson, 1988). Amicable relationships between them might, furthermore, lessen the child's loyalty conflicts, or the feeling of needing to "pick sides" (Hornstra et al., 2020), increasing the probability of their respective individual and joint presence at the child's birthday. We hypothesize that: H3A: The better the relationship between the focal parent and the ex-partner the more likely that the ex-partner and the ex-partner and current partner are jointly present at the birthday. H3B: The better the relationship between the focal parent and the current partner the more likely that the current partner and the ex-partner and current partner are jointly present at the child's birthday. H3C: The better the relationship between the ex-partner and the current partner the more likely that the ex-partner, the current partner, and both of them are jointly present at the child's birthday.
Additionally, per family systems theory, family relationships are interrelated (Allen & Henderson, 2017, p. 104;Jensen, 2017). Thus, the focal parent's relationship with the expartner can influence the ex-partner's as well as the current partner's presence at the child's birthday (and vice-versa). For example, a recent study on interparental relationships in diverse stepfamilies showed that when parents' relationship with their expartners is strained, they might encourage the stepparents' (i.e., the current partners') involvement with the child (Hornstra et al., 2020), by, in this case, celebrating the child's birthday with the current partner. Likewise, when the focal parent's relationship with the current partner is strained, they might prefer celebrating the birthday with the ex-partner. Thus, the better the relationship with one partner (ex-partner or current partner), the less likely it is that the other partner is present. On the other hand, good relationships with one partner might increase the probability of the other partner being present. The higher the quality of the relationship with the ex-partner, the more might the focal parent be able to convince the ex-partner to "allow" the current partner to be present, and the less might the ex-partner object or feel threatened by the presence of the current partner (and viceversa). Given these contrary expectations, we refrain from giving a hypothesis about such "cross-relationship" effects.

Data and sample
We used the second wave of the survey New Families in the Netherlands (NFN; 2015/2016); (Poortman & van Gaalen, 2019) which includes information about birthday attendance in postdivorce families not asked in wave 1. For wave 1 (2012/2013), a random sample of parents with minor children who divorced or separated in 2010 was drawn by Statistics Netherlands (Poortman et al., 2014). Both parents were approached by mail and invited to complete the survey online. The response rate for wave 1 was 39% on the individual level and 58% on the level of the former couple, yielding 4,481 responses. About 30% of the responses are from both ex-partners. These response rates are, despite the online mode and the potentially difficult-to-reach target group, comparable to similar Dutch surveys (e.g., Dykstra et al., 2005).
For Wave 2, participants of wave 1 were invited to complete a follow-up survey in 2015/2016 (Poortman et al., 2018). 63% of the original participants did so, yielding 2,544 responses (response rate on the level of the former couples 69%). An additional random sample (drawn identically as for wave 1) was approached to participate in wave 2: this "refreshment" sample had a response rate of 32% on the individual and 52% on the former couple level, yielding 920 responses. In total, wave 2 contains responses from 3,464 formerly married and formerly cohabiting parents in the Netherlands. 17% of these responses are from both former partners.
The sample is selective on several criteria. Women, the native Dutch, respondents with high incomes, formerly married parents, and older people are overrepresented. Panel attrition was selective: those with high socioeconomic status (highly educated and with paid work) were more likely and men, younger respondents, and those with lower life satisfaction were less likely to respond again in wave 2. The group of formerly cohabiting parents is somewhat more selective than that of formerly married parents, as men with (more than) two children and older children, as well as women from urbanized areas, are overrepresented.
Like in wave 1, in wave 2 respondents were asked to provide information about a "focal child". Those who completed wave 1 were asked to report on the same focal child. For the refreshment sample, and for re-approached respondents who could not recall the focal child, the focal child was determined similarly as for wave 1. In wave 1, the focal child's age was centered around the age of 10 years. In wave 2, collected about three years later, the focal child's age was centered around the age of 13 years. If all of the respondent's children were younger than 13, the oldest child was selected to be the focal child, otherwise the youngest child.
We excluded cases according to several criteria. First, we excluded cases in which the respondent did not celebrate the child's last birthday (N=481, 14%) because the question about their current partner's and ex-partner's attendance apply only to parents who had attended their child's last birthday themselves and the objective of our study was to discern with whom family rituals are celebrated, rather than if they were celebrated at all. This does not imply that in 14% of cases the child's birthday was not celebrated at all. We conducted additional analyses among a subsample limited to responses from both former partners (N=572) showing that in 74.7% of cases (N=427) both biological parents celebrated the child's birthday (but not necessarily together with each other), in 23.4% of cases (N=134) only one biological parent did so and only in 1.9% of cases (N=11) did neither biological parent celebrate their child's birthday. Second, we excluded cases where the child's residence arrangement was specified as "other" (N=220, 6.4%). Third, we excluded cases with missing values on the variables of interest (N=312, 7%). Missing values were low to moderate, ranging from 0 to 8.8% (on the variable "stepchildren of expartner", see below). In total, the "total" sample contained 2,451 responses from 2,134 former households and was used to investigate the ex-partner's presence at the child's last birthday. The presence of the current partner and the ex-partner's and current partner's joint presence can, logically, only be investigated among a sample of respondents who are currently in a relationship. We, thus, limited a second "repartnered sample" to only those respondents currently in a relationship (N=1,524 respondents from N=1,374 former households).

