Measures of children’s time use, particularly with parents and siblings, are used to evaluate three hypotheses in relation to the vocabulary and mathematical skills development: (1) the resource dilution hypothesis, which argues that parental and household resources are diluted in larger families; (2) the confluence hypothesis, which suggests that the intellectual milieu of families is lowered with additional children; and (3) the admixture (“no effect”) hypothesis, which suggests that the negative relationship between family size and achievement is an artifact of cross-sectional research resulting from unobserved heterogeneity. Each hypothesis is tested using within-child estimates of change in cognitive scores over time with the addition of new children to families.
Family Size and Achievement
One of the most robust findings related to the influence of family structure on children’s academic success is that children from larger families fare less well than children from smaller families. One explanatory hypothesis—the confluence model (Zajonc and Markus, 1975)—holds that the presence of younger siblings reduces intellectual stimulation for older children. A second hypothesis—resource dilution—suggests that parental resources available to any particular child are diluted when additional children are added to the family (Blake 1981, 1989; Downey 1995). A third hypothesis is that the relationship between family size and achievement may be at least partially spurious, resulting from unobserved heterogeneity at the family level either through genetic or environmental factors (Guo and VanWey 1999a; Page and Grandon 1979; Rodgers et al. 2000). Previous empirical analyses have failed to provide a comprehensive assessment of these competing hypotheses, and the debate about their relative merits remains unresolved (Conley 2005). What has heretofore been neglected is that a critical element of both the confluence and resource dilution hypotheses, as well as a potential source of unobserved heterogeneity between families, is the amount of time that children spend in activities with others. The confluence hypothesis suggests that children will perform less well on measures of cognitive development to the degree that they interact with younger siblings. The resource dilution hypothesis implies that with more children in the household, parents have less of all resources but, most importantly for intellectual development, less time and attention to allocate to any particular child. A unique element of the current research is the precise measurement of children’s time use in theoretically relevant activities, most significantly in interaction with parents and siblings.
This research presents tests of the resource dilution and confluence hypotheses concerning the relationship between family size and both symbolic learning and reading and mathematics skills as measured by the Woodcock-Johnson Revised Letter-Word Identification (LW-R) and Applied Problems (AP) tests (Woodcock et al. 1989). We first estimate cross-sectional models to assess the degree to which any negative association between family size and these assessments may be attributable to a negative association with time spent with siblings (relevant to the confluence model) or to positive associations between this measure and resources that may be diluted in larger families, including household resources, extrahousehold activities, and parental attention. We then turn to within-child fixed-effects models explicitly estimating the association between the addition of a marginal child to the family and change in these cognitive scores. Comparing these estimates with those from the cross-sectional results, we evaluate the extent to which any effect of family size may be due to unobserved heterogeneity.
Resource Dilution, Confluence, and Admixture Hypotheses
One of the most influential and replicated findings in the status attainment literature in the last 40 years is that children from larger families fare less well than children from smaller ones on a wide variety of outcomes (Blake 1989; Downey 1995; Steelman et al. 2002). The three main explanations in the sociological and psychological literatures that have been forwarded have been labeled the resource dilution, confluence, and admixture hypotheses.
This perspective holds that family resources are to some degree finite and that with additional children, resources available to a particular child are reduced. Three types of resources are central: household resources, parental attention, and opportunities to engage in extrahousehold activities (Blake 1981:422). Household resources include the “types of homes, necessities of life, cultural objects (like books, pictures, music and so on).” Parental attention includes “personal attention, intervention, and teaching.” The last type of resource relates to “specific chances to engage the outside world, or, as kids say, ‘to get to do things.’”
Previous research concerning the resource dilution hypothesis has been limited in operationalizing these three types of resources. The most comprehensive work to date has operationalized household resources with indexes for the presence of particular types of educational objects in the home; whether a child had attended classes in art, music, or dance; and an indicator for the presence of a computer in the home (Downey 1995). A second analysis measured change in an index of learning materials and toys available to children across time in the context of changes in family size (Baydar et al. 1997). Although the first analysis was weakly supportive of the resource dilution hypothesis, the second suggested the effect of the birth of a new child on vocabulary skills was positive—a result we that will return to later in consideration of the admixture hypothesis.
Such indexes are generally problematic because they simply aggregate presence or absence of disparate physical objects as a proxy for their use. They are further complicated by the high degree of potential shareability when measured at the household level (Downey 1995). The same number of books, for example, may yield an equal benefit for any number of children in the household and thus may not be diluted at all in their effect on cognitive development in larger sibships.
