Reshaping the Study of Recidivism: Exploring Variations in the Timing of Recidivism Following Release From Prison

Abstract

While hazard analyses allow researchers to identify distributional changes over time, this powerful benefit is often underutilized. This article incorporates the shape parameter—in addition to level—into lognormal hazard models to examine recidivism patterns for individuals returning home from prison. Using a sample of adults released in 1994 from 15 state prison facilities, the results indicate that factors influencing the shape are both individual-level (race, age, prior arrest history) and jurisdiction-driven (prison admission type and state). While targeting the highest “risk” individuals is a well-established best practice, the present study suggests that reentry planners may benefit from focusing on groups undergoing change in the postrelease period in addition to those experiencing the highest hazard levels on average. Future research would benefit from incorporating the shape parameter into recidivism studies and including additional factors in shape analyses, such as social indicators, to further contextualize the reentry–recidivism relationship.

Keywords

reentry recidivism criminal justice policy lognormal hazards shape

Introduction

In a classic book on recidivism, Maltz (1984/2001) argues that calculating static recidivism rates, or the percentage of individuals who have recidivated within a specified window of time, hinders our understanding of how recidivism rates change over time. In a 2001 preface to an online edition, Maltz observes that in the two decades since his book was first published, researchers have largely transitioned from recidivism rates to analyzing survival or hazard¹ models. Researchers have utilized hazard models for a variety of purposes, including theory testing (e.g., Kurlychek, Bushway, & Brame, 2012; Paternoster, Bachman, Kerrison, O’Connell, & Smith, 2016), studying the effectiveness of interventions and policy decisions (Hamilton & Campbell, 2013; Ostermann, 2015), and examining how legal and structural factors influence recidivism and deviance (Grattet, Lin, & Petersilia, 2011). In particular, the Cox proportional hazards model is currently regarded as the most popular model for analyzing survival data (Harrell, 2015).

However, the use of these more complex models by themselves does not necessarily mean that researchers are detailing change. More commonly, researchers describe survival analysis results in terms of average levels or the average time to failure. This approach assumes that individuals within the sample react similarly and that these effects are constant over time. Maltz (1984/2001) includes a cautionary note to researchers about the Cox proportional hazards model in particular, noting that it “makes the implicit assumption that the hazard rate of each individual is the same as for that of all other individuals, except for a multiplicative constant” (p. xv). In other words, although the overall hazard can take any shape, that same basic shape is then fixed for everyone in the sample.

The proportional hazards assumption might overlook important differences between groups of interest, such as men and women, younger and older individuals with criminal records, or groups with varying prior criminal histories. Distinguishing these groups is critical for policy purposes, and has been central to risk assessment research. For example, the risk–needs–responsivity (RNR) model is a widely used recidivism reduction strategy based on research that indicates targeting the highest “risk” individuals brings the largest gains (Andrews, Bonta, & Hoge, 1990; Bonta & Andrews, 2007). However, it is also possible that those typically considered “high risk” have more stable hazard levels and patterns relative to those in the “low risk” group. If lower risk individuals are experiencing more dynamic change upon release from prison, policymakers may have a new incentive to divert some resources to this group. By making the proportionality assumption in the hazard model selection process, researchers restrict the ability to observe important differences in how people experience the reentry process or respond to different policies.

The current study seeks to address this restriction by incorporating variation in shape more directly into hazard analyses. By doing so, I am able to examine heterogeneity in two ways: examining the overall fit of the hazard model and indications of underlying unobserved heterogeneity, and visualizing conceptually meaningful differences between observable characteristics within a sample (such as comparing Black and White individuals). Because shape is not currently a mainstream component of recidivism hazard analyses, this article begins by providing a strategy for understanding and using shape. The first section discusses theoretical explanations for certain recidivism shapes, and why these assumptions are important in the model selection process. This is followed by a discussion of the analytic strategy, sample, and indicators in this study. The main analyses include a descriptive demonstration of how covariates can influence the shape, how level and shape work together, and which specific covariates influence the shape parameter using a flexible hazard model.

These analyses generate three main findings. First, the results support and extend prior hazard-level research. Several of the covariates commonly emphasized in level studies, including race, age, and prior criminal history, also contribute to explaining the shape of the hazard, and the level results remain stable when the shape parameter is incorporated. Second, level and shape coefficients are typically in the same direction (i.e., both are positive or both are negative), which indicates that lower (higher) levels of risk are associated with steeper (decreased) slopes. A third finding relates to the importance of jurisdiction in explaining shape patterns, even when holding individual-level demographic and criminal justice indicators constant. Overall, these findings suggest that the primary contribution of the shape parameter is its ability to identify “riskiness” in terms of timing and change, instead of average levels across time. As the indicators in this study are often available in administrative corrections and risk assessment data, combining level and shape in analyses provides a complementary approach for informing policy interventions.

Why Recidivism Hazard Patterns Might Theoretically Differ

Rearrest hazards typically take one of two basic shapes, which has important theoretical implications. The first pattern is an “immediate decline,” where the peak or highest hazard point occurs in the first unit of time. The other common pattern is a “delayed peak,” where the hazard rises and experiences a turning point at some point after the first time period.

There are two theoretical explanations for the immediate decline pattern. First, individuals could be changing and lowering their risk as they become settled into the community. Individuals may be acquiring social capital and strengthening their social bonds (Berg & Huebner, 2011; Rocque, Bierie, Posick, & MacKenzie, 2013). The second possibility is that individuals are not changing, but the composition of the individuals in the sample is changing. In this scenario, the highest risk people simply fail first and are no longer used to calculate the hazard, creating an apparent declining pattern. This unobserved heterogeneity in risk levels is referred to as “offender sorting” (Kurlychek et al., 2012; Lattimore, MacDonald, Piquero, Linster, & Visher, 2004). Under this assumption, classifying individuals into risk groups for intervention targeting purposes supports the RNR approach (Andrews et al., 1990; Bonta & Andrews, 2007). Unfortunately, it is difficult to distinguish between true individual-level change and aggregate offender sorting with an immediate decline pattern (Allison, 2012).

