What You Can and Can’t Properly Do With Regression∗ Richard Berk Department of Statistics Department of Criminology University of Pennsylvania May 17, 2010

1

Introduction

Regression analysis, broadly construed, has over the past 60 years become the dominant statistical paradigm within the social sciences and criminology. In its most canonical and popular form, a regression analysis becomes a “structural equation model” from which “causal effects” can be estimated. Consider some examples from the two most recent issues of Criminology. Volume 47, Issue 4, has ten articles, seven of which employ some form of casual modeling. One exception (Mears and Bales, 2009) estimates causal effects using matching, another exception (Schultz and Tabanico, 2009) estimates causal effects using randomized experiments, and the final exception (Guerette and Bowers, 2009) estimates causal effects using meta-analysis.1 Volume 48, Issue 1, has nine articles of which eight employ some form of causal modeling. The one exception (Skardhamar, 2010) is a useful critique of a causal modeling special case: “semiparametric group-based modeling.”2 ∗

Thanks go to Alex Piquero and Jim Lynch for very help comments on earlier drafts of this paper. 1 Of course, much the research on which Guerette and Bowers rely employs causal modeling. For a discussion of well-known, but typically ignored, problems with metaanalysis, see Berk (2007) 2 See Berk et al. (2010) for a more formal and general discussion.

1

The many problems with causal modeling are well articulated in a very large literature (e.g., Box, 1976; Leamer, 1978; 1983; Lieberson, 1985; Holland, 1986; Rubin, 1986; Freedman, 1987; 2005; Kish, 1987; de Leeuw, 1994; Manski, 1995; Pearl, 2000; Heckman, 2000; Breiman, 2001; Berk, 2004; Imbens, 2009). Perhaps too simply put, before the data are analyzed, one must have a causal model that is nearly right.3 With the model specified, all that remains is to estimate the values of the regression parameters. In practice, one or two small flaws can sometimes be tolerated as long as they are easily identified and corrected with the data on hand. It is very difficult to find empirical research demonstrably based on nearly right models. In the absence of a nearly right model, the many desirable statistical properties of a causal model can be badly compromised. Omitted variables, for instance, can badly bias parameter estimates even in very large samples. Alternatively, using the data to inductively arrive at a nearly right model assumes that all of the required regressors are included in the data set. Even if they are (and how would you know?), badly biased estimates can result because the model and the parameter estimates are extracted from the same data (Leeb and P¨otscher, 2005; 2006; Berk et al., 2010a). There are five kinds of responses to the causal modeling critique. The first is rhetorical and is by far the least constructive. David Freedman compiled the following (but incomplete) list of ripostes (2005: 195). We all know that. Nothing is perfect. Linearity has to be a good first approximation. Log linearity has to be a good first approximation. The assumptions are reasonable. The assumptions don’t matter. The assumptions are conservative. You can’t prove the assumptions are wrong. The biases will cancel. We can model the biases. We’re only doing what everyone else does. Now we will use more sophisticated techniques. If we don’t do it, someone else will. What would you do? The decision-maker has to be better off with us than without us. We all have mental models, not using a model is still a model. The models aren’t totally useless. You have to do the best you can with the data. You have to make assumptions in order to make progress. You have to give the models the benefit of the doubt. Where’s the harm? 3

A model is the “right” model when it accurately represents how the data on hand were generated. A detailed discussion can be found in many formal expositions of causal modeling (e.g., Freedman, 2005).