Parental attention, although key to the resource dilution model, has also been operationalized through relatively poor proxies. The most frequent of these in the empirical literature has been parental aspirations for children’s educational attainment, which has been found to explain some of the negative association between family size and high school students’ educational expectations (Blake 1989) and middle school grades (Downey 1995). A significant problem with this measure is that even if it is possible that such aspirations are modified downward with increasing family size, it is equally if not more likely that they may be co-determined with family size by unobserved factors as suggested by the admixture hypothesis. Other operationalizations of parental attention have included frequency of parental conversations, whether parents know their children’s friends or their friends’ parents, observed parenting style, and maternal work hours (as a negative proxy) (Baydar et al. 1997; Downey 1995).1
Measuring and testing for effects related to the dilution of opportunities to engage the outside world have been almost completely neglected in the empirical literature. Only Downey’s (1995) work has attempted this by examining whether parents report children having visited art, science, or history museums.
As this summary suggests, despite their theoretical importance, systematic study of all three types of resources has been extremely rare in the empirical literature. Further, the three types of resources have never been tested in joint specifications. When measures of each have been tested separately, results have been weakly or equivocally supportive of the resources dilution hypothesis. Whether this is because such effects are not strong or because of their relatively poor measurement is an open question. We argue that none of these measures adequately captures what Blake originally suggested would be diluted with the addition of siblings: specifically, the utilization of resources by children.
Fortunately, recent data collected on children’s time use allow for measures related to resource dilution that may both more closely match the theoretical conceptualization of these resources and simultaneously produce more efficient estimators. Direct, continuous measures of the amount of time children use household resources (such as books, newspapers, and computers) better operationalize the concept than a summative index for the presence of some of these resources in the home. Measurement of time children spend with parents in a variety of activities—talking, eating meals together, engaged in activities, being helped with homework, discussing classwork or their friends—may be a better, more comprehensive operationalization of parental attention to a particular child than aspirations for future education or knowledge of their friends’ names. Measurement of time spent engaged in extrahousehold activities (e.g., at museums, civic associations, social groups, sports, parties) can likewise be seen as a better measure of engagement with the outside world than a simple measure of whether a child had visited one or more type of museum. Such measures are less prone to measurement error, encompass a wide variety of resource use, and are not prone to the shareability problem. Additionally, because time use can be aggregated to capture a variety of activities simultaneously and is measured continuously, it can produce more efficient estimates than single-item measures (or indexes constructed of a few of them) with lower variance.
The Confluence Hypothesis
In this theoretical framework, developed and championed by Robert Zajonc and colleagues over three decades, the influence of family size on children’s intellectual development operates through a confluence of two mechanisms associated with increasing family size. In the first, additional children are hypothesized to lower the level of intellectual stimulation that older siblings are exposed to through lower levels of interaction with parents (as in the resource dilution model) and increased interaction with, on average, younger siblings (Zajonc and Markus 1975). This negative effect, however, is hypothesized to be offset by a secondary “teaching” mechanism operating among children of lower birth orders who may benefit cognitively as they take on the role of caregivers and tutors to their younger siblings (Zajonc 1983, 2001; Zajonc and Markus 1975; Zajonc et al. 1979). These countervailing effects, although nuanced, have led to criticism that confluence theory is capable of accommodating most empirically observed patterns.
The degree of empirical support for the confluence model has been hotly contested almost since its inception. Debates have concerned correct model specification, the appropriate level of measurement, and the fit of reported results to particular data sets (Page and Grandon 1979; Retherford and Sewell 1991; Rodgers et al. 2000; Steelman 1985; Wichman et al. 2006; Zajonc and Bargh 1980; Zajonc and Mullally 1997; Zajonc et al. 1979). Responding to critiques, supporters of the confluence model have suggested that because of the potential for bias associated with a host of unobserved family-level factors, the only valid individual-level test of the model would entail the use of repeated measures for both cognitive outcomes and the actual interactions between children, their parents, and siblings—data that were not previously available (Zajonc and Mullally 1997; Zajonc et al. 1979).
The Admixture (“No Effect”) Hypothesis
Another group of scholars has suggested that negative associations between family size and children’s achievement are themselves an artifact of unmeasured between-family differences as measured in cross-sectional data. This idea—the admixture hypothesis—posits that the negative association may reflect unobserved heterogeneity in either genetic (often framed in terms of intelligence) or environmental factors to which children are exposed (Page and Grandon 1979; Rodgers et al. 2000).
Both resource dilution and confluence theories concern mechanisms taking place within families related to changes in resources and intellectual stimulation available to a given child with the addition of new siblings. For this reason, admixture theorists suggest that only within-family or within-child designs are appropriate to test them. The most prominent of these methods is the within-family sibling design. In such models, siblings from the same families are compared, controlling by design for environmental and family factors that can be assumed to be constant. This literature finds no association between birth order and children’s intelligence, supporting the hypothesis that remaining family size effects are due to unobserved differences between families (Retherford and Sewell 1991; Rodgers et al. 2000; Wichman et al. 2006). Although these results appear to be clear evidence against both resource dilution and confluence mechanisms, sibling models cannot address the effects of unobserved factors related to the addition of more siblings that may differ for children of different birth orders (and hence ages) within the family. These include, potentially, changing resources allocated to specific children as well as the intellectual and interactional environment of the family.
