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Regression to the Mean, Murder Rates, and Shall-Issue Laws [Political Science]

Patricia G RAMBSCH

implications. Plausible arguments have been made for either reductions or increases in violent crime due to these laws (Lott The relationship between state murder rates and the liberal- and Mustard 1997). Proponents of the laws claim that shallization of conditions under which a citizen can obtain a permit issue laws may deter crime if they increase concealed carry to carry a concealed weapon (shall-issue laws) is controversial in public places. This in turn will increase the probability that and important for policy. Many analyses have been done dur- the perpetrator’s intended victim would be armed. On the other ing the last decade, but regression to the mean has been ignored hand (Ayres and Donohue 2003; Snyder 1997), opponents of the with the exception of two papers which concluded that it did laws argue that they could encourage criminals to carry more not matter. We consider state murder rates for 1976–2001 and powerful guns and discharge them more readily. Also, accessicompare relative murder rate slopes (relative to the U.S. murder ble guns could produce lethal consequences in otherwise minor rate) for the five years following state adoption of shall-issue arguments. Finally, if shall-issue laws led to more guns in circulaws to the five years preceding for the 25 states becoming shall- lation, more guns would be available to criminals, because hunissue in 1981–1996. We find strong evidence for regression to dreds of thousands of guns are stolen yearly in the U.S. (Duggan the mean. Using both a random and a fixed effects model, we 2001). compare analyses ignoring the regression effect via a paired tTo provide context, Figure 1 shows the overall murder rate in test to those controlling for it by conditioning on the pre shall- the 25 states that adopted shall issue laws between 1981 and issue slopes. We find that controlling for regression to the mean 1996 and the overall murder rate in the remaining 25 states changes the sign of the estimated intervention effect on mur- plus the District of Columbia. With the exception of Washingder rate slopes from negative to positive, has strong impact on ton (adopted 1960), New Hampshire (adopted before 1930), and statistical significance, and gives no support to the hypothesis Vermont (no permit required), none of these states were shallthat shall-issue laws have beneficial effects in reducing murder rates. KEY WORDS: Concealed carry; Homicide; Poisson regression; Regression effect; Statistical regression. 1. INTRODUCTION

Recent decades have seen liberalization of state laws regarding permits to carry concealed weapons, typically handguns, in public places. Before 1980, almost all states either forbade civilian carry of concealed weapons (CCW) or granted permits on a discretionary basis. In discretionary states, a local law enforcement official must be convinced that the applicant has a justifiable need for concealed carry and/or is of good character. These terms are undefined in the law, leaving much latitude for discretion to the states. Between 1981 and 1996, 25 states (Table 1) adopted shall-issue (SI) laws, requiring that anyone meeting fairly objective criteria be issued a permit to carry a concealed weapon. These criteria include a minimum age, typically no prior felony conviction, no insanity, alcoholism or substance abuse, not a fugitive from justice, and often completion of a firearm use and safety course or demonstration of firearm competence. This article estimates the effect adoption of these laws had on murder rates, a controversial topic with important public health Patricia Grambsch is Associate Professor, School of Public Health, University of Minnesota, 420 Delaware St. SE, Minneapolis, MN. 55455 (E-mail: [email protected]). c

2008 American Statistical Association

DOI: 10.1198/000313008X362446

Table 1. Shall Issue States: name, year of shall-issue adoption, population (millions) for that year, murder rate (number of murder victims per 100,00 per year) averaged over the 11-year period from 5 years before to 5 years after shall issue, and which of the four U.S. census regions contained that state. State

Alaska Arizona Arkansas Florida Georgia Idaho Kentucky Louisiana Maine Mississippi Montana Nevada North Carolina North Dakota Oklahoma Oregon Pennsylvania South Carolina South Dakota Tennessee Texas Utah Virginia West Virginia Wyoming

