Wellford C.S., Pepper J.V. - Firearms and Violence[c] What Do We Know[q] (2005)(en)

168	FIREARMS AND VIOLENCE

cumstances, proxies reveal the sign but not the magnitude of the relationship of interest (Krasker and Pratt, 1986; Maddala, 1992). Azrael et al. (2004) attempt to provide some insight into this scale problem by running a simple linear regression of the form:

PREV = β0 + β1FS/S + U,

where PREV is the true ownership rate, FS/S is the observed proxy, β0 and β1 are unknown coefficients, and U is a mean zero unobserved random variable, conditional on FS/S. The estimated slope coefficient is near unity, suggesting that a one-unit increase in FS/S implies a one-unit increase in the expected prevalence rate. The authors take this result, coupled with the strong cross-sectional correlation coefficients, as evidence supporting the idea that the FS/S proxy leads to (nearly) unbiased estimators of both the sign and the magnitude of the relationships of interest.

This logic, however, could be misleading. In the classical omitted variable model described by Wooldridge (2000:284-286), a unit coefficient on β1 is sufficient. In other models, however, unbiased estimators may not exist. It is difficult to assess whether these conditions result in an unbiased estimator since Azrael et al. (2004) do not clearly describe the model they have in mind.6 This problem becomes particularly important when FS/S is being used as a proxy in the study of suicide, and it seems to be an important source of misunderstanding. For example, Miller et al. (2002a, 2002c) assess the potential biases created by the FS/S proxy in the study of suicide, using statistical simulations. These authors claim to demonstrate that FS/S is not, by construction, correlated with the overall suicide rate, so that FS/ S may be appropriately used as a measure of gun ownership in such a study. However, they do not explicitly describe their statistical model, and their description of the Monte Carlo simulation does not provide enough information to understand much about what was done. Furthermore, it is not

6No one, as far as we can tell, has investigated the actual linear or nonlinear shape of the relation between FS/S and gun ownership. Furthermore, Azrael et al. do not consider issues associated with the statistical error of the model. Suppose instead that we consider another linear model, in which the gun suicide rate is a function of the gun ownership prevalence: FS/S = g0 + g1PREV + V, with V being a mean zero unobserved random variable, conditional on PREV (see, for example, Duggan, 2003). Indeed, this model may be more plausible if one believes that gun ownership is a causal factor in firearm-related suicides. And, if this were correct, then in models of the relation between suicide and the FS/S proxy, the explanatory variable (FS/S) would be correlated with the regression “error,” a well-known cause of bias in regression analysis. In any case, the two models are not the same and do not have the same implications for the effects of using FS/S as a proxy. In the first model, the measurement errors are mean-independent of the proxy but not of the variable of interest, prevalence. In the second model, the measurement errors are independent of prevalence but not of the proxy.

FIREARMS AND SUICIDE

169

obvious why the simulation is at all relevant: the basic finding that proxies create biases is an analytical result that cannot be resolved by a simulation. It is very easy to create other plausible simulations that lead to substantial correlations between FS/S and suicide and, more importantly, substantial biases in the estimated relations of interest.

In Box 7-2, for example, we present the results of a simulation conducted by the committee. In this Monte Carlo simulation, we study the relation between the suicide rate and FS/S as a proxy for gun ownership, but we derive very different results than those reported by Miller et al. (2002a, 2002c). In particular, we find a negative association between the suicide rate and FS/S: in this simulation, if FS/S is a good proxy for ownership, gun owners are less likely than nonowners to commit suicide.

This exercise illustrates at least two things: (1) the design of the Monte Carlo simulation matters and (2) having suicide-related variables on both sides of the regression can produce perverse results. In the end, the biases created by proxy measures are application specific. Duggan (2003), for example, highlights the potential problems caused by using FS/S as an explanatory variable in a model whose dependent variable is also suicide-related. As demonstrated in the simulation above, unobserved factors associated with

BOX 7-2

Monte Carlo Experiment

There is not enough information available from the published Monte Carlo design (Miller et al., 2002a, 2002b) to enable someone to replicate it. However, the committee did a Monte Carlo experiment that implied quite different results. The

Monte Carlo simulates a study of the relation between the suicide rate and FS/S as a proxy for gun ownership. Let Z1, Z2, and Z3 denote unobserved independent standard normal variables, and let

FS = 10 + Z1;

NFS = 6 + Z2;

FS/S = FS/(FS + NFS);

POP = 50 + Z3; and

RATE = (FS +NFS)/POP,

where FS is the number of firearm suicides, NFS is the number of nonfirearm suicides, POP is the population size, and RATE is the total suicide rate for the population. With 1,000 replications, this design gave a mean value of FS/S in the neighborhood of 0.6 (similar to the fraction of suicides currently committed with a firearm in the United States). The correlation coefficient of FS/S and RATE was –0.29. The linear regression of RATE on FS/S gave a slope coefficient of –0.18 with a t-statistic of 9.6. So, according to this simulation, there is a negative association between the suicide rate and FS/S. In other words, if FS/S is a good proxy for ownership, gun owners are less likely than nonowners to commit suicide.

