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Disentangling Self-Control from Its Elements: A Bifactor Analysis

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Abstract

Objectives

Disentangle self-control from its elements and provide several new insights into the self-control dimensionality debate including: the proportion explained variance in scale items attributed to self-control and its elements, the viability of using total and individual scores to measure self-control and its elements in observed variable analyses, and the unique effects of general (self-control) and specific (elements) latent factors on crime and victimization.

Methods

The current study utilizes bifactor measurement and structural equation models to address the research objectives. The sample consists of Florida jail inmates and self-control and its elements are measured with the Grasmick et al. scale.

Results

Results indicate the elements exist above and beyond the general factor of self-control, and that these specific factors collectively account for nearly one-third of the total proportion explained variance in the scale items. Findings from omega reliability analyses provide evidence supporting the use of a total score to measure self-control, but discouraging the use of subscales to measure the individual elements, when measurement error is not taken into account. Results from a bifactor structural equation model predicting crime and victimization reveal that the effects of three latent specific factors (temper, risk-seeking, and self-centeredness) are substantially larger than the effects of the general factor (self-control).

Conclusions

Bifactor methods placed self-control and the elements on equal conceptual footing and found both to explain variation in Grasmick et al. item responses and both to influence crime and victimization. Future work should examine the origins and stability of self-control vis-à-vis the individual elements.

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Notes

  1. For example, more than 2,500 studies in the past 5 years (2009–2013) have cited Gottfredson and Hirschi’s (1990) original self-control theory and 470 have cited Grasmick et al.’s (1993) work. This is compared to 156 that have cited Hirschi’s (2004) redefinition. Citation counts generated using Google Scholar, current as of 5/7/2014. It should be noted that Google Scholar results are not necessarily reflective of only peer-reviewed journal articles; other sources indexed by Google Scholar include doctoral dissertations, web-accessible technical reports, and conference proceedings. Importantly, we also explored the ratio of citation counts for ‘G&H:Hirschi’ and ‘Grasmick:Hirschi’ for each individual year over this time period to assess whether there were patterns indicating a shift in influence. From 2009 to 2013, citation ratios for G&H:Hirschi were approximately 15.6, 18.7, 16.3, 16.0, and 15.4 and citation ratios for Grasmick:Hirschi were 2.6, 3.8, 2.7, 3.1, and 3.1, respectively. While an imperfect assessment, these findings are nevertheless suggestive that Gottfredson and Hirschi’s original statements continue to be steadily influential to criminologists.

  2. Self-control and the individual elements are appropriately measured using a “reflective” measurement model, which is the most frequently used approach and the one most familiar to researchers. As Coltman et al. (2008) explain, in a reflective model the latent construct is assumed to exist, causality runs from the construct to the items, items are manifested by the construct, items have high positive inter-correlations, items have similar sign and relationships with antecedents/consequences as the construct and, importantly, error terms can be identified.

  3. Recent simulation research encourages cautious interpretations of bifactor versus higher order models and suggests making decisions not solely based on Chi square difference testing. Other considerations for choosing between these alternative models include “substantive” changes in fit (e.g., ΔCFI ≥ 0.01) and, importantly, theory and the goals of the research (see Murray and Johnson 2013).

  4. When using general and specific factors to predict outcomes in a bifactor structural equation model, Chen et al. (2006) note conditions where at least one specific factor should be excluded to prevent exact linear dependency among the predictors. This need not be an issue when specific and general factors are uncorrelated. If the choice is made to remove a specific factor, a logical exclusion would be one that does not explain any, or is only minimally influential in explaining, common variance in scale items after partialling out the general factor.

  5. An equivalent model can be achieved using a higher-order factor structure, but this requires that one models the disturbances of first-order factors as latent constructs which is considerably more cumbersome and computationally demanding and may result in estimation difficulties (see Gustafsson and Balke 1993).

  6. We also estimated models employing robust weighted least squares (WLSMV) with listwise deletion and the results were substantively similar. We report using pairwise present, which is the default in Mplus for handling missing data when there are categorical outcomes.

  7. Because Chi square statistics are sample size dependent, it is traditional to rely on fit indices when evaluating the fit of measurement models. TLI/CFI values that are ≥0.90 indicate adequate model fit and values that are ≥0.95 indicate good fit. RMSEA values that are ≤0.05 indicate a good model fit.

