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The Impact of Measurement Error in Regression Models Using Police Recorded Crime Rates

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Abstract

Objectives

Assess the extent to which measurement error in police recorded crime rates impact the estimates of regression models exploring the causes and consequences of crime.

Methods

We focus on linear models where crime rates are included either as the response or as an explanatory variable, in their original scale or log-transformed. Two measurement error mechanisms are considered, systematic errors in the form of under-recorded crime, and random errors in the form of recording inconsistencies across areas. The extent to which such measurement error mechanisms impact model parameters is demonstrated algebraically using formal notation, and graphically using simulations.

Results

The impact of measurement error is highly variable across different settings. Depending on the crime type, the spatial resolution, but also where and how police recorded crime rates are introduced in the model, the measurement error induced biases could range from negligible to severe, affecting even estimates from explanatory variables free of measurement error. We also demonstrate how in models where crime rates are introduced as the response variable, the impact of measurement error could be eliminated using log-transformations.

Conclusions

The validity of a large share of the evidence base exploring the effects and consequences of crime is put into question. In interpreting findings from the literature relying on regression models and police recorded crime rates, we urge researchers to consider the biasing effects shown here. Future studies should also anticipate the impact in their findings and employ sensitivity analysis if the expected measurement error induced bias is non-negligible.

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Notes

  1. See for example Brantingham (2018), or Pepper et al. (2010), where modelling strategies to adjust for measurement error in police data are used for binary outcome models used in hotspot policing, or time-series analysis assessing changes in crime rates across time.

  2. The CSEW sampling approach is designed to enable the calculation of reliable victimisation estimates at the PFA level, with an average sample of 1,096 respondents in each area (min = 917, max = 4023). PFA is an UK spatial unit commonly used in the literature (Abramovaite et al. 2019; Han et al. 2013; Machin and Meghir 2004), encompassing 1.3 million people on average, which makes them similar to states and large counties in the US (Barnett 1981; Philipson and Posner 1996).

  3. Home Office data is available here: https://www.gov.uk/government/statistics/police-recorded-crime-open-data-tables.

  4. The City of London is primarily a business and financial centre with a small resident population of approximately 10,000 but a large day‐time population leading to artificially high crime rates.

  5. UCR data is available here: https://ucr.fbi.gov/crime-in-the-u.s/2019/crime-in-the-u.s.-2019/topic-pages/tables/table-20, NCHS data is available here: https://wonder.cdc.gov/controller/saved/D76/D99F056.

  6. Proof of the variance being unaffected by a change of origin:

    $$S_{{X^{*} }}^{2} = \frac{{\sum \left( {X^{*} - \overline{X}^{*} } \right)^{2} }}{n - 1} = \frac{{\sum \left( {X + u - \left( {\overline{X} + u} \right)} \right)^{2} }}{n - 1} = \frac{{\sum \left( {X - \overline{X}} \right)^{2} }}{n - 1} = S_{X}^{2} .$$

    Proof of the covariance being unaffected by a change of origin:

    $$S_{{X^{*} ,Y}} = \frac{{\Sigma (X^{*} - \overline{X}^{*} ) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = \frac{{\Sigma \left( {X + u - \left( {\overline{X} + u} \right)} \right) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = \frac{{\Sigma \left( {X - \overline{X}} \right) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = S_{X,Y} .$$

  7. Proof of the variance being affected by a change in scale:

    $$S_{{X^{*} }}^{2} = \frac{{\sum \left( {X^{*} - \overline{X}^{*} } \right)^{2} }}{n - 1} = \frac{{\sum \left( {Xu - \overline{X}u} \right)^{2} }}{n - 1} = \frac{{u^{2} \sum \left( {X - \overline{X}} \right)^{2} }}{n - 1} = u^{2} S_{X}^{2}$$

    Proof of the covariance being affected by a change in scale:

    $$S_{{X^{*} ,Y}} = \frac{{\Sigma (X^{*} - \overline{X}^{*} ) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = \frac{{\Sigma \left( {Xu - \left( {\overline{X}u} \right)} \right) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = \frac{{u\Sigma \left( {X - \overline{X}} \right) (Y^{*} - \overline{Y}^{*} )}}{n - 1} = uS_{X,Y} .$$

  8. The R code used can be found here, https://osf.io/kv3sc/.

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Funding

This work is supported by the Secondary Data Analysis Initiative of the Economic and Social Research Council (Grant Ref: ES/T015667/1).

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Authors and Affiliations

  1. School of Law, University of Leeds, Leeds, LS2 9JT, UK

    Jose Pina-Sánchez

  2. School of Social Sciences, University of Manchester, Manchester, UK

    David Buil-Gil & Alexandru Cernat

  3. Department of Sociology, University of Surrey, Guildford, UK

    Ian Brunton-Smith

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Supplementary Information

Appendix: Specific Offences Used to Define Broader Crime Types in Table 1

Appendix: Specific Offences Used to Define Broader Crime Types in Table 1

CSEW 2018–2019

NCVS 2017–2020*

Crime type

Cases reported in the interview

% known to police (weighted)

Crime type

Cases reported in the interview

% known to police (weighted)

Violent crime

1979

38.8

 Violent crime

516

46.6

 Hit with fists or weapon

538

46.6

 Assault

200

49.5

 Threaten to use force or violence on you

1319

36.4

 Attempted assault

299

44.7

 Sexually assaulted

95

28.2

 Rape

8

**

 Violent from household member

37

36.5

 Unwanted sexual contact from household member

9

**

Property crime

2035

36.7

Property crime

995

41.8

 Something stolen out of hands or pockets

304

46.2

 Larceny

927

40.7

 Other theft

360

24.8

   

 Tried to steal

203

11.7

 Attempt larceny

52

53.5

   

 Robbery

16

59.6

 Something stolen off car

796

40.0

   

 Bike theft

372

46.2

   

Burglary

719

59.5

Burglary

248

45.5

 Get in previous house to steal

38

69.0

Burglary

194

45.1

 Get in previous house and cause damage

10

79.3

   

 Get in house since moved in to steal

8

**

   

 Get in current house to steal

250

75.7

   

 Get in current house and cause damage

37

70.3

   

Try to get in previous house to steal/damage

21

15.4

Attempted burglary

  

 Try to get in current house to steal/damage

355

48.0

 

54

47.5

 Motor vehicle theft

130

89.7

Motor vehicle theft

33

73.5

  1. *Estimates from the NCVS are derived from a wider timeframe to obtain a larger sample size
  2. **Crime types with samples smaller than 10 are only used to calculate the overall proportion of cases known to the police, not to calculate their crime specific proportion

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Pina-Sánchez, J., Buil-Gil, D., Brunton-Smith, I. et al. The Impact of Measurement Error in Regression Models Using Police Recorded Crime Rates. J Quant Criminol 39, 975–1002 (2023). https://doi.org/10.1007/s10940-022-09557-6

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