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September 2015Volume 25, Issue 9, Pages 668–673.e2
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U.S. spousal homicide rates by racial composition of marriage

John M. Roberts Jr
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John M. Roberts
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Correspondence

  • Corresponding author. Department of Sociology, University of Wisconsin-Milwaukee, PO Box 413, Milwaukee, WI 53201. Tel.: +1-414-229-6944; fax: +1-414-229-4266.
, PhD'Correspondence information about the author PhD John M. Roberts
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John M. Roberts
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Correspondence

  • Corresponding author. Department of Sociology, University of Wisconsin-Milwaukee, PO Box 413, Milwaukee, WI 53201. Tel.: +1-414-229-6944; fax: +1-414-229-4266.
Department of Sociology, University of Wisconsin-Milwaukee, Milwaukee
U.S. spousal homicide rates by racial composition of marriage
DOI: https://doi.org/10.1016/j.annepidem.2015.04.004
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Abstract

Purpose

American spousal homicide rates persistently and substantially vary by racial composition of the married couple. Analyses examined different racial couple types' spousal homicide rates in light of nonspousal homicide victimization and offending rates and couple types' average social, demographic, and economic characteristics.

Methods

Analyses used 2003 to 2007 spousal homicide data from Supplementary Homicide Reports for which missing data have been multiply imputed. Current Population Survey data provided estimates of the number and average characteristics of different couple types. Log-linear models related couple types' differing spousal homicide rates to different race-sex groups' general rates of homicide victimization and offending and couple types' average characteristics.

Results

Among couple types with at least 50,000 couples, annual rates of male-on-female spousal homicide ranged from 0.95 to 8.76 per 100,000 couples; for female-on-male spousal homicide, this range was 0.13 to 2.29. Rates somewhat reflect different race-sex groups' nonspousal homicide activity, but with greater gender disparity and an excess of spousal homicide in some couple types. The association between victim's and offender's race is parsimoniously described by models using couple types' average characteristics (proportion with female's education exceeding the male's, proportion in central cities, and relative frequency).

Conclusions

General homicidal-violence reduction strategies may partly apply to spousal homicide, but specifically targeted efforts are required too. Interventions must address different couple types' particular social, economic, and cultural experiences.

Keywords:

Homicide, Spouses, Health status disparities, United States

Introduction

Homicide between heterosexual spouses is a long-standing international public health concern [[1], [2], [3], [4], [5], [6], [7]]. In the United States, spousal homicide is a substantial component of intimate partner homicide [[8], [9]], with differences between legal marriages and informal cohabitations in homicide rates and characteristics likely narrower than before [[10], [11]]. Recent decades have seen important changes in American spousal homicide's scope and nature, including reduced overall rates and increased ratios of female to male victims [12]. However, one persistent feature is that spousal homicide rates still vary substantially across different combinations of husband's and wife's race. For example, 1979 to 1981 estimates for Chicago and the entire United States indicated that women's spousal homicide victimization rate in both-black marriages was about four to seven times greater than women's victimization rate in both-white marriages [[1], [13]]. Newer data (2003–2007) in the following text show a similar difference. Understanding and responding to the spousal homicide problem requires deeper investigation of these disparities.

This article considers American rates of spousal homicide in couples of different racial compositions, describing these rates and reporting on statistical analyses addressing two key questions regarding spousal homicide's racial disparities. First, do homicide rates in different kinds of couples reflect general rates of homicide offending and victimization in the partners' groups? Race-sex groups differ substantially in their nonspousal homicide offending and victimization rates, and perhaps spousal homicide's observed variability by racial combination is simply a manifestation of these more general differences. If not, it is important to examine how spousal homicide rates in different couple types depart from overall patterns of homicide involvement for different race-sex groups. Second, what is the nature of the association between victim's and offender's race in spousal homicide? Women in different racial groups have different patterns of spousal homicide victimization and offending across the various categories of husband's race. How are these patterns structured, and do they reflect variability in the aggregate social, demographic, and economic characteristics of different racial couple types? Analyses for this research question explore whether such factors can account for the observed association between victim's and offender's race.

