Foster care reunification: An exploration of non-linear hierarchical modeling

Children and Youth Services Review 33 (2011) 705–714

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Children and Youth Services Review j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c h i l d yo u t h

Foster care reunification: An exploration of non-linear hierarchical modeling Emily Putnam-Hornstein a,⁎, Terry V. Shaw b a b

Center for Social Services Research, School of Social Welfare, University of California, Berkeley 16 Haviland Hall, #7400, Berkeley, CA 94720, United States School of Social Work, University of Maryland Baltimore, Louis K. Kaplan Hall, 525 West Redwood Street, Baltimore, MD 21201, United States

a r t i c l e

i n f o

Article history: Received 27 August 2010 Accepted 10 November 2010 Available online 21 November 2010 Keywords: Hierarchical models Reunification Clustered data Foster care

a b s t r a c t Hierarchical or multilevel models are based on the same fundamental concepts that apply to simple linear models. The linear forms of these models can be interpreted with relative ease as parameter estimates do not differ in magnitude or interpretation from standard non-hierarchical models. Non-linear hierarchical models, however, are more complex as the introduction of a random intercept means that parameter estimates must be interpreted as “subject-specific” rather than “population-averaged”. Depending on the specifics of the data being modeled, these parameters may be very different in magnitude. In this article we provide two examples of non-linear hierarchical modeling using administrative child welfare data. For each example, we estimate the odds of reunification for a cohort of children in California using both standard logistic regression models and random intercept models. © 2010 Published by Elsevier Ltd.

1. Introduction A fundamental assumption of most statistical models is that observations included in the analysis are independent. Hierarchical data often violate this assumption: since observations nested in higher level units may not be fully independent of one another, associated standard errors are frequently too small. Although researchers have long been aware of the problems associated with treating correlated data as if they were independent, it is more recent that statistical software has evolved to a point that specifying the nested structure of data can be accomplished with relative ease. Arising from these advances are hierarchical or multilevel model classes that reflect the consideration of higher order “contextual effects”, but are based on the same fundamental concepts that apply to standard linear models. At the first level, an outcome variable is modeled as a linear function of an intercept and one or more covariates. At higher levels, the level 1 intercept and slope(s), if appropriate, are modeled as the dependent variable(s) and predicted from level 2 covariates. The level 2 parameters can then be modeled as dependent variables of level 3 covariates and so forth. The resulting parameter estimates and associated inferences are thus purged of the shared variance that defines their membership in a given cluster or level. Multilevel approaches to modeling data often prove superior to more traditional aggregation and sampling methods historically used to address dependent data (Raudenbush & Bryk, 2002). Yet, the ⁎ Corresponding author. Tel.: +1 917 282 7861; fax: +1 510 642 1895. E-mail addresses: [email protected] (E. Putnam-Hornstein), [email protected] (T.V. Shaw). 0190-7409/$ – see front matter © 2010 Published by Elsevier Ltd. doi:10.1016/j.childyouth.2010.11.010

interpretation of non-linear hierarchical models is less than straightforward. In non-linear models, frequently used for the analysis of child welfare data (Courtney & Barth, 1996; Needell, Brookhart, & Lee, 2003; Shaw, 2006), coefficient estimates derived from hierarchical methods may be more extreme than estimates obtained from a standard model. A standard logistic regression model estimates a “population-averaged” coefficient, or the logged odds of the average effect of a covariate (Rabe-Hesketh & Skrondal, 2008). In contrast, a non-linear hierarchical model estimates a “subject-specific” coefficient, or the average of each subject's logged odds. Since the mean of the log of subject-specific odds is not equal to the log of mean odds, coefficient estimates may differ in magnitude between these two models. The extent to which differences in magnitude may be observed is driven largely by the specifics of the data being modeled. Generally, the smaller the clustering units the greater the within-cluster homogeneity (or greater between-group heterogeneity) one should expect the data to exhibit. This means that small clustering units will tend towards more extreme log odds covariate effects much more so than larger clustering units. These extreme cluster-specific log odds then translate into a more extreme average log odds covariate effect estimated across all clusters. From a technical standpoint, there is nothing that positions the more extreme estimates derived from a non-linear hierarchical model as “wrong” (or those less extreme estimates derived from a standard model as “right”). The two models are merely estimating different effects (population averaged vs. subject specific). From a practical standpoint, however, relying on the hierarchical estimates of subject specific effects may be problematic. Most consumers of research interpret coefficients as estimates of a covariate's effect on an

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“average” person (all else being equal). In a non-linear hierarchical model, this widely-accepted interpretation no longer holds. This nuanced distinction may lead to interpretational difficulties for audiences unfamiliar with non-linear multilevel modeling. Yet, one could (correctly) argue that presenting unbiased estimates (or less biased estimates) should trump ease of interpretation. One of the strengths of a multilevel modeling approach is the ability to identify, and then control for, omitted variables. But if the parameter estimates do not indicate bias, either because all members of the clustering unit have the same outcome (meaning that there is not enough within-group heterogeneity to model the cluster as an additional parameter) or because the clustering unit is uncorrelated with the covariates being modeled, justifications for multilevel modeling become far less compelling. As demonstrated by the two examples in the sections that follow, the utility of multilevel modeling is determined by both the specifics of the data and the objectives of the research. In this paper, we present two examples of non-linear multilevel modeling using a cohort of children in California who entered an outof-home foster care placement for the first time in 2001. In the first analysis, we focus on children nested within families; in the second analysis we examine families nested within counties. For both analyses, we compare parameters estimated using standard logistic regression models to those derived from a series of random intercept models which explicitly model the clustering variance. In the context of these analyses, we address the following two questions: Question #1. What are the estimation consequences of modeling “family” as the clustering unit for siblings? Question #2. What are the estimation consequences of modeling “county” as the clustering unit for families? The decision was made to present only basic random intercept models with covariates commonly included in reunification models. Within-level and cross-level interactions were not explored, nor were random slopes modeled. The purpose of such an approach was to explore basic model specifications to understand the extent to which hierarchical models proved a useful fit for these data, before introducing additional complexities. Thus, results should be considered in the context of a multilevel modeling exercise. It should be assumed that these models do not fully capture early reunification dynamics. 2. Background In this article, we pursue two research questions related by their common methodological concern (i.e., the clustered nature of the data). The first question explores the estimation consequences of two methods of adjusting for the correlation of siblings within families and has broad applicability as a majority of children in foster care are joined by siblings who are also in out-of-home care (Needell et al., 2010; Wulczyn & Zimmerman, 2005). The second question surrounds the estimation impact of adjusting for familial correlation at the county level. Although this second question is focused more narrowly on the circumstances presented by California's child welfare system, it provides an illustrative example of hierarchical modeling using geography as a clustering unit. The background for each question is briefly reviewed in the sections that follow. 2.1. Within family clustering Children from the same family usually share genetic ties through at least one common parent and typically reside with each other throughout childhood. Thus, the combined influence of shared genes and a shared environment strongly suggest that siblings within