170	FIREARMS AND VIOLENCE

the measure of gun and nongun suicide (e.g., measurement error) may lead to purely spurious correlations between suicide and FS/S. Since suicide, S, is on both sides of the estimated equation, the implicit model is often a complicated, nonlinear relation between S and FS, not the linear model that is assumed in the literature. These issues may or may not be problematic when using FS/S to estimate the relationship between gun ownership and homicide.

Another important issue is how the proxy affects inference from specific models that may include other explanatory variables. This depends, among other things, on how true firearms prevalence and FS/S are related to the other observed and unobserved explanatory variables. These issues are complicated, and most of them have not been recognized, much less investigated, in the suicide and firearms literature.

Ecological Bias

All empirical studies face difficulties with making causal inferences, but ecological studies face special sources of bias in dealing with exposures and confounders. These difficulties arise because of the aggregation of observations and because the data on exposures, confounders, and outcomes are from different sources. At the most basic level, the data on firearms ownership in these studies may not come from the persons who committed suicide. Thus, ecological studies cannot establish whether there is a relation between gun ownership by an individual or household and suicide by that individual or member of the household. This may seem like a small problem in the case of gun suicide; after all, the victims of a gun suicide have undeniably achieved access to a gun. But community-level rates of gun ownership may not reflect the rates of gun ownership among highly suicidal persons. If, for example, the relationship between gun access and gun suicide varies by age and sex or by psychiatric disorder, then the aggregate association may reflect differences in the prevalence of suicidal states among persons of different age and sex or psychiatric disorder in the population, rather than differences in access to firearms. The geographical level of aggregation in state-level or regional ecological studies may be so high that there is no way of knowing whether the gun homicides or gun suicides occurred in the same areas with high levels of gun ownership.

Thus, even if FS/S is found to be a valid proxy for state-level gun prevalence, something that is not yet established, ecological studies may lead to biased inferences. The proxy is not a substitute for good data on household-level ownership or even ownership at a smaller level of aggregation by age, sex, or geography. Rather, better individual-level studies exploring the relationship between gun ownership and suicide may be needed in order to further understanding of the overall relationship between firearms and the risk of suicide.

FIREARMS AND SUICIDE

171

INDIVIDUAL-LEVEL STUDIES OF THE ASSOCIATION

BETWEEN FIREARMS AND SUICIDE

Most individual-level studies use case-control or response-based study designs to study rare events, such as completed suicide. However, the strengths and weaknesses of this study design are not well understood by investigators outside the public health community, and in order to clarify the controversy surrounding some of these studies, it may be helpful to describe the most important features of the case-control study design. Studies of the rates and determinants of illness or behaviors can be classified as retrospective or prospective. Prospective studies usually select people on the basis of exposure and determine how many persons with the exposure, compared with persons without exposure, develop a certain outcome. In contrast, retrospective studies usually start by choosing persons according to whether an illness or behavior has already developed and seek to find the phenomena that might be associated with the development of the outcome. Intuitively, it makes sense that if one is studying a rare outcome, then a prospective design is inefficient because it may take a very large sample or a very long time to accumulate enough occurrences. In this case, the casecontrol sampling design is beneficial because it oversamples the behavior or outcome of interest.

To investigate suicide, for example, a case-control study might select as cases those persons who have committed suicide, and then randomly select as controls a certain predetermined number of subjects from the same population who did not commit suicide. The study design would seek to establish an association between the outcome (suicide) and an exposure (such as firearms or depression) by noting the proportions of cases and controls that have been exposed to the possible risk factor.

There are a number of important advantages to the case-control method that explain its common use in epidemiology. Because the outcomes have already happened, case-control studies require no costly fol- low-up waiting for the outcome to develop. Because case-control studies oversample the outcome of interest, they also require smaller samples sizes than prospective studies of comparable power; for this reason, the case-control sampling scheme is often the only feasible way to collect the information of interest. For example, although suicide is the most common cause of firearm-related deaths in the United States, the overall suicide rate is approximately 11 suicides per 100,000 persons per year. Very few prospectively collected data sets would be large enough to draw precise inferences about completed suicide.