2

The obvious problem with these and the many other rhetorical devices is that they are little more than a reflexive defense of the problematic status quo. The second response is to assert that the current assortment of regression diagnostics can find any serious problems in a regression model and that these problems can then be readily fixed. For example, model misspecification can be identified with “specification tests,” and a subsequent instrumental variable estimator can save the day (e.g., Greene, Section 5.5.). In principle, this view has merit. In practice, it usually stumbles badly. For both the diagnostics and the remedies, new and untestable assumptions are required even before one gets to a number of thorny technical complications. On the matter of diagnostics, Freedman observes (2009: 839) that , “...skepticism about diagnostics is warranted. As shown by the theorems presented here, a model can pass diagnostics with flying colors yet be ill-suited for the task at hand.” He is no more sanguine about about instrumental variable solutions (Freedman, 2005: Section 8.5) and has lots of company (e.g., Berk, 2004: Section 9.5; Kennedy, 2008: Chapter 9; Leamer, 2010). The third response is what I have elsewhere called “Model-Lite Causal Analysis” (Berk, 2009). The basic idea is apply the randomized experimental paradigm to observational data. One tries to make the case that conditional on a set of covariates, “nature” conducted a randomized experiment. This naturally leads to a variety of matching or post-stratification analysis methods that can be more robust than causal models (Rubin, 2008). Regression models can creep in, however, in the construction matching variables such propensity scores or in post-matching adjustments for sample imbalance (Rosenbaum, 2002b). Morgan and Winship (2007) provide an excellent overview of the issues. The primary literature can be found in the influential writings of William Cochran, Donald Rubin, Paul Rosenbaum, Judea Pearl and others. Unfortunately, the elephant in the room remains: there is no apparent way to overcome the impact of omitted variables although in some cases, sensitivity analyses can suggest that the omitted variables may not be important (Rosenbaum, 2002a, Chapter 4; 2010, chapter 14). The fourth response is to rely on research designs far better suited for causal inference than the usual observational approaches (Angrist and Pischke, 2010). Properly implemented randomized experiments are the reigning poster child. But strong quasi-experiments will often perform nearly as well (Cook et al., 2008). For example, recent advances in the regression discon3

tinuity design (Imbens and Lemieux, 2008) can make that approach very attractive when a randomized experiment is not feasible (Berk et al., 2010b). Capitalizing on stronger designs is clearly a useful suggestion when such designs are possible. The fifth response is to go back to the formal definition of a regression analysis and reconsider what can be learned depending on the assumptions that one can credibly make. We turn to that approach now. There will be some good news.

2

What is Regression Analysis?

Cook and Weisberg (1999: 27) provide a definition of regression analysis that comports well with standard statistical perspectives: “[to understand] as far as possible with the available data how the conditional distribution of the response y varies across subpopulations determined by the possible values of the predictor or predictors.” The definition makes the entire conditional distribution of y fair game although in practice, the conditional mean and/or conditional variance are the primary focus. A key point is that there is no mention of estimation, hypothesis tests, or confidence intervals nor any reference to causal inference. One can do a proper regression analysis without any effort to address the role of chance or to make causal statements.

2.1

Level I Regression Analysis

We have seen that by definition, regression analysis is a procedure by which conditional relationships in data may be described. One might consider, for example, how the homicide rate in a city varies with unemployment, police practices fixed. And one might consider how the homicide rate in a city varies with police practices, unemployment fixed. In effect, one is identifying interesting patterns in the data, which can be subtle, complicated and even rare. The patterns can be found over time, over space, and over observational units that can differ in complex ways. The analysis can be directed by existing theory or can be highly exploratory. One is not limited to conventional linear regression. For example, the definition includes response variables that are categorical, ordinal or counts. Categorical regressors are can be included. Nonlinear relationships can be

4

taken into account. One can consider several response variables at once such as a parolee’s arrests for new crimes and the charges for those new crimes. It follows that for description, one need not worry about all of the potentially problematic assumptions required if one is to undertake credible statistical or causal inference. One just characterizes associations in the data at hand. Thus, it does not matter, for instance, if the data are a probability sample or the product of well-defined stochastic process. Likewise, there is no concern about omitted variables. Indeed, it is not clear how to define an omitted variable in this context. It is the proverbial “what you see is what you get.” Consistent with a more expansive discussion elsewhere (Berk, 2003: Section 11.5), one can call descriptive regression a “Level I” regression analysis. Level I regression analyses are always formally appropriate when a regression analysis could be useful and do not depend on any of the assumptions required for statistical inference or causal inference. This is not a call to abandon advanced statistical procedures and complex quantitative reasoning. Description has long been at the center of state-ofthe-art statistical practice. Among the more popular approaches are multivariate statistics (Anderson, 1958; Gifi, 1990), exploratory data analysis (Tukey, 1977; Diaconis, 1985), and more recently, dynamic graphics (Cook and Swayne, 2007) and Machine/Statistical Learning (Hastie et al., 2009, Berk, 2008). The growing use of GIS studies in criminology is one example of graphical methods (Groff and La Vigne, 2001; Chainey and Ratcliffe, 2005). It is then a small step to build visualizations that take both time and space into account (Berk et al., 2009). Machine learning is also finding its way into the criminology journals (Berk et al., 2009). Making descriptive regression analysis the central approach to data analysis in criminology is not a radical suggestion. Despite the statistical framing that one finds in the criminology journals, description is usually the actual enterprise. And the reason is apparent: in criminology, and for most social science applications more generally, there are rarely any widely accepted, nearly right models that can be used with real data. By default, the true enterprise is description. Most everything else is puffery.

