Contextually Specific Effects and Other Generalizations of the Hierarchical Linear Model for Comparative Analysis

Beginning of article

1 INTRODUCTION

This article extends the hierarchical--or multilevel--linear model in ways that enhance its usefulness for quantitative comparative analysis based on relatively large numbers of contexts (e.g., countries, states, cities). The substantive focus is on comparative analysis of cohort human fertility during the 1970s in countries that were among the poorest during the 1970s and earlier. We address the important policy question of whether official government family planning programs in such countries had an impact on cohort fertility during the 1970s. To provide a better answer to this question than has heretofore been possible using cross-national information, it is necessary not only to model key aspects of cumulative fertility from a perspective in which policy considerations are one component, but also to develop further the analytic tools for quantitative comparative analysis.

In this article we extend the model of Mason, Wong, and Entwisle (1983) for comparative analysis to allow for contextually specific variables and for restrictions on coefficients and micro error variances. In its full generality, our extension has not previously been formulated. The notion of contextual specificity is central to the persistent debate in the social sciences over whether comparative analysis is possible between even two countries, much less among many countries. The argument against comparative analysis is that it cannot take account of factors that are truly noncomparable. If this argument were valid, the social sciences would be unable to draw conclusions beyond the bounds of a single context (e.g., a single society).

Ethnic group membership is potentially important for comparative analysis, yet it is not generally comparable across societies. Many phenomena--including fertility--are known or thought to vary in part as a function of ethnicity. The contextual specificity of ethnicity might, therefore, preclude comparative analysis of fertility. Thus the question, can ethnicity be incorporated formally into comparative analyses based on large numbers of countries? We propose treating ethnicity as contextually specific, which would permit the meaning of the ethnic dimension to vary between countries. Our approach is to incorporate the notion of contextually specific variables into a general linear hierarchical model. Contextual specificity can be used with other variables besides ethnicity and in situations in which noncomparability is not the primary issue. Our other extensions of the hierarchical linear model (restrictions on coefficients and micro error variances) are primarily for obtaining greater efficiency in estimation and do not involve fundamental alteration of the multilevel model.

We turn next to a treatment of contextually specific variables that is motivated entirely by the need to incorporate ethnicity into quantitative comparative analysis. We then present a substantive model of fertility that allows ethnicity to have different meanings in different societies and also addresses the policy question of whether government-sponsored family planning programs had an impact on cumulative fertility during their early years (primarily the 1970s). This is followed by presentation of the statistical model, presentation of findings, and further discussion.

2. ETHNICITY AND QUANTITATIVE COMPARATIVE ANALYSIS

Apart from small and relatively isolated population groupings, most, perhaps all, contemporary societies display ethnic or religious diversity (Yinger 1985). The importance of ethnic group identity (and religious group identity, which is sometimes fused with ethnicity) for societal functioning has varied historically, and varies now, within and between societies. Ethnicity and religion cannot, therefore, be ignored in comparative analysis. But how is the existence of ethnic or religious identification to be comprehended? Should such bases of identification define groups that become units of analysis in their own right, and if so, what are the implications of this view for formal models of comparative analysis? Alternatively, if societies or their subunits, or political units more generally, constitute the social systems to be compared, how does ethnic or religious group identification become an integral feature of the modeling framework on which comparative analysis is based? We describe several alternative approaches, all of them related to a multilevel model. For this purpose, it is helpful to examine a simple case.

Consider the following micro specification, in which individuals are nested within contexts and there are three variables, Y, a micro response variable; [X.sub.1]u a micro re-gressor; and Gt, a macro regressor:

[Y.sub.ij] = [[beta].sub.0j] + [[beta].sub.1j][X.sub.ij1] + [[member of].sub.ij], (2.1)

where [Y.sub.ij] is an observation on the dependent variable for the ith individual in the jth context, [X.sub.ij1] is a regressor evaluated for the same individual, and [[member of].sub.ij] ~ N(0, [[sigma].sub.j.sup.2]) is a random error term. We assume that all of the micro errors are independent. In this setup, there are [n.sub.j] individuals in the jth of J contexts.

At the context (macro) level, there are two equations:

[[beta].sub.0j] = [[eta].sub.00] + [[eta].sub.01][G.sub.j1] + [[alpha].sub.0j], (2.2)

and

[[beta].sub.1j] = [[eta].sub.10] + [[eta].sub.11][G.sub.j1] + [[alpha].sub.1j], (2.3)

for which we assume that [[alpha].sub.j] = ([[alpha].sub.0j], [a.sub.lj]) are bivariate normal vectors with mean (0, 0), V([[alpha].sub.0j]) = [[gamma].sub.00], V([[alpha].sub.1j]) = [[gamma].sub.11], and cov([[alpha].sub.oj], [[alpha].sub.ij]) = [[gamma].sub.01]. We also assume that the macro disturbances are independent of the micro errors and that they are independent across contexts, so that

cov([[alpha].sub.j], [[alpha].sub.j'] = 0, j [not equal to] j'. (2.4)

The assumption of between-context independence of the macro errors seems plausible in many applications, such as when contexts are widely dispersed societies. This independence assumption is not always valid, however. We next take up this point as well as others in a consideration of alternative ways in which to expand the basic multilevel model to include ethnic or religious group membership.

