Three Hypotheses About Community Effects on Social Outcomes

AJ. Douglas Willms [ * ]



This paper sets out three hypotheses relevant to differences among communities in their social outcomes, and the relationships between individuals’ social outcomes and their socio-economic status. It presents some of the recent evidence pertaining to these hypotheses and argues that they are central to understanding how social capital affects social outcomes. The three hypotheses can be embodied in a multilevel framework, and there are powerful statistical models for testing them. In discussing the evidence pertaining to these hypotheses in the field of education, some processes used to explain community differences are identified. It is argued that these may be a much better proxy for social capital than “trust” or “the size of people’s social networks,” which have been used in macro-level analyses. Finally, I speculate as to how social capital might contribute to the distribution of social outcomes, and discuss the implications of this research for conducting large-scale studies that could contribute to our understanding of the role of social capital.

ASIDE FROM THE PROBLEMS associated with the definition and measurement of social capital, which are discussed elsewhere in this volume, there are several problems confronting any assessment of the impact of social capital on social outcomes. First, social capital has to do with relationships among people in some “community,” such as a school, workplace, neighbourhood or some larger jurisdiction. To make any progress, a researcher must specify the units of analysis, and in some way define “community.” But any definition of community is easily challenged. Indeed, the notion that social capital embodies networks suggests that the boundary of what people call their “community” itself depends on their stock of social capital. Moreover, every individual participates in multiple and overlapping communities (e.g., family, neighbourhood, workplace, sports teams, church group).

Second, even when community is narrowly defined, as a school or workplace for example, it is rarely feasible to randomly assign individuals to communities. Moreover, social capital is undoubtedly correlated at the community level with the aspects of economic and human capital that are known to affect social outcomes, and all of these forms of capital are correlated with demographic characteristics of the community. In statistical terms, selection bias is exacerbated by the presence of confounding variables. Third, as one moves to higher levels of analysis, e.g., from comparisons within communities to comparisons across countries, the number of potential confounding variables may well multiply.

Fourth, the causal direction is unclear and may also interact with the type of individual: for some people, social capital may help them gain access to better jobs and schooling; for others, wealth and access to better schooling may help them develop and strengthen their social capital.

Finally, social capital may have latent effects. For example, many children have to cope with economic hardship and inadequate family support, yet some of these vulnerable children go on to have successful marriages and working careers. Studies of resilient children have suggested that a relationship with a strong mentor during childhood is one of the most important factors contributing to resiliency.[ 1 ]

To provide further insight into the relationship between social capital and social outcomes, this paper sets out three hypotheses relevant to differences among communities in their social outcomes, and the relationships between individuals’ social outcomes and their socio-economic status.

The first of the three hypotheses, the Hypothesis of Community Differences, is straightforward: it posits that communities differ in their social outcomes, even after accounting for people’s socio-economic status. The second hypothesis is concerned with the relationship between social outcomes and socio-economic status, which are referred to here as socio-economic “gradients.” The Hypothesis of Converging Gradients holds that gradients vary among communities and that they converge at higher levels of socio-economic status. Consequently, successful communities are those that have been successful in bolstering the social outcomes of their least advantaged citizens. Third, the Hypothesis of Double Jeopardy holds that people from less advantaged backgrounds are vulnerable, but people from less advantaged backgrounds who also live in less advantaged communities are especially vulnerable.

The examples presented here pertain mainly to the distribution of literacy skills during the period of formal schooling, and among youth aged 16 to 25. The term “literacy” is used in a very broad sense, as in the International Adult Literacy Survey (IALS), to describe an individual’s ability to: “us[e] printed and written information to function in society, to achieve one’s goals, and to develop one’s knowledge and potential.” Literacy is not viewed as a dichotomy of literate versus illiterate, but a skill continuum. Findings from the IALS suggest that a person’s position on that continuum has dramatic implications for his or her economic success, health and well-being.[ 2 ]

I focus on quantitative literacy because it is more closely related to the effects of schooling per se whereas literacy skills in the language arts are more strongly affected by family background; it is closely related to the acquisition of high-paying jobs and long-term employment; and it is more straightforward to assess.

Although literacy skills are normally thought of as a form of human capital, their acquisition has important implications for social capital: they must certainly affect the nature of the social networks in which people are included and engaged, and the extent to which people can transform social capital into economic capital. Moreover, compared with other social outcomes, literacy may have a particularly strong relationship with social capital. People become members of social networks by learning the language of the culture, and using it to engage in social relations.

