Finite mixture models are well known to have poorly behaved likelihood functions featuring singularities and multiple optima. Growth mixture models may suffer from fewer of these problems, potentially benefiting from the structure imposed on the estimated class means and covariances by the specified growth model. As demonstrated here, however, local solutions may still be problematic. Results from an empirical case study and a small Monte Carlo simulation show that failure to thoroughly consider the possible presence of local optima in the estimation of a growth mixture model can sometimes have serious consequences, possibly leading to adoption of an inferior solution that differs in substantively important ways from the actual maximum likelihood solution. Often, the defaults of current software need to be overridden to thoroughly evaluate the parameter space and obtain confidence that the maximum likelihood solution has in fact been obtained. Keywords: growth mixture models, start values, latent classesMany developmental theories posit that different subgroups of individuals follow qualitatively different developmental trajectories over time. Theories of this type are particularly common in the study of developmental psychopathology (e.g., Moffitt, 1993;Schulenberg, O'Malley, Bachman, Wadsworth, & Johnston, 1996), but they have also been advanced in domains as diverse as cognitive and language development (McCall, Appelbaum, & Hogarty, 1973;Rescorla, Mirak, & Singh, 2000) and health and aging (Aldwin, Spiro, Levenson, & Cupertino, 2001). What these theories hold in common is that the hypothesized subpopulations are largely defined by the presentation of behavior over time rather than by grouping variables that are known a priori. Growth mixture models (GMMs) provide an approach for evaluating such theories by identifying latent classes of individuals distinguished by different patterns of change through time.Mixture models in general are well known to present certain estimation difficulties; namely, there may be many local optima and, in the case of normal mixtures, also singularities on the likelihood surface (singularities are points where the likelihood function goes to infinity, causing model nonconvergence). Estimation of a mixture model with multiple sets of start values is thus often recommended to avoid these irregularities on the likelihood surface and to discriminate local optima from the global optimum (McLachlan & Peel, 2000;Molina, 2000;B. O. Muthén, 2001; B. O. Muthén & Muthén, 2001, p. 373;Solka, Wegman, Priebe, Poston, & Rogers, 1998). In the present article, we examine this issue with respect to the GMM, including the practical impact of these potential problems of estimation on the fitting and interpretation of GMMs in empirical research.We focus on two variants of the GMM that have appeared often in recent applied research. First, we consider the GMM advanced by B. O. Muthén and Shedden (1999), which generalizes the latent curve approach to analyzing growth trajectories within a structural eq...

Previous research suggests that minority residential areas have a disproportionate likelihood of hosting various environmental hazards. Some critics have responded that the contemporary correlation of race and hazards may reflect post-siting minority movein, perhaps because of a risk effect on housing costs, rather than discrimination in siting. This article examines the disproportionate siting and minority move-in hypotheses in Los Angeles County by reconciling tract geography and data over three decades with firm-level information on the initial siting dates for toxic storage and disposal facilities. Using simple t-tests, logit analysis, and a novel simultaneous model, we find that disproportionate siting matters more than disproportionate minority move-in in the sample area. Racial transition is also an important predictor of siting, suggesting a role for multiracial organizing in resisting new facilities.

