Abstract
In the previous chapter, we examined how to analyze data in a binary logistic regression model that included a dependent variable with two categories. This allowed us to overcome problems associated with using Ordinary Least Squares Regression in cases where the variable that is being explained is measured as a simple dichotomy. Accordingly, we have now described tools that allow the researcher to develop explanatory models with either an interval dependent variable or a dichotomous dependent variable.
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Notes
- 1.
See, for example, D. Nagin, Group-based M odeling of Develop ment (Cambridge, MA: Harvard University Press, 2005).
- 2.
You can verify this identity by using the fact that the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator: ln(x/y) =ln(x) - ln(y). Specifically, for this equality, we note that
$$\ln \left( \frac{P(Y = C1)}{P(Y = C2)}\right) = \ln(P(Y=C1)) - \ln(P(Y=C2))$$and
$$\ln \left( \frac{P(Y = C2)}{P(Y = C3)}\right) = \ln(P(Y=C2)) - \ln(P(Y=C3)).$$When we put these two pieces together in a single equation, we have
$$\begin{array}{lll}[\ln(P(Y & = & C1))-\ln(P(Y=C2))]+[\ln(P(Y=C2))-\ln(P(Y=C3))] \\ & = & \ln(P(Y=C1)) - \ln (P(Y=C2)) + \ln(P(Y=C2)) - \ln(P(Y=C3)) \\ & = & \ln(P(Y=C1)) - \ln(P(Y=C3)) \\ & = & \ln\left( \frac{\ln(P(Y=C1))}{\ln(P(Y=C3))} \right) \end{array}$$Which establishes the equality. We explain the practical implication of this equality below in our discussion of the interpretation of the coefficients from a multinomial logistic regression model.
- 3.
While it may appear odd at first glance that we have not included those individuals who were acquitted at a trial, there were very few individuals who fell into this category. Like most jurisdictions, courts in California acquit relatively few individuals through a trial – it was about 1% of all cases in the 1990s. What this means is that once the prosecutor has filed charges against a defendant, rather than dismiss the case, it will likely result in the conviction of the defendant through either a guilty plea or a trial conviction. This also implies that a dismissal of the case functions much like an acquittal, but one made by the prosecuting attorney rather than a judge or jury.
- 4.
It is worth pointing out that the binary logistic regression model presented in Chapter 18 is a special case of the multinomial logistic regression model, where m = 2. If you work through both Equations 19.2 and 19.3 above assuming that m = 2, you will be able to replicate the equations in the previous chapter.
- 5.
Recall from footnote # 13 in Chapter 18 that the Wald statistic is sensitive to small samples (e.g., less than 100), while the LR test is not.
- 6.
You should verify the statistical significance of each coefficient presented in Table 19.6 using a Wald test statistic with df = 1
- 7.
Brant, Rollin. 1990. “Assessing proportionality in the proportional odds model for ordinal logistic regression.” Biometrics 46: 1171–1178.
- 8.
There are a number of sources interested readers can consult, although most of these are much more technical than the material presented in this text. See, for example, Fu, Vincent. 1998. “Estimating generalized ordered logit models.” Stata Technical Bulletin 8:160–164. Lall, R., Walters, S.J., Morgan, K., and MRC CFAS Co-operative Institute of Public Health. 2002. “A review of ordinal regression models applied on health-related quality of life assessments.” Statistical Methods in Medical Research 11:49–67. O’Connell, Ann A. 2006. Logistic Regression Models for Ordinal Response Variables. Thousand Oaks, CA: Sage. Peterson, Bercedis and Harrell, Jr, Frank E. 1990. “Partial proportional odds models for ordinal response variables.” Applied Statistics 39: 205–217. Williams, Richard. 2006. “Generalized Ordered Logit/ Partial Proportional Odds Models for Ordinal Dependent Variables.” The Stata Journal 6(1):58–82.
- 9.
Steffensmeier, D., Ulmer, J. and Kramer, J. (1998), The Interaction of Race, Gender, and Age in Criminal Sentencing: The Punishment Cost of Being Young, Black, and Male. Criminology, 36: 763–798.
- 10.
Long, J.S. and J. Freese, 2006, Regression Models for Categorical Dependent Variables Using Stata, 2 ed., College Station, TX: Stata Press.
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Weisburd, D., Britt, C. (2014). Multivariate Regression with Multiple Category Nominal or Ordinal Measures. In: Statistics in Criminal Justice. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-9170-5_19
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