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Is the Magic Still There? The Use of the Heckman Two-Step Correction for Selection Bias in Criminology

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Abstract

Issues of selection bias pervade criminological research. Despite their ubiquity, considerable confusion surrounds various approaches for addressing sample selection. The most common approach for dealing with selection bias in criminology remains Heckman’s [(1976) Ann Econ Social Measure 5:475–492] two-step correction. This technique has often been misapplied in criminological research. This paper highlights some common problems with its application, including its use with dichotomous dependent variables, difficulties with calculating the hazard rate, misestimated standard error estimates, and collinearity between the correction term and other regressors in the substantive model of interest. We also discuss the fundamental importance of exclusion restrictions, or theoretically determined variables that affect selection but not the substantive problem of interest. Standard statistical software can readily address some of these common errors, but the real problem with selection bias is substantive, not technical. Any correction for selection bias requires that the researcher understand the source and magnitude of the bias. To illustrate this, we apply a diagnostic technique by Stolzenberg and Relles [(1997) Am Sociol Rev 62:494–507] to help develop intuition about selection bias in the context of criminal sentencing research. Our investigation suggests that while Heckman’s two-step correction can be an appropriate technique for addressing this bias, it is not a magic solution to the problem. Thoughtful consideration is therefore needed before employing this common but overused technique.

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Notes

  1. These topics range from those concerned with sample attrition or non-response (e.g. Gondolf 2000; Maxwell et al. 2002; Robertson et al. 2002; Worrall 2002) to studies of racial profiling in police stops (Lundman and Kaufman 2003), to research examining the effects of race on criminal justice outcomes (Klepper et al. 1983; Wooldredge 1998).

  2. We identified articles by electronically searching for papers in Criminology that cite the seminal article by Berk (1983) and/or included the word Heckman. This process resulted in 25 articles that use the Heckman in some shape or form (7 other studies cite Heckman or Berk but were not directly relevant).

  3. Part of the reason for the limited application of Heckman in the treatment equation approach is the availability of a number of other approaches for dealing with selection, including experiments, instrumental variable estimation and panel models. For useful in-depth discussions of alternative approaches to the treatment effects model, see Halaby (2004), Angrist (2001), and Heckman et al. (1999).

  4. It is important to distinguish between the terms conditional and marginal. A slope coefficient is an estimate of the marginal impact of x on y. We can usually estimate two different terms—the marginal effect of x in the conditional and unconditional models. As Greene (1993) makes clear, the estimate of interest will vary by case.

  5. An additional distinction can be made between the simple Two Part Model (TPM) and the censored two stage model (CTSM). Whereas the substantive regression in the TPM is based on the sub-sample of selected cases, for the CTSM, it includes all observations (including zeros for unobserved values). The TPM provides the conditional estimate given selection into the sub-sample of the population whereas the CTSM provides the unconditional estimate based on all observations. This distinction can be useful in certain applications, but it is often confusing because the CTSM requires that censored observations have meaningful 0 values (e.g. sentence lengths of 0 months of incarceration). This distinction is additionally muddled by the common convention of using the TPM to obtain estimates that ostensibly represent the unconditional population of interest (Puhani 2000). We constrain our discussion to the TPM given its focus in the literature, but further comparison of these two estimators offers an additional line of potentially interesting future research.

  6. While it is possible to estimate the TPM with alternative specifications, such as the logit instead of the probit, it is important to recognize that the estimator then assumes a different error structure (i.e. errors are distributed log normal).

  7. This distinction is relevant to the extent that slightly different modeling procedures are appropriate for the two types of samples. Truncated regression models are more appropriate for situations with truncated samples whereas limited-dependent-variable models or Tobit models are generally preferred for censored samples (see Fig. 3).

  8. The Tobit model also has some limitations. These limitations include restrictive normality assumptions regarding the dependent variable (Chay and Powell 2001; Chesher and Irish 1987) restrictive homoskeasticity assumptions regarding error terms (Wooldridge 2005), and the key assumption that the effects of independent variables are constant for the selection process and the outcome of interest (Smith and Brame 2003). See Osgood et al. (2002) for a recent overview of Tobit models.