  8. The correlated traits model is nested within the bifactor model (see Reise 2012). However, Chi square difference testing in Mplus requires not only that the more restrictive model (i.e., more degrees of freedom and fewer estimated parameters) is nested within the less restricted model, but also that the more restricted model has a worse fitting function (i.e., higher value). The more restricted model actually had a slightly better fitting function. We checked for the possibility that the results from either model were due to a local maximum solution by increasing random starts and this was found not to be the case. Therefore, we compare the fitting functions directly and also report supplemental fit analyses.

  9. We report the results obtained from setting the factor variances to 1 (i.e., the standardized solution) to simplify the mathematics as aforementioned and to obtain an estimate of the proportion explained variance by the general and specific factor for all items (which is necessary information for estimating PEV due to general and specific factors in scales and subscales). The unstandardized solution (estimated but not shown) is obtained by settling an arbitrary factor loading to 1 for the general and specific factors. It is prudent to look at the unstandardized solution to assist in determining whether a factor variance differs from zero. Results indicated that risk-seeking, temper, physical activity, self-centeredness, simple tasks, and self-control were all statistically significant (p < 0.001). Impulsivity was statistically significant but at the less stringent alpha level of significance of 0.05 (p = 0.03) and was the smallest of the factor variances.

  10. For comparison purposes, we also estimated a full bifactor structural equation model that included impulsivity (results not shown). The coefficients for general and specific factors were trivially influenced and all substantive conclusions about the general and five other specific factors remained identical. It is worth noting that the coefficient for impulsivity in this model was positive which, given the coding scheme, indicated that as impulsivity increases, delinquency and victimization decreases. At first glance, this finding might seem quite counterintuitive and inconsistent with the literature. But, assuming for the moment that loadings on the impulsivity factor were sizeable (in reality they were not) and stock should be placed in them, this finding might be understood by returning to an old discussion about two types of impulsivity. Dickman (1990) draws an important distinction between functional (e.g., being admired for thinking quickly such as reacting in an emergency) and dysfunctional impulsivity (e.g., getting into trouble due to inadequate forethought) and suggests that these traits can have differing effects on outcomes. Low self-control is very much aligned with the idea of dysfunctional impulsivity. On the other hand, functional impulsivity involves reacting with little forethought when it is optimal to do so and such a tendency might lead to many prosocial opportunities. While future research should explore this issue in further detail, we thought it most appropriate not to highlight these findings given the weak and inconsistent factor loadings on the impulsivity factor and the fact that almost 90 % of the common explained variance in these scale items was due to self-control.

  11. To supplement these comparisons, we also estimated a model where age, sex, and race were not controlled; we did this in order to easily calculate the proportion each factor contributes to the total explained variance in outcome variables when all predictors are orthogonal and there are no covariates (i.e., for a given factor, this value can be obtained by taking the squared standardized effect and dividing it by the sum of all squared standardized effects). Before reporting these supplemental analyses it should be noted that the pattern of coefficients for the self-control and elements on the outcome variables were substantively similar as compared to those reported in Table 4, though there were some relatively minor fluctuations (e.g., −23 vs. −20 for self-control’s effect on delinquency). Self-control, risk-seeking, physical activities, temper, self-centeredness, and simple tasks were responsible for approximately 8, 26, 7, 33, 24, and 2 % of the explained variance in delinquency and 5, 38, 12, 28, 14, 4 % of the explained variance in victimization, respectively.

  12. Interestingly, the idea of cost consideration has become a central focus of Hirschi’s (2004) revised definition of self-control (see Piquero and Bouffard 2007; Ward et al. Online first).

  13. Specialization in offending may also require a good deal of task persistence due the technical and interpersonal skills required for success (e.g., see Wright et al. 1995). We wonder whether this finding might implicate decision making processes for time discounting, in which consideration of benefits and costs follows a non-linear, hyperbolic trend (Loughran et al. 2012). Those with a greater preference for simple tasks may fail to have the necessary patience to wait for later rewards. That is, the offender with low self-control and high preference for simple tasks (low task persistence) may settle for less than ideal criminal opportunities making them more likely to be caught. Those with low self-control but more task persistence may be able to hold off for the better opportunity and possible bigger payout.

