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Published online by Cambridge University Press: 04 January 2016
This paper presents a new statistical method that is well-suited to building and testing theories of change. An exploratory, robust two-way analysis of New Hampshire’s county-level presidential election returns for the past eighty years illustrates how this general technique can help to clarify theoretical concepts as well as to reveal systematic regularities in empirical data.
Because two-way analysis involves a number of novel techniques, this paper is primarily a methodological exposition. It devotes much more attention to step-by-step details of how the analysis was carried out, and the accompanying exhibits contain much more detailed information than normally would be the case for reporting results of such research.
In contrast to classical methods, the techniques presented in this paper are intended primarily for exploration of data rather than for statistical inference. They require very few assumptions about the data, and they are not unduly influenced by extreme or missing values. Graphic and semi-graphic devices keep the analyst close to the data and make it possible to display many numbers simultaneously in contrast to simple single-number summaries which can conceal information. Removing obvious regularities in the data facilitates discovery of other underlying patterns.
An earlier version of this paper was presented at the First Annual Meeting of the Social Science History Association (Philadelphia, 1976). It has been prepared in part in connection with research at Princeton University, sponsored by the Army Research Office (Durham). Many people have made helpful comments at various stages of the work. Special thanks go to Susan Bridge, David Hoaglin, Morgan Kousser, David Mayhew, Tom Piazza, Ed Tufte, and Howard Wainer.
1 For a general, comprehensive discussion of the various new statistical methods used in this paper, see Tukey, John W., Exploratory Data Analysis, (Massachusetts, 1977)Google Scholar. An example using economic data is Hoaglin, David C. and Kuh, Edwin, “Exploration in Economic Data I: Average Annual Unemployment,” National Bureau of Economic Research, Report W0010 (February 1973).Google Scholar
2 Machine-readable data were provided by the Inter-University Consortium for Political and Social Research, which bears no responsibility for the results reported herein. In years such as 1912, 1924, and 1948, when there were a substantial number of votes cast for presidential candidates outside the two major parties, the Democratic percentage of the total vote that we use differs considerably from the Democratic percentage of the two-party vote. Hence Republican percentages of the total vote are not always one-hundred minus the Democratic percentage.
Flanigan, William and Zingale, Nancy, “The Measurement of Electoral Change,” Political Methodology, 1, (Summer 1974), 49–82Google Scholar notes the differences between correlations based on Democratic percentage of the total vote (used by Gerald Pomper) and those based on Republican percentage of the total vote (used by Walter Dean Burnham). Pomper, Gerald, Elections in America (New York, 1968), 267–69Google Scholar; Burnham, Walter Dean, “American Voting Behavior in the 1964 Election,” Midwest Journal of Political Science (1968), 1–40CrossRefGoogle Scholar. For more complete understanding, one needs to analyze each party’s percentage of the total vote separately.
3 Census Director Francis Walker and historian Frederick Jackson Turner made elaborate use of multi-colored maps to relate electoral and demographic data in the 1890s and 1880s. During the 1920s Charles Merriam, William Ogburn, and Stuart Rice began to use new statistical techniques such as time series curve fitting and Pearson’s product-moment correlation for electoral analysis. Harold Gosnell extended the election analyst’s tool kit to multiple regression and factor analysis in the 1930s. Jensen, Richard, “The Historian and the Political Scientist,” and “American Electoral Analysis, A Case History of Methodological Innovation and Diffusion,” in Lipset, Seymour Martin, ed., Politics and the Social Sciences (New York, 1969), 1–28 and 226–43Google Scholar. A recent paper that applies traditional methods to a similar set of data is Lindeen, James W., “Longitudinal Analysis of Republican Electoral Trends, 1896–1968,” Midwest Journal of Political Science, 16 (May 1972), 102–22.CrossRefGoogle Scholar
4 Outlying cases and other aspects of data that violate assumptions of the linear regression model can bias results substantially. Anscombe, F. J., “Graphs in Statistical Analysis,” American Statistician, 27 (February 1973), 17–21CrossRefGoogle Scholar; Daniel, Cuthbert and Wood, Fred, Fitting Equations to Data (New York, 1971), 25–26Google Scholar, and passim.