Measures of dependent variables
Ex-partner present (total sample). This variable captures whether the respondent celebrated the child's last birthday together with the ex-partner (0=no, 1=yes).
Current partner present (repartnered sample). Respondents who indicated currently being in a relationship indicated whether they celebrated the birthday with their current partner (0=no, 1=yes). These two dependent variables are not mutually exclusive: respondents who celebrated their child's birthday with both ex-partner and current partner are coded 'yes' on both variables.
Ex-partner and current partner present (repartnered sample). This variable captures whether both the ex-partner and current partner were present (0=no, 1=yes). The 'no' group is heterogeneous: it includes respondents who celebrated their child's birthday alone or only with either their ex-partner or current partner.

Measures of independent variables
Repartnering of the respondent. Respondents were asked whether they "currently [had] a steady partner" (0=no, 1=yes).
Type of union of the respondent. Respondents who indicated currently having a steady partner were asked about relationship status: "steady partner, but not living together or married (LAT)", "living together unmarried" or "married". For the total sample, we created three dummy variables indicating whether the respondent had a LAT partner, a cohabiting partner, or a new spouse, with "no partner" as the respective reference category. For the repartnered sample, we constructed two dummy variables indicating whether the respondent has a cohabiting partner or spouse, with "LAT partner" as the reference category.
Repartnering of the ex-partner. Respondents were asked whether their ex-partner currently had a steady partner (0=no, 1=yes).
Type of union of the ex-partner. If their ex-partners were currently in a relationship, respondents were asked whether their ex-partners had a "steady partner, but not living together or married" (i.e., LAT), was "living together unmarried" (i.e., cohabitation) or was "married". We recoded the responses into three dummy variables indicating whether the ex-partner had a LAT partner, a cohabiting partner, or a spouse, with "no partner" as the reference category.
Biological child of respondent and current partner. Repartnered respondents were asked whether they "had or adopted children with [their] current partner" (0=no, 1=yes). We assigned respondents without a current partner "0" on this variable.
Stepchildren of respondent. Repartnered respondents indicated whether their current partner had children from a previous relationship (0=no, 1=yes). We, again, assigned respondents without a current partner "0" on this variable.
Biological child of ex-partner and ex-partner's new partner. Respondents were asked if their "ex-partner and his/her new partner had or adopted children" (0=no, 1=yes). We assigned ex-partners without a current partner "0" on this variable.
Stepchildren of ex-partner. Respondents were asked whether "the new partner of [the] ex-partner has children from a previous relationship" (0=no, 1=yes). We assigned expartners without a current partner "0" on this variable.
Child's residence arrangements. Respondents were asked where the focal child mostly resided: "with me", "with my ex-partner", or "with both (approximately) equal". We coded these responses into dichotomous variables measuring whether the child resides with the respondent or with the ex-partner, with shared residence ("with both (approximately) equal") as the reference category.
Parental relationship quality. Respondents rated the quality of the respective relationship with their ex-partner, their current partner, and that between the current partner and the ex-partner on a scale from 1 (very poor) to 10 (excellent). Note that the variables capturing the relationship quality with the current partner and between the current partner and ex-partner depend on the respondent having a current partner. For the analyses of the ex-partner's presence at the child's birthday (i.e., for the total sample), we assigned respondents without a partner the respective mean on these variables. This means that the effect of the dummy for repartnering (see above) refers to the difference between respondents without a partner and repartnered respondents with average relationship qualities. The effect of the mean-imputed "relationship quality" variables refers to respondents with a new partner only, when a variable for repartnering is included (Poortman & Kalmijn, 2002).

