A Nonparametric Imputation Approach for Dealing With Missing Variables in SHR Data

http://hsx.sagepub.com/Homicide Studies

http://hsx.sagepub.com/content/8/3/255The online version of this article can be found at:

DOI: 10.1177/1088767904265361

2004 8: 255Homicide StudiesRobert L. Flewelling

A Nonparametric Imputation Approach for Dealing With Missing Variables in SHR Data

Published by:

http://www.sagepublications.com

On behalf of:

Homicide Research Working Group

can be found at:Homicide StudiesAdditional services and information for

http://hsx.sagepub.com/cgi/alertsEmail Alerts:

http://hsx.sagepub.com/subscriptionsSubscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://hsx.sagepub.com/content/8/3/255.refs.htmlCitations:

at TEXAS SOUTHERN UNIVERSITY on December 7, 2014hsx.sagepub.comDownloaded from at TEXAS SOUTHERN UNIVERSITY on December 7, 2014hsx.sagepub.comDownloaded from

http://hsx.sagepub.com/

http://hsx.sagepub.com/content/8/3/255

http://www.sagepublications.com

http://www.icpsr.umich.edu/HRWG

http://hsx.sagepub.com/cgi/alerts

http://hsx.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.com/journalsPermissions.nav

http://hsx.sagepub.com/content/8/3/255.refs.html



10.1177/1088767904265361HOMICIDE STUDIES / August 2004Flewelling / ANONPARAMETRIC IMPUTATION APPROACH

A Nonparametric ImputationApproach for Dealing With

Missing Variables in SHR Data

ROBERT L. FLEWELLINGPacific Institute for Research and Evaluation

Despite the considerable percentage of homicides recorded in the Supplementary Homi-cide Reports (SHR) that have missing information, there is no standard approach used byresearchers to adjust homicide rates to accommodate missing data. Yet, rates that aredefined on the basis of offender characteristics or victim-offender relationship will beseverely underestimated if no adjustments, or imputations, are made. Several approachesto imputation of SHR data have been described in the literature. This research notedescribes an imputation strategy that appears to offer a viable alternative approach,although further development and testing of this and other techniques are needed beforean optimal strategy can be determined.

Keywords: Supplementary Homicide Reports; missing data; imputation

It has now been more than 15 years since Kirk Williams and I pub-lished an article in Criminology (Williams & Flewelling, 1987)describing procedures we used to address missing data problemsin the Supplementary Homicide Reports (SHR). That effort waspart of a broader study to examine structural correlates of disag-gregated homicide rates in American cities. A central theme ofthat research, reiterated by Flewelling and Williams (1999), was

255

AUTHOR’S NOTE: This article was first submitted as a brief research note and com-mentary but expanded into a longer piece at the suggestion of the editors for this specialissue. The approach described still warrants further development and validation, but it ishoped that the article will stimulate consideration of this and other alternative approachesfor addressing missing data in the SHR. The author thanks Marc Riedel and WendyRegoeczi for their encouragement concerning this submission, and Kirk Williams for guid-ing early work in this area. The development of the methods described herein was fundedin part by a grant (95-CX-0114) from the National Institute of Justice.

HOMICIDE STUDIES, Vol. 8 No. 3, August 2004 255-266DOI: 10.1177/1088767904265361© 2004 Sage Publications

at TEXAS SOUTHERN UNIVERSITY on December 7, 2014hsx.sagepub.comDownloaded from


the critical importance of disaggregating homicides into concep-tually meaningful subgroups to advance understanding throughcomparative research. The 1987 article focused on two problemsinherent in SHR data: underreporting of homicide incidents andmissing fields of information within reported incidents. Missinginformation typically involves offender characteristics and thevictim-offender relationship and now affects more than one thirdof all SHR records. Disaggregated homicide rates defined on thebasis of offender characteristics, therefore, are subject to signifi-cant underestimation of varying degrees. The adjustment proce-dures described in the article were designed to improve the accu-racy of estimated homicide rates over calculations that ignoredmissing data or that made simple proportionate adjustmentswithout using additional information that was available in thehomicide record data.

