Neighborhoods, Communities and Child Maltreatment: A Global Perspective (Child Maltreatment, 15) 3030930955, 9783030930950

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Child Maltreatment: Contemporary Issues in Research and Policy 15

Kathryn Maguire-Jack Carmit Katz   Editors

Neighborhoods, Communities and Child Maltreatment A Global Perspective

Child Maltreatment Contemporary Issues in Research and Policy Volume 15

Series Editors Jill E. Korbin, Department of Anthropology, Case Western Reserve University, Cleveland, OH, USA Richard D. Krugman, University of Colorado School of Medicine, Aurora, CO, USA

This series provides a high-quality, cutting edge, and comprehensive source offering the current best knowledge on child maltreatment from multidisciplinary and multicultural perspectives. It consists of a core handbook that is followed by two or three edited volumes of original contributions per year. The core handbook will present a comprehensive view of the field. Each chapter will summarize current knowledge and suggest future directions in a specific area. It will also highlight controversial and contested issues in that area, thus moving the field forward. The handbook will be updated every five years. The edited volumes will focus on critical issues in the field from basic biology and neuroscience to practice and policy. Both the handbook and edited volumes will involve creative thinking about moving the field forward and will not be a recitation of past research. Both will also take multidisciplinary, multicultural and mixed methods approaches.

More information about this series at http://link.springer.com/series/8863

Kathryn Maguire-Jack • Carmit Katz Editors

Neighborhoods, Communities and Child Maltreatment A Global Perspective

Editors Kathryn Maguire-Jack School of Social Work University of Michigan Ann Arbor, MI, USA

Carmit Katz The Bob Shapell School of Social Work Tel Aviv University Tel Aviv, Israel

ISSN 2211-9701 ISSN 2211-971X (electronic) Child Maltreatment ISBN 978-3-030-93095-0 ISBN 978-3-030-93096-7 (eBook) https://doi.org/10.1007/978-3-030-93096-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Acknowledgment

The editors graciously acknowledge Dr. Jill Korbin and Dr. Richard Krugman. Thank you for your support and encouragement to complete this book. Your work inspired our interest and passion in understanding the role of neighborhoods in child maltreatment.

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Communities’ Essential Role in Protecting Children from Maltreatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carmit Katz and Kathryn Maguire-Jack

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The Spatio-temporal Epidemiology of Child Maltreatment: Using Bayesian Hierarchical Models to Assess Neighborhood Influences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Miriam Marco, Antonio López-Quílez, Enrique Gracia, and Kathryn Maguire-Jack

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The Impact of Place-Based Services on Child Maltreatment: Evaluation Through Big Data Linkage and Analytics . . . . . . . . . . . Ilan Katz, Judy Rose, Samantha Low-Choy, and Ross Homel

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Racism and the Racialization of U.S. Neighborhoods: Impacts on Child Maltreatment and Child Maltreatment Reporting . . . . . . Kristen A. Berg, Claudia J. Coulton, and Adam T. Perzynski

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Culture, Religion, and Spirituality in Understanding Child Maltreatment: Perceptions of Parents and Professionals in the Ultra-Orthodox Community . . . . . . . . . . . . . . . . . . . . . . . . . Yochay Nadan, Dafna Tener, Netanel Gemara, Nili Rozenfeld-Tzafar, and Maggi Sharabani

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Structural Inequalities, Neighbourhoods and Protecting Children . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Will Mason, Brid Featherstone, and Paul Bywaters

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Measuring Informal Social Control in Child Maltreatment: From “Whether” to “How,” from One-Off to Sustained, from Individual to Community . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Clifton R. Emery and Alhassan Abdullah

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Neighborhood Effects on Child Maltreatment in Rural Areas . . . . . 117 Kathryn Maguire-Jack, Brooke Jespersen, Jill E. Korbin, Derek Van Berkel, and James C. Spilsbury

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Community Interventions for Preventing Child Abuse and Neglect: Lessons for Expansion . . . . . . . . . . . . . . . . . . . . . . . . 131 Daphna Gross-Manos and Ayala Cohen

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Communities’ Lockdown and the Challenge of Keeping Children Safe: What Have We Learned from the COVID-19 Pandemic? . . . 155 Carmit Katz and Noa Cohen

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Neighborhoods and Child Maltreatment: Looking to the Future . . . 173 Kathryn Maguire-Jack and Carmit Katz

Chapter 1

Communities’ Essential Role in Protecting Children from Maltreatment Carmit Katz and Kathryn Maguire-Jack

1.1

Background

In 1988, the U.S. Child Abuse Prevention, and Treatment Act created an Advisory Board in relation to child abuse and neglect. This Advisory Board was intended to evaluate the implementation of the Act and provide recommendations regarding possible improvements to the executive and legislative branches. Despite prior recognition of issues related to the U.S. child protection system, reports from the Advisory Board brought about a national focus that went beyond traditional prevention methods to include a community-based approach as part of the first step in child maltreatment prevention. To tackle the national emergency and provide background for the recommendations, the Advisory Board described the existing and ongoing gaps that needed to be addressed in redesigning the child protection system: The most serious shortcoming of the nation’s system of intervention on behalf of children is that it depends upon a reporting and response process that has punitive connotations, and requires massive resources dedicated to the investigation of allegations. State and County child welfare programs have not been designed to get immediate help to families based on voluntary requests for assistance. As a result, it has become far easier to pick up the telephone to report one’s neighbor for child abuse than it is for that neighbor to pick up the telephone to request and receive help before the abuse happens (U.S. ABCAN 1993, p. 80).

