Journal of Quantitative Criminology

, Volume 30, Issue 4, pp 709–730

Investigating the Functional Form of the Self-control–Delinquency Relationship in a Sample of Serious Young Offenders

Original Paper

DOI: 10.1007/s10940-014-9220-y

Cite this article as:
Sullivan, C.J. & Loughran, T. J Quant Criminol (2014) 30: 709. doi:10.1007/s10940-014-9220-y

Abstract

Objective

This work further examines the functional form of the self-control–delinquency relationship as an extension of recent work by Mears et al. (J Quant Criminol, 2013). Given the importance of the authors’ conclusions regarding the nonlinear relationship between these two variables and the recognition that there are some potential limitations in the sample and assumptions required for the analytic methods used, we apply both similar and alternative techniques with a data set comprised of serious youthful offenders to determine whether key findings can be replicated.

Methods

Data from the Pathways to Desistance study, which comprise extensive individual and social history interviews with 1,354 offenders over multiple waves spread out over 84 months, is utilized in this analysis. These data are well-suited to investigating the questions of interest as the target population comprises youth with offending histories that are more extensive than those likely to be found in general surveys of adolescents. The analyses consider the self-control–delinquency relationship in an alternative sample with the previously used Generalized Propensity Score (GPS) procedure, which requires strong assumptions, as well as nonparametric regression which requires far weaker assumptions to consider the functional form of the self-control–delinquency relationship.

Results

The results generally show that the identified functional form of the self-control–delinquency relationship seems to be at least partly dependent on aspects of the modeling of dose–response associated with GPS procedures. When nonparametric general additive models are used with the same data, the relationship between self-control and delinquency seems to be approximately linear.

Conclusions

Identifying functional form relationships has importance for many criminological theories, but it is a task that requires that the balance of model assumptions to exploratory data analysis falls toward the latter. Nonparametric approaches to such questions may be a necessary first step in learning about the nature of mechanisms presumed to be at work in important explanations for crime and criminality.

Keywords

Functional form General theory of crime Generalized Propensity Score Nonparametric models 

Since its publication in 1990, a tremendous amount of empirical research has been directed at the “General Theory of Crime” proposed by Gottfredson and Hirschi. In considering the underlying mechanisms of the linkage between its two primary constructs of interest, these studies have, largely through omission, assumed and specified a linear relationship between self-control and delinquent/criminal behavior in their models. Taking a cue from other social and behavioral sciences, some criminologists have come to question assumptions of linearity implicit in explanations of criminal behavior like the General Theory of Crime (e.g., Agnew 2005). Recently, Mears et al. (2013) investigated the functional form of the self-control–delinquency relationship and found evidence of nonlinearity using data from the National Longitudinal Study of Adolescent Health (ADD Health). Specifically, using a novel variation of propensity score matching methods known as a Generalized Propensity Score (GPS; Hirano and Imbens 2004), the authors argue for evidence of two separate thresholds—one for high self-control, one for low—between which they observe an expected linear relationship for self-control and delinquency, but beyond either threshold no such association exists.

This finding is important as it has implications for “taking stock” of the vast body of literature that has found support for a linear relationship between self-control and criminal behavior. Furthermore, as noted, the identification of a nonlinear effect can further explicate the mechanism by which the former impacts the latter. Specifically, the identified functional form of an observed relationship carries insight as to the manner in which an individual trait or social circumstance actually impacts behavior. Consequently, knowledge of the functional form helps inform a better understanding of mechanisms at work in an observed association, meaning that these are important questions to consider in evaluating and building criminological theory. Theories are meant to illuminate the mechanisms that underlie important relationships and properties of linearity, or nonlinearity, are clearly relevant to understanding the means by which explanatory variables affect outcomes (Hedström 2005) as they more precisely demarcate when an effect may be more or less salient.

While the Mears et al. study has important implications for enhancing the field’s understanding of the effect of self-control on offending, there is a need to further consider this finding in light of its possible redefinition of a relationship that has garnered considerable attention among criminologists. First, the high school-based youth sample in ADD Health, though broadly generalizable for the most part, represents an uncertain fit to the current task as the important lower ranges of the self-control distribution may not be well populated. This is particularly important as it is in the lower range of the self-control continuum where Mears et al. look for evidence of a “lower threshold” effect. Second, the use of the GPS method seems to be reliant on strong and untestable assumptions that one must make when specifying the model to ‘learn’ the functional form. Under certain specific instances where the researcher is willing to make such strong assumptions and adequately defend and investigate them to demonstrate identification, we believe the GPS method is capable (Flores et al. 2012; Kluve et al. 2012). Yet, given the importance of the current application to decades of criminological theory development and research, the sensitivity of such assumptions in the Mears et al. application warrants further consideration.

The analyses presented in this note address these issues. First, we investigate the self-control–delinquent behavior relationship in a sample of serious juvenile offenders, for whom we might reasonably expect data support over the lower ranges of self-control.1 This allows us to potentially replicate prior analysis using the GPS model to test if the same pattern exists across individuals with more extensive offending histories who presumably more fully populate the lower ranges of the self-control continuum. Second, we use an alternative strategy of nonparametric regression where we are able to rely on fewer a priori assumptions about the relationship. Specifically, we study the relationship using a generalized additive model (GAM; Fox 2000b, 2008; Hastie and Tibshirani 1990), an alternative, nonparametric estimator that relies on fewer assumptions and is therefore perhaps better suited to initially ‘learn’ and describe the underlying functional form of this relationship. On this point, our analysis does not proffer that one method (GAM) is ‘better’ than another (GPS), but rather we wish to consider the nature of the relationship when it is estimated under the much weaker assumptions required by GAM. Together, the use of an alternate data set that includes more cases in the range of interest with respect to the key question, coupled with a model that makes fewer assumptions, helps to counteract “the law of decreasing credibility” that sometimes accompanies results derived from modeling approaches with more stringent assumptions and data demands (Manski 2007: 3).

