Sentencing Disparities

As for legal factors, prior research has confirmed that primary predictors of federal sentencing outcomes are offense seriousness, criminal history,

E.g., Oleson et al., supra note 11, at 323 tbl. 2; Rob Tillyer et al., Differential Treatment of Female Defendants: Does Criminal History Moderate the Effect of Gender on Sentence Length in Federal Narcotics Cases, 42 CRIM. JUST. & BEHAV. 703, 705 (2015) [hereinafter Tillyer et al., Gender] (citing studies); Jeffery T. Ulmer et al., Trial Penalties in Federal Sentencing: Extra-Guidelines Factors and District Variation, 27 JUST. Q. 560, 576 tbl. 2 (2010) [hereinafter Ulmer et al., Trial Penalties].

Much research has found that demographic characteristics, which are generally considered to be extralegal factors for punishment purposes, are still correlated with sentence length. As for race and ethnicity, multiple studies of federal sentencing show that whites receive sentences of shorter length than blacks

E.g., Oleson et al., supra note 11, at 323 tbl. 2; Doerner & Demuth, supra note 110, at 255 tbl. 2; Feldmeyer & Ulmer, supra note 39, at 252 tbl. 3; Amy Farrell et al., Intersections of Gender and Race in Federal Sentencing: Examining Court Contexts and the Effects of Representative Court Authorities, 14 J. GENDER RACE & JUST. 85, 115 tbl. 3 (2010).

Studies of sentencing rather consistently indicate that males are sentenced to longer periods of incarceration.

E.g., Tillyer et al., Gender, supra note 108, at 713 tbl. 2 (citing studies and reporting on study of drug offenses); RHODES ET AL., supra note 37, at 67; David B. Mustard, Racial, Ethnic, and Gender Disparities in Sentencing: Evidence from the U.S. Federal Courts, 44 J.L. & ECON. 285, 300 (2001).

In some studies, noncitizens are at a statistically significant greater likelihood of incarceration

Michael T. Light, The New Face of Legal Inequality: Noncitizens and the Long-Term Trends in Sentencing Disparities Across U.S. District Courts, 1992-2009, 48 LAW & SOC’Y REV. 447, 464 tbl. 2 (2014) [hereinafter Light, Noncitizens]; Johnson & Betsinger, supra note 38, at 1067 tbl. 3.

Studies commonly indicate that older offenders are treated more leniently than their younger counterparts.

E.g., Anita N. Blowers & Jill K. Doerner, Sentencing Outcomes of the Older Prison Population: An Exploration of the Age Leniency Argument, 38 J. CRIME & JUST. 1, 3-4 (2013) (citing studies); Johnson et al., Social Context, supra note 52, at 761 tbl. 5 (finding older age positively correlated with downward departure decisions); John D. Burrow & Barbara A. Koons-Witt, Elderly Status, Extraordinary Physical Impairments and Intercircuit Variation Under the Federal Sentencing Guidelines, 11 ELDER L.J. 273, 31213 tbl.3, 4 (2004) (finding that in a few districts defendants age 50 and over were more likely to receive downward departures).

Two case-processing factors are relevant to predicting sentencing decisions. The so-called trial penalty occurs when being found guilty at trial (rather than plead) is correlated with more serious punishments.

E.g., Kim et al., supra note 43, at 1084 tbl. 2; Andrew Chongseh Kim, Underestimating the Trial Penalty: An Empirical Analysis of the Federal Trial Penalty and Critique of the Abrams Study, 84 MISS. L.J. 1195, 1220 (2015); Breen, supra note 101, at 211 (citing studies).

As for the second case-processing factor, studies at the state and federal levels rather consistently show that pretrial detention is significantly and positively related to incarceration and sentence length.

E.g., Oleson et al., supra note 11, at 316-17 (citing studies); Franklin, Educational Attainment, supra note 38, at 151 tbl. 2; Wolfe et al., supra note 123, at 355 tbl. 2. There is one study to the contrary where being out on bail increased sentence length, which the authors did not expect and do not explain the result. Farrell et al., supra note 113, at 115 tbl. 3.

Studies which include district or circuit variables in their models have generally found geographic disparity in federal sentences.

E.g., Wu & Spohn, supra note 52, at 306 (finding differences in the likelihood of downward departures across three Midwestern districts); Ulmer, Localized Uses, supra note 63, at 269 (finding from a study of four districts significant variations in the likelihood of granting substantial assistance downward departures).

A significant majority of the foregoing studies on federal sentencing use the incarceration decision (in/out) and/or sentence length as their outcome of interest. Some researchers affirmatively, though, recognize the importance of investigating departure decisions. Almost all of the studies of federal departure decisions to date which model the dependent variable on departure outcomes address downward departures.

