DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 91125 LOG-LINEAR ANALYSIS OF CONTINGENCY TABLES: AN INTRODUCTION FOR HISTORIANS WITH AN APPLICATION TO THERNSTROM ON THE "FLOATING PROLETARIAT" Morgan Kousser J. California Institute of Technology Gary W. Cox California Institute of Technology David W. Galenson University of Chicago SOCIAL SCIENCE WORKING PAPER 417 February 1982 SUMMARY For historians or other social scientists who se data is available in discrete (nominal- or ordina l-level) form, recently developed "log-linear" mu ltivariate statistical techniques offer considerable advantages over commonsensical devices and are in many respects superior to such multivariate methods as mu ltiple classification analysis, weighted least-squares, and logit. Reana lyzing Thernstrom' s Boston data on geographic mobility. we explain the ideas beh ind and the procedures of log-linear analysis explicitly, step-by-step. Intended for people who are already somewhat familiar with Rtatistics ( say, through mul tiple regression) , the paper is self-contained and as simple as we could make it. Af ter reading it carefully , one should be well prepared to perform such an analysis himself. Substantively. we sketch a simple economic model which points to age as an important determinant of the decision to move or stay, and our results cast doubt on Thernstrom's tentatively offered notion of a "floa ting proletariat." All.ALYSIS OF CONTINGENCY TABLES: U>u-Ll11EJ\J: AN INTilODUCTION FOR HISTORIANS WITH APPLI CATION TO TIIERNSTROM ON TIIE AN • "FLOATING PROLETARIAT" Morgan Kous ser J, Gary W. Cox David W. Galenson Suppose a researche r has inform ation on several attributes of a collection of individuals, and that the data he has is avail able only in qualitativ e (synonyms are categorical, discrete, polytomous, or ordinal- or nominal-level ), as oppo sed to quantitativ e (continuous or interval-level) fo rm. Fo r in stance , imagine that his information is about yes or no votes, occupationa l classes, or age groups, but none is in the form of, say, the do llar amounts of property held (not broken into categorie s) or the length of residence, in months or years, at a pa rticular location. Then he might construct tables, such as Table 1, which show how many people have each se t of traits -- for example, how many young, unskilled, chil dless men in a sample were • An curlier version of this paper was presented at the Social Science History Association convention in 1981 . We wish to thank Stanl ey Enger01 an, Douglas Hibbs , Philip Hoffman, Co lin Loftin, Dou glas Riv ers , Stcvhan Thcrnstrom , Quang Vuong , Sally Ward and especially Robert McCaa for comments on various iterations of the piece, Kousser's research wa s partially supported by grant R0-20225-82 from the Na tiona l Endowment for the lhunanities, We take reponsibility for al l remaining error s. 2 3 found in both the Bost on census schedul e in 1880 and the city literature, encourage and prepare historians to make use of log-linear directory in 1890. When there is very litt le information available, analysis as we ll as to be able to go on to more advanced tr eatment s in say, data on only two or three variables, commonsensical methods of statistics texts . Those who de sire a brief overview of the subj ect analysis may suffice . But what should one do when one is confronted may wish to save section III , in which we lay out the algebra step by by such monsters as the eighty-celled "four-way" Table 1? st ep, for a second reading , while those al re ady famil iar with or The convent ional historical answer to this que stion has been indifferent to log-l ine ar methods may wish to skip sections II-V. We to combine the categories (or, to put it another way, to collapse the will al so attempt to show how social scientific theory can help to table) into what are called "marginal tables," relating two or perhaps guide data analysis and shal l emphasize a hitherto often overlooked three variables, as in the panels in Table 5, below. While this is facet of a much-studied historical problem, Thus , we hope to blend useful and often informative, the practice may hide information which substantive with methodological po ints. Written at a fairly is available in the ful l table, Fortunately, in the pa st fifteen elementary level and self-conta ined, the paper assumes only that one years statist icians have developed new methods for squeezing many more has a speaking acquaintance with such st atistical concepts as Chi­ conclusions out of such tables, Historians have made too littl e use Square te sts and regre ssion