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Modeling the Distribution of Sentence Length Decisions Under a Guidelines System: An Application of Quantile Regression Models

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Abstract

How should sentencing disparity be assessed when decisions are constrained under a sentencing guidelines system? Much of the debate over the measurement of sentence disparity under a guidelines system has focused primarily on using specific values from within the sentencing grid (e.g., minimum recommended sentence) or on using interaction terms in regression models to capture the non-additive effects of offense severity and prior record on length of sentence. In this paper, I propose an alternative method for assessing sentencing disparity that uses quantile regression models. These models offer several advantages over traditional OLS analyses (and related linear models) of sentence length, by allowing for an examination of the effects of case and offender characteristics across the full distribution of sentence lengths for a given sample of offenders. The analysis of the distribution of sentence lengths with quantile regression models allows for an examination of questions such as: Do offender characteristics, such as race or offense severity, have the same effect on sentence length for the 10% of offenders who receive the shortest sentences as they do for the 10% of offenders who receive the longest sentences? I illustrate the application and interpretation of these models using 1998 sentencing data from Pennsylvania. Key findings show that the effects of case and offender characteristics are variable across the distribution of sentence lengths, meaning that traditional linear models assuming a constant effect fail to capture important differences in how case and offender characteristics affect punishment decisions. I discuss the implications of these findings for understanding sentencing disparitites, as well as other possible applications of quantile regression models in the study of crime and the criminal justice system.

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Notes

  1. Similarly, a small but growing body of research has shown the effects of case and offender characteristics to differentially affect the jail v. prison decision (see, e.g., Holleran and Spohn 2004).

  2. Boot camp incarceration is also an option for the sentencing judge, but since a boot camp incarceration would not typically be included in an analysis of sentence length decisions, it is excluded from the range of sentence lengths noted here.

  3. Bushway and Piehl (2001) noted a similar issue in their analysis of data from Maryland.

  4. A quantile refers to a percentile in the distribution of a variable. For example, a quantile of 0.70 would refer to the same point in a distribution as the 70th percentile.

  5. This is a situation most sentencing researchers are also confronted with: Those offenders who are not incarcerated are effectively a “0” on the dependent variable and then excluded from the analysis. Quantile regression techniques could be used to address this concern without relying on various sample selection models that may be more or less appropriate for the analysis of sentencing decisions (Bushway et al. 2007).

  6. For a comprehensive discussion of the construction and interpretation of boxplots, see Tukey (1977).

  7. The choice of data set was somewhat arbitrary. My goal was to illustrate the application and the interpretation of quantile regression models with data that had been analyzed in other published studies. Hopefully, interested readers would then make comparisons between the results reported here and those in other published works using the same or similar data.

  8. The quantreg library (Koenker 2008), a component of the R system (R Development Core Team 2003), was used to estimate all quantile regression models. Readers interested in testing quantile regresison models for their research questions may also estimate these models in Stata and SAS. Stata’s version 10.0 (Stata Corporation 2007) includes quantile regression procedures that allow for the estimation of individual quantiles or a range of quantiles. SAS’s version 9.1 includes an experimental quantile regression procedure (PROC QUANTREG) that can be downloaded from the SAS web site (http://www.sas.com). PROC QUANTREG (updated in January 2009) also estimates individual quantiles as well as a range of quantiles. By using the ODS feature in SAS, it is possible to plot the quantile regression process in a manner consistent with the results displayed in Figs. 8 and 9 in this paper. However, at the time of this writing, neither Stata’s nor SAS’s quantile regression procedures allow for testing the location and location-scale hypotheses.

  9. All of the coefficients noted as not statistically significant have p > 0.10.

  10. Basic collinearity diagnostics did not indicate any problems with the inclusion of this set of case and offender characteristics.

  11. To the extent that there are minor differences between the results reported here and those published in prior research using the Pennsylvania data, the differences are due to the large number of cases with very short sentences (i.e., less than one month) that were excluded from the analyses reported here.

  12. A reviewer raised a concern about collinearity diagnostics for the quantile regression results. At this point, there are no unique ways of testing for collinearity in a multivariate quantile regression model. An indicator that collinearity is likely not a problem is reflected by the pattern of most case and offender characteristics remaining statistically significant across the range of quantiles.

  13. In much of the literature on quantile regression coefficients, the full range of effects plotted in these figures is referred to as the quantile regression process.

  14. Ulmer and Bradley’s (2006) analysis of Pennsylvania sentencing data included interaction effects for their jury trial variable with offense severity, prior record score, rape conviction offense, robbery conviction offense, and offender’s race (black). Although their inclusion of an interaction effect was a positive step forward in the analysis of the “trial penalty,” it still falls to the criticisms noted above regarding fixed effects of the independent variables across the distribution of sentence lengths.

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  1. College of Criminal Justice, Northeastern University, 360 Huntington Avenue, Boston, MA, 02115, USA

    Chester L. Britt

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Britt, C.L. Modeling the Distribution of Sentence Length Decisions Under a Guidelines System: An Application of Quantile Regression Models. J Quant Criminol 25, 341–370 (2009). https://doi.org/10.1007/s10940-009-9066-x

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