Studying the Self-control–Delinquency Relationship

The propositions within Gottfredson and Hirschi’s (1990) General Theory of Crime have been the subject of considerable research since the theory was initially introduced to the field (Goode 2008). This work has proceeded along both substantive and methodological tracks, touching multiple dimensions within each area over time. From a substantive standpoint, studies typically have found empirical support for the primary tenet of the theory: a positive, linear association between self-control and offending. In their meta-analysis, which included more than 100 estimates drawn from most major studies of the theory completed in the 1990s, Pratt and Cullen (2000) found a moderate, positive effect of self-control on a wide swath of anti-social behaviors. Specifically, lower self-control predicted more offending—regardless of controls for rival theories, measurement approach, and whether the outcome measure was crime or some analogous behavior. Research of this type informs Gottfredson’s (2006: 83) conclusion that there is “very significant empirical support” for the theory’s main claims, but there are some caveats to that summary conclusion (Goode 2008).

From the standpoint of assessing the theory, Gottfredson and Hirschi (1990) argued for generality and parsimony (see also, Hirschi and Gottfredson 2008), which has affected not only its perception and the empirical puzzles that it spawned (see Kuhn 1970), but also scholars’ practical approach to it from the standpoint of variable selection, measurement, model specification, and data analysis. Piquero (2008), for example, details the early investigations on how to best measure self-control by Grasmick et al. (1993), where they developed a survey-based measure of the key dimensions of self-control outlined in Gottfredson and Hirschi’s work. Hirschi and Gottfredson’s (1993) argued that behavioral measures of the construct were more appropriate, however. This precipitated a good deal of work on how self-control might be best measured and there are several questions still remaining in that regard (Piquero 2008).

A number of other aspects of the theory, including the presence of individual stability in self-control and its origins in socialization practices, have been investigated (Gottfredson 2006). This signals the progression of empirical research on the theory from an initial focus on the association between self-control and antisocial behavior to one that is more invested in explaining the mechanisms underlying offending with an eye toward where self-control fits. For instance, increasingly the means by which self-control might interact with opportunity and other socialization factors to affect delinquent behavior has been a subject of important research. Ousey and Wilcox (2007) considered how an “antisocial propensity” measure consistent with some dimensions of self-control might moderate influences from Social Bonding, Opportunity, and General Strain theories that ebb and flow across developmental periods. Similarly, but working from Self-Control and Routine Activities perspectives, Hay and Forrest (2008) studied the non-additive relationship between self-control and opportunity (measured via “time spent with peers,” and “parental supervision”) and found some interaction effects between the two. They conclude that the latter should receive a more prominent place alongside the propensity aspect of the General Theory in order to explain criminal behavior.

At the same time, it has become clear that theories and research outside of criminology work with constructs similar to Gottfredson and Hirschi’s notion of self-control and that these may lead to alternate conceptions and ways of assessing its effects on behavior (e.g., Kochanska and Knaack 2003; Posner and Rothbart 2000). As the theory has been elaborated, and different mechanisms considered in empirical testing, our understanding of the central relationships has grown, but further questions as to the best means of measuring and analyzing key constructs have arisen. This is certainly true in potentially moving from specification of a parsimonious, linear relationship, which has generally been the case to this point, to a more complex, nonlinear view of how self-control affects criminal behavior.

Nonlinearity in the Self-control–Delinquency Relationship?

Given the evolution of research from (a) determining the best way to measure and test the self-control–delinquency relationship, to (b) establishing empirical support for it, to (c) more recent consideration of its mechanics as it affects criminal behavior, the thrust of Mears, Cochran, and Beaver’s recent paper is an important one. Generally criminologists assume linearity in specifying models based on theory—even when there may be reason to believe that this assumption does not hold. This is important, as in the process of building, testing, and refining theory, criminologists first must explore and learn about relationships of interest rather than embed untested assumptions into the way in which they are studied and move forward with the analysis and consideration of results without attention to that underlying context.

Like many studies of self-control theory, Mears and colleagues rely on a general population sample (National Longitudinal Study of Adolescent Health) to address these questions. In one important respect, the ADD Health data is appropriate to study larger issues of self-control due to its broad generalizability. Unfortunately, this places some restrictions on the variation in the key independent variable and in turn presents an identification problem in fully estimating the functional form of the self-control–delinquency relationship, as the requisite support of the data is restricted over the important lower ranges of self-control.2 According to Manski (2003: 4), “[i]dentification problems are most severe when the researcher knows little about the population under study and the sampling process yields only weak data on the population.” One such identification problem discussed by Manski relevant to the present study is the extrapolation of regression predictions off of the support of the data, which one can only do by making untestable assumptions about the functional form over unobservable (or sparsely-populated) ranges. To be clear, the goal of identification of a lower ‘threshold’ in the self-control–delinquency relationship is not one that is readily achievable with a sample representative of the entire population; rather this task requires individuals who explicitly populate the lower ranges of the self-control continuum, the portion over which such a feature of the functional form would be identifiable.3

The authors do some checking of their results within the ADD Health data set to determine whether the same conclusions might be reached with different configurations of the key measures and subsamples of the data. Overall, the main conveyance of their conclusions comes through plots of the self-control–delinquency functions for the entire continuum (21–81) of low self-control, though. As the authors note, a large portion of the estimated response curves at either end of the distribution is based on limited support of observable data (<5 % of the observations fall below 36 or above 64), meaning that any functional form estimates over these ranges are necessarily based on limited information. Therefore, inferences about the curve over these ranges cannot be drawn from the data without some extrapolation. This is problematic since, consistent with their broader point of exploring this relationship in a different light, the functional form of the relationship should be learned based on the nature of the data as opposed to assumed as it must be when such extrapolation occurs (Manski 2007).