Methods

Data

The Supplementary Homicide Reports (SHRs) of the Federal Bureau of Investigation’s Uniform Crime Reporting System are a key source of incident-level homicide data in the United States [14]. Participating police agencies report information on each of their jurisdiction's homicides, including, if known, the victim's and offender's demographic characteristics and the victim-offender relationship, and incident circumstances. Agencies' participation in the Reporting System is voluntary, but very widespread: In 2007, participating police agencies covered about 95% of the U.S. population [15]. SHR data therefore intend to provide a virtually complete record of homicides known to police in the United States, and national homicide rates derived from SHR can be adjusted for this minor undercoverage [16]. In this respect, SHR can reasonably be regarded as population data rather than a sample, and its national time trend corresponds closely with the homicide trend implied by mortality records [17]. SHR has long been widely used in criminological research [18].

Still, it has also long been recognized that SHR data are imperfect [19]. Some apparently participating agencies report data to the Federal Bureau of Investigation only intermittently, or incompletely [[16], [18]]. Research in Massachusetts comparing SHR and other data sources found that some intimate partner homicides were not represented in SHR at all [20], although matching can also fail due to data entry discrepancies. This and other work also found instances of miscoding of victim-offender relationship, or of classification of intimate partner incidents as unknown victim-offender relationship, in raw SHR data [[20], [21], [22]]. Offender's race is also subject to misclassification or missingness in SHR [[20], [23]]. In data of national scope, there is no feasible strategy for identifying and correcting misclassification of victim-offender relationship or race, but missing (unknown) data can be addressed with imputation strategies. This is important because victim-offender relationship was missing in 45% of 2003 to 2007 SHR cases, and offender's race was missing in 34%.

In Fox and Swatt's version of 1976 to 2007 SHR data, missing data on all variables have been multiply imputed [24] to create five completed SHR data sets [[16], [25]]. Under the multiple imputation method, the imputed value for a given missing observation could differ across the different imputed data sets. As some incidents' data therefore vary across the five completed SHR data sets, analysis of multiply imputed data must address this variability [24]. I extracted completed data for 2003 to 2007, noting the victim's sex and the victim's and offender's race (white, black, Native American, Asian or Pacific Islander, or other; SHR data do not indicate Hispanic ethnicity) for homicides between heterosexual spouses. Fox and Swatt's imputed data categorized some variables more coarsely than in the original SHR, as successful imputation can be challenging when variables include relatively rare categories [16]. The coarser “intimate” category grouped “spouse” with categories such as “boyfriend/girlfriend” in the imputed data, and the coarser racial categorization was {white, black, other}. The present study required identification of spousal homicides in the different racial combinations, which in turn required some of this lost detail on relationship and race. I adjusted imputed values to distinguish spouses from others (boyfriends, girlfriends, or ex-spouses) in the “intimate” category and Asians from Native Americans in the “other” race category. (The Supplementary Material describes this in detail.) This yielded tables of spousal homicide counts classified by victim's and offender's sex and race; each of the five completed data sets yielded such a table, for five in all. The total numbers of spousal homicides in these tables ranged from 3983 to 4053. Imputed values of “spouse” for missing victim-offender relationships contributed about 19% of these totals, with the remaining 81% identified as spousal homicides in the original SHR data.

I obtained the numbers of married couples in different racial combinations by averaging national estimates from the March 2003, 2005, and 2007 Current Population Surveys (CPSs), excluding couples in which a spouse reported a multiracial identity. CPS data also provided estimates of aggregate characteristics of different couple types. I considered six characteristics suggested by previous research as potentially related to aspects of spousal or intimate partner homicide [[26], [27], [28]]. These included (i) the proportion of couples with male aged 30 years or younger, (ii) the proportion in which the male had not completed secondary education, (iii) the proportion in which the male's education was less than the female's, (iv) the proportion living in central cities, and, as an economic measure, (v) the proportion whose household's received public assistance food stamps. (I smoothed the estimated proportions from CPS because some racial combinations have few sampled couples.) Also, I measured (vi) a couple type's relative frequency by comparing its observed number of couples to that expected solely from the different racial groups' total numbers of married men and women. Excesses or deficits were represented by interaction terms in a saturated log-linear model [29] for the marriage data. From the entire 2003 to 2007 imputed SHR data, I counted nonspousal (i.e., excluding spousal incidents) homicide victimizations and offenses for each racial group's adult (18 years and older) men and women and transformed the counts into rates via census estimates of the various racial groups' adult (18 years and older) male and female populations. The Supplementary Material gives more details on these data.