Shall issue

Year

1994 1994 1995 1987 1989 1990 1996 1996 1981 1990 1991 1995 1995 1985 1995 1989 1989 1996 1985 1994 1995 1995 1995 1989 1994

Popn. 0.61 4.08 2.48 12.02 6.44 1.01 3.88 4.35 1.13 2.57 0.81 1.53 7.20 0.69 3.28 2.82 12.04 3.70 0.71 5.18 18.72 1.95 6.62 1.86 0.48

Murder rate 7.84 8.42 9.38 11.12 11.29 3.02 5.80 15.64 2.46 11.75 3.36 10.48 9.39 1.29 7.39 4.91 5.80 8.51 2.11 9.55 9.76 2.87 7.62 5.54 3.54

Region

West West South South South West South South Northeast South West West South Midwest South West Northeast South Midwest South South West South South West

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Murder Rate per 100,000

9

10

11

shall-issue could appear to reduce violent crime even if the two are unrelated. In the foregoing research and commentary, such a regression effect (regression to the mean) has been largely ignored. We have seen it mentioned only by Hepburn et al. (2004) and by Rosengart et al. (2005). Both analyzed changes in murder rate levels and concluded it was not present. To describe the regression effect, let Y1 and Y2 represent outcome measurements from a study unit before and after an intervention, respectively. Assume bivariate normality with respective means µ and µ + 4, common variance τ , and correlation ρ. A traditional estimate of the intervention effect is ˆ T = Y¯2 − Y¯1 , 4

8

the sample mean of Y2 − Y1 . It can be tested by the paired t-test. However, this approach requires that the Y1 ’s form a random sample. If at least some of the Y1 ’s are selected for large (or small) values, it can mislead. Normal distribution theory gives

5

6

7

E[Y2 |Y1 ] = µ + 4 + ρ(Y1 − µ).

1975

1980

1985

Year

1990

1995

2000

Figure 1. Murder rates of the 25 shall-issue states (solid line) and the remaining states (dashed line).

issue in the period 1976–2001 shown on the plot. The shallissue states start with much higher rates, but the two sets of states move closer together, although not monotonically, over time, suggesting a beneficial effect of shall-issue laws. However, the decline of the murder rate in the mid and late 1990s is less in the shall-issue states, suggesting a harmful effect of shall-issue laws. Extensive empirical research has not led to definitive conclusions. The first comprehensive analysis, by Lott and colleagues (Lott and Mustard 1997; Lott 2000), using state, county, and city level data for all major crimes reported by the FBI and many covariates found large, statistically significant reductions in the level or rate of change (i.e., slope) of murder and other violent crime rates after liberalization of concealed carry laws. Some independent researchers have confirmed these findings (Plassmann and Tideman 2001; Bartley and Cohen 1998; Moody 2001). However, many reanalyses of Lott’s and related data by other researchers (Ayres and Donohue 2003; Black and Nagin 1998; Duggan 2001; Helland and Tabarrok 2004; Hepburn et al. 2004; Rosengart et al. 2005) have found small, statistically insignificant effects. The various approaches, particularly Lott’s, have been critiqued notably by Manning (Manning 2003) and most recently by the National Research Council (Wellford et al. 2005). It is possible that shall-issue laws are adopted in response to transitory rises in violent crime which would have subsided without liberalized concealed carry. If so, 290 Interdisciplinary: Political Science

(1)

