Abstract
Analysis of crime and criminal justice data often requires the researcher to deal with a categorical outcome variable that may be ordered or unordered. Our focus in this chapter is a discussion on the type of logistic regression model best suited to an analysis of categorical outcome variables. We review binary logistic regression models for situations where the dependent variable has only two categories, and then build on this material to illustrate the application and interpretation of multinomial and two different ordinal logistic regression models (proportional odds and partial proportional odds) for situations where the dependent variable has at least three unordered or ordered categories, respectively. Our general approach in the discussion of each model will be to highlight some of the key technical details of the model, and then to emphasize the application and the interpretation of the model. We conclude by noting some alternative logistic regression models that extend the models discussed in this chapter and which may have interesting applications in the study of crime and criminal justice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Software packages differ in the default results that are reported. For example, SAS and SPSS report the Wald statistic, while Stata and LIMDEP report the z-statistic. Practically, it makes no difference which statistic (W or z) is reported, since the observed significance level of the coefficient will be the same.
- 2.
- 3.
These three logits can be linked in an identity equation that illustrates how knowledge of any two logits can produce the values of the third. The identity equation can be stated as
$$\ln \left (\frac{P\!\left (Y = C1\right )} {P\!\left (Y = C2\right )}\right ) +\ln \left (\frac{P\!\left (Y = C2\right )} {P\!\left (Y = C3\right )}\right ) =\ln \left (\frac{P\!\left (Y = C1\right )} {P\!\left (Y = C3\right )}\right ).$$ - 4.
As noted previously, the choice of the reference category is arbitrary. All possible comparisons of the outcome categories can be made based on single set of J−1 logits.
- 5.
It is worth pointing out that the binary logistic regression model presented above is a special case of the multinomial logistic regression model, where m=2.
- 6.
The vertical spread of the dots is simply a convenience to illustrate the placement of cases along the Agreement–Disagreement continuum.
- 7.
Long and Freese (2006) have written a procedure for Stata to compute the Brant Test.
- 8.
The Score Test similarly indicates a failure of the full model to satisfy the parallel slopes assumption (χ2=81.39, df =12, p<0.001).
- 9.
King and Zeng (2001a, b) also note that their corrective procedure is not applicable to situations where the researcher is studying a rare event in a small sample.
- 10.
Through some algebraic manipulation, the conditional logit model can be shown to be equivalent to the multinomial logistic regression model (see, for example, Long 1997).
References
Agresti A (1984) Analysis of ordinal categorical data. Wiley, New York
Agresti A (2002) Categorical data analysis, 2nd ed. Wiley, New York
Brant R (1990) Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics 46:1171–1178
Clogg C, Shihadeh E (1994) Statistical models for ordinal data. Sage, Thousand Oaks, CA
Fox J, Andersen R (2006) Effect displays for multinomial and proportional-odds logit models. Sociol Methodol 36:225–255
Fox J (2003) Effect displays in R for generalised linear models. J Stat Softw 8:1–27
Fox J (2008) Applied regression analysis and generalized linear models, 2nd ed. Sage, Thousand Oaks, CA
Fu V (1998) Estimating generalized ordered logit models. Stata Technical Bulletin 8:160–164
Greene W (2008) LIMDEP 9.0 Econometric modeling guide, volumes 1 and 2. Econometric Software, Plainville, NY
Holleran D, Sphon C (2004) On the use of the total incarceration variable in sentencing research. Criminology 42:211–240
Hosmer, and Lemeshow 2000. Applied logistic regression, 2nd ed. Wiley, New York
King G, Zeng L (2001a) Explaining rare events in international relations. International Organization 55:693–715
King G, Zeng L (2001b) Logistic regression in rare events data. Polit Anal 9:137–163
Lall R, Walters SJ, Morgan K, MRC CFAS Co-operative Institute of Public Health (2002) A review of ordinal regression models applied on health-related quality of life assessments. Stat Methods Med Res 11:49–67
Long JS (1987) A graphical method for the interpretation of multinomial logit analysis. Sociological Methods and Research 15:420–446
Long JS (1997) Regression models for categorical and limited dependent variables. Sage, Thousand Oaks, CA
Long JS, Freese J (2006) Regression models for categorical dependent variables using Stata, 2nd ed. Stata, College Station, TX
Maddala GS (1983) Limited-dependent and qualitative variables in econometrics. Cambridge University Press, New York
McFadden D (1973) Conditional logit analysis of qualitative choice behavior. In: Zarembka P (ed) Frontiers of econometrics. Academic, New York, pp 105–142
McFadden D (1978) Modelling the choice of residential location. In: Karlquist A, et al. (eds) Spatial interaction theory and residential location. North-Holland, Amsterdam, pp 75–96
McFadden D (1981) Econometric models of probabilistic choice. In: Manski CF, McFaden D (eds) Structural analysis of discrete data: with econometric applications. MIT, Cambridge, MA
Menard S (2002) Applied logistic regression analysis, 2nd ed. Sage, Thousand Oaks, CA
O’Connell AA (2006) Logistic regression models for ordinal response variables. Sage, Thousand Oaks, CA
Pampel F (2000) Logistic regression: a primer. Sage, Thousand Oaks, CA
Peterson B, Harrell FE (1990) Partial proportional odds models for ordinal response variables. Appl Stat 39:205–217
Phillips S (2003) Social structure of vengeance: a test of Black’s model. Criminology 41:673–708
Piquero AR, MacDonald J, Dobrin A, Daigle LH, Cullen FT (2005) Self-control, violent offending, and homicide victimization: assessing the general theory of crime. J Quant Criminol 21:55–71
Plotnick RD (1992) The effects of attitudes on teenage premarital pregnancy and its resolution. Am Sociol Rev 57:800–811
R Development Core Team (2003) R: A language and environment for statistical computing. http://www.R-project.org (accessed 16 November 2009)
SAS Institute (2009) SAS 9.2. SAS Institute, Cary, NC
South S, Baumer E (2001) Community effects on the resolution of adolescent premarital pregnancy. J Fam Issues 22:1025–1043
SPSS Inc (2007) SPSS 16.0 for Windows. SPSS, Chicago
Stata Corporation (2009) Stata 10. Stata Corporation, College Station, TX
Williams R (2005) Gologit2: a program for generalized logistic regression/partial proportional odds models for ordinal variables. http://www.nd.edu/~rwilliam/gologit2/gologit2.pdf, Accessed April, 2009
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Britt, C.L., Weisburd, D. (2010). Logistic Regression Models for Categorical Outcome Variables. In: Piquero, A., Weisburd, D. (eds) Handbook of Quantitative Criminology. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77650-7_31
Download citation
DOI: https://doi.org/10.1007/978-0-387-77650-7_31
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-77649-1
Online ISBN: 978-0-387-77650-7
eBook Packages: Humanities, Social Sciences and LawSocial Sciences (R0)