Ten Hypotheses about Socioeconomic Gradients and Community Differences in Children's Developmental Outcomes - February 2003

3. Ten Hypotheses about Socioeconomic Gradients and Community Differences

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3.1 The Hypothesis of a Socioeconomic Gradient

There is a significant relationship between social outcomes and socioeconomic status

The most basic hypothesis about socioeconomic gradients is that there is a significant bivariate relationship between a particular social outcome and SES. This can be tested in a straightforward manner using ordinary least squares regression for continuous outcomes, or logistic regression analyses for dichotomous outcomes. For example, the relationship between children's receptive vocabulary and SES is statistically significant — the slope is 4.57, which is greater than zero (p < 0.05). However, usually the interest is in the gradients for individual communities within a larger unit, such as cities or health regions within a state or province, or schools within a school district. The national gradient for children's vocabulary shown in Figure 1 can be decomposed into a within-community gradient and a between-community gradient. The relative strength of these components has implications for social policy which will be discussed later.

3.1.1 An Example

Figure 2 shows the socioeconomic gradients for children's receptive vocabulary for 18 Canadian cities (light lines) and five UEY "communities" (darker lines), based on data from the NLSCY and the UEY surveys. The 18 cities were selected because they had sample sizes of at least 80 children, and included at least one city from each of the ten Canadian provinces. The five UEY communities were the first five communities to participate in the UEY survey. They included a rural community in Southwest Newfoundland; Prince Edward Island; children served by a school district in the inner city of Winnipeg, Manitoba; Prince Albert, Saskatchewan; and an area of Coquitlam, British Columbia. The measures of children's vocabulary and socioeconomic status are the same as in Figure 1 with the national scaling for the NLSCY applied to the five UEY communities.

3.1.2 Statistical Analysis

In a multilevel framework, a separate regression equation is fit to the data for each community:

           Within-Community Equation (1)

where Y_i is a person's outcome score, X_i is their score on the SES measure. The intercept, ß₀, is the expected outcome score for a person who has a score of zero on X_i. In most multilevel models, X_i is "centred" on a particular value, such as the national mean, so that a value of zero on X refers to a hypothetical person with particular characteristics. In the example, SES is centred on the 1998 national mean, and thus the ß₀ for each community is the level of gradient. The parameter ß₁ is the slope of the socioeconomic gradient. It is an estimate of the expected change in the outcome score Y_i for a one-unit change in X_i. The parameters, r_i, are the residuals; that is, the deviation of people's scores from the regression line. The strength of the gradient, as gauged by the proportion of variance in the outcome measure explained by SES (i.e., R²), is the difference between the variance in Y_i and the variance of the residuals expressed as a fraction of the variance in Y_i.

With j communities (in the example, j = 23) one can write j such equations:

           A Set of Within-Community Equations (2)

where the subscript j has been added to each element of equation 1. Therefore, one now have j different ß₀'s, one for each community, and j different ß₁'s. The ß_0j's are the levels of the socioeconomic gradients, and the ß_1j's are the slopes of the socioeconomic gradients for the set of communities. The ß_0j's can be expressed as an average ß₀, called y₀₀, plus the deviation of each community's ß_0j from the average:

           Among-Community Equation for Levels of the Gradients (3)

where y₀₀ is the grand mean, or the mean of the community means, and u_0j is the deviation from each community's mean from the grand mean. Similarly, the slopes of the gradients vary among communities, and can be expressed as an average slope plus a deviation from the average slope:

           Among-Community Equation for Slopes of the Gradients (4)

where y₁₀ is the mean of the community slopes, and u_1j is the deviation from each community's slope from the mean slope.

The Gradient Hypothesis is that the average socioeconomic gradient across the communities is statistically significant; that is, that y₁₀ is significantly different from zero:

           The Gradient Hypothesis (5)

which is assessed with a t-test with j-1 degrees of freedom. Note that in this case, the slopes were allowed to vary; that is, there is a different slope for each community.

3.2 The Hypothesis of Community Differences

Communities vary in their social outcomes even after taking account of individual's socioeconomic background

This hypothesis arises from research on school effectiveness where researchers are interested in whether the outcomes of students with differing family backgrounds vary in their achievement scores across schools (Raudenbush & Willms, 1995). In the first instance, one can simply compare the mean vocabulary scores across the 23 communities. They indeed vary substantially: among the 18 cities from 97.8 for Vancouver to 105.2 for Quebec, and among the five UEY communities from 94.5 for Prince Albert to 105.5 for PEI.

The analysis of within-community gradients also allows one to determine whether communities differ in their outcomes after taking account of the socioeconomic status of the children's families. One can draw a vertical line at an SES value of zero. The point at which the line intersects a community's gradient is the expected outcome score for a child with average socioeconomic characteristics; that is, the levels of the socioeconomic gradients. In this example, the expected scores vary from 95.5 (Vancouver) to 104.0 (Quebec) among the cities, and from 99.9 (Prince Albert) to 106.4 (PEI) among the UEY communities. A formal test of the "hypothesis of community differences' can be tested within a multilevel framework. In this example, the community differences in their outcomes are statistically significant, even after taking account of SES.

