Statistical - University of North Carolina at Chapel Hill of 13.3, for a theoretical ... con taining...
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Transcript of Statistical - University of North Carolina at Chapel Hill of 13.3, for a theoretical ... con taining...
A Statistical Assessment of Buchanan's Vote in
Palm Beach County
Richard L. Smith
Department of Statistics
University of North Carolina
Chapel Hill, NC 27599-3260
November 20, 2000
Abstract
One of the strongest allegations of voting irregularities in the recent
Presidential election was that a faulty ballot design in Palm Beach County,
Florida, caused many votes to cast their ballots for the Reform Party can-
didate Patrick Buchanan, when they intended to vote for Al Gore. We ex-
amine this statistically, by performing a regression analysis of Buchanan's
vote over all 67 counties of Florida, using demographic variables (popula-
tion size, race, age distribution, education level and income) and votes for
other candidates as covariates. A critical point of statistical implemen-
tation is to choose a suitable transformation of the response variable to
achieve approximate homoscedasticity across counties. After considering
a number of alternatives, a cube root transformation (of the number of
votes cast for Buchanan) is chosen. Variable selection is performed using
either backward selection or the Mallows Cp criterion, leading to similar
models. The results con�rm that, if the regression model is �tted using all
67 counties, then Palm Beach is an enormous outlier, with a studentized
residual of 13.3, for a theoretical signi�cance level of around 10�67 under
the standard normal-theory assumptions. If the regression model is �tted
to the remaining 66 counties and used to predict the Buchanan vote in
Palm Beach, we obtain a point prediction of 326 and a 95% prediction
interval of (181,534), compared with the actual vote reported in initial
returns of 3,407. These results demonstrate conclusively that the Palm
Beach County vote was indeed anomalous.
1 Introduction
Oone of the most intense arguments to follow the disputed U.S. Presidential elec-tion of November 7, 2000, concerned the voting results in Palm Beach County,
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the second most populous county in the state of Florida. It was alleged that afaulty design of the ballot paper misled many supporters of the Democrat can-didate, Al Gore, into voting for the Reform Party candidate, Patrick Buchanan.Support for this view was provided by the large vote for Buchanan (3,407 inthe initial returns), whereas the state-wide average vote for Buchanan (of 0.3%)would have led one to expect about 1,300 votes. The alleged excess of about2,000 votes, if credited to Gore, would have won Gore the Florida vote andhence the White House.
Within a few days of the election, numerous papers appeared on variouswebsites, containing statistical analyses of the election results. Many of these
papers focussed speci�cally on the Buchanan vote in Palm Beach County, usingvarious political or demographic covariates to predict the number of votes heshould have received. Most such analyses in fact put the predicted number of
Buchanan votes at considerably less than 1,300, thus strengthening the casethat some irregularity occurred. However, these papers also raised a number ofquestions about the regression methodology employed.
The purpose of this paper is to present a detailed regression analysis ofthe number of Buchanan votes across all 67 Florida counties, using both thevotes for other candidates and demographic variables, obtained from the Cen-sus, as covariates. The analysis devotes particular attention to �nding a suit-able transformation of the response variable to make the data approximatelyhomoscedastic (meaning that the variances of the observations are the same |a key assumption of traditional regression analysis). The analysis also coversother standard regression issues such as the selection of covariates, testing nor-mality of the errors, and the use of outlier and in uence diagnostics. I �nd thatthe best transformation uses cube root of Buchanan votes as a response vari-able, though analyses based on either a square root transformation (suggestedby standard variance stabilizing arguments) or a logarithmic transformation ofproportions (suggested by several previous commentators) also appear to bevalid. All three analyses accept the null hypothesis of normally distributed er-rors provided Palm Beach County is excluded from the analysis. With PalmBeach included, however, it is an enormous outlier: for example, in the cube rootanalysis, Palm Beach has a studentized residual of 13.3, which if taken literally,would correspond to a nominal tail probability of around 10�67. A predictioninterval for the Palm Beach vote, based on the other 66 counties, using the cuberoot regression, shows a point prediction of 326, with a 95% prediction interval(181,534). Other regressions produce similar results. In contrast, the reportedvote for Buchanan in Palm Beach County was 3,407. Such results seems tocon�rm very clearly that the Palm Beach vote was indeed highly anomalous.
My motivation in writing this note is to draw attention to statistical, notpolitical, issues. To the best of my knowledge, there is no legal or constitutionalbasis for invalidating the result of an election on the basis that the resultswere inconsistent with some statistical prediction, and even if one acceptedthat Buchanan obtained more votes than were intended for him in Palm Beach
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County, there is no evidence in the present analysis that the excess votes camefrom Gore's supporters rather than Bush's (though some other commentatorshave tried to argue that point | see review in Section 2).
However, the enormous public interest in the outcome of the Florida votehas resulted in a number of statistical analyses of the results. In my view, thisis quite a challenging exercise which raises numerous issues over the correctapplication of regression methodology. I make no pretense that the analysispresented here is in any sense de�nitive, but I believe it sheds light on a numberof those issues.
An outline of the paper is as follows. Section 2 is a review of other analysis
that have appeared since the election. Section 3 describes the data used in thisanalysis. Section 4 goes through the various stages of building a regressionmodeland discusses in detail such issues as homoscedasticity, normality of the errors,
selection of variables and outlier and in uence diagnostics. Section 5 discussesthe actual predictions, and Section 6 presents a summary and conclusions.
The present paper is also available from my website, along with data and pro-grams (http://www.stat.unc.edu/postscript/rs/palmbeach.html). Other web-sites that contain detailed information include those of Greg Adams(http://madison.hss.cmu.edu.) and Jonathan O'Kee�e(http://www.bestbookmarks.com/election). Fox (2000) also has a detailed re-view of other analyses.
2 Review of Other Analyses
Too many instant analyses have appeared to make it feasible to review all ofthem, but here are a few, which have some overlap with the present study.
Adams and Fastnow (2000) presented a simple regression analysis of thenumber of Buchanan votes on either the number of Bush votes, the num-ber of Gore votes or the total number of votes cast, across all 67 counties ofFlorida. Their analysis is open to the objection that it ignores the question ofhomoscedasticity, which they acknowledged as a weakness but did not do any-thing to correct. They also also presented an analysis of Buchanan votes in the1996 Republican Primary, noting that in that case, Palm beach was not at allan outlier.
Fox (2000) used log(1 + yi) (where yi is the number of Buchanan votes incounty i) as the response variable, though he claimed it did not produce resultsvery much di�erent from a regression based on yi directly. He used multipleregression on the votes of the other candidates, noting that the proportion ofBush votes was the only signi�cant predictor.
In possibly the most comprehensive analysis so far published, Wand et al.
