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Heterogeneity in the Frequency Distribution of Crime Victimization

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Abstract

Objectives

Tests the idea that the frequency distribution typically observed in crosssectional crime victimization data sampled from surveys of general populations is a heterogeneously distributed result of the mixing of two latent processes associated, respectively, with each of the tails of the distribution.

Methods

Datasets are assembled from a number of samples taken from the British Crime Survey and the Scottish Crime Victimization Survey. Latent class analysis is used to explore the probable, latent distributions of individual property crime and personal crime victimization matrices that express the frequency and type of victimization that are self-reported by respondents over the survey recall period.

Results

The analysis obtains broadly similar solutions for both types of victimization across the respective datasets. It is demonstrated that a hypothesized mixing process will produce a heterogeneous set of local sub-distributions: a large sub-population that is predominantly not victimized, a very small ‘chronic’ sub-population that is frequently and consistently victimized across crime-type, and an ‘intermediate’ sub-population (whose granularity varies with sample size) to whom the bulk of victimization occurs. Additionally, attention is paid to the position of very high frequency victimization within these sub-populations.

Conclusions

The analysis supports the idea that crime victimization may be a function of two propensities: for immunity, and exposure. It demonstrates that zero-inflation is also a defining feature of the distribution that needs to be set alongside the significance that has been attached to the thickness of its right tail. The results suggest a new baseline model for investigating population distributions of crime victimization.

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Notes

  1. Definitions of the crime categories used in this paper are given in the “Appendix”.

  2. The SP model is a benchmark for counts data that consist of a number of discrete events occurring at the unit-level over a fixed time interval (Cameron and Trivedi 1998). It assumes that successive events for any individual case occur independently of each other over time at a constant rate, an assumption violated by over-dispersion (Nelson 1980: 871).

  3. NB2 assumes a Poisson distribution, with gamma-distributed unobserved individual heterogeneity (Cameron and Trivedi 1998: 71).

  4. Even the relatively few analyses of longitudinal crime victimization data to date have been unable to distinguish clearly and conclusively the mixing components of the DGP, or the relative importance of heterogeneity versus state-dependency (Bottoms and Costello 2009; Hope and Trickett 2004, 2008; Wittebrood and Nieuwbeerta 2000; Lauritsen and Davis Quinet 1995). Morgan (2007) provides a detailed discussion and analysis of time-dependent models of crime victimization.

  5. The NB model specification can support a variety of probability mechanisms, a range of mixtures that might produce it (Cameron and Trivedi 1998: 102) and a variety of Mixed-Poisson models to help explain it (Karlis and Xekalaki 2005).

  6. Specifically, the study utilized a bivariate probit regression model with censoring (Osborn et al. 1996).

  7. The categories of counts were: Non-victim (no incidents over the recall period); Low-level (1 victimization incident over the recall period); and High-level (2 or more incidents). The waves were measured by a panel embedded in two cross-sectional self-report sample surveys, with 12-month recall periods, separated by a period of 3 years between each sampling (Hope and Trickett 2008: 47).

  8. In addition to having been victimized over the past 5 years, contemporaneous multiple crime-type victims were significantly more likely to be younger adults, living with children, renting from the social or public housing authorities and living in poorer, urban areas; all characteristic of economically marginal and socially vulnerable sub-populations in the UK who might be more likely to be most often exposed to active offenders (Hope et al. 2001).

  9. Tseloni et al. (2010) also call attention to composite crimes, i.e. when more than one crime type coincides as part of a single event. This might be a further dimension to an individual’s crime victimization event matrix but is excluded from consideration here both on the grounds of a lack of accessible data on event-composition (Tseloni et al. 2010) and because its inclusion would increase the complexity of subsequent analysis (probably exponentially) to the detriment of exposition. Still, we should concede that all efforts to classify crime victimization events or states statistically are highly circumscribed in their phenomenology, due largely to the constraints of the incident-counting machinery, whether sample surveying or offence recording (Biderman 1981).

  10. These two hypothetical probability distributions may be thought of as propensities, in the sense of Popper’s propensity interpretation of probability (Popper 1983/1957); that is, as heuristics that give an idealized account of the objective, relational properties of the physical world (Gillies 2000).