3.2.1 Statistical Analysis

The statistical model described by equations 2 to 4 is also used to test this hypothesis. The Hypothesis of Community Differences holds that the levels of the gradients vary significantly, which is equivalent to stating that the variance of the

u_0j's is greater than zero. This between-community variance is referred to as tau (), and the hypothesis is:

           Hypothesis of Community Differences (6)

It is easier conceptually to formulate multilevel models as separate within- and between-community equations, as specified in equations 2 to 4. However, the estimation of these models entails the substitution of equations 3 and 4 into equation 2 to yield a model with both within- and between-community residuals. These can be easily estimated with available software such as HLM (Raudenbush, Bryk, Cheong, & Congdon, 2001) or MLN (Goldstein et al., 1998). Estimation of the model represented by equations 2 and 3 for the example yielded a grand mean, y₀₀ = 100.41 (s.e. = 0.79), and = 10.87. The test of whether is greater than 0 is assessed with a chi-square (X²) test, with j-1 degrees of freedom. For the example, the null hypothesis that is equal to zero is rejected, and therefore one can conclude that these communities differ in their level of receptive vocabulary scores, even after taking account of family SES.

3.3 The Hypothesis of Diminishing Returns

The relationship between social outcomes and socioeconomic status is weaker at higher levels of socioeconomic status

This hypothesis holds that the gradient rises steadily with increasing levels of SES, but gradually tapers off at a higher level of SES. This is an important hypothesis, because if there are diminishing returns above some level of SES, it would suggest that one could improve social outcomes for the least advantaged through policies which reduced inequalities in SES. For example, if one examined the separate components of SES — income, education, and occupation — and found that there was a curvilinear income gradient, with diminishing returns above a certain income, it would suggest that one could reduce inequalities through policies that redistributed wealth. This appears to be the case for certain health outcomes, at least in the U.S. Among US adults who earn less than $20,000, an increase in income is associated with markedly better health outcomes. However, above that threshold, increasing income has only a marginal effect on health status (Epelbaum, 1990; House et al., 1990; Mirowsky & Hu, 1996; Rogot et al., 1992). Similar research in Canada on health outcomes generally support the hypothesis, but the extent of curvilinearity is not as marked (Boyle & Willms, 1999; Wolfson et al., 1999), such that one can not easily identify an income threshold. The relationship between life expectancy and Gross National Product (GNP) at the level of countries is also curvilinear, with diminishing returns at levels above $5,000 (World Bank, 1993).

The hypothesis of diminishing returns has not received much attention with respect to children's cognitive or behavioural outcomes. Willms (2002a) reported small but statistically significant curvilinear income gradients for receptive vocabulary skills at ages 4 and 5, and for mathematics skills in the primary grades, but the extent of curvilinearity was negligible. The results of PISA for the reading scores of 15-year old youth suggest that there is slight curvilinearity supporting the diminishing returns hypothesis in a few countries, but in most countries the gradient is decidedly linear (OECD, 2001).

Willms and Somers (2001) found that the parental education gradients for children's reading and mathematics scores were curvilinear in several Latin American countries, but the curve increased with increasing levels of parental education. It appears that there is some minimum level of parental education necessary for children to benefit from elementary schooling in these countries. Similar findings for youth's reading performance were found for Mexico and Brazil in the PISA, but these may be attributable to a "floor effect" on the reading test.

In the example describing children's vocabulary skills, the national socioeconomic gradient is slightly curvilinear, indicating a diminishing return relationship (see Figure 1). The socioeconomic gradients for most of the communities are quite linear (see Figure 2). However, the socioeconomic gradient for Ottawa-Hull is also a good example — the slope of the gradient becomes increasing more gradual as SES increases, such that there is virtually no relationship for families with SES scores above 1.0.

3.3.1 Statistical Analysis

The most common approach to testing this hypothesis is simply to add a quadratic term for SES into the within-community model:

           Within-Community Equations

with Curvilinear Term (7)

where the ß_2j's are the curvilinear terms for the socioeconomic gradients. These can be expressed as an average effect, y₂₀, and the deviation of each community from the average effect:

           Among-Community Equation for Curvilinear Gradients (8)

where y₂₀ is the mean of the curvilinear effects, and u_2j is the deviation of each community's curvilinear effect from the mean effect. One can test whether the mean effect is statistically significant:

           Hypothesis of Diminishing Returns (9)

which is assessed with a t-test with j-1 degrees of freedom (in this case, with the slopes allowed to vary). One can also test whether the curvilinear effect varies significantly among communities:

          Hypothesis of Varying Diminishing Returns (10)

The hypothesis of diminishing returns does not hold for the data describing the 18 Canadian communities. The coefficient for SES-squared is -0.272, which is not statistically significant (p = 0.17). Also, the coefficients for SES-squared did not vary significantly (p = 0.20) across the 18 communities. Thus, a model with only the linear component of SES provides a better fit to these data.

3.4 The Hypothesis of Converging Gradients

Variation among communities in their social outcomes decreases with increasing levels of socioeconomic status

This hypothesis also has important implications for policy analysis. If socioeconomic gradients converge at higher levels of SES, it suggests that successful communities are those that have been successful in bolstering the social outcomes of their least advantaged citizens. At the national or provincial and state levels, evidence of converging gradients would suggest that jurisdictions with low and steep socioeconomic gradients should emphasize policies that are targeted at improving the outcomes of people from lower socioeconomic backgrounds, while those with high and gradual socioeconomic gradients should emphasize more universal interventions aimed at increasing levels of performance for all citizens.

