Bayesian models to adjust for response bias in survey data for estimating rape and domestic violence rates from the NCVS

The Annals of Applie d Statistics 2008, V ol. 2, No. 2, 665–686 DOI: 10.1214 /08-A OAS160 c  Institute of Mathematical Statistics , 2 008 BA YESIAN MODELS TO ADJUST F OR RESPONSE BIAS IN SUR VEY D A T A FOR ESTIMA TING RAPE A ND DOMESTIC VIOLENCE RA TE S FR OM THE NCVS 1 By Qingzhao Yu, Elizabeth A. St asny and Bin Li L ouisiana Sta te University and Ohio State U niversity It is difficult to accurately estimate the rates of rap e and domestic violence due to the sensitive nature of th ese crimes. There is evidence that bias in estimating the crime rates from surv ey d ata ma y arise b e- cause some w omen resp ondents are “gagged” in rep orting some types of crimes by the u se of a telephone rather than a personal interview , and by the p resence of a spou se during the interview. On t he other hand, as data on these crimes are collected every year, it would b e more efficient in data analysis if w e could identify and m ake use of information from previous data. In this pap er we propose a mo del to adjust the estimates of the rates of rap e and domestic v iolence to account for the resp onse bias du e to the “gag” factors. T o estimate parameters in the model, we iden tify the information that is not sensi- tive to time and incorp orate this into prior distributions. The strength of Bay esian estimators is their ability to combine information from long ob serva tional records in a sensible wa y . Within a Bay esian frame- w ork , we develop an Exp ectation-Maximization-Ba yesian (EMB) al- gorithm for computation in analyzing con t ingency t able and w e app ly the jackknife to estimate the accuracy of the estimates. Our approach is illustrated using t he yearly crime data from the N ational Crime Victimization Survey . The illustration show s th at compared with the classical metho d, our mod el leads to more efficient estimation but does not require more complicated compu tation. 1. In tro d uction. Rap e and d omestic violence rates are d ifficult to esti- mate b ecause of difficulties in colle cting data on these crimes. Ann u al rap e incidence r ates in the U.S. obtained from p olice s tatistics, rep orted thr ou gh Received March 2007; revised Jan uary 2008. 1 Supp orted in part by aw ard number 93-IJ-CX- 0050 from the National Institute of Justice, Office of Justice Programs, U.S. Department of Justice, and by the Cen ter for Survey Research Sum mer F ellow ship, OSU. Key wor ds and phr ases. Categorical data analysis, contingency table, EMB algorithm, incompletely classified data, Jac kknife method, panel survey, survey mo de. This is a n e lectronic repr int of the o riginal article published by the Institute of Ma thematical Statistics in The Annals of Applie d Statistics , 2008, V ol. 2, No. 2, 6 65–68 6 . This repr int differs from the origina l in paginatio n and typogra phic deta il. 1 2 Q. Y U, E. A. ST A SNY AN D B. LI the Unif orm Crime Rep ort (UCR), were estimated to b e 0.3 p er 1,000 p er- sons in females age 12 an d older [Bureau of Justice Statistics ( 2002 )]. But the ma jorit y of b oth rap e and domestic violence in ciden ts are not rep orted to p olice. Data f r om the National W omen’s Stud y , a longitudinal telephone sur- v ey of a n ational household pr obabilit y sample of wo men at least 18 yea r s of age, sh ow that 683,000 w omen we re f orcibly rap ed eac h y ear and th at 84% of rap e victims did not r ep ort the offense to the p olice (CDC’s National Center of Inju r y Prev en tion and C on trol web site). Thus, we b elieve that the UCR underestimates the r ates of su c h crimes. The National Crime Victimization Survey (NCVS) is the b est national source for estimates of rates of rap e and domestic violence in the Un ited States. It includes crimes r ep orted and not rep orted to the p olic e. But NCVS rates migh t still b e b iased b ecause: (a) the definition of “criminal” rap e or domestic violence w as left to the resp onden t who ma y defi n e rap e d ifferen tly relativ e to the legal defi nition; (b) p ersonal NCVS in terviews may not b e conducted confiden tially (with the r esp ondent alone); and (c) telephone interviews ma y not b e suffi cien tly priv ate. Leggett et al. ( 2003 ) rep orted that p eople are more likel y to r ep ort in a face-to-face in terview, compared with a telephone interview. In this p ap er we deal with resp ond en t bias caused by (a) pr iv acy concerns in telephone inte rviews and (b) the in-p erson in terviews not conducted confidentia lly . It is imp ortan t to ha ve accurate and reliable estimates of