(2000) analyzed Buchanan votes in 4,481 reporting districts (cities or counties)covering 46 states, claiming that Palm Beach was one of the �ve most irregularoverall, and the most irregular among reporting districts with over 25,000 vot-
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ers. They also looked at individual precinct data within Palm Beach County,pointing out that within the county, Buchanan votes are positively correlatedwith Gore votes | this is one of only two analyses I have seen (the other beingBrady (2000), who also looked at precinct data) which claimed statistical evi-dence that Buchanan in fact obtained his votes from Gore rather than Bush, theargument being that because Buchanan obtained a larger proportion of votesin the precincts favoring Gore, any confusion caused by the design of the ballotpaper must have been at the expense of Gore rather than Bush.
In other analyses considered by Wand et al., they also looked at votes forGoverner Jeb Bush in the 1998 gubernatorial election, and for defeated Repub-
lican Senate candidate Bennett McCollum, using both of these as covariatesfor predicting the Buchanan votes in 2000, with similar results to those usingother covariates. They also looked at demographic variables obtained from the
U.S. Census, as I do in the analysis to follow. The statistical methodology theyemployed is essentially a generalized linear models analysis in which Buchanan'svote is represented by a binomial distribution with either probit or logistic link.This analysis is somewhat similar to the linear regression analysis based on asquare root transformation. which I consider in Section 4. In particular, theyobtained a studentized residual for Palm Beach of about 17, similar to Fig. 5(a)below. They adjusted the analysis to allow for overdispersion but they did notcomment on the size of the overdispersion, which as we shall see in Section 4,is large enough to cast doubt on the validity of either a binomial or Poissonregression analysis.
Hansen (2000) also used demographic predictors derived from the U.S. Cen-sus, using the logarithm of the proportion of Buchanan votes as a responsevariable (this is another analysis which we shall replicate in Section 4), but ad-justing for heteroscedasticity by representing the variance as a function of logpopulation size.
Carroll (2000) developed a regression analysis based on the 1996 RepublicanPrimary, using log yi as the response variable, regressing it on the log Buchananvote in 1996, and including a separate indicator variable for the \Palm Beache�ect".
S-PLUS (2000) also posted a note demonstrating the use of robust regressionand Java Graphlets as an aid to interactive diagnostics, though based primarilyon the simple analysis of total Buchanan votes against total Bush or Gore votes,without considering other types of regression model.
All of the above analyses essentially con�rmed that Buchanan's Palm Beachvote was highly anomalous, most predicting a vote of well under 1,000. Someanalyses have, however, challenged these conclusions. Shimer (2000) argued thatis in inappropriate to use yi as a response variable, suggesting log proportion ofBuchanan votes as an alternative. From a plot of Buchanan votes share againstGore vote share, he argued that the Palm Beach vote is by no means anomalous(see Fig. 1(j) below). I agree with this point as far as it goes, but the analysis tofollow shows that with log proportion of Buchanan votes as a response, perform-
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ing a full multiple regression analysis including demographic variables, it stillappears that Buchanan's vote in Palm Beach was highly anomalous. Farrow(2000) raised some similar points, objecting on a number of counts to the anal-ysis of Adams and Fastnow (2000), and pointing out, as did Shimer, that a plotbased on proportion of Buchanan votes against proportion of Gore votes doesnot show up Palm Beach as an obvious outlier. I believe that Farrow's pointsabout methodology are correct, but they are answerable, and when placed inthe context of a full regression analysis, Palm Beach still shows up as a veryextreme data point.
3 Data Used in This Study
Two data sets have been assembled, one consisting of election returns from theFlorida Division of Elections, and the other of demographic data compiled fromthe U.S. Census Bureau. Tables 1 and 2 give the votes for the ten presidentialcandidates on the ballot in Florida, classi�ed by county. (Note: These are theprovisional results issued immediately after the election, not the �nal certi�edtotals.) In addition to the votes for Bush, Gore and Buchanan, we shall alsoconsider in our analysis the votes for the Libertarian candidate Browne, andthe Green Party candidate Nader. The other �ve candidates (Harris, Hagelin,McReynolds, Phillips and Moorehead) all achieved less than 0.1% of the voteand shall play no part in our analysis, though they may be of some interest fordetermining whether other minor-party candidates also showed unusual votingpatterns, an aspect we shall not consider here. The demographic data in tables3 and 4 include the following variables:
� Pop: county population in 1997,
� Whi: percentage of whites in 1996,
� Bla: percentage of blacks in 1996,
� Hisp: percentage of Hispanics in 1996 (note that the percentages of whites,blacks and Hispanics sometimes add up to more than 100, because Hispanicsinclude other races),
� � 65: percentage of the population aged 65 and over (actually calculated fromthe 1996 population aged 65 and over, divided by the 1997 total population),
� HS: percentage of the population graduating from high school (1990 census),
� Coll: percentage of the population graduating from college (1990 census),
� Inc: Mean personal income (1994).
As an initial analysis of the data, Fig. 1 plots the Buchanan percentagevote against 12 covariates. In each case, Palm Beach County is marked with an
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X. The plots shows that the percentage of Buchanan votes is overall decreas-ing with total population size; not obviously dependent on the percentages ofwhites and blacks but strongly negatively correlated with Hispanics; decreasingwith the percentage of population aged 65+; decreasing with mean high schoolgraduation rate; decreasing with mean college graduation rate; and decreasingwith mean personal income. There are also clear negative correlations with theproportions of Gore and Nader votes, and a positive correlation with the pro-portion of Bush votes. On several of these plots, Palm Beach County standsout as an outlier, in the sense of being inconsistent with the pattern of the restof the data points.
4 Building a Regression Model
4.1 Transformation of the response variable
Possibly the main technical diÆculty with �tting a regression to these data isthe enormous variation in county populations, ranging from under 10,000 toover 2 million in the case of Miami-Dade County. If we de�ne Ni to be thetotal number of votes cast and yi to be the number of votes for Buchananin county i, then the Binomial distribution would suggest that the varianceof yi is approximately Nipi(1 � pi), where pi is the proportion of voters whosupport Buchanan in county i. Since the overall value of pi is about 0.003, wesubsequently neglect 1� pi in this expression and write
Var(yi) � Nipi: (1)
Also, in view of the small pi, the Binomial distribution is very closely approx-imated by a Poisson distribution, for which (1) is exact. Thus, any direct at-tempt to perform least-squares regression analysis runs into the diÆculty thatthe data are heteroscedastic (non-constant variance), so contradicting one of thestandard assumptions of least-squares regression analysis.