Hence, the system in place to protect children is feared by the families who need support, due to the potential for accusations and allegations of abuse. As the focus of

C. Katz (*) The Bob Shapell School of Social Work, Tel Aviv University, Tel Aviv, Israel K. Maguire-Jack School of Social Work, University of Michigan, Ann Arbor, MI, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Maguire-Jack, C. Katz (eds.), Neighborhoods, Communities and Child Maltreatment, Child Maltreatment 15, https://doi.org/10.1007/978-3-030-93096-7_1

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child protection services is on investigations of abuse, it thereby weakens the sense of personal and community responsibility to aid neighbors in need. Hence, the responsibility of neighborly support has shifted from helping directly in times of distress to keeping distance and reporting suspected child maltreatment to the authorities. Consequently, child protection resources have been redirected to law enforcement, further minimizing the help that was available for families. The third report of the US Advisory Board advocated for a child-centered, neighbors helping neighbors child protection system. In this way, “all American adults. . . resolve to be good neighbors—to know, watch, and support their neighbors’ children and to offer help when needed to their neighbors’ families” (U.S. ABCAN 1993, p. 82). The report detailed a five-point strategy to accomplish this: (a) strengthening neighborhoods as places to encourage child development and family life; (b) reorienting services to focus on child maltreatment prevention and promotion of family well-being; (c) improving government involvement in child protection (e.g., developing comprehensive plans for child protection; restructuring the financing of government services to facilitate integration); (d) reshaping societal values that may contribute to CM; and (e) generating knowledge to encourage comprehensive community efforts to prevent child maltreatment. A series of commissioned papers partly informed the Advisory Board’s recommendation (in the edited volumes Melton and Barry 1994; Melton et al. 2002) were conducted by researchers who highlighted the need for a broader approach to child maltreatment prevention (e.g., Belsky 1980; Garbarino and Kostelny 1992) and the understanding that neighborhood characteristics and chronic poverty have significant roles in the causes of child maltreatment.

1.2

Neighborhoods and Child Maltreatment

As mentioned, neighborhood factors have been found to play a vital role in ensuring children’s safety. However, this responsibility becomes increasingly difficult when families are confronted with continuous stressors as well as a lack of social and economic resources. Previous studies found several neighborhood-level factors related to children’s safety and well-being (e.g., Leventhal and Brooks-Gunn 2000; Ross and Mirowsky 2009; Sampson et al. 2002), which are characterized as either structure- or process-oriented. The structure-oriented category includes aspects such as the number of single-parent households and households living in poverty. The process-oriented category includes elements such as collective efficacy (Sampson 2003; Sampson and Morenoff 2004; Sampson et al. 1997), social capital (Coleman 1988; Putnam 2001), and neighborhood cohesion (Coulton et al. 2007; Silk et al. 2004). Both orientations have been linked to outcomes for children including health (e.g., Browning and Cagney 2003) and youth development (e.g., Leventhal and Brooks-Gunn 2004), as well as safety and well-being. Furthermore, parental efficacy (Furstenberg et al. 1999) and utilization of effective parenting strategies (Leventhal and Brooks-Gunn 2000; Simons et al. 1997) have also been

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found to relate to neighborhood factors. For example, studies have found that parents could better care for their children when they had access to social supports (e.g., emotional, financial, in-kind assistance; Marra et al. 2009; Priel and Besser 2002). In line with this, the impact of neighborhoods on children’s safety is a topic that has been gaining attention in recent years (for reviews, see Coulton et al. 2007; MaguireJack 2014). As stated by Daro and Dodge (2009): . . .attention has shifted from directly improving the skills of parents to creating environments that facilitate a parent’s ability to do the right thing. It is increasingly recognized that environmental forces can overwhelm even well-intended parents, communities can support parents in their role, and public expenditures might be most cost-beneficial if directed toward community strategies (p. 68).

Collective efficacy is another area of interest for researchers due to its potential effect on outcomes for children. It has been noted that members’ beliefs in their joint ability to problem solve as a group plays a role in building strength within a community (Bandura 1995). Collective efficacy comes from social disorganization theory (Shaw and McKay 1942; Wilson 1987). This theory asserts that communities with residential instability, low economic status, and ethnic/racial heterogeneity have high distress and do not have the necessary resources to prevent neighborhood crime. However, in areas with high levels of collective efficacy, lower rates of violence were found, despite the neighborhood being considered disadvantaged (Sampson et al. 1997). The examination of whether there is a relationship between collective efficacy and child maltreatment has been the topic of several studies (e.g., Emery et al. 2015; Freisthler 2004; Guterman et al. 2009; Kim and Maguire-Jack 2015; Molnar et al. 2016; Sabol et al. 2004). One study of 3356 mothers from 20 U.S. cities found that perceived collective efficacy was negatively associated with psychological and physical aggression (Guterman et al. 2009). In another study, Molnar et al. (2016) looked at survey and administrative data from 1995 to 2005 from Chicago, Illinois. Their results showed that neighborhoods with a higher collective efficacy had lower rates of physical and sexual abuse and neglect than neighborhoods with lower scores. Generalizing from crime to child maltreatment in regard to neighborhood safety relates to findings that in socially cohesive neighborhoods, parents are better able to care for their children, thereby decreasing the likelihood of maltreatment. In Furstenberg’s (1993) study, parents were more prone to partake in effective caregiving roles when they lived in communities that saw raising children as a collective responsibility. Furthermore, Benson et al. (1998) found that healthy communities built internal relationships that accentuated support, opportunity, and a shared commitment to encouraging the healthy development of children and youth. In addition, individual factors play an essential role in the etiology of child maltreatment. Namely, this refers to psychological aspects (e.g., parental depression; perceived efficacy of parents as individuals, family members, and community members), which are also closely tied to social and economic variables (Pelton 1994). An example of this was found in one study where family support in moderate-

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to high-violence neighborhoods was shown to reduce the risk of child maltreatment by decreasing parents’ vulnerability to depression (Martin et al. 2012). Based on previous evidence, the Advisory Board theorized that an effective child protection system should be integrated into daily life. Meaning that child protection services should be present in the places where families live, work, study, worship, and play. In this way, the needs of families and children are seen in real-time and there is a greater ability to respond immediately, reciprocally, and practically. Moreover, incorporating reciprocal help into individuals’ and families’ everyday lives would help “normalize” receiving assistance and reaching out to child protection services. This would result in both minimizing stigmas and maximizing the usefulness and generalizability of such services (for reviews of the nature and effectiveness of informal social support, see Limber and Hashima 2002; Thompson 1994; for mutual assistance, including “self-help” groups, see Murphy-Berman and Melton 2002). The Advisory Board also further identified key components of normalizing assistance, specifically, by making it universal and inclusive. This means ensuring the inclusion of high-resource families, which would increase the availability of resources in a reciprocal system. Furthermore, there is an observed trend in most industrialized societies that all families, regardless of resource access, are becoming progressively more isolated. Even resource-rich families may lack support during times of crisis (e.g., parental illness). It is also imperative for a system of this kind to provide an inclusive environment for families who are often considered outsiders. For instance, although adult offenders are often removed from their communities due to their conduct, this punishment should not be put upon their children. The children and those caring for them during and after their parents’ incarceration should be able to remain within the community safety net. This should also be extended to others who might otherwise not be fully included or integrated into communities, such as new immigrants and ethnic minorities. Although strongly embedded in empirical studies, the idea of communities playing an essential role in the protection of children from maltreatment has had less visibility in policy and practice. Even before the onset of the COVID-19 pandemic, most of the efforts in the field of child maltreatment were dedicated to the parent-child dyad with considerably fewer resources targeting the communities in which these children and families live (Katz et al. 2019).