Measures of control variables
We control for various (social-demographic) factors that might influence who attends the child's birthday. Following previous studies on family rituals (or parental involvement in general), we control for, for example, parents' age and education levels and the geographical distance between the households of both former partners (e.g., Bakker et al., 2015;Costa, 2014;Meggiolaro & Ongaro, 2015).
Child's age. The child's age is important to control for as (joint) birthday celebrations might be less relevant for older children. The age of the focal child is measured in years.
Child's and respondent's gender. The gender of the respondent and the focal child were both coded with a dummy variable, with "male" as the reference category.
Respondent's and ex-partner's level of education. Respondents indicated, respectively, their and their ex-partner's highest attained level of education on a scale from 1 (incomplete elementary school) to 10 (post-graduate education). We treated these measures as quasi-continuous, as alternative specifications yielded similar results in the analyses.
Geographical distance between the respondent's and ex-partner's households. The distance between parents' homes might be a constraint for attending the child's birthday. Respondents indicated the travel time (in minutes) from their home to that of their expartners for a typical one-way journey. Values exceeding 600 were recoded to a maximum of 600 to prevent exceedingly large values from having too much influence on the results.
Former union type. This variable indicates whether the respondent and the ex-partner were previously cohabiting (0) or married (1).
Refreshment. This variable indicates whether the response came from the main sample (0) or the refreshment sample (1) (see above). Table 1 presents descriptive statistics for all independent and control variables, by total sample and by repartnered sample.

Analytical strategy
We, first, describe the presence of the ex-partner, the current partner, and their joint presence using descriptive statistics. To test the hypotheses about relationship quality and family structure, we used linear probability models (LPM). We did not opt for multinomial logistic regression as our dependent variables are not mutually exclusive categories with a common base. This is because we were interested in, for instance, what predicts the presence of the ex-partner, not what predicts "only the ex-partner being present, and not the current partner". Furthermore, LPM, unlike logistic regression, allows for comparisons of coefficients across models. We also ran all models as logistic regression models, which yielded the same conclusions. For analyzing the ex-partner's presence we used the total sample. We used the repartnered sample for analyzing the current partner's presence and joint presence. We estimated two models for every analysis. For analyzing the ex-partner's presence, model 1 includes the dummy variables for repartnering, and model 2 uses the categorical type-of-union variables. In the analyses of the current partner's presence and joint presence, models 3 and 5, respectively, include the respondent's type of union and the dummy variable for repartnering of the ex-partner, whereas models 4 and 6, respectively, include the categorical type-of-union variables for the respondent and ex-partner. We clustered the standard errors of all models on the level of the divorced/dissolved union to account for possible dependencies between observations due to the partial multi-actor design (see Rogers, 1993).

Descriptive findings
We, first, describe who was present at the child's last birthday. Figure 1 shows that 34.0% of all respondents celebrated the child's last birthday with the ex-partner. 87.3% of repartnered respondents celebrated the child's birthday together with their current partner. In 24.4% of cases were both the ex-partner and current partner jointly present at the child's birthday. Additional analyses among the "repartnered" sample (not shown) show that 7.9% of respondents celebrated the child's birthday "alone" (without either the ex-partner or current partner). 4.8% celebrated the child's birthday only with the expartner and 62.9% did so only with the current partner. Taken together with our explanations in the Data section, these findings imply that though most parents do celebrate their respective children's birthdays after divorce, they usually do so without their ex-partner. Next, we systematically explore the factors related to the presence of the ex-partner, current partner as well as their joint presence at the child's last birthday.

Figure 1:
Relative frequencies of ex-partner's, current partner's, and their joint presence at the child's last birthday, by group. Table 2 shows the results of two linear probability models estimating the probability of the ex-partner's presence at the child's last birthday.  Note: a The difference between residence with respondent and residence with ex-partner is statistically significant (p< .05). b The difference between LAT and marriage is statistically significant (p< .01). c The differences between LAT and cohabitation/marriage are statistically significant (p< .05). Robust standard errors clustered on the level of the former household. * p < .05, ** p < .01, *** p < .001 (two-tailed). Source: New Families in the Netherlands Wave 2, 2015-2016.