  9. There is also a large statistical literature that attempts to loosen the assumption of bivariate normality by estimating the first stage semi-parametrically, see especially Ahn and Powell (1993) and Blundell and Powell (2001) as well as Vella (1999) and Newey (1999).

  10. STATA 8.2 provides for both the FIML and Heckman two-step estimators, while LIMDEP 7.0 also provides for a third maximum likelihood estimator of the Heckman two-step, sometimes called the Limited Information Maximum Likelihood (LIML).

  11. We are unaware of any case in which the FIML (or the related LIML model) has been used in criminology. This is somewhat ironic because Heckman originally recommended using the two-step approach only to generate starting values for the FIML estimator (Heckman 1976).

  12. Considerable confusion surrounds the predicted probability, inverse Mills ratio, and what is often referred to as the “hazard rate”. The probability is simply the predicted value from the probit equation. To calculate the inverse Mills ratio, this value is multiplied by negative one and inserted into (Eq. 5). This inverse Mills ratio, then, represents the hazard rate, or the instantaneous probability of exclusion for each observation conditional on being at risk (see Berk 1983).

  13. Dubin and Rivers (1990) discuss alternative formulations of both a two-step and maximum likelihood adaptation of Heckman’s model to the case of dichotomous outcomes, but we are unaware of any criminological application of these procedures.

  14. As one example, Nobiling et al. (1998, p. 470) describe this convention as follows: “We used logistic regression to estimate the likelihood that the offender would be sentenced to prison. For each case, the logistic regression produced its predicted probability of exclusion from the sentence length model—the hazard rate.” An anonymous referee suggested this approach represents an altogether different estimator based on propensity scoring, and should therefore not be confused with the Heckman approach. Given that each study reviewed cited Heckman (1976, 1979) or Berk (1983) and no study made explicit reference to the literature on propensity scoring, we feel justified in associating this approach with the Heckman correction. It is possible that there is statistical merit in such an approach, but we leave it to other researchers to verify it in this context.

  15. In the treatment effects version of the Heckman estimator, there are actually two different formulas for the inverse Mills ratio depending on whether an individual case is selected. Failure to correctly specify the inverse Mills ratio can bias one’s findings in the face of selection. Spohn and Holleran (2002) is an example of an application of the treatment model where the probability was substituted for the inverse Mills ratio. Correspondence with the authors indicated that the coefficient on the inverse Mills ratio was not significant when the model was re-estimated correctly, suggesting an absence of selection bias in their study. We thank Cassia Spohn and David Holleran for their willingness to engage in a constructive discussion regarding their paper.

  16. The majority of studies in Table 1 use logit rather than probit, a feature not available in the standard software programs. Also, it is unlikely that any paper that describes using the probability rather than the IMR used standard software.

  17. Desirable estimators have three properties—they are unbiased, efficient and consistent. An unbiased estimate means the difference between the true parameter and the expected value of the estimator of the parameter is 0. This has to do with how well our point estimate represents the true population value of interest. Efficiency has to do with the standard error of the estimate. More efficient estimates have smaller standard errors, meaning any one estimate is more likely to approximate the true population value. Finally, consistent estimates are estimates that become unbiased as N approaches infinity. While the TPM estimates are biased, they will be more efficient, whereas the Heckman estimates will be unbiased and consistent, but less efficient. Which estimate better captures the true parameter of interest depends on the degree of bias and inefficiency that exists in each.

  18. This list includes articles outside of our Criminology sample. This list was compiled using a search of articles that cite Stolzenberg and Relles.

  19. Exclusion restrictions are very similar to instrumental variables (Angrist 2001). In each case, we try to identify factors that affect one variable but do not affect a second variable. The ultimate exclusion restriction would be experimental or random assignment. Clearly, if people are randomly assigned to prison, then there will be no correlation between the error terms of the prison and sentence length equations, and there will be no selection bias. In the absence of experiments, we need to find quasi-experiments or elements of the selection process that are uncorrelated with the substantive decision of interest.