To address these problems, a within-child repeated measures design as suggested by Zajonc and colleagues is most appropriate. In this design, change in a particular outcome with the addition of the marginal sibling(s) can be assessed net of factors introducing heterogeneity between siblings. To date, only two such studies have been conducted. The first, already discussed in the context of the resource dilution model, tested for changes in cognitive development among young children after the birth of a closely spaced sibling using data from the NLSY (Baydar et al. 1997). Although not explicitly framed for the purpose, this analysis found support for both admixture and (to a limited degree) resource dilution hypotheses. Using a within-child design the authors found a positive, although nonsignificant, association between the birth of a sibling and vocabulary test scores. When the authors controlled for the dilution of some household resources as part of their broader index measure, the positive sibling effect increased in magnitude and became statistically significant.
A second analysis by Guo and VanWey using a within-child design is directly framed in the literature on family size and achievement (1999a). Their analysis compared cross-sectional, within-family, and within-child estimates of cognitive scores associated with different family sizes using data from the NLSY. Although not explicitly modeling resource dilution or confluence effects, it does provide a relatively strong test of the admixture hypothesis. Negative associations between family size and children’s standardized vocabulary, reading, and math test scores identified in the cross-sectional model disappeared in the within-child child analysis; and in the case of applied math scores, the association became positive—a similar result to that reported by Baydar et al. (1997) concerning vocabulary scores.
The validity of the inferences drawn from Guo and VanWey’s within-child model has been challenged, however. One major critique is that such models, using widely spaced panels, structurally constrain estimation to children whose families added relatively widely spaced siblings (Downey et al. 1999). Because other (cross-sectional) research has found that negative associations between sibship size and cognitive achievement are weaker among more widely spaced siblings than among closely spaced ones (Downey et al. 1999; Powell and Steelman 1990), it was suggested that comparing these estimates with those derived from the cross-sectional analysis was inappropriate. Guo and VanWey argued (and presented supporting evidence) in response that spacing effects identified cross-sectionally may also be biased by the type of unobserved heterogeneity that the within-child design is meant to control for (Guo and VanWey 1999b). These critiques, however, and their seemingly anomalous results have led a wide body of scholars to reserve judgment on these findings until they can be replicated (Steelman et al. 2002).
In this article, we present what is to our knowledge the first comprehensive simultaneous test of the resource dilution, confluence, and admixture hypotheses. In the first part of our analysis, we test whether simple bivariate associations exist among Woodcock-Johnson Psycho-Educational Battery-Revised (WJ-R) letter-word scores of symbolic learning and reading skills (Woodcock et al. 1989), applied math problem–solving scores, family size, and key variables related to resource dilution and confluence models among U.S. children aged 3–18 in 2002. These variables include children’s time with parents, time spent using household resources and in extrahousehold activities, and children’s time with siblings. We then move to a multivariate analysis, using the same data to evaluate support for the confluence and resource dilution models cross-sectionally.
In the second part of the analysis, we replicate these tests using data on change in the dependent and independent variables of interest between 1997 and 2002, using within-child fixed-effects models. These models allow us to evaluate whether unobserved heterogeneity on the family or individual level is responsible for any association between sibship size, resource and confluence mechanisms, and standardized test scores identified in the first part of the analysis.
Methods and Data
The data used here come from two waves of the nationally representative Panel Study of Income Dynamics Child Development Supplement (PSID-CDS I and -CDS II). In 1997, the CDS I collected extensive data including cognitive assessments and detailed time diaries from up to two children aged 3–12 sampled from households in the core PSID for that year with children younger than age 12. In total, the CDS I interviewed 3,563 children, of whom 2,907 were eligible for and reinterviewed in the CDS II in 2002.
Analytic Sample and Estimation Strategy
The resource dilution and confluence hypotheses were developed during a period in the United States, where the frame of reference was the stable two-parent nuclear family. Potential interactions between confluence and resource dilution mechanisms and diverse family configurations (largely untheorized to date) may significantly influence the direction, magnitude, and efficiency of test statistics if suppressed. Differences in family structure, for example, have been shown to be associated with both cognitive outcomes and key independent variables related to both resource dilution and confluence frameworks, such as time with parents (Hofferth and Anderson 2001; Hofferth and Sandberg 2001). Such structural differences likely extend to time spent in household resource and extrahousehold activities as well as time spent with siblings. At the same time, living in a single-parent family or experiencing transitions in family structure can have negative impacts on reading scores and other academic indicators (Carlson and Corcoran 2001; Cooksey 1997; McLanahan and Sandefur 1994). If such structural differences remain uncontrolled, inferences concerning causal mechanisms in both resource dilution and confluence models will be biased.