  11. For example, Tseloni et al. test a Bivariate Zero-inflated Poisson (BZIP) model that assumes zero counts to arise in two ways: first, that “…no crime occurs, with probability p and produces only zeros, while the other state where crime exists, occurs with probability 1 − p and leads to a standard Poisson count” (2010: 334).

  12. A substantial, significant zero-inflation coefficient is identified in the BZIP model estimated by Tseloni et al. (2010).

  13. The idea that distributions (such as crime victimization) are multiplicative heterogeneous mixtures is supported by reference to the Two-Crossings Theorem (Mullahy 1997): “…Two Crossings Theorem. For the random variable y, continuous or discrete, let f (y | x, v) denote an exponential family conditional (on v) model density and let E [v] = 1, V [v] = σ2 > 0. Then the mixed (marginal with respect to v) distribution h (y | x) = Ev f (y | x, v) will have heavier tails than f (y | x, v) in the sense that the sign pattern of marginal minus the conditional h (y | x) − f (y | x, v) is {+ , −, +} as y increases on its support. That is, for the same mean, any marginal distribution must ‘cross’ the conditional distribution twice, first from above and then from below, the first crossing accounting for a relative excess of zeros, and the second for the thickness of the right tail” (Cameron and Trivedi 1998: 99).

  14. Pickles and Angold (2003) describe the approach as akin to Gaussian Quadrature, that represents “…smoothly varying densities of a distribution by estimating a limited number of spikes at a set of specific values…each spike is assigned a probability weight…the spikes are, mathematically speaking, identical to a set of ordered latent classes” (Pickles and Angold 2003: 540).

  15. Recently, both Survey series have undergone name changes, while the SCVS has had several changes of nomenclature. These names are retained however as those most appropriate at the time the surveys were conducted.

  16. Responses for England and Wales were taken from the 1992, 1996, 2001, 2003/4 and 2006/7 BCS. Responses for Scotland were taken from the 1993, 1996, 2000, 2003 and 2006 SCVS.

  17. Chiefly, this has been done to facilitate the development of performance indicators to assist the Home Office in its governance of the police service in England and Wales (Allen 2007).

  18. Further comfort that the different sample designs of the components of our dataset have not biased our results is provided by a recent and extensive investigation of this issue, concluding that, under all of its sample designs, the BCS has generated estimates of victimization with low levels of sampling error (Tipping et al. 2010).

  19. Methods based on simulation have been suggested as providing a more accurate indication as to how many classes are needed for an LCA model to accurately reflect an underlying dataset (Nylund et al. 2007). Of these, the most common is the Bootstrap Log-likelihood Ratio Test (BLR) (McLachlan and Peel 2000). This calculates a p value that indicates whether or not the inclusion of an additional class significantly improves the relationship between the data and the model. However, the use of these methods with weighted data is not well understood or widely implemented at this time.

  20. See Fn 14. Morgan (2007) makes an analogous point regarding the merits of event-hazard models for estimating stochastic crime victimization processes.

  21. Although the counts from series that are derived from victim forms are subsequently capped at 6, being the maximum count per series per victim form (Farrell and Pease 2007).

  22. For example, Tseloni (2006), Hope et al. (2001), Osborn and Tseloni (1998), Ellingworth et al. (1997), Osborn et al. (1992, 1996), Trickett et al. (1992).

  23. “…repeated victimizations cannot be looked at comprehensively through the victim forms, because of the constraint imposed on the number of victim forms completed per respondent, and the maximum number of events ‘permissible’ in the series…there is no alternative to the use of the main questionnaire for the purposes advanced here…The general pattern is robust across different limits to the number of victimizations allowable per person” (Ellingworth et al. 1995: 361).

  24. A table summarising the operation of capping is available from the authors, on request.

  25. Within the BCS and SCVS datasets, such responses are coded with a very high value, typically ‘97’, though this is usually understood as a missing value code (Bolling et al. 2007: 71; Brown 2007: 91). Although these can form a substantial proportion of the cases with values over six (the highest, at 52 %, being for the BCS property crime ‘entered property and committed theft’), the proportions reporting ‘too many to count’ vary not only between crime types but also between the BCS and SCVS samples, suggesting a complex outcome of a combination of reporting and coding processes. A tabulation of these cases is available from the authors on request (see also Fn. 23).