families should not be assumed independent — especially when the outcome being modeled is related to their common clustering unit, the family (Shlonsky, Webster, & Needell, 2003; Webster, Shlonsky, Shaw, & Brookhart, 2005). The non-independence of children within families is a particularly salient topic to child welfare researchers as roughly two-thirds of children in foster care have a sibling also in care (Needell et al., 2010) and there is evidence that the foster care population is increasingly comprised of “non-singleton” children (Wulczyn & Zimmerman, 2005). Traditionally, the issue of within-family dependence has either been ignored in child welfare research, or overcome by randomly selecting one sibling from a family group for inclusion in the analysis (Leathers, 2005). General estimating equations have also been employed as a means of adjusting standard errors (Webster et al., 2005; Shaw, 2006), as have traditional models with robust standard error adjustments. In the first analysis presented in this article, we examine the estimated within-family correlation (or the between-family variance) of the likelihood that an “early” (within 6 months of entry) reunification occurs, conditioned on other factors. We also examine the extent to which parameter estimates and associated inferences shift in the context of two different model types specified as follows: 1) a standard logistic regression model that utilizes the full cohort of children; and 2) a hierarchical model that also uses the full cohort of children, but employs a family-level random intercept. It should be noted that although reunifications are deemed to be a positive outcome and a preferred policy option, we place no normative value judgement on the 6 month reunification outcome as desirable or undesirable. In fact, evidence suggests that children who are reunified quickly fare worse in terms of re-entry rates than children who remain in care for longer periods of time (Courtney, 1995; Jonson-Reid, 2003; Shaw, 2006; Wells & Guo, 1999).

2.2. Within county clustering By relying on a county administered child welfare system, California finds itself in the company of only a handful of other states where supervision and administration has not been assumed by the state (California can be described as a state supervised, countyadministered, child welfare system). The California Department of Social Services is authorized to provide oversight in the management of federal funds and directs policy through state mandates, but the counties are still afforded fairly wide latitude in their work with families and children. Given California's county-administered system, differences in various performance outcomes (e.g., time to reunification, rates of adoptions) across the state's 58 counties have raised questions as to the underlying sources of this observed variability (SF Chronicle Editorial, 2006). There are many possible explanations for between county variability, including demographic differences, the manner in which resources are allocated, and distinct approaches to practice. In this article, we do not seek to explain the sources of this variability; instead we focus our attention on the degree to which county-specific dynamics may influence outcomes at the individual level. In the second analysis, we examine the estimated within-county correlation (or the between-county variance) of the likelihood that a reunification occurs within 6 months of entry, conditioned on other factors. We also consider the extent to which parameter estimates and associated inferences shift in the context of two different model types specified as follows: 1) a standard logistic regression model that utilizes a cohort of children randomly sampled from each sibling group, and 2) a hierarchical model utilizing this same random sample, but employing a county-level random intercept.

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3. Method 3.1. Data Data used for this analysis were drawn from a longitudinal extract of the California Child Welfare Services/Case Management System (CWS/CMS) housed at the Center for Social Services Research at the University of California at Berkeley's School of Social Welfare. This longitudinal database (1998–present) contains a unique record for each child who has had contact with California's child welfare system (i.e., has been alleged to have been maltreated) and then tracks that child over time through all subsequent points of contact (i.e., substantiation, entry to care, exit from care, and re-entry to care). Data were sorted and filtered using a unique child identifier variable and a constructed episode placement count variable to create a cohort of children first entering foster care in California during the 2001 calendar year. These data were then merged with a file of county level data compiled from publicly available statistical abstracts and other administrative data reports (available on request from the authors). Because a number of California's counties have extremely small populations, all county level variables reflect a rate averaged across three years of data spanning 2000–2002. The resulting dataset contained 26,963 unique children, 15,715 unique families (including single child family units), and 57 unique counties (one county did not have any first entries to foster care in 2001). 3.2. Hierarchical structure These data were conceptualized as hierarchical in nature. Although a three-level hierarchical structure certainly would have made sense – with children nested within families and these families then nested within counties – due to reasons discussed in the Limitations section, and a desire to provide illustrative examples of the more basic two-level random intercept models, two separate two-level analyses were specified instead. In the first analysis, the cohort of individual children entering foster care served as the base or first level for the hierarchical models. Because these children were nested within families, and children within a given family were assumed to be more similar and more likely to experience the same outcome than children from different families, the family unit was treated as the clustering unit. In the second analysis, a single child was randomly sampled from each family (thus removing the family as a higher level) and the clustering unit became the county in which the child was removed. The rationale for clustering at this level was based upon two assumptions. First, it was assumed that families self-select into counties and therefore families within a county share more characteristics than do families from different counties. Second, it was assumed that because California relies on a county administered child welfare service system, systematic differences in approaches to practice and local policy across counties would also influence a child's odds of reunification, independent of measured child and family covariates. 3.3. Variables The dependent variable in both analyses was a dichotomous measure of reunification within 6 months of entering foster care (1 = reunified and 0 = not reunified). Unique identifiers were established at the child-level, the family-level, and the county-level. Child and family covariates were chosen to reflect established factors associated with reunification outcomes, along with commonly employed demographic control variables (Courtney, 1994; Goerge, 1990; Webster et al., 2005; Wells & Guo, 1999). Child covariates examined in the models that follow were all coded as categorical dummy variables and included race (Black, White, Hispanic, Asian/ Pacific Islander, and Native American), gender (male and female), age at removal (infant, 1–2 years, 3–5 years, 6–10 years and 11–15 years),