An analysis of youth literacy skills based on data from the International Adult Literacy Study (IALS) revealed a marked pattern of converging gradients, not only among OECD countries, but also among provinces and states in Canada and the US (Willms, 1999a). Also, for the US, the analysis found that the literacy skill gaps between African American and white youth, and between Hispanic and white youth, were smaller in some states than in others, and the states that had high overall levels of literacy skills were those that had been successful in closing the achievement gap between minority and non-minority youth.

The converging gradient hypothesis was tested for a set of 31 elementary schools in one school district in British Columbia (Willms, 2001c), for which longitudinal data on children's achievement were available covering the period from grade 3 through to the end of grade 7, as well as data on children's general cognitive ability at the end of grade 3. The hypothesis also held for "ability" gradients on the rate at which children acquired literacy skills during the elementary years: children with high levels of cognitive ability tended to increase their cognitive skills at a relatively fast pace in all schools, whereas children with low cognitive ability tended to learn at a fast pace in some schools but not in others. The schools which were most successful, as gauged by the rate at which children acquired skills, were those that were successful with the least able pupils. These school-level results, although not based on a nationally representative sample, are important vis-à-vis the macro-level IALS results because they were based on children's growth trajectories in literacy skills over a five-year period, rather than on their level of skills at some particular point during their schooling careers.

However, the pattern of grade 3 and 4 reading and mathematics skills for Latin America, based on the UNESCO-OREALC study, does not support the converging gradient hypothesis (Willms & Somers, 2001). The analysis indicated that gradients were relatively low and gradual in most countries (Bolivia, Columbia, Dominican Republic, Honduras, Paraguay, Peru, and Venezuela) and somewhat higher but with steep slopes in other countries (Argentina, Brazil, and Chile, and to some extent Mexico). Cuba stood alone with a high, gradual socioeconomic gradient. This pattern was also evident for the PISA data — Brazil and Mexico had relatively low but gradual gradients compared with the OECD countries participating in the study (OECD, 2001). It may be that as countries develop their education systems, the performance levels of high SES students increase first, either because of uneven investments in the public sector or because of a growing private sector. Later, as the systems develop further, they are able to achieve relatively high levels of performance for all pupils, as demonstrated by the success of Cuba.

The results presented in Figure 2 for the SES gradients of receptive vocabulary skills across Canadian cities also do not support the converging gradients hypothesis. Indeed, the slopes are nearly parallel, and do not vary significantly across the 18 communities. The correlation between the levels of performance and the slopes is -0.26, which is not statistically significant.

3.4.1 Statistical Model

The model described by equations 2, 3, and 4 above is fitted to the data to test this hypothesis. The estimation of this model entails estimation also of the variance of the gradient levels [i.e., Var (u_0j)], the variance of the gradient slopes [i.e., Var (u_1j)], and the covariance of the levels and slopes [i.e., Cov (u_0j,u_1j)]. In the first instance, one tests whether the slopes vary significantly among communities:

           Hypothesis of Variable Slopes (11)

Assuming the slopes vary significantly among communities, one then wants to test also whether the covariance between levels and slopes is statistically significant:

           Hypothesis of Converging Gradients (12)

3.5 The Hypothesis of Contextual Effects or Double Jeopardy

The average level of socioeconomic status of a community has an effect on social outcomes over and above the effects associated with individuals' socioeconomic status

The presence of a socioeconomic gradient indicates that there is a relationship between social outcomes and SES. However, there is often an additional effect, called a "contextual effect", associated with the SES of the community. Research in the sociology of education has been concerned with contextual effects for at least twenty-five years, because they are directly relevant to issues concerning the manner in which students are allocated to schools, classrooms and instructional groups. This research has provided convincing evidence that there are contextual effects associated with the demographic characteristics of a classroom or school, over and above the effects associated with an individual's family background. The research indicates that when children are segregated, either between schools through residential segregation or the "creaming" of the most able pupils into selective schools (Brookover et al., 1978; Henderson, Mieszkowski, & Sauvageau, 1978; Rumberger & Willms, 1992; Shavit & Williams, 1985; Willms, 1986), or between classes through tracking or streaming (Gamoran, 1991, 1992; Kerckhoff, 1986), children from advantaged backgrounds do better, while those from disadvantaged backgrounds do worse. This is called the "hypothesis of double jeopardy" because it suggests that children from low SES families tend to be disadvantaged because of their circumstances at home, but when they are also segregated into low SES schools they are likely to fare even worse.

In most studies of school effects, the data are not sufficient for testing whether contextual effects are stronger for children with differing levels of SES, because it requires sufficiently large samples within groups to achieve accurate estimates of the slopes of the gradients for each school. The hypothesis is that contextual effects tend to be stronger for minority students or children from low SES backgrounds, which could be called the "hypothesis of triple jeopardy". The interactions between family SES and group mean SES may favour advantaged students because when high SES families are confronted with unfavourable contexts, they are able to draw on other resources (e.g., extra tutoring or more help with homework) to ensure their child does not suffer. Contextual effects were evident for every country examined in the PISA study (OECD, 2001). Moreover, they tended to be slightly larger for males and for youth from lower SES backgrounds. Similar results were found for the educational attainment of school-leavers for a small sample of Scottish schools (Willms, 1985).