focus on how neighborhoods change over time, how that change both affects and is affected by neighborhood crime, and the role networks and institutions play in that change. He approaches these questions using quantitative methods as well as social network analysis.Neighborhood nesting 3 Block. Tract, and Levels of Aggregation: Neighborhood Structure and Crime and Disorder as a Case in Point AbstractThis paper highlights the importance of seriously considering the proper level of aggregation when estimating neighborhood effects. Using a unique non-rural sub-sample from a large national survey (the American Housing Survey) at three time points that allows placing respondents in blocks and census tracts, this study tests the appropriate level of aggregation of the structural characteristics hypothesized to affect block-level perceived crime and disorder. A key finding is that structural characteristics differ in their effects based on the level of aggregation employed. While the effects of racial/ethnic heterogeneity were fairly robust to geographical level of aggregation, the stronger effects when measured at the level of the surrounding census tract suggest more far-flung networks are important for perceived crime and disorder. In contrast, economic resources showed a particularly localized effect only evident when aggregating to the block-level and differed based on the outcome: higher average income reduced disorder, but increased crime, likely by increasing the number of attractive targets. And the presence of broken households had a localized effect for social disorder, but a more diffuse effect for perceived crime. These findings suggest the need to consider the mechanisms involved when aggregating various structural characteristics in neighborhood studies of crime rates, as well as the broader neighborhood effects literature.Neighborhood nesting 1 Block. Tract, and Levels of Aggregation: Neighborhood Structure and Crime and Disorder as a Case in PointConsiderable social science research focuses on how structural characteristics affect various outcomes: indeed, this is arguably a linchpin of sociological scholarship. One form of this research tests whether structural characteristics of neighborhoods affect various aggregate outcomes, such as crime, economic vibrancy, cohesion, or even death from heat waves (Browning, Wallace, Feinberg, and Cagney 2006). Another form of this research employs multilevel models to test whether the structural or cultural characteristics of "neighborhoods" affect various individual-level outcomes such as educational achievement, psychological wellbeing, or residential satisfaction. Despite the variety of research paradigms focusing on the importance of neighborhoods, a commonality of many studies is that less attention is paid to the appropriate level of aggregation for such "neighborhood" effects. As one consequence, whereas a shared knowledge has developed that the size of such neighborhood effects tend to be relatively small compared to individual-level effects (Liska 1990), i...

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We studied a sample of reentering parolees in California in 2005–2006 to examine whether the social structural context of the census tract, as well as nearby tracts, along with the relative physical closeness of social service providers affects serious recidivism resulting in imprisonment. We found that a 1 standard deviation increase in the presence of nearby social service providers (within 2 miles) decreases the likelihood of recidivating 41 percent and that this protective effect was particularly strong for African American parolees. This protective effect was diminished by overtaxed services (as proxied by potential demand). We found that higher concentrated disadvantage and social disorder (as measured by bar and liquor store capacity) in the tract increases recidivism and that higher levels of disadvantage and disorder in nearby tracts increase recidivism. A 1 standard deviation increase in the concentrated disadvantage of the focal neighborhood and the surrounding neighborhoods increases the likelihood of recidivating by 26 percent. The findings suggest that the social context to which parolees return (both in their own neighborhood and in nearby neighborhoods), as well as the geographic accessibility of social service agencies, play important roles in their successful reintegration.

This study tests the effects of neighborhood inequality and heterogeneity on crime rates. The results of this study, which were obtained by using a large sample of census tracts in 19 cities in 2000, provide strong evidence of the importance of racial/ethnic heterogeneity for the amount of all types of crime generally committed by strangers, even controlling for the effects of income inequality. Consistent with predictions of several theories, greater overall inequality in the tract was associated with higher crime rates, particularly for violent types of crime. Strong evidence revealed that within racial/ethnic group inequality increases crime rates: Only the relative deprivation model predicted this association. An illuminating finding is that the effect of tract poverty on robbery and murder becomes nonsignificant when the level of income inequality is taken into account; this finding suggests that past studies that failed to take income inequality into account may have inappropriately attributed causal importance to poverty. This large sample also provides evidence that it is the presence of homeowners, rather than residential stability (as measured by the average length of residence), that significantly reduces the level of crime in neighborhoods.

Defining “neighborhoods” is a bedeviling challenge faced by all studies of neighborhood effects and ecological models of social processes. Although scholars frequently lament the inadequacies of the various existing definitions of “neighborhood,” we argue that previous strategies relying on nonoverlapping boundaries such as block groups and tracts are fundamentally flawed. The approach taken here instead builds on insights of the mental mapping literature, the social networks literature, the daily activities pattern literature, and the travel to crime literature to propose a new definition of neighborhoods: egohoods. These egohoods are conceptualized as waves washing across the surface of cities, as opposed to independent units with nonoverlapping boundaries. This approach is illustrated using crime data from nine cities: Buffalo, Chicago, Cincinnati, Cleveland, Dallas, Los Angeles, Sacramento, St. Louis, and Tucson. The results show that measures aggregated to our egohoods explain more of the variation in crime across the social environment than do models with measures aggregated to block groups or tracts. The results also suggest that measuring inequality in egohoods provides dramatically stronger positive effects on crime rates than when using the nonoverlapping boundary approach, highlighting the important new insights that can be obtained by using our egohood approach.