rap e and d o- mestic violence r ates. The crimin al justice communit y , f or example, needs accurate estimates of these rates to ev aluate ho w w ell it meets th e needs of victims, and whether criminal justice interv ent ions help reduce the rates of rap e or domestic violence. Since the NCVS is a p opulation-based data source for estimating crime rates, in terven tions to r ed uce r ap e (or domestic violence) can use the NCVS data to help ev aluate their efficacy . The NCVS survey can b e conducted by ph one or in -p erson. Other in di- viduals are allo wed to b e pr esen t during the interview. Ou r researc h on the NCVS data (see Section 2.2 ) su ggests that rap e and domestic violence inci- den ts are un der-rep orted by women in telephone in terviews, or if a sp ouse is present du ring the inte rview. W e refer to these as the gag factors in re- p orting rap e and domestic violence. Our research also indicates that the effect of gag f actors in un derrep orting crimes is relativ ely constan t o ve r time. In the follo wing w e deve lop a Bay esian mo del that allo ws the use of b oth curren t data and previous inf ormation to estimate rates of rap e and domestic violence wh ile taking in to account p ote n tial un der-rep orting of these crimes du e to the gag effect. This pap er presents a Ba y esian mo del for estimating classification probabilities and p robabilities of resp onden ts’ bias. Our mo dels tak e in to accoun t the resp onden t bias caused by kn o wn influentia l factors. Moreo v er, in estimating the distrib u tions of int erest, we iden tify from the previous data the information that is not sensitiv e to time and us e it to bu ild prior d istr ibutions for some of the parameters, so that BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 3 the previous in f ormation is efficien tly used in estimation. W e also pro vide computation metho d s that allo w complicated p osterior d istributions to b e explored through simple iterations. W e u se the jac kknife in the Ba yesia n en viron m en t to calculate the acc uracy of the estimates, and then compare the results to estimates obtained from the classical metho d. W e illus tr ate our metho dology by estimat ing rates of rap e and domestic violence usin g the p ublic NCVS d ata. The bac kground literature for our resp onse-bias-adjusted Ba yesia n mo d- els is based on three lines of researc h. First, extensive researc h h as b een done on metho d s to capture information on resp onden t bias. Our mo d el h as its r o ots in th e work of Stasny and Cok er ( 1997 ), who id en tified the pr ob- lem of resp onse bias in self-rep orted crime and d ev elop ed a mo d el to adjust for the b ias. In this pap er w e mo d ify the mo del and in corp orate historical information to impro v e estimation. The second line of bac kground researc h concerns the extension of the EM algorithm to fin d empirical Ba yesian es- timates in analyzing con tingency tables. S ee Little and Ru bin ( 2002 ) f or an in tro duction to EM algorithm; Carlin and Louis ( 2000 ) and Bishop, Fien- b erg and Holland ( 1975 ) f or an introd uction of emp irical-Ba y esian estima- tors of cell probabilities. W e extend the EM algorithm to find empir ical Ba y esian estimates b y addin g a third step, the B step, in w hic h we cal- culate empirical-Ba yesian estimates based on cur ren t maximum lik eliho o d estimates. W e call this an Ex p ectatio n-Maximizatio n -Ba y esian (EMB) al- gorithm. T o our knowledge , this is the first pr op osal for s uc h an algo rithm, although there are Ba ye sian ve rsions of EM b y Gelman et al. ( 2004 ). EMB is u seful in estimating surv ey classification and resp ondent bias caused b y an y infl uent ial factors when pr ior inf ormation is a v ailable. Third , we u s e the jac kknife metho d [Efr on ( 1987 )] in the Ba y esian en vir on m en t to measure the accuracy of the estimates. In the next s ection w e pro vide a brief description of the NCVS and an exploratory analysis of the d ata. Section 3 p resen ts a mo del that adjusts data for gag factors. Section 4 discus ses efficiency gains usin g the Ba y esian mo del and describ es buildin g prior distributions, the EMB alg orithm and ho w to use the jac kknife m etho d in the Ba y esian environmen t. Section 5 present s the results of analysis on NCVS. Finally , Section 6 p oints out some directions for future r esearc h. 2. The National Crime Victimization Surve y and data. 2.1. Survey design. The National Crime Victimization S urvey is adm in- istered by the US Census Bur eau on b ehalf of the Bureau of J ustice Statis- tics. The sur v ey has b een collecting data