There has been very little research aimed at uncovering the causal mechanisms associated with contextual effects. The early research argued that contextual effects were a proxy for peer effects that occur when bright and motivated students work together (Heath, 1984). However, schools or classrooms with high social-class or high ability intakes tend to have several advantages associated with their context (Willms, 1986; Willms & Somers, 2001; OECD, 2001): they are more likely to attract and retain talented and motivated teachers, and on average they are more likely to have greater support from parents, fewer disciplinary problems, and generally an atmosphere conducive to learning. Also, there has been relatively little research on the contextual effects for communities other than schools or classrooms, or for social outcomes other than school achievement. In research on health outcomes, the hypothesis would hold that people who are poor are more likely to have health problems, but suffer double jeopardy if they also live in a poor community. A few recent studies suggest there are contextual effects associated with neighbourhood deprivation (Sloggett & Joshi, 1998; Yen & Kaplan, 1999).

3.5.1 Statistical Model

The hypothesis of double jeopardy is tested in a multilevel analysis by asking whether school mean SES is related to the SES-adjusted level of outcomes. The model is identical to that described by equations 2 to 4, except that equation 3 is extended to include the mean level of SES as a predictor of the levels of the community gradients:

           Hypothesis of Double Jeopardy (13)

where y₀₀ is the mean of the community means adjusted for both individual level SES and the mean level of SES of the community, y₀₁ is the contextual effect associated with the mean SES of the community, and is the mean SES of the community. In estimating with individual-level data aggregated to such models, it is common practice to estimate the group level. However, in many cases it would be preferable to use census data, or reliable data from other sources. Moreover, one can also estimate the effects of other variables describing group context, such as the percentage of minorities in the community. The level of aggregation is also an issue. In studies of school effects, the school and classroom are appropriate levels for assessing contextual effects. In studies of child development, the local neighbourhood may be a more appropriate level.

The hypothesis that the contextual effect varies depending on the level of SES of the individual is tested as a "cross-level interaction", that is the slopes of the gradients are regressed on group mean SES:

           Hypothesis of Triple Jeopardy (14)

For the data in the example, the effect of mean SES is -5.30, which is statistically significant (p < .01). The direction of the contextual effect is opposite to the double jeopardy hypothesis, as it indicates that low SES children have better early vocabulary skills in low SES communities. In this instance, the negative contextual effect may be essentially an urban-rural effect, whereby children living in rural areas, which tend to have a lower SES, develop more quickly than would be expected given their family SES. The hypothesis pertaining to triple jeopardy cannot be tested with these data, as the results indicated that the slopes did not vary significantly across communities.

3.6 The Hypothesis of Relative Deprivation

Communities that are relatively homogeneous in their socioeconomic status tend to have superior social outcomes than those that are relatively heterogeneous in their socioeconomic status

This hypothesis stems mainly from research on health outcomes, particularly the work by Wilkinson (1992, 1996, 2000). He observed that the relationship between life expectancy and income was much weaker across countries than within them, and concluded that what matters most is a person's relative status or position within a society. He showed that a country's life expectancy was related more closely to the extent of income inequality, not its absolute level of income. This finding was supported by analyses of adults' life expectancy across states in the US, which also uncovered a strong relationship between mortality and income inequality (Kaplan et al., 1996; Kennedy, Kawashi, & Prothrow-Stith, 1996). Similar findings have been observed for US cities (Lynch et al., 1998), and local authorities in the UK (Ben-Schlomo, White & Marmot, 1996).

However, these findings have been based on aggregate data, and therefore are subject to criticism. The principal argument is that the relationship between mortality and income in many jurisdictions is curvilinear, with income having a stronger impact among poor people than among rich people, and because jurisdictions with greater income inequality have disproportionately more people at the extremes of the income distribution, societies with greater income inequality will on average have higher mortality rates (Gravelle, 1998). Deaton and Lubotsky (2001) have shown that a measure of racial inequality accounts for the effects of income inequality on mortality among US cities and states.

The argument underlying the relative deprivation hypothesis is that relative status is related to health because it is proxy for people's autonomy: those with a low sense of control over their lives suffer more stress which in turn affects their health (Marmot et al., 1997; Syme, 1996). The relative deprivation argument is supported by smaller-scale studies, including the Whitehall study (Marmot, et al., 1991), that have examined people's relative status. Ross et al. (2000) observed a weaker relationship between income inequality and mortality among Canadian cities compared with the US. They maintain that the relationships between income inequality and health stem mainly from an inadequate provision of goods and services to meet the needs of the poor in US cities. However, the extent of income inequality in Canadian cities is much smaller than most US cities, and the range in income inequality is much less. Thus, the Canada-US comparisons do not provide any purchase on the causal mechanisms contributing to the observed effect of income inequality.

These debates call attention to the need for analyses of socioeconomic gradients using individual and community-level data in a multilevel framework. In the first instance one needs individual-level data for cities, such that the within-city socioeconomic (and income) gradients can be estimated. These analyses would furnish estimates of the health outcomes of people at differing levels of SES. It is then possible to enter SES (or income) inequality at the city level, and estimate its effect. One could also discern whether the effect of inequality varied for people with differing levels of SES. Finally, one could enter other individual- and city-level factors, such as sense of control or service provision, to determine the extent to which they mediated the outcome-inequality relationship, or operated as independent factors.