One standard way to deal with this issue is to transform the response variablebefore �tting a linear regression model. Suppose one is considering a model ofform
h(yi) =Xj
xij�j + �i; (2)
where h is the transformation function, fxij ; 1 � i � n; 1 � j � pg are theregressors, and f�ig are random errors. A standard approximation formula gives
Varfh(yi)g � fh0(�i)g2�2i ; (3)
where �i is the mean of yi and �2 is its variance. Applying (3) with �i = �
2i =
nipi leads to, for example,
Varfpyig � 1
4; (4)
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County Bush Gore Brow Nade Har Hag Buc Mc Ph Mo
Alachua 34124 47365 658 3226 6 42 263 4 20 21Baker 5610 2392 17 53 0 3 73 0 3 3Bay 38637 18850 171 828 5 18 248 3 18 27Bradford 5414 3075 28 84 0 2 65 0 2 3Brevard 115185 97318 643 4470 11 39 570 11 72 76Broward 177323 386561 1212 7101 50 129 788 34 74 124Calhoun 2873 2155 10 39 0 1 90 1 2 3Charlotte 35426 29645 127 1462 6 15 182 3 18 12Citrus 29765 25525 194 1379 5 16 270 0 18 28Clay 41736 14632 204 562 1 14 186 3 6 9Collier 60433 29918 185 1399 7 34 122 4 10 29Columbia 10964 7047 127 258 1 7 89 2 8 5Desoto 4256 3320 23 157 0 0 36 3 8 2Dixie 2697 1826 32 75 0 2 29 0 3 2Duval 152098 107864 952 2757 37 162 652 15 58 41Escambia 73017 40943 296 1727 6 24 502 3 110 20Flagler 12613 13897 60 435 1 4 83 3 3 12Franklin 2454 2046 17 85 1 3 33 0 3 2Gadsden 4767 9735 24 139 3 4 38 4 7 6Gilchrist 3300 1910 52 97 0 1 29 0 2 4Glades 1841 1442 12 56 0 3 9 1 0 1Gulf 3550 2397 21 86 2 4 71 2 2 9Hamilton 2146 1722 12 37 4 1 23 8 7 4Hardee 3765 2339 17 75 0 2 30 0 2 3Hendry 4747 3240 11 103 3 1 22 2 7 2Hernando 30646 32644 116 1501 8 26 242 4 10 22Highlands 20206 14167 64 545 6 16 127 3 7 8Hillsborough 180760 169557 1138 7490 35 217 847 29 68 154Holmes 5011 2177 18 94 1 7 76 3 6 2
Indian River 28635 19768 122 950 4 13 105 2 13 10Jackson 9138 6868 40 138 0 2 102 1 4 7Je�erson 2478 3041 14 76 2 1 29 1 0 0Lafayette 1670 789 6 26 2 0 10 1 1 0Lake 50010 36571 204 1460 4 36 289 1 21 15Lee 106141 73560 538 3587 30 81 305 5 34 96Leon 39053 61425 330 1932 9 28 282 7 16 31Levy 6858 5398 92 284 1 1 67 1 10 12Liberty 1317 1017 12 19 0 3 39 0 1 2
Table 1: County Voting Data, Part I.
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County Bush Gore Brow Nade Har Hag Buc Mc Ph Mo
Madison 3038 3014 18 54 0 2 29 1 1 5Manatee 57952 49177 242 2491 5 35 271 3 19 26Marion 55141 44665 662 1809 13 26 563 6 22 49Martin 33970 26620 109 1118 14 29 112 7 20 14Miami-Dade 289492 328764 760 5352 87 119 560 35 69 124Monroe 16059 16483 162 1090 1 26 47 0 3 7Nassau 16280 6879 62 253 0 7 90 4 3 3Okaloosa 52093 16948 313 985 4 15 267 2 33 20Okeechobee 5057 4588 21 131 1 4 43 1 3 4Orange 134517 140220 891 3879 13 65 446 7 41 46Osceola 26212 28181 309 732 10 20 145 5 10 33Palm Beach 152846 268945 743 5564 45 143 3407 302 188 103Pasco 68582 69564 413 3393 19 83 570 14 16 77Pinellas 184823 200629 1230 10022 41 442 1013 27 72 170Polk 90180 75193 365 2062 8 59 532 5 46 36Putnam 13447 12102 114 377 2 7 148 3 10 12Santa Rosa 36274 12802 131 724 1 13 311 1 43 19Sarasota 83100 72853 431 4069 11 94 305 5 15 59Seminole 75677 59174 550 1946 6 38 194 5 18 26St. Johns 39546 19502 210 1217 4 11 229 2 12 13St. Lucie 34705 41559 165 1368 4 12 124 10 13 29Sumter 12127 9637 53 306 2 2 114 0 3 17Suwannee 8006 4075 52 180 2 4 108 0 9 5Taylor 4056 2649 4 59 0 3 27 1 8 1Union 2332 1407 15 33 1 0 37 0 1 0
Volusia 82214 97063 442 2903 8 36 496 5 20 69Wakulla 4512 3838 30 149 2 3 46 1 0 6Walton 12182 5642 68 265 3 11 120 2 7 18Washington 4994 2798 32 93 0 2 88 0 9 5
Table 2: County Voting Data, Part II.
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County Pop Whi Bla Hisp � 65 HS Coll Inc
Alachua 198326 74.4 21.8 4.7 9.4 82.7 34.6 19412Baker 20761 82.4 16.8 1.5 7.7 64.1 5.7 14859Bay 146223 84.2 12.4 2.4 11.9 74.7 15.7 17838Bradford 24646 76.1 22.9 2.6 11.8 65.0 8.1 13681Brevard 460977 88.3 9.2 4.1 16.5 82.3 20.4 19567Broward 1470758 80.3 17.5 10.9 20.3 76.8 18.8 24706Calhoun 12337 81.6 16.9 1.6 14.3 55.9 8.2 12570Charlotte 133681 94.3 4.4 3.4 33.4 75.7 13.4 18977Citrus 112454 96.2 2.8 2.5 30.7 68.6 10.4 16060Clay 135179 91.0 6.0 3.5 7.9 81.2 17.9 18598Collier 195731 93.3 5.7 17.1 21.5 79.0 22.3 30906Columbia 52856 78.3 20.5 1.9 12.3 69.0 11.0 15349Desoto 26259 80.6 18.1 12.1 18.0 54.5 7.6 16544Dixie 12563 89.8 9.5 1.2 14.4 57.7 6.2 12035Duval 732622 69.4 27.5 3.4 10.7 76.9 18.4 20686Escambia 282604 73.3 22.7 2.6 11.7 76.2 18.2 17661Flagler 46128 88.5 9.8 5.9 23.0 78.7 17.3 15613Franklin 10133 84.5 14.5 1.0 17.8 59.5 12.4 15735Gadsden 45441 37.6 61.8 2.9 11.6 59.9 11.2 14416Gilchrist 13367 90.0 9.3 2.1 13.0 63.0 7.4 12865Glades 9698 79.6 13.7 10.1 15.3 57.4 7.1 14789Gulf 13926 73.9 25.2 1.1 13.6 66.4 9.2 15482Hamilton 12521 56.3 43.0 3.6 10.9 58.4 7.0 12357Hardee 22113 93.1 5.9 28.4 13.3 54.8 8.6 16812Hendry 31634 78.2 18.8 26.6 9.9 56.6 10.0 17823Hernando 125537 94.4 4.6 4.0 29.6 70.5 9.7 16062Highlands 76854 87.1 11.6 6.7 32.4 68.2 10.9 17655Hillsborough 909444 82.8 14.9 16.0 12.3 75.6 20.2 20167Holmes 18382 91.7 6.5 1.7 15.5 57.1 7.4 12790Indian River 99215 89.2 9.9 3.9 26.6 76.5 19.1 28977Jackson 45706 69.5 29.6 3.5 14.4 61.6 10.9 15519Je�erson 13232 49.4 50.1 1.3 13.4 64.1 14.7 15574Lafayette 6289 83.0 16.4 5.1 10.7 58.2 5.2 13663Lake 196214 88.2 10.9 3.8 26.3 70.6 12.7 18269Lee 387091 91.1 7.8 5.9 24.4 76.9 16.4 22053Leon 215170 70.4 27.3 3.1 8.4 84.9 37.1 16705Levy 32254 84.4 14.2 2.6 17.6 62.8 8.3 13745Liberty 6703 78.1 20.9 3.1 10.7 56.7 7.3 14896
Table 3: County Demographic Data, Part I.
