Ann. Rev. Psychol. 1981. 32:191-241 Copyright © 1981 by Annual Reviews Inc. All rights reserved GRAPHICAL DATA ANALYSIS +344 Howard Wainerl Bureau of Social Science Research, Washington, D.C. 20036, and Educational Testing Service, Princeton, New Jersey 08540 David Thissen Department of Psychology, University of Kansas, Lawrence, Kansas 66044 CONTENTS INTRODUCTION ................................................................................... ....................... 192 Limits of this Chapter .......................... ......... ............... ........................... .......... ....... 192 HISTORY ...................................................................................................................... 194 GRAPHICS FOR DATA ANALYSIS ........................................................................ 196 One-Way Displays of One- Way Data ..................... ............................ ..... ................ 196 Two-Way Displays of One-Way Data .......... ........ ................................................ .... 199 Stem-and-lea! diagrams ......... ............ .......... ............ ........... .......................... .......... 199 Box plots ... ..... ....................... .................. .... ...... ................. .................. .................. 203 Suspended rootograms . .................. ..... ... ... ..... ........... ................. ... ....... ..................... 204 pop and Q-Q plots .......... ........... .................. ............. ............... .......... ..... ..... ........... 207 Two-Way Displays of (Mostly) Two-Way Data ........................................................ 211 Diagnostic plots ................. ;..................... ................................................................. 213 Enhanced scatterplots ................................................................................................ 214 Two-Way Displays of Multivariate Data ............... ................................................... 219 Tables .................................................................................................................... 220 Access provided by Columbia University on 01/22/15. For personal use only. Glyphs .................................................................................................................... 221 Annu. Rev. Psychol. 1981.32:191-241. Downloaded from www.annualreviews.org Inside-out plots ........................................................................................................ 221 Faces ...................................................................................................................... 222 Other suggestions ................................... ................................... ......... ..................... 223 A multivariate example .................................................................................... :....... 223 Three-and-More-Way Displays of Multivariate Data ............... ............................... 231 EVERYTHING ELSE .................................................................................................. 233 Complexity vs Simplicity ........................................................... ..... ................. ......... 233 Standards .................................... .......... ......... ..................... ......... ... .......... ................ 234 Experimental Evidence ................................ ... ............ ...... ......................................... 234 SUMMARY .................................................................................................................... 236 'Research for this chapter was supported in part by a grant from the National Science Foundation (SOC-76-l7768), Albert D. Biderman and Howard Wainer, principal investiga­ tors. 191 0066-4308/81/0201-0191$01.00 192 WAINER & THISSEN INTRODUCTION Psychology is full of a facile complexity. Physical sciences gain in truth when most quantitative, whereas psychology, when most quantitative is of least scope, though the sweetness of elegance lingers on. Over the century since Wundt's laboratory started modem experimental psychology, its characteristic paradigms have emerged in fits and starts. In an attempt to deal with the frequent crudeness of its data, psychology has adopted, adapted, and polished a set of statistical procedures and models that are second in sophistication and complexity only to those employed by Car­ lyle'S "dismal science." The often self-conscious attempt to make paradigms statistically rigorous when experimental controls lacked rigor was in keep­ ing with comments of statisticians such as Wallis (1949), who argued that "So-called 'high-powered,' 'refined,' or 'elaborate' statistical techniques are generally called for when the data are crude and inadequate-exactly the opposite, if I may be permitted an obiter dictum, of what crude and inade­ quate statisticians usually think." Of course, there were many reasons other than self-conscious defensive­ ness for the development and use of "high-powered" statistical techniques. Often such techniques were desperately required. This need provided the impetus for the development of rigorous, formal statistical models, but deemphasized the need for descriptive methods: methods whose aims were exploratory rather than confirmatory. The past 10 years have seen a vigorous development of methods, by the foremost of statisticians, which have as their very nature an informality that was unheard of in the social sciences for a half century. These techniques are primarily graphical in form, flexible and robust in character, and infor­ mative in a wide variety of problems and so are of general applicability. It is toward a review of these techniques that this chapter is directed. The signal event in this area was the 1977 publication of Tukey's Explor­ atory Data AnalYSis (EDA). Prepublication versions of this document Access provided by Columbia University on 01/22/15. For personal use only. Annu. Rev. Psychol. 1981.32:191-241. Downloaded from www.annualreviews.org abounded, and so dating the start of the movement of informal data analysis by that book's publication is a bit misleading. Consequently we shall refer to Tukey as the modem progenitor of the movement which dates back at least to 1962 and the publication of Tukey's prophetic "Future of data analysis." Other recent works of importance on this topic include: Everitt (1978), Gnanadesikan (1977), Hartwig & Dearing (1979), Monkhouse & Wilkinson (1971), Mosteller & Tukey (1968, 1977), and Wang (1978). Limits of this Ch ap ter In this chapter we shall not discuss directly issues of robustness. although that is one aim of many of the procedures which will be discussed. Some- GRAPHICAL DATA ANALYSIS 193 times graphical methods allow the easy computation ofa statistical test (e.g. binomial probability paper, Mosteller & Tukey 1949) or a statistic (comput­ ing a robust correlation graphically, Sandiford 1932; or model parameters, Bock and Jones 1968); we shall generally not discuss this aspect, although occasional reference toward such things will be made. Further, although we recognize that graphics have many uses, we shall generally limit ourselves to data analysis. Graphs for communicative purposes or nomographs for computational ones we shall leave to other accounts. Occasionally some rules for good exploratory graphs carry over into presentational rules, and when this occurs we shall make mention of it. We view this chapter as didactic, and so it will contain many examples of the sorts of plots we will be discussing. This will avoid the anomalous situation of a chapter on graphs describing its subject in prose. We intend to "show" rather than "tell" whenever possible. We shall exclude color for purely economic reasons. It is the purpose of a review article to start where the last review article left off and to contain as much of the intervening work as space permits. This allows the researcher to read the most recent review and be more or less current with the work done at the time that review was written. Since this is the first review of graphics to appear in the Annual Review of Psychology, we will have to go back a bit further than most such chapters and have a bit spottier coverage. We shall attempt to increase the depth of coverage as the work becomes more current. The role of graphs intheory development is oftenmisunderstood. Tilling (1975), in a recent history of experimental graphs, restates this misconcep­ tion: Clearly an ability to plot an experimental graph necessarily precedesan ability to analyse it. However, although any map may be considered as a graph, and carefully constructed maps had been in use long before the eighteenth century, we do not expect the shape of a coast-line to follow a mathematical law. Further, although there are a great many physical phenomena that we do expect to follow mathematical laws, they are"in general Access provided by Columbia University on 01/22/15. For personal use only. Annu. Rev. Psychol. 1981.32:191-241. Downloaded from www.annualreviews.org so complex in nature that direct plotting will reveal little about the nature of those laws ... (p. 193) Attitudes like these have hindered appropriately serious regard for such theories as that of continental drift, whose initial evidence (noticed by every school child) is solely graphical. The point of this chapter is to introduce many of the developments in graphical data analysis to psychology, as well as