The distribution of self-control in the ADD Health data suggests that there are very few data points at the extreme highs and lows of the distribution; this presents an identification problem in terms of understanding the functional form of the self-control/delinquency relationship because the data do not allow for credible learning about theory at those points without strong and ultimately untestable assumptions (Manski 2007). Manski (p. 28) points out that “failures of a theory on the support are detectable … failures off the support are inherently not detectable.” In the present case, the analyst cannot develop a good sense of whether observed thresholds provide an accurate depiction of the relationship because there is limited data at those upper and lower areas of the distribution. Therefore, the theory and the analytic approach carry more of the inferential burden than do the data. In short, the data and analytic approach are questionable in their fit to learning about and describing the self-control–delinquency relationship.

Mears et al. used a novel, but relatively untested, analytic approach to analyzing the self-control–delinquency relationship: Dose Response GPS modeling (see Bia and Mattei 2008; Hirano and Imbens 2004; Imbens et al. 2001). Whereas traditional propensity score methods consider binary treatment exposure, GPS considers self-control as a continuous treatment variable which youth “select into” based on a host of pre-existing covariates across several relevant theoretical domains (e.g., family, peer relationships). This ‘propensity’ is then included in a conditional expectation function which relates the outcome to the treatment, capturing the effect of self-control on delinquency across a continuum of self-control. The GPS methodology as presented by Hirano and Imbens (2004) requires the researcher to make assumptions about the specification of the conditional expectation function (which describes this ‘effect’ over the continuum). The estimation of that function results from a linear model including the treatment variable, the GPS, and an interaction of the two. There are no theoretical reasons apparent as to why this particular functional form is optimal, nor is it clear why this is the best means of estimating that function (particularly the interaction term). Other researchers have used a more flexible regression specification alongside the GPS approach to identify a dose–response relationship (see e.g., Hirano and Imbens 2004; Flores et al. 2012; Kluve et al. 2012).

Given this foundation, and the recognition that extensive sensitivity analysis around variation in sample and model conditions is an important aspect of the process of propensity score analysis (see, e.g., Guo and Fraser 2010; Rosenbaum 1995), some further analysis of the ADD Health data was undertaken with the same covariates and outcome measures utilized in the Mears et al. (2013) study. As noted above, the bulk of the support for estimating the functional form of the data lies in the center of the distribution where the dose–response function appears to be linear in their Fig. 2. This drove the selection of that range for sensitivity analysis as this is where Mears et al. found the most support for the linear relationship identified in prior research. When the data are limited across a number of conditions related to the critique mentioned above [e.g., between a score of 45 and 55 on the self-control measure where approximately half of the cases lie (53 %)], all of these reveal dose–response curves similar to their Fig. 2. For example, Fig. 1 in this note, which considers only the data for self-control scores between 45 and 55 and a dichotomous nonviolent delinquency measure, illustrates that the functional form appears to be nonlinear—even in a portion of the distribution that had a linear relationship in the Mears et al. analysis.4 This would seem to suggest that the identified functional form is at least partly a product of the assumptions about the conditional expectation function. Additionally, the ‘flattening-out’ near the ends of the distribution, something the authors observe in the original figure when using the entire continuum, is in fact expected as the (sometimes artificial) product of the boundary bias common in local averaging methods (see Fox 2000a). In other words, the deviations from nonlinearity near the boundaries of the continuum—even when arbitrarily defined—can be an artifact of the modeling process.
Fig. 1

Functional form of self-control–nonviolent delinquency relationship in ADD Health (45–55 range)

As noted above, estimation of the dose–response function appears to be carried out through a parametric regression modeling approach with covariates where the conditional expectation function is specified using the self-control score, the GPS, and an interaction term. In other words, this assumes that the conditional response function has a specific parametric form governing it. The use of this particular parameterization is not theoretically transparent nor is it explained in detail. Still, it would seem that the true functional form of the relationship would be robust to slight alterations in this specification. To that end, we estimated the dose response function without the interaction term as it was unclear where that fit with the GPS approach to understanding the relationship of interest, which suggests that it is a relevant parameter for sensitivity testing.5 Figure 2 presents the result of that analysis for one of the two dependent variables utilized by Mears et al. It shows an overall pattern that could be seen as approximately linear. Again, this suggests that the parametric assumptions used in this method may, at least in part, be driving the observed functional form of the self-control–delinquency relationship in this case.6
Fig. 2

Functional form of self–control-nonviolent delinquency relationship in ADD Health (no interaction)

Furthermore, unlike traditional propensity score methods, which allow for more explicit validation of common support between treated and control groups, the GPS methodology relies on a more implicit assumption that common support has been achieved across the entire (continuous) range of the treatment variable. Thus, a strong amount of support data at each level of the predictor is necessary to ‘learn’ the true functional form, or else the relationship will necessarily be based on strong and ultimately untestable functional form assumptions (in particular, where the functional form is off support of the data). This is particularly limiting in the case of the general population ADD Health dataset, as the lower ranges of self-control, an essential part of the distribution over which criminologists would most like to understand the functional form of this relationship, are under-populated. This implies the need for a sample more representative of the offender population, i.e., an offender sample.