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removal reason (neglect, physical abuse, sexual abuse, and other abuse), and predominant placement type (kin and non-kin). Family-level covariates included a dummy variable for instances where there was a recommendation of parent drug or alcohol treatment in the case plan (drug treatment and no drug treatment) and categorical groupings for the current count of siblings also placed in the foster care system (no placed siblings, one placed sibling, two placed siblings, and 3+ placed siblings). The count of siblings included both siblings who first entered foster care in 2001 and those who were already in foster care stemming from an earlier placement. County-level variables included the county's foster care entry rate, teen birth rate, and the percentage of the population that was Black. Foster care entry rates serve as an indicator of a county's propensity to remove children and may be related to the subsequent service needs of the out-of-home population. If one county removes many children who fall along the margin, this same county may have a higher reunification rate as the safety concerns that need to be addressed before a child returns home represent lesser hurdles. In contrast, a county that removes children in only those most serious cases of abuse or neglect may have a much more difficult population of families to work with and may experience lower reunification rates as a result. The rate of teen births was included as a proxy for county-level demographic service needs as young maternal age is a risk factor strongly associated with child maltreatment (Putnam-Hornstein & Needell, in press; Lee & Goerge, 1999; Sidebotham & Heron, 2006). Counties with higher teen birth rates may also be faced with child welfare populations with greater service needs, making quick reunifications less probable. Since Black children are more likely to be placed in foster care and less likely to reunify than children of other races (Shaw, Putnam-Hornstein, Magruder, & Needell, 2008; Wells & Guo, 1999), while also experiencing a fairly constant rate of foster care entry in both low and high poverty counties (Wulczyn & Lery, 2007), a variable capturing the percentage of the county population that was Black was also included. A number of other county level covariates were explored in earlier bivariate analyses (e.g., population density, child poverty rates) but these alternative variables demonstrated weaker predictive power and were highly correlated with the three variables described earlier. All county-level variables were averaged across three years of data (2000–2002) as a means of rate stabilization. These variables were also mean-centered and scaled to ease interpretation (Enders & Tofighi, 2007). 3.4. Programming All statistical analyses were conducted using StataSE (v.10 StatCorp, College Station, TX, USA; 2007). All hierarchical models were estimated using the GLLAMM (Generalized Linear Latent and Mixed Models) procedure with the logit link function and a binomial specification (Rabe-Hesketh, Skrondal, & Pickles, 2005). Standard logistic regression models were estimated using Stata's logistic function which fits a maximum likelihood function for binary outcomes and reports exponentiated coefficients, or odds ratios. Huber–White robust standard error adjustments were made in all standard logistic models. 3.5. Intraclass correlation To assess the between and within group variance, residual intraclass correlations were computed for all hierarchical models using the following latent-response formula for dichotomous outcomes (Rabe-Hesketh & Skrondal, 2008):


ρ ≡ Corr yijk ; yi′ jk jxijk ; xi′ jk =

φ φ + π2 = 3

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This formula relates the estimated variance between different clustering units (φ) as a function of the total variance according to a logistic distribution (φ + π2/3). The ratio of the between cluster variance to the total variation serves as a measure of the within cluster correlation. As specified above the correlation is estimated for children (i and i′) from the same family (j) (and thus the same county (k)), conditioned on a vector of covariates (x). Similarly, this same formula was used to estimate the within county correlation of families in the second analysis by relating the estimated variance between counties to the total variance observed. In the second analysis, the captured correlation is between children from different families (j and j′) since a single child was randomly sampled from each family. 3.6. Models Eight models were iteratively specified and examined in the context of each of the two analyses. The first analysis utilized the full entry cohort of children (26,963); the second analysis utilized a sample based on a single child randomly selected from each sibling group. 3.6.1. Standard logistic regression models Model 1. Unconditional logistic regression model. Model 2. Logistic regression model conditioned on child covariates. Model 3. Logistic regression model conditioned on child and family covariates. Model 4. Logistic regression model conditioned on child, family, and county covariates. The final (full) model specification (Model 4) using standard logistic regression takes the following form: logit fPrðy = 1jX Þg = β0 + β1 x + … + β20 x In this model, X ≡ (x1,…,x20)′ is a vector containing child, family, and country covariates. 3.6.2. Random intercept logistic regression models Model 5. Unconditional logistic regression model with random intercept. Model 6. Logistic regression model with random intercept, conditioned on child covariates. Model 7. Logistic regression model with random intercept, conditioned on child and family covariates. Model 8. Logistic regression model with random intercept, conditioned on child, family, and county covariates.

4. Results and discussion 4.1. Descriptive statistics Three tables of descriptive statistics are provided. Table 1 presents the full cohort of children first entering care (n = 26,963), along with the subset of children who were reunified within 6 months of entry (n=6,492; 24% of the cohort). Hispanic children were slightly less likely to have reunified within 6 months than suggested by their presence among all first entries. Infants made-up 20.8% of the full cohort and yet just 15.5% of children who reunified. Children removed for reasons of neglect were less likely to have reunified than children removed for sexual or physical abuse. Children placed with kin were less likely than those placed with non-kin to have reunified within 6 months. Descriptive statistics for family covariates are presented in Table 2. There were 15,715 family units included in these data, slightly more than half of which were comprised of multiple children placed in foster care (i.e., a family unit for the purposes of this analysis could be a single child). An indication of parental drug or alcohol use was negatively associated with a reunification within 6 months of entry. Children without any placed siblings were overrepresented among reunifications; children with two or more siblings also placed in foster care were underrepresented. Finally, county-level covariates are reported in Table 3. The mean foster care entry rate was 3.8 children per 1000 with a standard deviation of 2. The percentage of live births to teenage mothers was 11% and ranged from a low of 0% to a high of 19%. The percentage of the total county population that was Black ranged from 0.2% to 16%, with a mean of 3.3.

4.2. Sample comparisons Table 4 presents proportions and unadjusted odds ratios for both the full cohort of children entering foster care in California in 2001 (used in the first analysis) and children randomly sampled from each family unit in this entry cohort (used in the second analysis). Only modest differences in the proportion of children reunified and the corresponding magnitudes of the crude odds ratios are observed between the full cohort and the sample based on a randomly selected child.

Table 1 Characteristics of children first entering care in 2001 and reunified within 6 m of entry.

The final (full) model specification (Model 8) is shown as follows: n
o logit Pr yij = 1jXij ; ςj = β0 + β1 xij + … + β20 xj + ςj In this model, Xij ≡ (x1ij,…,x20j)′ is a vector containing child, family, and country covariates, while ςj ∼ N(0,φ) is a random intercept varying over clusters (j), with cluster j fixed for each family in analysis 1 and for each county in analysis 2. In each of the random intercept models (Models 5 through 8), the assumption of independent observations within the clustering unit was relaxed through the inclusion of a family-specific (analysis 1) or county-specific (analysis 2) random intercept (ςj). This random intercept was assumed to be normally distributed with a mean of zero and a variance of φ. Covariates at the child, family, and county level were added iteratively to observe their impact on the respective “between family” and “between county” variance estimates in the two analyses.