The delayed peak pattern, on the contrary, is typically complicated by measurement issues rather than theoretical ones. Recidivism researchers have expressed concern that hazard patterns may be artificial due to delays in certain parts of the criminal justice system such as a lag between the time an individual is arrested and later convicted (Schmidt & Witte, 1988). Therefore, the pattern only appears delayed due to criminal justice system processing. However, by using rearrests, which occur earlier in the criminal justice system process (and are closer in time to the offense of interest), this issue is largely avoided.²

Under the assumption that delayed rearrest hazard patterns are not a reflection of artificial institutional delays, there may be several criminological explanations for the delayed peak pattern. Anytime a pattern is in decline—even when following an increase in the hazard pattern—this could be a reflection of unobserved heterogeneity. However, sorting cannot explain an increasing pattern. Instead, this reflects an increase in risk, at least for some group or groups of individuals. Although this article is not testing a particular theory, two relevant theoretical explanations are explored here: one that involves reactions to the criminal justice system and one that is driven by other motivations to disengage from crime.

One possibility for the delayed peak pattern is that individuals are deterred from engaging in crime but only temporarily. In its most basic form, deterrence theory posits that if the threat of punishment is appropriately severe, certain, and swift, individuals will refrain from crime (Beccaria, 1764/1986; Bentham, 1789/1988). One popular notion is that the individuals most likely to engage in crime and be unreceptive to threats of punishment share certain static characteristics such as impulsivity or low self-control (Gottfredson & Hirschi, 1990). Other researchers emphasize that surrounding situations and circumstances may be significant (Taylor, Walton, & Young, 1973), and there may be certain individuals who are particularly influenced by deterrence strategies (Wright, Caspi, Moffitt, & Paternoster, 2004). These “marginal offenders” are those neither completely likely nor completely unlikely to recidivate, but linger somewhere in between (Zimring & Hawkins, 1968, 1973). This is the group of interest to policymakers, and who theoretically may be contributing to a delayed peak hazard pattern. For example, parole requirements or a mandated rehabilitation program may successfully deter a marginal group, but only as long as the policy is in effect. Once the policy is removed or practitioners reduce the intensity of an intervention, the marginal group may experience a heightened risk of recidivism.

A second potential theoretical explanation for the delayed peak pattern is that some individuals are temporarily motivated to desist from crime immediately upon release, but this motivation weakens over time or individuals encounter derailments or setbacks (Paternoster & Bushway, 2009; Shilling, 1999). While desistance is typically discussed as an individual-level phenomenon, change in the immediate postrelease time frame may be visible at the aggregate level if individuals are experiencing similar challenges (and therefore, setbacks) in the reintegration process. In addition to the discrimination individuals with criminal records face in the labor and housing markets (e.g., Evans & Porter, 2015; Pager, 2007), setbacks may be further exacerbated by formal collateral consequences. The American Bar Association (2016) has documented tens of thousands of federal, state, and local laws and regulations that block individuals with criminal convictions from accessing various employment, housing, and educational opportunities and civic rights. Longitudinal reentry research indicates that while prisoners tend to have high levels of optimism about the future before release, this often converts to high levels of stress and frustration in the transition process postrelease (Visher & O’Connell, 2012; Visher, Yahner, & La Vigne, 2010). These attitudes and self-perceptions upon release from prison can also influence recidivism outcomes (LeBel, Burnett, Maruna, & Bushway, 2008). Delayed hazard peaks may be a reflection of setbacks in the desistance process, where the risk of rearrest increases after a few weeks or months.

In addition, programs that assist returning prisoners in this transition process can be a beneficial source of intervention but sometimes only have short-term effects. For example, comparing individuals who successfully complete an in-prison General Education Development (GED) program to similar inmates not enrolled in the course, Tyler and Kling (2007) find a notable bump upon release—around a 15% increase in postprison earnings—but a decline in this benefit over time. Specifically, these increases in earnings apply to minorities in their sample but not to White individuals, and there is a “virtual disappearance”of the postrelease earnings bump after the second year passes (Tyler & Kling, 2007, p. 250). When considering postprison release interventions, a transitional jobs program for returning prisoners run by the Center for Employment Opportunities (CEO) also indicates promising short-term impacts. Redcross, Millensky, Rudd, and Levshin (2012) find that the program reduces conviction and reincarceration outcomes for participants, but that these effects are driven by first-year postrelease outcomes and diminish over time. Furthermore, there may be variation in who is able to access available programs. For example, individuals may need a parole officer reference to be considered for a program (e.g., Redcross et al., 2012), and certain types of individuals, such as those convicted of sex offenses or violent crimes, may not have the same programmatic options due to crime-type restrictions.

While some studies depict immediate decline patterns when using aggregate time units³ or for large diverse samples (e.g., see the full sample hazard ratios in Table 12 of Beck & Shipley, 1989), the general consensus in the recidivism hazard literature is a delayed peak pattern. In studies that use official rearrest hazard patterns, the delayed peak pattern finding holds for different “starting points,” including police contacts (Kurlychek, Brame, & Bushway, 2006), boot camp graduation (Kurlychek & Kempinen, 2006), and juvenile facility release (Caudill, 2010; Lattimore, Visher, & Linster, 1995). The delayed peak pattern is also consistent for adult pretrial defendants (DeJong, 1997; Visher & Linster, 1990), adults with felony convictions (Kurlychek et al., 2012), and adults incarcerated in the past 6 months (Uggen, 2000: Figure 3). These samples also vary by geographical locations. Different geographic areas may contribute to variations in recidivism patterns due to localized laws, sentencing practices, and supervision/technical violation policies (Frase, 2005; Travis and Lawrence, 2002), varying labor market conditions (Yang, 2016), and collateral consequences (American Bar Association, 2016).