3.6.1 Statistical Model

The hypothesis of relative deprivation is tested in a multilevel analysis by adding into the second level a term describing the within-group variation of SES:

           Hypothesis of Relative Deprivation (15)

where y₀₂ is the effect associated with SES inequality. Z_j is a measure of SES inequality, such as the standard deviation of SES. There are several measures of income inequality, and they tend to yield similar results in analyses of health outcomes (Kennedy et al., 1996). It is preferable to include also a measure of the mean level of SES, as in equation 14. This is not essential if the measure of relative deprivation is uncorrelated with the mean level of SES, but in most cases the theory underlying the analysis would call for it.

For the example pertaining to children's early vocabulary, a measure of the standard deviation SES was used as an indicator of inequality. The standard deviation of SES is negatively related to receptive vocabulary scores (y₀₂ = -6.80, se = 8.36), which is consistent with the hypothesis of relative deprivation. However, the effect is not statistically significant (p=0.43).

3.7 The Hypothesis of SES by Group Status Interactions

The gradients for low-status groups tend to be lower and steeper than the gradients for high-status groups

3.7.1 Moderators and Mediators

Before discussing this hypothesis, it would be useful to discuss the concepts of moderator and mediator with respect to socioeconomic gradients. In psychological and health research, the term "moderator" has been used to indicate that the magnitude of an effect varies across levels of another variable, whereas the term, "mediator" refers to variables which explain how or why one variable is related to another (Baron & Kenny, 1986; Chaplin, 1991; Rothman & Greenland, 1998). However, the distinction between moderators and mediators can be confusing, as it depends largely on whether one can maintain there is a causal relationship between the predictors and the social outcome, or at least specify the temporal sequence of the variables (Kraemer, Stice, Kazdin, Offord, & Kupfer, 2001). Kraemer et al. (2001) have brought some precision to the terms moderator and mediator by distinguishing them from proxy, overlapping, and independent risk factors, and by clarifying the role of each kind of factor in data analysis. With respect to socioeconomic gradients, these terms are as follows:

1. A proxy risk factor would be a strong correlate of SES, which has a relationship with a social outcome mainly through its association with SES. The proxy factor may or may not precede SES temporally, and generally the strength of the SES relationship with the outcome would dominate that of the proxy factor. For example, in many large-scale educational surveys, data are collected on "possessions in the home" as an indicator of wealth. Whether a family "owns a dish-washer" may be related to a child's achievement score, but this only occurs because of its correlation with SES. One is not usually interested in such factors, unless they are used in a composite scale to represent income when direct assessment is impossible.
2. An over-lapping risk factor would be one that is correlated with SES, and occurs contemporaneously with SES (or at least the temporal sequence cannot be determined or is irrelevant to the causal link between SES and the outcome). Thus, it has the same status as SES as a potential risk factor. Ethnicity, family structure and maternal age could be considered as over-lapping risk factors, and treated in analysis as factors which are correlated with SES, and have effects that overlap with SES.
3. An independent risk factor is one that also occurs contemporaneously with SES, or is at least irrelevant with respect to the causal link, and is uncorrelated with SES. The child's gender is a good example, as family SES does not cause gender, and SES and gender are uncorrelated. Gender is a risk factor for many childhood outcomes, favouring girls in some cases, and boys in others.
4. A moderator is a factor which affects the relationship between another variable and the social outcome. With respect to gradients one is concerned with whether SES is a moderator. One could conceive of two sub-populations, for example, children from high SES and low SES families. If some other factor of interest, for example "positive parenting", has a different relationship with the social outcome for the two sub-populations, one would say that SES moderates the effects of positive parenting. This is an important kind of interaction, because if certain factors are risk factors for high SES groups but not low SES groups, or vice versa, one is closer to understanding the underlying causal mechanisms. Kraemer et al. (2001) require that for SES to be a moderator, it must precede the moderated risk factor, and be uncorrelated with it.
5. A mediator is a factor which is influenced by SES directly, and helps to explain why there is a relationship between SES and the social outcome. For example, parents' income, education, and occupational status may have an effect on parents' ability to be warm and nurturing parents, to be engaged with their child, or to function as a cohesive family unit. They may also affect the mental health of the parents. If so, one would expect to observe a correlation between SES and these factors. Based on previous research, one would also expect them to be significantly related to many childhood outcomes. One would say that these variables — positive parenting, engagement, family functioning, and parental depression — mediated the socioeconomic gradient if (i) they were significantly related to the social outcome, (ii) they were correlated with SES, and (iii) they either partially or totally accounted for the SES-outcome relationship.

The hypothesis of SES-by-group-status interactions maintains that there is an interaction between SES and group status in their effects on social outcomes, and more specifically that the gap in social outcomes between high- and low-status groups is greater at lower levels of SES. Previously, one might have considered SES as a moderator of the group status effect, but as Kraemer et al. (2001) point out, this concept is not useful when SES does not precede the factor of interest. In their sense, then, this hypothesis is about the dominance or co-dominance of two over-lapping factors. Generally, one wants to discern whether there are significant interactions between SES and gender, SES and ethnicity, and SES and other factors describing group status.