2.2

Level II Regression Analysis

Level I regression analysis does not require any assumptions about how the data were generated. If one wants more from the data analysis, assumptions are required. For a Level II regression analysis, the added feature is statistical 5

inference: estimation, hypothesis tests and confidence intervals. When the data are produced by probability sampling from a well-defined population, estimation, hypothesis tests and confidence intervals are on the table. A random sample of inmates from the set of all inmates in a state’s prison system might be properly used to estimate, for example, the number of gang members in state’s overall prison system. Hypothesis tests and confidence intervals might also be usefully employed. In addition, one might estimate, for instance, the distribution of in-prison misconduct committed by men compared to the in-prison misconduct committed by women, holding age fixed. Hypothesis tests or confidence intervals could again follow naturally. The key assumption is that each inmate in the population has a known probability of selection. If the probability sampling is implemented largely as designed, statistical inference can rest on reasonably sound footing. Note that there is no talk of causal effects and no causal model. Description is combined with statistical inference. In the absence of probability sampling, the case for Level II regression analysis is far more difficult to make. There are several options addressed in some depth elsewhere (Berk, 2003: 39-58). 1. Treating the data as a population — In effect, this is a fallback to Level I regression analysis in which there is no statistical inference. 2. Treating the data as if it were a probability sample from a well-defined population — For example, a convenience sample of big-city police departments might be treated as if it were a simple random sample of police departments from cities of over 100,000 residents. One would have to argue convincingly that despite the lack of formal probability sampling, the way in which the data were generated is de facto a close approximation. And that, in turn, would lead to an in-depth discussion of why one should believe that each department in the sample was, in effect, drawn independently of all others with a known probability of selection. If access was obtained by a sequence of referrals, the as-if approach would fail. It would also fail if the sample was limited to the subset of departments for which the requisite data were available. The as-if approach is rarely credible in practice. 3. Treating the data as a random realization from an imaginary “superpopulation” — Inferences to imaginary populations are imaginary. However, there is an important distinction between an imaginary popu6

lation that could exist and an imaginary population that could not. Inferences to the former might not be imaginary, but are even more difficult to justify than inferences from the as-if strategy. The result can be a mind numbing exercise in circular reasoning. The population is defined as the population that would exist if the data were a random sample from that population. 4. Model based sampling — For this strategy, one needs a credible stochastic model of how the data were generated. For example, with conventional linear regression, one assumes y i = β T Xi + ε i ,

(1)

where εi ∼ N IID(0, σ 2 ). If the model really captures how the data were generated, statistical inference can easily follow. But, inferences are made to the data generation process, not to a real population in the usual sense.4 Model-based sampling leads one back into much of the critical literature cited earlier. Why should one believe the model? An omitted variable, for instance, will mean that the assumed properties of the disturbance term do not hold, and statistical inference can be badly compromised. The same can follow when the wrong functional forms are assumed. In short, a Level II regression analysis depends on how the data were generated. If the data are a well-implemented probability sample, Level I description can be supplemented by Level II statistical inference. Without probability sampling, Level II regression analysis can be difficult to justify.

2.3

Level III Regression Analysis

The goal in a Level III regression analysis is to supplement Level I description and Level II statistical inference with causal inference. In conventional regression, for instance, one needs a nearly right model like equation 1, but one must also be able to argue credibly that manipulation of one or more regressors alters the expected conditional distribution of the response. Moreover, any given causal variable can be manipulated independently of any other 4

Alternatively, one might define the population as all possible realizations of the given stochastic process. This population is imaginary.

7

causal variable and independently of the disturbances. There is nothing in the data itself that can speak to these requirements. The case will rest on how the data were actually produced. For example, if there was a real intervention, a good argument for manipulability might well be made. Thus, an explicit change in police patrolling practices ordered by the local Chief will perhaps pass the manipulability sniff test. Changes in the demographic mix of a neighborhood will probably not. Here too, there is extensive discussion elsewhere (e.g., Berk, 2003, Chapter 5). Suffice it to say, it is very difficult to find in criminology credible examples of a Level III regression analysis. Either the parallel to equation 1 is not credible, claims of manipulability are not credible, or both.