on p ersonal and household vic- timizatio ns since July 19 72. It w as formerly kno wn as the National Crime Survey b efore its redesign in 1989 w hen the curr en t su rv ey metho dology 4 Q. Y U, E. A. ST A SNY AN D B. LI b egan systematic field testing. T he first annual resu lts fr om the redesigned surve y were pub lished in 1993 [Bureau of Justice S tatistics ( 1995 )]. There w ere some fu r ther changes to the sur v ey after that, but the data collection pro cedur e and instru men t that pr o vides imp ortant in formation to our anal- ysis w ere consistent ov er these y ears. W e u se d ata from 199 3 to the most currentl y av ailable data online. The NCVS is the primary source of n ational- lev el information on victimizations, includin g n ot only crimes rep orted to the p olice, bu t also those not rep orted to la w enf orcement authorities. W e briefly describ e the NCVS in the rest of this su bsection. Add itional in f or- mation on the design and history of the NCVS is pro vid ed, for example, b y the U.S. Department of J ustice and Bureau of Ju stice Statistics ( 2001 ) and Stasn y and Cok er ( 1997 ). The NCVS is compr ised of a str atified, multi-stag e, cluster sample of hous- ing u nits (HUs). This ongoing s u rv ey seeks to obtain a repr esen tativ e sample of individuals 12 yea rs of age and older living in hou s eholds or group quarters within the United States. Semi-annual data on th e frequency , c haracteristics and consequences of crimin al victimizatio n are collected fr om app ro ximately 49,000 hous eh olds comprising ab out 100,000 p ersons. Th e NCVS u ses a ro- tating panel design whereby a samp led HU is maintai ned in th e sample for sev en panels with inte rviews conducted at six-month interv als. The fir st in- terview conducted within a household is considered a b ounding inte rview, whic h is not p ublished b ut is used as a con tr ol to av oid duplicate rep orting of an inciden t. New h ouseholds r otate in to the sample on an ongoing basis. During the interview, in dividuals are asked ab out crimes committed against them or against the h ousehold (HH) in the past six months. The crimes are catego rized as p ers onal (whic h includ es rap e/sexual assault, robb ery , ag- gra v ated/simple assault and p ersonal larceny) or prop erty (whic h includes burglary , auto or motor ve hicle theft, th eft and v andalism) related. Crimes not cov ered include kidnapp ing, murder, shoplifting and crimes that o ccur at places of busin ess. Th e survey instru m en t is comp osed of a scr eenin g sec- tion and an incident rep ort. A single HH resp ond en t is asked a series of six screening questions to elicit inform ation on crimes committed against the HH (e.g ., bur glary , larcen y , m otor v ehicle theft). Next, an elev en-question screener is used to elici t information from eac h in dividual in the HH con- cerning p ersonal crimes committed against th at individual. If any screening question elicits a p ositiv e resp onse, an incident rep ort is filled out. The r ep ort is designed to obtain detailed data on the charac teristics and circums tances of the crime, s u c h as the mon th , time, lo cation of the in cidence, relationships b et w een victim and offender, offender charac teristics, self-pr otectiv e actio ns, t yp e of pr op ert y lost, wh ether crime was rep orted to the p olice, consequences of the victimizat ion and offender use of we ap ons, dr ugs or alcohol. The initial NCVS interview at a housing u nit must b e conducted in p er- son. The su b sequent survey conta cts at the same add ress could b e condu cted BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 5 either thr ough telephone or f ace-to -face. Primarily for cost reasons, phone con tacts are emp hasized in the later in terviews; a face-to-face in terview is conducted only when it is inefficien t or infeasible to mak e con tact b y ph one. Since r esp ond ents are ask ed to describ e the victimization, the lac k of priv acy can influen ce resp onses d uring a telephone interview. Id eally , a p ersonal in- terview is conducted and the in terview er and r esp ondent are alone dur ing the in terview. In our data, appro x im ately 45% of p ersonal int erviews are conducted alone with the resp ond en t. Ho wev er, this is not alw a ys p ossible. In neither the phone nor p ersonal su r v eys are in terview ers instructed to es- tablish a priv ate inte rview setting. Dur in g the face-to-face int erview, if th e resp ond en t is not alone, the interview er indicates on the questionnaire who else is pr esent. When a telephone in terview is conducted, the resp onden t is not ask ed so we h a ve n o information ab out whether other individuals are present . Although there are limitations in using the NCVS to estimate rates of b oth rap e and domestic violence , this d ata set is the only on-going large and nationally repr esen tativ e sur v ey to ask individuals directly whether they ha ve b een victims of sp ecific crimes. 