Presence of the ex-partner
Model 1 shows that, as expected (see H1A), repartnering of the focal parent and the ex-partner reduced the probability of the ex-partner being present at the child's birthday.
In model 2, we tested if these effects differ between marriage, cohabitation, and LAT. For both the focal parent and ex-partner the effects follow the expected order, with marriage having the strongest negative effect, followed by cohabitation, and, lastly, LAT. However, compared to being single, the effect of LAT was not statistically significant, while those of cohabitation and marriage were. Additionally, for the focal parent, the difference between LAT and marriage was statistically significant and for the ex-partner that between LAT and marriage or cohabitation.
Our expectations regarding the presence of (step)children (see H1C) were partly confirmed. In models 1 and 2, we found a statistically significant negative effect of the expartner having biological children, but not of having stepchildren. The respondent having (step)children, however, did not statistically significantly affect the ex-partner's presence.
Considering the child's residence arrangement, we expected shared residence to be associated with a higher probability of the ex-partner's presence (H2A). However, not shared residence but residence with ex-partner stands out in that this arrangement reduces the probability of the ex-partner's presence compared to shared residence and compared to residence with focal parent. Residence with the focal parent, compared to shared residence, did not reduce the probability of the ex-partner being present.
We found that relationship qualities influence the ex-partner's presence, but different from how we expected. As expected (see H3A and H3C), models 1 and 2 show that the better the focal parent's relationship with the ex-partner, and the better the relationship between the ex-partner and the current partner, the more likely that the ex-partner was present at the child's birthday. Contrary to our expectations (see H3B), there was no statistically significant effect of the focal parent's relationship quality with the current partner. Additionally, the child's age and the respondent's and ex-partner's education level are positively related to the ex-partner's presence, while previous marriage (compared to cohabitation) and increased geographical distance reduced the probability of the expartner's presence. Table 3 presents two linear probability models estimating the probability of the current partner being present at the child's birthday.

Presence of the current partner
Regarding repartnering, the results confirmed our expectation that married and cohabiting current partners have higher probabilities of being present at the child's birthday than LAT parents -the difference between cohabitation and marriage was, however, not statistically significant. Model 3 additionally shows that the current partner is more likely to be present when the ex-partner of the respondent is repartnered. Regarding the effects of the ex-partner's type of union, model 4 shows that the ex-partner having a cohabiting or LAT partner increased the probability of the current partner being present (compared to being single), whereas being married did not. Contrary to H1C, the presence of (step)children did not affect the current partner's presence. Note: a The difference between residence with the respondent and residence with the ex-partner is statistically significant (p< .05). Robust standard errors clustered on the level of the former household. * p < .05, ** p < .01, *** p < .001 (two-tailed). Source: New Families in the Netherlands Wave 2, 2015-2016. Sole residence of the child with the ex-partner stands out in models 3 and 4, as it reduces the probability of the current partner being present, compared to shared residence, as well as residence with the focal parent. Contrary to our expectations (see H2B), we found no statistically significant effect for residence with the focal parent vis-àvis shared residence.
Additionally, we found that the better the relationship quality between the focal parent and ex-partner, the less probable it is that the current partner was present at the child's last birthday. We found no statistically significant effect of the focal parent's relationship quality with the current partner or of the relationship quality between the current partner and the ex-partner, contrary to H3B/C. None of the control variables statistically significantly affected the outcome.

Joint presence of ex-partner and current partner
Lastly, Table 4 presents two linear probability models estimating the joint presence of the ex-partner and current partner.
In neither model did the focal parent's or the ex-partner's union type affect the probability of joint presence. Regarding our expectations about having (step)children, models 5 and 6 show that the ex-partner having a biological child reduces the probability of joint presence, whereas having a stepchild does not. Congruent with all previous models, the respondent having (step)children with the current partner did not affect joint presence.
Our expectation that joint presence is most common in shared vis-à-vis sole residence was not met -in neither model did sole-residence of the child with either biological parent affect the probability of joint presence. Note that, though not hypothesized, we did find a statistically significant negative effect for residence of the child with the ex-partner, compared to residence with the focal parent, in models 5 and 6 (b=-0.09, p<.05, not shown in Table 4).
Models 5 and 6 show that the better the focal parent's relationship with the ex-partner, and the better the relationship between the ex-partner and the current partner, the more likely joint presence is. Again, the relationship quality with the current partner had no statistically significant effect on the outcome. Of the control variables, the child's age and both parent's education levels positively related to joint presence, while the respondent being previously married (compared to cohabiting) with the ex-partner and geographical distance reduced joint presence. Note: a The difference between residence with the ex-partner and residence with the respondent is statistically significant (p<.05). b The difference between LAT and cohabitation/marriage is statistically significant (p<.05). Robust standard errors clustered on the level of the former household. * p < .05, ** p < .01, *** p < .001 (twotailed). Source: New Families in the Netherlands wave 2.