1.3

An Orientation to This Book

Given the importance of neighborhoods and communities to the lives of children and families, this book spotlights advances in knowledge with respect to neighborhoods, communities and child maltreatment, in order to advance future research and policy worldwide. The chapters that follow are comprised of a variety of topics related to understanding communities and child maltreatment. Specifically, in Chap. 2, Miriam Marco, Antonio López-Quílez, Enrique Gracia, and Kathryn Maguire-Jack review

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advanced methods for studying neighborhoods and maltreatment. Ilan Katz, Judy Rose, Sama Low-Choy and Ross Homel identify methods for evaluating effects on maltreatment in place-based initiatives in Chap. 3. In Chap. 4, Adam Perzynski, Kristen Berg, and Claudia Coulton examine the changing construct of neighborhoods and how their importance has changed over time. Yochay Nadan, Dafna Tener, Netanel Gemara, and Nili Rozenfeld-Tzafar and Maggi Sharabani uncover culture and religion as important concepts of community in understanding maltreatment among an ultra-orthodox community in Chap. 5. In Chap. 6, Will Mason, Brid Featherstone and Paul Bywaters explore concentrated disadvantage and child maltreatment by examining the in-depth lived experiences of parents living in neighborhoods characterized by high levels of structural disadvantage. Clifton Emery and Alhassan Abdullah examine the measurement of informal social control and offer innovative solutions for measuring this neighborhood process in relation to family violence in Chap. 7. In Chap. 8, Kathryn Maguire-Jack, Brooke Jespersen, and Jill Korbin, Derek VanBerkel, and James Spilsbury pinpoint the urban bias within neighborhood-based research and delve into the transferability of these findings to rural areas. Daphna Gross Manos and Ayala Cohen summarize neighborhood-based programs designed to prevent child maltreatment in Chap. 9. In Chap. 10, Noa Cohen and Carmit Katz analyze the unique ways in which the COVID-19 pandemic has influenced the relationship between communities and child maltreatment. Finally, in Chap. 11, Kathryn Maguire-Jack and Carmit Katz conclude by assessing next steps for research and policy related to neighborhoods and child maltreatment.

References Bandura, A. (1995). Exercise of personal and collective efficacy in changing societies. Self-efficacy in changing societies, 15, 334. Belsky, J. (1980). Child maltreatment: An ecological integration. American Psychologist, 35(4), 320. Benson, P. L., Leffert, N., Scales, P. C., & Blyth, D. A. (1998). Beyond the “village” rhetoric: Creating healthy communities for children and adolescents. Applied Developmental Science, 2, 138–159. Browning, C. R., & Cagney, K. A. (2003). Moving beyond poverty: Neighborhood structure, social processes, and health. Journal of Health and Social Behavior, 44, 552–571. Coleman, J. S. (1988). Social capital in the creation of human capital. American Journal of Sociology, 94, S95–S120. Coulton, C. J., Crampton, D. S., Irwin, M., Spilsbury, J. C., & Korbin, J. E. (2007). How neighborhoods influence child maltreatment: A review of the literature and alternative pathways. Child Abuse & Neglect, 31(11–12), 1117–1142. Daro, D., & Dodge, K. A. (2009). Creating community responsibility for child protection possibilities and challenges. The Future of Children, 19(20), 67–93. Emery, C. R., Trung, H. N., & Wu, S. (2015). Neighborhood informal social control and child maltreatment: A comparison of protective and punitive approaches. Child Abuse & Neglect, 41, 158–169. Freisthler, B. (2004). A spatial analysis of social disorganization, alcohol access, and rates of child maltreatment in neighborhoods. Children and Youth Services Review, 26(9), 803–819.

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Furstenberg, F. F. (1993). How families manage risk and opportunity in dangerous neighborhoods. In W. J. Wilson (Ed.), Sociology and the public agenda (pp. 231–258). Sage. Furstenberg, F. F., Jr., Cook, T. D., Eccles, J., Elder, G. H., Jr., & Sameroff, A. (Eds.). (1999). Managing to make it: Urban families and adolescent success. University of Chicago Press. Garbarino, J., & Kostelny, K. (1992). Child maltreatment as a community problem. Child Abuse & Neglect, 16(4), 455–464. Guterman, N. B., Lee, S. J., Taylor, C. A., & Rathouz, P. J. (2009). Parental perceptions of neighborhood processes, stress, personal control, and risk for physical child abuse and neglect. Child Abuse & Neglect, 33, 897–906. Katz, C., McLeigh, J., & Arieh, A. B. (2019). Reflections on the traditional role of social workers in child protection: Lessons learned from the Strong Communities initiative in Israel. Kim, B., & Maguire-Jack, K. (2015). Community interaction and child maltreatment. Child Abuse & Neglect, 41, 146–157. Leventhal, T., & Brooks-Gunn, J. (2000). The neighborhoods they live in: The effects of neighborhood resident upon child and adolescent outcomes. Psychological Bulletin, 126, 309–337. Leventhal, T., & Brooks-Gunn, J. (2004). Diversity in developmental trajectories across adolescence: Neighborhood influences. In R. M. Lerner & L. Steinberg (Eds.), Handbook of adolescent psychology (pp. 451–486). Wiley. Limber, S. P., & Hashima, P. Y. (2002). The social context: What comes naturally in child protection. Maguire-Jack, K. (2014). Multilevel investigation into the community context of child maltreatment. Journal of Aggression, Maltreatment, and Trauma, 23(3), 229–248. Marra, J. V., McCarthy, E., Lin, H. J., Ford, J., Rodis, E., & Frisman, L. K. (2009). Effects of social support and conflict on parenting among homeless mothers. American Journal of Orthopsychiatry, 79, 348–356. Martin, A., Gardner, M., & Brooks-Gunn, J. (2012). The mediated and moderated effects of family support on child maltreatment. Journal of Family Issues, 33(7), 920–941. Melton, G. B., & Barry, F. D. (Eds.). (1994). Protecting children from abuse and neglect: Foundations for a new national strategy (p. 131). Guilford Press. Melton, G. B., Thompson, R. A., & Small, M. A. (2002). Toward a child-centered, neighborhoodbased child protection system: A report of the consortium on children, families, and the law. Praeger Publishers/Greenwood Publishing Group. Molnar, B. E., Goerge, R. M., Gilsanz, P., Hill, A., Subramanian, S. V., Holton, J. K., Duncan, D. T., Beatriz, E. D., & Beardslee, W. R. (2016). Neighborhood-level social processes and substantiated cases of child maltreatment. Child Abuse & Neglect, 51, 41–53. Murphy-Berman, V., & Melton, G. B. (2002). The self-help movement and neighborhood support for troubled families. Pelton, L. H. (1994). The role of material factors in child abuse and neglect. Priel, B., & Besser, A. (2002). Perceptions of early relationships during the transition to motherhood: The mediating role of social support. Infant Mental Health Journal, 23, 343–360. Putnam, R. (2001). Social capital: Measurement and consequences. Canadian Journal of Policy Research, 2(1), 41–51. Ross, C., & Mirowsky, J. (2009). Neighborhood disorder, subjective alienation, and Distress. Journal of Health and Social Behavior, 50, 49–64. Sabol, W., Coulton, C., & Polousky, E. (2004). Measuring child maltreatment risk in communities: A life table approach. Child Abuse & Neglect, 28(9), 967–983. Sampson, R. J. (2003). The neighborhood context of well-being. Perspectives in Biology and Medicine, 46, 53–64. Sampson, R. J., & Morenoff, J. D. (2004). Spatial (dis)advantage and homicide in Chicago neighborhoods. In M. Goodchild & D. Janelle (Eds.), Spatially integrated social science (pp. 145–170). Oxford University Press. Sampson, R. J., Morenoff, J. D., & Gannon-Rowley, T. (2002). Assessing ‘neighborhood effects’: Social processes and new directions in research. Annual Review of Sociology, 28, 443–478.