Statistical analysis

I produced a descriptive table of spousal homicide rates in different couple types, based on average counts in the five data sets created by multiple imputation of missing data. For statistical analyses, I applied the Poisson log rate models [29] in the following text to spousal homicide count tables. Models address the two main research goals by relating nonspousal homicide victimization and offending rates to different couple types' spousal homicide rates and by examining the association between victim's and offender's race.

Analyses must consider not one but five tables of spousal homicide counts, from the five data sets created by multiple imputation. I assessed model fit via Meng and Rubin's method [30] for P-values from analysis of the five completed data sets, also adapting it for comparing nested models. (Tables report models' average likelihood ratios across these five data sets.) Unlike the usual case of incompletely classified data, here the imputed tables have slightly varying totals, from variability in multiple imputation of missing victim-offender relationships. However, an adjustment for this (Supplementary Material) made no practical difference.

Models for the first research question consider spousal homicide patterns in light of race-sex groups' general homicide victimization and offending rates. hij represents the expected number of spousal homicides involving a (race-sex) category i victim and a category j offender, whereas Nij represents the total number of type {i, j} married couples (Nij = Nji). The exposure (offset) term log Nij implies a “log rate” interpretation of the models [29]. Baseline model 0 unrealistically expects the same rate in every couple type:

(0)Math Eq

More plausibly, spousal homicide patterns might reflect groups' general homicide rates. Let vi and oj represent estimated nonspousal homicide victimization and offending rates, for adults in race-sex categories i and j, with log(vi oj) the logged product of these rates and d(vi oj) the deviation of log(vi oj) from the mean over all heterosexual combinations. In model 1a, parameter Math Eq relates d(vi oj) to spousal homicide counts:

(1a)Math Eq

Model 1a should fit the data (with Math Eq > 0) if spousal homicide patterns essentially reflect groups' general homicide victimization and offending, but it may inadequately account for spousal homicide's sex direction disparity. Model 1b's sex direction parameter addresses disparity beyond that implied by general victimization and offending rates alone (ordering categories as in Table 1, so i > j for male-on-female cells and i < j for female-on-male):

(1b)Math Eq

Table 1U.S. annual spousal homicide rates (per 100,000 couples), 2003 to 2007
VictimOffender
MaleFemale
WhiteBlackNative AmericanAsianWhiteBlackNative AmericanAsian
Male
 White————0.272.291.010.13
 Black————1.381.522.15∗0∗
 Native American————0.140∗1.054.68∗
 Asian————0.133.76∗0∗0.19
Female
 White0.988.760.951.48————
 Black4.813.811.91∗0.86∗————
 Native American1.423.00∗2.0616.04∗————
 Asian0.984.64∗0.84∗0.75————
View Table in HTML
∗Couple types estimated to have fewer than 50,000 married couples.

If model 1b does not fit, spousal homicide patterns go beyond general homicide rates and sex direction disparity. Absent specific theoretical guidance, one could consider model 1b's cell residuals, and highlight cells in which model 1b substantially underpredicts spousal homicide. Model 1c adds parameters for both sex directions in the following four offender-on-victim racial combinations: black-on-white, white-on-black, black-on-Asian, and Asian-on-Native American. With categories numbered to follow Table 1's order, this model is:

(1c)Math Eq

Finally, model 1c's residuals motivated model 1d, whose additional term Math Eq highlights sex direction difference in black-on-black cells.