Race/ethnicity

Gender Age at entry

Removal reason

Placement type

Hispanic Black White Asian/Pacific Islander Native American Female Male b 1 year 1–2 years 3–5 years 6–10 years 11–15 years Neglect Physical abuse Sexual abuse Other abuse Non-kin Kin

First entries (n = 26,963)

Reunified (n = 6492)

n

%

n

%

11,626 5293 8767 853 319 13,706 13,255 5601 4007 4694 6876 5785 19,822 4018 1715 1405 16,539 10,424

43.3 19.7 32.6 3.2 1.2 50.8 49.2 20.8 14.9 17.4 25.5 21.5 73.5 14.9 6.4 5.2 61.3 38.7

2614 1189 2326 275 88 3343 3229 1020 930 1132 1863 1628 4372 1301 600 300 5105 1468

40.3 18.3 35.8 4.2 1.4 50.9 49.1 15.5 14.2 17.2 28.3 24.8 66.5 19.8 9.1 4.6 77.7 22.3

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Table 2 Characteristics of families with children first entering care in 2001 and reunified within 6 m of entry. Reunifieda (n = 3566)

First entries (n = 15,715)

Substance use Number of placed siblings

a b

None Drugs/alcohol None One sibling Two siblings Three + siblings

Mixed outcomeb (n = 558)

n

%

n

%

n

%

7358 8357 7552 3728 2359 2076

46.8 53.2 48.1 23.7 15.0 13.2

2147 1419 1986 824 449 307

60.2% 39.8% 55.7% 23.1% 12.6% 8.6%

246 312 – 165 197 196

44.1 55.9 29.6 35.3 35.1

All children were reunified if a multichild family. Mixed reunification outcome among multichild families.

4.3. Models: Analysis 1 Models specified for the first analysis are presented in Table 5. The goal of the first analysis was to address the question: What are the estimation consequences of modeling “family” as the clustering unit for siblings? All eight models utilized the full sample of children (n = 26,963) with Models 1 through 4 adjusted for potential withinfamily dependence through the use of robust standard error adjustments while Models 5 through 8 introduced a family-specific random intercept to explicitly account for this clustering unit. 4.3.1. Modeling results The first thing to note is that the odds ratios estimated in the random intercept models are notably more extreme (i.e., further from 1) than those estimated using the standard logistic regression models. This difference is observed because the standard logistic models are estimating the population-averaged (or marginal) odds of reunification while the random intercept models are estimating the subjectspecific (or conditional) odds of reunification (Rabe Hesketh & Skrondal, 2008). As noted earlier, the differences observed in the covariate values between the standard logistic models and the random intercept logistic models are due to the fact that in the random intercept models the logit coefficients (from which the odds ratios were computed) reflect the average of each family's logged odds of reunification, whereas in the standard models, the logit coefficients reflect the logged odds of the average of all children's reunification odds. Since “the average of a non-linear function is not the same as the nonlinear function of the average” (Rabe-Hesketh & Skrondal, 2008, p. 255), differences are observed in the magnitudes of the odds ratios between the two model types. Another difference between the standard logistic regression models with robust standard errors and the random intercept models is that the former merely adjusts for the possible dependence of observations while the latter provides an additional parameter estimate that can be used to derive an estimate of the within family correlation. In the unconditional random intercept model (Model 5), the between family variance is estimated to be 85.45 while the total residual variance amounts to the sum of the between family variance and the variance of an ordinary logistic regression model (approximately 3.29). This

Table 3 County characteristics of children first entering care in 2001. Supervising counties (n = 57)

Foster care entry rate (per 1000 children) Percentage of live births to teen mothers Percentage population black

Mean

Std dev

Min

Max

3.8 11.0 3.3

2.0 4.1 4.0

1.0 0.0 0.2

9.7 19.4 16.2

translates into an exceptionally high residual intraclass (or within family) correlation of 0.96, meaning that 96% of the residual variance in reunification outcomes at 6 months was estimated to be between families, while only 4% was observed within families. In Model 6, all of the child covariates were introduced. Although the inclusion of these variables decreased the residual variation in reunification outcomes (reducing the between family variance by nearly 25%), the between family variation in reunification outcomes dropped by only one percentage point (to 95% of the total variation) since the magnitude of the between family variation was still so large relative to the variance associated with a logit distribution. The introduction of family and county covariates in Models 7 and 8 resulted in insignificant shifts to the estimates of residual variance. 4.3.2. Modeling discussion All told, the exceptionally high within family uniformity of reunification outcomes (i.e., either all children in a sibling group were reunified or none were) suggests that modeling the family unit as a random intercept introduces unjustified complexities, trading off ease of interpretation while getting little in return. In other words, it would appear that there is little residual variation to be captured within families when the outcome of interest is reunification within 6 months. This finding is also consistent with what is arguably the most pragmatic practice approach to reunifications: the home environment is either deemed safe enough for all, or none, of the children within a family. In terms of the child, family, and county covariates, several significant relationships emerged that proved robust across all eight models specified. It is important to note that the sheer size of this dataset means that qualitatively unimportant differences may well appear statistically significant (Gelman & Stern, 2006). Therefore, we focus on those odds ratios that are 1) large in magnitude, 2) estimated differently in the context of the standard logistic model versus the random intercept models, or 3) exhibit a demonstrable shift in upon the inclusion of additional variables in the model. The findings in this first analysis are largely consistent with prior reunification research (Courtney, 1994; Webster et al., 2005; Wells & Guo, 1999). When only child covariates were included, the subject-specific odds of reunification within 6 months for a child removed for physical or sexual abuse were significantly greater than those for a child removed for neglect. Although the populationaveraged odds were far lower in the comparable traditional logistic regression model, the same patterns emerged across both specifications. The heightened odds of early reunification for abused children versus neglected children decreased with the inclusion of family covariates, but remained fairly constant in magnitude with the inclusion of county covariates and were highly significant across all models. As was the case with removal reason, similar age patterns emerged across the standard and the random intercept models. The general

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Table 4 Full cohort versus a sampled child: comparisons of distributions and crude odds ratios. Full cohort (n = 26,963)

Race/ethnicity

Gender Age at entry

Removal reason

Placement type Substance use Number of placed siblings

Hispanic (referent) Black White Asian/Pacific Islander Native American Female (referent) Male b 1 year (referent) 1–2 years 3–5 years 6–10 years 11–15 years Neglect (referent) Physical abuse Sexual abuse Other abuse Non-kin (referent) Kin None (referent) Drugs/alcohol None (referent) One sibling Two siblings Three + siblings