Additional analyses and robustness checks
Though previous, related, literature on (step)parents' postdivorce involvement with their (step)children paints an inconclusive picture regarding gender differences (e.g., Koster et al., 2021;Meggiolaro & Ongaro, 2015), we nonetheless explored the effect of gender in more detail by estimating separate linear probability models for men and women and testing for statistically significant differences (using "suest" in Stata; see Online Appendix Table 1). To the extent that we found differences between men and women, they referred to the presence of the ex-partner. First, the effect of the repartnering of the ex-partner was more strongly negative for men. Second, the negative effect of the respondent having biological children with the current partner only applied to men. We observed almost the mirror image of this in the negative effect of the ex-partner having a biological child or stepchild, which was more strongly negative for women (i.e., when the ex-partner was male). Our interpretation of these findings is that repartnering plays a stronger role for women than men when it comes to no longer celebrating the child's birthday together with the ex-partner, whereas for men, this is only so when they also have new biological or stepchildren after repartnering. We performed several robustness checks. First, we restricted the sample to minor children, as birthday celebrations with (step)parents might be less relevant for young adults. The findings for these analyses were similar to the ones presented in the paper. Second, as having (step)children and repartnering are logically associated, we calculated each model by including only either the repartnering or (step)children variables before computing the "full" models presented in this paper. The only substantial difference compared to the full models was that, without controlling for focal parent's union type, the effect of the focal parent having biological children with the current partner on the probability of the current partner's presence was statistically significant (b=0.03, p<.05). Third, we controlled for parents' work hours (i.e., potential work-family conflict) and religiosity (i.e., family norms); both were unrelated to birthday attendance. Lastly, to rule out bias from dependency between the responses from both former partners even after clustering standard errors, we randomly included only one observation from each former household. This did not yield different results than those presented in the paper.