E.g., Tillyer & Hartley, supra note 7, at 635 tbl. 2; Kimberly A. Kaiser & Cassia Spohn, Fundamentally Flawed? Exploring the Use of Policy Disagreements in Judicial Downward Departures for Child Pornography Sentences, 13 CRIMINOLOGY & PUB. POL’Y 241 (2014); Melissa A. Logue, Downward Departures in US Federal Courts: Do Family Ties, Sex, and Race/Ethnicity Matter?, 34 ETHNIC & RACIAL STUD. 683 (2011); Johnson et al., Social Context, supra note 52.

This is curiously true, despite upward departures arguably being more substantial, such as leading to longer sentences in the face of the federal prison overpopulation. Plus, their relative rarity renders upward departures more symbolic in nature, perhaps perceived therefore as arbitrary. Almost all the studies to date which consider the upward departure decision as a variable at all simply add it as a control without further discussion of its significance because their interests concerned other aspects of sentencing.

E.g., Franklin et al., Intermediate Sanctions, supra note 5, at 870 n.12; Tillyer et al., Gender, supra note 108, at 713 tbl. 2; Ulmer & Light, supra note 39, at 336 tbl. 2; Feldmeyer & Ulmer, supra note 39, at 252 tbl. 3.

It appears that only three studies (two of them by the same author) have so far utilized the upward departure decision as an outcome variable. Nevertheless, in these trio of studies the upward departure decision was one of multiple outcomes in single-level regressions and the authors did not spend too much space delving into the upward departure’s importance in federal sentencing outcomes.

Crystal S. Yang, Free at Last? Judicial Discretion and Racial Disparities in Federal Sentencing, 44 J. LEG. STUD. 75, 95-98 (2015) [hereinafter Yang, Discretion]; Crystal S. Yang, Have Interjudge Sentencing Disparities Increased in an Advisory Guidelines Regime? Evidence from Booker, 89 N.YU. L. REV. 1268, 1314 (2014) [hereinafter Yang, Interjudge]; Mustard, supra note 117, at 305-09.

Due to the paucity of research with a concentration on the upward departure decision, the importance of it in the results of sentencing outcomes in terms of severity of sentence, and the symbolic nature of the discretionary decision with respect to potentially reflecting gaps in the Guidelines, the opportunity to fill the void was compelling. Then the recent availability of more aggressive computing resources to permit employing a sophisticated research design known as multilevel modeling would allow this study to also be able to test for possible regional disparities. Hence, the next Section offers such a study.

The results for the fixed effects (i.e., individual defendant predictors) will be addressed first. All of the legal factors achieved statistical significance in their individual effects on upward departures. The final offense level was negatively associated with the odds of an upward departure: the odds of an upward departure decreased 7% for every one level increase in the final offense level. The criminal history score had the opposite effect in being positively associated with an upward departure: the odds of an upward departure increased 6% for each one unit increase in criminal history category. The presence of multiple counts of conviction were associated with increased odds of an upward departure. Regarding crime type, compared to drug offenders as the reference category, the other offense types were more likely to receive upward departures. Violent offenders faced almost five times the odds of an upward departure while the odds for firearm offenders doubled. Only immigration offenses did not result in statistical significance. Acceptance of responsibility lowered the odds of an upward departure by a factor of two.

As the coefficient is less than 1.00, we can interpret the effect on the odds by taking the reciprocal of the odds ratio = 1/.483.

Demographic variables were also modeled as fixed effects. Females were significantly less likely to receive upward departures than males, even after controlling for multiple factors: an upward departure for males was almost two times the odds as for females. U.S. citizens were more likely to be assigned upward departures, with the odds of citizens receiving upward departures being 66% greater as compared to noncitizens. There was also an age effect, with those age 50 and over being more likely to receive an upward departure compared to their younger counterparts.

Minorities were at higher risk of upward departures. The odds of a minority defendant receiving an upward departure increased 5% when controlling for the other legal and nonlegal variables. However, the result at the individual case level (Level-1) for the minority variable was not statistically significant. Still, as will be addressed further below, the minority factor was retained as there was a statistically significant random effect (districts at Level-2) for it, indicating that the lack of significance at the individual case level does not mean there is not a minority effect on increasing the odds of an upward departure in at least some districts.

Both case-processing factors were statistically significant. Custody status exhibited a large effect, increasing the odds of an upward departure by a factor of four for those in custody at sentencing. The trial penalty was not statistically significant at the individual level. However, the trial versus plea factor was retained because, as also addressed below, the random effect coefficient for the trial penalty at the district level indicated statistical significance, signifying that there are trial penalties in at least some districts.

The random effects (i.e., variations among districts) of the variables in the far right columns of Table 3 indicate whether the effect of each predictor varied across districts (except offense type which was excluded for statistical reasons per the Appendix). All but two of the predictor factors with random effects (being gender and age over 50) were found to vary across districts to a statistically significant degree.