The second set of models examines association between victim's and offender's race. hij is considered in light of category i's total number of spousal homicide victimizations and category j's total number of spousal homicide offenses (and the exposure term log Nij). Model 2 implies independence (no association) between victim's and offender's race:

(2)Math Eq

Its χ2 value represents total observed association between victim's and offender's race.

This association's nature could be the same for male-on-female and female-on-male spousal homicide, even with the former's greater frequency. Under quasi-symmetry (model 3), the association (net of marginal totals) is completely symmetric with respect to sex direction:

(3)Math Eq

Parameters Math Eq and Math Eq fit the table's marginal totals, and interaction parameters are symmetric: Math Eq (with appropriate constraints) [29]. The symmetry means that net of the marginal totals, the relative frequency of, say, Asian-female-on-white-male spousal homicide (cell 1, 8) is equivalent to that of white-male-on-Asian-female (cell 8, 1). Under model 3, a couple type's net spousal homicide excess or deficit is evident in both sex directions.

The structure of this symmetric association may reflect couple types' average characteristics. Suppose Math Eq represents average characteristic k of couples of type (i, j) in the general population, centered to show deviations from the characteristic's mean across all couple types (note that Math Eq). Model 4a uses the following six such centered characteristics: proportion with male under 30 years old, proportion with male not completing secondary school, proportion with male less educated than female, proportion living in central city areas, proportion receiving food stamps, and the couple type's relative frequency.

(4a)Math Eq

Model 4b is empirically motivated and more parsimonious, using only the male-female education, central city, and relative frequency measures.

Results

Table 1 summarizes observed (unsmoothed) annual spousal homicide rates, 2003 to 2007. (Rates include minor adjustments for SHR undercoverage [16]; analyses in the following text did not.) The lower left block reports male-on-female rates, and the upper right female-on-male. Certain uncommon couple types had no observed or imputed spousal homicides, and in two types—{Native American male, Asian female}, {Asian male, black female}—female-on-male rates exceeded male-on-female, although these rare types' (6433 and 13,085 estimated couples, respectively) rates involved small numerators and denominators.

Table 2 summarizes results of models linking spousal homicide counts to race-sex groups' general homicide victimization and offending rates. Model 0 naturally fit poorly (P < .001; model “fit” refers to comparing the named model [H0] to a saturated model [H1] that exactly reproduces the observed data [29]). Model 1a substantially and significantly (P < .001) improved on model 0. (In nested model comparison, H0 represents the model with fewer parameters, and H1 the model with more [29].) Still, model 1a fit poorly (P < .001). Although spousal homicide patterns do partly reflect general homicide rates (Math Eq = 0.47), model 1a's nonfit indicates that such rates are insufficient to satisfactorily account for spousal homicide patterns.

Table 2Fit of models relating spousal homicide to general homicidal violence
ModelMean G2 over five imputed data setsDegrees of freedomModel fit∗Comparison to preceding model†
0: Expected rates same in all couple types and sex directions2993.631P < .001; does not fit—
1a: Expected rates reflect nonspousal homicide offending and victimization rates949.430P < .001; does not fitP < .001; prefer model 1a
1b: Adds a parameter to model 1a accounting for excess of male-on-female spousal homicides289.929P < .001; does not fitP < .001; prefer model 1b
1c: Adds parameters to model 1b accounting for excess of black-on-white, white-on-black, black-on-Asian, Asian-on-Native American spousal homicides51.425P = .22; does fitP < .001; prefer model 1c
1d: Adds a parameter to model 1c for sex direction in black-on-black spousal homicide31.724P = .86; does fitP < .001; prefer model 1d
View Table in HTML
∗Compares named model [H0] to saturated model [H1].
†[H0] corresponds to model with fewer parameters, [H1] to model with more, so comparisons are model 0 [H0] versus model 1a [H1], model 1a [H0] versus model 1b [H1], model 1b [H0] versus model 1c [H1], and model 1c [H0] versus model 1d [H1].