Discussion and conclusions
Family rituals serve vital functions, such as showing that one cares about one another. This is particularly so for children's birthdays -a family ritual that children and parents alike frequently look forward to celebrating (Costa, 2014;Pett et al., 1992). After divorce, it is ambiguous who practices family rituals together -because of the different types of parental figures (i.e., biological and cohabiting or LAT stepparents) and the advent of newer postdivorce residence arrangements (e.g., shared residence). This study contributed to the limited prior research by investigating with whom divorced parents celebrated their children's last birthday: the ex-partner, the current partner, or jointly with both. To our knowledge, this study is the first large-scale, quantitative, study investigating how these more recent and emergent types of postdivorce families practice family rituals like birthdays.
We found that family rituals in postdivorce families mostly concern the new family: only 30% of parents celebrated their child's birthday with their ex-partner, but nearly 90% did so with their current partner. About 25% celebrated the birthday jointly with both. For the majority of divorced parents, celebrating the child's birthday with the current partnerthe child's stepparent -may be the "default" (Costa, 2014). This does not mean that a substantial portion of parents does not celebrate the child's birthday at all: in most cases, both parents did -just separately from each other. Also after divorce, birthday celebrations are nearly-universal family rituals (see Costa, 2014).
Parents' new family context proved critical for understanding who celebrates the child's birthday together (Braithwaite et al., 1998). The tendency to celebrate the child's birthday without the ex-partner was strongest when parents had a new partner and biological or stepchildren with him or her -showing that the advent of a new family shifts parents' focus away from their old family (Braithwaite et al., 1998;Costa, 2014;Whiteside, 1989). The effect of repartnering differed between relationship types: married and cohabiting partners were more likely to be present at the child's birthday than LAT partners. This shows that the more formalized parents' current unions are, the more likely that they practice family rituals with the new than the old family. Our additional analyses showed that the effects of repartnering of the ex-partner and that of having biological children with the current partner applied particularly to men, whereas the effects of the ex-partner having biological children or stepchildren applied particularly to women. This may imply that mothers more readily shift to practicing family rituals without their expartners (i.e., already upon repartnering), than fathers (i.e., only when also having children with the new partner).
Another major finding is that parents' willingness to interact with each other is an important precondition for who is present at the child's birthday. The better the quality of relationships between the members of the households of the child's two biological parents, especially between the former partners, the more likely that the ex-partner, and that the ex-partner and current partner are jointly present at the child's birthday (see Costa, 2014). Interestingly, the current partner's presence was negatively affected by the quality of the relationship between the biological parents, suggesting that parents may facilitate the stepparent (i.e., the current partner) being more involved when they are not on good terms with their ex-partner. To our knowledge, this effect has not been mentioned regarding family rituals, but aligns with findings from research on interparental relationships in stepfamilies (Hornstra et al., 2020). Notably, the quality of the relationship with the current partner was unrelated to the current partner's presence at the child's birthday. This may be a result of celebrating together with the current partner being the default, which is not dependent on relationship quality, or a consequence of our respondents having high relationship qualities with their current partners, with little variation.
Our last major finding is that living together with the child, by offering structural opportunities for engagement and bonding, may, par excellence, create a new family that practices family rituals together: current partners who reside with the child and the focal parent are more likely to be present at the child's birthday than those who do not. When looking at the ex-partner's presence, the role of structural opportunities is less clear. Besides parents who opted for shared residence perhaps having stronger preferences to celebrate birthdays together with their ex-partner (Bakker et al., 2015), we assumed that sole residence would entail fewer opportunities for non-resident parents to be involved, compared to shared residence. Although effects differed between models, we found support for our hypothesis that the ex-partner is less likely to be present in case of sole residence vis-à-vis shared residence, but only when the ex-partner is the resident parent (and not the focal parent). While we are, from a theoretical perspective, unsure about why this effect does not seem to apply when the focal parent is the resident parent, there could be several empirical explanations for this finding, such as recall bias and social desirability vis-à-vis birthday celebrations. For example, when the child lives with the ex-partner of the responding focal parent, respondents might be more inclined to admit or recall that they celebrated the birthday without the ex-partner. We also assumed that shared residence would increase joint presence. This was not the case, challenging the assumption that shared residence facilitates joint family rituals (Bakker et al., 2015;Smart & Neale, 1999).
Our results need to be interpreted with some caveats in mind. First, as we used crosssectional data, we cannot make causal claims. For example, the relationship quality with the ex-partner might be lower because of his or her lack of willingness to jointly practice family rituals. Future research would benefit from using panel data. Second, our analytical sample is by definition restricted to parents who celebrated their child's last birthday. Although the parents who did not celebrate might have had practical reasons for doing so (e.g., being ill), it could also indicate that they are less involved in their children's lives, possibly limiting the generalizability of our results. Studying the reasons why parents do (not) celebrate their child's birthday would be based on different theoretical notions which are beyond the scope of the present study. For example, it seems plausible that the strength of the parent-child relationship would be a key factor in determining whether parents and their children decide to celebrate the child's birthday together. We encourage further research into this under-studied aspect of family rituals. Third, our sample was somewhat selective in terms of, amongst others, socioeconomic status, urbanization, and age, which might limit generalizability. Fourth, our findings regarding the different union types (LAT, cohabitation, and remarriage) might be confounded by the respective length of the relationship as LAT relationships are, for example, usually the first step before cohabitation. As NFN includes no information on the length of parents' new relationship, our findings might at least partially reflect relationship duration. Fifth, we could not account for the quality of the relationship between the different child-(step)parent dyads, as this information is not contained in NFN. Future research should consider this, as the quality of the child-parent relationship might be an important determinant of who attends birthdays. Lastly, though we controlled for major sources of logistical reasons vis-à-vis being present at the child's birthday (e.g., work, and geographical distance), we cannot entirely rule out the role of logistical reasons for (not) being present at the child's birthday.
Overall, this study shows that nearly every child's birthday in postdivorce families is celebrated, but most biological parents do not celebrate children's birthdays together. Parents' loyalties appear to shift to the new family, especially when the relationship with the ex-partner is strained. Structural opportunities for access to the child (such as coresidence) as well as relationship qualities and structural aspects of parents' new relationships seem to matter for with whom the child's birthday is celebrated after divorce. Given that it is rather the new than the ex-partner with whom the birthday is celebrated, an open question remains whether this is in the interest of the child. Limited prior research suggests that children define their families more inclusively (Castrén & Widmer, 2015), and want both of their parents present at family events (Zartler, 2014); our findings may reflect parents' efforts to reify what should constitute their new families (Whiteside, 1989). For the child, celebrating together with parents who do not get along with each other may be stressful and disappointing, but so might be having two separate birthdays (see Baxter & Braithwaite, 2006). Given the importance of family rituals throughout the life course, we encourage researchers to investigate family rituals and the consequences for child wellbeing.

Data availability
For purposes of scientific research, the New Families in the Netherlands (NFN) data is available at DANS. Poortman