Although my own areas of research have since been refocusedonto other topics, I have always believed that those initial, andadmittedly still crude, adjustment procedures developed in themid-1980s could be further developed and substantiallyenhanced. What we designed then served the purposes of theunderlying study in which we were engaged. At the same time,we stressed the importance of further efforts to assess and refinethese procedures, which would include identifying both theirstrengths and their limitations. Indeed, other investigators haveidentified concerns in applying our strategy to other researchquestions. For example, Langford, Isaac, and Kabat (1998)reported significant overestimation of intimate partner victimiza-tions when applying the procedures to SHR data from the state ofMassachusetts.

Although the percentage of homicides that have missing infor-mation has increased significantly over the past two decades, acursory review of research published in Homicide Studies suggeststhat until very recently there had not been substantial furtherprogress in developing techniques to compensate for missinginformation through adjustment or imputation strategies. Onenotable exception is the procedure designed by Fox (1997, 2001)that employs a weighting technique that essentially replacesrecords with missing offender information with incidents (or,

256 HOMICIDE STUDIES / August 2004



more precisely, weighted averages of incidents) from within thesame state and year that are similar with respect to victim charac-teristics. This approach is noteworthy because it uses additionalinformation (victim characteristics) known about the incidents inquestion to help infer the values of missing data elements. It alsoprovides users of public use files with a standard approach tomissing data adjustment that is easy to apply.

Two very recently published articles describe the use of formalstatistical modeling techniques to impute the values of missingvictim-offender relationship categories. Regoeczi and Riedel(2003) employed an expectation-maximization (E-M) algorithmin imputing the missing values for homicides recorded in Chi-cago and Los Angeles, and Messner, Deane, and Beaulieu (2002)used a log-multiplicative modeling approach with data fromthe 1996 and 1997 SHR files. These methods employ statisticalmodels based on the relationships between victim-offender rela-tionship and other variables in the homicide records that aremuch less likely to be missing to estimate the value of the victim-offender relationship for incidents in which it is not reported.Using statistically related information from records that havemissing information on offender characteristics (includingvictim- offender relationship) to help impute those characteristicsmakes a great deal of sense, and there are a number of potentiallyviable approaches for doing so.

Several years ago, I developed a strategy that differs from boththe weighting procedure developed by Fox and the statisticalmodeling approaches described above, yet it also was designed touse the information that is known about incidents with missinginformation to improve the accuracy of imputation for thosefields that are missing. Conceptually, it was an extension of theapproach described by Williams and Flewelling (1987) but tookadvantage of substantially more information than just the circum-stance code in making the imputations. The procedure was neverfully operationalized or documented but was carried far enoughalong to suggest that it could be a viable and effective techniquefor helping deal with missing data in the SHR (Flewelling, 1997a).The remainder of this research note briefly summarizes thatapproach and comments on its potential utility.

Flewelling / A NONPARAMETRIC IMPUTATION APPROACH 257



DESCRIPTION OF PROCEDURE

In the procedures developed by Williams and Flewelling(1987), the distributional assumptions concerning variables forwhich there are missing data were inferred from known cases inthe same jurisdiction but done separately within several catego-ries of circumstance. The rationale for this strategy was based onthe fact that certain types of homicide, as defined by their circum-stance code, were disproportionately represented among inci-dents with missing offender information. Furthermore, thesehomicide types exhibited significantly different distributions onoffender characteristics among the cases where offender informa-tion was present. The accuracy of imputation was presumablyimproved, therefore, by allowing for different adjustments to bemade for missing data based on the circumstance codes of thecases with the missing data. By way of example, it was observedthat felony-type homicides were much more likely to have miss-ing offender information than other homicide types. Felonyhomicides also had a considerably higher likelihood of beingstranger homicides than did homicides associated with othercircumstances. Therefore, a greater number of homicides withan unknown victim-offender relationship were imputed to bestranger homicides than would have been the case if circum-stance was not considered in making the imputation.