Model 1b's additional parameter for capturing the excess of male-on-female incidents significantly improved on model 1a (P < .001), but model 1b still did not fit (P < .001). Model 1c added parameters applying to both sex directions in four racial combinations in which model 1b underpredicted spousal homicide; it improved on model 1b (P < .001) and appeared to fit the data (P = .22). Adding parameters to model 1c may thus be overfitting; still model 1d's highlighting of sex direction difference in black-on-black cells improved fit (P = .001). (This parameter's estimate indicated more black-female-on-black-male spousal homicide than otherwise expected, although the whole model still implied a much higher black-male-on-black-female rate.) Although nonspousal homicide rates help account for spousal homicide, the remaining excess of (i) male-on-female incidents and (ii) spousal homicides in certain racial combinations must also be considered. In assessing model fit, note the caveat of some small cell counts [29].

Table 3 reports results for the second set of models examining association between victim's and offender's race. Model 2's poor fit (P < .001) suggested meaningful association in the table. Model 3, which fully accounts for symmetric association, immensely reduced model 2's chi-square value, and fit well (P = .50), indicating that the table's association is mostly symmetric with respect to sex direction. A couple type's excess (or deficit) of spousal homicide appears in both sex directions.

Table 3Fit of models for association between victim's and offender's race in spousal homicide
ModelMean G2 over five imputed data setsDegrees of freedomModel fit∗Comparison to preceding model†
2: No association (independence) between victim's and offender's race138.018P < .001; does not fit—
3: Sex-symmetric association accounted for completely (quasi-symmetry); net of margins, same racial pattern for both sex directions18.09P = .50; does fitP < .001; prefer model 3
4a: Sex-symmetric association, but via terms for six aggregate characteristics of couple types‡19.512P = .72; does fitP = .96; prefer model 4a
4b: Sex-symmetric association, but via terms for only three aggregate characteristics of couple types§26.315P = .66; does fitP = .33; prefer model 4b
View Table in HTML
∗Compares named model [H0] to saturated model [H1].
†[H0] corresponds to model with fewer parameters, [H1] to model with more, so comparisons are model 2 [H0] versus model 3 [H1], model 4a [H0] versus model 3 [H1], and model 4b [H0] versus model 4a [H1].
‡The six characteristics were proportion in which the male is under 30 years old, proportion in which the male did not complete secondary school, proportion in which the male's education is less than the female's, proportion receiving food stamps, proportion living in central city areas, and the couple type's relative frequency.
§The three characteristics were proportion in which the male's education is less than the female's, proportion living in central city areas, and the couple type's relative frequency.

Model 4a used the six couple-type characteristics. It is more parsimonious than quasi-symmetry (model 3), yet fit as well (P = .96 for comparing model 4a [H0] to model 3 [H1]). These six couple-type characteristics accounted well for the spousal homicide table's sex-symmetric association. In fact model 4b, using only the male-female education, central city, and couple-type relative frequency measures, fit as well as model 4a (P = .33 for comparing model 4b [H0] to model 4a [H1]) and was preferred for its parsimony. Parameter estimates indicated higher spousal homicide rates, net of the marginal totals, among couple types with greater proportions of marriages in which the female's education exceeds the male's, greater proportions living in central cities, and lower relative frequency. Model 4b's residuals did indicate underprediction of observed Asian-male-on-Native American-female spousal homicides, but recall the rarity of this couple type. That rarity means that even a small number of homicides creates a high observed rate.

