Abstract
Ordinary least squares regression is a very useful tool for identifying how one or a series of independent variables affects an interval-level dependent variable. As noted in Chapter 16, this method may also be used—though with caution—to explain dependent variables that are measured at an ordinal level. But what should the researcher do when faced with a binary or dichotomous dependent variable? Such situations are common in criminology and criminal justice. For example, in examining sentencing practices, the researcher may want to explain why certain defendants get a prison sentence while others do not. In assessing the success of a drug treatment program, the researcher may be interested in whether offenders failed a drug test or whether they returned to prison within a fixed follow-up period. In each of these examples, the variable that the researcher seeks to explain is a simple binary outcome. It is not appropriate to examine binary dependent variables using the regression methods that we have reviewed thus far.
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Notes
- 1.
A method called generalized least squares might also be used to deal with violations of our assumptions, though logistic regression analysis is generally the preferred method. See E. A. Hanushek and J. E. Jackson, Statistical Methods for Social Scientists (New York: Academic Press, 1977) for a comparison of these approaches. See also David W. Hosmer and Stanley Lemeshow, Applied Regression Analysis, 2nd ed. (New York: Wiley, 2000). Another method, probit regression analysis, is very similar to that presented here, though it is based on the standard normal distribution rather than the logistic model curve. The estimates gained from probit regression are likely to be very similar to those gained from logistic regression. Because logistic regression analysis has become much more widely used and is available in most statistical software packages, we focus on logistic regression in this chapter.
- 2.
Your calculator likely has a button labeled "e x," which performs this operation. If there is no e x button, then you should be able to locate a button labeled "INV" and another for the natural logarithm, ln. By pushing "INV" and then "ln" (the inverse or antilog), you will be able to perform this operation.
- 3.
It should be noted, however, that maximum likelihood techniques do not always require an iterative process.
- 4.
See John Tukey, Report to the Special Master, p. 5; Report to the New Jersey Supreme Court 27 (1997).
- 5.
For a description of this study, see David Weisburd, Stephen Mastrofski, Ann Marie McNally, and Rosann Greenspan, Compstat and Organizational Change (Washington, DC: The Police Foundation, 2001).
- 6.
Departments with 1,300 or more officers were coded in our example as 1,300 officers. This transformation was used in order to take into account the fact that only 5% of the departments surveyed had more than this number of officers and their totals varied very widely relative to the overall distribution. Another solution that could be used to address the problem of outliers is to define the measure as the logarithm of the number of sworn officers, rather than the raw scores. We relied on the former solution for our example because interpretation of the coefficients is more straightforward. In an analysis of this problem, a researcher would ordinarily want to compare different transformations of the dependent variable in order to define the one that best fit the data being examined.
- 7.
See Stanton Wheeler, David Weisburd, and Nancy Bode, "Sentencing the White Collar Offender: Rhetoric and Reality," American Sociological Review 47 (1982): 641–659.
- 8.
Some researchers have proposed alternative ways of calculating standardized logistic regression coefficients that allow for interpretations related to changes in probabilities. See, for example, Robert L. Kaufman, "Comparing Effects in Dichotomous Logistic Regression: A Variety of Standardized Coefficients," Social Science Quarterly 77 (1996): 90–109.
- 9.
For example, see Andy Field, Discovering Statistics Using SPSS for Windows (London: Sage Publications, 2000).
- 10.
As with standardized regression coefficients in OLS regression, you should not compare standardized logistic regression coefficients across models. Moreover, while we report standardized regression coefficients for the dummy variables included in the model, you should use caution in interpreting standardized coefficients for dummy variables. See Chapter 16, pages 493–494, for a discussion of this problem.
- 11.
D. R. Cox and E. J. Snell, The Analysis of Binary Data, 2nd ed. (London: Chapman and Hall, 1989).
- 12.
See N. J. D. Nagelkerke, "A Note on a General Definition of the Coefficient of Determination, Biometrika 78 (1991): 691–692.
- 13.
We discuss the Wald statistic in detail here, because it is the most common test of statistical significance reported in many statistical software applications. it should be noted, however, that some researchers have noted that the Wald statistic is sensitive to small sample sizes (e.g., less than 100 cases). The likelihood-ratio test discussed later in this chapter and in more detail in Chapter 19 offers an alternative test for statistical significance that is appropriate to both small and large samples (see J. Scott Long, Regression Models for Categorical and Limited Dependent Variables (Thousand Oaks, CA, Sage, 1997).
- 14.
The difference between our result and that shown in Table 18.12 is due to rounding error.
- 15.
This printout is identical to that in Table 18.5. It is reproduced here for easy reference as you work through the computations presented in this section.
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Weisburd, D., Britt, C. (2014). Logistic Regression. In: Statistics in Criminal Justice. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9170-5_18
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