3

Conclusions

Level I regression analysis is always formally appropriate and does not depend on any assumptions about how the data were generated. Truth be told, description is really the de facto product of most regression analyses in criminology. Description is not just a legitimate scientific activity, but corresponds well to the developmental stage in which criminology finds itself. Level II regression analysis adds to description, estimation, hypothesis tests and confidence intervals. When the data are a probability sample from a well-defined population, statistical inference can follow easily. When the data are not, statistical inference can be difficult to justify. Level III regression analysis adds to description and statistical inference, causal inference. One requires not just a nearly right model of how the data were generated, but good information justifying any claims that all causal variables are independently manipulable. In the absence of a nearly right model and one or more regressors whose values can be “set” independently of other regressors and the disturbances, causal inferences cannot not make much sense. The implications for practice in criminology are clear but somewhat daunting. With rare exceptions, regression analyses of observational data are best undertaken at Level I. With proper sampling, a Level II analysis can be helpful. The goal is to characterize associations in the data, perhaps taking uncertainty into account. The daunting part is getting the analysis past criminology gatekeepers. Reviewers and journal editors typically equate proper statistical practice with Level III. 8

References Anderson, Theodore. 1958. An introduction to multivariate statistical analysis, First Edition. New York: John Wiley and Sons. Angrist, Joshua, and Steven Pischke. 2010. The credibility revolution in Empirical Economics: how better research design is taking the con out of econometrics. Journal of Economic Perspectives 24(2). Berk, Richard A. 2004. Regression analysis: a constructive critique. Newbury Park: Sage Publications. Berk, Richard A. 2007. Meta-analysis and statistical inference (with commentary), Journal of Experimental Criminology 3(3): 247- 297. Berk, Richard A. 2008 Statistical learning from a regression perspective. New York: Springer. Berk, Richard A. 2009a. Cant tell: comments on “Does the Death Penalty Save Lives?” Criminology & Public Policy 8(4): 843-849. Berk, Richard A. 2009b. The role of race in forecasts of violent crime. Race and Social Problems 1: 131-242. Berk, Richard A., Lawrence Brown, and Linda Zhao. 2010a Statistical inference after model selection. Journal of Quantitative Criminology, forthcoming. Berk, Richard A., Lawrence Sherman, Geoffrey Barnes, Ellen Kurtz, and Lindsay Ahlman. 2009. Forecasting murder within a population of probationers and parolees: a high stakes application of statistical forecasting. Journal of the Royal Statistical Society (Series A) 172, Part 1: 191-211. Berk, Richard A., and John MacDonald 2009. The dynamics of crime regimes. Criminology 47(3): 971-1008 Berk, Richard A., Geoffrey Barnes, Lindsay Ahlman, and Ellen Kurtz 2010b. When second best is good enough: a comparison between a true experiment and a regression discontinuity quasi-experiment. Journal of Experimental Criminology, forthcoming. 9

Box, George E.P. 1976. Science and statistics. Journal of the American Statistical Association 71: 791-799. Breiman, Leo. 2001. Statistical moldeling: two cultures, (with discussion). Statistical Science 16: 199-231. Campbell, Donald T., and Julian Stanley 1963. Experimental and quasiexperimental designs for research. New York: Wadsworth Publishing. Chainey, Spencer and Jerry Ratcliffe (2005) GIS and crime mapping. New York: John Wiley & Sons. Cook, Dianne, and Deborah F. Swayne. 2007. Interactive dynamc graphics and data analysis. New York: Springer. Cook, Thomas D., William R. Shadish, Vivian C. Wong 2008. Three conditions under which experiments and observational studies produce comparable causal estimates: new findings from within-study comparisons. Journal of Policy Analysis and Management 727(4):724-750. de Leeuw, Jan. 1994. Statistics and the sciences, in Trends and Perspectives in Empirical Social Science, I. Borg and P.P. Mohler (eds.), New York: Walter de Gruyter. Diaconis, Persi. 1985 Theories of data analysis: from magical thinking through classical statistics.” In Exploring data tables, trends, and shapes, D.C. Hoaglin, F. Mosteller, and J. Tukey (eds.). New York: John Wiley and Sons. Freedman, David A. 1987. As others see us: a case study in path analysis. (with discussion). Journal of Educational Statistics 12: 101-223. Freedman, David A. 2005. Statistical models: theory and practice. Cambridge: Cambridge University Press. Freedman, David A. 2009 Diagnostics cannot have much power against general alternatives. International Journal of Forecasting 25 (4): 833-839. Gifi, Albert., Nonlinear multivariate analysis. 1990. New York: John Wiley and Sons.