  20. One possible exclusion restriction for sentencing research is strength of evidence. Early studies suggest that this factor affects the likelihood of conviction, but not the severity of the sentence post-conviction (Albonetti 1991).

  21. To be more comprehensive, we attempted to identify additional papers outside the journal Criminology that used the Heckman with exclusion restrictions. We did find four other examples of models using the Heckman correction and incorporating exclusion restrictions (Albonetti 1991; Maxwell et al. 2002; Robertson et al. 2002; Worrall 2002), but none of the articles we reviewed defended or discussed their exclusion restrictions in any detail.

  22. In Table 1, we counted those articles which arbitrarily excluded variables that were highly correlated with the correction factors in order to estimate the model as not having exclusion restrictions.

  23. There is a substantial literature on alternative estimation frameworks, including non-parametric frameworks, which avoid some of the problems associated with the restrictive parametric assumptions required by the bivariate normal models (Ahn and Powell 1993, Chay and Lee 2001). While we do not deny the potential usefulness of these alternative frameworks, we have chosen to remain within a framework known to most criminologists.

  24. The assumption of a two-stage decision making process in criminal sentencing largely grew out of early research on white collar offenders in federal districts that argued “the first and hardest decision the judge makes is whether the person will go to prison or not,” which is “experienced as qualitatively different from the decision as to how long an inmate should stay in prison” (Wheeler et al. 1982, pp. 642, 652). This assumption has been challenged recently and may be less valid for cases sentenced in states with presumptive sentencing guidelines (Bushway and Piehl 2001).

  25. The STATA do file used to estimate these coefficients is available from the authors by request.

  26. Statistically, the condition number is the ratio of the maximum Eigenvalue to the minimum Eigenvalue for the matrix of independent variables in an analysis (see Belsley et al. 1980, pp. 100–104, for a technical treatment of the condition number). It can be calculated in STATA 8.2 with the collin() function and is also provided in standard output for collinearity diagnostics in SPSS and other statistical software packages.

  27. This is commonly understood by criminologists in another context. The logit and probit models will provide very similar effect size estimates when the probability of the dependent variable being a success is not close to 1 or 0. However, as the proportion of successes approaches 1 or 0, which requires extreme values of Z, the two models can provide very different results because of the different parametric assumptions.

  28. A similar problem is endemic in comparisons of coefficients across different models affected by selection. In sentencing, key modeling decisions, such as the decision to examine incarceration and sentence length as two distinct outcomes, rest on the finding that coefficients exert different effects on the two outcomes. However, if the error terms in the two models are correlated, the coefficients for sentence length will be biased. This will lead to flawed comparisons if a simple TPM is utilized or if the Heckman is implemented incorrectly. This observation may also apply to the current debate about whether jail and prison can be treated as the same outcome (Holleran and Spohn 2004). To the extent that selection bias exists in the coefficients for jail and prison sentences (i.e. non-random selection processes determine conviction and therefore offender placement), any comparison of these effects across models without good controls for selection must be conducted with caution.

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Acknowledgements

The authors would like to thank Justin McCrary, Wayne Osgood, Anne Piehl, Jeff Smith and seminar participants at the Economics and Crime summer workshop for helpful comments. We would also like to thank Gary Sweeten for research assistance and useful discussions. Most of the research for the paper was completed while Bushway was a professor at the University of Maryland. The paper is dedicated to the memory of our friend, mentor, and former colleague Douglas A. Smith. Doug helped us to understand the relevant selection issues, and encouraged us to pursue the paper in its formative stages. He is sorely missed.

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  1. School of Criminal Justice, State University of New York at Albany, Albany, NY, USA

    Shawn Bushway

  2. Department of Criminology and Criminal Justice, University of Maryland, 2220 LeFrak Hall, College Park, MD, 20742, USA

    Brian D. Johnson & Lee Ann Slocum

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Correspondence to Brian D. Johnson.

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Bushway, S., Johnson, B.D. & Slocum, L.A. Is the Magic Still There? The Use of the Heckman Two-Step Correction for Selection Bias in Criminology. J Quant Criminol 23, 151–178 (2007). https://doi.org/10.1007/s10940-007-9024-4

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