The Hypothesis of Community Differences

The first hypothesis asks whether communities vary in their outcomes, after taking account of individuals’ socio-economic status and other characteristics. A useful starting point, however, is to ask first, “To what extent do communities vary in their outcomes?”

There have been some attempts by the Council of Ministers of Education, Canada (CMEC) to monitor performance at the national level, and provide comparative data. Frempong and I have assembled these data, as well as data from three national and international studies, to discern whether provinces do indeed vary in their achievement scores, and to estimate the extent of variation among communities within provinces.[ 3 ] The data were garnered from the first wave of the National Longitudinal Study of Children and Youth (NLSCY), the Third International Mathematics and Science Study (TIMSS)[ 4 ] and the International Adult Literacy Study (IALS). Together these studies provide a useful portrait of successful schools and schooling systems in Canada. Figure 1 presents a summary of our findings pertaining to interprovincial variation in mathematics achievement.


When children enter school, there is considerable variation in their cognitive capacity, and their potential to benefit from formal schooling — what is often loosely called “readiness to learn.” Analyses of children’s receptive vocabulary at ages 4 and 5 suggest that much of this variation is among schools (and communities defined in other ways) within provinces, and relatively little variation is between provinces.[ 5 ] However, by the end of grade 2, the variation among provinces, at least in mathematics results, is discernible and statistically significant. Moreover, the extent of variation among provinces increases as children progress through the schooling system. The results for Quebec are particularly intriguing: it clearly emerges as the top-performing province by the end of grade 4, and it maintains its advantage through to the end of secondary school. In contrast, Ontario, which is Canada’s largest and most affluent province, anchors the bottom end of the distribution. The figure also depicts a widening east-west divide: as children progress through the system, British Columbia and the three prairie provinces tend to have scores that are above the national average, while the average scores of the four Atlantic provinces fall below the national average.

Some of the differences among the Canadian provinces in their quantitative literacy skills have been evident for nearly two decades.[ 6 ] They are not attributable to variation in children’s socio-economic backgrounds or their race or ethnicity; in fact, controlling for socio-economic status and minority status yields estimates of an even wider gap between Ontario and Quebec. Understanding why these differences persist is clearly relevant to the economic growth and well-being of Canadians. But they also have an important lesson for the study of human and social capital. These results indicate that we can identify successful communities as early as the second grade. We believe that at least some, and perhaps a large proportion, of the variation among jurisdictions is rooted in the early years and determined by the ability of communities to develop children’s literacy skills during the period from conception to age 5.[ 7 ]

The Hypothesis of Converging Gradients

Figure 2 displays the socio-economic gradients for youth aged 16 to 25 for quantitative literacy skills across 8 of the 12 countries that had participated in the IALS by 1997.[ 8 ]


Clearly, countries vary considerably in both their levels of literacy scores and in their socio-economic gradients. However, perhaps more important, at least with respect to the discussion on social capital, gradients also converge at higher levels of socio-economic status: there is a strong inverse relationship between the level of skills for a country and its socio-economic gradient. This means that youth from relatively advantaged backgrounds tend to have high literacy scores in every country, whereas the average levels of skills of youth from less advantaged backgrounds vary considerably among countries.

A similar pattern was found for states within the United States, and provinces within Canada. In this analysis, there was also a relationship between gradients and latitude: states that were further north tended to have shallower gradients and higher scores.[ 9 ] Also, the gaps between minorities and non-minorities in literacy scores were smaller in more northerly states. The results indicated that some of the inter-jurisdiction variation was attributable to the amount of time youth spent watching television rather than participating in literacy activities at home and at work.

As a result of these and other analyses, I maintain that the hypothesis of converging gradients is worth testing to achieve a better purchase on the nature of human capital formation and the role of social capital. In some situations, I have found that the hypothesis cannot be rejected. For example, I examined the gradients in literacy skills for youth in Poland across 49 administrative areas.[ 10 ] The results indicated that these local communities varied substantially in their literacy skills, but the hypothesis of converging gradients did not hold. Similarly, Marie-André Somers and I have examined the socio-economic gradients in reading and mathematics scores for 11 countries in Latin America.[ 11 ] Here also, countries varied in the level of their performance, but the gradients did not converge. We did find, however, that the gradients in some countries were non-linear, and that there appeared to be a “premium” associated with completing secondary school. The results for mathematics for nine of those countries are shown in Figure 3.