Further information on the variability of each predictor factor that was modeled with fixed and random effects can be provided. Computations adding and subtracting one and two standard deviations indicated by each predictor variable’s random effect from the same variable’s fixed effect coefficient show whether the variability between districts concerns the strength of the correlation with the outcome and if the direction of the correction is positive in some districts yet negative in others.

See generally JOOP, supra note 148, at 19.

For six of the random effects, the size of the effect across two standard deviations varied between districts (i.e., across 95% of the districts), but not the direction. The number of counts of conviction, age over 50, and being in custody at sentencing were each positively correlated with upward departures in at least 95% of districts. The final offense level, acceptance of responsibility, and being female were negative predictors of upward departures in at least 95% of districts.

In contrast, the effect of each of criminal history score, minority status, and trial penalty showed that the strength and the direction of its influence changed across just one standard deviation (i.e., two-thirds of districts). This means that not only the size of the effect of these three variables varied amongst districts but that each held a positive effect in at least some districts while indicating a negative impact in others. U.S. citizenship held a positive association with upward departures in one standard deviation, but across two standard deviations the effect was observed to be negative in at least a few districts.

A supplemental data analysis provides further information about the reasons for upward departure decisions derived from the judges’ Statement of Reasons forms filed with sentencing paperwork in individual cases. Table 4 contains the top ten cited reasons for upward departures capture through frequency analyses of the Commission’s data, along with their prevalence.

Table 4

Specific Reasons Given by Judges for Upward Departures

Importantly, considering the title of this Article, unwarranted disparities in upward departures as an external consequence was among the top ten rationales as observed in Table 4. Judges cited disparity issues in one out of twelve upward departure decisions. This result indicates that numerous judges remain cognizant of the potential downsides of the appearance of disparities in sentencing practices. It is also suggestive of gaps in the Guidelines to the extent these judges perceive that the Guidelines calculations in the instant cases failed to achieve proportionality with sentences for similarly-situated defendants. The other reasons judges gave as indicated in Table 4 as justifications for upward departures will be explored further in the context of the general discussion of the results that follows.

The first research question queried whether there existed significant variation between district courts in the use of upward departures. The answer is in the affirmative. Bivariate results that were the result of additional statistical analyses indicated a differential of twelve times the rate of upward departures between the lowest rate district and the highest. Significant variation was confirmed in a null multilevel model (see the Appendix) which indicated that 8% of the total variance in upward departure outcomes is explained at the district court level. This rate was statistically significant at the .001 level. In other words, this means that eight percent of the differences in upward departure decisions are accounted for by district court practices. This result of district differences was expected from the courtroom workgroup perspective in that cultures unique to certain districts may influence sentencing outcomes that contrast with outcomes from other cultures/ districts.

The second general research question asked to what extent legal, extralegal, and case-processing factors accounted for upward departures in individual cases. Generally the results support the influence of the focal concerns (concerning the defendant’s culpability and future risk and the consequences of the sentence) on individual outcomes with respect to upward departures.

The legal variables supported the focal concerns expectation that perceptions of the defendant’s blameworthiness are highly relevant to individual penalties. The results indicated an increased likelihood of an upward departure for a higher criminal history score, multiple counts of conviction, and violent and firearms offenses (compared to drug offenders). Criminal history and additional counts signify multiple crimes and perhaps perpetrated on multiple occasions, possibly demonstrating greater culpability and harm. The increased odds for violent and firearms offenses reveal culpability concerns in that crimes posing a risk to human life likely are considered more egregious than many nonviolent offences.

The decreased likelihood of an upward departure for acceptance of responsibility is also consistent with a concern for the defendant’s blameworthiness as well as with the focal concern of future risk. Accepting responsibility by admitting guilt at an early stage in the proceeding may be perceived to reduce one’s culpability while predicting positive rehabilitation potential. The negative correlation of acceptance of responsibility with upward departures was consistent across at least 95% of districts.

Curiously, the final offense level was negatively correlated with the upward departure decision. This result seems to be somewhat contradictory to the focal concern with greater offender culpability predicting more severe sentences. It may instead, then, suggest that in these cases judges find the Guidelines calculations to be more than sufficiently proportional to reasonable sentences as adjudging offense severity. This explanation is likely because stakeholders tend to find Guidelines recommendations are overly punitive as a general rule.

See resources cited supra note 43.