To address these potential sources of bias and to produce a clean test of these hypotheses, we restrict our analytic sample to those children who were the biological sons or daughters of the head of the household in 1997 and who lived with both biological parents from 1997–2003. Of the 2,907 children reinterviewed in the PSID-CDS II, 48.1 % (1,413) came from such families. Children in families with biological children living in institutions and those with biological siblings but none residing in the family unit (29 in total) were also excluded.
To avoid influence resulting from the relatively small numbers of sample children residing in large families, we also restrict our analytic sample to children of up to birth order 3 living in families with up to three additional siblings, eliminating a total of 162 children from the sample. Both resource dilution and confluence models are concerned with increasing numbers of children. Unlike the resource dilution hypothesis, however, the confluence hypothesis is not symmetric with regard to the direction of changes in family size. For this reason, we impose a further restriction that sample children not come from a family where size of the coresident sibship had seen a gross decrease between 1997 and 2002, thus eliminating 56 children.
A final minor restriction here is that children must have recorded five or more distinct activities on either day (all but 111 children in 2002), below which the accuracy of the time diary data may be questioned. Considering all these restrictions, we use an analytic sample in the cross-sectional analysis of 1,041 children between 3 and 18 years of age in 2002. Of these, 41 and 43 children, respectively, did not have valid outcomes for the letter-word scores and for the applied problems scores, used as the dependent variables here. This leaves final sample sizes of 1,000 children for the letter-word analysis and 998 applied problems analysis. Among these, 41 children were missing information on their mothers’ educational attainment, and 2 were missing information from the family listing file from which the measures for the mean age and standard deviation of sibship size (two important control variables) were derived. Employing listwise deletion of cases with missing data on these independent variables, we are left with analytic samples of 951 and 949 children in total. Overall, this represents approximately one-third of all children and 67 % of all children living in stable, two-parent families in the United States in 2002. We use a fully balanced design for the within-child models, restricting the analytic sample to those who met these criteria in both 1997 and 2002 and had valid assessments in both years, for final samples of 536 children for the letter word scores, and 534 for the applied problems scores.
Dependent and Independent Variables
The WJ-R Letter-Word test assesses symbolic learning and reading skills; the WJ-R Applied Problems test measures math knowledge, calculation skills, and quantitative reasoning. Total raw scores are standardized to average national scores for children of the same age. Changes in both scores between 1997 and 2002 were normally distributed with a mean of 0. Family or sibship size is measured in the cross-sectional analysis as the number of biological siblings living with the sample child in 2002. In the fixed-effects models, the change in family size between 1997 and 2002 is measured as a binary indicator for the addition of at least one marginal child. This measure was chosen because only three children had more than one additional sibling. In total, 12.5 % (67) children had a new sibling between the panels.
Resource Dilution Measures
A significant contribution of this research is the measurement of all three types of resources that may be diluted with the addition of children to a family in terms of children’s time use. Children’s time use with parents and siblings, as well as in activities related to the use of household resources and engagement with the outside world, are measured with detailed 24-hour time diary data. Diaries were collected in the PSID-CDS for one random weekday and one random weekend day in both 1997 and 2002. Respondents provided information on each activity in which children were engaged in over a 24-hour period, when they began and ended, who else was participating with the child, and who else may have been accessible to the child but not directly engaged in the activity.
We operationalize household resources with aggregate weekly time use in activities such as computer usage, reading, or art activities. Extrahousehold activity, or engagement with the outside world, is operationalized with weekly hours in activities such as attending sporting and cultural events, theater, and other events.
To operationalize dilution of parental attention, we use four measures of aggregate time children spend with parents. The first two of these are gross measures of weekly time directly engaged in activities with any parent and with at least one parent accessible to the child but not directly participating in his or her activities. The kinds of things that parents do to foster children’s development at a young age most often (although not exclusively) involve spending time with them in interaction or, in activities in which children have relatively more self-direction, being available to monitor them when they need support or direction (Sandberg and Hofferth 2001). Our measure of time in direct interaction includes time spent talking with parents, eating meals together, discussing classwork or their children’s friends, as well as every other activity a parent does with the child. An important caveat here is that some children may receive more attention from parents in attempts to remediate cognitive or behavioral difficulties. This sort of endogeneity may result in a negative relationship between children’s time with parents and cognitive outcomes in the cross-sectional analysis. To control for this, we also test specifications with a measure of aggregate weekly time spent engaged with parents in teaching or achievement-related activities. When used in specifications that also include the measure of gross time in interaction with parents, this measure isolates some of the endogenous behavior related to remedial attention.
The final measure is weekly time with parents in play and leisure activities. This is a broad category including unstructured play as well as numerous types of leisure pursuits (excluding television watching), such as sporting and cultural events and also participating in non-organized team and individual sports. A complete list of discrete activities in all aggregate time-use categories is available in Online Resource 1.