One child random sample (n = 15,715)

% Reunified

Crude OR (95% CI)

% Reunified

Crude OR (95% CI)

22.5 22.5 26.5 32.2 27.6 24.4 24.4 18.2 23.2 24.1 27.1 28.1 22.1 32.4 35.0 21.3 30.9 14.1 31.4 18.7 26.3 24.9 24.4 21.8

– 1.00 1.24 1.64 1.31 – 1.00 – 1.36 1.43 1.67 1.76 – 1.69 1.90 0.96 – 0.37 – 0.50 – 0.93 0.90 0.78

22.6 21.9 26.5 32.5 27.3 24.4 24.5 18.1 23.5 26.3 27.1 29.4 22.1 33.5 36.1 21.5 30.7 14.0 31.0 18.7 26.3 24.1 23.5 19.6

– 0.95 1.23 1.64 1.28 – 1.00 – 1.39 1.61 1.68 1.88 – 1.78 2.01 0.96 – 0.37 – 0.51 – 0.89 0.86 0.68

trend is one of heightened odds of reunification with increasing age (compared with infants) through the age of 10. Odds ratios for all age groups compared to infants were reduced in the random intercept models upon the inclusion of family covariates, but increased (slightly) in the standard logistic regression models. The odds of reunification within 6 months were significantly lower for children placed with kin than for children placed with non-kin. Significantly lowered odds were observed across both the standard and random intercept models and no changes in the estimates were observed within model-type upon the inclusion of higher order covariates. No significant gender differences emerged in any of the models. An indication of parental substance abuse significantly reduced the odds of reunification in both the standard and random intercept specifications. Increasing sibling group size demonstrated a directionally consistent pattern (i.e., the more children in the family, the lower the odds of reunification), but only those children with 3 or more siblings placed in care exhibited significantly reduced odds of reunification compared to singleton children. Two of the three county covariates were found to be significantly associated with 6 month reunifications. County teen birth rates were negatively related to the odds of reunification. In contrast, countylevel entry rates were associated with significantly increased odds of reunification within 6 months. This finding is consistent with arguments that front-end removal practices may significantly impact subsequent outcomes. Although Black children were consistently less likely than Hispanic children (the reference group in these models) to reunify within 6 months, it was only upon the inclusion of the county covariates that this relationship emerged as significant in either the random intercept or standard logistic regression models. Meanwhile, the odds that a White child reunified within 6 months were no longer significantly different than Hispanic children upon the addition of county covariates. Further exploration is needed, yet this finding is consistent with other recent research suggesting that the contextual effects of the county of removal may be related to foster care outcomes, and may differ by race (Drake, Lee, & Jonson-Reid, 2009; Wulczyn & Lery, 2008).

(n.s.) (1.17–1.33) (1.41–1.91) (1.02–1.68) (n.s.) (1.23–1.50) (1.30–1.57) (1.53–1.82) (1.61–1.92) (1.57–1.82) (1.71–2.11) (n.s) (0.34–0.39) (0.48–0.53) (0.86–0.99) (0.83–0.98) (0.72–0.84)

(n.s.) (1.13–1.34) (1.35–1.99) (0.92–1.79) (n.s.) (1.22–1.56) (1.43–1.82) (1.51–1.88) (1.69–2.09) (1.62–1.96) (1.74–2.33) (n.s.) (0.34–0.40) (0.47–0.55) (0.81–0.97) (0.77–0.96) (0.60–0.77)

4.4. Models: Analysis 2 Models specified for the second analysis are presented in Table 6. The goal of this analysis was to address the question: What are the estimation consequences of modeling “county” as the clustering unit? All eight models in this analysis relied on a randomly sampled child from each family (n = 15,715). Models 1 through 4 adjusted for potential within-county dependence through the use of robust standard error adjustments; a county-specific random intercept was introduced in Models 5 through 8. 4.4.1. Modeling results As already discussed in the context of the first analysis, parameter estimates must be interpreted as population-averaged in the standard logistic regression models (Models 1–4), but as subject-specific in the random intercept models (Models 5–8). Yet, while the odds ratios estimated in the random intercept models were demonstrably more extreme than those estimated in the standard logistic models for the first analysis, the difference was far less pronounced (even negligible) in this second analysis. The reason lies in the differing higher level clustering units that were explicitly modeled in the context of these two analyses. In this second analysis the clustering unit was far larger and the total variance was much closer to the fixed variance of a logit distribution. As a result, the subject-specific estimates were much closer to the marginal or population-averaged estimates. Whereas a very high between family variance (and therefore within family correlation) was observed in the first analysis, the magnitude of the between county variance was far smaller. In the unconditional model the between county variance was estimated to be 0.279 and the within county correlation estimated at 0.078. This means that before conditioning on any covariates, the observed correlation of reunification outcomes within county estimated to be 7.8%. This within-county correlation changed little upon the addition of the child-level covariates, while the inclusion of family-level covariates increased the between county variance and thus the within county correlation to 8.4%. This increase in the within-county correlation upon inclusion of family-level covariates is consistent

Table 5 Analysis 1 models (fixed estimates reported as odds ratios with 95% confidence intervals). Standard logistic regression models (n = 26,963)

Random Between family variance Total variance Within family correlation Log likelihood a b c

Model 1: Unconditional

Model 2: Child covariates

Model 3: Family covariates

Model 4: County covariates

Model 5: Unconditional

Model 6: Child covariates

Model 7: Family covariates

Model 8: County covariates

– – – – – – – – – – – – –

0.99 1.22 1.37 1.46 1.03 1.40 1.43 1.57 1.45 1.59 1.64 0.91 0.37

0.98 1.21 1.49 1.31 1.03 1.45 1.50 1.59 1.37 1.40 1.41 0.82 0.39

(n.s.) (1.13–1.30) (1.15–1.92) (1.13–1.53) (n.s.) (1.31–1.61) (1.36–1.66) (1.45–1.75) (1.25–1.51) (1.30–1.52) (1.26–1.58) (0.71–0.94) (0.37–0.42)

0.90 1.03 1.02 1.22 1.04 1.32 1.35 1.42 1.22 1.41 1.42 0.82 0.39

(0.83–0.98) (n.s.) (n.s.) (1.04–1.43) (n.s.) (1.16–1.51) (1.19–1.55) (1.24–1.62) (1.06–1.39) (1.30–1.53) (1.27–1.60) (0.71–0.94) (0.37–0.42)