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Sampson, R. J., Raudenbush, S. W., & Earls, R. F. (1997). Neighborhoods and violent crime: A multilevel study of collective efficacy. Science, 277, 918–924. Shaw, C. R., & McKay, H. D. (1942). Juvenile delinquency and urban areas: A study of rates of delinquents in relation to differential characteristics of local communities in American cities. University of Chicago Press. Silk, J. S., Sessa, F. M., Sheffield, M. A., Steinberg, L., & Avenevoli, S. (2004). Neighborhood cohesion as a buffer against hostile maternal parenting. Journal of Family Psychology, 18(1), 135–146. Simons, R. L., Johnson, C., Conger, R. D., & Lorenz, F. O. (1997). Linking community context to quality of parenting: A study of rural families. Rural Sociology, 62, 207–230. Thompson, R. A. (1994). Social support and the prevention of child maltreatment. U.S. Advisory Board on Child Abuse and Neglect [U.S. ABCAN]. (1993). Neighbors helping neighbors: A new national strategy for the protection of children. U.S. Government Printing Office. Wilson, W. J. (1987). The truly disadvantaged. University of Chicago.

Chapter 2

The Spatio-temporal Epidemiology of Child Maltreatment: Using Bayesian Hierarchical Models to Assess Neighborhood Influences Miriam Marco , Antonio López-Quílez and Kathryn Maguire-Jack

2.1 2.1.1

, Enrique Gracia

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Spatio-temporals Models in the Study of Social Problems: A Brief Landscape Origins of Spatial and Spatio-temporal Models

The history and development of spatio-temporal approaches spans epidemiology, public health, and disease mapping disciplines (Lawson 2018; Waller and Gotway 2004). The first research sought to evaluate the geographical or spatial variations of health-related outcomes, examine risk maps, and analyze the characteristics of the environment that influence diseases such as cancer or infectious diseases (Elliot et al. 2000; Mollie and Richardson 1991). Some of these studies based on environmental exposures such as pollution, lead or NO2 (Biggeri et al. 1996), while others focused on social and spatial inequalities in health issues such as poverty or migration (MacKinnon et al. 2007). The main characteristic of spatial models is that they examine a geocoded outcome, in which its spatial placement is considered relevant to explain and understand the phenomenon. Thus, they are linked to ecological models, which study the interrelationships of outcomes and their environment, and suggest that the context (for example, the neighborhood where the outcome takes place) has an important influence beyond the individual characteristics (Waller and Gotway 2004).

M. Marco (*) · E. Gracia Department of Social Psychology, University of Valencia, Valencia, Spain e-mail: [email protected] A. López-Quílez Department of Statistics and Operational Research, University of Valencia, Burjassot, Spain K. Maguire-Jack School of Social Work, University of Michigan, Ann Arbor, MI, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 K. Maguire-Jack, C. Katz (eds.), Neighborhoods, Communities and Child Maltreatment, Child Maltreatment 15, https://doi.org/10.1007/978-3-030-93096-7_2

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Space-time studies are a more complex approach to spatial models. When data have a geographic component and are collected over different time periods, it is necessary to conduct a spatio-temporal data analysis. These models analyze the spatial distribution and spatial correlates of an outcome, as well as examine whether it changes or maintain the same spatial patterns and contextual influences over time. For example, it may be interesting to measure the temporal evolution of influenza in space to calculate the risk of new cases in the future and to provide health services with statistical tools to control the epidemic (Amorós et al. 2020).

2.1.2

Spatio-temporal Models and Social Problems

Although their origin and development took place in medicine and health studies, spatio-temporal models are increasingly used for the study of social problems. Some theoretical approaches, such as social disorganization theory (Shaw and McKay 1942), have highlighted the importance of context beyond individual characteristics, and spatial and spatio-temporal models represents an advanced methodological approach to evaluate these influences on social outcomes. Particularly relevant is the study of the spatio-temporal distribution of crime (Law and Quick 2013; Law et al. 2014; Matthews et al. 2010; Sparks 2011). This research has been used to evaluate the hotspots and spatial influences in different types of crime, among which violent crime and homicides, vehicle crashes, drug-related outcomes, burglary, or juvenile delinquency stand out (Groff et al. 2009; Haining et al. 2009; Law et al. 2014; Matthews et al. 2010; Sparks 2011). These studies focus on the prediction of crime, detecting clusters of conflict areas and how they have moved over time to study future cases. Other problems have been incorporated into this line of research. There are currently studies that assess the spatio-temporal distribution of outcomes such as drug-related problems (Lum 2008; Marco et al. 2017a, b), intimate partner violence (Cunradi et al. 2011; Gracia et al. 2015, 2021), suicide (Congdon 2011; Helbich et al. 2017; Marco et al. 2018), and child abuse and neglect (Barboza 2019; Freisthler and Weiss 2008; Gracia et al. 2017).

