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117 NOTE S L 1Usage of the term "model" varies widely within and between sciences. m e model of this chapter consists, at a minimum, of a set of structural equations. A more encompassing use of the term "model" would include not only the structural equations, but the verbal definitions of the vari- ables, and the reasoning behind assignments of exogeneity, endogeneity, and mixed status (endogenous with respect to some variables, exogenous with respect to others). A stir' more encompassing use of the term "model" would include the reasoning and hypotheses about coeff icient variability across contexts, or the lack thereof . 20ur use of terms such as "cause,n Causality," and "causal links" is compatible with longstanding tradition in the study of problems not generally susceptible to experimental manipulation. Recognizing that there will always be philosophical dispute in science about causality, it should be noted that the intellectual position manifested in this report can be traced at least as far back as the work of Durkheim (1897~. Other relevant contributors to this general stance include, among many others, Wright {1921, 1923), Simon (1952, 1953r 1954~; Tukey (1954~; Blalock (1961~; Fisher (19611; and Duncan (1975~. 3More formally, we exploit Nelder and Wedderburn's (1972) distinction between systematic and error components in generalized linear models. Our presentation is in terms of the systematic component of each struc- tural equation. In actual empirical research, estimation of the full set of structural equations will be problematic because the nature of the error components will not necessarily be the same for each equation. However, recent statistical progress, summarized and synthesized by Winship and Mare (1981), should eventually mitigate estimation problems. Since the term "linear" in generalized linear models" may be confusing to some, it should be noted that ~linear" here means linear in the Parameters; explanatory variables may have nonlinear effects on the response variables. Where we postulate linearity and additivity, we do so for two reasons. First, for discrete response variables, we wish to think of the functional relationships in linear model terms (rather than "multiplicativen). Thus, for example, we prefer to treat a discrete response variable in terms of a "log-linear. equation in which the ~response" is a logit, as distinguished from a multiplicative equation in
118 which the "response" is a probability. Second, and more important, when we postulate linearity and additivity, we do so because we think it is substantively appropriate. Certain, but not all, equations contain nonlinearity and interaction terms. Where this occurs, the increased complexity is always justified by substantive and theoretical considerations. 4Although it is tempting to think of this profile as monotonically increasing with time, this need not be the case. Trends in knowledge and practice can diverge in the short if not in the long run because of such factors as changes in the political climate. ith appropriate assumptions about the errors, equations (1.16) and (2.1) form a mixed model. Of particular relevance here is that the l-term can be thought of either as arising from uncertainty in the specification of (2.1) (see, e.g., Lindley and Smith, 1972), or as random error occasioned by the sampling of countries from a super-population. For the purposes of the present discussion of substantive, theoretical hypotheses about the systematic effects of macro variables, the distinc- tion is not critical. It does matter for estimation. Due to limitations of time, funding, the pace of statistical developments, and availability of appropriate computer programs, the estimation results to be presented later in this chapter are by no means optimal with respect to either view of macro error. Future research on this subject will take advantage of some estimation progress for mixed models of the type this chapter rests on (Wong and Mason, forthcoming). 6Note the connection between our hypothesis about the lack of systematic setting variability in the SES and AFB effects on EF, and the hypothesis developed here about setting variability in the EF intercept. If the former hypothesis is wrong, this should be most clearly and meaningfully demonstrable for S. given the validity of what we have argued about the strategies of family planning programs in the Third World. For this reason, we examine the macro data for a potential relationship between the micro socioeconomic coefficients of the EF equation and level of socioeconomic development (for which Table 2.2 contains the only zero entry) . 7The abbreviations used in this chapter appear in the Glossary. 8This is not necessarily the most extreme contrast. Its use is nonetheless valid and appropriate for present purposes, if not optimal. This is addressed further in subsequent discussion. 9Tables 2.6 to 2.8 deliberately do not include standard errors of regression coefficients, although they report three levels of p-values. The reasons for this are as follows. First, most of the socioeconomic explanatory variables are polytomous, and the standard errors would refer to contrasts. The contrasts in this instance would be those between the included categories of a polytomy and the category excluded for purposes of estimation (see, e.g., Johnston, 1972:1781. Standard errors for these
119 particular contrasts are not necessarily relevant for present substantive purposes. Instead of presenting such standard errors, Tables 2.6 to 2.8 report the results of logically prior global tests of the null hypothesis that all coefficients for a polytomy are simultaneously zero, against the alternative that at least one of them differs from zero. A second reason for not reporting standard errors is that there is no cross-country or within-country inference we wish to make which depends crucially on reporting point estimates of standard errors. A third reason for not presenting standard errors is that efficient estimates of them are not currently available base Wbng and Mason, forthoom~), given that our purview is ultimately that of the mixed (hyperparameter} model. A fourth reason for not reporting standard errors is specif i ~ to Table 2 ~ 8 ~ which contains interactions involving polytomies. In this instance, the stan- dard errorsto£ specific contrasts are of little value because it is the entire pattern of interaction that must be apprehended, and not whether one particular coefficient differs "significantly. from zero--i.e., from the effect of being in the omitted category. 10Purging the intercept for Korea of its religion component is essentially the first step in renormalizing the religion coefficients into the form of weighted deviations from the grand mean. Since it would be confusing to present the coefficients for religion or ethnicity in this normalization while maintaining the coefficients for other variables in the dummy coding normalization, we do not show the religion or ethnicity coefficients. Tn any case, these are nuisance parameters for immediate purposes, and are insignificant in the AFB, EF, and LF regressions for Korea. 11TO illustrate the potential relevance of other macro variables, it may be noted that Peru does appear to have shorter average breastfeeding than Korea; this difference could account for the difference in EF intercepts between the countries (World Fertility Survey, 1979, 1980~. 12Th is result may be due to imperfect control for the pace of fertility and may arise, in effect, from an omitted variable. A better specifica- tion might well produce the expected results, and our later analyses will address this issue. 13The term "conditional coef f ic lent" may be new to some readers; the following example may help to clarify it. The regressions presented in Table 2.8 specify that EF interacts with both WED and HOCC. This means that there is an EF coeff icient for each cell in the cross-classif ication of WED by HOCC. Each of these coeff icients is a conditional coeff icient. 14That the cell sizes are small for occupational categories, given that EF = 0, can be seen by noting that the standard deviation for EF is 1.3 4 . Thus EF = 0 is more than two standard deviations from the mean of EF. 15Sharp diminution of socioeconomic effects on fertility was found in Taiwan for the parents of couples interviewed in the early 1970s. This is compatible with our expectations of cohort differences in a society
120 moving from the more traditional to the more transitional end of the setting continuum. See Hermalin et al. {1982:262, fin. 17~. Although it is possible and desirable to indicate the magnitudes of the standard errors of the estimated coefficients as part of these plots, this has not been done due to software, time, and funding limitations. Future research will present such information, and will do so for coefficients estimated using more efficient procedures than were available when the work reported here was in progress. 17Mauldin et al. (1978) report this correlation to be .58, computed over many more countries. The bivariate distribution of countries is: Prouram Strength Strong Other Social Setting High Social Setting Other 6 4 3 2 The association between these two dichotomized variables in the Mauldin et al. (1978) data is, however, positive.
