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Neighbourhood characteristics and crime^{1}Descriptive results: a comparison of high and lowcrime neighbourhoods Since the 1940s (Shaw and McKay 1942), numerous studies have documented the relationship between neighbourhood characteristics and crime rates. However, these studies have differed with respect to the importance they place on factors such as low income, residential mobility, ethnocultural composition, opportunities for criminal behaviour, collective efficacy (or the level of trust and reciprocity in a neighbourhood), and social disorganization (or a decrease in the influence of social rules over behaviour) (Cohen and Felson 1979; Brantingham and Brantingham 1982; Ronek and Maier 1991; Sampson and Lauritsen 1994; Sampson et al. 1997). This section explores the relationship between many of these factors and 2001 rates of violent and property crime in Montréal neighbourhoods. The analysis makes use of total violent and property crime rates rather than rates for individual offence types to maximize the number of incidents being considered. For reasons of data confidentiality and reliability, Statistics Canada requires that when using individual, family or household income data, the population size for any Canadian geographic area being considered must be least 250 people living in at least 40 private households.As a result, only 506 of the 521 CTs are included. A map is appended showing the coverage of the 506 CTs over the territory of the Island of Montréal. Characteristics considered in this analysis are taken from the 2001 Census and land use data of the Communauté métropolitaine de Montréal. They are outlined in the section entitled Description of variables. Descriptive results: a comparison of high and lowcrime neighbourhoodsTo examine the relationship between violent and property crime rates and selected neighbourhood characteristics, the 506 CTs have been divided into two groups for each crime type. The first group contains the 126 CTs that recorded the highest property and violent crime rates (25%), and the second group contains the remaining 380 CTs, representing 75% of the total.^{2} Before controlling for other factors, significant differences are noted in selected characteristics when comparing neighbourhoods with higher crime rates with their lower crime rate counterparts. These differences in crime rates are consistent across a number of socioeconomic, demographic, land use and dwelling characteristics for both violent and property crimes. Figure 5 shows that when compared with census tracts that recorded fewer violent incidents, census tracts with the highest rates had, on average, a significantly greater proportion of single residents (53% and 43%), loneparent families (27% and 19%) and greater residential mobility (20% and 16%, respectively), meaning that a larger proportion of people were not living at the same address one year prior to the 2001 Census. The percentages were also higher for recent immigrants, meaning those who arrived in Canada between 1991 and 2001 (45% and 34%, respectively), in highcrime neighbourhoods. Another difference, small but statistically significant, was that neighbourhoods with high levels of violent crime had a higher maletofemale ratio and a larger proportion of young males aged 15 to 24. Figure 6 shows similar differences with respect to property crimes. Figure 5. Demographic characteristics in neighbourhoods with high and lower rates of violent incidents, Montréal, 2001 Figure 6. Demographic characteristics in neighbourhoods with high and lower rates of property incidents, Montréal, 2001 Finally, Figure 7 and Figure 8 show that there are many disparities in the socioeconomic variables in neighbourhoods with higher violent and property crime rates. These disparities reflect much larger proportions of residents who have incomes below the lowincome cutoff (42% vs. 26%), receive government transfers (21% vs. 14%) or are unemployed (12% vs. 9%). Also, these neighbourhoods have a substantially lower median household income than CTs with low violent crime rates ($28,000 vs. $44,000). The proportion of persons aged 20 and over who have a bachelor's degree is significantly different, namely 26% in neighbourhoods with low violent crime compared to 17%. A similar observation may be made with respect to the proportion of persons in professional occupations, which is higher in neighbourhoods where violent crime is low (61%) than in those with high crime (50%). Figure 7. Socioeconomic characteristics in neighbourhoods with high and lower rates of violent offences, Montréal, 2001 Figure 8. Socioeconomic characteristics in neighbourhoods with high and lower rates of property offences, Montréal, 2001 Figure 8 indicates similar results for property crime rates, except for the proportion of persons holding a bachelor's degree and the proportion with a professional occupation, which are not statistically significant. The Winnipeg study showed that there was a significant difference in the percentages of persons with no high school diploma in lowcrime and highcrime neighbourhoods (Fitzgerald, Wisener and Savoie 2004). This difference is not statistically significant in Montréal neighbourhoods in the case of property crime. The differences in land use and housing characteristics were greater in the case of property offences than in that of violent offences (Figure 9 and Figure 10). The proportion of commercial zoning was greater in neighbourhoods with higher property crime rates than in other neighbourhoods (10% and 5% respectively); the proportion of multifamily zoning was also greater in the former neighbourhoods (29% vs. 23%), while the proportion in singlefamily zoning was smaller (7% vs. 17%) (Figure 10). The density of bars was much greater in neighbourhoods with high property crime, at 14 per km^{2} compared to 3 per km^{2}. Neighbourhoods where the property crime rate was higher also recorded a larger proportion of dwellings constructed before 1961 (59% vs. 48%). There were also lower proportions of owneroccupied dwellings in these same high property crime neighbourhoods (22% vs. 41%), and greater proportions of unaffordable housing, represented by households spending more than 30% of their income on shelter (35% vs. 29%). Figure 