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County Pop Whi Bla Hisp � 65 HS Coll Inc
Madison 17558 53.9 45.6 1.9 13.8 56.5 9.7 13002Manatee 237159 89.8 9.0 5.8 27.8 75.6 15.5 23031Marion 237308 84.3 14.6 4.0 21.4 69.6 11.5 14502Martin 116087 91.8 6.9 6.2 26.6 79.7 20.3 31996Miami-Dade 2044600 77.0 21.2 54.4 14.4 65.0 18.8 20014Monroe 81919 92.3 6.2 15.8 15.9 79.7 20.3 25160Nassau 54096 87.3 11.9 1.5 9.8 71.2 21.5 20874Okaloosa 167580 85.3 10.3 4.2 9.1 83.8 21.0 18959Okeechobee 33102 91.1 7.5 14.8 14.6 59.1 9.8 15162Orange 783974 79.1 17.5 12.3 10.4 78.8 21.2 20469Osceola 142128 90.7 6.6 15.3 13.2 73.7 11.2 16256Palm Beach 1018524 83.9 14.4 9.8 23.7 78.8 22.1 33518Pasco 320253 96.5 2.3 4.4 32.0 66.9 9.1 16924Pinellas 871766 89.1 9.0 3.1 26.6 78.1 18.5 24796Polk 448646 83.3 15.4 5.3 18.2 68.0 12.9 17824Putnam 70430 78.1 20.9 3.4 17.6 64.3 8.3 14250Santa Rosa 114481 92.6 4.6 2.0 8.9 78.5 18.6 17127Sarasota 301644 94.0 5.1 2.8 32.3 81.3 21.9 30205Seminole 344729 87.4 9.8 8.4 10.1 84.6 26.3 21815St. Johns 112707 88.7 10.1 3.0 15.6 79.9 23.6 25637St. Lucie 179559 79.6 19.0 5.2 20.1 71.7 13.1 16483Sumter 39428 81.0 18.1 3.1 20.3 64.3 7.8 14606Suwannee 33077 82.2 16.9 2.0 15.8 63.8 8.2 14773Taylor 18718 77.1 21.5 1.3 12.7 62.1 9.8 15459Union 12359 71.0 27.8 4.8 7.0 67.7 7.9 10783Volusia 419797 88.0 10.5 5.0 22.7 75.4 14.8 17778Wakulla 19172 83.9 14.9 .9 10.9 71.6 10.9 15570Walton 37914 88.9 8.6 1.2 14.9 66.5 11.9 14866Washington 20221 79.7 17.6 1.5 16.4 60.9 7.4 13732
Table 4: County Demographic Data, Part II.
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Varflog yig � 1
nipi: (5)
From (5) we see that using the logarithm of either the number or the proportionof Buchanan votes as a response variable, as some other analysts have suggested,runs into the theoretical diÆculty that the variance of the response is large inthe very small counties, whereas according to (4), a square root transformationshould stabilize the variances. These (well-known) considerations led me, ini-tially, to consider a square root transformation as a suitable response variablefor the regression. However, we shall see that matters are not quite as simpleas that, as it is far from clear that the relation (1) is satis�ed.
These transformations may all be considered special cases of the Box and
Cox (1964) transformation family, usually written
h(yi) = C�
y�i � 1
�; (6)
where C� is a scaling constant and � may, at least in theory, be any real number,the case � ! 0 corresponding to a logarithmic transformation. The scalingconstant C� is chosen to satsify the relationshipY
i
jh0(yi)j = 1; (7)
where the product runs over all observations i. Combining (6) and (7) leads to
C� = _y1�� (8)
where _y is the geometric mean, i.e. (Q
i yi)1=n where n is the number of obser-
vations.The rationale behind the scaling constant is that it makes sums of squares
comparisons of di�erent transformations equivalent to likelihood comparisons.For example, the maximum likelihood estimator of � is the value which results
in the smallest residual sum of squares after �tting the regression model (2) byordinary least squares.
For the initial stages of the analysis, three response variables were used: yiitself,
pyi and log(yi=Ni). According to (6){(8), these should be multiplied by
scaling constants which are, respectively, 1, 2p_y and _y. For this particular data
set, _y, the geometric mean of Buchanan votes across all counties, is 127.2651.Aside from transformation and scaling issues associated with the response
variable, one must also consider the scaling issue with respect to the covariates.Consider income, for example, a signi�cant covariate in all the models to bediscussed. It does not make sense that the in uence of income on the total
Buchanan vote in Miami-Dade County (population 2,044,600) would be thesame as that in, say, Lafayette County (population 6,289). Logically, if \income"a�ects votes at all, then the level of the e�ect is on the probability that an
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individual voter supports Buchanan, and if one wants to know the in uence onthe yi, one must multiply by the total number of votes cast in the county. Thesame argument applies to all the covariates we are considering, and by the sameargument, if the response variable is proportional to y�i for some other value of�, then the covariates should be scaled by N
�i . This rescaling of the covariates
has been made in all the analysis which are presented below.For minor technical reasons associated with the way SAS handles missing
data, the actual de�nition of Ni has been the sum of all votes cast excluding
Buchanan. (For some analyses we want to treat yi, but not Ni, as a missingvalue in Palm Beach County.) Given that the overall percentage of Buchanan's
vote is so small, this should make no di�erence to the results.
4.2 Variable selection
After both the covariates and the response variable were suitably transformedand rescaled, regression analyses were performed to determine which covariateswere statistically signi�cant. Noting from Fig. 1 that some of the covariates givemore meaningful responses when plotted on a logarithmic scale, a logarithmictransformation was applied in those cases. The full list of covariates consideredis shown in Table 5. It will be noted that the proportion of Gore votes has notbeen included in this analysis, but the reason for that is that it is so highlycorrelated with the proportion of Bush votes (in most counties, the two add upto very nearly 100%) that it makes no sense to consider both in the analysis |we would get very similar results to the following if we deleted the proportionof Bush voters from the analysis, and replaced it with the proportion of Gorevoters (with opposite signs in the corresponding regression coeÆcients).
Because of our strong prior suspicion, supported by Fig. 1, that Palm BeachCounty was a very strong outlier, this has been omitted from all analyses usedto determine variable selection.