– – – – – – – – – – – – –

0.80 1.98 1.08 3.90 1.09 3.25 3.36 4.08 2.95 6.72 6.82 1.12 0.02

0.75 1.92 1.30 2.50 1.08 2.27 2.29 2.54 1.53 4.16 4.04 0.77 0.02

(n.s.) (1.33–2.80) (n.s.) (n.s.) (n.s.) (1.48–3.49) (1.48–3.54) (1.59–4.06) (n.s.) (2.54–6.83) (1.99–8.20) (n.s.) (0.01–0.04)

0.56 1.18 0.53 1.83 1.11 2.34 2.29 2.52 1.49 4.13 4.17 0.74 0.03

(0.35–0.89) (n.s.) (n.s.) (n.s.) (n.s.) (1.52–3.59) (1.48–3.55) (1.57–4.02) (n.s.) (2.54–6.72) (2.09–8.33) (n.s.) (0.02–0.04)

– – – –

– – – –

0.60 0.98 0.94 0.83

(0.57–0.64) (n.s.) (n.s.) (0.76–0.91)

0.56 0.96 0.93 0.82

(0.52–0.59) (n.s.) (n.s.) (0.75–0.90)

– – – –

– – – –

0.12 0.75 0.62 0.27

(0.08–0.17) (n.s.) (n.s.) (0.15–0.47)

0.09 0.73 0.59 0.29

(0.06–0.13) (n.s.) (n.s.) (0.16–0.49)

– – –

– – –

– – –

1.47 (1.42–1.51) 0.91 (0.90–0.92) 0.99 (n.s.)

– – –

– – –

– – –

5.20 (4.16–6.50) 0.63 (0.58–0.68) 0.95 (n.s.)

– – – −14,975.15

– – – −14,730.85

– – – −14,084.51

– – – −13,730.33

85.45 88.74 0.96 −11,091.30

64.28 67.57 0.95 −10,616.34

64.35 67.64 0.95 −10,507.39

61.17 64.46 0.95 −10,304.33

(n.s.) (1.14–1.31) (1.07–1.76) (1.26–1.70) (n.s.) (1.27–1.55) (1.22–1.48) (1.43–1.71) (1.32–1.60) (1.47–1.71) (1.47–1.84) (n.s.) (0.35–0.39)

(n.s.) (1.36–2.90) (n.s.) (1.46–10.34) (n.s.) (2.24–4.71) (2.33–4.84) (2.79–5.98) (1.94–4.49) (4.07–11.09) (3.28–14.19) (n.s.) (0.01–0.03)

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Fixed Child level covariates Blacka Whitea Native Americana Asian/Pacific Islandera Male 1–2 yearsb 3–5 yearsb 6–10 yearsb 11–15 yearsb Physical abusec Sexual abusec Other abusec Kinship placement Family level covariates Substance use Child has one placed sibling Child has two placed siblings Child has 3+ placed siblings County level covariates Entry rate (unit = 1 per 1000) Teen births (unit = 5 pctage pts) Pop. Black (unit = 5 pctage pts)

Family-level random intercept models (n = 26,963)

Reference group: Hispanic. Reference group: infants. Reference group: neglect.

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712

Table 6 Analysis 2 models (fixed estimates reported as odds ratios with 95% confidence intervals). Standard logistic regression models (n = 15,715)

County-level random intercept models (n = 15,715)

Model 2: Child covariates

Model 3: Family covariates:

Model 4: County covariates

Model 5: Unconditional

Model 6: Child covariates

Model 7: Family covariates

Model 8: County covariates

– – – – – – – – – – – – –

0.95 1.17 1.32 1.41 1.05 1.47 1.62 1.56 1.51 1.63 1.70 0.90 0.37

0.93 1.17 1.35 1.28 1.05 1.49 1.65 1.56 1.36 1.47 1.52 0.82 0.39

(n.s.) (1.07–1.28) (n.s.) (1.05–1.56) (n.s.) (1.31–1.70) (1.45–1.87) (1.39–1.75) (1.21–1.52) (1.33–1.63) (1.30–1.77) (0.68–0.97) (0.36–0.42)

0.86 1.03 1.00 1.18 1.05 1.51 1.66 1.56 1.34 1.48 1.54 0.81 0.39

(0.77–0.95) (n.s.) (n.s.) (n.s.) (n.s.) (1.32–1.72) (1.46–1.89) (1.38–1.75) (1.19–1.50) (1.33–1.64) (1.31–1.80) (0.67–0.96) (0.36–0.43)

– – – – – – – – – – – – –

0.81 0.92 0.90 1.27 1.05 1.46 1.56 1.57 1.41 1.69 1.83 0.99 0.38

0.80 (0.71–0.89) 0.93 (n.s.) 0.94 (n.s.) 1.14 (n.s.) 1.05 (n.s.) 1.33 (1.08–1.62) 1.42 (1.16–1.74) 1.39 (1.12–1.70) 1.12 (n.s.) 1.50 (1.34–1.67) 1.59 (1.36–1.87) 0.89 (n.s.) 0.41 (0.37–0.45)

0.79 0.92 0.91 1.14 1.04 1.33 1.42 1.39 1.12 1.50 1.59 0.88 0.41

(0.71–0.88) (n.s.) (n.s.) (n.s.) (n.s.) (1.09–1.63) (1.16–1.74) (1.12–1.71) (n.s.) (1.34–1.67) (1.36–1.87) (n.s.) (0.37–0.44)

(0.58–0.68) (n.s.) (n.s.) (0.66–0.86)

0.59 0.92 0.89 0.76

(0.54–0.64) (n.s.) (n.s.) (0.66–0.86)

– – – –

– – – –

0.57 (0.52–0.62) 0.98 (n.s.) 0.92 (n.s.) 0.79 (0.69–0.90)

0.56 0.97 0.92 0.79

(0.52–0.61) (n.s.) (n.s.) (0.69–0.90)

Fixed Child level covariates Blacka Whitea Native Americana Asian/Pacific Islandera Male 1–2 yearsb 3–5 yearsb 6–10 yearsb 11–15 yearsb Physical abusec Sexual abusec Other abusec Kinship placement Family level covariates Substance use Child has one placed sibling Child has two placed siblings Child has 3+ placed siblings County level covariates Entry rate (unit = 1 per 1000) Teen births (unit = 5 pctage pts) Pop. Black (unit = 5 pctage pts)

– – – –

– – – –

0.63 0.93 0.91 0.76

– – –

– – –

– – –

1.42 (1.37–1.48) 0.90 (0.89–0.92) 0.99 (n.s.)