In spite of these limitations, Mears and colleagues’ work opens an important line of investigation in its questioning of the way in which criminologists think about the mechanisms underlying offending behavior (Wikström 2008)—using one of its most prominent theories as a focal point. However, the newness of the analytic methods, coupled with the nature of the data involved and the amount of empirical research called into question by the key findings, do suggest the need for further investigation. Specifically, the question of fit between data, analysis, and research question points to the need for possible replication across studies. In this particular case, given that self-control has been operationalized in a variety of ways and tested in a range of samples in past research (Pratt and Cullen 2000), it is even more important to thoroughly consider the relationship under multiple conditions.

The use of multiple, somewhat similar data sources, to attempt replication can be fruitful in illuminating the degree to which important conclusions are robust and might hold (or not) across sample, measures, or method (see, e.g., Osgood and Schreck 2007)—particularly when some extrapolation is necessary in the initial finding. Specifically, this helps in determining whether samples, measures, and methods are irrelevant to the finding of interest such that it can be generalized more broadly (Shadish et al. 2002). This is particularly important when key findings are derived mainly from a subset of the population for which a more broadly representative sample such as the ADD Health might be limited. From the standpoint of future empirical research on the topic, further inquiry through the use of an alternate data source and methods that use fewer assumptions provides a sense of how the field should adjust to a novel conclusion like that identified by Mears et al. (2013).

Current Study

This paper further considers the relationship between self-control and delinquency with data from the Pathways to Desistance study (Pathways), a longitudinal investigation of serious adolescent offenders transitioning from adolescence to young adulthood. Participants in the Pathways study are adolescents who were found guilty of a serious offense (almost entirely felony offenses) in the juvenile or adult court systems in Maricopa County, AZ or Philadelphia County, PA. These youth were ages 14–17 at the time of the study index offense. A total of 1,354 adolescents are enrolled in the study, representing approximately one in three adolescents adjudicated on the enumerated charges in each locale during the recruitment period (November, 2000–January, 2003). General descriptive statistics on the analytic sample and the key study measures are presented in Table 1 (bivariate correlations are presented in the “Appendix”). The study sample is predominately minority (41.3 % African American, 33.5 % Hispanic) and male (86.4 %). These data, which comprise extensive individual and social history interviews over ten waves spread out over 84 months, are well-suited to this task as the target population comprises youth with offending records that are more extensive than those likely to be found in general surveys of adolescents such as ADD Health.7 Likewise, these cases fall at a different point on the self-control scale (see Fig. 3), which allows for consideration of the extent to which previous findings hold across a broader range of its distribution. Specifically, the entire range of the self-control continuum is populated. The extensive individual and social histories of the Pathways study participants also allowed for matching procedures on many relevant variables that might confound the self-control–offending relationship.
Table 1

Descriptive statistics for key study measures: pathways data (n = 1,261)

 

Mean

SD

Range

Treatment and response measures

Self-control (Weinberger adjustment inventory)

2.87

0.85

1–5

Violent delinquency (self-reported)

0.56

0.50

0–1

Property delinquency (self-reported)

0.36

0.48

0–1

Covariates

Age

16.04

1.14

14–19

Gender (1 = male)

0.86

0–1

Race

 White (ref)

0.20

0–1

 Black

0.41

0–1

 Hispanic

0.34

0–1

 Other race/ethnicity

0.05

0–1

Parental warmth

3.06

0.69

1–4

Parental hostility

1.59

0.42

1–4

Parental knowledge

2.70

0.81

1–4

Parental monitoring

2.80

0.86

1–4

Parent SES

51.41

12.3

11–77

Social support

6.64

1.75

0–8

School performance

4.80

1.90

1–8

Religiosity

3.28

1.21

1–5

Fig. 3

Distribution of WAI self-control measure in Pathways to Desistance data

We utilized measures as similar as possible to those in the Mears et al. study. For self-control, we use Baseline measures from the Weinberger Adjustment Inventory (WAI; Weinberger and Schwartz 1990), an assessment of an individual’s social-emotional adjustment in terms of self regulation. Specifically, we use the temperance subscale, which comprises 15 items from the dimension of impulse control (e.g., “I say the first thing that comes into my mind without thinking enough about it”) and suppression of aggression (e.g. “People who get me angry better watch out”). The measure used here was assessed by the original Pathways researchers with a second-order confirmatory factor analysis (CFA) model. Temperance was the second-order factor and impulse control and suppression of aggression were the first-order latent variables. The model showed acceptable fit (Normed Fit Index = 0.91; Comparative Fit Index = 0.93; Root Mean Square Error of Approximation = 0.06) on most conventional criteria (Mulvey et al. 2010). Higher scores on each of the subscales indicate more prosocial self-regulatory behavior (scored 1-False to 5-True) (α = 0.87). WAI scales have previously been studied with regard to their association with adolescent behavioral problems (Feldman and Weinberger 1994).

The delinquency measures, which were measured in the Wave 1 interview reflecting self-reported offending (SRO) in the previous 6 months, come from two sub-categories: Aggressive (e.g., “Been in a fight?”) and Income offending (e.g., “Used checks or credit cards illegally?”). “Aggressive,” or violent, offending comprised 11 items and “Income,” or nonviolent, offending consisted of a total of 10 items. The initial Cronbach alpha values were 0.74 for the Aggressive offenses and 0.80 for Income offenses. Knight et al. (2004) reported CFA results for the SRO measures in the Pathways data, finding Comparative Fit Indices between 0.87 and 0.90 in various tests and RMSEA values of around 0.04 in the same analyses. Their item-level analyses of factor loadings and criterion-related validity checks also suggest that the Pathways SRO items formed a reasonably-good measure of delinquent behavior. Given their distributions, these measures were dichotomized for the purposes of this analysis.8 As shown in Table 1, 31 % of the sample engaged in a nonviolent act at the first wave, while 52.2 % engage in a violent act at the first wave.