In reviewing those adjustment procedures, and in applyingthem to a study of trends in youth-perpetrated homicide rates(Flewelling, 1996, 1997b), several enhancements to this approachwere developed. First, and consistent with other imputationapproaches described above, it was observed that among therecords with no missing data, victim variables (e.g., age, gender,and race/ethnicity of the victims) are moderately related withoffender characteristics and the victim-offender relationship.Therefore, these variables provide potentially useful additionalor alternative pieces of information that could be used to infer thevalues for the variables that have missing information. For exam-ple, age of victim and age of offender exhibit a fairly strong associ-ation based on cases in which both of these data elements are pres-ent. As shown in Table 1, whereas slightly more than 50% of theoffenders in incidents from 1995 to 1999 involving victims aged 15




to 19 were also in that age group, only 19% of those who murderedindividuals aged 25 to 34 were in the 15 to 19 age group. It is alsoapparent that the association is not strictly a linear one, as evi-denced by the high percentage of very young victims murderedby persons in the 25 to 34 age group. This association, along withthose between other victim and offender characteristics (e.g., seeAllen & Buckner, 1997), provides the basis for an adjustment pro-cedure that makes use of several pieces of known information tomore accurately infer the values (or, more specifically, theprobabilities) of variables that contain missing data.

A second modification to the original procedure involvedexpanding the number of incidents used to extrapolate the distri-butions of the missing variables. Previously, the adjustmentswere made entirely within each geographic unit for which rateswere being calculated. This may be problematic for geographicunits that have relatively small numbers of homicides and thattherefore may produce markedly skewed estimates. Therefore,the new procedure specifies that the distributions used as thebasis for imputation be derived from all homicide events withinthe U.S. Census-designated geographic division (i.e., the nine-category subregional classification provided in the SHR) of theincident in question, aggregated over a 5-year period.

The mechanics of the procedure are also somewhat differentfrom either the proportional adjustment technique used in theWilliams and Flewelling approach or the other approaches (i.e.,differential weighting and E-M) mentioned earlier. An offender-level file is first created that contains one record for each offenderidentified in the SHR data. Records are created even when thecharacteristics of the offender are unknown. Victim and offendercharacteristics, relationship, circumstance, and other variables ofinterest are then collapsed into new variables with smaller num-bers of discrete values. Based on these recoded variables, 90 possi-ble unique combinations of offender age (5 categories), gender (2categories), race/ethnicity (3 categories), and relationship withthe victim (3 categories) are defined. Dummy variables are cre-ated to correspond with these combinations. For incidents inwhich the offender characteristics are reported, a 1 is assigned tothe specific dummy variable associated with the combination forthat incident, and the other 89 dummy variables are assigned azero.




For every offender record that has missing information withregard to offender characteristics and victim-offender relation-ship, the dummy variables are assigned values that reflect proba-bilities, ranging from zero to one. For each such case, the sum ofthe probabilities across all 90 dummy variables equals 1.0. Theprobabilities are empirically derived from all incidents within thesame geographic division and 5-year time period that have thesame exact combination of values for circumstance, victim age,gender, and race/ethnicity and that have valid data for offendercharacteristics and victim-offender relationship. Specifically, theyare calculated as the relative proportions in the distribution ofthese cases across the 90 possible combinations of offender char-acteristics and victim-offender relationship. These proportionsare calculated prior to the imputation process, through the analy-sis of cases that have valid data on offender characteristics, andstored in matrices that can be referenced as the missing dummyvariable values are assigned. By collapsing across categories ofcircumstance and victim characteristics in a similar fashion as wasdone with the offender characteristics, 90 possible combinationsof circumstance and victim characteristics, or “imputationclasses,” were defined for this purpose.

Any set of homicide perpetration counts required may then becalculated by aggregating the values from individual records inthe reconstructed file to the geographic level desired. Because thedummy variables contain zeroes, ones, or fractions indicating a


TABLE 1Age Distribution of Homicide Offenders by Age of Victim: 1995-1999 (in percentages)

Age of Offender

Age of Victim N 10-14 15-19 20-24 25-34 35+

0-9 3,141 2.3 17.0 29.7 35.9 15.110-14 912 12.6 45.2 17.5 11.4 13.315-19 8,619 2.1 50.6 30.2 12.0 5.220-24 10,403 .8 30.3 38.2 22.4 8.225-34 14,514 .8 19.0 25.4 34.6 20.335+ 21,564 1.2 17.2 15.4 23.0 43.2Total 59,153 1.4 25.2 24.8 24.7 23.9