Before encountering the Latin American results, I had concluded that the success of a society, as gauged by these types of indicators, depends on the extent to which it is successful in reducing inequalities. It may be that societies progress from relatively flat gradients with low levels of social outcomes, to steep gradients with average levels of outcomes, and finally to shallow gradients with high levels of social outcomes, and that progression depends on how social and human capital are invested. Nevertheless, both the examples and the counter-examples provide evidence that it is possible to achieve both high levels of social outcomes and equality of social outcomes among low- and high-status groups.

The research indicating that gradients do converge in some cases has important implications for how we think about social capital. It suggests that there are social, economic and historical factors associated with the culture of a society which shape and constrain people’s behaviours in ways that determine its socio-economic gradient. Thus, raising and flattening gradients may be a difficult and long-term process. We require a better understanding of the structural and contextual features of societies and local communities that lead to greater equality. In high-income countries, success depends on investments in human and social capital that improve the social outcomes for its most vulnerable citizens.

The Hypothesis of Double Jeopardy

Research on schooling in several countries has suggested that while there is a positive effect associated with an individual’s socio-economic status, there is also an additional positive effect associated with the socio-economic status of the school to which the individual belongs. This occurs when the average gradient within communities is shallower than the overall gradient between communities.

Figure 4 provides an example. It shows the school mean reading achievement plotted against the school mean socio-economic status for nearly 1000 schools that participated in the U.S. National Educational Longitudinal Study. The longer black line is the between-school gradient. Schools that scored above this line, on average, were performing better than expected, given the socio-economic status of the students they served, whereas schools that scored below this line were performing worse than expected. The average within-school gradient is somewhat shallower. It has been depicted for two schools that are on the between-school gradient — on the longer line. Note that the expected score for a child with nationally average socio-economic status (a score of zero on the X – axis) is higher in the school with the higher average socio-economic status. In this example, the effect is similar for students with low or high socio-economic status — on average both advantaged and disadvantaged students achieved better results when they attended schools with high average socio-economic status.


The Hypothesis of Double Jeopardy holds that people from less advantaged backgrounds are vulnerable, but people from less advantaged backgrounds who also live in less advantaged communities are particularly vulnerable. There is strong evidence that this hypothesis holds for school achievement when children are segregated, either between schools through residential segregation or by the “creaming” of the most able pupils into selective schools (e.g., private schools or charter schools),[ 12 ] between classes through tracking or streaming[ 13 ] or within classes through ability grouping:[ 14 ] children from advantaged backgrounds do better, while those from disadvantaged backgrounds do worse. Whether the contextual effects associated with school mean socio-economic status tend to be stronger for low socio-economic groups than for high socio-economic groups is still an open question, but in cases where there is an interaction between school mean socio-economic status and individual-level socio-economic status, it suggests that disadvantaged students fare worse. Consequently, segregation seems to be especially harmful for disadvantaged students — thus the term, “double jeopardy.”

Sui-Chu Ho and I used data from the National Educational Longitudinal Study to examine whether contextual effects were partially mediated by parents’ involvement in school.[ 15 ] We came up with three important findings: (1) the schools with high levels of parental involvement tend to be high socio-economic status schools, and vice-versa; (2) parental involvement has an overall positive effect on achievement (this is evident by comparing schools which have a mean socio-economic status near the national mean); and (3) the gradients tend to be shallower in high involvement schools than in low involvement schools. Thus, increased parental involvement in the school seems to not only raise achievement levels, but it also flattens the gradient.

If we consider parental involvement in school as a potent form of social capital, these cross-sectional findings illustrate two important points with respect to the formation of social capital. First, when people are segregated, either within or between communities, it is difficult for them to generate social capital. Second, in communities where there is a high level of social capital, outcomes are improved and inequalities reduced.

A multi-level framework for testing the three hypotheses

In most cases, the hypotheses presented in the examples above have been tested formally using multi-level regression models. Multi-level modeling, or hierarchical linear modeling (HLM), is a particular regression technique designed to take into account the hierarchical structure of nested data, such as when students are nested within schools, patients within hospitals, or citizens within communities.[ 16 ] An assumption underlying traditional regression approaches is that the observations are independent; that is, the observations of any one individual are not in any way systematically related to the observations of any other individual. This assumption is violated, for example, if some of the observed subjects are from the same family or, as in the examples above, from the same schools or communities. The use of traditional approaches usually yields biased estimates of the relationships among variables, and standard errors that are too small.