The covariates utilized in developing the GPS were chosen from the Pathways data to be as consistent as possible with the Mears et al. work, providing a “virtual” replication (Finifter 1975).9 Specifically, Baseline measures for age, gender, race (dummy variables for Black, Hispanic, Other with White as reference), the degree to which youth were exposed to caring adults (social support), parent index of social position to capture SES, parental hostility, parental warmth, parental monitoring and knowledge of whereabouts, academic performance in school, and self-reported importance of spirituality and religion were included in the model.10

Nonparametric Regression

The GPS approach, which was originally demonstrated by Hirano and Imbens (2004), is useful in a variety of circumstances. Still, it is a complex method and there are assumptions inherent in the specification of the dose response portion of the model. Specifically, when using the GPS approach, delinquency is expected to be a linear function of the covariates. This is, of course, a common assumption of parametric regression (Fox 2008). In that sense, at least at the early stages of exploring this particular functional form question, it may be preferable to use a nonparametric method to better ‘learn’ the true functional form without concern for a causal interpretation or parametric assumptions, the objective of other methods (such as GPS) for studying dose–response relationships.11 Fox (2000a: 78) indicates that “the potential gain [in using nonparametric methods] is fidelity to the data” and the desire to learn about functional form is one scenario where letting the data show through unfiltered is an essential goal of the analytic process. Again, this not to say that GPS is incorrect or unequivocally adjudicate between methods, but rather to suggest the importance of uncovering basic descriptive information about this relationship with a method that requires weaker assumptions.

The primary goal of any regression model is to estimate the population regression function, or conditional expectation function, E[yi | xi]. Whereas traditional regression makes strong assumptions (e.g., linearity between the regressors and parameters), nonparametric regression relaxes this and is assumption-free in terms of expected functional form. This is a useful first step to take given the uncertainty about the nature of the self-control–delinquency relationship in the population. Instead of relying on the parametric assumption that delinquency is a linear function of model covariates (as is the case in GPS), whether standard covariates or those derived from GPS analysis, the dependent variable is specified as a smooth function of the covariates (Cleveland 1979; Fox 2000b, 2008; Hastie and Tibshirani 1990), a potentially more appropriate way to learn about the relationship. If the main objective is to learn about the data in general, and the functional form in particular, this approach helps that endeavor without as many strings attached as other methods (including GPS).12 Smoothing with a nonparametric approach allows for an exploration of the data without assumptions to learn about properties like functional form (Manski 2007). In other words, instead of assuming the appropriate functional form, we would like the data to yield it for us.

As a case in point, Hirano and Imbens (2004) originally specified their model with higher order polynomials to ensure as much flexibility as possible in using the GPS to understand the relationship of interest in their study. Additionally, Kluve et al. (2012) used a nonparametric modeling approach as part of their investigation of the dose–response relationship for training and employment outcomes in the context of GPS (see also Flores et al. 2012; Imai and Van Dyke 2004) and Bia et al. (2013) have developed semi-parametric approaches to estimating the dose–response function that builds on their previous work with GPS. Here we use a specific type of nonparametric regression model, the GAM (Hastie and Tibshirani 1990), which is a simple smoothing function that fits a locally-weighted mean of some outcome y across the entire range of the predictor x.13 Unlike estimators that rely on binning and hence are prone to bias near the boundaries, smoothing splines (and, by extension, GAM) are not as susceptible to this problem.

Study Results

GPS Results Using the Offending Sample

We first went through the GPS modeling process with the Pathways data, using the covariates, treatment, and outcome variable described above. The results of the covariate balancing are shown in Table 2.14 These differences are formally assessed via t tests. The results in Table 2 indicate that there are some residual imbalances after use of the GPS procedure, but only 4 of 39 possible comparisons reveal statistically significant differences. Furthermore, in most cases where the pre-GPS differences were large and/or statistically significant, those relationships tend to be rendered non-significant or are at least strongly attenuated following the GPS procedure.
Table 2

Test for covariate balance across intervals after Generalized Propensity Score estimation

Covariates

Pre-GPS

Post-GPS

Diff

SE

t-valuea

Diff

SD

t-value

Self-control interval 1 (1.02.4)

Age

−0.036

0.066

−0.54

−0.048

0.064

−0.741

Gender (1 = male)

−0.008

0.020

−0.39

−0.005

0.021

−0.271

Black

0.127

0.028

4.46*

0.020

0.027

0.725

Hispanic

−0.067

0.028

−2.47*

−0.021

0.026

−0.787

Other race/ethnicity

−0.022

0.014

−1.81

−0.005

0.011

−0.480

Parental warmth

0.200

0.040

5.01*

0.018

0.036

0.500

Parental hostility

−0.204

0.024

−8.57*

−0.020

0.015

−1.36

Parental knowledge

0.239

0.048

4.98*

0.006

0.041

0.159

Parental monitoring

0.188

0.053

3.56*

−0.034

0.044

−0.772

Parent SES

−0.071

0.715

−0.10

0.511

0.700

0.733

Social support

0.336

0.101

3.33*

0.170

0.094

1.82

School performance

−0.376

0.110

−3.43*

−0.139

0.106

−1.31

Religiosity

0.253

0.070

3.60*

0.009

0.066

0.136

Self-control interval 2 (2.53.2)

Age

0.053

0.066

0.80

0.045

0.060

0.752

Gender (1 = male)