9 shows similar differences in the case of violent crimes. Figure 9. Land use and housing characteristics in neighbourhoods with high and lower rates of violent incidents, Montréal, 2001 Figure 10. Land use and housing characteristics in neighbourhoods with high and lower rates of property incidents Montréal, 2001 Results of multivariate analysisWhen considered individually, the neighbourhood characteristics discussed above are associated with higher rates of violent and property crime, but the strength of this association can vary when multiple characteristics are considered together. Multivariate analysis in this section is used to examine the interrelationships among variables and to observe how they relate to the level of crime after taking other factors into account. Ordinary least squares (OLS) regression is used to examine the distribution of violent and property crime rates as a function of the set of explanatory factors. The use of this method requires a continuous or quantitative outcome variable that has a normal distribution, in this case the crime rate. Since the distribution of crime rates is often skewed, with a small proportion of neighbourhoods accounting for a larger proportion of reported incidents, it was necessary to log transform the crime variables. Many of the neighbourhood characteristics in this study are highly correlated with each other or convey essentially the same information. This multicollinearity between factors stems from the strong association among many factors that are individually linked to crime (Land, McCall and Cohen 1990). To avoid the problem of multicollinearity, which may distort the results, highly correlated variables were eliminated from the analysis.^{3} Thus, Table 3 shows that median household income (r=0.76), the proportion of persons receiving government transfers (r=0.74) and the percentages of persons unemployed (r=0.76), owneroccupied dwellings (r=0.80) and unaffordable housing (r=0.79) are all highly correlated with the low income cutoff, and hence had to be eliminated. Table 3 shows that the proportion of people with a bachelor's degree is closely related to occupation. Therefore the occupation variable, which was also highly correlated with median household income, was dropped from the analysis. Another aspect that must be taken into account in a spatial analysis of data such as crime data is spatial autocorrelation (see the modelling and spatial autocorrection text box). The presence of a strong autocorrelation is detected in the residuals of the OLS regression models for Montréal, that is a Moran's I statistic of 0.14 (p<0.001) in the case of violent crimes and 0.24 (p<0.001) in the case of property crime. Therefore, in modelling relationships between neighbourhoods, it is appropriate to take their relative geographic position into account. The use of a spatial autoregressive model is required. To assess the relative contribution of neighbourhood characteristics to the explanation of crime, the set of variables was regressed separately on violent and property crime rates. The results are shown in Table 4. The modeling process reveals a set of six explanatory variables for the variation in violent crimes and a set of five variables in the case of property crimes. The spatial autoregressive model gives a squared correlation coefficient of 0.60 (p<0.05) between actual values for the neighbourhood crime rates and the predicted values in the case of violent crimes, and of 0.61 (p<0.05) in the case of property crimes. The estimated regression coefficients provide an indication of the relative contribution of each variable after controlling for the other variables in the model.^{4} Table 4. Spatial autoregressive models for violent and property crime rates, Montréal neighbourhoods, 2001 The violent crime rate model shows that the proportion of neighbourhood residents aged 20 and older with a bachelor's degree has the greatest explanatory power when the other variables are held constant. Thus the violent crime rates are lower in neighbourhoods where a larger proportion of residents aged 20 and older have a bachelor's degree (b=0.22). This factor appears to offer protection with respect to violent crime. In contrast, violent crime rates are highest where the proportions of lowincome persons (b=0.20) and single persons (b=0.16) are highest. The reported rate of violent crime also increases in residential neighbourhoods, whether the housing be singlefamily (b=0.11) or multifamily (b=0.10). The proportion of a neighbourhood that is zoned commercial also contributes to the explanatory model, although its contribution is smaller (b=0.07). The results of the spatial regressive model applied to property crime provide a slightly different picture. Commercial land use on the Island of Montréal has the greatest explanatory power for the variation in property crime (b=0.12). Property crime rates are also the highest where the proportions of low income persons (b=0.10) and single persons (b=0.11) are the largest. Drinking place density is also related to higher crime (b=0.05), but its contribution to the explanatory model is the smallest. To a lesser but statistically significant degree, the percentage of the neighbourhood population belonging to a visible minority (b=0.5) is a protection factor in the case of property crime. In other words, the greater the proportion of persons belonging to a visible minority, the lower the level of property crime. Modelling and spatial autocorrelationSpatial autocorrelation reflects a relationship or dependence between two different units of observation owing to their geographic location (Anselin and Bera 1998). The presence of spatial autocorrelation substantially changes the properties of the OLS estimators and the statistical inference based on these estimators. If spatial autocorrelation is present, these estimators may be biased or inefficient. In detecting autocorrelation in geographic data, the task is to model the relationships between the units taking account of their relative position in the geographic area being studied. In the spatial analysis of crime, spatial autocorrelation is characterized by the geographic clustering of similar crime rates, and it may also be a result of how natural neighbourhoods are defined. Map 4 and Map 6 