Multi-level modeling also provides a useful framework for incorporating aspects of human and social capital at more than one level. For example, when individuals participate in social clubs and form networks, this social capital may lead to a collective action that affects all members of a community, but it may also contribute to improving individuals’ efficacy and sense of belonging, resulting in increased participation at home and at work. Multi-level models provide a structure for thinking about such effects at different levels, and a means for testing relevant hypotheses. In educational research, researchers used to debate whether the student, the classroom, or the school was the appropriate level for analysis. But they realized that this was the wrong question, and called for techniques that explicitly modeled the multi-level structure of the data.[ 17 ] This “level-of-analysis” problem has been solved through advances in statistical theory and computing, and now computer programs that can be used to analyze multi-level data are widely accessible. With respect to social capital and its effects on sustained economic growth and well-being, these methods allow us to explicitly model different forms of social and human capital, conceptualized and measured at different levels of aggregation to estimate their effects on individuals’ social outcomes.

Evidence of community effects relevant to social capital in the area of education

If we are to understand the role of social capital on children’s development, we need to understand how it relates to some of the more proximal variables affecting children’s achievement. An important point, relevant to the Hypothesis of Community Differences, is that most of the action is at the classroom level. For example, in a study of children’s schooling outcomes in New Brunswick, I partitioned an array of schooling outcomes into district-, school-, classroom- and student-level components. The majority of variation was among students within classrooms, which is consistent with several studies of school effectiveness. However, for every outcome measure examined, there was considerably more variation among classrooms within schools than among schools, or among school districts. For example, 7 percent of the variation in mathematics scores was among classrooms, compared with only 4.7 percent among schools, and 1.8 percent among school districts. The results for reading science and writing scores indicated even greater variation among classrooms and less variation among schools. The same results were evident for affective outcomes describing children’s self-esteem, sense of belonging, general well-being and general health. Thus, in trying to understand the role of social capital, we might look first at classroom “communities.”

Research on schooling that has emphasized the importance of the learning environment in the classroom has identified several factors relevant to the role of networks and norms. A review of this literature by Scheerens identified “structured teaching” and “effective learning time” as the most important factors.[ 18 ] These two aspects of successful schools are captured by the term “academic press,” which is used in the literature to describe schools where principals and teachers project the belief that all students can master the curriculum.[ 19 ] Their high expectations are manifest in a number of teaching practices and school routines, including homework practices, the content and pace of the curriculum and how time and resources are used in the classroom.[ 20 ]

The research has also emphasized the importance of parental involvement, as discussed in the examples above. However, there has been little emphasis on the role that social capital might play on children’s behaviour. One of the most significant factors associated with classroom achievement is the disciplinary climate of the classroom,[ 21 ] but usually this is treated as having to do with the teacher’s management skills rather than peer networks or parents’ support of school norms. Also, we know relatively little about how social capital is distributed in segregated schooling systems, such as those where there is tracking or streaming.

Researchers have not paid much attention to variation among schools in their socio-economic gradients or the hypothesis of converging gradients. Lee and Bryk found that U.S. secondary schools differed significantly in their socio-economic gradients and in the achievement gap between minority and non-minority students.[ 22 ] They attributed the variation to various aspects of academic organization, including the extent to which schools differentiated students into various course-taking patterns. Small schools with less differentiation, on average, had shallower gradients. In our analysis of the Canadian TIMSS data, FremÂong and I found that classrooms varied significantly in their socio-economic gradients. Higher achievement was found in classrooms where there was less ability grouping and smaller class sizes.[ 23 ] We found a significant but modest negative correlation (-0.14) between adjusted levels of achievement and gradients. To summarize, there is strong evidence that gradients vary among classrooms, schools and school districts, but there have been only a few efforts to test the Hypothesis of Converging Gradients at various levels of the schooling system. One of the problems is that it is difficult to achieve a powerful enough research design to discern why gradients are steep or shallow in certain classrooms or schools.