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Groff, Elizabeth, and Nancy G. La Vigne. 2002. Forecasting the future of predictive crime mapping. In N. Tilley (ed.), Analysis for Crime Prevention 13: 29-58). Criminal Justice Press: Monsey, NY. Guerette, Rob T., and Kate J. Bowers 2009. Assessing the extent of crime displacement and diffusion of benefits: a review of situational crime prevention evaluations. Criminology 47(4): 1331-1368. Greene, William H. 2003. Econometric Analysis, fifth edition. New York: Prentice Hall. Hastie, Trevor, Robert Tibshirani and Jerome Friedman (2009) The elements of statistical learning, Second Edition. New York: Springer. Heckman, James J. 2000. Causal parameters and policy analysis in economics: a twentieth century retrospective. Quarterly Journal of Economics 45-97. Holland, Pail W. 1986. Statistics and causal inference. Journal of the American Statistical Association 81: 945-960. Imbens, Guido. 2004. Nonparametric estimation of average treatment effects under exogeneity: a review. The Review of Economics and Statistics 86(1): 4-29. Imbens, Guido. 2009. Better late than nothing: some comments on Deaton (2009) and Heckman and Urzua (2009). Working Paper, Department of Economics, Harvard University. Kennedy, Peter. (1979) A Guide to Econometrics. Boston: MIT Press. Kish, Leslie. 1987. Statistical design for research. New York: John Wiley & Sons. Leamer, Edward E. 1978. Specification searches: ad hoc inference with nonexperimental Data. New York, John Wiley & Sons. Leamer, Edward E. 1983. Let’s take the con of econometrics. American Economics Review 73: 31–43. Leamer, Edward E. 2010. Tantalus on the road to asymptopia. Journal of Economic Perspectives 24(2): 1-16. 11

Leeb, Hannes and Benedikt M. P¨otscher. 2005. Model selection and inference: facts and fiction,” Econometric Theory 21: 21–59. Leeb, Hannes and Benedikt .M. P¨otscher 2006. Can one estimate the conditional distribution of post-model-selection estimators?” The Annals of Statistics 34(5): 2554–2591. Lieberson, Stanley. 1985. Making it count: the improvement of social research and theory. Berkeley, University of California Press. Lohr, Steven. 2009. For today’s graduate, just one word: statistics. New York Times, August 5, 2009. Manski, Charles. 1995. Identification problems in the social sciences. Cambridge: Cambridge University Press. Mears, Daniel P. and William D. Bales. 2009. Supermax incarceration amd recodovism. Criminology 47(4): 1131-1166. Morgan, Stephen L, and Christopher Winship. 2007. Counterfactuals and causal inference: methods and principles for social research. Cambridge: Cambridge University Press. Pearl, Judea. 2000. Causality: models, reasoning and inference. Cambridge: Cambridge University Press. Rosenbaum, Paul 2002a. Observational studies, second edition. New York: Springer-Verlag. Rosenbaum, Paul. 2002b. Covariance adjustments in randomized experiments and observational studies. Statistical Science 17(3): 286-327. Rosenbaum, Paul, 2010. The Design of Observational studies, New York: Springer-Verlag. Rubin, Donald B. 1986. Which ifs have causal answers. Journal of the American Statistical Association 81: 961-962. Rubin, Donald B. 2008. For objective causal inference, design trumps analysis. The Annals of Applied Statistics 2(3): 808-840.

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Schultz,Thomas D., and Jennifer J. Tabanico. 2009. Criminal beware: a social norms perspective on posting public warning signs. Criminology 47(4): 1201-1222. Skardhamar, Torbjørn. 2010. “Distinguishing facts from artifacts in groupbased modeling.” Criminology 48(1): 295-320. Tukey, John. 1977. Exploratory data analysis. Reading, MA: Addison -Wesley.

13

1

Introduction

Regression analysis, broadly construed, has over the past 60 years become the dominant statistical paradigm within the social sciences and criminology. In its most canonical and popular form, a regression analysis becomes a “structural equation model” from which “causal effects” can be estimated. Consider some examples from the two most recent issues of Criminology. Volume 47, Issue 4, has ten articles, seven of which employ some form of casual modeling. One exception (Mears and Bales, 2009) estimates causal effects using matching, another exception (Schultz and Tabanico, 2009) estimates causal effects using randomized experiments, and the final exception (Guerette and Bowers, 2009) estimates causal effects using meta-analysis.1 Volume 48, Issue 1, has nine articles of which eight employ some form of causal modeling. The one exception (Skardhamar, 2010) is a useful critique of a causal modeling special case: “semiparametric group-based modeling.”2 ∗

Thanks go to Alex Piquero and Jim Lynch for very help comments on earlier drafts of this paper. 1 Of course, much the research on which Guerette and Bowers rely employs causal modeling. For a discussion of well-known, but typically ignored, problems with metaanalysis, see Berk (2007) 2 See Berk et al. (2010) for a more formal and general discussion.