5 Robust methods are resistant to distortion from aberrant or missing data values. For futher references, see Andrews, D.F., et. al., Robust Estimates of Location (New Jersey, 1972)Google Scholar; Hogg, R.V., “Adaptive Robust Procedures: A Partial Review and Some Suggestions for Further Application and Theory,” Journal of the American Statistical Association, 69 (1974), 909–27CrossRefGoogle Scholar; Huber, P.J., “Robust Statistics: A Review,” Annals of Mathematical Statistics, 43 (1972), 1041–67.CrossRefGoogle Scholar
6 Snedecor, G.W. and Cochran, W.G., Statistical Methods, 6th ed. (Iowa, 1967)Google Scholar is an outstanding exception, as is the Multiple Classification Analysis Section of the SPSS Analysis of Variance Module. A few political scientists have used analysis of variance to separate national, state and local election effects as well as to discover critical elections: Stokes, Donald E., “A Variance Components Model of Political Effects,” in Claunch, John M., ed., Mathematical Applications in Political Science, (Dallas, 1965), 1:61–85Google Scholar; Stokes, Donald E., “Parties and the Nationalization of Electoral Forces,” in Chambers, W.N. and Burnham, W.D., eds., The American Party Systems (New York, 1968), 182–202Google Scholar; Flanigan, William H. and Zingale, Nancy, op. cit.Google Scholar; Burnham, W.D., Clubb, J.M., Flanigan, W.H., “Partisan Realignment: A Systematic Perspective,” in Silbey, J.H., Bogue, A.G., and Flanigans, W.H., eds. The History of American Electoral Behavior (New Jersey, 1978), 45–77Google Scholar; Katz, Richard S., “The Attribution of Variance in Election Returns: An Alternative Measurement Technique,” American Political Science Review, 67 (September 1973), 817–32.CrossRefGoogle Scholar
7 Since a mean can change substantially due to a single value, it is not a very good summary measure for rows and columns containing missing data. The median does not change substantially depending on the presence or absence of a single value. Other robust measures of central tendency, such as trimmed means, can also be used; some may do a more thorough and efficient job of finding fits, while leaving fluctuations in the residuals, than medians, but the process is the same. See D.F. Andrews, et. al., Robust Estimates of Location.
8 The implications of perceptive ablities for data analysis are discussed in Quenouille, M.H., Associated Measurements (New York, 1952)Google Scholar; Tukey, John W., “The Future of Data Analysis,” Annals of Mathematical Statistics, 33 (March 1962), 1–67Google Scholar; Singer, Burton, “Examples of Exploratory Questions about Mass Behavior” (Paper delivered at the Symposium on Exploratory Data Analysis, American Association for the Advancement of Science Annual Meeting, Washington, D.C., December 1972)Google Scholar; and Miller, G.A., “The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information,” The Psychological Review, 63 (1956), 81–97.Google Scholar
9 Variance and standard deviation are even more sensitive to outliers than the mean because they involve squares of deviations about the mean. In order to calculate a midspread, arrange the batch of numbers in order from highest to lowest and divide the number of observations by four in order to come up with a value for the rank (or depth) that would be one quarter of the way in from one end of the batch. If the batch of numbers is evenly divisible by four (and thus the quarter point falls on a particular observation), one “hinge” will be the median between that observation and the next observation toward the center of the batch. If the batch of numbers is not evenly divisible by four, the “hinges” will be those two observations whose depths are the next largest integers in from either end of the batch. An alternate way to find the depth of the hinges is to take the depth (rank) of the median, and round it down if it is not an integer; then add 1 and divide by 2. Thus for a batch of five numbers, the hinges are the second and fourth observations; for a batch of eight numbers, one hinge would be the median of the second and third largest observations, while the other hinge would be the median of the sixth and seventh largest. The midspread is the absolute value of the difference between the two hinges.