Researchers have devoted considerable effort to testing the Hypothesis of Double Jeopardy, because it is relevant to questions about how students are allocated to schools, classrooms and instructional groups. There is unequivocal evidence that the average socio-economic status of a child’s class or school has an effect on his or her outcomes, even after taking account of individual-level ability and socio-economic status.[ 24 ] Sociologists have attributed contextual effects to peer interactions, and one could easily extend the idea to stress the importance of social capital. I have a relatively simple explanation. Suppose that roughly one quarter of the students in a community are vulnerable because of cognitive or behavioural problems. If one segregates the majority of these students into one side of the system through residential segregation, streaming, special programs for gifted students, or charter schools and private schools, then for teachers in that side of the schooling system, about one half of their students (about 12 to 15 students in a class with 24 to 30 students) will have special needs. In such circumstances, it is more difficult to effectively use support from parents, maintain high expectations, establish a positive disciplinary climate and have positive student-teacher interactions — all of the factors embodied in the concept of social capital.

Implications for OECD research on indicators and the study of social capital

There are at least six themes running through this paper relevant to our understanding of how social capital might affect sustained economic growth and well-being. First, it is a multi-level problem. Social capital is about relationships among people, and these directly affect the distribution of social outcomes at the micro level. Thus, before we can make much progress at the macro level, we need to understand how investments in social capital affect the social outcomes of individuals within the family, classroom, workplace and neighbourhood. But social capital is also about collective actions derived from relationships, and these affect the distribution of social outcomes at micro and macro levels. Bringing the two perspectives together requires a multi-level framework.

Second, children’s outcomes during the early years are the foundation of social and human capital for a society. Differences among communities in children’s cognitive and behavioural outcomes can be discerned as early as age 7, and probably earlier. We need a better understanding of how investments in social capital can be used to strengthen this foundation.

Third, successful societies are those that are successful in improving the social outcomes of their most vulnerable citizens. We need a better understanding of how investments in social capital are related to raising and flattening socio-economic gradients.

Fourth, the segregation of people along social class lines, or among racial and ethnic groups, affects the distribution of social outcomes. Given that social capital is about relationships among people, we need a better understanding of how it is formed and used in segregated and desegregated societies.

Fifth, the quality of social relationships appears to be more important than quantity. An understanding of the role of social capital requires an assessment of how social networks affect the processes that are proximal to social outcomes, such as social integration, social support, family functioning, intergenerational closure and micro-level personality variables (e.g., self-efficacy and self-esteem).

Sixth, social capital is embedded in the culture of a society and, therefore, affected by social, economic and historical factors. Achieving some purchase on the effects of social capital will require us to incorporate these factors into analyses. Progress in this vein would likely be furthered by assessments that enable us to understand how social capital and its relationship with social outcomes are distributed geographically within and between communities.

The macro-level analyses of the effects of social capital on economic growth and well-being have used rather crude indicators of social capital, such as “trust” and “transience,” and have been based mainly on data aggregated at a macro level (e.g., states and countries). My concern is that such indicators are highly correlated at these levels with other constructs that could give us a better purchase on how social and human capital affect economic growth and well-being. If we believe that social networks and collective actions affect social outcomes by increasing social support and social integration, or by reducing alienation and giving people a greater sense of control, then these are the constructs we need to measure. Moreover, the macro-level analyses do not capture the important processes at the levels of family and community where social capital is invested and transformed into other forms of capital that bear on social outcomes.

I believe that there are several ways that the OECD and its member countries can strengthen their large-scale assessments and monitoring programs to address this issue. Most of these are not expensive. First, we require an integrated set of longitudinal surveys that cover the life span from conception to old age. We are close to this in Canada with a set of about four or five longitudinal surveys being conducted by HRDC and Statistics Canada. Second, we need studies that also track “communities,” defined in different ways, so that we can discern whether changes in intercepts and gradients are related to changes in social capital, at the level of local communities. Third, we need to better integrate geography into our analyses. In virtually all of the research on school effectiveness we have treated schools as independent entities, without attention to their relationship to other schools in the community. I believe we could make a giant leap forward in this area if we had sufficient geographical data to conduct two kinds of analyses. One involves incorporating geography into the analysis to estimate spatial auto-correlation. The second entails estimating regressions at the local level to assess the extent of spatial non-stationarity, essentially by fitting a regression model separately within each local area.[ 25 ] For example, imagine the power of a map of Canada and the United States that displayed the relationship between social capital and health status, adjusted for socio-economic status, across local areas.