Although these studies present delayed peak patterns descriptively (i.e., the overall baseline hazard patterns), the hazard model analyses in this prior work overwhelmingly emphasize level. The methodological contribution of this article is to extend the popular and flexible lognormal hazard model to include a shape parameter, which will allow specific covariates to vary not only in level but also in shape. Motivated by the theoretical explanations explored for the delayed peak pattern here, the next section describes the hazard model selection process.

Hazard Models, Assumptions, and Theory

Nonparametric approaches, such as life tables, Kaplan–Meier estimates, and baseline hazard graphs, are often the starting point for survival analysis studies. These are informative descriptive methods that require few distributional assumptions. This flexibility comes at a cost, however. Nonparametric models can be sensitive to subtle changes in the data (such as how time intervals are constructed, particularly with small sample sizes) and are often less reliable at the tail ends of the distributions (Blossfeld, Golsch, & Rohwer, 2007; Cleves, Gould, Gutierrez, & Marchenko, 2010). A more critical issue for the present study is that all nonparametric models require the “proportional hazards” assumption. Under this assumption, covariates can move the shape up or down but are not allowed to affect the shape itself. While there are clear benefits to not requiring parametric assumptions, the proportional hazards assumption removes the option of nonparametric models for researchers interested in studying variation in shape.

Another class of hazard models has semiparametric requirements, which do not assume a functional form but still allow covariates to influence the hazard. One of the main advantages of the Cox proportional hazards (Cox, 1972) model, one of the most popular models more broadly and in criminology specifically, is that it does not assume a shape. This model allows for whatever form the distribution takes, but still requires the proportional hazards assumption. The Cox model is particularly useful if theory indicates that certain covariates are important predictors of time to recidivism, and that these covariates do not change over time. However, while researchers can incorporate interaction terms and/or stratify by a few covariates of interest, the variables are still not allowed to affect the shape of the curve and act instead as controls (Allison, 2012). Because this study is focused on understanding how covariates affect the shape of the hazard, the Cox model is still too restrictive.

Different parametric models have direct substantive connections to the underlying behavior under study. For example, criminal career researchers comfortable with a stable, Poisson rate of offending might use an exponential model (Blumstein, Cohen, Roth, & Visher, 1986). However, if researchers believe that the pattern may increase or decrease in a monotonic way, the Weibull model not only allows for a constant hazard rate but also provides this added flexibility. If there is a compelling theoretical reason to believe that the hazard pattern is nonmonotonic (e.g., increases and decreases), the lognormal (or similarly, the loglogistic) model provides a promising alternative. The lognormal does not force the covariates to have a proportional influence on the hazard (as the Cox and other models do); it is a very flexible model that allows the baseline hazard to take on a wide variety of shapes. Context-specific research suggests that the lognormal pattern aligns with what researchers would generally expect for individuals returning home from prison, and this pattern accommodates the theoretical notions of temporary deterrence or temporary desistance. While both Weibull and lognormal models are reasonable options, I prefer the lognormal model, which allows for the turning point that prior research suggests is important (Kurlychek et al., 2012; Schmidt & Witte, 1988).

A final option, which shares similar goals with the current study, involves a latent grouping strategy to tease out sample heterogeneity. The most popular method in survival analyses, split population (or cure) models, incorporates the assumption that not everyone will eventually recidivate (e.g., Kurlychek et al., 2012; Maltz, 1984/2001; Schmidt & Witte, 1988, 1989). Another recent approach involves defining different trajectories for groups based on certain developmental characteristics (Nagin & Tremblay, 2005). For example, Bachman, Kerrison, O’Connell, and Paternoster (2013) and Martin, O’Connell, Paternoster, and Bachman (2011) recently used trajectory analyses for a sample of drug-involved returning prisoners to study the impact of an intervention program on five groups with different developmental trajectories. Morris, Barnes, Worrall, and Orrick (2013) also recently combined survival analysis techniques with a trajectory analysis approach to separate out classes of individuals based on offending patterns (immediate, delayed, and low-risk recidivists). These approaches essentially loosen the restriction of proportionality by only requiring proportionality within each group. In other words, each group can (and does) have a different shape for the baseline hazard.

Both lognormal hazards and grouping strategies are solutions to understanding underlying patterns within a sample, but they approach the formation of the groups under examination differently. In the case of split population models, groups are defined by the outcome of interest, eventual recidivism. Trajectory researchers use both theory-based (Huebner & Berg, 2011) and empirically driven (Morris et al., 2013) approaches to defining groups in trajectory analyses, but ultimately, once the groups are fixed, within-group differences are assumed to be proportional. While the characteristics of a particular trajectory group can be described, it is more difficult to demonstrate which covariates influence the particular shape of the hazard, as the comparison is between all features of one group and all features of the other group. If level is the main difference between the two hazards, the different covariates might be explaining level, rather than changes in shape. While parametric, the lognormal model is very flexible and avoids this ambiguity by explicitly allowing covariates to affect both level and shape. For this reason, the present study reintroduces the lognormal hazard as a way to examine and understand shape in recidivism analyses. The next section discusses the dataset, measures, and methods used in this study before turning to a discussion of which factors influence rearrest hazard shapes.