Feasible and efficient as the case-control design may seem, it also suffers from important limitations arising from the nonrandom selection of cases or controls and from misclassification of the outcome or exposure.

172	FIREARMS AND VIOLENCE

For example, case-control studies are particularly susceptible to recall bias— a bias resulting from differential recall among case respondents compared with control respondents. The likelihood of recall bias may be directly influenced by the respondent’s motivation to explain the illness (or outcome) itself. In a study of suicide, the victim’s past history of depression might be more salient to the relatives of a person who has committed suicide compared with the relatives of a control subject, so that case-control studies of completed suicide might overstate the risk of psychopathology or of gun ownership among persons who have committed suicide, compared with controls.

Furthermore, relatives may follow a “stopping rule”: once the family has found a “sufficient” explanation for the occurrence of the suicide— whether it is a gun in the home or psychopathology—they may be less likely to admit the presence of other, less socially acceptable risk factors; such ascertainment bias can lead to the underreporting of co-morbidity among risk factors and could explain reports of a greater frequency of gun ownership among suicides with no reported history of psychopathology. In the case of gun suicides, ascertainment bias may also arise because the outcome itself provides evidence of access to a gun. For example, family members are not always aware that firearms are kept in the home. If a subject has killed himself with a gun, family members would not be able to deny the gun’s existence, even if they have first learned of its existence because the suicide has occurred. In contrast, the relatives of a living control subject may not know with certainty whether a gun is present in the household (Ludwig et al., 1998). Family awareness of suicidal risks could lead them to take steps to prevent the suicide of family members known to be at risk. In this case, the absence of firearms would be a sign of appropriate family responsiveness, and a nonexperimental study design would be unable to distinguish the protective effects of gun removal from the protective effects of other steps that the family may have undertaken at the same time.

Other limitations of case-control studies include nonrandom selection of cases or controls; it is often difficult to design a sample selection procedure that ensures that controls are, in fact, representative of the same population from which the cases were drawn. Even if the data are accurate and the sampling scheme is well defined, case-control studies, like other nonexperimental study designs, have a limited ability to distinguish causal from noncausal connections. In the case of firearms, individuals who own guns might have unobserved attributes that are associated with increased suicide risk, or, just as important, some individuals may seek to purchase guns because of a specific plan to commit suicide. These possibilities have very different implications from the point of view of preventive intervention.

FIREARMS AND SUICIDE

173

Finally, the parameter reported in many case-control studies, termed the odds ratio, is often not the parameter of interest for policy. Presumably, policy makers are interested in the expected number of lives saved or lost because of firearms or other factors. The odds ratio, which is roughly the suicide probability with firearms divided by the suicide probability without firearms, can translate into many or few lives, depending on the suicide probabilities that are involved. Thus, a large odds ratio does not necessarily translate into a large number of lives, and a small odds ratio does not necessarily translate into a small number of lives. To see the problem, consider two populations, one in which the suicide probability conditional on owning a firearm is 0.02 per person per year and the suicide probability conditional on not owning a firearm is 0.01 per person per year, and another in which these two probabilities are 0.0002 and 0.0001, respectively. The odds ratio and the relative risk are the same in both scenarios, but if guns are causal, then removal of guns from the population might avert 0.01 deaths per person per year in the first scenario, but only 0.0001 deaths per person per year in the second. Policy makers would usually like to know the attributable risk, which can be defined as the difference between the incidence of the outcome among the exposed and the incidence of the outcome among the unexposed. For the odds ratio or relative risk to inform policy, it must therefore be considered in light of additional information. The appendix to this chapter provides a detailed discussion of the measures of association in case-control designs, illustrating the strengths and weaknesses of the odds ratio as a measure of association and explaining the information needed to estimate attributable risk.

Psychological Autopsy Studies

A number of studies have now been published that compare the prevalence of firearms in the homes of suicide victims with the prevalence of firearms in the homes of living controls; these studies, most of which make use of a “psychological autopsy” case-control design, are summarized in Table 7-3. Psychological autopsy studies are retrospective studies using interviews with relatives, neighbors, coworkers, or other close contacts of a deceased person (or of a living control subject) seeking to reconstruct the presence or absence of behavioral or psychological risk factors that may have predisposed the death. All of the studies that the committee reviewed have found a positive association between household gun ownership and suicide risk, although the magnitude of the estimated association varies. Although more recent studies have used better data collection strategies and more appropriate study samples (e.g., Conwell et al., 2002; Beautrais et al., 1996), the earlier studies suffer from methodological problems—ranging from sample selection problems to measurement bias, small samples, and