  26. Models were also created based on negative binomial distributions without zero inflation and, in the case of the BCS data, using a censored distribution capped at 6. While the exact model fit statistics for these different distributions varied, the number of groups identified as optimal for capturing patterns of victimization was the same as when using zero-inflated distributions.

  27. Extensive introductions to LCA can be found in Magidson and Vermunt (2004), and McCutcheon (1987).

  28. The LCA models in this analysis were estimated using MPlus Version 5.1 (Muthen and Muthen 1998–2008).

  29. Although the BCS analysis optimizes one more group than the SCVS for both property crime (six compared with five) and personal crime (five compared with four), probably this is a ‘degrees of freedom’ phenomenon, due to differences in sample size and in numbers of variables (see Table 1). To test the effect of sample size on optimization capacity, an identical LCA model was run on sub-samples of diminishing size, each selected randomly from the overall pooled sample of BCS property crime. Results clearly confirm the importance of sample size; while twenty per cent of the overall pooled sample could support a minimum three-class solution, it would need around eighty per cent to support the optimum six-class solution, according to the ABIC.

  30. The labels used to describe the classes in this and other solutions are merely descriptive and are employed for illustrative purposes only. As discussed earlier, the point of this analysis is to establish that the distribution is composed of ordered latent classes, rather than to identify groups. To reflect this, the following labeling convention is used throughout: Non-Victims > Intermediate Victims (A > B > C > D) > Chronic Victims, ordered by class size, where Non-Victims + Intermediate Victims + Chronic Victims = \( \sum\nolimits_{j = 1}^{C} {\pi_{j} } \) (Eq. 1).

  31. See Table 1, and Fn 28.

  32. In England and Wales, 16 % of the population fall into the Victim classes; while the corresponding figure for Scotland is 11.6 %.

  33. Specifically, the size of the ‘non-victim’ group reduces from 88.3 % (2-class), 82.8 % (3-class), 81.3 % (4-class), 81.0 % (5-class), to 79.8 % (6-class).

  34. Similar patterns were found for BCS personal crime.

  35. Likewise, the dip at the count of 11 in the probability of being a chronic victim (p = 0.642) is attributable mainly to the probabilities of being an Intermediate C (p = 0.292) or Intermediate A Victim (p = 0.064); while the dip at the count of 12 (p = 0.486) is attributable to the probabilities of being an Intermediate C (p = 0.363), Intermediate D (p = 0.109) or Intermediate A (p = 0.042) Victim. These occurrences may be in part a consequence of the capping procedure employed (see section “The Effect of Uncapped Frequencies”). Specifically, respondents associated with the chronic class typically report high levels of victimization across the range of crime types, which might be subject to capping, while Intermediate Victim respondents may only experience frequent victimization in one or two types of victimization, which would be the ones that were subjected to the capping procedure.

  36. While negligible numbers of victims with a frequency of 14 victimizations or more were classified as anything other than Chronic Victims, below that level, substantial numbers were placed in other classes as well.

  37. With the caveat that the population we are considering here is characterized only by its type and frequency of crime victimization.

  38. The Non-Victim class also captures substantial numbers of low-frequency victims. Victims with a frequency (f) of 1 victimization have a probability (p) of 0.514 of being classed in the Non-Victim category, and subsequently f = 2, p = 0.142, and f = 3, p = .03 (Fig. 11).

  39. This consistency did not quite hold for the SCVS dataset (see Figs. 6, 8), though it is difficult to know whether this is due to substantive differences or artefactual differences between the surveys (see Fn. 25).

  40. Nevertheless, it should be remembered that this is a methodological precept. As Karl Popper remarked “…we must always clearly distinguish between an ‘essentialist’ explanation with appeals to the nature of things and a ‘descriptive’ explanation which appeals to a Law of Nature, i.e. to the description of an observed regularity. Of these two kinds of explanation only the latter is admissible in physical science” (1969: 169).