There are at least five different ways of contending with ethnic or religious group membership. Although each has its appeal, one seems most desirable for unrestricted comparisons. We next present these alternatives, reserving treatment of our preferred strategy for last. In the following discussion we focus on ethnic groups, although our points apply to religious groups and other contextually specific bases of differentiation, such as regions.

Two of the approaches incorporate group ties into the macro level. The first treats ethnic groups as contexts. The second uses political definitions (e.g., societies) to identify contexts and then characterizes contexts by their ethnic composition. The remaining three approaches deal with ethnic group membership at the micro level.

2.1 Groups as Contexts

One response to the question of ethnicity is to treat the ethnic groups themselves as contexts (e.g., Hirschman and Rindfuss 1982; Lesthaeghe and Eelens 1989). There are two basic problems with this approach. One of them is statistical; the other concerns availability of data. First, assumption (2.4)--which is an assumption of no spatial autocorrelation--will be invalid, since a pair of ethnic groups within the same society will be more alike than a random pairing from different societies. The cross-group disturbance co-variances are inestimable unless heavily constrained. It is not obvious, however, what constraints would be optimal or how they could be validated in actual applications. Second, there may be problems with obtaining economic, public health, and other kinds of macro information for ethnic groups. Wong and Mason (1989) discussed these and other problems in more detail.

2.2 Societies as Contexts, Ethnicity Measured by Composition Variables

What might be termed a traditional approach to the incorporation of ethnicity is to use individuals and societies as the two basic units of analysis and insert compositional variables (i.e., aggregated micro variables) into the macro equations. Thus, for example, Equations (2.2)-(2.3) might contain a set of terms describing the percent of observations in the first ethnic group, the second ethnic group, and so on. This approach suffers from two problems. First, ethnic detail is completely absent from the micro level, so that the effects of composition are subject to classical aggregation bias. Second, the approach presumes that it is possible to define an adequate ethnic classification that applies to all contexts. We have been unable to do this in our substantive research and question the feasibility of developing such a classification. Certainly, for the sociodemographic application that is the basis for the work reported here, we have been unable to arrive at a useful classification scheme. For these reasons, we next consider micro-level approaches for coping analytically with the existence of ethnic group ties.

2.3 Purging Ethnicity From the Micro Model

At the micro level, there are several ways to take ethnicity into account. The simplest of these is to purge it explicitly from the multilevel model in a context-by-context fashion. This is not the same as ignoring ethnicity. Rather, one simply partials Y and [X..sub.1], for ethnicity, so that the micro coefficient linking Y and [X.sub.1] is net of ethnicity. This is problematic. First, the procedure does not incorporate ethnicity into the full multilevel model and is thus ad hoc; in fact, maximum likelihood ethnicity estimates for a given context depend on both micro and macro data from other contexts, as we shall show in Section 6. Second, such an ad hoc procedure is defined for a linear regression model, but it is not general; for instance, it cannot be extended to the logistic response multilevel model (Wong and Mason 1985).

2.4 General Ethnicity Classification

A second approach to micro-level incorporation of ethnicity into a multilevel model is to develop a classification of ethnic groups that can be applied to each society. With this universal classification, each micro observation would be uniquely assigned to a category in the exhaustive scheme. In fact, this strategy is the basis of the approach described in Section 2.2 for composition indicator variables at the macro level, since it, too, requires a set of ethnic categories common to all societies. We reject this approach at both the macro and the micro levels for the same reason: We have yet to encounter, nor have we been able to devise, a satisfactory pan-ethnic or religious classification. Thus we accept a fundamental point in the argument against comparative analysis. It may sometimes be inappropriate to treat sociodemographic phenomena as having invariant meaning or definition across societies, particularly when such phenomena are closely linked to culture, as in the case of ethnicity. This does not, however, invalidate comparative analysis, as we argue next. [Mason (1991) discussed the problem of "meaning invariance" in quantitative comparative analysis in more detail.]

2.5 Ethnicity as a Contextually Specific Variable

The last approach we consider for incorporating ethnicity into a multilevel model, and the one we develop in detail, is to treat ethnicity as a contextually specific dimension and to define the units of analysis as individuals within societies. Using this strategy, we define ethnic groups in as much detail as needed, subject to the inevitable restrictions imposed by sample size, and allow ethnicity to appear additively or interactively in combination with the other micro regressors [just [X.sub.1]in the case of Eq. (2.1)]. At the same time, ethnically specific parameters at the micro level are not modeled at the macro level. Thus we continue to model the [[beta].sub.0j] and [[beta].sub.1j], controlling where necessary for ethnic group membership.