Data

This study analyzes data collected by the Bureau of Justice Statistics (BJS) in their second major recidivism study (Langan & Levin, 2002). BJS collected criminal justice information for adults released from prison in 1994 in 15 states⁴ based on a stratified sampling strategy, including official records for 3 years postrelease. I incorporate BJS’ assigned weights, which represent the inverse probability of selection, into the analyses.

The main advantage of this dataset is that it contains comprehensive official criminal history information, and the states represented in the weighted data file account for approximately two thirds of all individuals released in 1994.⁵ The main disadvantage is the limited number of indicators outside of criminal history data. The BJS dataset does not contain social indicators (e.g., family or peer relationships, marital status, employment history, drug/alcohol abuse or mental health issues, or preprison and postrelease housing) or other indicators that may be of interest, such as misconduct, while incarcerated. However, the included criminal history variables are traditionally available and powerful predictors of recidivism.

The final sample includes 270,723 individuals (33,675 unweighted cases), and the sample characteristics are described in Table 1. Available demographic information reflects measures commonly used in prior recidivism studies, including sex, race, and age at release.⁶ Ninety-one percent of this sample is male, and the majority of prisoners are either White (49%) or Black (47%). Age at release ranges from 18 to 88, with a mean of 32 years.

Table 1.

Descriptive Statistics.

Variable	Released prisoners (%)
Demographics
Age at release (years)
18-24	21.1%
25-29	22.9%
30-34	22.8%
35-39	16.3%
40-44	9.5%
45+	7.5%
Male (0 = no/1 = yes)	91.3%
White (0 = no/1 = yes)	49.3%
Black (0 = no/1 = yes)	47.4%
Other race (0 = no/1 = yes)	1.0%
Missing race (0 = no/1 = yes)	2.3%
Criminal history
Age at first arrest: Mean (years)	21.7
Number of prior arrests: Mean	8.8
Number of prior convictions: Mean	3.8
Admission type
Court commitment/transfer (0 = no/1 = yes)	57.3%
Some form of revocation (0 = no/1 = yes)	32.0%
Missing admission type (0 = no/1 = yes)	10.7%
Instant offense type (0 = no/1 = yes)
Violent	22.5%
Property	33.5%
Drugs	32.6%
Public order/other	11.5%
Release type (0 = no/1 = yes)
Parole board	25.3%
Mandatory parole	56.6%
Other release types	18.0%
Missing release type	0.05%
Recidivism rate (0 = no/1 = yes)
Rearrested during data collection time frame	77.0%
Rearrested within 3 years	67.4%
Number of released individuals	270,723

Note. The table uses weighted data. Due to missing data, the sample used in this analysis slightly differs from the sample used in the BJS 1994 report. BJS = Bureau of Justice Statistics.

Prior criminal history has the most consistent and dependable relationship with recidivism, with criminal history measures serving as the best predictors of future criminal justice system involvement (Gendreau, Little, & Goggin, 1996). On average, individuals in this sample have 8.8 prior arrests and 3.8 prior convictions. Prior arrests are used in this study to reduce discretionary criminal justice decisions in estimating criminal propensity. Here, the number of prior arrests is categorized into five groups (0, 1, 2-3, 4-5, and 6+) for the regression models. In addition to measures of prior frequency of engaging in crime, Rosenfeld, Wallman, and Fornango (2005) argue that the admission type for the instant offense of interest is important. Specifically, in their study using the 1994 BJS data, they find that “prior recidivism as distinct from prior arrests (or prior incarceration) influences subsequent arrests” (Rosenfeld et al., 2005, p. 95). Here, I code admission type as one of three dummy variables: new court commitment or transfer, some form of revocation (parole, mandatory parole, probation, or escapee, regardless of whether a new sentence was issued or not), and missing (“other” or “unknown”). Approximately 57% of cases in this sample are new court commitments/transfers, 32% are revocations, and 11% are included in the missing dummy. Age at first arrest is also correlated with the propensity to engage in crime (Farrington et al., 1990; Wolfgang, Figlio, & Sellin, 1972), and career criminologists have argued that it is one of the strongest predictors of future offending patterns (Blumstein, Farrington, & Moitra, 1985). Age at first arrest is a continuous variable, with a mean age of 22 years and a range of 10 to 79.⁷

In addition, individuals committing property offenses (or what BJS describes as crimes for monetary purposes) tend to recidivate at higher rates than other crime types (Durose, Cooper, & Snyder, 2014; Langan & Levin, 2002; Rosenfeld, 2007 as cited in National Research Council, 2007). Here, four offense type categories (violent, property, drug, and public order/other) are included. Approximately 23% are classified as violent offenses, 34% property, 33% drug, and 12% public order/other.

In addition to these individual characteristics and preprison factors, whether individuals leave prison with supervision—and whether they leave under discretionary or mandatory parole supervision—theoretically has important implications for recidivism rates (Ireland & Prause, 2005; Petersilia, 2003). To further explore whether release type influences the shape of the hazard, this policy decision is also included. In this sample, release types include parole board decisions where the person did not serve a minimum, mandatory parole release, probation release/shock probation, other conditional release, expired sentence, other unspecified release types, and “other.” A handful of commutations/pardons are excluded from the analysis. When controlling for release types in the regression models, I include parole board decision (25%), mandatory parole release (57%), and all other release types (18%).

Finally, using an earlier BJS dataset of prisoners released in 1983, Bierens and Carvalho (2007) find substantial differences across states in their hazard models, noting the challenge in deriving interpretations and policy implications due to sample heterogeneity. To account for varying state policies and practices, state dummy variables are included and standard errors are clustered by state in the models. California is the reference state in these models.

