Abstract
While the linear regression model is common to Political Science, many of the outcome measures researchers wish to study are binary, ordinal, nominal, or count variables. When we study these limited dependent variables, we turn to techniques such as logistic regression, probit regression, ordered logit (and probit) regression, multinomial logit (and probit) regression, Poisson regression, and negative binomial regression. A review of these and several other methods can be seen in volumes such as KingĀ (1989) and LongĀ (1997).
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Notes
- 1.
In this case, the coefficient estimates we obtain are similar to those reported by SinghĀ (2014a). However, our standard errors are smaller (and hence z and p values are bigger) because Singh clusters the standard errors. This is a useful idea because the respondents are nested within elections, though multilevel models (which Singh also reports) address this issue as wellāsee Sect.ā8.1
- 2.
LaTeXĀ users can create a table similar to this quickly by typing: library(xtable); xtable(inc.linear).
- 3.
An explanation of how the inferential properties of this model are derived can be found in EliasonĀ (1993, pp.Ā 26ā27).
- 4.
Deviance is calculated as ā2 times the logged ratio of the fitted likelihood to the saturated likelihood. Formally, \(-2\log \frac{L_{1}} {L_{2}}\), where L 1 is the fitted likelihood and L 2 is the saturated likelihood. RĀ reports two quantities: the null deviance computes this for an intercept-only model that always predicts the modal value, and the residual deviance calculates this for the reported model.
- 5.
Additionally, in advanced settings for which we need to develop a multivariate distribution for multiple outcome variables, the normal distribution is relatively easy to work with.
- 6.
This is because the logit link function is the log-odds, or logarithm of the odds of the event.
- 7.
This and the next example do not exactly replicate the original results, which also include random effects by country-year. Also, the next example illustrates ordered probit regression, instead of the ordered logistic model from the original article. Both of the examples are based on models found in the online supporting material at the European Journal of Political Research website.
- 8.
If a user does need to install the package, install.packages("MASS") will do the job.
- 9.
An equivalent specification would have been to includeĀ voted_ideo+winner+ winnerXvoted_ideo as three separate terms from the data.
- 10.
Unfortunately the xtable command does not produce ready-made LaTeXĀ tables for results from polr. By creating a matrix with the relevant results, though, LaTeXĀ users can produce a table faster than hand coding, though some revisions of the final product are necessary. Try the following:
coef<-c(ideol.satisfaction$coefficients,ideol.satisfaction$zeta)
se<-sqrt(diag(vcov(ideol.satisfaction)))
z<-coef/se
p<-2*(1-pnorm(abs(z)))
xtable(cbind(coef,se,z,p),digits=4)
- 11.
The footnote should read: āFor users who do not have the file handy from Chapter 3, please download the file from the Dataverse linked on page vii or the chapter content linked on page 97.
- 12.
Note that the presidential speech terms are coded 1 only in the month of the speech, and 0 in all other months. The terms for the oil embargo and hostage crisis were coded 1 while these events were ongoing and 0 otherwise.
- 13.
Besides this approach of making predictions using central values of control variables, Hanmer and KalkanĀ (2013) make the case that forecasting outcomes based on the observed values of control variables in the data set is preferable. Readers are encouraged to consult their article for further advice on this issue.
- 14.
As a side note, by using Rās matrix algebra commands, described further in Chap.ā10, the user can compute predicted counts easily with alternate syntax. For instance, for the negative binomial model, we could have typed: forecast.nb<-exp(as.matrix(inputs.4)%*%energy.nb$coefficients)ā.
- 15.
Just as in the example from the chapter, these are time series data, so methods from Chap.ā9 are more appropriate.
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Monogan, J.E. (2015). Generalized Linear Models. In: Political Analysis Using R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-23446-5_7
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