2.2. The survey data. Our analysis d ata set includes all wome n 16 y ears of age or older in the NCVS d ata base, with the exception of p ro xy int erviews for th e yea rs from 1993 to 2004. W e use the NCVS data from 1993 to 1997 as pr ior information and condu ct the analysis on d ata fr om 1998 to 2004. It is n ecessary to com b ine inf orm ation from a n um b er of yea r s b ecause rap e and domestic violence are relativ ely rare ev en ts. W e combine the data f or the y ears 1993 to 199 7 and for the y ears 199 8 to 2004 since a descriptiv e analysis sho ws that although the crime r ates increases fr om th e first p erio d to the second p eriod , the crime rates in eac h p erio d are almost constan t. W e recognize that there exists a p oten tial correlation in resp onses f rom the same wo m an o v er time. But this corr elation should not p r esen t a significan t problem in our analysis sin ce: (a) the survey is con trolled so that n o crime is rep eatedly recorded; (b) th e data are collec ted so that p eople ha ve the same c hance of b eing in cluded in the samp le; and (c) an analysis is also p erformed u sing we igh ted data, wh ic h should b etter represen t the p opulatio n of interest. The ra w, unw eigh ted data from 1998 to 2004 b y type of crime, typ e of in terview and who was presen t during p ersonal inte rviews are present ed in T able 1 . W e divide the crimes into four group s: rap e, domestic violence, other assault and p ersonal larcen y . Rap e, attempted rap e and s exu al as- sault are categorized as rap e. The t yp es of crimes in cluded in the domestic violence and other assault categ ories are exactly th e same. If the offender is an in timate, then the assault is categorized as domestic violence. Other- wise, it is categorized as other assault. An inti mate is d efined as a sp ouse, 6 Q. Y U, E. A. ST A SNY AN D B. LI ex-sp ouse, b oyfriend or ex-b o y f riend. Pe rsonal larcen y in cludes p urse sn atc h- ing and p o ck et pic king. W e exclude verbal crimes in our analysis b ecause these kinds of crimes are hard to d efine. T able 1 lists frequ en cies and rates of crimes rep orted by in terview ers. F our categories are used to describ e who w as p r esen t durin g the p ersonal in terview: (1) a sp ouse and no one else (lab eled Sp ouse), (2) a s p ouse and at least one other p erson (Sp ouse and Other), (3) at least one p erson but no sp ouse (Other), and (4) no one else present (Alone). As m en tioned p r e- viously , we do not kn o w who is presen t with the r esp ondin g w oman d u ring telephone in terviews. Note that the ra w data rep ort cr im e rates p er 1,000 w omen inte rview ed as follo ws: 0 . 79 rap es, 1 . 66 incidence of domestic vio- lence, 4 . 14 other assaults and 0 . 60 incidence of p ers onal larcen y . If we consider the rates of v arious crimes in T able 1 b y t yp e of in terview, w e fin d that there are some large differences as shown in Figure 1 . Except for p ersonal larcen y , more crimes we re rep orted in p ersonal in terviews than in telephone interviews. F or example, rap e w as rep orted at a r ate 1.45 times higher in p ersonal in terviews compared to telephone in terviews; domestic violence wa s rep orted at a rate 1.33 times h igher and other assault w as rep orted at a rate 1.11 times higher. Th us, the telephone in terview app ears to hav e a gag effect in the rep orting of crimes. F rom T able 1 , we also note that rates of rep orted crimes in p ersonal in terviews dep end on who wa s present d uring the int erview. F or p ersonal T able 1 F r e quencies and r ates of crimes r ep orte d by settings of the interviews: NCVS 1998–2004 Numbers of in cidents rep orted by type of p ersonal Interviews crimes (rates p e r 1000 interviews) Ty p e of Who Domestic Other P ersonal No crime Interview present Number Rape violence assault larceny rep orted T elephone Unknown 412339 288 516 1539 270 4 09726 (0 . 70) (1 . 25) (3 . 73) (0 . 65) (993 . 66) Sp ouse 24063 4 12 45 10 23992 (0 . 17) (0 . 50) (1 . 87) (0 . 42) (997 . 05) Sp ouse 14322 5 2 29 2 14284 P ersonal and Other (0 . 35) (0 . 14) (2 . 02) (0 . 14) (997 . 35) Other 48916 6 162 350 18 48320 (1 . 35) (3 . 31) (7 . 16) (0 . 37) (987 . 82) Alone 71708 86 256 403 41 70922 (1 . 20) (3 . 57) (5 . 62) (0 . 57) (989 . 04) All p ersonal 159009 161 432 827 71 157518 (1 . 