Variable selection may be applied using any of the standard techniques usedfor multiple regression: the two criteria considered here are Mallows' Cp andbackward selection. Mallows' Cp statistic is an estimate of the total meansquared prediction error across all data points; the model with smallest Cp is\best" in this sense. Backward selection is a technique which starts with all the
variables in the model and then drops one at a time until all remaining variablesare statistically signi�cant. The variables selected by these two methods areshown in Table 6. The inclusion of 3
pyi as a response variable in this table will
emerge later on in the analysis.It can be seen that the models selected by backward selection and Mallows'
Cp di�er somewhat, but some comparisons of the models (not reported here)suggest that it makes very little di�erence to the eventual results which of thetwo model selection strategies are employed. However, the di�erences amongthe models �tted to the three response variables are far more signifcant, sowe concentrate on that aspect in subsequent discussion. The residual sums of
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Covariate De�nition
lpop Log total population sizewhit Proportion of whiteslblac Log proportion of blackslhisp Log proportion of Hispanicso65 Proportion of population aged 65 and overhsed Proportion graduated high schoolcoll Proportion graduated college
inco Mean personal incomepbush Proportion voting for Bushpbrow Proportion voting for Brownepnade Proportion voting for Nader
Table 5: List of covariates used in the analysis
Response Selection Variables selectedVariable Method
yi Cp lpop, whit, lblac, lhisp, o65, hsed, inco, pbrowyi Backward whit, lblac, lhisp, hsed, inco, pbrowpyi Cp lpop, whit, lhisp, hsed, inco, pbush, pbrowpyi Backward lpop, whit, lhisp, inco, pbush
log(yi=Ni) Cp lpop, lhisp, hsed, inco, pbushlog(yi=Ni) Backward lhisp, hsed, inco, pbush
3pyi Cp lpop, whit, lhisp, inco, pbush
3pyi Backward lpop, whit, lhisp, inco, pbush
Table 6: Covariates selected by either Mallows' Cp or backward selection
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squares (RSS) for the three models selected by Cp are 151,312 (57 df), 74,772 (58df) and 91,933 (60 df) for response variables yi,
pyi and log(yi=Ni) respectively,
after rescaling as mentioned earlier. This implies a clear preference for thesquare root transformation among the three considered, but it also shows thata logarithmic transformation is much better than using yi directly as a response.
However, after adjusting the residual sums of squares for the rescaling, theresidual variance derived from the square root transformation model is 2.53,compared with an approximate variance of 0.25 for
pyi if yi is Poisson. This
implies a very considerable amount of overdispersion compared with the Poissondistribution, and casts doubt upon the justi�cation for the square root trans-
formation.
4.3 Testing the �t
After tentatively identifying a model, we can now make various tests to decidewhether the model is appropriate. Given the strong emphasis we have placed sofar on the homoscedasticity assumption, it seems natural to examine this �rst.
Suppose a model of the form of (2) has been �tted. After constructing
estimates b�j of the parameters �j , we form residuals
ei = h(yi)�Xj
xijb�j ; 1 � i � n: (9)
One way to examine the data for homoscedasticity is to plot the absolute valuesof the residuals against some other variable of interest | if the variability of theresiduals increased or decreased over the range of the plot, that would indicatea problem with the model.
In Fig. 2, such plots are constructed using residuals from a model in which weused yi (number of Buchanan votes in county i) directly as a response variable.Absolute residuals are plotted against (a) �tted values, (b) number of votesNi, (c) proportion of votes pi. Also shown on the plot is a smoothed curverepresenting the local estimate of standard deviation | this was obtained byapplying the S-PLUS \lowess" procedure to the squared residuals, and thentaking square roots to transform back to the scale of the plot. The results show
clearly that standard deviation is increasing in plots (a) and (b), and decreasingin (c).
Corresponding plots are shown for the square root transformation in Fig.3, and for the log transformation in Fig. 4. In each case there is much lessevidence, than in Fig. 2, for the variation in standard deviation across the dataset. However, it is far from obvious that Fig. 3 is superior to Fig. 4, which iswhat our prior analysis, see equations (4) and (5), led us to expect. Thus at thisstage, the square root transformation appears easily superior to the regressionbased directly on yi, but it is not clear that it is superior to using a logarithmictransformation.
14
For the remaining diagnostic procedures, it is convenient to work, not withordinary residuals as in (9), but with studentized residuals, de�ned as follows:for the ith observation, yi is deleted from the analysis and the regression modelre�tted to the rest of the data. From this, a prediction
Pj xij
b�j(i) is calculated,where the notation (i) in the suÆces is intended to denote that they are basedon deletion of observation i. Also calculated is an estimated residual standarddeviation s(i), also based on the data with yi deleted. We then calculate a scaledresidual
d�
i =h(yi)�
Pj xij
b�j(i)s(i)
: (10)
Then d�
i is the ith studentized residual. What has been described is not theactual calculation method, but it is the simplest way of describing the concept.One advantage of the studentized residual is that if the standard assumptions
of a linear model are satis�ed, including the assumption that the errors �i; 1 �i � n in (2) are independent normally distributed with mean 0 and commonvariance �
2, then the distribution of d�i is exactly tn�p�1 for each i, i.e. thestudent's t distribution with n�p�1 degrees of freedom, where p is the numberof parameters in the model (including the intercept term if there is one). Thisfact can be very useful in testing for outlying values, as we shall see later.
Another useful diagnostic device is the DFFITS statistic, which is one ofseveral measures that have been devised for measuring the in uence of a par-ticular observation. We shall not give the de�nition here, but refer to any oneof several standard sources (e.g. Belsley et al. 1980, Cook and Weisberg 1982,Atkinson 1985, Neter et al. 1996) for discussion and comparison with alter-native measures. Loosely, DFFITSi measures the in uence of observation i onpredictions generated by the model. Observations that are outliers are generallyin uential as well, but the converse may not be true, so it is worth consideringboth concepts.
Atkinson (1985) proposed a plotting technique in which the absolute valuesof either the studentized residual or the DFFITS statistic are �rst arrangedin increasing order, and then plotted as a half-normal plot, i.e. if there are nobservations, a quantile-quantile plot is constructed as if they were the largest nobservations from 2n+1 normal observations. As a measure of how much sucha plot di�ers from what would be expected under normal-theory assumptions,Atkinson also proposed constructing con�dence bands by simulation. The entireregression model �t, up to the calculation of studentized residuals or DFFITS,is repeated but with response variables generated at random from a normaldistribution. In this way, probability bands for the order statistics of studentizedresiduals or DFFITS are constructed (separately for each order statistic). InAtkinson's own examples he used just 19 simulations to construct approximate95% con�dence bands, but for the experiments shown here, 1000 simulationshave been used and the 50th largest and smallest values marked by dashes,corresponding to approximate 5% tail probabilities in each tail.