Method

This study uses a lognormal hazard model to explore the relationship between shared characteristics and the shape of rearrest patterns. In a two-parameter lognormal distribution where $x$ is a random variable with a normal probability distribution, $y = \ln (x)$ is normally distributed.⁸ More formally, following the notation in Blossfeld and Rohwer (2002), the lognormal hazard rate is as follows:

r (t) = \frac{1}{b t} \frac{ϕ (z_{t})}{1 - Φ (z_{t})},

z_{t} = \frac{\log (t) - a}{b} .

Here, $t$ is the unit of time, $a$ is the level parameter (mean), b is the shape parameter (standard deviation), $ϕ$ is the standard normal density, and $Φ$ is the normal distribution function. In a model without covariates, $a = \propto_{0}, b = \exp (β_{0})$ (Blossfeld & Rohwer, 2002). By including covariates in a particular parameter, the researcher makes assumptions about the influence of the covariates (Blossfeld et al., 2007; Cleves et al., 2010).⁹

As a brief example of how level and shape work in this lognormal equation, Figure 1 provides an example of how a smaller and larger shape parameter value changes the hazard pattern when holding the mean constant. Specifically, I alter $b$ (or sigma) and maintain the same value of $a$ for weeks $t$ in Equation 2. The shape or sigma value that is 0.30 larger has a higher risk level and more of an immediate decline pattern, while the sigma that is 0.30 smaller has a lower level but a longer period of increasing risk. From a policy perspective, the varying front ends in these distributions may lead to different recommendations. If certain factors are correlated with a quicker time to failure, for example, policymakers might want to target groups with these shared characteristics for intervention.

Figure 1.

Changing sigma values.

To further demonstrate how shape works for a sample of individuals returning home from prison, the next section first provides the baseline estimate. Covariates can be included in the level and/or shape parameter as an extension to the traditional lognormal model, depending on which parameters the researcher believes the covariates may influence (Blossfeld & Rohwer, 2002). As a comparison, three separate models are reported for level, shape, and level + shape. To make the model interpretations more concrete, I provide visual examples comparing Black and White individuals and different state patterns. In all of the analyses, available data are used in the model estimations, although the graphs only display the immediate postrelease period. Select graphs throughout are also displayed with 95% confidence intervals using the delta method.

Results

As a starting point, I display the baseline hazard for the full sample in Figure 2 using a lognormal model without any covariates. As Blossfeld et al. (2007) note, time-dependent parametric models should be interpreted with caution as they do not account for heterogeneity. As this sample includes individuals from 15 states—each with their own laws, release policies, and reentry strategies—I might not expect to see the traditional delayed peak pattern commonly detected with more homogeneous samples in prior studies. In other words, while the prior literature finds similar patterns across different types of samples, the samples themselves were relatively homogeneous. While this baseline was estimated using all of the available data, in the graph I focus on the first 6 months postrelease and I measure time in weeks on the x-axis.

Figure 2.

Baseline hazard.

The lognormal hazard, which does not force a delayed peak shape but will accommodate one, displays a very small, quick turning point here. The sample peaks in Week 2 (with a hazard ratio of 0.027). However, even with this small delayed peak pattern visible, unobserved heterogeneity may still be affecting the distribution (Zorn, 2000). In reality, any time a distribution is declining, even when occurring after a period of increased risk, there may be unobserved heterogeneity (Allison, 2012). In addition, data points near the boundaries are always less reliable than the rest of the distribution, particularly when describing the pattern without considering covariates.

The main advantage of the lognormal model is the ability to see how the shape of the hazard varies by specific demographic, context, and criminal history covariates. Table 2 displays the covariates for three models: level only, shape only, and a combined level/shape model. The standard errors are clustered by the state from which the prisoner was released in 1994.

Table 2.

Model Parameters.

Parameter	No covariates	Level only	Shape only	Level and shape
M (constant)	4.19	5.40	4.13	5.55
ln_sigma	0.77	0.69	1.11	1.03
Sigma	2.15	1.99	3.03	2.80

Note. Every value displayed above was significant at the .001 level. Sigma = exp(ln_sigma).

Compared with the model without any covariates, the constant increases when covariates are included in just the level parameter or in both the level and shape models. The sigma values without covariates (2.15) or with only the level parameter (1.99) are both smaller than the shape only (3.03) and combined level/shape (2.80) models.¹⁰ This increase in the sigma value suggests that the full model distribution is steeper (or more resembles an immediate decline pattern) compared with models without covariates or with only level indicators (Blossfeld et al., 2007). Next, I present results for how individual covariates influence the value of sigma.

The results in Table 3 suggest that certain demographic and criminal history indicators have a significant influence on both the level and the shape of the distribution regardless of which model parameters I include. For the level, positive coefficients indicate a slower time to recidivism in the postrelease period, given that it will occur, and negative coefficients indicate a quicker time to failure (DeJong, 1997; Schmidt & Witte, 1988; Witte & Schmidt, 1977). Here, Black individuals (compared with White), younger individuals, males, those with more prior arrests, those entering prison for the 1994 release offense because of a revocation (compared with a new court commitment), those with property instant offenses (compared with violent), and people with another release type besides mandatory release (when compared with parole board release) are more likely to have a statistically significant quicker time to failure.¹¹ In addition, in comparison with California, released individuals from eight states (Arizona, Michigan, Minnesota, New Jersey, North Carolina, Ohio, Texas, and Virginia) consistently have statistically significantly slower failures in the level parameters.

Table 3.

Covariates in Level and Shape Parameters.