illustrate the concentration of crime hot spots in selected areas of the Island of Montréal, these data having been aggregated to the CT level in order to model them. The spatial modelling of crime at the level of neighbourhoods requires that hot spots (high density nodes) be distributed randomly among the CTs and have no influence or spillover effect on neighbouring CTs. Thus, where neighbouring CTs have similar levels of crime, it is possible that the key characteristic in the distribution of crime is location or proximity to another CT with a high density of crime (or other characteristics of the CT), rather than characteristics that are specific to it (Anselin, Cohen, Cook and Tita 2000). To determine the presence of spatial dependence in the data, a statistical test, Moran's I, is performed to determine whether the data are distributed randomly over the area studied. For computing Moran's I, neighbouring locations were defined as sharing a common boundary, point or vector. The significance of the Moran's I test is determined by a permutation approach, where a significant result indicates that there is spatial autocorrelation in the data. The value of the Moran's I statistic ranges between 1 and 1. A value approaching 1 indicates the presence of positive autocorrelation whereas a negative value indicates the presence of negative autocorrelation, and a value of zero, the absence of spatial autocorrelation. The value of the Moran's I statistic is 0.47 (p<0.001) in the case of violent offences and 0.61 (p<0.001) for property offence rates, indicating a spatially dependent structure in the data. After spatial autocorrelation is observed in the data, the residual values from the OLS regression models are analysed to determine whether the characteristics of the different sets of variables served to eliminate the autocorrelation. Once again the Moran's I test indicates the presence of spatial autocorrelation in the residuals. The value of this test is 0.14 (p<0.001) for the violent offences model and 0.24 (p<0.001) for property offences. These results indicate that explanatory variables or neighbourhood characteristics do not explain the entire spatial structure in the data and that therefore the location of the neighbourhood is having an underlying effect. When autocorrelation is present in the residuals of a regression model, the use of a spatial autoregressive model is strongly recommended to ensure that the regression coefficients and their associated variances are valid. The spatial autoregressive model offers the same explanatory analysis of neighbourhood characteristics as the standard linear model, but it controls for the effect of location. To do this, an extra variable called the spatial lag, representing the average crime rate from all neighbouring locations, is added to the other variables in the standard linear model and thus the spatial effects are filtered out of the model. The use of coefficient of determination (R2) in spatial autoregressive models is not recommended, since the variation introduced by the location effect cannot be separated from the variation in the predicted values and error terms. Alternatively, the squared correlation coefficient between actual crime rate values at the neighbourhood level and the values predicted using the coefficients of the spatial autoregressive model can be used. The squared correlation coefficient also allows comparison between different communities. Additionally, the coefficient of the spatial lag variable is not interpreted in the same way as those of other variables in the autoregressive model. The value of this parameter represents in part the effect of location in the area, but it also takes account of measurement error in the way the neighbourhoods are defined. Thus the spatial lag cannot be interpreted directly; it is only retained in the model to make the results accurate. Even with these two distinctions, the results of the spatial autoregressive model are essentially the same as those of other regressive models. For example, in Table 4, regression coefficients of the neighbourhood characteristics represent their relative contributions to the explanatory model for crime. The residuals from the Montréal spatial autoregressive models are again checked for the presence of spatial autocorrelation using Moran's I. They indicate a value of 0.02 (p>0.1), which is not statistically significant, both in the set of violent offence variables and the set of property offence variables. Thus, the models have succeeded in controlling for the effect of spatial autocorrelation, and the parameter estimates are therefore of greater accuracy and free of biases caused by the location of the neighbourhood. Notes1. It should not be concluded from the results this study that some neighbourhood characteristics are the cause of crime; rather the results show that these factors are associated with or cooccur with higher crime rates in neighbourhoods. 2. Dichotomous variables are used only for the descriptive or bivariate analysis. The multivariate analysis that follows is based on continuous dependent variables: violent and property crime rates. The differences are significant at p<0.001 unless otherwise indicated, based on a two sample Ttest. 3. The correlation between two variables reflects the degree to which the variables are related. The most common measure of correlation is Pearson's correlation coefficient (r), which reflects the degree of linear relationship between two variables. It ranges from +1 to 1. A correlation of +1 means that there is a perfect positive linear relationship between variables, while a correlation of 1 means that there is a perfect negative relationship. 4. Since the independent variables are initially transformed into zscores, the unstandardized regression coefficients provide a means of assessing the relative importance of the different predictor variables in the multiple regression models. The coefficients indicate the expected change, in standard deviation units, of the dependent variable per one standard deviation unit increase in the independent variable, after controlling for the other variables. The maximum possible values are +1 and 1, with coefficient values closer to 0 indicating a weaker contribution to the explanation of the dependent variable. 