0.003

0.020

0.15

0.009

0.018

0.476

Black

−0.015

0.028

−0.54

−0.018

0.026

−0.671

Hispanic

0.030

0.027

1.12

0.027

0.025

1.09

Other race/ethnicity

−0.004

0.012

−0.29

−0.004

0.011

−0.367

Parental warmth

0.097

0.040

2.44*

0.010

0.036

2.75*

Parental hostility

−0.024

0.024

−0.99

−0.028

0.021

−1.30

Parental knowledge

0.079

0.048

1.65

0.080

0.042

1.89

Parental monitoring

0.123

0.053

2.41*

0.113

0.046

2.47*

Parent SES

−0.103

0.709

−0.15

−0.476

0.648

−0.736

Social support

−0.100

0.101

−1.00

−0.099

0.089

−1.11

School performance

0.075

0.109

0.69

0.032

0.101

0.314

Religiosity

0.071

0.070

1.02

0.022

0.065

0.346

Self-control interval 3 (3.35.0)

Age

−0.011

0.066

−0.17

−0.003

0.066

−0.040

Gender (1 = male)

0.006

0.020

0.29

−0.018

0.020

−0.920

Black

−0.107

0.029

−3.77*

0.010

0.027

0.392

Hispanic

0.034

0.027

1.25

−0.006

0.028

−0.205

Other race/ethnicity

0.025

0.011

2.06*

0.007

0.013

0.539

Parental warmth

−0.300

0.039

−7.55*

−0.104

0.037

−2.80*

Parental hostility

0.227

0.024

9.65*

0.048

0.021

2.32*

Parental knowledge

−0.318

0.048

−6.68*

−0.056

0.042

−1.33

Parental monitoring

−0.309

0.052

−5.94*

−0.049

0.047

−1.04

Parent SES

0.154

0.713

0.22

0.183

0.728

0.252

Social support

−0.231

0.101

−2.28*

0.023

0.099

0.231

School performance

0.300

0.110

2.72*

0.123

0.112

1.10

Religiosity

−0.323

0.070

−4.64*

0.028

0.068

0.409

p < .05

az tests used for baseline test of proportions for gender, race/ethnicity variables

Table 3 shows that the regression equation for the functional form of the self-control–delinquency relationship was specified with polynomial terms for self-control as those were the best fitting configuration of the model (see Hirano and Imbens 2004; Kluve et al. 2012). As in the analysis presented in Fig. 2, we again considered these relationships with and without the interaction term for self-control and GPS in order to observe any possible differences associated with an alternate parameterization of the dose–response model. For the current study, the most important aspects of the GPS process are presented in Fig. 4a–d. The top panels of Fig. 4 show the functional form of the relationship between self-control and violent delinquency with and without the interaction term. Figure 4a, which includes the interaction term, shows multiple inflections in the relationship between self-control and the probability of violent delinquency. In particular, as observed in the Mears et al. analysis, there are up and down ticks at either end of the distribution that may reflect the boundary estimation issue mentioned above. Otherwise, the middle of the distribution seems to show a modest linear relationship between the two variables. It is unclear what this distribution would imply substantively about the self-control–delinquency relationship. Figure 4b in the top-right panel presents an analysis without the interaction or polynomial terms and it is clear that the relationship looks to approximate linearity much more closely than in Fig. 4a.
Table 3

Parametric and nonparametric estimates for self-control–delinquency relationship (logistic function)

Parametric, GPS

Violent delinquency

Nonviolent delinquency

Est. (SE)

Est. (SE)

Est. (SE)

Est. (SE)

Constant

4.03 (2.06)

1.26 (0.27)

4.83 (2.13)

0.79 (0.27)

Self-control

−4.99 (2.64)

−0.47* (0.07)

−7.94* (2.81)

−0.61* (0.08)

Generalized Propensity Score

7.27* (2.43)

0.907* (0.43)

12.90* (2.55)

0.95* (0.47)

Self-control, GPS interaction

−2.25* (0.82)

−4.46* (0.91)

Self-control2

1.85 (0.98)

3.19* (1.08)

Self-control3

−0.21 (0.11)

−0.38* (0.12)

Nonparametric, GAM

Est. f (df)

Est. f (df)

Constant

0.22 (0.06)

−0.65 (0.07)

Self-control

1.001* (1.001)

4.01* (5.15)

Generalized Propensity Score

1.12 (1.23)

1.00 (1.00)

Hypothesis tests for individual estimates in nonparametric GAM are based on tests with Chi Square distributions

Fig. 4

ad Functional form of self-control and violent/nonviolent delinquency relationships (interaction, no interaction term), GPS

Figure 4c, d show the results for the estimation of functional form with nonviolent delinquency. As was the case with violence, the model with the interaction and polynomial terms shows multiple inflections, suggesting nonlinearity—but not necessarily in line with any clear explanatory perspective. Figure 4d again shows that this same relationship does not hold when the model specification is changed slightly (i.e., the interaction term is removed). Even more than for violent delinquency, the association between self-control and nonviolence is approximately linear.

GAM Estimation of the Self-control–Offending Relationship

Figure 5 shows the relationship between self-control and violent offending using GAM estimation.15 The graph in Fig. 5 essentially shows a linear relationship between the two measures. Again, note that we make no a priori assumptions about the shape of this functional form, instead the data alone yield this linear relationship. Importantly, this function is learned over the lowest ranges of the data, or in other words, there is support of the data over the entire continuum of the self-control measure. This can be seen by examining the perpendicular lines in the “rug plot” at the base of the X-axis of the figure, which indicates the data coverage across the range of the distribution. Figure 6 shows the GAM estimates for the relationship between self-control and nonviolent offending. Again, this observed relationship looks to be largely linear, even over the lower ranges of the continuum.
Fig. 5