Suppose no intervention effect:4 = 0. Let Y1 > µ. If 1 > ρ > 0, then Equation (1) implies µ < E[Y2 |Y1 ] < Y1 . A large pre-intervention value is followed on average by a smaller postintervention value, closer to the mean. This is the source of the term regression to the mean. With ρ < 0, −Y1 < E[Y2 |Y1 ] < µ, regression “through” the mean occurs. The generic term “regression effect” encompasses both these cases. If selection on Y1 makes Y1 ’s sample mean exceed µ, then on average Y2 ’s sample mean is less than Y1 ’s. A statistically significant postminus pre-difference could be misinterpreted as a decrease due to the intervention, whereas it is merely an artifact of selection for large Y1 . This difficulty can be avoided. Mee and Chua (1991) (henceforth MC) suggested a way to assess any intervention controlling for the regression effect. They required known µ, the population mean in the absence of the intervention. Let X 1 = Y1 − µ and X 2 = Y2 − µ. Equation (1) becomes E[X 2 |X 1 ] = 4 + ρ X 1 ,

the well-known formula for simple linear regression. The usual least squares slope estimates ρ and the intercept estimates 4, the intervention effect. ˆ MC = X¯ 2 − ρˆ X¯ 1 = Y¯2 − µ − ρ( 4 ˆ Y¯1 − µ).

Standard regression methods give a test of H0 : 4 = 0 and a confidence interval. MC’s approach controls for selection by conditioning on Y1 rather than subtracting it as in the paired t-test approach which ignores selection. The size of the regression effect can be estimated by the difference in the two estimates of 4: ˆ MC − 4 ˆ T = (Y¯1 − µ)(1 − ρ). 4 ˆ

(2)

The estimated regression effect is 0 when Y¯1 = µ and there is no selection on Y1 or when ρˆ = 1. It increases as Y¯1 moves away from µ or as ρˆ decreases, particularly to negative values. Our aim is to assess the impact of the regression effect on comparison of state murder rate trends before and after the adoption of shall-issue laws, extending MC’s approach to our more complicated dataset.

7.5

0.95 0.90 0.80

6.5

0.85

Mean Murder Rate

7

Mean Relative Murder Rate

6

0.75

Murder rate Rel rate

−4

−2

0

Year from SI

2

4

Figure 2. Average murder rates and average relative murder rates for the 25 shall-issue states.

2. PRELIMINARY GRAPHICAL ANALYSIS In this section, we display a plot of average murder rates and average relative murder rates for adopting states in an 11year period surrounding the year of adoption. State murder rates from 1976 through 2001 come from the FBI’s yearly Uniform Crime Reports (National Archive of Criminal Justice Data 1976–2001). The rate is the number of victims of homicides occurring in the state in a calendar year divided by the state’s population in units of 100,000, on July 1 as estimated by the Census Bureau. The intervention data are the calender year that each state’s government passed the law. In all but two of the 25 shall-issue states, the law went into effect the same year or January 1 of the following year. This information (Table 1) was compiled from Lott (2000, pp. 43, 169) and Vernick and Hepburn (2003, Table 9A-5), consulting state session laws and state code to resolve disagreements. All 25 states have at least five years of data both before and after the law was passed. The year the law passed is coded as year 0. An obvious starting point is to plot each state’s murder rates over time, including years both before and after law passage. However, this could be misleading. Crime rates in different regions of the United States move roughly in tandem. If overall murder rates were decreasing just after law passage, for example, the plot would suggest a beneficial effect for the shall-issue law even if it had no effect. To control for overall trends, we computed relative murder rates, the ratio of each state’s murder rate to that of the entire United States for each year. Figure 2 shows the average mur-

der rate (dashed) and the average relative murder rate (solid) for the 25 shall-issue adopting states from five years before to five years after shall-issue law passage. Both averages suggest that shall-issue laws had a beneficial effect on murder rates, but are otherwise quite different. The average rate plot suggests a fairly horizontal, fluctuating, slightly increasing trend before law passage, followed by a precipitous drop with a nearly linear decrease after law passage. The average relative murder rate increases monotonically before passage and is nearly horizontal with substantial fluctuations after passage. The relative rate plot implies that the big drop in the rate plot is an artifact of overall trends. It also suggests a regression to the mean interpretation. Shall issue laws were passed when the state’s murder rate had been rising relative to other states, selecting out positive trends. The following horizontal trend could just be a regression to the mean artifact, rather than a beneficial effect of the law. Figure 2 motivates concentrating on relative murder rate slopes.