Variable	Separate models		Combined model
	(1)	(2)	(3)
	Level	Shape	Level	Shape
Black	−0.376*** (0.049)	−0.127*** (0.037)	−0.446*** (0.038)	−0.158*** (0.048)
Other race	0.468 (0.248)	−0.088 (0.062)	0.347 (0.285)	−0.029 (0.058)
Missing race	0.041 (0.110)	−0.131** (0.041)	−0.069 (0.087)	−0.128** (0.040)
Age 25-29	0.504*** (0.052)	0.044** (0.014)	0.492*** (0.057)	0.056*** (0.014)
Age 30-34	0.722*** (0.067)	0.003 (0.020)	0.695*** (0.074)	0.031 (0.016)
Age 35-39	0.905*** (0.067)	−0.005 (0.027)	0.869*** (0.068)	0.015 (0.043)
Age 40-44	1.151*** (0.065)	0.055* (0.026)	1.145*** (0.050)	0.094*** (0.023)
Age 45+	1.676*** (0.096)	0.151*** (0.033)	1.656*** (0.098)	0.111** (0.037)
Male	−0.459*** (0.115)	−0.004 (0.033)	−0.398*** (0.112)	0.004 (0.024)
Age at first arrest	−0.004 (0.006)	0.003 (0.003)	−0.000 (0.009)	0.001 (0.003)
1 prior arrest	−0.142 (0.144)	−0.175*** (0.043)	−0.305 (0.168)	−0.149*** (0.040)
2-3 prior arrests	−0.516** (0.163)	−0.211** (0.068)	−0.652*** (0.185)	−0.118 (0.067)
4-5 prior arrests	−0.862*** (0.165)	−0.271*** (0.054)	−1.008*** (0.176)	−0.153** (0.050)
6+ prior arrests	−1.418*** (0.199)	−0.265*** (0.049)	−1.594*** (0.189)	−0.188*** (0.041)
Admission type: Revocation	−0.321*** (0.097)	−0.114*** (0.034)	−0.341** (0.114)	−0.122*** (0.032)
Admission type: Missing	−0.487*** (0.105)	−0.207*** (0.034)	−0.536*** (0.140)	−0.253*** (0.040)
Property	−0.361*** (0.094)	−0.031 (0.016)	−0.360*** (0.070)	−0.029 (0.015)
Drug	−0.087 (0.118)	−0.005 (0.024)	−0.050 (0.087)	0.001 (0.027)
Public order/other	−0.015 (0.059)	−0.056* (0.026)	−0.021 (0.069)	−0.065* (0.026)
Mandatory release	−0.056 (0.112)	−0.034 (0.056)	−0.088 (0.123)	−0.059 (0.053)
Other release	−0.598* (0.269)	0.291* (0.135)	−0.309 (0.178)	0.317 (0.163)
Release-type missing	−0.056 (0.333)	−0.022 (0.191)	0.477 (0.259)	0.254 (0.197)
Arizona	0.652*** (0.161)	−0.445*** (0.119)	0.370** (0.121)	−0.484*** (0.122)
Delaware	0.389 (0.252)	−0.667*** (0.163)	0.057 (0.213)	−0.778*** (0.192)
Florida	−0.165 (0.251)	−0.171 (0.162)	−0.413* (0.208)	−0.247 (0.185)
Illinois	0.053 (0.058)	−0.191*** (0.014)	−0.033 (0.062)	−0.269*** (0.021)
Maryland	0.205* (0.082)	−0.225*** (0.037)	0.054 (0.097)	−0.244*** (0.047)
Michigan	1.136*** (0.108)	−0.191 (0.104)	0.944*** (0.121)	−0.339*** (0.056)
Minnesota	0.520*** (0.090)	−0.265*** (0.027)	0.229* (0.097)	−0.296*** (0.025)
New Jersey	0.697*** (0.105)	−0.350*** (0.081)	0.480*** (0.133)	−0.377*** (0.072)
New York	0.198 (0.121)	−0.148 (0.082)	0.124 (0.144)	−0.176* (0.083)
North Carolina	0.680*** (0.037)	−−0.154** (0.048)	0.609*** (0.045)	−0.166*** (0.036)
Ohio	1.292*** (0.228)	−0.090 (0.085)	0.951*** (0.125)	−0.099 (0.077)
Oregon	−0.024 (0.068)	−0.046 (0.036)	−0.193* (0.079)	−0.100** (0.034)
Texas	0.609*** (0.076)	−0.071 (0.090)	0.656*** (0.070)	−0.058 (0.104)
Virginia	0.481*** (0.064)	−−0.295*** (0.044)	0.350*** (0.079)	−0.279*** (0.035)
Level constant	5.405*** (0.301)	4.125*** (0.164)	5.546*** (0.311)	—
Shape constant	0.693*** (0.037)	1.112*** (0.109)	—	1.033*** (0.078)

Note. The models all contain 33,675 observations representing 270,723 cases. The covariates can only vary by level in the first model (column 1), by only shape in the second model (column 2), and by both level and shape in the third model (columns 3 and 4). A constant is still estimated when covariates are not specified in a parameter, leading to multiple constants in the separate models. Reference categories include White, age 18 to 24, zero prior arrests, court commitment admission type, violent instant offenses, and parole board release.

p < .05. **p < .01. ***p < .001.

The interpretation for the shape coefficients is somewhat less intuitive but follows the same basic interpretation demonstrated in the model sigma comparisons (see Figure 1). Factors that increase the shape parameter value create a steeper incline (i.e., more of an immediate decline), whereas factors that decrease sigma lead to a longer incline period, and therefore a more delayed peak or slower turning point (Blossfeld & Rohwer, 2002). In the political science literature, changes in the shape parameter are described in terms of an increasing (positive coefficient) or decreasing (negative coefficient) slope (Zorn, 2000). In other words, a more “positively sloped” delayed peak hazard is related to a higher sigma value, and steeper/more immediate decline pattern, than a negatively sloped hazard pattern (Zorn, 2000).