10 For alternate symbols see Tukey, Exploratory Data Analysis, 351.
11 A New Hampshire Republican party official estimated that over 50 percent of New Hampshire’s foreign-stock population deserted the Republican party because of the issues of prohibition and immigration, according to an unpublished Republican Party Report in the Herbert Hoover papers quoted by Lichtman, Allan, “A Quantitative Study of the Presidential Election of 1928,” (Ph.D. dissertation, Harvard University, 1973)Google Scholar. For further background on New Hampshire, see Lockard, Duane, New England State Politics (New Jersey, 1959), 46–78CrossRefGoogle Scholar. V.O. Key, Jr., discusses some of the different types of realignment that occurred in New Hampshire and other New England states in his two seminal articles: “A Theory of Critical Elections,” Journal of Politics, 17 (February 1955), 3–18: and “Secular Realignment and the Party System,” Journal of Politics, 21 (May 1959), 21:198-210. Walter Dean Burnham has also noted somewhat different realignment periods for certain parts of Pennsylvania in his Critical Elections and the Mainsprings of American Politics (New York, 1970).
12 For purposes of analysis, one would usually have separate tables for actual numerical values and coded residuals, as in Exhibits 4 and 5. We have combined the two in order to conserve space. For those familiar with regression analysis, what we have done in breaking the analysis into two separate tables is functionally equivalent to adding ten dummy variables for individual county changes in Democratic voting as well as one for the change in common value from 1896–1924 to 1928–1972. This procedure is not completely arbitrary and ad hoc. Other election analysts noted above have viewed the two periods as distinct. Moreover, division of the data at any other point yields larger residuals—i.e., a less close “fit” to the data. On the other hand, a case can also be made for treating such elections as a single series—see Tukey, Exploratory Data Analysis, 238–41.
13 In order to fit a resistant line to the residuals one takes the midpoint—here between 1948 and 1952—as zero and assigns numbers to elections on either side so that each election “step” is 1. Thus for the data in Exhibit 8, 1948 = -.05; 1944 = -1.5; 1956 = 1.5; 1960 = 2.5; etc. Then take the first and last thirds of the elections (here 1928–1940 and 1960–1972) and calculate medians for the residuals and election numbers in those thirds. Here the median election numbers are always -4 and +4, while the residual medians vary from county to county. In Cheshire county, for example, the early third’s median residual is 
and the late third’s median residual 
Then find a line from the two points (-4, -3.9) and (+4, 4.4), calculate residuals from the line, and repeat the same process on the new residuals until changes in the slope and intercept of the line are small—an iterative process similar to median polish. Like median polish and the midspread, this robust form of linear regression is much more resistant to changes in estimates caused by outliers and missing data than is the usual least squares procedure. This is only one of several useful procedures for robust regression; for some other relatively simple ones, see Tukey, Exploratory Data Analysis. The intercept of the line is added to its county’s effect because it applies equally across elections, and the slope gives the “step” or change in Democratic percentage from one election to the next—with 1950 as the zero point.
14 That is, if the percentage defection from Republican to Democrat is the same in every county, the absolute number of Republican defectors (and hence the county’s new Democratic percentage) depends upon how many Republican voters there were to begin with. Butler, David and Stokes, Donald in Political Change in Britain (New York, 1971), 86–94CrossRefGoogle Scholar, present a good discussion of this notion.
15 An alternative explanation for residuals that vary with county or election effects might be that it is harder to move from 90 to 95 percent than from 50 to 55 percent. The problem with such an explanation is that it may not account for systematic interactions with opposite signs in elections whose effects are roughly comparable. Interpretations relating to defection rates also are consistent with qualitative knowledge about individual counties and elections.
16 “County Effects” used to calculate election accordion effects may or may not include a time trend term for the election in question. For simplicity’s sake and because addition of such terms makes little difference in the present case, we have used the county effects without time trend terms to calculate accordion effects. The line-fitting procedure is described in note 13.
17 Our two-way plot may seem unusual in that it has no horizontal scale, but it is easy to construct. Using ordinary graph paper, draw lines for row effects along one axis and lines for column effects along the other axis, using the same scale for each axis. Rotating the picture 45 degrees puts points of equal fit on horizontal lines. It can be demonstrated that the sum of the old vertical and horizontal coordinates is constant along any 45 degree line. Label the vertical scale using 45 degree lines where the old horizontal and vertical coordinates plus general typical term sum to 10, 20, 30 percent, etc., and transfer the plot itself to tracing paper. Suppressing the grid makes it easier to grasp what is going on in any graph.