1

The many problems with causal modeling are well articulated in a very large literature (e.g., Box, 1976; Leamer, 1978; 1983; Lieberson, 1985; Holland, 1986; Rubin, 1986; Freedman, 1987; 2005; Kish, 1987; de Leeuw, 1994; Manski, 1995; Pearl, 2000; Heckman, 2000; Breiman, 2001; Berk, 2004; Imbens, 2009). Perhaps too simply put, before the data are analyzed, one must have a causal model that is nearly right.3 With the model specified, all that remains is to estimate the values of the regression parameters. In practice, one or two small flaws can sometimes be tolerated as long as they are easily identified and corrected with the data on hand. It is very difficult to find empirical research demonstrably based on nearly right models. In the absence of a nearly right model, the many desirable statistical properties of a causal model can be badly compromised. Omitted variables, for instance, can badly bias parameter estimates even in very large samples. Alternatively, using the data to inductively arrive at a nearly right model assumes that all of the required regressors are included in the data set. Even if they are (and how would you know?), badly biased estimates can result because the model and the parameter estimates are extracted from the same data (Leeb and P¨otscher, 2005; 2006; Berk et al., 2010a). There are five kinds of responses to the causal modeling critique. The first is rhetorical and is by far the least constructive. David Freedman compiled the following (but incomplete) list of ripostes (2005: 195). We all know that. Nothing is perfect. Linearity has to be a good first approximation. Log linearity has to be a good first approximation. The assumptions are reasonable. The assumptions don’t matter. The assumptions are conservative. You can’t prove the assumptions are wrong. The biases will cancel. We can model the biases. We’re only doing what everyone else does. Now we will use more sophisticated techniques. If we don’t do it, someone else will. What would you do? The decision-maker has to be better off with us than without us. We all have mental models, not using a model is still a model. The models aren’t totally useless. You have to do the best you can with the data. You have to make assumptions in order to make progress. You have to give the models the benefit of the doubt. Where’s the harm? 3

A model is the “right” model when it accurately represents how the data on hand were generated. A detailed discussion can be found in many formal expositions of causal modeling (e.g., Freedman, 2005).

2

The obvious problem with these and the many other rhetorical devices is that they are little more than a reflexive defense of the problematic status quo. The second response is to assert that the current assortment of regression diagnostics can find any serious problems in a regression model and that these problems can then be readily fixed. For example, model misspecification can be identified with “specification tests,” and a subsequent instrumental variable estimator can save the day (e.g., Greene, Section 5.5.). In principle, this view has merit. In practice, it usually stumbles badly. For both the diagnostics and the remedies, new and untestable assumptions are required even before one gets to a number of thorny technical complications. On the matter of diagnostics, Freedman observes (2009: 839) that , “...skepticism about diagnostics is warranted. As shown by the theorems presented here, a model can pass diagnostics with flying colors yet be ill-suited for the task at hand.” He is no more sanguine about about instrumental variable solutions (Freedman, 2005: Section 8.5) and has lots of company (e.g., Berk, 2004: Section 9.5; Kennedy, 2008: Chapter 9; Leamer, 2010). The third response is what I have elsewhere called “Model-Lite Causal Analysis” (Berk, 2009). The basic idea is apply the randomized experimental paradigm to observational data. One tries to make the case that conditional on a set of covariates, “nature” conducted a randomized experiment. This naturally leads to a variety of matching or post-stratification analysis methods that can be more robust than causal models (Rubin, 2008). Regression models can creep in, however, in the construction matching variables such propensity scores or in post-matching adjustments for sample imbalance (Rosenbaum, 2002b). Morgan and Winship (2007) provide an excellent overview of the issues. The primary literature can be found in the influential writings of William Cochran, Donald Rubin, Paul Rosenbaum, Judea Pearl and others. Unfortunately, the elephant in the room remains: there is no apparent way to overcome the impact of omitted variables although in some cases, sensitivity analyses can suggest that the omitted variables may not be important (Rosenbaum, 2002a, Chapter 4; 2010, chapter 14). The fourth response is to rely on research designs far better suited for causal inference than the usual observational approaches (Angrist and Pischke, 2010). Properly implemented randomized experiments are the reigning poster child. But strong quasi-experiments will often perform nearly as well (Cook et al., 2008). For example, recent advances in the regression discon3

tinuity design (Imbens and Lemieux, 2008) can make that approach very attractive when a randomized experiment is not feasible (Berk et al., 2010b). Capitalizing on stronger designs is clearly a useful suggestion when such designs are possible. The fifth response is to go back to the formal definition of a regression analysis and reconsider what can be learned depending on the assumptions that one can credibly make. We turn to that approach now. There will be some good news.