Confluence Measures of Time in Interaction With Siblings
We use four main measures of sibling interaction. The first two are measures of total weekly time in direct, engaged interaction with siblings and total time with siblings present but not directly participating. We also use measures of time spent with siblings in teaching or achievement-related activities, such as computer use, educational activities outside school, or reading and in games and unstructured play. This latter category—including playing card and board games, pretend, dress-up, and unspecified play—measures interaction in activities that are potentially crucial sites of symbolic learning but are also differentially cognitively stimulating for older and younger children when engaged in together. For this reason, participation in such activities may also capture a less formal teaching effect. As with all the measures of children’s time use here, these aggregates are imputed to weekly hours by multiplying weekday time by 5 and multiplying weekend time by 2.
In the multivariate cross-sectional analysis, we control for a number of variables that may simultaneously be associated with cognitive development and family size, thus biasing inferences. These include the sex of the child (coded 0 for males and 1 for females), the child’s race (coded 1 for non-Hispanic whites, and 0 for all others), and continuous measures for maternal age (in years) and weekly hours spent watching television. We also control for total family income (measured as a natural log) and maternal education (an indicator coded 1 for a college degree) as measures of socioeconomic status, long thought to be a potential source of heterogeneity between families driving negative sibship size effects (Powell and Steelman 1993).
Two additional controls are included in all specifications: the mean age of of the child’s sibship and the standard deviation of sibship age have substantive implications for confluence and resource dilution mechanisms. The mean age of a sibship can be seen as a rough control for variability in the intellectual environment in the family, as suggested by the confluence model. At the same time, a younger sibship likely means lower levels of parental and household resources to share among children because the family is in an early developmental stage. More widely spaced sibships are also expected to be positively associated with cognitive outcomes from both resource dilution and confluence perspectives. Under the confluence model, more closely spaced sibships diminish the potential for a “teaching” effect among older children. As noted earlier, the potential for greater dilution effects in more closely spaced sibships is a principle critique that has been leveled against Guo and VanWey’s comparison of cross-sectional and within-child estimates. The standard deviation of sibship size as specified here serves to control for such effects in both the cross-section and within-child models presented.
Finally, birth order is included as a control because of its important role in confluence theory and because it is often believed to be associated with cognitive development and achievement more generally, although the empirical literature is still divided (Black et al. 2005, 2010; Conley and Glauber 2006; Price 2008; Steelman et al. 2002). Birth order is operationalized as two indicators for second and third children, with the reference category being the first child.
The within-child fixed-effects analysis controls only for family income, time spent watching television, mean age of the sibship, and standard deviation of sibship age, given that the other controls are time-invariant. It should be noted that child age is not included as a control in either the cross-sectional or fixed-effects models. Although there is obvious and substantial variability by age associated with cognitive development, the WJ-R scores are age-standardized, accounting for developmental differences both between children in the cross-sectional analysis and over time in the within-child analysis.2
In all analyses presented here, standard errors have been adjusted for multiple children in the household and weighted to be representative of the U.S. population in 2002. Table 1 presents the means and standard deviations (unadjusted for clustering at the household level) for the dependent, independent, and control variables in the 2002 cross-sectional analysis as well as for the full unrestricted sample of children from the PSID-CDS II for comparative purposes. This table also presents relevant descriptive statistics for other models to follow.
Table 2 presents simple cross-sectional bivariate regressions for the 2002 sample to test the associations among sibship size, the cognitive measures, and the key independent variables. The results reveal, as expected, a significant negative zero-order association between sibship size and letter-word scores, as well as a negative but nonsignificant association (p = .123) with applied problems scores. Each additional sibling in the zero-order letter-word score model is predicted to be associated with a two-point lower standardized letter-word score. In addition, Table 2 demonstrates significant zero-order associations between family size and children’s time with parents and siblings that are congruent with the proposed interactional mechanisms underlying both the confluence and resource dilution models. Contrary to expectations of the resource dilution hypothesis, neither weekly hours spent in household resource activities nor extrahousehold activities are significantly associated with family size.
Multivariate Cross-Sectional Analysis
In each table, Model 1 presents the baseline specification with only controls. Girls score higher than boys on the letter-word test, whites do better than nonwhites on the applied problems test, and children of older and more highly educated mothers score better on both. Family income does not have a significant association with letter-word scores, but it is positively associated with the applied problems measure. Weekly hours of television viewing is negatively associated with both. In Tables 3 and 4, the coefficients for the mean age of the sibship are negative and statistically significant. The coefficients for the standard deviation of sibship age in both tables are positive, suggestive of spacing effects, but small and statistically insignificant.
Model 2 introduces the measure of family size. The association here is negative as expected for both letter-word and applied problem scores. Although neither is significant by conventional criteria, the estimate for letter-word scores comes close (p = .07) and remains about the same magnitude seen in the bivariate results. This suggests that none of the controls included here explain the zero-order sibship size association well. When sibship size is controlled for in this model, however, the magnitude of the association between birth order and letter-word scores is attenuated, dramatically so in the case of third children.