Discussion

Parsimonious models accounted well for the variability in spousal homicide across different couple types and sex directions. Results indicate that spousal homicide rates partly reflect different sex-race groups' rates of nonspousal homicide victimization and offending, and modeling this factor goes far in accounting for observed spousal homicide patterns. But general rates of homicidal violence importantly fail to capture the observed preponderance of male-on-female spousal homicide, and certain couple types' rates meaningfully depart from what would be expected from nonspousal homicide rates. These results speak to debates on spousal (more broadly, intimate partner) homicide's distinctiveness from general homicide [[31], [32], [33], [34]]. The similarity of spousal homicide's race-sex patterns to those of other homicidal violence was substantial but incomplete, so spousal homicide patterns retained important distinctiveness. This mixed similarity and distinctiveness suggests that policy measures intended to reduce general violence may help address spousal homicide, but efforts specifically targeted toward spousal (intimate partner) homicide are needed as well. Although there are inherent challenges in evaluating intimate partner violence prevention programs [35], traditional “batterer intervention” programs show little effectiveness; counseling and other family interventions, improved victim services, education promoting positive relationships, and even macrointerventions such as advertising campaigns are likely more promising [[36], [37], [38], [39]].

Second, analyses showed that the association between victim's and offender's race in spousal homicide was substantially symmetric with respect to sex direction (male-on-female or female-on-male). The table's marginal totals reflect male-female differences and different groups' differing overall involvement in this crime. But beyond the marginals' impact, the remaining pattern of racial combinations' high or low spousal homicide rates is essentially equivalent for the two sex directions. Furthermore, a parsimonious specification of aggregate characteristics (here, including measures of education difference, typical location, and marriage relative frequency) can account for this pattern.

Findings for the education and relative frequency measures were particularly notable. For education, male “status frustration” has long been suggested as a factor in male-on-female intimate partner violence [[8], [40]], but recall that the model here implied a sex-symmetric relationship with the education measure, so that couple types in which the wife's education more often exceeds the husband's have relatively more spousal homicide in both sex directions. Criminological theory emphasizes strain in explaining criminal behavior [[41], [42]]; here, life in couples with this education profile may be subject to strains that produce higher spousal homicide rates. For the relative frequency measure, rarer (relative to group sizes) racial combinations experienced higher spousal homicide rates. If a couple type's rarity indicates a kind of inconsistency with the dominant culture's expectations, such couples may experience more violence through mechanisms like those suggested in anthropological studies linking culturally “consonant” or “inconsonant” lifestyles to health [[43], [44]]. Urban residence may be associated with various criminogenic factors targeted by general social or economic policy initiatives. But results for educational difference and relative frequency suggest that different couple types' specific social and cultural experiences also must be addressed in interventions attacking spousal (and other intimate partner [45]) violence. This reinforces the importance of effective “cultural tailoring” as a current focus of intervention research [36].

Limitations

Analyses rely on the quality of the underlying SHR data, so the SHR concerns previously mentioned are particularly important. Of course, these concerns would not necessarily affect different couple types and/or sex directions in a way that distorts the analyses here. A related limitation is the mismatch of SHR's single-race classification of victims and offenders with the multiracial CPS and census scheme. The descriptive Table 1 and the models' exposure terms used CPS marriage estimates for single-race individuals, and this could disproportionately affect apparent spousal homicide rates in different couple types. For example, if SHR data tend to categorize biracial individuals whose identity includes black as “black,” rather than by their racial identity's other component, this would inflate estimated spousal homicide rates in couples that include a black partner relative to couples that do not. CPS estimates used here indicated relatively few multiracial spouses, but this will be increasingly important in future data. SHR's failure to report on Hispanic ethnicity is also unfortunate.

The six characteristics used in analyzing association in the spousal homicide table clearly do not exhaust the potentially relevant factors. For instance, other possible dimensions of socioeconomic standing include aggregate measures of income, employment, occupational prestige, and household poverty status in different couple types. In practice, many of these aggregate measures may be sufficiently similar as to produce essentially equivalent results to those reported previously, but in any case CPS data offer a wide variety of measures for analysts to explore.

These limitations suggest important directions for new data collection and research on spousal homicide. A variety of other log-linear models could be fit to these data, and different statistical perspectives can inform other modeling approaches that would support other research questions related to race and spousal homicide. Nonetheless, results here provide important insights into the nature of America's spousal homicide disparities.

Supplementary material

More detailed imputation of victim-offender relationship and race

As discussed in the text, the imputed data placed spouse in a more inclusive “intimate” category, and race was coarsely categorized as {white, black, other}. However, my study required that the imputed data distinguish spouses from other relations in the “intimate” category (such as boyfriend-girlfriend) and distinguish Asians and Native Americans within the “other” race category.