  41. See Cohen and Felson (1979) and Hindelang et al. (1978) respectively, and for further elaboration, inter alia, Cohen et al. (1981), Miethe and Meier (1994) and Osgood et al. (1996).

  42. That is, “…the probability that a violation will occur at any specific time and place might be taken as a function of the convergence of likely offenders and suitable targets in the absence of capable guardians” (Cohen and Felson 1979: 590).

  43. We are grateful to one of our anonymous reviewers for posing this question.

  44. Where measurement is by positive integers, a two-group solution is the inverse of the frequency distribution, replicating the cut-point between all positive values and zero.

  45. If the number of victims = V, the number in the population = P, and the number of victimization events = C, then Prevalence  = V/P, Concentration = C/V, and Incidence = C/P = (V/P)(C/V) (Hope 2007b: 103).

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Acknowledgments

The research for this paper was carried out while the authors were, respectively, Senior Visiting Research Fellow and Research Fellow at the Scottish Centre for Crime and Justice Research (SCCJR), University of Edinburgh, whose support and encouragement is gratefully acknowledged, especially that of Susan McVie and Richard Sparks. The UK Data Archive made data available. We also wish to acknowledge valuable discussions with Kauko Aromaa and Frank Morgan and comments from the anonymous reviewers that have helped to improve the paper.

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Authors and Affiliations

  1. Centre for Social Research, University of Salford, Manchester, UK

    Tim Hope

  2. School of Social and Political Science, University of Edinburgh, Edinburgh, Scotland, UK

    Paul A. Norris

  3. School of Humanities, Languages and Social Sciences, University of Salford, Salford, Manchester, M5 4WT, UK

    Tim Hope

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  1. Tim Hope
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Correspondence to Tim Hope.

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This paper is dedicated to the memory of Alan Trickett.

Appendix: Definitions of Property and Personal Crime

Appendix: Definitions of Property and Personal Crime

A1. Questions Used to Measure Property Crime (Taken From the Scottish Crime Victimization Survey 2006)

Since [1st April 2005], (apart from anything you have already mentioned,) has anyone GOT INTO your home without permission and STOLEN or TRIED TO STEAL anything?

(Apart from anything you have already mentioned,) in that time did anyone GET INTO your home without permission and CAUSE DAMAGE?

(Apart from anything you have already mentioned) in that time have you had any evidence that someone has TRIED to get in without permission to STEAL or to CAUSE DAMAGE?

(Apart from anything you have already mentioned), in that time was anything (else) stolen OUT OF your home? For example, by a guest, a workman or anyone else there with your permission.

And (apart from anything you have already mentioned), in that time was anything (else) that belonged to someone in your household stolen FROM OUTSIDE YOUR HOME—for example, from the doorstep, the garden, a shed, outhouse or garage? Please don t include milk bottles or newspapers.

And again, (apart from anything you have already mentioned), in that time has anyone deliberately DAMAGED or DEFACED your home or anything outside it (APART FROM A MOTOR VEHICLE) that belonged to YOU or ANYONE ELSE in your household?

A2. Questions Used to Measure Personal Crime (Taken from the Scottish Crime Victimization Survey 2006)

Since [1st April 2005], (apart from anything you have already mentioned), was anything you were CARRYING STOLEN out of your hands or from your pockets or from a bag or case you were carrying?

(Apart from anything you have already mentioned), in that time has anyone TRIED to STEAL something you were carrying out of your hands or from your pockets or from a bag or case you were carrying?

And again, (apart from anything you have already mentioned), since [1st April 2005] has anyone, including people you know well, DELIBERATELY HIT YOU with their fists, or with a weapon of any sort, or kicked you, or used force or violence on you in any other way?

And (apart from anything you have already mentioned), in that time, has anyone THREATENED to damage things of yours or THREATENED to use force or violence on you in any way that actually frightened you?

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Hope, T., Norris, P.A. Heterogeneity in the Frequency Distribution of Crime Victimization. J Quant Criminol 29, 543–578 (2013). https://doi.org/10.1007/s10940-012-9190-x

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