2

What is Regression Analysis?

Cook and Weisberg (1999: 27) provide a definition of regression analysis that comports well with standard statistical perspectives: “[to understand] as far as possible with the available data how the conditional distribution of the response y varies across subpopulations determined by the possible values of the predictor or predictors.” The definition makes the entire conditional distribution of y fair game although in practice, the conditional mean and/or conditional variance are the primary focus. A key point is that there is no mention of estimation, hypothesis tests, or confidence intervals nor any reference to causal inference. One can do a proper regression analysis without any effort to address the role of chance or to make causal statements.

2.1

Level I Regression Analysis

We have seen that by definition, regression analysis is a procedure by which conditional relationships in data may be described. One might consider, for example, how the homicide rate in a city varies with unemployment, police practices fixed. And one might consider how the homicide rate in a city varies with police practices, unemployment fixed. In effect, one is identifying interesting patterns in the data, which can be subtle, complicated and even rare. The patterns can be found over time, over space, and over observational units that can differ in complex ways. The analysis can be directed by existing theory or can be highly exploratory. One is not limited to conventional linear regression. For example, the definition includes response variables that are categorical, ordinal or counts. Categorical regressors are can be included. Nonlinear relationships can be

4

taken into account. One can consider several response variables at once such as a parolee’s arrests for new crimes and the charges for those new crimes. It follows that for description, one need not worry about all of the potentially problematic assumptions required if one is to undertake credible statistical or causal inference. One just characterizes associations in the data at hand. Thus, it does not matter, for instance, if the data are a probability sample or the product of well-defined stochastic process. Likewise, there is no concern about omitted variables. Indeed, it is not clear how to define an omitted variable in this context. It is the proverbial “what you see is what you get.” Consistent with a more expansive discussion elsewhere (Berk, 2003: Section 11.5), one can call descriptive regression a “Level I” regression analysis. Level I regression analyses are always formally appropriate when a regression analysis could be useful and do not depend on any of the assumptions required for statistical inference or causal inference. This is not a call to abandon advanced statistical procedures and complex quantitative reasoning. Description has long been at the center of state-ofthe-art statistical practice. Among the more popular approaches are multivariate statistics (Anderson, 1958; Gifi, 1990), exploratory data analysis (Tukey, 1977; Diaconis, 1985), and more recently, dynamic graphics (Cook and Swayne, 2007) and Machine/Statistical Learning (Hastie et al., 2009, Berk, 2008). The growing use of GIS studies in criminology is one example of graphical methods (Groff and La Vigne, 2001; Chainey and Ratcliffe, 2005). It is then a small step to build visualizations that take both time and space into account (Berk et al., 2009). Machine learning is also finding its way into the criminology journals (Berk et al., 2009). Making descriptive regression analysis the central approach to data analysis in criminology is not a radical suggestion. Despite the statistical framing that one finds in the criminology journals, description is usually the actual enterprise. And the reason is apparent: in criminology, and for most social science applications more generally, there are rarely any widely accepted, nearly right models that can be used with real data. By default, the true enterprise is description. Most everything else is puffery.

2.2

Level II Regression Analysis

Level I regression analysis does not require any assumptions about how the data were generated. If one wants more from the data analysis, assumptions are required. For a Level II regression analysis, the added feature is statistical 5