2.2 2.2.1

Modeling Spatial Influences in the Study of Child Maltreatment Why Measure Spatial Structure?

Child maltreatment is a complex phenomenon that must be considered beyond individual and relational factors to include the broader social context (Garbarino 1977). Among those factors related to social context that have received the most

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attention, research has highlighted the neighborhoods where families reside. Analyzing the relationships between neighborhood characteristics and the risk of child maltreatment and the likelihood of a child or a family to be involved in Child Protective Services (CPS) has a long tradition dating back to the pioneering research carried out by Garbarino and his colleagues (Garbarino and Sherman 1980). Starting from these studies, more recent research has linked the structural and demographic characteristics of neighborhoods (mainly impoverishment and social disorganization) with risk of child maltreatment (Coulton et al. 2007; Freisthler et al. 2006; Maguire-Jack 2014; Sampson et al. 1997). This body of research concludes that ‘place’ and its characteristics are significant to explain the variations in the risk of child maltreatment. Many studies that have analyzed neighborhood influences on child maltreatment have focused on multilevel models, such as hierarchical linear modeling (Coulton et al. 2007; Freisthler et al. 2006; Maguire-Jack 2014). These studies have provided important results by mixing individual and contextual risk factors, confirming that both are relevant and complementary, and suggesting some contextual characteristics are related to child abuse (Coulton et al. 2007). While multilevel models allow for estimating an individual’s likelihood of maltreating based on aspects of the neighborhood and control for clustering of individuals within neighborhoods, these models do not capture the spatial element, and do not allow mapping the spatial risk (Coulton et al. 2007). Spatial and spatio-temporal modeling are advanced methodologies to analyze spatial structures in child abuse and neglect and assess specific spatial risks that could guide community prevention policies and provide complementary scientifical evidence to the study of the ecology of child maltreatment. Specifically, the approach allows for mapping area-specific risk estimates and can contribute to planning and evaluating prevention strategies of formal social agencies including CPS, the City Council, or the Municipal Police Department. Assessing the areas where there is a greater risk of child maltreatment can be very useful to guide local actions, make a better distribution and management of resources, and develop preventive strategies for neighborhoods with greater risk (Gracia et al. 2015). In addition, the approach allows for analyzing neighborhoods where risk has increased or decreased in recent years and exploring the neighborhood-level covariates that could explain those changes over time. This information can provide a high-quality quantitative approach for evaluating the impact of preventive strategies. This approach may be used to detect and map whether maltreatment risk is reduced when different prevention and intervention policies have been implemented.

2.2.2

The Selection of Study Area: A Complex Issue

One of the key elements when conducting a spatial study is to select the most appropriate study area. There are different possibilities, ranging from a large area such as a country, to a small area, such as the street number. In between, we find a

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wide range of spatial units: states, counties, cities, districts, neighborhoods, zip codes, census blocks, census block groups, streets, or street segments. Selecting the most suitable unit for the study is not an easy task, and it largely depends on the problem and the availability of data. Small-area studies have shown some strengths compared to others (Gracia et al. 2017), because they minimize the aggregation bias intrinsic to spatial modeling. This bias occurs when there is heterogeneity within the study area: for example, the socioeconomic characteristics of a city can be very different in its Southern area compared to the North, and considering the city average can lead to erroneous conclusions. However, too small units may not fit adequately the social reality behind child maltreatment or may be impossible to determine due to lack of information. For example, crime studies have assessed streets or segments streets (Groff and Lockwood 2014; Steenbeek and Weisburd 2016); however, in the case of child abuse this measure would be excessively small, since most streets would show zero child maltreatment reports. The previous studies in this line come mostly from the United States and have commonly used counties, zip codes, or census tracts (Freisthler and Weiss 2008; Freisthler et al. 2012; Morris et al. 2019a, b). On the other hand, European studies have used census block groups, within a large city (Gracia et al. 2017, 2018). Both alternatives may be adequate, according to the aim of the study: the first studies may be interesting to carry out a geographical and political perspective of the areas with special needs regarding child abuse and neglect (i.e., areas with excess risk), while the second ones are of great interest to evaluate neighborhood characteristics associated with child maltreatment. Although spatial studies have many strengths, it is critical to avoid the ecological fallacy. This fallacy is due to the extrapolation of the results of a contextual analysis to conclude at the individual level (Martínez-Beneito and Botella-Rocamora 2019). Conclusions based on spatial studies must always be very cautious, and avoid establishing inferences at the individual level. For example, if a spatial model suggests that immigration is related to child maltreatment risk at the neighborhood level, the model is just assuming a contextual relationship (in areas with higher immigration there are a higher risk of child maltreatment, regardless of whether the family is immigrant or not) but does not mean that immigrant population are more likely to commit acts of child abuse and neglect.

2.2.3

Bayesian Hierarchical Models: BYM Models

Spatial models can be conducted from a frequentist or Bayesian perspective. The frequentist methodology, also known as classical statistics, considers parameters as fixed but unknown values which must be estimated. This approach relies on data solely from the current study and calculates the probabilities considering the experiment as one of an infinite sequence of repetitions under the same conditions. In this chapter, we will focus on the Bayesian methodology, which has shown important advantages to model the spatial structure.