Further discussion of criminal history is warranted as it played a strong role throughout the results. There were multiple indications that judges perceive inadequacies in the criminal history calculations. As previously indicated, a higher Guidelines-calculated criminal history score increased the odds of an upward departure despite multiple controls. This result implies that judges in these cases do not believe the criminal history calculation is sufficiently proportional to prior offending evidence, at least when the defendant already has a substantial criminal history as officially calculated pursuant to Guidelines rules. This observation is buttressed by the reasons judges listed in explaining upward departures. In the list of rationales judges gave for upward departures from the frequency distributions provided in Table 4, the role of criminal background is salient. Criminal history calculation issues were expressly cited in 60% of the cases, earning the top ranked reason overall for upward departures. Relatedly, as a separately coded reason, evidence of dismissed and acquitted conduct was listed as an explanation for upwardly departing in 8% of upward departures. Further, past offending may be part of the second ranked reason, which includes the history and character of the defendant, cited in over half of the upward departures. Because of the broad nature of that particular reason as including the nature and circumstances of the offense, though, it is difficult to parse what portion of the fifty percent was for prior offending specifically. Still, the failure of the formal criminal history calculation to adequately account for prior offending was evident in a significant majority of upward departures.

It is of particular note that judges candidly admitted the role of dismissed and acquitted conduct in their decisions to upwardly depart in one out of 12 (8%) cases. This finding might be of concern to critics of the real offense system in which individuals are penalized for conduct that is not the subject of conviction. Critics may be even more offended by increases in punishment for acquitted conduct. Here, it is not possible to tell exactly what percentage of those cases represented acquitted conduct, but it is likely that acquitted conduct played a role in at least some of them. These 8% of upward departure cases may also imply there are instances in which judges are countering plea bargaining to the extent that some percentage of these cases may represent increased penalties due to offenses dismissed as part of plea bargain deals. Perhaps this reflects judges acting as a check on prosecutorial authority in cases in which they view the plea bargains as overly lenient.

Overall, the salience of criminal history is theoretically important for another reason. The function of the defendant’s criminal history in the various results implicates the focal concern regarding the defendant’s future risk. The inclusion of criminal history in the Guidelines as a principal factor in the recommended sentence is often viewed as the Commission’s proxy to adjudge dangerousness.

MARJORIE A. MEYERS, CRIMINAL HISTORY: CALCULATION AND VARIANCE 1 (2012), available at http://www.ussc.gov/sites/default/files/pdf/training/annual-national-training-seminar/2012/3_Criminal_History-Calculation_and_Variance.pdf (presentation at U.S.Sent’g Comm’n Annual Training Conference).

Regarding future risk as a focal concern, other reasons in Table 4 more directly address dangerousness. The inclusion of the character of the defendant within the second ranked reason may well include assessments of past antisocial behavior as reflective of future risk. Ranked fifth in the top reasons given, the need to protect the public, clearly a future risk rationale, represented 41% of the upward departures. In sum, the relevance of the focal concern of future risk to severity in sentences is strongly confirmed in the data.

The multilevel results concerning offense type likewise provide interesting information about compliance with Guidelines’ proportionality judgments. The dummy series for offense type indicated that all other offense types, except for immigration offenses, were more likely to receive upwards departures than drug cases as the comparator. This implies that district judges as a general rule tend to believe the Guidelines are sufficiently punitive for drug offenses and immigration offenses. As drug and immigration cases combined are the bulk of federal sentencing in percentage terms, this particularly result situate the Guidelines in a positive light in terms of proportionality, at least with respect to generally being sufficiently punitive for a majority of crimes. However, the greater likelihood of upward departures for violent and firearms offenses implies that the judges may perceive the Guidelines as insufficiently punitive in those cases.

Moving onto the impact of extralegal variables, demographic characteristics presented with some expected results, while others were more surprising. There was support for gender leniency as women were far less likely to receive upward departures than men at the individual case level. Plus, gender leniency for women did not vary among districts, even after controlling for a host of other variables. This was the case even though gender is an extralegal factor and a prohibited rationale for sentencing outcomes per the Guidelines. Overall, then, the results indicate gender disparities, possibly even gender discrimination in favor of women, in upward departures.

Contrary to many studies, the results here indicate there was no individual-level minority discrimination in upward departure decisions. While the odds for minorities were 5% greater than whites, the result was not statistically significant. Indeed, minority status was the weakest individual predictor overall.

This result derives from F statistic comparisons.

It was surprising that noncitizenship was not a positive predictor of upward departures. Perhaps the explanation for the statistically greater likelihood of United States citizens to receive upward departures is that (according to a supplemental data analysis) two-thirds of the noncitizens in federal sentencing during the period of study (fiscal 2008-2015) were immigration offenders. Noncitizen immigration violators are likely to be subject to deportation. Deportation as an incapacitating gesture may impact an assessment of future risk at least regarding the danger to U.S. residents. Thus, it is possible that for noncitizen immigration offenders, prosecutors typically did not request upward departures in those cases and/or judges may have perceived them as unnecessary because of the deportation option. Still, the random effect of citizenship was statistically significant, indicating that the strength of the effect of citizenship significantly varied between districts. At two standard deviations, the effect of noncitizenship shows that it is actually positive (i.e., noncitizens were at higher odds of upward departures) in at least some districts.