The measures of time with siblings are introduced in Model 3 as a first test of the confluence model. None are statistically significant in either analysis, although all are negative in Table 3 concerning letter-word scores. The inclusion of these measures reduces the magnitude of the sibship size coefficient in the letter-word analysis by about 20 % and increases its standard error as well—a pattern that is consistent with confluence effects but is not apparent with regard to applied problems.
A more nuanced test of the confluence model is presented in each table in Model 4, which interacts birth order and the measures of time with siblings. In these models, we see a substantive diminution in the negative family size coefficient suggestive of a general confluence effect, and this model represents a significant improvement in fit relative to Model 2 in the letter-word analysis. In the letter-word score analysis, time in teaching/achievement activities appears to be associated with higher scores for third-born (hence younger) children relative to firstborns (p = .07) and negative for older siblings. In the applied problems analysis, time engaged with siblings is also associated with higher scores for second- and third-born than firstborn children, significantly so in the case of second-born children. These results suggest the lack of an easily identifiable teaching effect.
Model 5 presents the specification testing the resource dilution hypothesis. Time spent by children in household resource activities has a small positive association with both letter-word and applied problem scores. For the applied problem scores, this is statistically significant, and it is close to being so for the letter-word scores (p = .115). Coefficients for extrahousehold activities are not statistically significant for either assessment. Associations of these cognitive measures with time spent with parents are mixed. Controlling for time spent in teaching and achievement activities (which is negative and significant for both assessments, indicative of endogeneity related to remedial attention), there is no significant association between time children spent with parents and either outcome, although the positive associations with parental accessibility come close for letter-word scores (p = .07) and parental engagement for applied problems (p = .09). Most importantly, for the letter-word scores, specification of the resource dilution variables increases the magnitude of the negative family size coefficient and raises it to statistical significance. A similar pattern is seen with applied problem scores, where this model represents a significant improvement over Model 2, and the family size comes much closer to statistical significance (p = .16). This suggests some (again, weak) support for the resource dilution hypothesis: in particular, higher levels of participation in household resources suppress part of the negative association between family size and these cognitive assessments.
Model 6 in each table specifies both the resource dilution and sibling covariates simultaneously. Model 7 does the same, including the sibling-time birth order interactions. The negative associations of time with siblings accessible in Models 3 and 4 (and time with siblings in unstructured play in the letter-word table) are amplified in Models 6 and 7, becoming significant in Model 6 for applied problems and close to significant for letter-word scores (p = .06). On the resource dilution side, we see the coefficient for time with parents accessible almost doubled when controlling for time in sibling interaction in both tables; this becomes statistically significant in the letter-word score analysis. Although the coefficients across these models remain unchanged for the most part (indicating largely independent mechanisms), these two results suggest that some negative effects of sibling interaction may be suppressed by higher household and parental resources, while some positive effects of parental attention are potentially suppressed by an offsetting negative association related to sibling interaction.
Within-Child Fixed-Effects Models of Change in Sibship Size
As discussed earlier, cross-sectional associations such as these do not necessarily imply that the addition of children to a family will disadvantage previous children’s cognitive development. The admixture hypothesis suggests that there may be unobserved family factors causally related to cognitive development, family size, and time children spend in interaction with parents and siblings that produce the associations seen when comparing children between families. To address this hypothesis, we now turn to the within-child models.
Tables 5 and 6 present the fixed-effects regressions of letter-word and applied problems scores, respectively, on changes in sibship size and controls for potential sources of time-varying heterogeneity, including log income, weekly television hours, mean age of sibship, and standard deviation of sibship age. Fit statistics included for model evaluation in this table include the proportion of within-child variance explained by each model (R2 within) and the proportion of variance in assessments attributable to unobserved individual level factors (rho). The zero-order regression of letter-word scores and applied problems on a one-child change in the number of biological siblings in the family are large and positive: 1.37 for applied problems, and 4.90 for letter-word scores. The latter statistically significant at the .10 alpha level (p = .06; results not shown in Table 4).
Model 1 in Tables 5 and 6 serves as the baseline for the following analysis, estimating only the effects of changes in total family income, television viewing hours, and the mean and standard deviation of sibship age. Except for change in television hours, which is significantly (negatively) associated with change in applied problem scores, none of the other controls in either analysis are significant. Adding change in family size to this specification in Model 2, we see the associations identified in the bivariate analysis are still large and positive. Although neither family size coefficient is statistically significant at conventional levels, both would have been close (p = .181 for the letter-word scores, and p = .096 for applied problems), had we hypothesized this result a priori, which may have been justified given the results of Baydar et al. (1997) and Guo and VanWey (1999a), as discussed earlier.
Model 3 adds to this specification measures of weekly hours in interaction with siblings. For both letter-word and applied problems scores, coefficients for time engaged with siblings are positive, which again would be close to statistically significant using a one-tailed test (p = .06 and p = .11, respectively). For letter-word scores, the coefficient for increased time in unstructured play with siblings, however, is negative as predicted by the confluence model and also close to statistical significance (p = .08). In both models, the magnitude of the family size coefficient is diminished with inclusion of the sibling participation measures, suggesting that part of the positive association between family size and these assessments is accounted for by the positive association with time in sibling interaction.