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The Bayesian approach, based on Bayes’ Theorem, combines the construction of complex models with the inclusion of previously known information about the parameters (Banerjee et al. 2004). In other words, not only are the data used to carry out the analyses, but the results are also based on sources of prior information, known before the data was sampled (Lindley and Smith 1972). This prior information is transformed into posterior probabilities, and these probabilities are used to make the inference. This methodological approach is increasingly used in different disciplines, mainly due to the advantages of including this prior information, which frequentist statistics cannot include in its models and thus they are very limited to data. Likewise, Bayesian inference techniques are based on simulation methods, which allow calculating the posterior distributions of the parameters when it is not possible to sample directly (Gilks et al. 1996). Specifically, hierarchical Bayesian spatio-temporal models (Bernardinelli et al. 1995) are very effective in modeling prior information, decomposing it into different levels or layers, which allows for differentiating between structural parameters of the model and elements related to external information (Lawson 2018). Hierarchical models allow for the inclusion of random effects in any of their layers, such as, in our case, the spatial and temporal structure (Law et al. 2014). The first layer models the data conditioned by the random vector of the parameters, which have an associated prior distribution defined in the second layer. If the parameters of the model depend on other parameters (known as hyperparameters), the prior distributions of these are defined in the third layer. This approach is still in its early stages in the study of family violence, with a small number of publications to date (Barboza 2019; Cunradi et al. 2011; Freisthler and Weiss 2008; Freisthler et al. 2012; Gracia et al. 2017, 2018; Marco et al. 2018, 2020, Morris et al. 2019a, b), and even less in the field of child maltreatment (Freisthler and Weiss 2008; Freisthler et al. 2012; Gracia et al. 2017; Morris et al. 2019a, b). One of the most commonly used models within the spatial Bayesian hierarchical approach is the Besag, York and Mollié (BYM) model (Besag et al. 1991). This perspective assumes that the outcome is based on observations that are distributed as a conditionally independent Poisson in the following way: yi j ηi  PoðE i exp ðηi ÞÞ,

i ¼ 1, . . . , I

where Ei is a quantity which accounts for the expected number of observations (such as reports of child maltreatment or families involved in CPS) in proportion to the targeted population (the total of reports or families in CPS) in each area i, and ηi is the log relative risk of the outcome. In this model, the spatial effect is introduced by adding a random spatial autocorrelation effect to the relative risk. Spatial autocorrelation refers to the risk rates that we find in nearby areas are more similar than those from more distant areas. Detecting this spatial dependence can be very useful to provide information about

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the unobserved spatial structure of the data, which may be because neighboring areas have more similar characteristics, either at a social, economic, or cultural level (Banerjee et al. 2004; Bernardinelli et al. 1995; Waller and Gotway 2004). In addition, spatial models often include another random effect, known as heterogeneity or overdispersion. This random effect refers to the spatial differentiation of geographic units. When working with spatial units, the outcome is typically distributed differently in space, that is, the data are not homogeneous. However, due to the small values that the data usually take (sometimes there is a large presence of zeros, especially when a small-area study is conducted) it is important to correct and smooth the differences that may appear between areas, which are not produced by a real difference between them, but rather, by an overdispersion effect (Haining et al. 2009). Finally, fixed effects that reflect the influence of different neighborhood-level covariates in the outcome can be also incorporated into the relative risk modeling. Thus, the common structure of the spatial Bayesian hierarchical model is at follows: ηi ¼ μ þ X i β þ φi þ θi where μ is the intercept, β is the regression coefficients vector, X represents the matrix of covariates, φi is the spatially structured term, and θi is the unstructured term. To model the spatial effect φi, BYM model usually follows a Conditional Autoregressive (CAR) distribution, which reflects spatial neighborhood relationships. This distribution is defined as: φi j φi

σ 2φ 1X N φJ , ni ji ni

!

where ni is the number of neighboring areas of areas i, Si indicates the values of the S vector except for the i th component, j~i refers to all units j that are neighbors of areas i, and σ S is the standard deviation parameter. One of the main advantages of this model is that the log relative risk and the spatial effect can be mapped. The map of the log relative risk represents the areas with higher or lower levels of risk based of the impact of both random spatial effects and neighborhood-level covariates. The values of risk are calculated as the exponential of ηi, and a value of 1 represents the average risk, while areas with risks greater and lower than 1 indicate an above-average and below-average probability, respectively. Mapping relative risk is a useful tool to geographically explore the differences in risk between areas. In addition, the map of the spatial effect φi represents the posterior mean of the spatial random effect, i.e., the underlying spatial structure beyond contextual covariates. This map is especially interesting in order to propose alternative hypothesis of other neighborhood-level variables which can be explaining the spatial patterns.

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Assessing the Shared Component Between Outcomes: A Bayesian Joint Modeling

A more recent approach has focused on studying the common spatial patterns of different outcomes related to family violence. These models are also based on the CAR models, but they introduce a new structure in order to understand separately the potentially common spatial component and the two specific spatial patterns. These studies have analyzed the spatial overlap of substantiated and unsubstantiated child maltreatment and mapping the common spatial pattern (Marco et al. 2020), as well as analyzing the common spatial distribution of child maltreatment and intimate partner violence (Gracia et al. 2018). One of the joint modeling is the proposed by Knorr-Held and Best (Knorr-Held and Best 2001), which is defined as follows: yik j ηik  PoðEi exp ðηik ÞÞ,

i ¼ 1, . . . , I k ¼ 1, 2

ηi1 ¼ α1 þ ϕi  δ þ ψi1 ηi2 ¼ α2 þ ϕi =δ þ ψi2 where αk is the intercept, δ represents the scaling factor which allows the risk gradient for the shared component to be different for each outcome k; ϕ is the shared component, and ψ i1 and ψ i2 represent the two specific components for each outcome k. ϕ and ψ are composed of unstructured and structured spatial components. The shared component is logarithmically transformed, and the spatially structured term is modeled as a CAR model. Traditional Bayesian inference are performed by Markov Chain Monte Carlo (MCMC) methods (Gamerman and Lopes 2006). Another alternative method is the integrated nested Laplace approximation (INLA), which is increasingly gaining attention (Blangiardo and Cameletti 2015).

2.2.5

A Practical Example of Bayesian Hierarchical Spatial Models

In this section, an example of the models discussed in the previous section are shown. To illustrate them, we will use data from a study conducted in the city of Valencia, Spain (Marco et al. 2020). Specifically, we carried out a spatial Bayesian hierarchical spatial analysis of child maltreatment in the city. The census block group was used as the spatial unit, which is the smallest unit with available information provided by the City Council. This unit was considered the most appropriate because of the small aggregation of data and it can be considered as a proxy of neighborhoods. The study area is composed of 552 census block groups, with a total population of 800,215 inhabitants (2020 data).