Fourth, we need to think harder about opportunities for natural experiments and case studies that borrow strength from and build upon the findings of our large-scale studies. For example, given the large disparities in mathematics achievement between Quebec and the rest of the country, I am curious whether these differences would be evident if we compared schools in close proximity but on opposite sides of the Quebec — New Brunswick and the Quebec — Ontario borders. Over-sampling these schools would enable a more powerful analysis, but we would probably learn more through case studies of particular communities.


*   J. Douglas Willms is with the Canadian Research Institute for Social Policy, University of New Brunswick. The author is grateful to Statistics Canada and Human Resources Development Canada for their support of research on child development and adult literacy, to the U.S. Spencer Foundation for its support of the research project, School and Community Effects on Children’s Educational and Health Outcomes; and the Canadian Institute for Advanced Research, which funds the NB/CIBC Chair in Human Development at the University of New Brunswick. The opinions expressed in this paper are attributable to the author, and do not necessarily reflect those of OECD, HRDC or the other agencies supporting this research.

 1.  E.E. Werner and R.S. Smith, Vulnerable but invincible: A longitudinal study of resilient children and youth (New York: McGraw-Hill, 1982).

 2.  OECD and Statistics Canada, Literacy, economy, & society: Results of the first international adult literacy survey (Paris, France: OECD and Ottawa, ON: Minister of Industry, 1995); Human Resources and Development Canada, OECD and Statistics Canada, Literacy skills for the knowledge of society: Further results from the International Adult Literacy Survey (Paris, France: OECD and Ottawa, ON: HRDC and Statistics Canada, 1997).

 3.  G. Frempong and J.D. Willms, “Can School Compensate for Socioeconomic Disadvantage?” in J.D. Willms (ed.), Vulnerable children: Findings from Canada’s National Longitudinal Survey of Children and Youth (Edmonton, Alberta: University of Alberta Press, forthcoming).

 4.  A.E. Beaton, I.V.S. Mullis, M.O. Martin, E.J. Gonzalez, D.L. Kelly and T.A. Smith, Mathematics achievement in the middle school years: IEA’s Third International Mathematics and Science Study (TIMSS) (Chestnut Hill, MA: Boston College, 1996).

 5.  J.D. Willms, Inequalities in literacy skills among youth in Canada and the United States, International Adult Literacy Survey no. 6. (Ottawa, ON: Human Resources Development Canada and National Literacy Secretariat, 1999).

 6.  J.D. Willms, “Indicators of mathematics achievement in Canadian elementary schools,” in Growing up in Canada: National Longitudinal Study of Children and Youth (Ottawa, ON: Human Resources Development Canada and Statistics Canada, 1996), pp. 69–82.

 7.  Hon. M. McCain and F. Mustard, Reversing the real brain drain: Early years study (Children’s Secretariat, Toronto, ON, 1999).

 8.  Adapted from Willms, Inequalities in literacy skills among youth in Canada and the United States, op cit.

 9.  Willms, ibid.

 10.  J.D. Willms, Literacy skills in Poland, report prepared for Statistics Canada and the World Bank (1999).

 11.  J.D. Willms and M.A. Somers, Schooling outcomes in Latin America, report prepared for UNESCO (1999).

 12.  W.B. Brookover, J.H. Schweitzer, J.M. Schneider, C.H. Beady, P.K. Flood and J.M. Wisenbaker, “Elementary school social climate and school achievement,” American Educational Research Journal, Vol. 15 (1978), pp. 301–318; V. Henderson, P. Mieszkowski and Y. Sauvageau, “Peer group effects and educational production functions,” Journal of Public Economic, Vol. 10 (1978), pp. 97–106; R. Rumberger and J.D. Willms, “The impact of racial and ethnic segregation on the achievement gap in California high schools,” Educational Evaluation and Policy Analysis, Vol. 14, no. 4 (1992), pp. 377–396; Y. Shavit and R.A. Williams, “Ability grouping and contextual determinants of educational expectations in Israel,” American Sociological Review, Vol. 50 (1985), pp. 62–73; A.A. Summers and B.L. Wolfe, “Do schools make a difference?” American Economic Review, Vol. 67 (1977), pp. 639–52.