Any such positive associations stemming from interaction with siblings may, of course, vary by birth order. The birth order sibling interaction specification is presented in each table in Model 4. For third-born (younger) children, increased time in interaction with siblings after the birth of a marginal child is associated with significantly lower improvements in applied problems scores relative to firstborn children, suggestive of a general teaching effect not seen in the cross-sectional analysis for this measure. Increased time in unstructured play has a close to significant negative association with letter-word scores (p = .06), however, suggesting that older children do not benefit from this type of interaction as much as younger siblings in this analysis. Although neither model improves in fit relative to Model 3, they each increase the percentage of variance in change in scores over time by 3 percentage points.
Model 5 in each table adds the resource dilution variables to the baseline specification. In both tables, we see that increased time spent in interaction with parents is significantly negatively associated with the change in the assessment measure. This is most likely indicative of a more general endogeneity effect with regards to teaching previously discussed. In general, excluding cases of significant cognitive or developmental difficulties, we would expect children moving into their preteen and teen years to spend less time overall with their parents. We see that by controlling for this, these models indicate that increased time engaged with parents in unstructured play and in teaching activities is positively associated with these scores. This is predicted by the resource dilution hypothesis but could not be seen in the cross-sectional analyses. For the applied problems scores, the former coefficient is statistically significant, and the latter is close to being so (p = .104).
Models 6 and 7 again jointly specify measures related to both resource dilution and confluence mechanisms, showing improvements in model fit that are roughly proportional to each of their separate specifications. The first major difference from their analogous individual specifications is that after the resource dilution variables are controlled for, the positive association of increased time in sibling engagement increases in magnitude and precision for both assessments, becoming significant for the letter-word scores and almost so (p = .07) for applied problems. The positive association of time spent with siblings in unstructured play also increases in magnitude and crosses the threshold for statistical significance in the letter-word analysis.
Why should we find negative associations of sibship size in the cross-sectional analysis that can be at least partially explained by a combination of resource dilution and confluence mechanisms but positive associations in the fixed-effects models that can be explained by neither and potentially suggest a generally positive family-size effect? There are at least two potential explanations, which are not mutually exclusive. The first is that the admixture hypothesis is correct: there is unobserved heterogeneity that simultaneously structures these cognitive scores and family size that is further associated with differential time use with parents and siblings. That such unobserved heterogeneity exists is highlighted dramatically in Tables 5 and 6, where the proportion of variance in changes in the assessments because of unobserved characteristics (rho) is approximately two-thirds in all models. The second possibility is that the comparison we are making here is incorrect because of differences between the 2002 cross-sectional sample and the fixed-effects subsample. The most obvious difference of relevance here is that because of the restriction to cases with valid letter-word scores in both 1997 and 2002, children in the fixed-effects subsample are, on average, significantly older than those who were not (11.3 years vs. 9.7 years). Returning to the third column in Table 1, the consequences of this age difference stand out. The mean age of children’s sibships is significantly higher in the fixed-effects sample, as is mothers’ age. The fixed-effects children spend significantly less time engaged with siblings, more time using household resources, and somewhat less time engaged with parents. What is not different, however, is equally important. The children in the fixed-effects sample do not come from significantly more widely spaced sibships (measured through the standard deviation of sibship age)—a factor that Downey et al. (1999) suggested may have biased inferences in Guo and VanWey’s within-child analysis of cognitive assessments. One may argue, however, that this measure of spacing does not adequately address potential differences in the estimates specifically resulting from selection of more widely spaced younger siblings. More generally, because a number of unobserved factors may influence serial observation across panels, we present ordinary least squares (OLS) regressions identical to those in Tables 3 and 4 restricted to the within-child subsample in Tables 7 and 8 for comparison and to contextualize the fixed-effects results presented earlier.
Looking first at Table 7 for the letter-word scores, we see largely the same pattern of results seen in Table 3 in terms of magnitude, with precision of the estimates reduced, no doubt in part because of the smaller sample size. The exceptions are somewhat stronger negative effects of direct engagement with siblings (becoming much stronger for firstborns in Model 4). The sibling interaction models here largely explain the negative family-size coefficients. Taken together, these are indicative of a potentially stronger confluence effect in this subsample relative to the full sample. Turning to Table 8 for the applied problems scores, we see something quite different. The effect of family size here is still negative, but in contrast to the estimates from the full sample, it is much larger in magnitude and statistically significant in all models, except those jointly specifying confluence and resource dilution mechanisms. Other differences relative to the analysis using the full sample include negative coefficients for time with siblings in unstructured play that are somewhat stronger in Models 3 and 6 and greater importance of time in extrahousehold activities relative to household resource activities in the resource dilution models. Despite these differences, identified associations with confluence and resource dilution mechanisms remain robust in both analyses. Although differences resulting from sample restrictions may therefore be responsible for a somewhat diminished negative sibship-size effect in the cross-sectional letter-word analysis (which is likely due in part to reduced efficiency with the smaller sample), they cannot explain the positive association seen in the fixed-effects analysis. In the applied problems analysis, differences between the full and fixed-effects samples can be definitively ruled out as a contributor of the switch from a negative to positive association with family size between the two designs.