No age leniency was observed at least to the extent it means less punishment for older offenders. Indeed, those age 50 and above were more likely to receive an upward departure and, like gender, the strength of the effect did not vary across districts. This could be evidence of a policy dispute with the Commission’s rule that age should typically not be a relevant sentencing factor. An alternative explanation, and one more likely considering the existence of other studies affirming age leniency,

Seesupra note 124 and accompanying text.

In terms of case-processing variables, the failure to find a trial penalty at level-1 is inconsistent with much other research.

Seesupra text and sources accompanying notes 127-129.

Still, the random effects coefficient was significant, and at one standard deviation, the results indicate a trial penalty in at least some districts, which is in line with prior research.

As the last predictor variable to be discussed, custody status was the strongest factor in elevating the odds of an upward departure among the predictor variables.

This result derives from F statistic comparisons.

The third focal concern should also be mentioned regarding consequences of the penalty. Several of the top reasons judges indicated on the Statement of Reasons for upward departures (listed in Table 4) implicate external consequences. The third highest ranking justification includes respect for the law, which likely entails respect by the defendant individually and more broadly. The fourth reason cites a general deterrence function as a reason for the upward departure, being triggered in 43% of cases. Both reasons reflect upon the consequences of the penalty in its deterring potential offenders and promoting community safety. Another community consequence present among the top ten reasons relates to the rehabilitation of the offender. The frequency of the rehabilitation motive to justify an upward departure, present in 9% of cases, is curious as federal law specifically dictates that “imprisonment is not an appropriate means of promoting correction or rehabilitation.”

18 U.S.C. §3582(a).

Additional evidence exists that upward departure decisions are quite often about proportionality concerns. Rounding out the top ten reasons listed for upward departures are two categories that expressly indicate judicial perceptions that the Guidelines have gaps. Judges cited general guideline issues or general adequacy issues in up to 10% of upward departure cases.

The third broad research question queried whether district courts vary from each other in the extent to which they weigh each of the legal, extralegal, and caseprocessing factors when issuing upward departures. The results found numerous such variations, as has already been partly covered when discussing the second research question. Overall, significant random effects were observed for all but two of the predictor variables (excluding offense type which could not be modeled as random effects). The strengths of the effect of leniency for women and the lack of lenience for older offenders were consistent across districts. In contrast, minority status and the trial penalty, which were not statistically significant in individual cases (after controlling for other variables), achieved significance in their random effects. In general, these random effect results support the courtroom communities’ perspective which theoretically accounts for different regional sentencing patterns.

To cite two examples, criminal history score and U.S. citizenship were both significant positive predictors of upward departures in individual cases, yet they also held significant random effects, meaning that their relationship to upward departures varied between districts. Moreover, standard deviation computations indicated that criminal history and the citizenship effect were actually negative predictors in some regions.

The discussion shall end on an empirical note. Overall, the results provide strong reinforcement for modeling sentencing decisions with both fixed and random effects in a multilevel model to observe individual-and group-based factors. The statistical significance of multiple explanatory variables in fixed and random effects is itself informative. Then it is also of practical and empirical import that the statistical significance of four variables posed contrasts between their fixed and random effects. In sum, females and age over 50 were statistically significant at their fixed effects, with females and defendants under age 50 far less likely to be issued upward departures (controlling for other explanatory factors). However, there were no significant random effects for those two variables, meaning that the leniency to females and the lack of leniency for those over 50 years-of-age were consistent between districts. The fixed and random effects for two other variables were in the opposite directions. Minority status and going to trial, indicated no significant fixed effects, but their random effects were significant. For minorities and the trial penalty, this means that there are at least a few districts in which minority status is correlated with upward departures and that the trial penalty exists to some extent in at least some districts. The mixed multilevel model employed here was uniquely able to parse those contrasts between individual-level and group-level effects for these four explanatory variables.

Multilevel analyses, when suitable for the data, are able to provide numerous benefits over single-level regression models. First, multilevel methods can account for the lack of independence when individuals are nested in groups.

Id. (noting in single-level regressions the lack of independence may exaggerate the significance of the parameter estimate).

Third, multilevel models are not limited to two levels; they can accommodate additional levels. As an illustration, multilevel regressions are popular in educational research where students are nested in classrooms which are nested in schools. The current challenge of including multiple levels is the substantial increase in computer resource capacity that is necessary to run a model with numerous explanatory factors included. An attractive feature is that there need not be the same number of units at each level. Nor must the levels be strictly hierarchical in nature. They may merely be nested. Thus, a multilevel model can be cross-level, such as defendants nested in years and nested in districts. Such a design would account, then, for both annual and regional variables.

Fourth, multilevel models partition the overall variance in the outcome of interest among the levels of analysis (e.g., at the individual level and then at the group level). The result indicates how much of the variation in the outcome is accounted for by the grouping.