SOURCE: Supplementary Homicide Reports data, 1995-1999 (Fox, 2001).



probability between zero and one, they may be aggregated sim-ply by summing across records. The sums provide counts (or esti-mated counts) of each of the 90 different combinations of homi-cide types as characterized by offender age, race, gender, andrelationship to the victim. Counts may be collapsed across combi-nations—for example, if race-specific counts are not needed, thecounts from the appropriate combinations are simply combinedinto an overall (i.e., non-race-specific) count. Counts for combina-tions of offender characteristics with other attributes of each inci-dent, including victim characteristics, may also be generated bycomputing the products of the appropriate dummy variable com-binations before aggregation. Similar procedures may be appliedto victim-level files when calculating homicide victimizationrates that are disaggregated by offender characteristics, includingvictim-offender relationship.

One aspect of this procedure that may at first seem problem-atic is the assignment of probabilities, rather than specific values,to cases with missing data. Other imputation procedures assignactual values. Although this may be important when furtheranalyses are to be conducted at the level of individual records, it isnot necessary when the data are to be aggregated across records togenerate counts and/or rates for defined geographic or adminis-trative units. In fact, the use of probabilities would seem prefera-ble to assigning an actual value of a variable based on its esti-mated likelihood. For example, if 10 cases all had an estimatedprobability of .1 for being a “stranger” homicide, none would beassigned to that category based on their probability, thus yieldingan estimated count of zero. Yet, the expected number of strangerhomicides from within that set would be 1 (which, of course, is thesum of the 10 probability values).

Assessment of this approach has so far been limited. Asexpected, however, by using information about the victim and cir-cumstance, the adjustments generated higher rates of youthhomicide perpetration than obtained if the unknown incidentswere assumed to have the same distribution as the known cases,and they also indicated that the magnitude of the increase in thehomicide perpetration rate by persons aged 15 to 19 between 1980and 1993 was even more pronounced than what otherwise couldbe detected.




IMPLICATIONS

The preceding section describes an approach to dealing withmissing data in the SHR that, to my knowledge, has not yet beenimplemented in any published research but that appears to offerseveral advantages over various alternatives. The readily appar-ent advantage over the Williams and Flewelling procedure is itsuse of considerably more information from each incident, whichpresumably results in more accurate inferences with regard to thevalues of the missing variables. In addition, it bases the inferenceson a larger (although possibly more heterogeneous) set of inci-dents with known values. An apparent advantage over Fox’s pro-cedure is that all of the nonmissing information within eachrecord is retained—only the variables that are actually missingare imputed. This could be important when homicide subtyperates based on combinations of variables (e.g., offender agecrossed with weapon, circumstance, or victim age) are desired, orwhen generating rates for geographic units smaller than states.The use of a larger and more stable pool of incidents from which tobase inferences is also a relevant consideration.

The procedure described here bears certain conceptual similar-ities to the E-M approach used by Regoeczi and Riedel. In bothcases, multiple variables (or “covariates”) from each incident areused to inform the imputation of the unknown characteristics,and in both cases all the nonmissing information in these recordsis retained. Whereas the E-M procedure uses a statistical modelwith explicit distributional assumptions concerning the joint dis-tributions of the variables involved, however, the imputations inmy approach are guided only by the actual distributions that existin the data set, hence the descriptor “nonparametric.” Althoughthe computational procedures used for the two approaches arevery different, I would not be surprised if the estimated counts(for the geographic units of interest) produced by the twotechniques were similar, if the same set of covariates (all definedand/or categorized in the same manner) were used to guide theimputations. With regard to the covariates, it is important to keepin mind that the nonparametric approach described in this articledifferentiates every possible combination of values formed by thecollapsed versions of circumstance and victim age, race, and gen-der, in defining the classes within which the imputations are




conducted. This differs from the approach described by Regoecziand Riedel, in which the covariates appear to be included as maineffects only in the imputation model. Their model could, ofcourse, be adapted to incorporate specific combinations of predic-tors through the appropriate construction of interaction termsand/or dummy variables.