The obvious advantage of treating ethnicity as a contextually specific dimension within societies is that ethnicity is included in the multilevel model in as much detail as needed. If ethnicity interacts with [X.sub.1] in the determination of Y, interactions are included; otherwise they are not. At the same time, variability across societies in intercepts and slopes (of [X.sub.1]) is modeled. No information is lost in this formulation. Moreover, it provides a convenient format in which to determine the nature and extent of ethnic differentiation in each society, for the dependent variable of interest. We next present a detailed exposition of this perspective on the incorporation of ethnicity at the micro level.

3. INCORPORATING ETHNICITY INTO THE MICRO SPECIFICATION

To understand how ethnicity can be incorporated into the micro level of a multilevel model, uniquely within each society, consider a context in which there are two ethnic groups, denoted as a and b. If, conditional on ethnic group identity, Equation (2.1) is a valid representation of the process to be modeled, then, for the jth context, the maximal specification is

[Y.sub.ija] = [[beta].sub.0ja] + [[beta].sub.1ja][X.sub.ij1a] + [[member of].sub.ija] (3.1)

and

[Y.sub.ijb] = [[beta].sub.0jb] + [[beta].sub.1ja][X.sub.ij1b] + [[member of].sub.ijb]. (3.2)

Now, how are we to extract from (3.1)-(3.2) a single intercept and a single X-effect in the presence of contextually specific ethnic groups? That is, how is contextually specific ethnicity to be operationalized in a multilevel model? We answer this question in stages.

To begin with, we propose decomposing the [beta]'s as

[[beta].sub.0ja] = [[beta].sub.0j] + [[xi].sub.0ja], (3.3a)

[[beta].sub.0jb] = [[beta].sub.0j] + [[xi].sub.0jb], (3.3b)

and

[[beta].sub.1ja] = [[beta].sub.1j] + [[xi].sub.1ja], (3.4a)

[[beta].sub.1jb] = [[beta].sub.1j] + [[xi].sub.1jb], (3.4b)

so that [[beta].sub.0j] and [[beta].sub.1j] enter Equations (2.2) and (2.3), respectively, and not the ethnically specific [[xi].sub.j] That is, we propose defining for each context an intercept and slope around which the coefficients of the ethnic groups are dispersed. Since the ethnic groups are in general defined uniquely for each context, contrasts in their intercepts and slopes cannot be conceptualized as varying across contexts.

The reparameterization of the ethnically specific intercepts and slopes requires constraints on the [[xi].sub.j] so that [[beta].sub.0j] and [[beta].sub.1j] have meaning. To keep from linking [[beta].sub.0j] and [[beta].sub.1j] to a specific ethnic group, it is necessary to avoid a boundary point normalization (e.g., setting [[xi].sub.oja] = [[xi].sub.1ja] = 0). We thus choose the restrictions

[w.sub.ja][[xi].sub.0ja] + [w.sub.jb][[xi].sub.0ja] = 0

and

[w.sub.ja][[xi].sub.1ja] + [w.sub.jb][[xi].sub.1jb] + 0,

where the [w.sub.j] are known weights of the [[xi].sub.j]. From (3.3a) and (3.3b),

[w.sub.ja][[beta].sub.0ja] = [w.sub.ja][[beta].sub.0j] + [w.sub.ja][[beta].sub.0j] + [w.sub.ja][[xi].sub.0ja

and

[w.sub.jb][[beta].sub.0jb] = [w.sub.jb][[beta].sub.0j] + [w.sub.jb][[xi].sub.0jb];

from (3.4a) and (3.4b),

[w.sub.ja][[beta].sub.1ja] = [w.sub.ja][[beta].sub.1j] + [w.sub.ja][[xi].sub.1ja]

and

[w.sub.jb][[beta].sub.1jb] = [w.sub.jb][[beta].sub.1j] + [w.sub.jb][[xi].sub.1jb].

Adding separately these equations for intercepts and slopes yields

[[beta].sub.0j] = [[[w.sub.ja][[beta].sub.0ja] + [w.sub.jb][[beta].sub.0jb]]]/[[w.sub.ja] + [w.sub.jb]],

and

[[beta].sub.1j] = [[[w.sub.ja][[beta].sub.1ja] + [w.sub.jb][[beta].sub.1jb]]]/[[w.sub.ja] + [w.sub.jb]].

It is natural to choose [w.sub.ja] = [n.sub.ja]/[n.sub.j] and [w.sub.jb] = …