Читать книгу Handbook of Regression Analysis With Applications in R - Samprit Chatterjee - Страница 37

The 2000 US presidential election matching Republican George W. Bush against Democrat Al Gore attracted worldwide attention because of its close and controversial results, particularly in the state of Florida. The 2004 election, pitting the incumbent Bush against John Kerry, is less discussed, but was also controversial, in part because of the introduction of electronic voting machines in some polling places across the country (such machines were introduced in part because of the irregularities in paper balloting that occurred in Florida in the 2000 election). Some of the manufacturers of electronic voting machines were strong supporters of President Bush, and this, along with the fact that the machines did not produce a paper trail, led to speculation about whether the machines could be manipulated to favor one candidate over the other.

FIGURE 2.4: Plots for the 2004 election data. (a) Plot of percentage change in Bush vote versus 2000 Bush vote. (b) Side‐by‐side boxplots of percentage change in Bush vote by whether or not the county employed electronic voting in 2004.

This analysis is based on data from Hout et al. (2004) (see also Theus and Urbanek, 2009). The observations are the counties of Florida. Although this is not a sample of Florida counties (it is actually a census of all of them), these counties can be considered a sample of all of the counties in the country, making inferences drawn about the larger population of counties based on this set of counties meaningful. The target variable is the change in the percentage of votes cast for Bush from 2000 to 2004 (a positive number meaning a higher percentage in 2004). We start with the simple regression model relating the change in Bush percentage to the percentage of votes Bush took in 2000, with corresponding scatter plot given in the left plot of Figure 2.4. It can be seen that most of the changes are positive, reflecting that Bush carried the state by more than votes in 2004, compared with the very close result (a vote margin) in 2000.

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.9968 2.0253 -1.480 0.14379 Bush.pct.2000 0.1190 0.0355 3.352 0.00134 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.693 on 65 degrees of freedom Multiple R-squared: 0.1474, Adjusted R-squared: 0.1343 F-statistic: 11.24 on 1 and 65 DF, p-value: 0.00134

There is a weak, but statistically significant, relationship between 2000 Bush vote and the change in vote to 2004, with counties that went more strongly for Bush in 2000 gaining more in 2004. The constant shift model now adds an indicator variable for whether a county used electronic voting in 2004. The side‐by‐side boxplots in the right plot in Figure 2.4 show that overall the counties that used electronic voting had smaller gains for Bush than the that did not, but that of course does not take the 2000 Bush vote into account. There are also signs of nonconstant variance, as the variability is smaller among the counties that used electronic voting.

Coefficients: Estimate Std. Error t value Pr(>|t|) VIF (Intercept) -2.12713 2.10315 -1.011 0.31563 Bush.pct.2000 0.10804 0.03609 2.994 0.00391 1.049 ** e.Voting -1.12840 0.80218 -1.407 0.16437 1.049 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.672 on 64 degrees of freedom Multiple R-squared: 0.173, Adjusted R-squared: 0.1471 F-statistic: 6.692 on 2 and 64 DF, p-value: 0.002295

It can be seen that there is only weak (if any) evidence that the constant shift model provides improved performance over the pooled model. This does not mean that electronic voting is irrelevant, however, as it could be that two separate (unrestricted) lines are preferred.

Coefficients: Estimate Std.Error t value Pr(>|t|) VIF (Intercept) -5.23862 2.35084 -2.228 0.029431 * Bush.pct.2000 0.16228 0.04051 4.006 0.000166 1.44 *** e.Voting 9.67236 4.26530 2.268 0.026787 32.26 * Bush.2000 X e.Voting -0.20051 0.07789 -2.574 0.012403 31.10 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.562 on 63 degrees of freedom Multiple R-squared: 0.2517, Adjusted R-squared: 0.2161 F-statistic: 7.063 on 3 and 63 DF, p-value: 0.0003626

The ‐test for the product variable indicates that the model with two unrestricted lines is preferred over the model with two parallel lines. A partial ‐test comparing this model to the pooled model, which is (), also supports two distinct lines,

for counties that did not use electronic voting in 2004, and

for counties that did use electronic voting. This is represented in Figure 2.5. This relationship implies that in counties that did not use electronic voting the more Republican a county was in 2000, the larger the gain for Bush in 2004, while in counties with electronic voting, the opposite pattern held true.

FIGURE 2.5: Regression lines for election data separated by whether the county used electronic voting in 2004.