15
Because our objective now is to examine how much Palm Beach Countyreally is either an outlier or an in uential observation, the regression model hasbeen re�tted including Palm Beach, for the model with
pyi as the response
and lpop, whit, lhisp, hsed, inco, pbush and pbrow as regressors (the Cp-bestmodel of Table 6). Ordered values of the studentized residuals and DFFITSare plotted in Fig. 5, with simulated con�dence bands as just described. Ineach case there is an enormous outlier, which does correspond to Palm Beach.This contains the �rst clear evidence in this paper that Palm Beach is indeedextremely inconsistent with the rest of the data.
We can repeat the exercise using log(yi=Ni) as the response variable, with
covariates selected by Cp for that response variable, with results in Fig. 6.Although Palm Beach does not now appear as quite such an extreme outlier(studentized residual about 6, compared with 17 for Fig. 5(a)), it is still very
highly signi�cant.The question naturally arises whether Palm Beach is the only outlier | in
Fig. 5, it can be seen that some other large values of the studentized residualor DFFITS are outside the simulation bands, but it is not clear whether theserepresent other outliers, or whether this is simply part of the distorting e�ectof Palm Beach on the rest of the analysis. One way to examine this aspect isto delete Palm Beach altogether, and to repeat the entire analysis, includingthe plots for studentized residuals and DFFITS, on the reduced data set. Theresults are in Figs. 7 and 8, respectively for the square root transformation andthe log transformation. The results are much better, though from Fig. 7(b)we can see that there appear to be several observations with large values ofDFFITS.
One possible explanation for that is that, even in the absence of outliers,large cities are still in uential. As an indicator of that, DFFITS has been plottedagain in Fig. 9, against Ni. From Fig. 9(a), it does indeed appear that largecounties are in uential in the analysis using the square root transformation, butbased on Fig. 9(b), using the log transformation, it appears to be the other wayround, i.e. in this case, small cities are more in uential than large ones.
These comments about in uential observations do not necessarily invalidatethe analysis, but they provide further indication that, perhaps, at this stage wehave not yet found the best model.
4.4 Re-examination of the transformation
So far, we have seen that both the square root and logarithmic transformationsappear to work reasonably, but there is no clear preference between the two,and the theoretical argument in favor of a square root transformation is unlearbecause of the large overdispersion. Therefore, we are motivated to look furtherat the question of transformations.
One motivation behind the Box and Cox (1964) transformation (6) was thepossibility of searching through di�erent values of � to obtain the best regression
16
model. As noted earlier, if \best" is de�ned in terms of minimum residual sumof squares, then it is essential to adopt the scaling (8). In Fig. 10, this has beendone, plotting the RSS against �, for 0 < � < 1, for the scaled transformation.The model used for this calculation included the same covariates as for the
pyi
regression earlier, though very similar results are obtained with other selectionsof covariates. For this calculation, Palm Beach has been omitted from theanalysis | we get a very di�erent curve from Fig. 10 if we include Palm Beach,but we have already provided very strong evidence that Palm Beach is an outlier,so it is appropriate to omit it from this part of the analysis.
Fig. 10 shows that the smallest RSS is obtained very near � = 13, so the entire
analysis has been repeated using that value of �. In this case, the best-�ttingregression model, selected without using Palm Beach, contains the variableslpop, whit, lhisp, inco, pbush under both the Cp and backward selection criteria
(last two rows of Table 6).The studentized residuals and DFFITS statistics have been recalculated and
plotted using the same methods as described earlier, based on the new regressionmodel with 3
pyi as the response variable. Results are shown in Fig. 11 for the
model including Palm Beach, and Fig. 12 for the model excluding Palm Beach.Once again Palm Beach appears as an enormous outlier, but there appears tobe no problem with the analysis if Palm Beach is omitted.
Fig. 13 shows a plot of estimated local standard deviation, similar to Figs.2{4. There is still some evidence of heteroscedasticity; comparing Fig. 13 withFig. 3, it may be that the square root transformation is still the best from thisparticular point of view, even though Fig. 10 suggests that � = 1
3is a better
�t overall. Fig. 14 shows a plot of DFFITS against county size as measured byNi, to be compared with the two plots in Fig. 9. Although there is still comeevidence for the largest cities to be the most in uential, there is much morehomogeneity of DFFITS across di�erent county sizes than in either of the plotsof Fig. 9.
I am not aware of any physical explanation for the cube root transformation.A possible explanation might be that it results from some form of spatial clus-tering. This possibility might be investigated in future research | in the presentpaper, all the analyses have assumed that the results from di�erent counties arestatistically independent.
The clearest indication that Palm Beach is an outlier lies in the largeststudentized residual in Fig. 11(a) | 13.3. Under its nominal t60 distribution,the tail probability associated with that is of the order of 13:3�60 � 10�67. Ofcourse, a tail probability cannot be interpreted literally in such a situation (forexample, it is unlikely that the assumption of normal errors holds exactly), thisdoes provide a measure of just how extreme an outlier this data point is.
17
4.5 Testing normality
The last sentence touched brie y on the appropriateness of the normal errorsassumption and this is something else that can be tested. A number of stan-dard tests for normality are known | for example, the Shapiro-Wilk test, theKolmogorov-Smirnov test, the Cram�er-vonMises test and the Anderson-Darlingtest. All four are implemented as part of PROC UNIVARIATE in SAS, and thebook by D'Agostino and Stephens (1986) can be consulted as a general referencefor the whole issue of testing �t of distributional assumptions.
The form of the Shapiro-Wilk statistic used here is simply the correlationcoeÆcient between the observed and expected values of the ordered residuals.The other three test statistics are all based on how the empirical distribution
function
Fn(y) =1
nf Number of residuals � yg
matches up to the postulated true distribution function, F0(y) say. The Kolmogorov-Smirnov statistic is
D = max�1<y<1
jFn(y)� F0(y)j ;
the Cram�er-von Mises statistic is
W2 = n
Z1
�1
fFn(y)� F0(y)g2 dF0(y);
and the Anderson-Darling statistic is
A2 = n
Z1
�1
fFn(y)� F0(y)g2F0(y)f1� F0(y)g
dF0(y):
In practice, all three are evaluated through computational formulae that reduceto discrete maxima or sums, for which we refer to D'Agostino and Stephens(1986).
In the present study, the test statistics are evaluated using the studentizedresiduals but the test distribution F0 is still assumed to be standard normal.Evaluation of percentage points is by simulation, a direct extension of the sim-ulation technique used in the preceding subsection. One thousand simulationsof the model were used to calculate empirical p-values. The use of simulationto calculate the test statistics is superior, in the present context, to relying ontables of critical values, because the latter typically ignore the in uence of es-timating the regression coeÆcients on the critical points of the tests (and also,since we have chosen to use studentized residuals here, the di�erence betweenstudentized and ordinary residuals).
As it turns out, the results of these tests are remarkably consistent across thedi�erent test statistics and models. For each of the square root, cube root andlogarithmic transformations, at signi�cance level 0.05, the tests reject normality
18
if Palm Beach is included in the regression, and accept normality if Palm Beachis excluded. The only exceptions are that for the log(yi=Ni) response variable,using the Kolmogorov-Smirnov or Cram�er-von Mises tests, the null hypothesisis accepted (with p-values 0.13 and 0.06 respectively) even if Palm Beach isincluded in the analysis. However, both tests are known to be less sensitive toextreme observations than either the Shapiro-Wilk or Anderson-Darling tests,and both of those tests, applied to the same model, decisively reject the nullhypothesis (empirical p-values 0 and .013 respectively).