3.1

3. MATERIALS AND METHODS FOR FORMAL ANALYSIS Covariates

We have five state-level yearly exogenous regressors: unemployment rate, percent of population below the official U.S. poverty line, percent living in each state’s largest metropolitan county (measuring urbanization), percent Black, and percent young adults (15 to 24 years old). These regressors or variants of them are commonly used in research on the impact of concealed carry laws on crime. State-level poverty rates were unavailable for 1976–1978, but poverty rates were available for the four U.S. regions for those and surrounding years. We used the 1975 and 1980 data to estimate the relationship between the state and relevant region poverty rate and used that to impute state poverty rates for 1976–1978 by means of the iterative proportional fitting algorithm (Bishop et al. 1976). Demographic data for the 1970s, 1980s, and 1990s used intercensal estimates for the non-decennial years. We used the Center for Disease Control and Prevention’s race-bridging algorithm to make the 2000 and 2001 new racial and ethnic data compatible with prior censuses. 3.2

Statistical Model

Let Yi j be the number of murders in state i, i = 1, . . . , 25 for year j relative to shall-issue law passage, j = −5, . . . , 0, . . . , 5; Ni j be the population of state i in year j; RT i j be the overall U.S. murder rate for calendar year T corresponding to year j relative to law passage for state i; and Z i jk be the value of covariate k, k = 1 . . . , 5 for state i in year j. The model contains a linear spline for regression on years from law passage with a single knot at year 0. The spline design matrix transposed is   1 1 1 1 1 1 1 1 1 1 1 X T =  −5 −4 −3 −2 −1 0 0 0 0 0 0  . 0 0 0 0 0 0 +1 +2 +3 +4 +5 (3) The American Statistician, November 2008, Vol. 62, No. 4

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Let X j denote the jth row of X . Let µi j be state i’s expected number of murders in year j. We model Yi j as a Poisson random variable with log-linear mean specification: ! X αk log(Z i jk ) + X j (β0i , β1i , β2i )T . µi j = Ni j RT i j exp

Table 2. Estimated before shall-issue mean relative murder rate slopes under different analysis methods. RAN stands for the random effects method and FIX for the fixed effects method. Method RAN FIX

k

(4) The β1i ’s and β2i ’s are key parameters, denoting the slopes before and after shall-issue, respectively, of each state’s log expected relative murder rates, controlling for the covariates. A one-term Taylor expansion gives eβ ≈ 1 + β; so, for example, a state with β1 = 0.04 and β2 = −0.01 has its relative murder rate increasing about 4% per year in the five years before the shall-issue law and decreasing about 1% per year in the five years after. To apply MC, we need a value for µ, the mean slope in the absence of shall-issue laws. We do not know µ, but a reasonable value is µ = 0. Thus, in the absence of shall-issue laws, the murder rates in the adopting states follow the overall U.S. rate, modulated by the covariate influences and random fluctuations. We considered two methods for using our model for inference, a random effects and a fixed effects approach. The Appendix provides some goodness-of-fit statistics and plots. 3.3

Random Effects Inference

This method considers the vectors (β0i , β1i , β2i )T to be the realized values of a multivariate normal distribution with mean (β0 , β1 , β2 ) and variance matrix 6 = {σlm }. The coefficients αk are considered fixed. The model was fit to the data using PROC NLMIXED in SAS v. 9. To test for selection for large slopes before shall-issue adoption, we compared βˆ1 to its standard error. To assess the impact of shall-issue laws, ignoring the regression ˆ T = βˆ2 − βˆ1 to its standard error. To apeffect, we compared 4 2 (assuming σ ply MC, we computed ρˆ = σˆ 12 /σˆ 11 11 = σ22 and ˆ MC = βˆ2 − ρˆ βˆ1 , which we compared to its standard error. then 4