– – –

– – –

– – –

1.21 (1.10–1.33) 0.94 (0.90–0.98) 1.00 (n.s.)

Random Between county variance Total variance Within county correlation Log likelihood

– – – −8745.02

– – – −8301.60

– – – −8216.82

– – – −8031.80

0.28 3.57 0.078 −8282.04

0.28 3.85 0.078 −7899.57

0.30 3.59 0.084 −7795.18

0.22 3.51 0.071 −7787.94

a b c

Reference group: Hispanic. Reference group: infants. Reference group: neglect.

(n.s.) (1.07–1.28) (n.s.) (1.15–1.71) (n.s.) (1.30–1.67) (1.43–1.84) (1.39–1.75) (1.35–1.69) (1.47–1.81) (1.46–1.99) (n.s.) (0.34–0.40)

(0.75–0.91) (n.s.) (n.s.) (1.03–1.57) (n.s.) (1.28–1.66) (1.37–1.77) (1.39–1.76) (1.26–1.59) (1.52–1.88) (1.56–2.14) (n.s.) (0.35–0.42)

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Model 1: Unconditional

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with the assumption that families of differing demographics selfselect into counties. Finally, the addition of the three county level covariates (entry rate, teen birth rate, and Black population rate) led to the absorption of some of the residual differences observed between counties, causing the residual intraclass correlation to decrease from 8.4% to 7.1%. 4.4.2. Modeling discussion Although there is no perfect – or even widely adopted – heuristic for determining whether or not the within-cluster correlation necessitates an adjustment, if clustering units are sufficiently large, even a small degree of correlation can lead to biased estimates and inference (Guo, 2005; Kerry & Bland, 1998a,b; Ukoumunne, Gulliford, & Chinn, 2002). While the 7–8% within-county correlation appears superficially small (especially when compared with the high level of within-family correlation) this is still a notable correlation at the county level, which is a relatively high level of geographic clustering. It also bears mentioning that we conducted earlier analyses modeling reunification within 24 months, rather than 6 months, of entry. In these 24 month models the estimated within-county correlation was nearly zero. We attributed this to federal and state mandates to work quickly towards reunifications, with a two-year time frame presenting an ample window for counties to achieve this goal with relative uniformity, after adjusting for other child and family differences. The findings in this analysis suggest, however, that there may exist systematic county differences when it comes to early reunifications (within 6 months of an entry). As was the case in the first analysis, many of the same trends emerged across the standard logistic regression and the random intercept models. And, although the random intercept models estimated in the first analysis and the second analysis were conditioned on different clustering units (i.e., the family versus the county), covariate associations were generally robust to these two specifications. Controlling for other factors, children removed for reasons of physical or sexual abuse experienced heightened odds of reunification within 6 months compared with children removed for neglect. As was observed in the first analysis, these odds ratios dropped in magnitude upon the inclusion of family covariates but were not impacted by the introduction of county variables. Children placed with kin were less likely to reunify within 6 months than were children placed with non-kin. No significant gender differences were observed. An indication of parental substance use was associated with a 35–40% decreased odds of reunification across all models. Older children generally experienced increased odds of reunification within 6 months versus their infant counterparts. Across all models, children with only one or two placed siblings were no less likely to reunify than were singleton children, but children with three or more placed siblings faced a 21–24% decreased odds of reunification compared with children who had no siblings placed in foster care. In this second analysis, Black children were found to have significantly reduced odds of reunification compared with Hispanic children only when either county covariates were added in the context of the standard logistic model (Model 4) or the county was explicitly modeled as a random intercept (Models 6, 7, and 8). Although raising more questions than can be answered by this analysis, this finding suggests that county-level contextual effects may influence estimates of child-level racial dynamics. 4.5. Limitations Several limitations bear mentioning. The first involves the manner in which siblings were coded to construct family units. There are several different classification schemes which could have been used to define families in these data. Siblings could have been identified by the unique presence of a common mother, a common father, or a

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common home removal address. In this analysis, however, child links were made: if a sibling link was identified between child A and child B (as coded in the data upon entry by the caseworker), and then child B was also found to share a sibling link with child C, then child A and child C were linked as siblings. Although prior research involving these data has established this as a strong means to the identification of sibling groups (Lery, Shaw, & Magruder, 2005), certainly arguments can be made for using another means of classification. With that said, the strong within-family correlation found in the first analysis lends support to this method. A second limitation surrounds our ability to model these data fully. Ideally, these data would have been examined as a three-level hierarchical model structured with siblings nested within families and families nested within counties. While such a model is theoretically possible (Bryk & Raudenbush, 1988; Rabe-Hesketh & Skrondal, 2008), the binary nature of the outcome variable, the extremely high level of correlation within families (0.95–0.96), and the small cluster size of family units (median family size: 2 children) created numerous computational hurdles. The program we used (GLLAMM) relies on numerical integration by adaptive Guass–Hermite quadrature which has been shown to perform well under a number of modeling conditions (Skrondal & Rabe-Hesketh, 2003). This method seeks to achieve the maximized likelihood estimate by adaptively adjusting its future estimates based on past observations. However, a very high residual intraclass correlation in the context of a small clustering unit, as was encountered in the first analysis, is known to cause estimation problems (Rabe-Hesketh et al., 2005). Although all of the models reported here successfully converged, the run-time was quite lengthy (several days for each of the random intercept models). Despite multiple adjustments to the number of integration points used, we were unable to successfully model these data using both a random family intercept and a random county intercept. Finally, it should (again) be noted that the models specified are not well-positioned to fully explore the child-, family-, and county-level dynamics underlying reunifications within 6 months. Despite this, as summarized in the section that follows, several important points emerged regarding the use of random intercepts to model two commonly encountered clustering units in child welfare research: the family and the county. 5. Conclusion In child welfare research, the greatest contributors to reunification and a variety of other outcomes of interest are factors observed at the level of the individual child and his or her family. Yet the practice and policy context defined by the case worker, supervising county, or even state, may also exert important influences. Since no social science model will be perfectly deterministic (especially when it includes higher level units in which the influences are unlikely to be precisely captured) the unexplained contextual-level variation, or random residual variability, will often remain unmeasured and show-up in the error terms of lower-level units. In turn, lower-level units found in the same cluster are likely to have correlated errors, violating a key assumption of most statistical models. In this paper, we modeled reunifications within 6 months in the context of two separate analyses. In the first analysis we captured the unobserved family influence as a random family-level intercept; in the second analysis the unobserved county influence was modeled with a random county-level intercept. Three main points emerged from this research. First, while hierarchical modeling has many useful applications, modeling within-family correlations in the context of child welfare outcomes may not be one of them. In other words, “within family” variance may be a questionable parameter to estimate in light of the relative uniformity of reunification (and other) outcomes that are often observed within families.