inference: estimation, hypothesis tests and confidence intervals. When the data are produced by probability sampling from a well-defined population, estimation, hypothesis tests and confidence intervals are on the table. A random sample of inmates from the set of all inmates in a state’s prison system might be properly used to estimate, for example, the number of gang members in state’s overall prison system. Hypothesis tests and confidence intervals might also be usefully employed. In addition, one might estimate, for instance, the distribution of in-prison misconduct committed by men compared to the in-prison misconduct committed by women, holding age fixed. Hypothesis tests or confidence intervals could again follow naturally. The key assumption is that each inmate in the population has a known probability of selection. If the probability sampling is implemented largely as designed, statistical inference can rest on reasonably sound footing. Note that there is no talk of causal effects and no causal model. Description is combined with statistical inference. In the absence of probability sampling, the case for Level II regression analysis is far more difficult to make. There are several options addressed in some depth elsewhere (Berk, 2003: 39-58). 1. Treating the data as a population — In effect, this is a fallback to Level I regression analysis in which there is no statistical inference. 2. Treating the data as if it were a probability sample from a well-defined population — For example, a convenience sample of big-city police departments might be treated as if it were a simple random sample of police departments from cities of over 100,000 residents. One would have to argue convincingly that despite the lack of formal probability sampling, the way in which the data were generated is de facto a close approximation. And that, in turn, would lead to an in-depth discussion of why one should believe that each department in the sample was, in effect, drawn independently of all others with a known probability of selection. If access was obtained by a sequence of referrals, the as-if approach would fail. It would also fail if the sample was limited to the subset of departments for which the requisite data were available. The as-if approach is rarely credible in practice. 3. Treating the data as a random realization from an imaginary “superpopulation” — Inferences to imaginary populations are imaginary. However, there is an important distinction between an imaginary popu6

lation that could exist and an imaginary population that could not. Inferences to the former might not be imaginary, but are even more difficult to justify than inferences from the as-if strategy. The result can be a mind numbing exercise in circular reasoning. The population is defined as the population that would exist if the data were a random sample from that population. 4. Model based sampling — For this strategy, one needs a credible stochastic model of how the data were generated. For example, with conventional linear regression, one assumes y i = β T Xi + ε i ,

(1)

where εi ∼ N IID(0, σ 2 ). If the model really captures how the data were generated, statistical inference can easily follow. But, inferences are made to the data generation process, not to a real population in the usual sense.4 Model-based sampling leads one back into much of the critical literature cited earlier. Why should one believe the model? An omitted variable, for instance, will mean that the assumed properties of the disturbance term do not hold, and statistical inference can be badly compromised. The same can follow when the wrong functional forms are assumed. In short, a Level II regression analysis depends on how the data were generated. If the data are a well-implemented probability sample, Level I description can be supplemented by Level II statistical inference. Without probability sampling, Level II regression analysis can be difficult to justify.

2.3

Level III Regression Analysis

The goal in a Level III regression analysis is to supplement Level I description and Level II statistical inference with causal inference. In conventional regression, for instance, one needs a nearly right model like equation 1, but one must also be able to argue credibly that manipulation of one or more regressors alters the expected conditional distribution of the response. Moreover, any given causal variable can be manipulated independently of any other 4

Alternatively, one might define the population as all possible realizations of the given stochastic process. This population is imaginary.

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causal variable and independently of the disturbances. There is nothing in the data itself that can speak to these requirements. The case will rest on how the data were actually produced. For example, if there was a real intervention, a good argument for manipulability might well be made. Thus, an explicit change in police patrolling practices ordered by the local Chief will perhaps pass the manipulability sniff test. Changes in the demographic mix of a neighborhood will probably not. Here too, there is extensive discussion elsewhere (e.g., Berk, 2003, Chapter 5). Suffice it to say, it is very difficult to find in criminology credible examples of a Level III regression analysis. Either the parallel to equation 1 is not credible, claims of manipulability are not credible, or both.

3

Conclusions

Level I regression analysis is always formally appropriate and does not depend on any assumptions about how the data were generated. Truth be told, description is really the de facto product of most regression analyses in criminology. Description is not just a legitimate scientific activity, but corresponds well to the developmental stage in which criminology finds itself. Level II regression analysis adds to description, estimation, hypothesis tests and confidence intervals. When the data are a probability sample from a well-defined population, statistical inference can follow easily. When the data are not, statistical inference can be difficult to justify. Level III regression analysis adds to description and statistical inference, causal inference. One requires not just a nearly right model of how the data were generated, but good information justifying any claims that all causal variables are independently manipulable. In the absence of a nearly right model and one or more regressors whose values can be “set” independently of other regressors and the disturbances, causal inferences cannot not make much sense. The implications for practice in criminology are clear but somewhat daunting. With rare exceptions, regression analyses of observational data are best undertaken at Level I. With proper sampling, a Level II analysis can be helpful. The goal is to characterize associations in the data, perhaps taking uncertainty into account. The daunting part is getting the analysis past criminology gatekeepers. Reviewers and journal editors typically equate proper statistical practice with Level III. 8

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