The initial step in a multilevel model project is to run a null model. The null model is also referred to as an unconditional model because it has no explanatory factors included. The purpose is to statistically obtain the intraclass correlation coefficient (“ICC”) to determine if multilevel modeling is appropriate for the data. The ICC provides the proportion of the total variance in the outcome that is accounted for by the clustering at the nested group level. In other words, for purposes of this study, the statistic is a measure of how much of the differences in upward departure decisions are attributable to variations in district court practices. If the ICC indicates that intraclass correlation exists with statistical significance, the assumption of independence required by the single-level regression model may be rejected and the data are appropriate for multilevel modeling.

J. Kyle Roberts, An Introductory Primer on Multilevel and Hierarchical Linear Models, 2 LEARNING DISABILITIES 30, 32 (2004).

Multilevel models, like single-level regression models, are commonly tested on continuous dependent variables. But when the outcome of interest is binary in nature, different modeling must be employed because a binary dependent variable means that the normal assumptions of a normally distributed response variable and homoscedatic errors are violated.

Joop J. Hox & Cora J.M. Maas, Multilevel Analysis, in ENCYCLOPEDIA OF SOCIAL MEASUREMENT 785, 790 (Kimberly Kempf-Leonard ed., vol. 2, 2005).

A statistical model to fit data with a binary dependent variable is called a generalized linear model with three components: (1) a linear regression equation, (2) a specific error distribution, and (3) a nonlinear link function that transforms the predicted values for the dependent variable to the observed values.

Id.

For the study herein, the binary response variable for the ith defendant in district j, is:

Yij1for upward departure0for no upward departure$$\begin{array}{} \displaystyle {\rm Y_{ij}}\left\{\begin{array}{}1\,\, \text{for upward departure} \\ 0\,\, \text{for no upward departure}\end{array}\right. \end{array} $$

The transformation of the dichotomous dependent variable for an upward departure presented herein utilizes the logit link function.

η<!-- ? -->ij=ln(p1−<!-- - -->p)$\begin{array}{} \eta_{ij}=ln(\frac{p}{1-p}) \end{array} $Logit Link Function

RONALD H. HECK ET AL., MULTILEVEL MODELING OF CATEGORICAL OUTCOMES USING IBM SPSS 151 (2o12).

In the logit link function, the Greek letter eta (η) represents the transformed linear predictor. Exponentiating the resulting η parameter provides the odds ratio. The p is the probability of the outcome occurring and the denominator (1 – p) is the probability of the outcome not occurring. The equation represents the odds of the outcome.

At the outset of this study, it was considered that a three-level model might be appropriate considering district courts are nested within the higher level circuit courts of appeal and/or within years, with the latter perhaps accounting for changes in sentencing patterns over time and using annual time periods as the temporal division.

A few statistical notes should be briefly mentioned before addressing the models. The software utilized for the study presented herein, including the three-level models that follow, was SPSS version 24. Further, there is no issue of selection bias and therefore no need for the so-called Heckman correction. Selection bias may occur when the researcher obtains data from a non-random sub-sample of the population of interest.

Shawn Bushway et al., Is the Magic Still There? The Use of the Heckman Two-Step Correction for Selection Bias in Criminology, 23 J. QUANTITATIVE CRIMINOLOGY 151, 152 (2oo7).

In any event, the specification for a three-level null model is as follows:

η_ij = β_0jkLevel-1β_0jk = γ_00k + μ_0jkLevel-2γ_00k = γ₀₀₀ + μ_00kLevel-3

HECK ET AL., supra note 194, at 183.

It was of interest, then, to test for whether the final model ought to account for serious nesting patterns which may introduce bias from the circuit courts of appeal as Level-3. The initial step in creating a multilevel model with three levels is to estimate the null model, which is provided in Table 5.

Table 5

Null Model for Upward Departures with Districts Nested in Circuits.

From Table 5 it is estimated that 7% of the variation in upward departures is between district courts and almost 2% of the variation is between circuit courts of appeal. The ICC was statistically significant for Level-2 district courts, yet was not significant for the Level-3 circuit courts. Practically, it was not surprising that there was not shown to be statistical significance with circuit courts. An earlier scan of bivariate data for the proportion of upward departures in the districts did not reveal consistencies for districts nested in circuits. Instead, the circuits tended to encompass a mix of low and high use of upward departures within their nested districts. For example, while three of the districts within the Fifth Circuit yielded the highest proportions of upward departures (Northern District of Texas at 6.5%, Western District of Louisiana at 5.7%, and Eastern District of Louisiana at 4.8%), the Fifth Circuit also included one district with a below-average rate of upward departures (Southern District of Texas at 1.5%). Overall, the Fifth Circuit ranked as the fifth highest among the 12 circuits in its total proportion of upward departures. The First Circuit ranked first overall, with a total of 3.3% of sentences with upward departures. But the First Circuit also presented with vastly different practices within its district court outcomes, as well. Most of the upward departures in the First Circuit were issued in the District of Puerto Rico (at 4.4%), yet this circuit also included the District of Rhode Island which issued one of the lowest rates of upward departures (at 0.5%).