To address this, I first identified all incidents with a known (not imputed) intimate relationship between victim and offender. Combining cases from all five completed data sets for this subset of incidents, a logistic regression modeled the odds that the victim-offender relationship was a spousal rather than a nonspousal intimate relationship. Predictors of spousal relationship in this model included victim's and offender's age {under 25, 25 to 34, or over 34}, sex, and race {white, African-American, other}, whether the incident took place in an urban area, its region {E, MW, S, W}, whether a firearm was used, and the incident's circumstances {felony, argument, other}. These predictor variables included both known and imputed values, and for some variables, certain categories were combined to avoid estimation difficulties. The next step turned to the incidents in each completed data that had missing victim-offender relationship in the original data and an imputed intimate relationship requiring further detail. For each such incident, I calculated an estimated probability of the victim and offender having a spousal rather than other intimate relationship, based on the estimated parameters from the logistic regression. Finally, I used this estimated probability when randomly designating each of this set of incidents as spousal or not. I used a similar procedure for cases in which victim's or offender's race was imputed as “other,” with appropriate changes in the predictors listed previously.

Values imputed this way should still be drawn from the approximately correct joint posterior distribution for the multiple imputation [1], because of the use of imputed as well as observed values in the creation of the estimated probabilities. Note that the number of spousal homicides imputed through this approach was not trivial: Across imputations, an average of 779.8 spousal homicides were imputed this way, compared with 3247 known spousal homicides, so that roughly one in five of the spousal homicide cases analyzed here were designated as such from imputation. This is consistent with research indicating that classification of actual family homicides as “unknown relationship” in SHR does occur [[2], [3], [4]].

Smoothed proportions of various characteristics in different types of couples

Analyses used aggregate characteristics of married couples in different racial combinations, such as the proportion of couples in which the male was 30 years old or younger. Such proportions can be estimated from CPS data, with weights incorporated to obtain population estimates. Some of these estimated proportions rely on observations from very few sampled couples, given the rarity of some racial combinations among married people in the CPS samples. Therefore, I smoothed estimated proportions by beta-binomial shrinkage, in which binomial success probabilities are assumed to be drawn from a beta distribution [5]. For this smoothing, the racial combinations' observed (including weights) proportions were viewed as independent binomial proportions, and the (unweighted) number of couples in the different combinations as the sample sizes. The smoothing used an empirical Bayes approach in which parameters of the beta distribution were estimated from the mean and variance of the set of observed proportions. The implied posterior expected values for the binomial parameters provided the smoothed estimates. The smoothing pulls the observed proportions toward the mean of the set, with a stronger pull for the combinations that are rarer in the CPS sample.

Relative frequency of different couple types

Models 4a and 4b in the main text included a measure of a couple type's relative frequency in contemporary American society. The raw numbers of marriages observed (or estimated) in different couple types are not satisfactory because the different racial groups vary substantially in size, and numbers of different types of marriages naturally reflect these group size differences to a great extent. This means that a relative frequency measure needs to account for the overall number of married men and women from different groups. One such measure is given by the interaction terms in a saturated log-linear model for the number of marriages in the different couple types. The saturated model uses as many parameters as there are meaningful cells in the data table, and so reproduces the observed data exactly and leaves no degrees of freedom.

Writing Nij for the CPS estimate of the number of marriages of type {i, j}, the saturated model is

(1)Math Eq

With analysis of variance–type constraints on these parameters (and no zero values among the Nij), the Math Eq terms report the deviations of average logged cell counts in each row from the whole table's average logged cell count Math Eq, and Math Eq terms do the same for the columns. In this way the Math Eq and Math Eq terms reflect the overall number of married men and married women in each racial group. The interaction terms Math Eq therefore indicate relative frequency of different couple types, as they show differences between log Nij and the combination of the table, row i, and column j averages of logged cell values (i.e., differences between the observed–here, estimated from CPS–number of marriages in the various couple types and the group size terms). This construction of the interaction terms means that they will sum to zero and so are similar to the deviation form used for other couple-type characteristics Xij in models 4a and 4b.