analysis, of the new techniques, generally denominated "log-line ar contingency table analysis," probably because the initial articles and books explaining them were somewhat ob scurely phrased and were not easily I. HISTORICAL MOBILITY STUDIES acce ssible to those who lacked fairly advanced statistical training , Now that there are simp ler texts on the market and several computer Before beginning the statistical discussion, let us briefly programs available, it is time that many more historians took introduce the substant ive problem with which we shall be concerned advantage of them. throughout this article. During the past two decade s, many historians Th is paper is intended to prov ide both an intuitive and a have investigated geographic and social mob il ity in nineteenth-century practical mathemat ical understanding of the log-linear te chnique , Am erican cities, Aimed primarily at systemat ically de scribing the demonstrate it s use fulness by reexam ining an important historica l characteristics of indiv idual s which were associated with changes in topic, and, by mak ing every st ep in the development and appl ication of residence and in occupat ional or social rank, the ir works have rel ated the technique explicit, using the notati on now coLtmon in the both type s of mob il ity to such variables as age, occupat ion, family 4 5 soci al status , property holdings, ethnic origins, and gene rati on of From Thernstrom' s data set, we have chosen three factors which 1 residence in America, Drawing on such previously unexploited sourcos all pl ausibly bore on the 1880 Bostonian's de cision to move or stay: as federal and state manuscript ce nsuses, ci ty directori es, and tax family status, which we will call "S", and which we cut into two assessment rolls, the "new social hi stori an s" have attracted a good groups -- the first , sin gle or married but childless, and the se cond, de al of attention by taking advan tage of the chance these source s married with chil dren; occupation, or "O," whi ch we broke into five offer to study the live s of large numbers of individual s who have classes -- hi gh whi te collar, low whi te col lar, skilled, unskilled, previously eluded the view of hi storians. and unemployed; and age, or "A," which we cut in to four sets -- 14-20 , 3 Ye t these schol ar s have failed to make use of the available 20-30, 31-60 , and over 60 , The number of pe ople in the sample with statistical me thods and social scientific theory as fully as they have each set of traits in both 1880 and 1890 is di spl ayed in the four ransacked the sources. More specifically, they have generally related panels of Table 1, Since it is probable that some of those who were only two or three vari ables to each other, thereby in effect assuming not listed in the 1890 di rectory we re simply overlooked by the that the numerous "independent variables" in their mobility analyses canvassers but were still present in the Boston area, we wil l were un correlated with each other; their implici t statistical models, hereafter refer to the divi sion of the sample as those who were furthermore, generally assume, wi thout testing, that the relationship s "found" and "not found" rather than as "stayer s" and "movers" or 4 they seek are linear. By focusing on the di fferent correlates of "per si sters" and "nonpersisters" , We wil l label the variable "M" . mobility only one or two at a time, they have generally settled for The relationship of migration with occupation plays a large mere description, instead of confronting directly the prob lem of role in Thernstrom's explanation, that wi th age is stressed in the building a cohe sive explanation. And their an alyse s have been less economics li terature , and that with children taps the notion that wel l informed by social scientific theory , particul arly economic familial re sponsibilities constrain . We also initially included theory , than they might have been. We sh all illustrate how the se measure s of homeownership, the number of generations a man had been in problem s might have been largely obviated by reana ly zing data on America , and ethnicity in order to attempt to me asure some of the geographic mobility ga thered by Stephan Thernstrom for his study The effects, re spectively, of different levels of transactions costs Other Bostonians, which trace s individuals from the 1880 federal involved in the de ci sion to move, rootedness, an d a po ssible high manuscript census to the 1890 city directories of Boston and its de gree of employmen t di scrimina tion (e specially again st the Irish) in 2 suburbs .