While circuit court variation was not statistically significant, it alternatively was likely that there might be variations by time. Thus, a three-level null model was run for district courts nested in fiscal years, which is presented in Table 6.

Table 6

Null Model for Upward Departures for Districts Nested in Years.

This null model with district courts nested in fiscal years demonstrated that 8% of the variation in upward departures is between district courts. It was also found that there is a statistically significant variation with Level-3 being an annual indicator. Yet, for several reasons, the nesting of upward departure outcomes at a level with years was dropped to proceed with a more developed two-level model. The ICC for years was, in practical terms, indicating a low degree of variation by year at less than 3%. As multiple explanatory variables were expected to be included in the final model with both fixed and random effects, a three-level model including years would present as an extremely complicated model from a computing resource perspective. Indeed, as will be indicated below, even in a two-level design with district courts at the higher grouping, the final model had to be curtailed a bit because of convergence issues when attempting to model all independent variables as both fixed and random effects. An additional concern is that there were only 8 groups involved for years (i.e., eight consecutive fiscal years), an extremely low number for multilevel modeling purposes. In any event, as a primary interest for this study was regional variations in discretionary sentencing decisions, the Level-3 variation with years was dropped. Still, the three-level model indicated in Table 6 was presented herein for informational purposes.

As the three-level designs just summarized were vetoed, a null model with two levels to account for nesting in districts could be run. The null model for two-level design with a dichotomous dependent is specified with the following equations.

η_ij = β_0jLevel-1 Null Modelβ_0j = γ₀₀ + μ_0jLevel-2 Null Model

In these null models for this study, the term β_0j is the intercept, which is the average log odds of an upward departure in group j. At Level-2, the term γ₀₀ represents the fixed intercept, being the log odds of an upward departure in a typical district for the average individual. The variance parameter μ_0j is the random intercept and signifies the variability of the outcome across Level-2 groups.

HECK ET AL., supra note 194, at 151.

In a generalized linear multilevel model using a logit link because of a binary response variable, the Level-1 residuals are assumed to follow the standard logistic distribution, with a mean of 0 and a variance (σ̑²) set to π²/3. which is equal to 3.29. For a dichotomous outcome, the intraclass correlation coefficient (i.e., a statistic that indicates the proportion of total variability in outcomes which arises at the higher level) is computed in a two-level model as:

τ00τ00+3.29$\begin{array}{} \frac{\tau_{00}}{\tau_{00}+3.29}\end{array} $Intraclass Correlation Coefficient (ICC)

The term τ₀₀ represents the between-group variance at Level-2.

Id. at 94.

Table 7 provides for the null model results for upward departures where Level-1 are individual defendants and Level-2 are district courts. Table 7 is the basis for the final model contained in Table 3 in the main body of this Article.

Table 7

Null Model for Upward Departures Nested in Districts.

The ICC computed for the two-level null model means that 8% of the variability in upward departures is accounted for by districts.

This leaves 92% of the variability to be accounted for at the individual case level (or other unknown factors).

As expected from the courtroom communities’ perspective, the Level-2 random effect is significant at the .001 level, which indicates that the probability of an upward departure significantly varies between districts. Indeed, in a separate analysis to compare district means, wide variation in proportions were observed. The proportion of upward departures at the district court level ranges from a low of 0.5% (Northern District of Oklahoma, District of New Mexico, and District of Rhode Island) to a high of 6.5% (Northern District of Texas). Thus, the district with the greatest proportion of upward departures is more than twelve times that of the district with the lowest percentage, indicating a stark district level differential.

The intercept in the two-level null model represents an estimate that can be converted to the overall probability of an upward departure. The random effect represents the degree to which the outcome varies across federal districts. The estimated probability of a defendant receiving an upward departure in the average district is approximately 2%.

The formula to obtain the overall expected proportion is an inverse of the logit link function: [(1/(1 + e’^η)) x 100%]. Plugging in the coefficient for the fixed effect coefficient, the formula becomes [(1/(1 + e^3.921)) x 100%], which is equal to 1.94%.

Once the researcher chooses the null model with the appropriate higher level(s), the researcher can add explanatory factors. In a very simple model, we can add a Level-1 explanatory variable and a Level-2 predictor, such as the following equation illustrates.

η_ij = β_0j + β_1jX_1ijLevel-1β_0j = γ₀₀ + γ₀₁W_j + μ_0jLevel-2β_1j = γ₁₀ + μ_1j

Now γ₀₀ is the log odds that the outcome = 1 when explanatory variable X = 0 and μ = 0. β₁, is the log odds effect that the outcome is = 1 for every one unit increase in the variable X in group j. To get a more interpretable result for the effect of X, we can exponentiate β₁, to obtain the odds ratio to compare the odds for individuals spaced one unit apart on X. Then Wj represents the random effect of that predictor variable in group j.

