Biografía de los integrantes

A considerable number of social control variables, economic strain variables, and agency policy variables have been extensively examined in aggregate level research. Unfortunately, the findings of prior research reached few consistencies. Kaminski

(2008:351) commented that “more than two decades of research on police homicides has failed to identify virtually any regressor that is statistically significant and of the same sign across all models or studies.” The differences in the regression models, the nature of the units of analysis (national, states, counties, cities) and the nature of the data and design contribute to this problem. A brief discussion on these differences may reveal the limitations of prior research and suggest possible improvement in the present study.

(1) The choices of regression model. Early research mainly used the Ordinary Least Squares (OLS) regression to model the risk of police murder (Bailey, 1982; Bailey & Peterson,1987; 1994; Chamlin, 1989; Fridell & Pate; 1995; Lott, 2000; Peterson & Bailey,1988; Southwick, 1998). However, OLS regression is not appropriate for modelling the felonious murders of police officers, because the killings of officers are rare events, which do not follow a normal distribution. Later studies have widely applied Poisson or negative binomial models to estimate the fatal victimization of officers (Jacobs, & Carmichael, 2002; Kaminski, 2002, 2004; Kaminski, 2008; Kaminski & Stucky, 2009Kent, 2010; Moody et al, 2002; Mustard, 2001). Poisson is good at modelling low count outcomes, and negative binomial is an alternative to Poisson if overdispersion is an issue, which is very common in practice. If a likelihood-ratio test on the overdispersion parameter is significant, negative binomial models should be used. Still, there is an issue, excess zero counts, that needs to be solved. Recall that police homicides are rare. Usually, most enumeration units report zero police killing incidents, yielding zero-dominant data. The presence of excess zeroes severely distorts the

produce inaccurate estimates. Unfortunately, most studies have not paid enough attention to this issue (Kaminski, 2002, 2004, 2008; Kaminski & Stucky, 2009).

(2) The unit of analysis. The influence of the choices of unit of analysis may be another reason for these studies’ conflicting findings. In spatial analysis, researchers find that choosing different scales or shapes of aggregated spatial units could yield different statistical relationships between the outcome and predictors (Gehlke & Biehl, 1934; Openshaw & Taylor, 1979). This phenomenon is called the modifiable area unit problem (MAUP), which is closely related to the concept of the ecological fallacy (O'Sullivan & Unwin, 2010). Police murder studies have often aggregated police homicide incidents across space and over time to yield enough observations. Different levels of spatial unit are used. Smaller spatial units tend to have more homogeneity in some within-unit characteristics, such as social-economic structures, community resources, or agency policing cultures, than larger geographical units, thus appearing more appealing in police homicide research. However, due to the scarcity of the observations, the studies based on county or city-level data have to aggregate the outcomes over a very long time period (i.e., over a decade), or only sample areas with large populations (Jacobs, & Carmichael, 2002; Kaminski, 2002; 2008; Kent, 2010). A long temporal aggregation, which brings in tremendous temporal heterogeneity, hampers the ability to estimate effects of the

covariates accurately. Restricting the research only to areas with high populations, which excludes incidents that happened in areas with small populations, raises concerns about the findings’ generalizability. In comparison, state-level data, despite having more within-unit spatial variation, introduces much less within-unit temporal heterogeneity, as these data can produce acceptable counts of observations over a relatively short time

period. In fact, grouping state-level observations over an appropriate period of time can also reduce excess zeroes in the data.

(3) Data structures. In the studies adopting Poisson or negative binomial models, some (Jacobs, & Carmichael, 2002; Kaminski, 2008; Kent, 2010) have used

cross-sectional data. A cross-sectional design is easy to implement and analyze, but it has very limited inference ability due to its static nature. It cannot detect the historical effects of factors since it does not allow temporal variabilities across research units (Kramer, 1983). This limitation severely confines cross-sectional research’s usage. Time-series design brings in the variations in historical context, but for only one observation unit. Such a design could be used to identify temporal trends of the variable of interest and shed insight onto the temporal relationship between risk factors and the outcome. However, the lack of variations in spatial units in time-series analysis means that the generalizability of such an analysis is limited. Bailey and Peterson (1994), Southwick (1998), and Kaminski and Marvell (2002) all analyzed national time-series data, but they did not employ count models (i.e., Poisson), which are more appropriate for rare event data. A longitudinal study which repeatedly collects observations from multiple units at different time points can provide more information about the temporal effects of the independent variable. Depending on whether the number of units (N) exceeds the number of time periods (T), longitudinal data could be further classified as panel data (if N>T) and pooled time series cross-sectional (TSCS) data (if N<T) (Stimson, 1985). These longitudinal designs are more efficient to identify the effect of temporally varying variables than the cross-sectional or time series designs, thus providing a clearer picture of the relationship between the predictors and the outcome. However, the analysis of

longitudinal data of police homicide needs to consider the autocorrelations in time and/or space. Moody et al. (2002) estimated fixed-effect Poisson models of killings of officers, using state-level panel data from 1973-1998, but did not fully address spatial and temporal autocorrelation. Kaminski (2002, 2004) estimated a panel model in addition to cross-sectional models in four time periods, but the sample was restricted to 190 large municipal police departments.

There has been growing interest in examining the distribution and the factors that affect the variations of hazard outcomes involving both spatial and temporal information. A spatial-temporal mapping analysis can not only illustrate the trend patterns over time and spatial pattern across regions simultaneously, but also depict clear relationships between the response variable and related risk factors. It can accurately estimates the risks of outcome incidents after taking into account important risk factors and variation in space and time. Moreover, it can discern certain patterns from residuals due to

unmeasured or unobservable covariates, such as local community characteristics, agency practices, and regional subcultures. These residual patterns can guide the direction of future research.

Following the discussion above, state-level aggregated data in which observations are grouped by relatively short time periods will be considered in the present study. Such data can provide enough observations and reduce excess zero counts, with an acceptable level of within-unit heterogeneity. A pooled time series structure of the data can

efficiently detect the effect of temporally varying factors. A spatial-temporal mapping analysis can be used to control for the excess variations due to the autocorrelations in space and time.

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