18 Time trend terms would have to be set aside in order to see the accordion terms, but time trends cannot be pictured in a two-way plot because elections are not in chronological order. Both election and county accordion terms can be pictured in a two-way plot by twisting each individual county and election line about its midpoint. For more examples of two-way plots, see Tukey, Exploratory Data Analysis, 374–442.
19 For example, see V.O. Key, Jr., “Secular Realignment and the Party System,” Donald E. Stokes, “A Variance Components Model of Political Effects,” and Walter Dean Burnham. Critical Elections and the Mainsprings of American Politics.
20 In this respect, the divergence of normal county Democratic percentages takes the form of “opening scissors,” to use the analogy of V.O. Key, Jr., “A Theory of Critical Elections.”
21 Fitted constants can be thought of as coefficients for dummy variables in a regression equation. Put another way, each term in the overall fit (i.e., county effect, election year effect, election accordion term, county accordion term, general typical, etc.) can take on one or more specified values; the “number of fitted constants” refers to the number of specific values that a given term can take on. For example, since the data covers twenty elections, the election terms can have any one of twenty different values.
22 Realignment effects for 1924–1928 could also be calculated, taking time trends into consideration, which makes the 1924–1928 break smaller for some counties like Rockingham and Cheshire, but larger for other counties like Hillsborough. The bent plot also suggests that one might wish to consider the change in slope and the change in level as two different components of realignment.
23 In statistical terms, the question is how many degrees of freedom are necessary for meaningful results.
24 For futher discussions of this goodness of fit measure and its use, see Tukey, John W., Exploratory Data Analysis (Limited Preliminary Edition, 1970), chap. 18x, 19–21.Google Scholar
25 Residuals were examined by plotting their symbolic codes on county maps of the state for each election, as well as by looking at a table rearranged to order counties roughly from north to south and east to west. Maps often reveal a good deal of information, even though they did not do so in this case; they are natural to use for almost all types of aggregate data. Fitted terms, such as county effects or county accordion terms, can also be mapped in order to seek clues about what causal variables might underlie geographic patterns. Like time, geography is usually a surrogate for some other variable or set of variables.
26 The only economic data that are generally available for counties are those compiled in the decennial Population Census, Census of Manufactures, and Census of Agriculture. Employment data by state became available after 1947 as a result of the Full Employment Act of 1946 and are now available for major metropolitan areas.
27 The residuals for each county were smoothed using Princeton’s Interactive Statistical Processor (ISP) under the UNIX operating system on PDP 11/70 computer at Berkeley. The smoothing techniques are discussed in Tukey, Exploratory Data Analysis, 205–36 and 523–42.
28 Alford, Robert, Party and Society (Chicago, 1963), 219–341Google Scholar passim, especially 249–46; Odegard, Peter H., “Catholicism and Elections in the United States,” in Odegard, P.H., ed., Religion and Politics (New Jersey, 1960), 120–21Google Scholar; Converse, Philip E., Campbell, A., Miller, W.E., and Stokes, D.E., “Stability and Change in 1960: A Reinstating Election,”Google Scholar; and Converse, Philip E., “Religion and Politics: The 1960 Election,” in Campbell, Angus, et. al., Elections and the Political Order (New York, 1966), chaps. 5 and 6, 78–124Google Scholar. National survey results from the 1952 and 1956 elections suggest that the voting behavior of farmers and others from rural areas tends to be more volatile than that of their urban counterparts. Campbell, Angus, et. al., The American Voter, abridged edition (New York, 1964), 211–13Google Scholar. Another reason for greater variability might be the smaller number of voters; smaller numbers of cases tend to be less stable statistically.
29 Note the difference between the ten-county election year fits and the statewide Democratic percentages in Exhibit 4 above. The classic discussion of how “normal” aggregate voting patters relate to individual-level behavior is in Converse, Philip E., “The Concept of the Normal Vote,” chap. 1 in Elections in the Political Order, 9–39Google Scholar. See also Sellers, Charles, “The Equilibrium Cycle in Two-Party Politics,” Public Opinion Quarterly, 29 (Spring 1965), 16–38.CrossRefGoogle Scholar
30 Miller, W.L., “Measures of Electoral Change Using Aggregate Data,” Journal of the Royal Statistical Society, Series A, 135 (1972), 122–142CrossRefGoogle Scholar. The “Floating Voter” model is closely related to the “Mover-Stayer” model developed by Blumen and associates in a study of job-switching. Blumen, I., Kogan, M., and McCarthy, P. J., The Industrial Mobility of Labour as a Probability Process (Ithaca, 1955)Google Scholar.