In this study, the null model with district courts at Level-2 was the choice and the independent variables that survived into the final model are provided in Table 3 in the main body of the text. In Table 3, the ICC statistic indicates that 2% of the overall variance remains with district courts. The intraclass coefficient is no longer statistically significant when accounting for multiple fixed and random effects. Nonetheless, the substantial reduction in the -2 Log-Likelihood statistic between the null model and the full model indicates a significantly better fit of the full model for this dataset. Further discussion on methodological choices along the way to the final model is next.

Some variables were transformed for the final model as explained below. In addition, other factors were tested yet eliminated in the end for the reasons ascribed to them herein.

For purposes of the descriptive statistics in Table 2, the variables for final offense level, criminal history, and number of counts are in their original metrics. For the multilevel model in Table 3, these three variables are each grand mean centered for ease of interpretation as none of them can have zero as a real value. In federal sentencing, defendants must have at least one count of conviction, the lowest criminal history category is I (i.e., 1), and the minimum offense severity level is 1. In a logistic model, the intercept is interpreted to mean the value of the outcome when all predictors are equal to 0. This has no practical meaning for variables that cannot actually have a real world value of 0, which is the case for these three variables. Grand mean centering is the statistical convention for adjusting the metrics to have a more interpretable intercept in such a case.

The number of counts (of conviction) variable was transformed for statistical purposes. In the original data, the number of counts variable was skewed to the right. This variable was first centered at the grand mean. Then to enable a natural log transformation to adjust for the skew and more closely approximate a normal distribution, the value of .1 was added to the mean centered variable because log transformations are not possible on values of 0.

Race/ethnicity was originally coded as dummy variables of black, Hispanic, and other, with white as the reference category. In a full multilevel model with such coding with all fixed effects, the only statistically significant result was for the category of other as compared to whites. This result is practically meaningless because the grouping of “other’ includes a heterogeneous mix of native Alaskan, native American, non-U.S. American Indians, Asian, Pacific Islander, multi-racial, and a smaller subset of other.

U.S. SENT’G COMM’N, VARIABLE CODEBOOK FOR INDIVIDUAL OFFENDERS 31 (2015).

The full model includes all 94 district courts. This is mentioned because many studies that incorporate district courts in their variables exclude the districts that are in the U.S. territories (Puerto Rico, Virgin Islands, Guam, North Mariana Islands). These researchers argue the territories are viewed as different because states enjoy greater rights than them and, thus, the inclusion of the territories may introduce nonrandom bias.

E.g., Farrell et al., supra note 113, at 103 n. 75; Kautt, supra note 90, at 648. See also Light, Noncitizens, supra note 120, at 456 (excluding the territories without stating reason).

The general offense type was excluded from the random effects due to the complexity of the algorithm necessary to compute a multilevel model with them included. In other words, the model with the offense type having random effects was overly complicated for computational iterations, resulting in a failure of convergence. Convergence was achieved after excluding offense types at Level-2, while still retaining their Level-1 fixed effects.

It is noted that four additional independent variables were tested but removed before the final model for reasons of parsimony and specific statistical challenges. The applicability of a mandatory minimum statute was not statistically significant (at the .001 level) at Level-1 in any model and thus was removed as there was no theoretical justification to retain it as a factor in a study on upward departure outcomes. A variable tied to the Guidelines-recommended sentence was removed because of multicollinearity concerns with the final offense level and criminal history score variables. Notably, all independent variables attempted in any model were tested for multicollinearity. For the independent variables retained and shown in the final model in Table 3, results indicated no significant collinearity problems. All variance inflation factor scores resided within an acceptable level (VIFs < 3). A variable regarding the guideline recommended sentence had previously triggered multicollinearity concerns (with some VIFs greater than 5) and was therefore removed.

A series of dummy variables to distinguish fiscal years of sentencing were also dropped. While the annual rates of upward departures were statistically significant compared to 2008 as the dummy, the overall statistical impact (according to F statistic results) on explaining upward variances for the timing factor was among the weakest among the various explanatory variables. The statistical resources necessary to account for the seven dummy variables for years did not then seem worthwhile.

Another variable was tested and also dropped. No statistically significant effects of education level on upward departures were observed in any tested model. Without any pressing need to focus on educational level as it does not represent the most egregious type of discriminatory category, it was discarded as an explanatory factor.

As a final methodological note, the results here may advise other researchers that it might be preferable to model the main Guidelines proxies for crime severity and criminal background with the two separate factors of final offense level and final criminal history category, respectively, rather than their combination as indicated by the Guidelines’ minimum sentence recommendation. As shown herein, the two variables may actually have the opposite effect on the outcome of interest, which would unfortunately be indiscernible when using the minimum sentence combination instead.