This research addresses a contentious issue in the status attainment literature: that of the relationship between family size and children’s achievement. This relationship has important implications for our understanding of the generation and reproduction of social stratification, child development, and relationships within families. As Steelman et al. (2002) pointed out, structural constraints related to family size influencing intergenerational transmission of economic, social, cultural, and human capital to children continue to warrant particular attention in refining our understanding of differential status attainment. One of the mechanisms through which family size may influence children’s long-term human capital accumulation operates through their cognitive development. Exactly what influences, or is associated with higher cognitive scores—for example, parental attention, sibling interaction, or broader and unmeasured elements of the family context and environment—is critical to understanding developmental trajectories.
Although we use unique data for our analysis—a nationally representative sample including extensive measurement of children’s time use in addition to cognitive outcomes—our study has some limitations. We have restricted our attention to a very select analytic sample. These restrictions, necessary to isolate potential confounding influences of family structure and changes in it, render it impossible to make broad generalizations about the relationships we consider to all U.S. children. It should be noted, however, that these results are generalizable to the significant subpopulation of children in stable two-parent families, or about one-third of all children in the United States in 2002.
Despite these limitations, our research makes three important contributions to the literature on family size and achievement. The first is recognizing that children’s time use with parents, in household resource, and in extrahousehold activities are key mechanisms in the resource dilution model and that time with siblings is the key mechanism in the confluence model. This has allowed us to measure the underlying theoretical mechanisms more accurately than has ever before been possible.
The second contribution of this research is in jointly testing resource dilution and confluence hypotheses in conventional cross-sectional data for the first time. The cross-sectional models reveal some relatively weak support for both confluence (although without a teaching effect) and resource dilution hypotheses for both assessments. The joint models provide support for both confluence and resource dilution mechanisms simultaneously, operating through differential accessibility of parents and siblings, at least cross-sectionally. The support for the confluence interpretation is perhaps stronger as the dilution of parental time is also a component of confluence theory, but only as part of a tradeoff with augmentation of time with siblings.
Our final contribution is the comparison of these results with those for the within-child models of change in cognitive scores with the addition of new siblings. Although it has long been noted that a within-family or within-child strategy is preferable in this case (Rodgers et al. 2000), this is only the second time to our knowledge that a within-child design has been applied deliberately to the problem. It is also the first time that resource dilution and confluence mechanisms have been explicitly modeled using this estimation strategy. Our analysis indicates that not only is all of the negative association between family size and these cognitive assessments explained by time-constant unmeasured factors, but this unobserved heterogeneity actually distorts the within-child association dramatically. After we control for time-constant unobserved factors, the coefficients associated with the addition of a marginal child to the family are positive, in the zero-order and control-only models close to statistical significance had we predicted such a positive effect. Our results suggest a positive association between the addition of a marginal child to the family for both assessments, in part explained by positive associations with time in sibling interaction (although less so for older children) and potentially (although less importantly) the amount of time with parents in unstructured play in the analysis of applied problems.
What tentative conclusions may we draw from this last result? Empirical work tying family size to negative consequences for children’s achievement covers a wide variety of outcomes from childhood to old age, many with much stronger effects than are commonly seen related to cognitive measures (Downey et al. 1999). We make no claim that the results here have direct application to other measures of achievement, such as school completion, grades, or economic well-being later in life. That said, the clearest result here is supportive of the admixture hypothesis. There are substantial unobserved differences between children’s families that are associated with both family size and these cognitive assessments (underlined by the high proportion of panel-level variance), causing a negative relationship between the two cross-sectionally.
In comparing conventional cross-sectional and within-child models, we have replicated the pattern of negative cross-sectional associations between cognitive outcomes and family size that is reversed when considering change over time in children’s development seen in the work of Guo and VanWey (1999a) and Baydar et al. (1997), using a different nationally representative data set and different cognitive assessments. Previous findings that are potentially indicative of cognitive benefits associated with larger family sizes after unobserved heterogeneity between families is controlled have been received skeptically at times. We believe that our results support placing this directional relationship and the mechanisms that potentially produce it as central to further research in the area.
Proxying parental attention with maternal work hours is problematic given recent literature suggesting that working mothers do not compromise time with children as a function of their labor market participation (Bianchi 2000; Sandberg and Hofferth 2001; Sayer et al. 2004).
As a test of the validity of the standardization, children’s age was included as a covariate in all specifications presented here and in no case was this coefficient significantly different from zero (analyses not shown).