As can be seen from the VIFs, the predictor and the product variable are collinear. This isn't very surprising, since one is a function of the other, and such collinearity is more likely to occur if one of the subgroups is much larger than the other, or if group membership is related to the level or variability of the predictor variable. Given that using the product variable is just a computational construction that allows the fitting of two separate regression lines, this is not a problem in this context.

This model is probably underspecified, as it does not include control variables that would be expected to be related to voting percentage. Figure 2.6 gives scatter plots of the percentage change in Bush votes versus (a) the total county voter turnouts in 2000 and (b) 2004, (c) median income, and (d) percentage of the voters being Hispanic. None of the marginal relationships are very strong, but in the multiple regression summarized below, median income does seem to add important predictive power without changing the previous relationships between change in Bush voting percentage and 2000 Bush percentage very much.

Coefficients: Estimate Std.Error t val P(>|t|) VIF (Intercept) 1.166e+00 2.55e+00 0.46 0.650 Bush.pct.2000 1.639e-01 3.69e-02 4.45 3.9e-5 1.55 *** e.Voting 1.426e+01 4.84e+00 2.95 0.005 54.08 ** Bush.2000 X e.Voting -2.545e-01 8.47e-02 -3.01 0.004 47.91 ** Vote.turn.2000 -5.957e-06 3.10e-05 -0.19 0.848 210.66 Vote.turn.2004 1.413e-06 2.49e-05 0.06 0.955 205.81 Median.income -1.745e-04 5.61e-05 -3.11 0.003 1.66 ** Hispan.pop.pct -4.127e-02 3.18e-02 -1.30 0.200 1.32 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.244 on 59 degrees of freedom Multiple R-squared: 0.4624, Adjusted R-squared: 0.3986 F-statistic: 7.25 on 7 and 59 DF, p-value: 2.936e-06

FIGURE 2.6: Plots for the 2004 election data. (a) Plot of percentage change in Bush vote versus 2000 voter turnout. (b) Plot of percentage change in Bush vote versus 2004 voter turnout. (c) Plot of percentage change in Bush vote versus median income. (d) Plot of percentage change in Bush vote versus percentage Hispanic voters.

We could consider simplifying the model here, but often researchers prefer to not remove control variables, even if they do not add to the fit, so that they can be sure that the potential effect is accounted for. This is generally not unreasonable if collinearity is not a problem, but control variables that do not provide additional significant predictive power, but are collinear with the variables that are of direct interest, might be worth removing so they don't obscure the relationships involving the more important variables. In these data the two voter turnout variables are (not surprisingly) highly collinear, but a potential simplification to consider (particularly given that the target variable is the change in Bush voting percentage from 2000 to 2004) is to consider the change in voter turnout as a predictor (the fact that the estimated slope coefficients for 2000 and 2004 voter turnout are of opposite signs and not very different also supports this idea). The model using change in voter turnout is a subset of the model using 2000 and 2004 voter turnout separately (corresponding to restriction ), so the two models can be compared using a partial ‐test. As can be seen below, the fit of the simpler model is similar to that of the more complicated one, collinearity is no longer a problem, and it turns out that the partial ‐test (, ) supports that the simpler model fits well enough compared to the more complicated model to be preferred (although voter turnout is still apparently not important).

Coefficients: Estimate Std.Error t val P(>|t|) VIF (Intercept) 1.157e+00 2.54e+00 0.46 0.651 Bush.pct.2000 1.633e-01 3.67e-02 4.46 3.7e-05 1.55 *** e.Voting 1.272e+01 4.20e+00 3.03 0.004 41.25 ** Bush.2000 X e.Voting -2.297e-01 7.53e-02 -3.05 0.003 38.25 ** Change.turnout -1.223e-05 1.36e-05 -0.90 0.370 2.44 Median.income -1.718e-04 5.57e-05 -3.08 0.003 1.65 ** Hispan.pop.pct -4.892e-02 2.94e-02 -1.66 0.102 1.14 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.233 on 60 degrees of freedom Multiple R-squared: 0.4585, Adjusted R-squared: 0.4044 F-statistic: 8.468 on 6 and 60 DF, p-value: 1.145e-06

Residual plots given in Figure 2.7 do not indicate any obvious problems, although the potential nonconstant variance related to whether a county used electronic voting or not noted in Figure 2.4 is still indicated. We will not address that issue here, but correction of nonconstant variance related to subgroups in the data will be discussed in Section 6.3.3.

FIGURE 2.7: Residual plots for the 2004 election data.