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Second, these analyses highlight the potential for parameters of notably different magnitudes to be estimated by standard logistic regression models when compared with non-linear hierarchical models. Since no such differences (in either magnitude or interpretation) occur in linear models, many disciplines which rely primarily on hierarchical linear modeling have been able to avoid the potential confusions that arise in multilevel logistic models. Yet, because child and family research questions are often posed in terms of dichotomous outcome variables, this is an infrequently discussed topic that must be better understood by the field. Third, at least in California, the self-selection of families into counties, coupled with different contextual environments found across counties, lead to notable within-county correlations. This finding suggests that accounting for this level of geography may be of potential importance for understanding individual-level reunification outcomes. Acknowledgements The authors would like to thank Barbara Needell, Jane Mauldon, and Sophia Rabe-Hesketh, as well as hierarchical modeling groups at both the University of California at Berkeley and the University of Maryland, for providing thoughtful critiques and helpful suggestions on earlier drafts of this article. Additionally, the authors would like to acknowledge colleagues at the California Department of Social Services and the Center for Social Services for their contribution to data programming underlying these analyses. References Bryk, A. S., & Raudenbush, S. W. (1988). Toward a more appropriate conceptualization of research on school effects: A three-level hierarchical linear model. American Journal of Education, 97(1), 65−108. Courtney, M. (1994). Factors associated with reunification of foster children with their families. The Social Service Review, 68(1), 81−108. Courtney, M. (1995). Reentry to foster care of children returned to their families. Social Services Review, 69, 228−241. Courtney, M., & Barth, R. (1996). Pathways of older adolescents out of foster care: Implications for independent living services. Social Work, 41(75–83). Drake, B., Lee, S. M., & Jonson-Reid, M. (2009). Race and child maltreatment: Are Blacks overrepresented? Children and Youth Services Review, 31, 309−316. Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12(2), 121−138. Gelman, A., & Stern, H. (2006). The difference between “Significant” and “Not Significant” is not itself statistically significant. American Statistician, 60(4), 328−331. Goerge, R. M. (1990). The reunification process in substitute care. The Social Service Review, 64(3), 422−457. Guo, S. (2005). Analyzing grouped data with hierarchical linear modeling. Children and Youth Services Review, 27, 637−652.

Hu, F. B., Goldberg, J., Hedeker, D., Flay, B. R., & Pentz, M. A. (1998). Comparison of population-averaged and subject-specific approaches for analyzing repeated binary outcomes. American Journal of Epidemiology, 147(7), 694−703. Jonson-Reid, M. (2003). Foster care and future risk of maltreatment. Children and Youth Services Review, 25(4), 271−294. Kerry, S. M., & Bland, J. M. (1998a). The intracluster correlation coefficient in cluster randomisation. BMJ (British Medical Journal), 316, 1455. Kerry, S. M., & Bland, J. M. (1998b). Sample size in cluster randomisation. BMJ (British Medical Journal), 316, 549. Leathers, S. J. (2005). Separation from siblings: Associations with placement adaptation and outcomes among adolescents in long-term foster care. Children and Youth Services Review, 27(7), 793−819. Lee, B. J., & Goerge, R. (1999). Poverty, early childbearing, and child maltreatment: A multinomial analysis. Children and Youth Services Review, 21(9/10), 755−780. Lery, B., Shaw, T. V., & Magruder, J. (2005). Using administrative child welfare data to identify sibling groups. Children and Youth Services Review, 27, 783−791. Needell, B., Brookhart, M. A., & Lee, S. (2003). Black children and foster care placement in California. Children and Youth Services Review, 25(5), 393−408. Needell, B., Webster, D., Armijo, M., Lee, S., Dawson, W., Magruder, J., et al. (2010). Child Welfare Services Reports for California, retrieved May 15, 2010, from The University of California at Berkeley Center for Social Services Research website. URL:. http://cssr.berkeley.edu/ucb_childwelfare Putnam-Hornstein, E., Needell, B., in press. Predictors of child welfare contact between birth and age five: An examination of California's 2002 birth cohort. Children and Youth Services Review. Rabe-Hesketh, S., & Skrondal, A. (2008). Multilevel and longitudinal modelling using Stata (2nd ed.). College Station, TX: Stata Press. Rabe-Hesketh, S., Skrondal, A., & Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128(2), 301−323. Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (2nd ed.). Thousand Oaks, CA: Sage Publications, Inc. SF Chronicle Editorial (March 8, 2006). Accidents of geography. San Francisco Chronicle. Shaw, T. (2006). Reentry into the foster care system after reunification. Children and Youth Services Review, 28(11), 1375−1390. Shaw, T., Putnam-Hornstein, E., Magruder, J., & Needell, B. (2008). Measuring racial disparity in child welfare. Child Welfare, 87(2), 23−36. Shlonsky, A., Webster, D., & Needell, B. (2003). The ties that bind: A cross-sectional analysis of siblings in foster care. Journal of Social Service Research, 29(3), 27−52. Sidebotham, P., & Heron, J. (2006). Child maltreatment in the “children of the nineties”: A cohort study of risk factors. Child Abuse & Neglect, 30, 497−522. Skrondal, A., & Rabe-Hesketh, S. (2003). Some applications of generalized linear latent and mixed models in epidemiology. Norsk Epidemiologi, 13(2), 265−278. Ukoumunne, O. C., Gulliford, M. C., & Chinn, S. (2002). A note on the variance inflation factor for determining sample size in cluster randomized trials. The Statistician, 51 (4), 479−484. Webster, D., Shlonsky, A., Shaw, T. V., & Brookhart, M. A. (2005). The ties that bind II: Reunification for siblings in out-of-home care using a statistical technique for examining non-independent observations. Children and Youth Services Review, 27, 765−782. Wells, K., & Guo, S. (1999). Reunification and reentry of foster children. Children and Youth Services Review, 21(4), 273−294. Wulczyn, F., & Lery, B. (2007). Racial disparity in admissions to foster care. : Chapin Hall Center for Children at the University of Chicago. Wulczyn, F., & Lery, B. (2008). Social context and racial disparities in foster care admission. Chapin Hall Center for Children: University of Chicago. Wulczyn, F., & Zimmerman, E. (2005). Sibling placements in longitudinal perspective. Children and Youth Services Review, 27, 741−763.