Alimitation of the described procedure is that the optimal set ofcovariates, including how many there are and how they arerecoded and combined, is unknown. Also unknown is the opti-mal criteria for defining the pools of cases, with respect to geo-graphic unit and year, on which the assigned probabilities arebased. Although it is important that imputations be based onenough cases to provide some stability in the estimates derived—hence the choice to use regional data spanning 5 years—this mustbe balanced against the potentially greater imputation accuracythat might be gained by basing the imputation on only those casesthat are most similar to the one being imputed. These parametershave not been adequately studied, although Regoeczi and Miethe(2003) have warned against spanning across too many years indefining the imputation classes. It is certainly possible that usingfewer than 90 combinations of victim characteristics and circum-stance would produce satisfactory estimates with less likelihoodof extreme values that could be caused by small cell sizes. On theother hand, perhaps even more than 90 categories would enhancethe accuracy of imputation, especially if coupled with a decisionto not differentiate the reference cases by the nine categories ofgeographic region (or perhaps use only an urban/suburban/rural variable rather than region). These considerations are notonly pertinent to the method described here. Rather, they areimportant considerations in any potentially viable imputationstrategy and warrant careful examination in developing andtesting such procedures.

Another important limitation of the adjustment proceduredescribed above, and most imputation strategies in general, isthat the characteristic of having missing data does not itself influ-ence the imputed value(s). Rather, these values are assumed tohave the same distributions as those from records that have nomissing information. In reality, homicides for which offendercharacteristics are unknown are likely to be qualitatively differenttypes of events from those in which the offender is known at the




time the incident is reported. Stated another way, the true valuesof the missing variables are likely to be distributed differentlyamong cases for which this information is missing relative to thecases in which all information is reported. The extent to whichthese distributions differ (i.e., the missing data are “nonignorable”)has been the focus of some debate in the research literature (seeRegoeczi & Miethe, 2003) and is also the potential fly in the oint-ment of any imputation procedure that generates adjustments formissing data that are based on distributions and/or patterns ofrelationship among the cases without missing data. Although thedebate has focused primarily on the victim-offender relationship,this issue is clearly relevant to the imputation of offender charac-teristics in general. Based on the limited and somewhat conflict-ing research findings on this topic, I believe it is prudent toassume that extrapolation of information from incidents whereall information is available to those for which information such asoffender characteristics is missing will almost always introducesome systematic distortion in adjusted counts and rates. The useof as much supplemental information as possible about eachevent should help to reduce the bias in extrapolating from theknown to the unknown, perhaps to a negligible level, and isalmost certainly preferable to ignoring the missing value problemaltogether. But it is unlikely that such efforts alone can completelyovercome inherent differences between homicides that are basi-cally solved at the time the report is prepared and ones that arenot.

The log-multiplicative procedure described by Messner et al.(2002) employs a model that explicitly recognizes the nonignor-ability of missing victim-offender relationship and models thiseffect by scaling relationship categories on a continuum thatincludes the “relationship unknown” category at one end of thecontinuum. Given the concerns that have been raised about non-ignorability of missing homicide data, there is conceptual appealto that approach. An assessment of the appropriateness andvalidity of the statistics underlying the approach, however, isbeyond my level of expertise. I expect that further research anddebate on all the alternative approaches will be necessary to iden-tify those that appear to have the greatest validity across the wid-est spectrum of applications. To the extent that some convergenceis reached in the results of various imputation strategies,




decisions concerning which approach is preferable in any givensituation could depend largely on their ease of implementationand transparency to other researchers and their audiences.Although the imputation procedure proposed here does requiresome cumbersome programming, the underlying strategy is intu-itive and manipulations required to implement it are readilyapparent and do not require specialized statistical software.Another advantage is that imputations for missing offender char-acteristics, in addition to victim-offender relationship, are allmade simultaneously within the same operation.

