Addressing the Zeros Problem: Regression Models for Outcomes with a Large Proportion of Zeros, with an Application to Trial Outcomes

• Published date: 01 March 2015
• Author: Thomas Eisenberg,Theodore Eisenberg,Martin T. Wells,Min Zhang
• DOI: http://doi.org/10.1111/jels.12068
• Date: 01 March 2015

Addressing the Zeros Problem: Regression

Models for Outcomes with a Large

Proportion of Zeros, with an Application

to Trial Outcomes

Theodore Eisenberg, Thomas Eisenberg, Martin T. Wells, and Min Zhang*

In law-related and other social science contexts, researchers need to account for data with an

excess number of zeros. In addition, dollar damages in legal cases also often are skewed. This

article reviews various strategies for dealing with this data type. Tobit models are often

applied to deal with the excess number of zeros, but these are more appropriate in cases of

true censoring (e.g., when all negative values are recorded as zeros) and less appropriate

when zeros are in fact often observed as the amount awarded. Heckman selection models are

another methodology that is applied in this setting, yet they were developed for potential

outcomes rather than actual ones. Two-part models account for actual outcomes and avoid

the collinearity problems that often attend selection models. A two-part hierarchical model

is developed here that accounts for both the skewed, zero-inflated nature of damages data

and the fact that punitive damage awards may be correlated within case type, jurisdiction, or

time. Inference is conducted using a Markov chain Monte Carlo sampling scheme. Tobit

models, selection models, and two-part models are fit to two punitive damage awards data

sets and the results are compared. We illustrate that the nonsignificance of coefficients in a

selection model can be a consequence of collinearity, whereas that does not occur with

two-part models.

I. Introduction

Many legal system and other social science outcomes raise the question of how to model

phenomena with multiple zeros. Trial outcomes generally result in verdicts for plaintiffs or

defendants. A verdict for a plaintiff in an action involving money damages will lead to a

positive award. A verdict for a defendant will lead to a zero award. For some purposes, the

*Address correspondence to Martin T. Wells, email: mtw1@cornell.edu. Theodore Eisenberg was Henry Allen Mark

Professor, Cornell Law School and Adjunct Professor of Statistical Sciences, Cornell University; Thomas Eisenberg is

a gradate student, Department of Economics, Cornell University; Wells is Charles A. Alexander Professor of Statistical

Sciences, Professor of Biostatistics and Epidemiology, Weill Medical College, and Elected Member of the Law Faculty,

Cornell University; Zhang is Associate Professor Department of Statistics, Purdue University.

Earlier versions of this material were presented at the First Annual Conference on Empirical Legal Studies,

University of Texas and the International Conference on Empirical Legal Studies, conducted by the Cegla Center for

Interdisciplinary Research of the Law, Tel Aviv University, Buchmann Faculty of Law.

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Journal of Empirical Legal Studies

Volume 12, Issue 1, 161–186, March 2015

161

mass of zeros represented by defendant victories need not be accounted for. The research

question of interest might be, conditional on plaintiff winning at trial, how much was

recovered? In that case, the zero-award outcomes representing defendant trial wins might

be ignored.

If, however, the researcher wished to include defendant wins in the analysis, ignoring

the zero-award outcomes would not be satisfactory. This might be the case if one were

computing the expected value of a possible lawsuit and wanted to account for both the

probability of winning and the amount of any monetary award if the plaintiff prevailed.

Similarly, if one were interested in the amount of punitive damages awarded to a plaintiff

who won at trial and received a punitive damages award, the cases with punitive damages

awards of zero could be excluded. But if one were interested in the expected punitive

damages recovery in cases won by plaintiffs, it would be necessary to include the cases in

which punitive damages awards were zero in cases won by plaintiffs.

Many empirical legal researchers realize that simple ordinary least squares methods

may be unsatisfactory in the presence of many observations for which the dependent

variable equals zero. Tobit models often are regarded as appropriate when data have a

lower boundary of zero to avoid possibly biased and inconsistent ordinary least squares

estimates (Tobin 1958). A common approach in the presence of many zero values is

therefore to use Tobit models, which account for censoring of the dependent variable (e.g.,

De Ruijter & Braat 2008; Fehr & Gächter 2000; Hersch & Viscusi 2004). A possible problem

is that Tobit models do not account for the skewed, zero-inflated nature of damages data.

Tobit models assume censoring of the dependent variable rather than the dependent

variable often being observably equal to zero. This article reviews techniques for modeling

skewed damages awards in the presence of many zero values. This article develops a

two-part hierarchical model that accounts for the skewed, zero-inflated, and clustered

nature of damages data. It compares the results of the Tobit, selection, and two-part models

for two punitive damages data sets.

The underlying topics and statistical models that are discussed in this article have

appeared in various econometric and statistics literatures. Each of the models we discuss

below makes implicit distributional assumptions that need to be understood and vali-

dated. The goal of this article is to give an overview of the statistical models and issues

that face a legal researcher when analyzing data with many zeros. The rest of the article

is organized as follows. Section II reviews various models for analyzing data with a

large proportion of the continuous outcome variable being zero. Section III applies the

model to real data. We close the article with some discussion in Section IV and conclu-

sions. The Appendix gives Rand OpenBugs code useful for fitting the two-part hierarchi-

cal model.

II. Models for Analyzing a Continuous Outcome

Variable with a Large Proportion of Zeros

In empirical legal literature and other social science literature, there have been misunder-

standings about the proper modeling of a continuous outcome variable with a large

162 Eisenberg et al.