The hypothesis of SES-by-group-status interactions has been stated in a specific direction, positing that low-status groups are likely to have especially poor social outcomes if they are of low SES. The argument is that children from low-status groups often experience racial and ethnic prejudice, which has a negative effect on their social outcomes. This effect is likely to be greater in families with low SES, as they do not have the economic and social capital to help their children overcome these effects. Research on sector differences in literacy skills between Catholic and Protestant youth supports this hypothesis: Catholic youth had poorer literacy skills than Protestant youth, and the gap was greater for youth with lower SES backgrounds (Willms, 1998). Generally, however, one is interested in whether there is an interaction, whatever its direction.

3.7.2 Statistical Model

The hypothesis of SES by group-status interactions is tested by adding to the level 1 model (equation 3) a dummy variable denoting group status, and a variable constructed as the product of SES times group status:

           (16)

where X_1ij is SES, X_2ij is a dummy variable denoting group status (e.g., minority = 1; non-minority = 0), and X_1ij X_2ij is the SES by group-status interaction. The ß_2j's can be expressed as an average minority gap, y₂₀, plus the deviation of each community's ß_2j from the average; and similarly the ß_3j's can be expressed as an average interaction, y₃₀, plus the deviation of each community's ß_3j from the average. The hypothesis of SES by group-status interactions is then:

           Hypothesis of SES by Group-Status Interactions (17)

An Example. For the example in this study, the analysis examined whether children whose parents had immigrated to Canada within the previous five years, and children whose parents had immigrated more than five years previously, had lower vocabulary skills than those whose parents were non-immigrants. The analysis indicated that on average, within communities, the vocabulary scores of children of recent immigrants were 14.6 points lower, and those of established immigrants were 4.2 points lower, than non-immigrants. Figure 3 displays the average within-community gradients for the two groups. Note that the range of SES is truncated at the 5^th and 95^th percentiles of SES for each group. The estimate of the SES-by-immigrant interactions were 2.7, which is in the direction hypothesized, but not statistically significant (p = 0.06), and 0.8 (also non-significant).

3.8 The Hypothesis of Family- and Community-Level Mediators

Variation within and among communities in their social outcomes is attributable to the independent actions of individuals. Variation among communities is also attributable to features of the community which direct and shape individuals' actions

Coleman (1988) describes two dominant approaches to research in the social sciences, which are evident in the study of childhood development. One approach, commonly taken by psychologists and economists, emphasizes the independent actions of individuals, particularly parents. It assumes that parents make independent decisions to achieve what they perceive to be best for their family — what economists call "maximizing utility". Research on child development has strived to identify "risk factors", such as poverty or inadequate parenting, that are associated with undesirable childhood outcomes. Another approach, more characteristic of the work of sociologists, stresses the importance of social context in shaping, constraining, and redirecting individuals' actions. Researchers following this line of inquiry maintain that people's individual choices depend on the norms of their immediate community, and the kind of social support available to them. Until recently, most of the research on how social context affects children's development has been at the micro-level. There has been relatively little work concerned with the effects of community factors, and much of it has been directed at understanding the effects of neighbourhood poverty (see Brooks-Gunn, Duncan, & Aber, 1997).

The application of hierarchical models to the study of gradients and community differences provides a means for researchers to bring these two approaches together. The approach is straightforward: one simply introduces potentially mediating individual-level factors into the (level 1) model describing within-community relationships (equation 3), and community-level factors into the (level 2) model describing between-community relationships (equation 4).

3.8.1 Statistical Model

The hypothesis of an individual-level mediator is tested by adding the potential mediator to the level 1 model (equation 3):

          (18)

where X_1ij is SES, X_2ij is the potentially mediating factor. The ß_2j's can be expressed as an average effect across all communities, y₂₀, plus the deviation of each community's ß_2j from the average, U_2j (as per equation 4). The primary criterion for a mediator is that it be related to the outcome, even after controlling for SES:

           Hypothesis of Individual-Level Mediator (19)

The hypothesis of a community-level mediator is tested by adding the potential mediator to the second-level equation for the ß_0j's:

          (20)

where y₀₀ is the mean of the adjusted community means, and u_0j is the deviation from each community's adjusted mean from the grand mean. One is primarily interested in the magnitude and statistical significance of the coefficient, y₁₀:

          Hypothesis of Community-Level Mediator (21)

3.8.2 An Example

For the NLSCY and UEY data there are four individual-level factors that are potential mediators of the socioeconomic gradient. They include measures describing positive parenting practices and family functioning, which were scaled to range from zero to 10; a measure of the amount of time parents spent reading to their child, measured in occasions per week; and a dichotomous measure denoting whether the mother suffered from depression. The potential community-level mediators include measures of social support, the quality of the neighbourhood, and stability of the neighbourhood. Support and quality were also measured on scales which ranged from 0 to 10 at the individual level. Neighbourhood stability is the average number of years the people in the neighbourhood had lived at their current address. The model also included a measure of the mean SES of the community, as in the contextual effects model described above. The results are displayed in Table 1.³

Table 1 The Relationship Between Children's Receptive Vocabulary and Socioeconomic Status and Family and Community Factors

	Model I	Model II	Model III
Child's Sex and Family Background
Socioeconomic Status	4.85	4.87	4.20
Female		0.65	0.49
Number of Brothers and Sisters		-1.83	-1.74
Single Parent Family		0.19	0.62
Immigrated within Last Five Years		-13.10	-12.43
Family Processes
Positive Parenting Practices			-0.24
Reads to the Child			0.62
Family Functioning			0.41
Maternal Mental Health			-0.34
Community Factors
Mean Socioeconomic Status	-3.46	-3.52	-3.76
Social Support			0.70
Neighbourhood Quality			0.27
Neighbourhood Stability			0.14
Note. Figures in bold text are statistically significant at p < .05.