As a contrast, when the same tests were applied to the untransformed data,with the number of Buchanan votes yi used directly as the response variable,
the results were a decisive rejection of the null hypothesis, under all four tests,both including and excluding Palm Beach in the analysis.
Of course, acceptance of a null hypothesis does not prove that the hypothesis
is correct, but these results strengthen the case that while regression based onyi itself is not acceptable, any of the models based on
pyi, 3
pyi or log(yi=Ni)
gives an acceptable �t, provided Palm Beach is excluded from the analysis.
5 Predicting the Palm Beach Vote
We now return to the original purpose of the analysis: predicting the Palm Beachvote from a regression model in which Palm Beach itself has been omitted fromthe �t. In accordance with the results of the previous section, we consider anyone of
pyi, 3
pyi or log(yi=Ni) as a response variable, with corresponding model
selection using either the Cp criterion or backward selection, as in Table 6.Based on the model �tted to the other 66 counties, point estimates and 95%
prediction intervals were calculated for the values of the covariates correspondingto Palm Beach County. The intervals calculated are prediction intervals ratherthan con�dence intervals, i.e. they are intended to re ect the result of a singlevote rather than a long-run average over a series of votes. Predictions arecalculated on the transformed scale, then transformed back to the original scalefor the results reported here. It will be recalled that the actual Buchanan votesin Palm Beach County were 3,407.
Table 7 shows point predictions and prediction intervals for �ve versions ofthe model, corresponding to three di�erent response variables and two di�erentvariable selections (Cp or backward selection). In the case of the regressionbased on 3
pyi, the Cp and backward selection models are the same.
In all �ve cases, the point prediction is under 400, and the widest boundsfor the prediction interval are 148 at the lower end and 758 at the upper end.
19
Response Variable Point PredictionVariable Selection Predictor Intervalp
yi Cp 301 (162,483)pyi Backward 287 (148,470)
log(yi=Ni) Cp 364 (180,735)log(yi=Ni) Backward 371 (182,758)
3pyi Cp or backward 326 (181,534)
Table 7: Point predictions and prediction intervals under �ve versions of linearmodel.
6 Summary and Conclusions
Previous analyses published since the election have used either yi itself orlog(yi=Ni) as a response variable, with several authors claiming that the lat-ter is superior, and some claiming that there is no evidence of a \Palm Beache�ect" in this case. The present analysis con�rms that the analysis based onyi as a response fails on two counts, (a) lack of homoscedasticity, (b) failureto �t the normal distribution. A model with log(yi=Ni) as a response appearssatisfactory from the point of view of �t of the model, provided Palm Beach isexcluded from the regression, but I have argued that either
pyi or 3
pyi leads
to a better �t. In all three cases, however, the outlier and in uence diagnos-tics con�rm that Palm Beach is a very signi�cant outlier. Point predictions forBuchanan's vote in Palm Beach, based on the other 66 counties, are all under400 in the analyses presented here, with upper bounds to a 95% prediction in-terval of under 800 even for anaysis based on log proportions, for which somecommentators have claimed Palm Beach was not an outlier.
Overall, I believe the analysis con�rms that something very unusual hap-pened in Palm Beach County, and while this does not prove that Gore shouldhave won the election, I believe it is evidence that the ballot design in PalmBeach County may have in uenced the result, which suggests that it should notbe used in future elections.
7 References
Adams, G.D. and Fastnet, C. (2000), A note on the voting irregularities in PalmBeach, Florida. http://madison.hss.cmu.edu.
Atkinson, A.C. (1985), Plots, Transformations and Regression. Oxford Uni-versity Press.
Belsley, D., Kuh, E. and Welsch, R.E. (1980), Regression Diagnostics. Wiley,New York.
20
Box, G.E.P. and Cox, D.R. (1964), An analysis of transformations (withdiscussion). J.R. Statist. Soc. B 26, 211{246.
Brady, H.E. (2000), What happened in Palm Beach County?http://socrates.berkeley.edu/�ucdtpums.
Carroll, C.D. (2000), How many Buchanan votes in Palm Beach County wereerroneous? http://www.econ.jhu.edu/people/ccarroll/carroll.html.
Cook, R.D. and Weisberg, S. (1982), Residuals and In uence in Regression.Chapman and Hall, London.
D'Agostino, R.B. and Stephens, M.A. (1986), Goodness-Of-Fit Techniques.Marcel Dekker, New York.
Farrow, S. (2000), Untitled note in response to Adams and Fastnet (2000).http://www.andrew.cmu.edu/user/sf08/Adams.pdf.
Fox, C.R. (2000), A vote for Buchanan is a vote for Gore? An analysis of
the 2000 presidential election results in Palm Beach, Florida.http://faculty.fuqua.duke.edu/�cfox/Bio/election2000note.pdf.
Hansen, B.E. (2000), Who won Florida? Are the Palm Beach votes irregular?http://www.ssc.wisc.edu/�bhansen/vote/vote.html.
Neter, J., Kutner, M.H., Nachtsheim, C.J. and Wasserman, W. (1996), Ap-plied Linear Statistical Models. Fourth Edition: Irwin, Chicago.
Shimer, R. (2000), Election 2000: Size matters.http://www.princeton.edu/�shimer.
S-Plus (2000), Too many Buchanan votes in Palm Beach?http://www.s-plus.com/graphlets/votes.html.
Wand, J.N., Shotts, K.W., Sekhon, J.S., Mebane, W.R. and Herron, M.C.(2000), Voting irregularities in Palm Beach County.http://elections.fas.harvard.edu.
21
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10 15 20 25 30
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Percentage HS Grad
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55 60 65 70 75 80 85
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Figure 1. Percentage of Buchanan vote against 12 covariates. Palm BeachCounty is marked with an X.
22
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o
o
o
o
oo
ooo
ooo
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
(c)
Buchanan Vote
Abs
olut
e R
esid
ual
0.1 0.5 1.0
0
50
100
150
Figure 2. Plots of absolute values of studentized residuals, for the model withuntransformed Buchanan votes as the response variable, against (a) �tted val-ues, (b) total votes in county, (c) proportion of Buchanan votes. The smoothedcurves show approximate local standard deviation. Palm Beach County hasbeen omitted from the analysis.
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
ooo
o o
o
o
o
o
o
o
o
o
o
o o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
(a)
Fitted Value
Abs
olut
e R
esid
ual
5 7 9 20 30
0.0
0.5
1.0
1.5
2.0
2.5
3.0
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
ooo
o o
o
o
o
o
o
o
o
o
o
o o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
(b)
Total Vote
Abs
olut
e R
esid
ual
5000 50000 500000
0.0
0.5
1.0
1.5
2.0
2.5
3.0
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o oo
oo
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
(c)
Buchanan Vote
Abs
olut
e R
esid
ual
0.1 0.5 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Figure 3. Similar to Fig. 2, but for the model based on square roots ofBuchanan votes as the response variable.