4.2

βˆ1

0.025 0.050

se

0.011 0.017

p

0.035 0.008

Ignoring Versus Controlling for Regression Effect

Table 3 compares the traditional analysis that does not take into account a regression effect to our extension of MC’s conditional approach that does control for it. Results for each of the two analysis methods are presented. Under both methods ˆ implying that ignoring the regression effect gives a negative 4, relative murder rate slopes are less after shall-issue law passage, although neither estimate is statistically significant at conventional levels. However, controlling for the regression effect ˆ implying an increase in slopes. This is particgives a positive 4, ularly marked for the fixed effects method whose large, highly ˆ of 0.60 shows that the relative murder rate is insignificant 4 creasing about 6% per year more rapidly after passing the law than before. The random effects method has a very small nonsignificant estimate. The difference between the two methods ˆ MC − 4 ˆT can best be understood by examining the difference 4 as in Equation (2). The fixed method has both a larger βˆ1 (Table 2) (which plays the role of Y¯1 in Equation (2)) and a more negative ρˆ than the random effects method. Therefore, it must ˆ T. ˆ MC − 4 have a commensurably bigger difference for 4 5. DISCUSSION

We have presented a case study of regression to the mean in the topical, controversial area of gun control. It is statistically interesting because there is negative correlation between the pre and post shall-issue law adoption slopes. Usually, discussions 3.4 Fixed Effects Inference of regression to the mean assume positive correlation between This method estimates the vectors (β0i , β1i , β2i )T by fitting the pre and post intervention measures. However, as we have the model in Equation (4) to the data by Poisson regression, us- shown both theoretically (Equation (2)) and in our application ing PROC GENMOD in SAS v. 9. To test for selection, we did (Table 3), the regression effect is particularly large when the a one-sample t-test of the βˆ1i ’s, testing the null hypothesis of 0 correlation is large negative (and also when the pre-intervention mean. To assess the impact of shall-issue laws, ignoring the re- mean is far from the mean in the absence of intervention). Controlling for a variety of exogenous covariates, we found gression effect, we did a two-sample t-test comparing the βˆ1i ’s and βˆ2i ’s To apply MC, we regressed the βˆ2i ’s on the βˆ1i ’s us- that in the five-year period before law passage, the 25 states ˆ MC , ing PROC REG. The slope gave ρˆ and the intercept gave 4 which we compared to its standard error. 4.1

4. RESULTS

Selection on Slopes Before Shall-Issue

On average, the before shall-issue slopes were significantly positive (Table 2). The fixed effects method gave a bigger average slope than the random effects method. Selection occurred for large slopes before shall-issue and we expect a regression effect to return the slopes to lower values, regardless of the law being passed. 292 Interdisciplinary: Political Science

Table 3. Estimated change in relative murder rate slopes accompanying passage of shall-issue laws showing impact of analysis method. Method abbreviations as in Table 2 Method RAN FIX