Model I is the contextual effects model, which includes only SES and the mean level of SES at the city level. It provides an estimate of the average within-community SES slope, which is 4.85. The effect associated with mean SES is -3.46, which indicates that each one standard deviation increase in SES is associated with about a 3.5 point decrease in average receptive vocabulary scores.⁴

Model II includes variables denoting the child's sex, the number of siblings the child has, whether it is a single or two-parent family, and whether the family had immigrated within the past five years. The number of brothers and sisters has a large and significant negative relationship with receptive vocabulary scores: each additional sibling is associated with a decrease in scores of about 1.8 points. Also, as was evident in Figure 3 above, children from immigrant families have somewhat lower receptive vocabulary scores than their non-immigrant counterparts. The gap is about 13 points. The results also indicate that girls scored slightly higher than boys, but the difference was not statistically significant. The effects of living in a single parent family were negligible.

Model III introduces the seven factors describing family processes and community factors. The results indicate that reading to the child is an important determinant of receptive vocabulary scores: an increase of one extra reading session per week is associated with a 0.62 point increase in receptive vocabulary. Also, a one-point increase on the family functioning scale is associated with a 0.41 point increase in receptive vocabulary scores. The effects of maternal depression and positive parenting practices were not statistically significant. The effect associated with social support, measured at the community level, was also large and statistically significant: each one-point increase on the ten-point scale was associated with an increase of 0.70 points in receptive vocabulary. The effects of neighbourhood quality were not as large, and the effect was not statistically significant (p = 0.08). The effects of neighbourhood stability were also large and statistically significant: an increase of one year in the average time people had lived at their current address was associated with a 0.14 increase in receptive vocabulary scores.

The coefficient for SES when these seven potential mediating variables were not in the model was 4.87 (s.e. = 0.24). The SES effect was reduced to 4.20 (s.e. = 0.21) after inclusion of these variables. This is a statistically significant reduction in the slope of the SES gradient, and therefore one can claim that these factors partially mediated the SES gradient. However, the extent of mediation is slight — less than 20 percent — and therefore one could conclude that these factors operate largely as independent risk factors.

3.9 The Hypothesis of Spatial Auto-Correlation

Successful communities tend to be in close proximity to other successful communities

This hypothesis holds that the level of a socioeconomic gradient for a community is correlated with the levels of socioeconomic gradients for neighbouring communities. This is called spatial auto-correlation (Cliff & Ord, 1973; Haining, 1997). The same hypothesis would hold for the slopes of the gradients. For example, suppose a school were successful in its mathematics achievement, given the SES of its pupils; that is, the level of its gradient were relatively high compared with other schools in the province. One might expect that neighbouring schools would also be relatively effective for several reasons. The students attending the school would likely be living in areas with similar socioeconomic circumstances to students in neighbouring areas. There may be a "diffusion of best practice" because teachers at the school discuss practices with teachers in neighbouring schools. The students' parents might share similar attitudes towards schooling as the parents of students in neighbouring schools. However, despite the potential for school results to be spatially related, there are no studies in the literature on school effectiveness that examines these relationships. Indeed, one of the assumptions underlying models for estimating school effects (e.g., Raudenbush & Willms, 1995) is that the school-level residuals are independent (e.g., the u_oj in equation 3). If there is spatial auto-correlation, this assumption is violated, and the model will yield biased estimates.

A simple test for the presence of spatial auto-correlation can be conducted as follows: (1) Determine which communities are "neighbours" for each community with available data. In the case of schools, this could be defined as, say, the three closest schools, or perhaps all schools within a certain radius of a school. For communities, it is common practice to consider all communities that are geographically contiguous to a community to be its "neighbours". This is called the first level of contiguity. One could also consider as neighbours all the first-level contiguous communities and those communities that were contiguous with them — this is called the second level of contiguity. (2) Estimate the socioeconomic gradients for each community, applying the techniques described above. (3) For each community, estimate the average level and the average slope for its neighbouring communities. (4) Estimate the correlation of the levels of the gradients with the average levels for neighbouring communities. Do the same for the slopes. If the hypothesis of converging gradients holds, the correlations would be positive and statistically significant.

If there is auto-correlation, one might try to "fix" it. There are a number of spatial regression approaches which incorporate spatial information, with the aim of producing unbiased estimates of the desired coefficients (Cliff & Ord, 1973; Ord, 1975).

However, one can also view the auto-correlation as symptomatic of a mis-specified model (Miron, 1984). Taking the schooling example above, one could introduce measures of neighbourhood socioeconomic characteristics, schooling practices, and parents' attitudes, and examine the extent to which these variables accounted for the spatial auto-correlation. If these covariates were themselves spatially correlated, and related to the social outcome of interest, they would reduce the extent of auto-correlation among the levels and slopes of the gradients. From this perspective, spatial auto-correlation is welcome, as it opens up the possibility of identifying new risk and protective factors, and testing hypotheses about causal mechanisms in a more powerful way. Some of the most important factors related to social outcomes may have explicit spatial characteristics, such as a family's proximity to a neighbourhood park, or the distance a child must travel to a local school. One of the assumptions underlying the relative deprivation hypothesis in population health research is that an individual's feeling of being relatively deprived, and the accompanying feelings of lacking control over one's life (Syme, 1996) or being socially isolated (House, Williams, & Kessler, 1987), contribute to the relative deprivation effect. If this is the case, then one could also test hypotheses about the effects of relative deprivation measured locally (Gatrell, 1997); for example, by estimating the effects of a measure of the difference between the SES of a community and the average SES of its neighbouring communities. It is these kinds of analyses that are required to resolve the relative deprivation debate discussed above.

