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If you have issues viewing or accessing this file contact us at NCJRS.gov. '!Jr- • ~,-.. -- ____ •_____ ~-.--.;~~--, ~.-....., .'~-----~~"- .~ ..... ~~~_-'~ ...... ,-.-.-.,---t.\...-_~_ " . 4 .,. I National Criminal Justice Reference Service I-----------------~~----------------------------------------------------nCJrs This microfiche was produced from documents received for inclusion in the NCJRS data base. Since NCJRS cannot exercise .. MEASURING ASSOCIATIONS WITH control over the physical condition of the documents submitted, RANDOMIZED RESPONSE* the individual frame quality will vary. The resolution (;'hart on this frame may be used to evaluate the docum~nt quality. 1.0 :~ I~II~ IIIII~ w~ ~M 2.2 James Alan Fox L\l D36 W ~ I:iu u_n~ 2.0 College of Criminal Justice ... ~ 1.1 ... " .. Northeastern University ~I 'f" ! . '" and " i '~ , 111111.25 1I1111.~ 111111.6 ( ) . 1- I Paul E. Tracy Center for Studies in Criminology and Criminal Law II' University of Pennsylvania MICROCOPY RESOLUTION TEST CHART 11 NATIONAL BUR(AU OF STANDARDS.1963.A f 00 vi· ' !I' ' - l (April, 1982) II Microfilming procedures used to create this fiche comply with the standards set forth in 41CFR 101-11.504. I! 1/ cr , , Points of view or opinions stated in this document are those of the author(s) and do not represent the official position or policies of the U. S. Department of Justice. *Order of authors names determined by lottery. Direct communications to National Institute of Justice ! James Alan Fox, College of Criminal Ju'stice, Northeaste'r:n Unj.versity, United States Department of Justice f,,,> Boston, Massachusetts 02115. The aut.hors gratefully acknowledge Wesley Washington, D. C. 20531 Skogan of Northwestern University whose inquiry stimulated this note. : 0 '0 ' :3/27/84\ ) " ~. "1~------ ') MEASURING ASSOCIATIONS WITH RANDOMIZED RESPONSE Since its introduction by W'arner (1965) the randomized response Abstract ., approach has been the focus of considerable theoretical literature but has not been readily welcomed by survey researchers. The reluctance to The vi~w of many in the soc.ial science research community, it utilize randomized response is perhaps rooted in the fact that the de- appears, underestimates the scope and potential of the randomized response cision to employ the technique is irreversible. That is, unlike analytic technique, particularly with regard to its analytic capabilities. That innovations which can be experimented with and then discarded by the re- is, there seems to be some concern that this data collection strategy is, searcher without altering the quantity or quality of survey response data, by its nature, analytically restrictive. However, in this note we demon- the researcher cannot recapture information that is lost or altered by strate how the quantitative, unrelated question model of randomized response using a randomized response data collection strategy. It is easily under- can be reformulated into a measurement error model, which in turn allows a t stood, therefore, why randomized response is considered by some to be more wide range of multivariate approaches. Last, we illustrate through a f ) of a validation check than a primary method of collecting data (see, e.g., simulated correlation experiement the type of adjustments that need only Penick and Owens, 1976). be made to 'manipulate randomized response data more powerfully than has Skepticism over the randomized response technique appears to sur- been the practice in the past. round two principal issues: whether the reduction in bias earned through randomized response outweighs the inefficiency of the technique and the extent to which the technique limits analyt:f.c capabilities. The former U.S. Depertment ofJuatlee NatlOMIlnlttltute of Ju.tlce ThIs document hal bMn reproduced 8)(actly as reoefl/od ftom tho issue has been the topic of a number of validation efforts (see, e.g., ptI'IOn or 6fganlzaUon orIgoInaUng II. Polntl) of view or oplnluns 8111100 In thla documtnt 110 thoto of IIle authm and do oot oocessorlly t~r"'ol ltIe QlJc:ia/ potIUoo 01' poIIeIet ollhe NaUoo1I1 Institute of Folsom, 1974; Locander, 1974; Locander et. al., 1976; Bradburn and Sudman, ~~oo. ~t1on 10 reproducalhll ~ materia' has boon 1979; and Tracy and Fox, 1981) which seem to suggest that the randomized granted by • Public Danain,/Natl.onal Institute of Justice ----- response technique has value in reducing systematic response error. The ---------------------.----------- to the Nr.lonal Criminal Juab Retll'lII1Ce Setvloe (NCJRS). second concern, more partict~arly, that multivariate statistics are not Fur1hlw reproduction outllde of tt;. NOJRS a)'Stem requlro/i permlD' IkJ.n of Ihe~ 0WMr. possible with data ga.thered through randomized response methods, is still an issue around which there is, considet'able confusion. The application . , of univariate summary measures (e,g., proportions and means) and the 1 comparison of these measures across suhgroups are ohvious. HO't"ever, the richer ~, multivariate approaches (e.g., correlations, ANOVA; logit analysis) require t individual level data ...