To further assess the validity of current missing dataadjustment/imputation approaches and improve the accuracy ofthese techniques, it would be useful to conduct and/or furtheranalyze some large-scale follow-up studies to determine theactual characteristics of events that are initially unknown andtherefore not recorded in the SHR. Information gained from suchefforts could be extremely helpful in empirically characterizingthe inherent differences that may exist between cases that initiallycontain missing data and those that do not. This informationcould be incorporated into whatever adjustment procedures areused (whether they involve imputation, reweighting, or othertypes of statistical adjustment) to reduce any systematic biascaused by these differences. For the procedure described above,this would most likely involve adjusting the matrix of probabili-ties used to impute missing data so that it would more accuratelyreflect the characteristics of incidents that tend to initially containmissing information.1 Another step that would be useful, and hasbeen advocated for some time (e.g., Williams & Flewelling, 1987)but without any apparent success, is a more concerted effort toupdate SHR records with new information as it becomes availablefrom the reporting jurisdictions.

NOTE

1. Such an adjustment could be implemented by applying differential weights to thehomicide records prior to calculating the distributions used to specify the conditionalprobabilities. Specifically, cases with combinations of offender characteristics and victim-offender relationship that are more prevalent among records that have missing data wouldreceive higher weights.




REFERENCES

Allen, T., & Buckner, T. (1997). A graphical approach to analyzing relationships betweenoffenders and victims using the Supplementary Homicide Reports. Homicide Studies, 1,129-140.

Flewelling, R. L. (1996). Exploring the recent surge in youth homicide rates: Geographicvariations. In Proceedings of the 1996 Meeting of the Homicide Research Working Group.Washington, DC: National Institute of Justice.

Flewelling, R. L. (1997a, April 24-25). Issues of non-reporting agencies and missing data in theSupplemental Homicide Report. Presented at the Workshop on UCR Imputation Proce-dures, Bureau of Justice Statistics.

Flewelling, R. L. (1997b, November). Recent trends in youth homicide perpetration: An explor-atory analysis of underlying trends and geographic variability. Presented at the annual meet-ing of the American Society of Criminology.

Flewelling, R. L., & Williams, K. R. (1999). Categorizing homicides: The use of dis-aggregated data in homicide research. In D. Smith & M. Zahn (Eds.), Homicide studies: Asourcebook of social research (pp. 96-106). Thousand Oaks, CA: Sage.

Fox, J. A. (1997, April 24-25). Missing data problems in the Supplementary Homicide Reports: Anoffender imputation strategy based on victim characteristics. Paper presented at the Work-shop on UCR Imputation Procedures, Bureau of Justice Statistics.

Fox, J. A. (2001). Uniform Crime Reports: Supplementary Homicide Reports, 1976-1999(ICPSR Version) [Data file]. Boston: Northeastern University, College of Criminal Jus-tice [Producer]. Ann Arbor, MI: Inter-university Consortium for Political and SocialResearch [Distributor].

Langford, L., Isaac, N., & Kabat, S. (1998). Homicides related to intimate partner violence inMassachusetts: Examining case ascertainment and validity of the SHR. Homicide Stud-ies, 2, 353-277.

Messner, S. F., Deane, G., & Beaulieu, M. (2002). A log-multiplicative association model forallocating homicides with unknown victim-offender relationships. Criminology, 40,457-480.

Regoeczi, W. C., & Miethe, T. D. (2003). Talking on the unknown: Aqualitative comparativeanalysis of unknown relationship homicides. Homicide Studies, 7, 211-234.

Regoeczi, W. C., & Riedel, M. (2003). The application of missing data estimation models tothe problem of unknown victim/offender relationships in homicide cases. Journal ofQuantitative Criminology, 19, 155-183.

Williams, K. R., & Flewelling, R. L. (1987). Family, acquaintance, and stranger homicide:Alternative procedures for rate calculations. Criminology, 25, 543-560.

Robert L. Flewelling, Ph.D., is a senior research scientist with the Pacific Insti-tute for Research and Evaluation (PIRE) and is located in PIRE’s Chapel HillResearch Center in North Carolina. He received his doctorate in public health fromthe University of North Carolina in 1991. Although previous research pursuitshave included methodological issues in comparative homicide research, his cur-rent focus is primarily on epidemiological studies of adolescent high-risk behav-iors, prevention program evaluation and community needs assessment, and theapplication of research findings to prevention system development and programselection.




Documents

A Nonparametric Imputation Approach for Dealing With Missing Variables in SHR Data