You must be logged in to Tag Records. The Decomposition of Sociology 2. Disenthralling Sociology 3. Sociology and Subjectivity 4. Fascism, Communism, and Social Theory 5. From Socialism to Sociology 6. Scientific Access and Political Constraints 7. Public Choice and the Sociological Imagination 8. Social Contexts and Cultural Canons 9.
Reconstructing the Social Sciences Policy Research in a Post-Sociological Environment Prediction and Paradox in Society Freedom, Planning, and the Moral Order Social Disputations and Moral Implications Equation 3 can then be expressed as 12 where. The quantities R M , R T , and are all scalars. Based on equation 13 , the k th weight component for E is obtained as 14 Similarly, the k th weight component for C is given by 15 where. Thus, the composition weights reflect the contribution of the k th covariate to the Taylor approximation of E E T as determined by the magnitude of the group difference in means weighted by the reference group's effect.
Similarly, the coefficient weights reflect covariate k 's contribution to C T as determined by the magnitude of the group difference in the effects weighted by the comparison group's mean. The weights are invariant to change in the scale of the covariates. The raw difference can now be expressed in terms of the overall components as a total of weighted sums of the unique contributions: 16 This weighting method gives results that are nearly identical to the sampling and randomization procedures outlined earlier as long as enough samples are drawn.
Many applications ignore the sampling variability of the decomposition components e. The characteristics and effects components do not provide information about the precision of the contributions to group differences per se. For this reason, it is important to gauge the sampling variability of E and C in substantive applications. Because the components used in the decomposition are functions of maximum likelihood estimates, the delta method described by Rao —23 can be used to derive asymptotic standard errors of the detailed contributions.
Interval estimation and significance testing can be done in the usual way e.
This approach utilizes expressions for the first derivatives i. E and C , along with the detailed contributions, E k and C k , are nonlinear functions of the maximum likelihood estimates b. For the models considered here 19 which has a convenient form owing to the assumption of Poisson sampling. Following the same logic, the coefficient component can be written as the sum of individual contributions as 21 Each covariate's contribution to the overall coefficient component depends on the parameter vectors, b B and b W.
Significance tests on individual components, blocks of components, or the overall decomposition as a whole can be carried out using Wald tests by redefining E and C to include a subset of the original set of terms along with the corresponding submatrices of var b B and var b W.
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The variance estimates derived above assume that the independent variables are fixed and that groups are independent. They will underestimate the true variances if this is not the case. It would also be possible to obtain a bootstrapped distribution of the components by applying a repeated modeling approach. An alternative to bootstrapping or a full Bayesian approach is to simulate the distributions of each component by drawing M parameter vectors for each group, carrying out the decomposition on the simulated parameter vectors, and obtaining means and variances of the resulting distributions of the decomposition components.
Under this approach, the decomposition is carried out M times, resulting in a posterior predictive distribution for each quantity in the decomposition e. Statistical inference can be carried on the quantities from the resulting distributions. In particular, to what extent is the racial difference in rates attributable to compositional differences in predictors—such as what might be reflected by group differences in socioeconomic resources and family structure—and to differences in the effects of these predictors i.
The decomposition is carried out at the aggregate and detailed levels, thus allowing an assessment of the contribution of each model predictor to the racial gap. For research on first nonmarital fertility transitions, this type of analysis provides a way to assess the contributions of socioeconomic background and family structure, whose effects and distributions differ by race.
We adopt a parsimonious model specification using covariates that have been widely used in past research including 1 family background characteristics mother's education, adjusted family income, 7 and number of older siblings and 2 family structure characteristics mother's age at respondent's birth, proportion of years living in single mother family, and number of family changes up to the time of the event or before age 18, whichever occurs first.
To facilitate the presentation of results, we express this difference as Table 1 presents covariate means and model estimates for each group as well as the crude rates per and race differences in rates. Table 2 provides the detailed decomposition obtained by averaging the results of separate decompositions with interchanged reference and comparison groups.
Under the current model, compositional differences between blacks and whites i. We first discuss the contributions of the substantive predictors to the overall premarital birth rate gap. We discuss the baseline hazard contribution later. Table 2 shows the detailed decomposition for the family background and family structure variables. However, whites experience fewer of these transitions on average than blacks, with means of 0. The results in Table 2 show that with respect to the white reference group, this compositional disadvantage for black women contributes 0.