  14. For example, the mean age in the current sample was 32.07 years, similar to the adult offender samples from Longshore et al. (1996) and DeLisi et al. (2003) (30.8 and 29 years, respectively); race/ethnicity was also similar (35.6 % white, 64.4 % nonwhite) compared to Longshore et al. (37 % white and 63 % nonwhite); sex was also similar (74.5 % male, 25.5 % female) compared to Longshore et al. (75 % female and 25 % female). In addition, we previously noted nearly identical first-factor loadings (PCA) and self-control reliability estimates as compared to DeLisi et al.’s sample.

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Correspondence to Jeffrey T. Ward.

Appendices

Appendix 1

See Fig. 3.

Fig. 3
figure 3

Three alternative factor structures for the self-control construct. a Unidimensional factor structure, b correlated traits factor structure, c higher-order factor structure. Notes error paths for all items omitted for pictorial simplicity. To be clear, Structure B shows all factors correlated with all other factors (i.e., 15 correlations)

Appendix 2: Equations for Calculating Proportion Explained Variance and Omega Statistics

PEV-Global equations

$$PEV_{G} = \frac{{\varSigma \left( {\lambda_{i Self}^{2} } \right)}}{23}$$
$$PEV_{S} = \frac{{\varSigma \left( {\lambda_{i rs}^{2} } \right) + \varSigma \left( {\lambda_{i im}^{2} } \right) + \varSigma \left( {\lambda_{i pa}^{2} } \right) + \varSigma \left( {\lambda_{i tp}^{2} } \right) + \varSigma \left( {\lambda_{i sc}^{2} } \right) + \varSigma \left( {\lambda_{i st}^{2} } \right)}}{23}$$
$$PEV_{GS} = PEV_{G} + PEV_{S}$$

LPEV-Local equations (risk seeking example)

$$LPEV_{G (rs)} = \frac{{\varSigma \left( {\lambda_{i Self}^{2} } \right)}}{4}$$
$$LPEV_{S (rs)} = \frac{{\varSigma \left( {\lambda_{i rs}^{2} } \right)}}{4}$$
$$LPEV_{GS(rs)} = LPEV_{G (rs)} + LPEV_{S (rs)}$$

Model-based Omega for total scale:

$$\omega = \frac{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} + (\varSigma \lambda_{i im} )^{2} + (\varSigma \lambda_{i tp} )^{2} + (\varSigma \lambda_{i pa} )^{2} + (\varSigma \lambda_{i sc} )^{2} + (\varSigma \lambda_{i st} )^{2} }}{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} + (\varSigma \lambda_{i im} )^{2} + (\varSigma \lambda_{i tp} )^{2} + (\varSigma \lambda_{i pa} )^{2} + (\varSigma \lambda_{i sc} )^{2} + (\varSigma \lambda_{i st} )^{2} + \varSigma (\theta_{i}^{2} )}}$$

Omega Hierarchical:

$$\omega_{h} = \frac{{(\varSigma \lambda_{i Self} )^{2} }}{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} + (\varSigma \lambda_{i im} )^{2} + (\varSigma \lambda_{i tp} )^{2} + (\varSigma \lambda_{i pa} )^{2} + (\varSigma \lambda_{i sc} )^{2} + (\varSigma \lambda_{i st} )^{2} + \varSigma (\theta_{i}^{2} )}}$$

Model-based Omega for subscale (risk seeking example):

$$\omega = \frac{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} }}{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} + \varSigma (\theta_{i}^{2} )}}$$

Omega subscale (risk seeking example):

$$\omega_{s} = \frac{{(\varSigma \lambda_{i rs} )^{2} }}{{(\varSigma \lambda_{i Self} )^{2} + (\varSigma \lambda_{i rs} )^{2} + \varSigma (\theta_{i}^{2} )}}$$

Notes: We use the following notation: factor loadings (λ), error variances (θ2), item (i), self-control (Self), risk-seeking (rs), impulsivity (im), physical activities (pa), temper (tp), self-centeredness (sc), simple tasks (st), general (G) and specific (S) factors, proportion explained variance (PEV), and local proportion explained variance (LPEV). Please note that these equations cannot be used for other models (e.g., continuous items) or for unstandardized solutions; slight modifications are necessary to arrive at correct values in these cases.

Appendix 3

See Table 5.

Table 5 Summary of modeling steps in the bifactor analysis

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Ward, J.T., Nobles, M.R. & Fox, K.A. Disentangling Self-Control from Its Elements: A Bifactor Analysis. J Quant Criminol 31, 595–627 (2015). https://doi.org/10.1007/s10940-014-9241-6

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