3.9.1 An Example

For the example used in this study, it is not particularly interesting to examine the spatial auto-correlation hypothesis with respect to the location of the 23 communities. However, the UEY data provide an opportunity to examine the hypothesis as it pertains to local neighbourhoods. For these data, the enumeration area (EA) where each child resided was available. The EA is a geographical unit which on average comprises about 400 families. For each of the five UEY sites, the extent of auto-correlation among the levels of the socioeconomic gradients was estimated, following the procedures above. (There were insufficient data within each EA to obtain accurate estimates of the within-EA slopes, and thus these were treated as fixed effects.) The correlations between estimated levels (the ß₀'s of equation 3) and the average of the estimated levels of the contiguous EAs for each community were as follows: Southwest Newfoundland: -0.043; Prince Edward Island: 0.018; children served by a school district in the inner city of Winnipeg, Manitoba: 0.050; Prince Albert, Saskatchewan: -0.049; and the area of Coquitlam, British Columbia: -0.038. In all cases, the correlation coefficients were not statistically significant. These rather disappointing findings are discussed in the concluding section.

3.10 The Hypothesis of Stable Gradients

Socioeconomic gradients tend to be stable over time

This hypothesis is that societies establish certain tolerable equilibria for inequalities in social outcomes, which are maintained by powerful economic and political forces. Research on the relationship between health outcomes and wealth has shown that certain diseases, including lung cancer, heart disease, and HIV infection, were initially diseases of the rich, but over time became diseases of the poor, with a socioeconomic gradient consistent with other diseases (Deaton, 2002; Preston, 1974). Research on schooling suggests that there are similar forces at play which result in a stability of socioeconomic gradients. Heath (1990) argued that the gap in educational attainment in Britain had been relatively constant throughout the twentieth century. However, McPherson and Willms (1986) showed that the comprehensive school reforms of the late 1960s and early 1970s, which called for the abolishment of selective schooling in favour of comprehensive schools, had the effect of raising and flattening socioeconomic gradients. The analyses were based on longitudinal data for Scotland where the reform was embraced by all local educational authorities (McPherson & Willms, 1986). Despite the success of comprehensive reforms, the Government of the United Kingdom introduced parent choice of schools in 1980 which allowed parents to choose schools outside their designated catchment areas. They observed that middle class parents were more likely to exercise choice, and that they disproportionately chose schools with high social-class intake (Echols, McPherson, & Willms, 1990; Willms & Echols, 1992). This may well have pulled socioeconomic gradients back towards the equilibria described by Heath.

An assessment of the stability of socioeconomic gradients requires data describing the same communities over time (see Willms & Raudenbush, 1989). For example, school districts often collect achievement data annually or biennially for all students at particular grade levels. Similarly, it will be possible to assess changes in socioeconomic gradients for school performance for countries participating in the OECD PISA, as comparable data are being collected triennially (OECD, 2001).

3.10.1 Statistical Analysis

The statistical analysis for data collected at two time points is straightforward. One simply creates a dummy variable at the individual level denoting whether the individual was sampled at time 1 or time 2. One then simply stacks the data for the two cohorts, with the appropriate community-level identifier to link data by community. At the individual level, one then sets out a model similar to equation 15:

           (21)

where X_ij is SES, W_ij is a dummy variable denoting year (e.g., base year = 0; follow-up = 1), and X_ij W_ij is the SES-by-year interaction. The ß_2j's represent the changes in the level of the gradient from baseline to follow-up, and are expressed as an average increase (or decrease), y₂₀, plus the deviation of each community's ß_2j from the average. The ß_3j's represent the changes in the slope of the socioeconomic gradient, and are expressed as an average change, y₃₀, plus the deviation of each community's ß_3j from the average. The hypothesis of stable gradients is then:

           Hypothesis of Stable Levels (22)

and

          Hypothesis of Stable Slopes (23)

When data for three or more successive cohorts are available, one can introduce cohort as a level in the analysis, such that one has children within communities, communities over time, and communities. These models can afford very powerful tests of the effects of community level factors, as one can then estimate whether changes in community-level factors are related to changes in childhood outcomes. These models are discussed by Willms & Raudenbush (1989). An example of the test for stable gradients based on the NLSCY data is provided by Willms (2001d).

• ³The community-level indicators of social support, neighbourhood quality, and stable neighbourhood were measured at the level of enumeration area (EA), which is a geographical unit comprising on average about 400 families. Consequently, the analysis for this example required a three-level hierarchical model (communities, EAs, and children). This is not particularly relevant to the expository aspect of this paper, and therefore to avoid distracting the reader from the central issue, the three sets of equations are not specified. These can be seen in Bryk and Raudenbush (2002, Chapter 8).
• ⁴This estimate differs from the results of the two-level model presented in the section on contextual effects presented above, because the information contained in the EA level data accounts for some of the contextual effect.