 13.  J.D. Willms, “The balance thesis: Contextual effects of ability on pupils’ O-grade examination results,” Oxford Review of Education, Vol. 11, no. 1 (1985), pp. 33–41; J.D. Willms, “Social class segregation and its relationship to pupils’ examination results in Scotland,” American Sociological Review, Vol. 51 (1986), pp. 224–241; A. Gamoran, “Schools, classrooms, and pupils: International studies of schooling from a multilevel perspective,” in A. Gamoran (ed.), Schooling and achievement: Additive versus interactive models (San Diego, CA: Academic Press, 1991); A. Gamoran, “The variable effects of high school tracking,” Sociology of Education, Vol. 57 (1992), pp. 812–828; A.C. Kerckhoff, “Effects of ability grouping,” American Sociological Review, Vol. 51, no. 6 (1986), pp. 842-58; A.C. Kerckhoff, Diverging pathways: Social structure and career deflection (New York: Cambridge University Press, 1993).

 14.  Y. Dar and N. Resh, “Classroom intellectual composition and academic achievement,” American Educational Research Journal, Vol. 23 (1986), pp. 357–74; R. Dreeben and A. Gamoran, “Race, instruction, and learning,” American Sociological Review, Vol. 51 (1986), pp. 660–69; B. Rowan and A.W. Jr. Miracle, “Systems of ability grouping and the stratification of achievement in elementary schools,” Sociology of Education, Vol. 56, no. 2 (1983), pp. 133–144; R.E. Slavin, “Ability grouping and student achievement in elementary schools: A best-evidence synthesis,” Review of Educational Research, Vol. 57, no. 3 (1987), pp. 293–336; A.B. Sørenson and M. Hallinan, “Effects of race on assignment to ability groups,” in P.L. Peterson, L.C. Wilkinson and M. Hallinan (eds.), The social context of education (New York: Academic Press, 1984); J.D. Willms and M. Chen, “The effects of ability grouping on the ethnic achievement gap in Israeli elementary schools,” American Journal of Education, Vol. 97, no. 3 (1989), pp. 237–257.

 15.  E. Ho and J.D. Willms, “The effects of parental involvement on eighth grade achievement,” Sociology of Education, vol. 69 (1969), pp. 126-141.

 16.  A.S. Bryk and S.W. Raudenbush, Hierarchical linear models for social and behavioral research: Applications and data analysis methods. (Newbury Park, CA: Sage; H. Goldstein, Multilevel statistical models (2nd ed.) (London: Arnold, 1996)

 17.  L. Burstein, “The analysis of multilevel data in educational research and evaluation”, in D. C. Berliner (Ed.), Review of research in education, (Washington, DC: American Educational Research Association, 1980); L.J. Cronbach, J.E. Deken and N. Webb. Research on classrooms and schools: Formulation of questions, design, and analysis (Stanford, CA: Stanford University, 1976)

 18.  J. Scheerens, Effective schooling: Research, theory, and practice (London: Cassell, 1992).

 19.  C.S. Anderson, “The investigation of school climate,” in G.R. Austin and H. Garber (eds.), Research on exemplary schools (Orlando: Academic Press, 1985).

 20.  Anderson, op cit.; Dreeben and Gamoran, op cit.; I. Plewis, “Using multilevel models to link educational progress with curriculum coverage,” in S.W. Raudenbush and J.D. Willms, (eds.), Schools, classrooms, and pupils: International studies of schooling from a multilevel perspective (San Diego: Academic Press, 1991), pp. 149–166.

 21.  Willms and Somers, op cit.

 22.  V.E. Lee and A.S. Bryk, “A multilevel model of the social distribution of high school achievement,” Sociology of Education, Vol. 62, no. 3 (1989), pp. 172–192.

 23.  Frempong and Willms, op cit.

 24.  See op cit.: Brookover et al.; Henderson, Mieszkowski and Sauvageau; Rumberger and Willms; Shavit and Williams; Summers and Wolfe; Willms, 1985, 1986; Gamoran, 1991, 1992; Kerckhoff, 1986, 1993; Dar and Resh; Dreeben and Gamoran; Rowan and Miracle; Slavin; Sorenson and Hallinan; Willms and Chen.

 25.  A.S. Fotheringham, M. Charlton and C. Brunsdon, “Measuring spatial variation in relationships with geographically weighted regression,” in M.M. Fischer and A. Getis (eds.), Recent developments in spatial analysis (New York: Springer-Verlag, 1997), pp. 60–82.