23
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
oo
oo
o
oo
o
o
o
o
o
o
o
o
o
oo
o
(a)
Fitted Value
Abs
olut
e R
esid
ual
-2.0 -1.0 0.0
0.0
0.2
0.4
0.6
0.8
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
oo
oo
o
o o
o
o
o
o
o
o
o
o
o
oo
o
(b)
Total Vote
Abs
olut
e R
esid
ual
5000 50000 500000
0.0
0.2
0.4
0.6
0.8
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
oo
oo
o
oo
o
o
o
o
o
o
o
o
o
oo
o
(c)
Buchanan Vote
Abs
olut
e R
esid
ual
0.1 0.5 1.0
0.0
0.2
0.4
0.6
0.8
Figure 4. Similar to Fig. 2, but for the model based on log percentages ofBuchanan votes as the response variable.
ooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooo o
o
o
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
5
10
15
----------------------------------------------
------------- - - - - - - - -
---------------------------------------
-------------------- - - - - - - -
-
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo o
oo
o
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
5
10
15
------------------------------------------------------------ - - - - - - -
----------------------------------------------------------- - - - - - - -
-
Figure 5. (a) Half-normal plot of ordered studentized residuals, for the modelbased on square root Buchanan votes, with pointwise 90% simulation bounds.(b) Half-normal plot of ordered DFFITS, for the model based on square rootBuchanan votes, with pointwise 90% simulation bounds. Palm Beach Countyis the large outlier on both plots.
24
ooooooooooooooooooooooo
ooooooooooooooooooo
oooooooooooo
oooooo
ooo oo
o
o
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
4
5
6
----------------------
----------------
------------
---------
- - - - - - --
--------------
----------------
-------------
-----------
----- - - - - -
--
-
ooooooooooooooooooooooooooooooooooooo
oooooooooooooooooo
oooooooo o o
o
o
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
------------------------------------
-------------------
---- - - - - - - --
----------------------------
---------------------
----------
- - - - --
-
-
Figure 6. Same as Fig. 5, but for model based on log percentages of Buchananvotes.
oooooooooo
ooooooooo
ooooooooooo
ooooooooooo
oooooooo
ooooo
oooooo
oo
o o
o
o
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
--------------
----------
----------
--------
------
------
----
- - - - --
--
----------------
---------
--------
--------
-------
-----
----
- - --
--
-
-
-
oooooooooooooooooooooooooooooooooo
ooooooooooooooo
oooooooo
oooo
oo
o
o
o
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
-----------------------------------
------------------
------ - - - - -
--
----------------------------
-------------------
--------
--- - -
--
-
-
-
-
Figure 7. Same as Fig. 5, but omitting Palm Beach altogether.
25
ooooooooooo
ooooooooooooooo
oooooooooo
ooooooo
oooooo
oooooo
ooooo
ooo o
o
o
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
--------------
-----------
---------
--------
------
------
----
- - - --
--
-
---------
---------
----------
--------
--------
------
----
----
---
--
-
-
-
ooooooooooooooooo
ooooooooooooooo
oooooooo
oooooooo
oooooooooo
oooo o o
oo
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
-------------------
-------------
-----------
-------
------
--- - - -
--
--
-------------
------------
-----------
--------
-------
----
--- -
--
-
-
-
-
-
Figure 8. Same as Fig. 6, but omitting Palm Beach altogether.
o
o oo
o
o
o
oo
o
o
ooo
o
o
ooooo
oooo
oo
o
o oo
oo
o
o
o
o
oo
o
o
o
o
oo
oo
oo
o
o
oo
o
o
o
o
o
oo
o
oooo
o
(a)
Total Votes
DF
FIT
S
0 100000 300000 500000
-3
-2
-1
0
1
2
o
o
o
oo
o
o
o
o
o
oo
o
o
o
oo
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
oo
o
oo
o
o
o
o
o
o
o
o
o
o
oo
o
(b)
Total Votes
DF
FIT
S
0 100000 300000 500000
-0.5
0.0
0.5
1.0
Figure 9. In uence diagnostics: plotting the DFFITS statistic against totalvotes in county. (a) Square roots of Buchanan votes as the response variable.(b) Log percentages of Buchanan votes as the response variable.
26
Lambda
Res
cale
d R
SS
(/1
0000
)
0.0 0.2 0.4 0.6 0.8 1.0
8
10
12
14
16
Figure 10. Selecting the transformation: plotting the residual sum of squaresagainst transformation parameter �, for the normalized transformation.
ooooooooooooooooooooooooooooooooooooooooooooo
oooooooooooooooooo
o oo
o
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
2
4
6
8
10
12
--------------------------------------
--------------------
- - - - - - - - -
-----------------------------
-----------------------
-------- - - - - -
--
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo o
o
o
o
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
2
4
6
8
10
12
----------------------------------------------------------- - - - - - - - -
----------------------------------------------------------- - - - - - -
-
-
Figure 11. Same as Fig. 5, but for model based on cube roots of Buchananvotes.
27
oooooooo
oooooooooooooooo
oooooooooooooo
oooooooo
ooooooo
oooooo
o
oo
o o oo
(a)
Expected Value
Obs
erve
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0
1
2
3
--------------
----------
---------
---------
-------
-----
----
- - - --
--
-
-----------------
---------
--------
--------
------
------
----
- - --
--
-
-
ooooooooooooooooooooooooooooo
ooooooooooooooo
oooooooo
oooooooo
oo oo
o
o
(b)
Observed Value
Exp
ecte
d V
alue
0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
1.5
2.0
-----------------------------
-----------------
----------
-- - - - - - --
-
---------------------
------------------
-----------
-------
- - --
--
-
-
-
Figure 12. Same as Fig. 12, but omitting Palm Beach.
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
oo
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
(a)
Fitted Value
Abs
olut
e R
esid
ual
3 4 5 6 7 9
0.0
0.2
0.4
0.6
0.8
1.0
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
oo
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
(b)
Total Vote
Abs
olut
e R
esid
ual
5000 50000 500000
0.0
0.2
0.4
0.6
0.8
1.0
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
oo
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
(c)
Buchanan Vote
Abs
olut
e R
esid
ual
0.1 0.5 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Figure 13. Similar to Fig. 2, but for the model based on square roots ofBuchanan votes as the response variable.
28
o
oo
o
o
o
o
o
o
o
o
o
o
o
o
o
oooo
o
o
o
o
oo
o
o
oo
o
o
o
o
o
oo
o
o
o
o
o
o
oo
oo
o
o
o
ooo
o
o
o
o
o
oo
o
o
oo
o
o
Total Votes
DF
FIT
S
0 100000 300000 500000
-1.0
-0.5
0.0
0.5
Figure 14. In uence diagnostics: plotting the DFFITS statistic against totalvotes in county, cube roots of Buchanan votes as the response variable.
29