Estimating nonspousal homicide victimization and offending rates

Some analyses used nonspousal homicide victimization and offending rates for adults (18 years and older) in the different sex-race groups for the period 2003 to 2007. To calculate such victimization rates, I first excluded spousal homicide victims (including those in which the relationship was imputed as spousal, as described previously) as well as all nonadult victims in SHR. I identified the remaining nonspousal homicide victims by race and sex, including the imputation discussed previously of Asian or Native American race for any victims whose imputed race was “other.” The resulting counts of nonspousal homicide victims by race and sex were averaged over the five imputed SHR data sets.

For denominators in the victimization rates, I averaged annual published census estimates of the size of the 18-and-over population in each race-sex group for 2003 to 2007. Only single-race populations were counted, with the Hawaiian and Pacific Islander category merged with Asian, and the two-race category of “Asian and Hawaiian/Pacific Islander” also included in this merged category. Published census materials do not directly report age-sex population estimates for multirace groups, but undifferentiated (by age and sex) estimates for the size of the two-race “Asian and Hawaiian/Pacific Islander” group and the single-race “Hawaiian/Pacific Islander” group are available. I applied the ratio of these two group sizes to the age-sex differentiated estimates for “Hawaiian/Pacific Islander” and took these as age-sex differentiated estimates for the “Asian and Hawaiian/Pacific Islander” group. I added the resulting figures added to those for the merged Asian category.

A similar procedure gave nonspousal adult offending rates in the different race-sex groups, again using imputed SHR data and census population estimates.

Adjusting P-values for differing totals in imputed data tables

In the cross-classified table context, imputation theory focuses on incompletely classified data [[6], [7]]. That is, some observations have missing values for certain variables, so an imputation method distributes these incompletely classified observations to cells. When multiple imputation is applied in that situation, some cell counts in the resulting cross-classification tables will likely differ across imputations, but the tables' overall totals will be the same. For the tables here, however, that is not the case. Fox and Swatt's imputation [1] was carried out on the original homicide data involving all victim-offender relationships, but the present analyses included only the subset of spousal homicides. Different imputed data sets will include different numbers of incidents in which a missing victim-offender relationship was imputed as “spouse,” and therefore, totals in the spousal homicide data tables will vary across the different imputed data sets. This could potentially affect the application of methods designed for the standard situation, and I am not aware of any formal work by statisticians addressing this possibility.

For the tables here, this is likely not an important matter, as the difference between the largest and smallest totals in the five imputed spousal homicide tables was quite small (4053 vs. 3983, or <2% difference). Still, it seemed valuable to at least explore some ad hoc adjustment of P-values for this situation. Meng and Rubin's method [7] is based on likelihood-ratio model fit statistics. It is well known that these statistics are affected by a table's total count; for instance, doubling all the cell counts in a table will double the obtained model fit statistics, even though the table's pattern of association does not change. This suggests an ad hoc adjustment for the present case: Before using the different imputed tables' fit statistics in Meng and Rubin's method, deflate each imputed table's fit statistic by the ratio of that table's total to the minimum total (here 3983) of the five imputed tables. This adjustment lacks a formal justification, but still embodies an intuitive sense that the differing totals should be equalized at their lowest value.

As already noted, here the imputed tables actually had very similar totals, so it is not surprising that this ad hoc adjustment had essentially no impact. No conclusions from assessment of model fit or comparison of nested models changed, and any differences in calculated P-values under the adjustment were quite small. Still, this situation may be an interesting area for future investigation by theorists of missing data imputation (again assuming that there is no existing treatment of it in the literature), so that a fully principled approach could replace the ad hoc adjustment here.

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References

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The author received no financial support for this research, and there are no potential conflicts of interest.

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September 2015Volume 25, Issue 9, Pages 668–673.e2