GAM estimation of self-control–violent delinquency relationship

Fig. 6

GAM estimation of self-control–nonviolent delinquency relationship

Table 3 shows the basic nonparametric regression estimates for models involving violent and nonviolent delinquency. This provides the foundation for the GAM estimation of the response functions shown in Figs. 7 and 8. These GAM models include the self-control and GPS measures.16 Figure 7 shows that the relationship between self-control and violent delinquency looks approximately linear with the inclusion of the GPS score. Figure 8 shows comparable results for the nonviolent delinquency measure. Comparison of these figures to those in GPS also clearly shows that the degree of nonlinearity observed in Fig. 4a–d is not present when estimating functional form for self-control–delinquency with a more flexible nonparametric regression model.
Fig. 7

Functional form of self-control and violent delinquency relationship with GPS variable, GAM

Fig. 8

Functional form of self-control and nonviolent delinquency relationship with GPS variable, GAM

Conclusion

This analysis takes the premise and findings of the recent study by Mears et al. (2013) as a starting point to look further at functional form in the self-control–delinquency relationship. Their work emphasized the importance of considering potential nonlinearities in expected empirical relationships as a means of more thoroughly investigating theoretical propositions, which is essential for understanding causal mechanisms inherent in key explanations of crime and delinquency. Still, there are important assumptions at work in the GPS modeling approach used in their study that, when further elaborated, indicate some limitations in considering the functional form question—particularly if any causal inferences are being drawn. A brief reanalysis of the ADD Health sample and measures suggested that (a) the nonlinearity observed in GPS seemed to hold even in parts of the distribution where the functional form of the self-control–delinquency relationship was initially linear and (b) the relationship seemed to be linear when an interaction term for the propensity score and treatment variable was removed. Together, these analyses, coupled with the general importance of establishing external validity, suggested that the technique and its assumptions as applied to ADD Health might influence the observed results in a way that makes it important to consider the question with other techniques and data sets. In this sense, our goal here is the continued investigation of the functional form question from alternative methodological angles as opposed to a referendum on any particular methodology or data sets.

Using data from the Pathways to Desistance study, we found that similar results seemed to emerge when moving from the default GPS dose response estimation to a model that makes an alteration to the parameterization by removing the interaction term. Consequently, it is unclear whether the GPS approach is sufficiently devoid of assumptions to make it a neutral medium through which functional form can be learned. Existing accounts in the literature using GPS suggest that although the dose–response function can be estimated in a flexible fashion, there are still some remaining distributional assumptions inherent in the parametric methods used for estimating the relationship (see, more generally, Cook and Weisberg 1999). Furthermore, the ‘causal’ dimension of the GPS approach may serve to confound the goal of learning the functional form as it immediately moves beyond the exploratory data analysis stage to something more formal.

Though choice of methods tends to be more a matter of achieving intersubjective agreement than reaching a clear, unequivocal solution, it is nevertheless important that research findings be considered and reconsidered in light of their potential sensitivities to the assumptions inherent in the broad (i.e., models and estimators used) and narrower decisions (e.g., variable inclusion/exclusions) made during the process. They also should be guided by the underlying substantive theory. Granted, no method can be absolutely credible in terms of being applicable to a certain problem each time—rather the choice of methodology depends heavily on the assumptions that one is willing to make.17 In this case, if the intent is preliminary learning about a relationship that has widely been considered to be linear, we believe that Manski’s (2007) advice on proceeding with methods that first allow us to learn about our data in the context of relatively weak assumptions is most trustworthy. After having learned from the data initially, we should then proceed to specify models with greater complexity (and attendant assumptions), but consider their credibility within the data both prior to and during the analytic process. In this way, the original descriptive analysis might inform later research and identify data needs for a stronger, more credible test of a theoretically-informed relationship (Berk 2004). For example, it is important to know whether there is supporting data to test specific expectations about a relationship at the ends of a distribution, which is inherently the case when considering nonlinear thresholds. In scenarios where such an approach is not undertaken, we risk prematurely asking and answering questions in ways that may have profound implications for later research and theory (Lieberson 1985).

Given these concerns, we presented an alternative way of looking at the current problem through the application of nonparametric regression procedures with the Pathways to Desistance data which have a different distribution of the key explanatory measure and one more representative of areas of the population over which the original Mears et al. analysis found null associations. Those results showed that the same degree of nonlinearity was not present when the data were analyzed with GAMs, which make fewer assumptions than a standard parametric regression analysis. Although the curve for nonviolent delinquency showed a slight inflection in one case, the trends that emerged in these analyses tended to show that the relationship between self-control and delinquency lacks the nonlinear features observed in the GPS estimation of dose–response.

The analyses presented in this note suggest that the nonlinearities observed earlier are likely to be at least partly a byproduct of the methods used as the key findings tended to not fully generalize to a different data set with more serious offenders and different measures of self-control and offending. Furthermore the use of nonparametric methods that take on fewer assumptions revealed evidence of an approximately linear relationship. Mears, Cochran and Beaver have clearly identified an important dimension in the study of self-control and offending, and the questioning of inherent functional form relationships has great importance for many criminological theories. We would also advocate for continued study in the area of theory refinement through a better understanding of the underlying functional forms between key criminological predictors and outcomes. Still, we would urge caution in the use of complicated and assumption-heavy methods to derive general results—particularly at an early stage of assessing a particular substantive question. When focused on research questions related to the nature of effects—especially in initial forays into a given problem—it is important to make maximal use of approaches that do not “tell” the data much about the relationship of interest through assumptions but instead “ask” that they show it to us with few strings attached.

Footnotes
1

To be clear, it is likely that the ADD Health data have generalizability over the larger range of the self-control continuum, yet to more precisely identify a lower threshold, one would be interested in a very restricted range of this continuum, which would presumably be well-populated in an offender sample.

 
2

An identification problem occurs when reasonable inferences cannot be drawn even when the researcher has access to very large (or infinite) sample sizes. This is in contrast to questions of statistical inference, which aim to draw weaker conclusions from finite samples.