Regression effect ignore control for ignore control for

ˆ 4 −0.026 0.005 −0.012 0.060

se 0.016 0.011 0.028 0.015

p 0.116 0.656 0.676 0.001

ρˆ −0.237 −0.450

passing shall-issue laws between 1981 and 1996 had an increasing trend on average in murder rates relative to the U.S. murder rate. The magnitude varied from 2.5 to 5.0% per year, depending on whether the analysis method was random or fixed effect. In all cases it was statistically significant. Thus, rather than a random samples of slopes, we have selection for positive slopes. State governments tended to pass shall-issue laws when murder rates were relatively increasing. When selection is inappropriately ignored we found that, on average, relative murder rates increase more slowly after passage of the shall-issue law than before, although the effect size and statistical significance depend on the analysis method. The results controlling for selection and the resulting regression effect by an extension of MC’s approach varied greatly by method. However, in no instance did they give a significantly negative value. We conclude that shall-issue laws are not associated with decreases in relative murder rate trends, post minus pre. They are associated with increases in relative murder rate trends prior to shall-issue law passage. Therefore, research on gun control cannot ignore shall-issue laws since they do have an effect. An important limitation to this work is the limited set of covariates. To some extent, this was forced on us by practical considerations. We have only 25 states adopting shall-issue and thus cannot reliably estimate coefficients for a large number of covariates. Also, when we added the percent of young black men, suggested by a reviewer or the estimated consumption of ethanol in gallons per capita, used by Hepburn et al. (2004) there were problems with the Hessian for PROC NLMIXED. Most research in this area, certainly all the articles cited above, use demographics, economic indicators (typically unemployment rates and percent below poverty) and measures of urbanization. However, the FBI’s Crime in the United States has a section titled “Variables Affecting Crime” which includes these variables and also, inter alia, stability of the population, job availability, educational level, economic dependence on nonresidents (i.e., tourists and convention attendees), divorces and family cohesiveness, and weather. None of the research cited here mentions these variables, perhaps because they are not as readily available. To our knowledge, this work is the first to find a regression effect in the impact of shall-issue laws on murder rates. Only two previous references looked for it. Rosengart et al. (2005) and Hepburn et al. (2004) found no evidence of a selection effect, although they also used yearly state homicide rates (Rosengart et al. considered only firearm homicides), similar covariates, and loglinear models. We conjecture that the most important difference between this work and theirs was their use of relative murder rate levels, rather than rates of change. Levels are a natural starting point and have the advantage of a relative risk interpretation, but do not show regression to the mean. Other differences include these authors’ use of National Center for Health Statistics mortality files rather than FBI data, a slightly different time period, 1979–1998, and a slightly different coding of when states changed their laws, closer to Vernick and Hepburn than to ours. They controlled for overall U.S. murder rates by including 0-1 dummy indicators for each year in the model, whereas we divided by overall U.S. murder rates. These differences are likely to be fairly minor in impact. Like

our analyses ignoring the regression effect and including all adopting states, their analyses found no statistically significant differences before and after passage of the shall-issue laws. It is surprising that so little has been done with regression to the mean for these data. Back in 1997, Lott and Mustard (1997) noted that “For most violent crimes, the time trend leading up to the adoption of the [shall-issue] laws indicates that crime was rising prior to the laws being enacted,” a statement with obvious implications for regression to the mean. It is only recently that those implications have been developed. We believe that methods for detecting and controlling for regression to the mean should be an important part of the statistical toolkit for investigating the relationship between crime rates and crime control measures like shall-issue laws. APPENDIX

To assess the goodness of fit for our two methods, we used an interpretable measure of error, the average relative error for each state 5 1 X |yi j − yˆi j | Ei = . yi j 11 j=−5

The errors for the two methods were very similar. The fix method had a minimum error of 0.020, median of 0.072, interquartile range of 0.067, and maximum of 0.406. For the ran method, the minimum was 0.016, the median was 0.068, the interquartile range was 0.066, and the maximum was 0.491. The Pearson correlation was 0.995. Therefore, we used the average of the two methods as the error measure for each state. Figure A.1 shows the raw data, number of murders, with the fitted values for each of the two methods superimposed for a selection of states. We have the state with the largest error, the state with the smallest error, and the error quintiles in between. As the errors get smaller, the two methods become more similar. The lowest quintile shows very good fit. The plots show that except for the highest quintile, the fits are reasonable. The sixth highest state, Oklahoma, has an error of 0.128. Thus, below the highest quintile, on average, the fitted values are within 13% of the actual number of murders. The states in the highest quintile with their error measures are North Dakota, 0.448; South Dakota, 0.360; Wyoming, 0.196; Idaho, 0.182; and Montana, 0.177. The lowest quintile has Virginia, 0.032; Florida, 0.031, Texas, 0.028; Georgia, 0.023 and North Carolina, 0.018. [Received December 2006. Revised June 2008.]

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