In this analysis, factors that are statistically significant and consistently indicate a decreased slope in the shape parameters include Black (compared with White), those with more prior arrests, those with revocation admission types, and most of the states when compared with California. The oldest age category (45+) compared with the youngest (18-24) and those with another release type besides mandatory release (when compared with parole board release) have positive shape coefficients and increased slopes.

These level and shape findings may seem counterintuitive. Almost all of the demographic and criminal history indicators that are significant in both the level and shape share the same direction (positive level/positive shape; negative level/negative shape) pattern. For example, compared with White individuals, Black individuals have a negative-level coefficient and negative shape coefficient, which represents a higher hazard level and a decreased hazard slope (or less of an immediate decline/steep slope). In other words, groups experiencing rearrest at higher hazard levels appear to undergo change differently than lower level groups do. However, a decreased hazard slope does not necessarily mean that the shape is a delayed peak pattern, and a group with a higher level is not necessarily higher in every time period. The best way to fully grasp these interpretations involves graphical comparisons. Using all of the same controls in the full level + shape model shown in Table 3, I separate out Black individuals and White individuals as an example in Figure 3.

Figure 3.

Rearrest hazard patterns by race.

This helps to illustrate the steeper slope experienced by White (compared with Black) released prisoners, while also demonstrating that the group with the higher hazard level over time may not be the group with the highest peak hazard point. Incorporating the shape parameter can provide insight into changes occurring, and visual depictions can provide information about the front end of a distribution in addition to these averages over time.

Visual interpretations may also be helpful for understanding macro-level indicators such as jurisdiction. With the exception of Oregon (which reflects the individual-level covariate pattern described above), all of the states in the full level + shape model shown in Table 3 that are significantly different from California in both the level and shape have positive-level coefficients and negative shape coefficients. However, the interpretation is not immediately clear from the model coefficients. To illustrate this jurisdiction-level/shape relationship, I compare three states: Arizona, California, and Florida. I estimate a separate model for each state, and do not include admission or release-type information because these are intertwined with state-level policies.¹² I estimate these models for the full follow-up period but only display the first 6 months postrelease in the Figure 4 graph.

Figure 4.

State-level lognormal model comparisons.

When plotting the graphs in state-specific models, a few observations emerge. The graphs confirm the level comparisons from Table 3, which indicate that Arizona is lower (positive coefficient) and Florida is higher (negative coefficient) than California (the reference category in the full model). As California is known for its high technical violation rate (Petersilia, 2008), other states might have comparatively lower recidivism levels, at least in the immediate postrelease period. As Fischer (2005) points out in his state comparison, compared with Florida, California has higher reconviction (49%vs.45%) and return to prison on technical violation (39%vs.26%) rates but actually has a lower rearrest rate (70%vs.79%). This further emphasizes that careful definitions of “recidivism” matter greatly (Maltz, 1984/2001), particularly as state reputations are often affected by recidivism comparisons (Petersilia, 2008, p. 263).

However, due to the steep nature of Florida’s pattern and the wide confidence intervals that accompany it (see Appendix D), the shape coefficient in the model is not statistically significant. Arizona’s significantly negative shape coefficient, on the contrary, is reflected in the graph with the slow hazard incline over a several-week period. By pairing hazard models with graphical interpretations, researchers can identify where jurisdictions vary by hazard level and shape.

A related theme that emerges from this state comparison involves sample heterogeneity. After controlling for individual-level demographics and criminal history information, the overall rearrest hazard pattern is the solid line shown in Figure 4. By running the models separately by state and graphing the hazard patterns, underlying differences emerge and the states driving the overall pattern become clearer. Overall, nine states (Arizona, Delaware, Illinois, Maryland, Michigan, Minnesota, New Jersey, North Carolina, and Virginia) have delayed peak patterns, two states have immediate decline patterns (Florida and Texas¹³), and four states have somewhat ambiguous patterns due to the confidence intervals (California, New York, Ohio, and Oregon). Although researchers increasingly discuss the importance of acknowledging and examining state differences in recidivism analyses, national statistics on recidivism (e.g., Langan & Levin, 2002) are the most widely reported and often without contextual prefaces.

Discussion

Maltz (1984/2001, 1994) cautions that in an effort to describe findings for a particular sample, researchers (and consequently, policymakers) may unintentionally condense or restrict results in a way that obscures other potentially relevant information. With a few exceptions (e.g., Visher & Linster, 1990; Windzio, 2006; Zorn, 2000), researchers typically make level comparisons in hazard analyses. For example, when Witte and Schmidt (1977) explain that people with certain characteristics are more likely to “go longer” before experiencing an arrest that leads to a conviction, they are describing a positive-level coefficient. By focusing on specific points in time and comparing differences in means, researchers often overlook useful pieces of information about the distribution, or “the very features that make research an exciting art rather than a formulaic ‘science’” (Maltz, 1994, p. 443). Survival analysis researchers caution that while models such as the Cox proportional hazards have several important advantages, there are also trade-offs in using models with less shape flexibility. This article addresses Maltz’s still common concerns by modeling under the assumption that individual hazard rates are not necessarily proportionate to other individuals in the sample and by incorporating shape into analyses that have previously been dominated by level. The methodological contribution of this article is the recognition that by allowing covariates to influence the shape parameter, researchers are able to obtain additional information on recidivism patterns that is not readily available in traditional level analyses.