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Information about child maltreatment was collected as the number of families with child maltreatment referrals. Data of all families involved in CPS from 2004 to 2015 were collected. The sample was divided in two types of reports: cases of substantiated child maltreatment (1799 families), and cases of unsubstantiated child maltreatment (1638 families). Child maltreatment refers to any type of child maltreatment, including physical, psychological, or sexual abuse, as well as neglect. Substantiated cases had associated a protective measure (i.e., home visiting, family support programs, family or residential foster care, or adoption), while unsubstantiated cases are those which CPS investigated but were closed without measures because being considered less severe. The address of the families was geocoded in the 552 census block groups of the city. Following the social disorganization framework, different neighborhood-level variables were collected at the census block group: three indicators to assess concentrated neighborhood disadvantage (neighborhood economic status, neighborhood education level, and policing activity, as a proxy of neighborhood public disorder and crime), an indicator of ethnic heterogeneity (immigrant concentration), and an indicator of residential instability. A BYM model was performed for each outcome (substantiated and unsubstantiated child maltreatment). In this case, Ei i was a quantity that accounts for the expected number of families in census block group i, that is: E i ¼ Families living at census block group i 

Total families with referrals Total families in Valencia

The unstructured spatial effect θ was modeled as independent identically distributed Gaussian random variables N(0, σ θ2), while the structured spatial effect ϕ follows a CAR model reflecting the spatial neighborhood relationships. An improper uniform distribution was assigned for μ, while for the hyperparameters of σ θ and σ ϕ, prior distributions of standard deviations are uniform distributions σ ϕ, σ θ  U(0, 1). In this case, we used MCMC methods. According to these methods, a number of simulations of the parameters was performed. Specifically, we generated 100,000 iterations, where the first 10,000 were discarded as burn-in, in order to obtain convergent results. All the models were performed using the software WinBUGS and run in R (using R2WinBUGS package). Results of the models show that risk of child maltreatment referrals, regardless of whether they were substantiated or unsubstantiated, are higher in neighborhoods with low economic status and education level, and with high levels of immigrant concentration, residential instability, and policing activity. In addition, results showed that the spatial effect is relevant to the model, and the relative risk is not randomly distributed in the city, but in some areas the risks are more than twice the average, reflecting very high-risk levels of child maltreatment. Figure 2.1 shows the maps of log relative risk for substantiated and unsubstantiated child maltreatment. As we can observe, both maps are very similar, suggesting a common spatial structure between both types of child maltreatment. Thus, we conducted a Bayesian

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Fig. 2.1 Maps of log relative risks for substantiated (left) and unsubstantiated (right) child maltreatment

Joint Modeling in order to study the spatial overlap between substantiated and unsubstantiated child maltreatment cases. Therefore, we introduced a shared component in the model with k ¼ 1 for substantiated referrals, and k ¼ 2 for unsubstantiated referrals. The unstructured term is modeled by an independent identically distributed Gaussian random variables. Additionally, an improper uniform distribution is used for α1 and α2. We obtained the proportion of shared variance for each outcome. The same number of iterations was generated (100,000 iterations and 10,000 of burn-in, using MCMC methods). The results show a common spatial distribution of substantiated and unsubstantiated child maltreatment cases. A large percentage of the variation in both types of referrals across city areas was explained by a common spatial component (90% for substantiated child maltreatment, 88% for unsubstantiated child maltreatment). This result indicates that both types share almost the same spatial structure, with a small specific and differential spatial pattern. Fig. 2.2 shows the map of the shared component (See Marco et al. 2020 for a more detailed description of these results).

2.3 2.3.1

Introducing Time in the Spatial Modeling of Child Maltreatment The Advantages of Incorporating Space and Time

As part of the aggregation bias, spatial models can also have the problem of mixing temporal information which could have different patterns. Studies that do not consider the temporal effect may misinterpret risk estimations, especially if there

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Fig. 2.2 Map of the shared component

was an increasing or decreasing risk pattern in some areas. In addition, the correlation between variables may also have a temporal effect and aggregating them in a single temporal period may mask these relationship (Lawson et al. 2003). Therefore, ‘time’ also becomes a key element to analyze variations in child maltreatment risk. Spatio-temporal analyses allow for the identification of areas where child maltreatment risk increased or decreased over time. These analyses can help detect whether those areas where there is a high risk of child maltreatment reduce their risk once different prevention and intervention policies have been applied (Gracia et al. 2017). These advantages have led to an increasing interest in applying a spatiotemporal approach to analyze neighbourhood-level child maltreatment risks (Barboza 2019, 2020; Freisthler and Weiss 2008; Gracia et al. 2017; Morris et al. 2019a, b).

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Dealing with the Time Period

As the spatial unit is essential in spatial modeling, the temporal unit acquires special relevance in the spatio-temporal models. Usually, spatio-temporal studies employ the year as the temporal period, for which data is most often available. However, different time periods may be more appropriate depending on the aim of the study and the outcome analyzed: for example, some studies have used quarters to analyze police-related outcomes because of the potential influence of seasonality (Marco et al. 2017a, b, 2018; Williams et al. 2018); others, especially from criminology, have focused on the monthly spatio-temporal distribution of crime (Hu et al. 2018; Mahfoud et al. 2020) while other research has employed hour time period to explore spatio-temporal structures of crime and predict future observations (Luan et al. 2016). In the case of child maltreatment, the existing literature has mainly focused on change across years (Barboza 2019, 2020; Gracia et al. 2017; Freisthler and Weiss 2008; Freisthler et al. 2012; Morris et al. 2019a, b; Thurston et al. 2017). In the small-area studies, there are usually many areas with zero cases of child maltreatment per year. Sometimes, a zero-inflated Poisson regression approach is used to avoid biases related to these excess zeros. If we split the temporal unit, the number of areas with zero counts would increase considerably. In addition, theoretical approaches have not highlighted the relevant difference of prevalence of child maltreatment among months or other smaller-than-year temporal units, as they do in other outcomes such as police calls for service, crime rates or suicide (Andresen and Malleson 2013; Woo et al. 2012). Therefore, most studies have found the year period as the most reasonable measure. However, the new research challenges that are emerging as a result of the onset of the COVID-19 pandemic are leading to the development of models where the time period is shortened, with studies that analyze, for example, child abuse per day, to assess the impact of COVID-19 in family processes (Barboza et al. 2020).