·hich are seemingly unavailahle 'Idth the randomized \ \ Distribution of Measurement Error in Randomized Responses response technique. Sudman and Bradhurn's (1982:81) recent comment reflects this \ ~ I, Consider the quantitative, unrelated question model employing an By using this procedure, you can estimate the undesirahle hehavior of a group; and, at the same time, the respo.ndent's I alternative question having a known distribution (see Greenberg et a1. , a.nonymity is fully protected. With this method, ho ...·ever, you I cannot relate individual characteristics of respondents to in­ 1971). More particularly, a survey respondent is instructed to manipulate dividual behavior. That is, standard regression procedures are not possihle at an individual level. If you have a very large a randomizing device following a bernoulli distribution with known para­ sample, group characteristics can he related to the estimates I ohtained from randomized response. For example, you could look I meter p, and then either to answer a sensitive question having unknown at all the ans ...·ers of young ~'omen and compare them to all the answers of men and older age groups. On the whole, however, much mean and variance, J.I and 0'2, or to report innocuously a random digit information is lost '1"hen randomized response is used. x x I (or nonsensitive response) having known location and scale parameters, In this note ~'e sho~T hm,' unkno ...n individual level data can he estimated for and 0' 2 , depending on the outcome of the randomizing device. use in multivariate analyses ...·ithout loss of information, explicating mathematically Y Let represent the verbal (observed) response given by the ith an approach recommended conceptually by Boruch (1972:410) (see also Boruch and respondent, and let and y' be the underlying (unobserved) sensitive Cecil, 1979:142). MOre precisely, we demonstrate that randomized response ohser- i and nonsensitive scores. As dictated by the randomizing device with vations can he treated as individual level scores that are contaminated hy se1ec tion probability p, random measurement error and, consequently, that ~ultivariate measures need only he corrected for or purged of the effects of this error. Xi with probability p As is true of most of the theoretical and applied literature concerning z (1) i = randomized response, efforts to develop bivariate and multivariate approaches Yi with probability 1-p for randomized response data have focused ~ainly on dichotomous (or polytomous) Conditional on fixed xi' data in multh'ay contingency tahles (see Boruch, 1971; Barksdale, 1971; Drane, 1976; Cl icker ?-nd Igl e'\l'icz, 1976; Chen, 1979; Kraemer, 1980; and Tamhane, 1981) (2) with only limited attention to quantitative responses (see Rosenherg, 1979; Thus, although individual scores on the sensitive question (x) are un- Kraemer, 1980; and Himmelfarb and Edgell, 1981). The "errors in variahles" known, we can estimate approach to quantitative randomized response data presented here, like Chen's (1979) misclassification perspective in the qualitative case, is sufficiently (3) gene~al to apply to a variety of data collection strategies and analytic techniques. l 3 ~.. .........,~ ---- Thus, We ~an define, further, the measurement error introduced by random- ized response as ~ = (l-p) (~ ... ~ ) r p x y' (4) 2 o == r Alternatively, the estimated individual scores derived from Equation (3) can be viewed as the actual score plus a disturbance term, i.e., ...:nd (5) \.is = ~ yx- u Considering next properties of the disturbance u ' we sub- i 2 222 o = C1 /p + 0 • stitute Equation (3) into Equation (4) , s Y x· u Substituting rand s back into Equation (6), it can be shown that i = ~ = P Jl (l-p) \.i 0 (7) From Equation (1) u r + s'= and [x - (l-p) ~ l/p - x w.p. p i y i ~ u i = (6) w.p. l-p i ~ For simplicity, set + (l-p) o2 /p2 + 0 2 + (~ ... ~ ) 2J j [ y !f 1. x x r = [Xi ... (l-p) ~y]/p ... xi ' i I • (l-p) 2 ! f02 + 0 / + (~ _ ~ )2] (8) • (1-p) i p Lx y p x y p (Xi - ~y) ; >~ ~, i producing the desired expressions for the mean a~'ld variance of the dis- ~ and 1 turbance term •• :j Next, the covariance between Xi and u can be obtained in a simi­ :ir i 'I <,~ ,I lar fashion. We have ./ 4 5 f'' , possible so long as the summary statistics (e.g., correlations) are G = E(xu) - E(x) E(u) = E(xu) xu corrected for attenuation produced by the randomized response procedure. because E(u) = O. Multiplying Equation (6) by xi' and collecting That is, the randomized response estimated scores are unbiased and, 2 xi terms, although containing measurement error, the effects of this unreliability can be corrected, as we demonstrate in the next sp-ction. 2 )/ (1 -p ) ( xi - xi ~y P w.p. p w.p. 1-p Next, we take expected values and collect covariance terms, (l-p) (a2 + ~2 _ ~ ~) w.p.
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