From Table 2 we see that the difference in family income by race accounts for 4. Among the compositional factors considered here, making family incomes and number of older siblings in the comparison population blacks equal to that of the reference population whites would produce the largest reductions in the racial gap in the premarital birth rate. A similar interpretation applies to the effects component, C k. For example, whites and blacks experience different returns to maternal education.
Based on the results from the proportional hazards model in Table 1 , each additional year of mother's schooling reduces the risk of premarital birth by Differences in returns to maternal education, as well as differences in the effects of maternal age at respondent's birth, are the largest contributors to the overall gap. It is probably safe to say that changing behavioral responses presents a more challenging task from a policy perspective than equalizing socioeconomic resources across groups.
Compositional components involving the dummy variables for the age intervals that define the baseline hazard play the same role as the constant term in a standard multivariate decomposition. The piecewise constant hazard model effectively partitions the constant term into several pieces, with individuals differing on the number of pieces they contribute.
In standard models, the mean value of the constant is always 1 and the difference in means across groups is always 0. For the decomposition of the piecewise constant hazard model, the characteristics effects associated with the pieces of the baseline hazard reflect race differences in the distribution of exposure, which in turn is a function of race differences in the age distribution of events and censoring.
The fact that the characteristics effects of the baseline hazard reported in the first panel of Table 2 are at first negative and then positive reflects the fact that the age distribution of events is centered at a younger age for black women and at an older age for white women.
The usefulness and uselessness of the decomposition of tobit coefficients
Taken together, group differences in the logged baseline hazards i. We find that the largest contributors to the differential are attributed to differences in the baseline hazards pertaining to the first three age intervals, which reflect the race differences in the underlying rates for teenagers. Adding these types of interactions presents no additional difficulty in the decomposition procedure per se.
However, the characteristics effects for an interaction term will reflect differences in exposure in age interval l in addition to differences in characteristics for those at risk in age interval l. However, the coefficient effects have a straightforward interpretation. We refer to this as Model 2. Table 3 provides the relevant income effects for both groups as well as the decomposition results from Model 2 and from the original model Model 1. For black women, increased income has no significant effect on the risk of premarital births at older ages. Race differences in effects of income at older ages account for a small portion of the total gap, and equalizing income effects for older women would be expected to increase the gap in the premarital birth rate by less than 1 birth per women.
Under Model 2, differences in characteristics account for These results are similar to those from Model 1 Multivariate decomposition provides a way to partition an observed group difference in statistics into portions that can be attributed to group differences in characteristics, or endowments, and to group differences in effects, or coefficients from a multivariate model. The statistics of interest could be means, proportions, counts, and as shown here rates.
Multivariate decomposition of a difference in means involves a straightforward substitution technique based on results from OLS regression models, and has been widely used over the past several decades. Techniques for multivariate decomposition for nonlinear models have been developed more recently, but have not thus far been extended to hazard rate models.
Our proposed method involves a simple parametric model with a flexible functional form for the baseline hazard. Due to its popularity, a multivariate decomposition technique based on the Cox regression model would seem particularly attractive. However, applying our decomposition approach to the Cox model poses several problems. Because the baseline hazard is unspecified in Cox regression, it is not directly available as a decomposable part of the model. Moreover, although we could retrieve the baseline hazard using numerical methods, there is no guarantee that a difference in the hazard at age t across groups would exist due to group differences in the ages that events occur.
For example, in the NLSY, the youngest and oldest event ages are Therefore, applying this approach to a Cox regression model would require that the number of unique event times be the same and that the event times are equal for both groups. Despite these difficulties, there may be alternative decomposition techniques that are better suited to the Cox model. Here we offer a practical alternative to approximate the baseline hazard of the Cox regression model using a step function defined along a common set of cut points.
The decomposition of differences in predictions from discrete time logit hazard models is straightforward. The empirical probabilities to be differenced take the form 28 Some minor changes to the formulas for the standard errors of the decomposition quantities are needed. List this Seller's Books. Payment Methods accepted by seller. Bookseller: Wormhole Books Address: Mt. Items related to The Decomposition of Sociology. Stock Image.
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