The first main finding from this study relates to the explanatory power of covariates on the shape and the stability of level results when the shape parameter is included. Race, certain age categories, prior arrest history, prison admission type, and state of prison release have statistically different effects on shape relative to their respective comparison categories. Out of this group, three of these indicators are also among important factors in recidivism hazard studies that analyze the level. The level findings for age and prior criminal history support Witte and Schmidt’s (1977) conviction and DeJong’s (1997) rearrest results, which both found that those with fewer priors and who were older recidivated more slowly than their counterparts. The finding for race in this study also contributes to existing mixed results in prior research.¹⁴ Although incorporating the shape parameter changes the magnitude some of the level covariates have, unless the coefficients are insignificant and close to zero, the direction and overall interpretation of the level results are stable.

The second finding describes the relationship between level and shape. Typically when the level and shape coefficients are both statistically significant, the direction of the two coefficients matches. The combined level/shape model indicates that lower (higher) levels of risk are typically associated with steeper (decreased) slopes. Specifically, Black individuals, younger individuals (those below age 25 compared with those in the 40-44 and 45+ age groups), those with more prior arrests (compared with 0), and those with revocation admission types (compared with court commitments) experience both higher levels in the rearrest hazard and decreased slopes (or less drastic change). Returning to the theoretical and methodological perspectives described earlier, the results suggest that the individuals criminologists typically think of as “high risk” may have more stable hazard levels and patterns over time, and within the “low risk” group, individuals may be sorting (i.e., a portion of this group experiences a rearrest) immediately upon release while the remaining group retains low, stable probabilities of rearrest.

In addition to the individual-level indicators typically found in recidivism studies comparing hazard levels, jurisdiction matters in explaining the hazard shape. The majority of states in this study have some form of the popular delayed peak pattern found in prior research (Caudill, 2010; DeJong, 1997; Kurlychek et al., 2006; Kurlychek & Kempinen, 2006; Lattimore et al., 1995; Visher & Linster, 1990). At the macro level, there could be a variety of explanations for varying shape patterns, including policing and sentencing practices, reentry partnerships and funding, and parole resources and caseloads. Grattet et al. (2011), for example, find that both individual-level (including personal characteristics and criminal record histories) and macro forces (such as the level of intensity and tolerance of supervision practices) influence parole violation rates. States have wide discretion in designing their state prison reentry strategies and huge opportunities to engage individuals in successful reintegration processes. In moving forward with this type of research, distinguishing between recidivism explanations (e.g., changes in criminal justice practitioner policies) and desistance explanations (such as an individual adopting a crime-free identity) will be particularly important.

Along with differentiating between theoretical explanations for the hazard shape in future research, there are heterogeneity considerations. While visualizing the shape the hazard pattern takes is useful in interpreting shape model coefficients, when comparing decreased slopes where the groups both have immediate decline patterns (as in Figure 3), it is possible that there is unobserved heterogeneity (or sorting) or that the declines represent real change. While this study used the full BJS dataset to explore the shape parameter, future studies could further explore contextual issues within certain jurisdictions. California, for example, is a more homogeneous sample than the 15-state sample in a variety of ways; one easily observable feature is that 100% of the individuals released from California state prisons in 1994 were released through mandatory parole. As in the broader 15-state sample, White individuals have a larger sigma value compared with Black individuals in California. However, within this more homogeneous sample, a more clearly defined delayed peak pattern appears for Black individuals (Figure 5), even as this group retains higher overall levels over time. A delayed peak pattern provides a window of time to target resources and supervision to released prisoners, and applying this method to jurisdiction-level policies and interventions could enable researchers to identify some of the timing complexities in recidivism research.

Figure 5.

California hazard patterns by race.

The major policy implication from this study is that groups with higher hazard levels over time may not necessarily be the groups needing attention immediately upon release. As prior work typically focuses on variations in the level and often treats shape as parallel or fixed across groups, there were no concrete a priori expectations for whether (and if so, how) shape plays a meaningful role in recidivism patterns. However, the results from the present study raise questions about RNR research, which has repeatedly concluded that the highest risk¹⁵ group is the group worth targeting (see Lowenkamp & Latessa, 2004, for an overview of RNR meta-analyses). The present study emphasizes the role timing plays in defining the probability of a new criminal justice system event. Prior research points to the first weeks or months home as a critical period of adjustment for individuals returning home from prison (National Research Council, 2007; Nelson, Deess, & Allen, 2011). While those with sustained high hazard levels are certainly worthy of attention and high doses of programming, reentry planners or correctional staff in prerelease programs may also benefit from focusing attention to the groups in their jurisdictions that are changing in the immediate postrelease period, whether they are traditionally “high risk” (i.e., high level) individuals or not. Future research could also explore incorporating hazard patterns and trends into assessments of risk. For example, state reentry planners could coordinate with researchers to build level + shape lognormal hazard models, using available covariates of interest, and identify key factors associated with change. Appendix E contains example Stata code and notes for interested jurisdictions.

This study is intended to generate a more in-depth conversation about shape in recidivism hazards. While studies focusing exclusively on level are useful for understanding differences among groups in a sample more broadly, this information is limited. Including the shape parameter in survival analyses not only adds another layer of complexity to the model interpretation but also provides new information to better understand complicated issues like recidivism. Further research on macro-level indicators driving these patterns, such as variations in labor markets, political ideologies, and correctional spending, would be useful for reentry planners and state and local policymakers. Future research could also benefit from comparing more specific crime types, testing the model presented here on other data, and including dynamic factors in level and shape analyses such as employment, marriage, major life events, drug and alcohol use, and social bonds.¹⁶

Footnotes

Appendix A

Appendix B

Appendix C

Appendix D

Appendix E

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Notes

Author Biography

Megan Denver is a PhD candidate in the School of Criminal Justice at the University at Albany, SUNY. Her work examines perceptions of and policy responses to criminal records, employment and recidivism, and desistance.

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