31 Philip E. Converse, “The Concept of a Normal Vote,” 30.
32 Key, V.O. Jr., “A Theory of Critical Elections”Google Scholar; MacRae, Duncan and Meldrum, James, “Critical Elections in Illinois, 1888–1958,” American Political Science Review (1960), 669–83CrossRefGoogle Scholar; Pomper, Gerald, Elections in America, 101–11Google Scholar, and “Classification of Presidential Elections,” Journal of Politics (1967), 535–66; Burnam, Walter Dean, Critical Elections and the Mainsprings of American PoliticsGoogle Scholar; Campbell, Angus, “A Classification of Presidential Elections” in Campbell, , et. al., Elections and the Political Order, 63–77.Google Scholar
33 Gerald Pomper, Elections in America, 104.
34 Campbell, Angus, et. al., The American Voter, 531–38Google Scholar; Campbell, Angus, “A Classification of Presidential Elections.”Google Scholar
35 This definition of “deviating” elections differs slightly from that of Pomper and Campbell, et. al., in that it is not comfined to situations when control of government passes to the minority party.
36 The distinction between global changes and changes in the bases of party support has been suggested by Lichtman, Allan J., “A Quantitative Study of the Presidential Election of 1928,”Google Scholar and Flanigan, William H. and Zingale, Nancy, “The Measurement of Electoral Change.”Google Scholar
37 Converse, Philip E., Campbell, Angus, Miller, Warren P., and Stokes, Donald E., “Stability and Change in 1960: A Reinstating Election,” American Political Science Review, 55 (June, 1961)CrossRefGoogle Scholar, reprinted as Chapter 5 in Campbell, et. al., Elections and the Political Order, 78–95.
38 Key, V.O. Jr., “Interpreting the Election Results,” chap. 6 in David, Paul, ed., The Presidential Election and Transition, 1960–1961 (Washington, D.C., 1961), 162Google Scholar; Afford, Robert, Party and Society, 219–49 and 326–41.Google Scholar
39 Econometricians have devoted considerable attention to the problem of specification error. For useful discussions of the problem in relation to models of political behavior, see Tufte, Edward R., Data Analysis for Politics and Policy (New Jersey, 1974)Google Scholar; and Hanushek, Eric A., Jackson, John E., and Kain, John F., “Model Specification, Use of Aggregate Data, and the Ecological Correlation Fallacy,” Political Methodology, 1 (1974), 89–107.Google Scholar
40 Computer programs for two-way analysis are becoming available. Source code for APL and FORTRAN programs that do median polish for row and column fits is given in McNeil, Interactive Data Analysis (New York, 1977), 116–17. Similar routines are available as part of the Princeton Statistics Department’s UNDC-based “Interactive Statistical Processor” package, developed by Professor Peter Bloomfield. Interactive and batch FORTRAN programs for more extensive two-way analysis as described in this paper are under development for future release by John McCarthy at Berkeley’s Survey Research Center.
41 Another approach suggested by Gudmund Iversen, “Recovering Individual Data in the Presence of Group and Individual Effects,” (mimeo, Ann Arbor, 1971) would be to conduct sample surveys stratified by geographic areas that seem to behave similarly, and then to use estimates of proportions obtained from the surveys to construct ecological regression models for the entire population. Here two-way analysis could play a role in defining “similar” areas as well as in specifying the models.
42 Tukey, Exploratory Data Analysis, 443–65. Certain types of data call for some modification of the analysis—such as treating main diagonal effects separately in migration and social mobility tables. See also McNeil, D. R. and Tukey, J. W., “Higher-Order Diagnosis of Two-way Tables, Illustrated on Two Sets of Demographic Empirical Distributions,” Biometrics, 31 (June 1975), 487–510.CrossRefGoogle ScholarPubMed