 
3

Some prior research has identified important distinctions in the self-control-delinquency relationship in offender samples as opposed to youth from general population samples suggesting a need to consider generalizability in the measurement and observed effects across subpopulations in a substantive sense (see, e.g., DeLisi et al. 2003).

 
4

This is an extension of the analysis Mears and colleagues' describe where they removed 5 % of cases at either end of the distribution, which identified a similar trend to their Fig. 2. While it is somewhat unexpected, it is intuitive that a near-identical trend would still be seen when a relatively small proportion of cases is removed at either end of the distribution. However, reaching that same conclusion when restricting the analysis to the portion of the data that displayed a linear trend previously is counterintuitive as the process should not produce a similar trend if we are capturing an unfiltered picture of the relationship between self-control and delinquency without boundary bias.

 
5

This sensitivity analysis approach in some sense parallels the approach taken by Kluve et al. (2012) in the “Robustness” section of their paper as far as considering the impact of removing certain covariates representing second-order moments from the dose–response model.

 
6

Ostensibly, the use of more terms in the parametric specification would seem to imply greater flexibility and fewer assumptions (since the high degree of parameterization would presumably allow flexible results). Yet we seem to be observing the opposite, namely that the results are sensitive to the parameterization assumptions.

 
7

Information regarding the rationale and overall design of the study can be found in Mulvey et al. (2004), while details regarding recruitment, a description of the full sample, and the study methodology are discussed in Schubert et al. (2004). All data used in this analysis are publicly available through Mulvey (2012) and our syntax is available upon request.

 
8

Although there are limitations in dichotomizing key variables (MacCallum et al. 2002), the Mears et al. work was based on limited dependent variables (a truncated count and a dichotomy) so that approach was utilized here to ensure as much comparability as possible.

 
9

Given the question of temporal order and the fact that Gottfredson and Hirschi (1990) view substance use as an “analogous behavior” and thus an outcome measure in their theory, we did not include alcohol use as in the Mears et al. study (see Caliendo and Kopeinig 2008; Rosenbaum 1984).

 
10

For more detailed information on measures and constructs, interested individuals are encouraged to visit the study website (http://www.pathwaysstudy.pitt.edu).

 
11

This parallels Berk’s (2004) suggested best practices for using multiple regression as a tool for description and learning about relationships as opposed to a method for causal inference.

 
12

For further information on nonparametric regression, see Fox (2000a, 2000b).

 
13

GAM generalizes the linear model by modeling E[Y|X] = s0 + s1(X), where s1(·) is any smooth function, here estimated using smoothing splines.

 
14

Balance is evaluated at three cut points in the GPS process such that the propensity score, comprising the potential treatment values, is segmented into intervals and then the median of the GPS for each of those groups is evaluated relative to the others in terms of the differences in covariate values for those that fall into the focal treatment interval and those that are assigned to that interval by virtue of their propensity score but actually belong to a different treatment interval (Bia and Mattei 2008; Hirano and Imbens 2004). Other authors stress the importance of assessing whether common support conditions are met in the context of GPS (Flores et al. 2012; Kluve et al. 2012) and recent work has attempted to integrate those tests into the Stata program as well (Bia et al. 2013).

 
15

Two technical notes on estimation bear mentioning. First, since the outcome is dichotomized, we use a standard logit link function in the estimator. Second, a typical concern with many nonparametric estimators is over-fitting or under-fitting the model, either of which may obscure results. We address this issue by using cross-validation to ensure optimal fit in the R package, Mixed GAM Computation Vehicle with GCV/AIC/REML smoothness estimation and GAMs by REML/PQL (MGCV), developed by Wood (2006).

 
16

In an effort to estimate GAM models that were similar to the parametric specification utilized in GPS, we did also analyze variants of the nonviolent and violent delinquency models that included an interaction term. That variable was treated parametrically in GAM due to the fact that it is a nonadditive term (see Fox 2008). The patterns in those plots were similar to those shown in Figs. 5, 6 and 7, except that there is a slight inflection for the upper range of self-control for nonviolent offending. That pattern becomes nearly linear when the interaction term is removed from the model (as shown in Fig. 7).

 
17

When it is appropriate to the question, researchers applying a GPS-like method might look at the works by Flores et al. (2012) and Kluve et al. (2012) for advice on checking key assumptions and examining sensitivity.

 

Acknowledgments

This study employs data from Mulvey, Edward P. Research on Pathways to Desistance (Maricopa County, AZ and Philadelphia County, PA): Subject Measures, 2000–2010. ICPSR29961-v2. Ann Arbor, MI: Inter-university Consortium for Political and Social Research [distributor], 2013-01-07. doi:10.3886/ICPSR29961.v2. We wish to thank Carol Schubert for her assistance in compiling and transferring the data and members of the Pathways to Desistance research group for helpful comments on an early concept paper and Ray Paternoster and the anonymous reviewers for useful feedback on later versions of this work. All interpretations and conclusions reached in this work are those of the authors. Additionally, a portion of this article uses data from ADD Health, a program project designed by J. Richard Udry, Peter S. Bearman, and Kathleen Mullan Harris, and funded by a Grant P01-HD31921 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Special acknowledgment is due to Ronald R. Rindfuss and Barbara Entwisle for assistance in the original design. Persons interested in obtaining data files from ADD Health should contact ADD Health, Carolina Population Center, 123 W. Franklin Street, Chapel Hill, NC 27516-2524 (addhealth@unc.edu). No direct support was received from Grant P01-HD31921 for this analysis.

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.University of CincinnatiCincinnatiUSA
  2. 2.University of MarylandCollege ParkUSA

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