174		FIREARMS AND VIOLENCE
TABLE 7-3 Psychological Autopsy Studies of Firearm Prevalence
and Suicide
	Cases	Controls
Source	N	n
Conwell et al.	Older adult	Community
(2002)	suicides	controls
	N = 86	n = 86
Shah et al.	Adolescent	School-selected
(2000)	gun suicides	controls
	N = 36
		n = 36
Brent et al.	Adolescent	Community
(1999)	suicides	controls
	N = 140a	n = 131
Bailey et al.	Female	Community
(1997)	homicides and	controls
	suicides in
	the home
	N = 123	n = 266 pairs
	suicides; 143
	homicidesa
Beautrais et al.	Suicides	Community
(1996)		controls
	N = 197	n = 1,028 normal
		controls
Brent et al.	Adolescent	Community
(1994)	suicides with	controls with
	affective	affective disorder
	disorder	n = 23
	N = 63a

FIREARMS AND SUICIDE

175

		Result: Gun
Gun	Covariates, Matching	Access and Overall
Measure	Factors	Suicide Risk
Firearm in	Education, living situation,	+: any gun, handgun
home	psychiatric illness	0: long gun
	Matching: age, race, sex,
	county of residence
Firearm in	Previous mental health	n/a: no information
the home	problems, alcohol use,	about overall suicide
	conduct disorder
		(although gun is +
	Matching: age, sex, school	associated with risk
		of gun suicide)
Firearm in	Psychiatric diagnosis,	+: any gun
the home	family history, life
	stressors, history of abuse
	Matching by sex; age, race,
	county of origin,
	socioeconomic status
Firearm in	Mental illness; history of	+: any gun
the home	domestic violence; alcohol
	use, alcohol problems,
	prior arrest; illicit drug use;
	home security
	Matching: neighborhood,
	sex, race, age
Firearm in	Age, gender, ethnicity,	0: gun not
the home	psychiatric diagnosis	associated with
		overall risk of
		suicide
		(although gun is
		associated with risk
		of gun suicide)
Firearm in	Psychiatric diagnosis,	+: any gun, handgun
the home	family history, stressful	0: not long gun
	life events, past treatment
	Matching: age, sex, county
	of origin, socioeconomic
	status

continued

176

TABLE 7-3 Continued

FIREARMS AND VIOLENCE

	Cases	Controls
Source	N	n
Bukstein et al.	Adolescent	Community
(1993)	suicides with	controls with
	substance	substance abuse
	abuse
	N = 23a
		n = 12
Brent et al.	Adolescent	Community
(1993a)	suicides	controls
	N = 67a	n = 67
Brent et al.	Adolescent	Community
(1993b)	suicides	controls without
	N = 67a	psychiatric
		disorder
		n = 38
Kellermann et al.	Suicides in	Community
(1992)	the home	controls
	N = 438b	n = 438

Brent et al.	Adolescent	Inpatient controls
(1991)	suicides
	N = 47a	n = 94
		47	attempters,
		47	never-suicidal
Brent et al.	Adolescent	Inpatient controls
(1988)	suicides	n = 56
	N = 27

aOverlapping samples, western Pennsylvania.

FIREARMS AND SUICIDE		177
		Result: Gun
Gun	Covariates, Matching	Access and Overall
Measure	Factors	Suicide Risk
Firearm in	Psychiatric diagnosis,	+: any gun, handgun
the home	family history, stressful	0: not long gun
	life events, past treatment	0: not gun storage
	Matching: age, race, sex,
	socioeconomic status,
	county of residence
Firearm in	Psychiatric diagnosis	+: any gun,
the home		handgun
	Matching: age, sex,
	socioeconomic status,	•particularly when
	county of origin	no psychiatric
		disorder is present
Firearm in	Psychiatric diagnosis,	+: any gun, loaded
the home	family history, stressful	gun
	life events	•particularly when
	Matching: age, sex, county	no psychiatric
	of origin, socioeconomic	disorder is present
	status
Firearm in	Alcohol use, illicit drug	+: any gun
the home	use, domestic violence,
	living alone, education,	•particularly when
	previous hospitalization	no psychopathology
	due to alcohol, current	is reported
	psychiatric medication.
	Matching: age, race, sex,
	neighborhood
Firearm in	Psychiatric diagnosis,	+: any gun
the home	family history; female
	headed household, treatment	0: Not gun storage
	history
	Matching: age, sex, county
	of origin
Firearm in	Precipitants, psychiatric	+: any gun
the home	diagnosis, family history,
	exposure to suicidal contact

bOverlapping samples, King County, Washington, and Shelby County, Tennessee.