2.4.1

Bayesian Hierarchical Spatio-temporal Modeling

There are different approaches to capture the spatio-temporal distribution of an outcome. In this chapter we will focus on two of the contributions that have been most used in the field of child maltreatment. Again, we will focus on a Bayesian approach, due to the advantages that have been explained previously. Both spatiotemporal approaches are based on CAR models. In the same way as in the spatial approach, a temporal structure can be introduced into the model to detect the temporal dependence of data. When the number of periods available is limited, it is reasonable to expect a linear time effect on the data. The first model follows the structure of the CAR model and incorporates a parameter to capture the linear time trend (Lawson et al. 2003). This

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approach has been used in different research (Freisthler et al. 2012; Marco et al. 2017b). Specifically, the log-relative risk is now defined as follows: ηit ¼ μ þ X it β þ ϕi þ θi þ γ  t þ δi  t where a spatio-temporal structure is added, and specifically γ  t refers to a fixed linear time trend for t years, and δi  t is a random spatio-temporal interaction. Again, the structured spatial effect ϕ is modelled as a CAR model, and the spatio-temporal term δi is also modeled as a CAR model following the same distribution. The spatial CAR prior on the space-time interaction term assumes that nearby areas exhibit similar linear time trends. This model allows for mapping the estimated time trend as the exponential of γ + δi, and detect areas with an increasing and a decreasing time trend. The sign of γ indicates the trend, with positive values representing an increase over time, and negative values, a decrease over time. In cases in which the time period is longer, there may be the existence of a nonlinear temporal structure, that could not be captured by the previous model. To this end, it may be more appropriate to apply more complex approaches. One of them is the autoregressive model (Martínez-Beneito et al. 2008), which combines autoregressive time series and spatial modeling. This model has been increasingly used to study the geographical and temporal patterns of child maltreatment (Gracia et al. 2017, 2021; Morris et al. 2019a, b). A spatio-temporal structure in which the relative risks are both spatially and temporally dependent are defined as follows:  1=2 ηi1 ¼ μ þ X i β þ α1 þ 1  ρ2  ðϕi1 þ θi1 Þ   ηit ¼ μ þ αt þ Xi β þ ρ ηiðt1Þ  μ  αt1 þ ϕit þ θit The first equation represents the log-relative risk for the first time period observed, and the second equation defines the log-relative risk for the following time periods. This model incorporates αt, which is the mean deviation of the risk in the period t, and ρ, which refers to the temporal correlation between those periods. The structured spatial effect is specified as a CAR model. One of the main contributions of this approach is that it allows for mapping relative risk at the different periods and monitor risk changing areas over time, as well as detailing those changes, which in turn can be useful for the implementation (or assessment) of prevention strategies.

2.4.2

A Practical Example of Bayesian Hierarchical Spatio-temporal Modeling

Following with the illustrative example used for the spatial model, we applied two spatio-temporal models for substantiated child maltreatment using the data from a

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study in the city of Valencia (Gracia et al. 2017): a linear time trend model, and an autoregressive approach. The temporal unit used was the 12 years of study (2004–2015), and the same covariates as in the spatial model were used: economic status, education level, policing activity, residential instability, and immigrant concentration. Once again, assigning the priors is the last step to complete the models, namely, vague Gaussian distributions for the fixed effects β and the time-trend coefficient γ; μ is an improper uniform distribution; the autoregressive term ρ is modeled as a uniform over the whole space U(1, 1). The unstructured spatial effect θ is modeled as an independent identically distributed Gaussian random variable N(0, σ θ2). Finally, uniform distributions are used for the three hyperparameters σ δ, σ α, σ ϕ, σ θ~U(0, 1). We used MCMC methods, generating 100,000 iterations, and a burn-in of 10,000 iterations. The results of both models can be compared following the Deviance Information Criterion (DIC). This value indicates the model fit, where models with a lower DIC value show a better performance that models with higher values (Spiegelhalter et al. 2002). Linear space-time model shows a DIC of 8166.8, while autoregressive model presents a DIC of 8126.1. These models can be also compared with spatial approach, which shows a DIC of 8159.8. The results of these comparative analyses indicate that spatio-temporal models are more appropriate than spatial modeling (with a relevant difference of DIC values), and the autoregressive model performs better than the linear spatio-temporal approach for this outcome, probably because of the large period (12 years). The results of both spatio-temporal models are shown in Table 2.1. In both cases, the relevant variables are economic status, education level, and policing activity, i.e., areas with higher policing activity and lower economic status and education level show higher risks of substantiated child maltreatment. Figure 2.3 shows the map of change in child maltreatment rates from 2004 to 2015 using the estimated time trend. This map is based on the γ parameter, which has a negative value, which would indicate a decreasing time trend, however, it was not found relevant to the model (the 95% credible interval includes zero). The map shows why this linear time trend is not relevant, as almost all areas present no changes over the time period. We find few areas with linear changes: some northern areas have slightly increased their risks over the years, while some areas in the west and east of the city have decreased their risks, with just 5 census block groups with more than a 5% of changes. Thus, results suggest the presence of nonlinear temporal trends, and the autoregressive model has been found more appropriate. Figure 2.4 shows the maps of the log relative risk for five of the years of study (2004, 2007, 2010, 2013 and 2015) using this autoregressive model. Taken as a whole, our results show chronic spatial patterns of risk of substantiated child maltreatment over the years. The temporal correlation ρ has a high value (ρ ¼ .90) which indicates a high correlation between a particular year and the previous one. Specifically, some peripheral areas of the city show stable high risks that are more than twice the city average, while the city center presents a stable low

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Table 2.1 Results of the spatio-temporal regression Bayesian models

Intercept Economic statusa Education level Policing activity Residential instability Immigrant concentration Spatial heterogeneity (σθ) Spatial structure σϕ Year (γ) Temporal heterogeneity (σα) Temporal correlation (ρ) DIC

Linear time trend model Mean SD 95% CrI 4.325 .520 3.323, 5.328 .013 .004 .023, .008 1.441 .159 1.752, 1.140 .035 .011 .013, .056 .000 .001 .001, .001 .008 .005 .003, .018

Autoregressive model Mean SD 95% CrI 4.135 .500 3.274, 5.127 .016 .004 .023, .008 1.391 .157 1.690, 1.122 .031 .011 .009, .053 .000 .001 .001, .001 .009 .006 .003, .020

.788

.114

.560, .984

.234

.045

.162, .333

.313 .001

.083 .007

.126, .458 .016, .012

.257

.062

.149, .391

.021

.019

.001, .070

.903 8126.1

.031

.827, .946

8159.8

Posterior mean, standard deviation (SD), and the 95% credible interval (CrI) of all parameters this variable was divided by 1000 to solve computational problems with its prior distribution

a

Fig. 2.3 Changes of child maltreatment rates from 2004 to 2015

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2007

2004

2013

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2015