01) (2 . 72) (5 . 20) (0 . 45) (990 . 62) All interviews 571348 449 948 2366 341 5 67244 (0 . 79) (1 . 66) (4 . 14) (0 . 60) (992 . 82) BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 7 in terviews, we compare other situ ations to the case in wh ic h the w oman wa s in terviewed alone, since that is considered the ideal case. Comp ared with a w oman wh o was in terviewed alone, rap e wa s rep orted ab out one-fifth as frequent ly when a sp ouse w as pr esen t (either with or without others). Con- sidering d omestic violence, th e in ciden t was rep orted approxi mately one- ten th as frequen tly if a sp ouse was p resen t. Th e other assault category w as rep orted 1 / 2 . 93 as fr equen tly , and p ersonal larcen y was r ep orted 1 / 1 . 84 as frequent ly if a sp ouse w as presen t d uring the in terview. Pa r t of this rep orting differen tial ma y b e explained b y th e pr otectio n offered by ha ving a sp ouse in the household. All rates of rep orted crimes we re lo wer when a sp ouse w as present , ho wev er, the rep orted r ates for rap es and domestic violence are sig- nifican tly lo wer than those obtained when the w oman was int erview ed alone. W e therefore b eliev e that the sp ouse b eing p resen t du ring an in terview has a differen tial infl uence on the rep orting of rap e and domestic violence. Rap e is under-rep orted b ecause it is a sensitive crime, while domestic violence is under-rep orted not only b ecause it is a s ensitiv e crime b ut also b ecause the offender ma y b e present. Fig. 1. Comp arison of crime r ates r ep orte d by women by typ e of interview and who was pr esent during interview. 8 Q. Y U, E. A. ST A SNY AN D B. LI T able 2 Observe d data (fr e quencies) f or r ep orting r ap e, domestic violenc e and other crimes fr om 1998 to 2004 Pers onal interview T elep hone interview Sp ouse present Spouse n ot present Rap e 9 152 288 Domestic violence 14 418 516 Other assault 74 753 1539 P ersonal larceny 12 59 270 No crime 38276 119242 409726 This exploratory analysis of the raw d ata suggests th at there is resp onse bias in the NCVS related to t yp e of interview (p ersonal versus telephone) and wh o w as pr esen t durin g the in terview. In the later case, the bias is par- ticularly large in the rep orting of the sensitiv e crimes of rap e and domestic violence. Thus, w e consider a mo d el that allo ws u s to adju s t the estimates for those crime rates according to the p ossible r esp onse bias. 3. A mo d el for resp onse b ias adjus tmen t. In our mo del w e classify the data according to th e t y p e of crime, pr esence of the sp ous e (either with or without others) dur ing the in terview an d whether the in terview w as con- ducted by telephone or face-to-fac e. T he crimes are classified into five cat- egories: (1) r ap e and p ossibly some other crime, (2) domestic violence, n ot rap e but p ossibly some other crime, (3) other assault except for rap e and domestic violence, (4) p ersonal larceny except for all kind s of assault, and (5) n o p ers on al crime rep orted. T he unw eigh ted and weigh ted data are sum- marized in T able 2 and T able 3 resp ectiv ely . W eigh ted data are used in the analysis to accoun t for the sample design and so cio-economic indicators. T ables 2 and 3 do not present the “truth.” S ome women prefer not rep ort- ing crimes u nder certa in circumstances. W e wa n t to build a mo del taking T able 3 Weighte d-adjuste d data (fr e quencies) for r ep orting r ap e, domestic vi olenc e and other crimes fr om 1998 to 2004 Pers onal interview T elep hone interview Sp ouse present Spouse n ot present Rap e 8 . 93 177 . 65 338 . 66 Domestic violence 13 . 19 492 . 70 565 . 67 Other assault 77 . 08 867 . 77 1673 . 10 P ersonal larceny 10 . 28 57 . 99 263 . 93 No crime 37234 . 9 1 121675 . 28 407890 . 86 BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 9 in to acco unt the circu m stances of the interview to obtain more accurate estimates of th e rates of rap e and domestic violence. W e susp ect that our estimates still un derestimate th e true r ates since some w omen would nev er rep ort some incident s under an y circums tances. W e use the information a v ailable on factors b elieve d to provi de gag effects to impro ve estimation. In our mo d el we assume a w oman may b e gagged from rep orting a crime for t wo reasons: a sp ouse wa s present , or the interview was conducted ov er the telephone. W e fur ther assume that the sp ouse’s p r esence only influences the rep orting of rap e and domestic violence, wh er eas condu cting the in ter- view o v er the ph on e m a y influence all crimes except for p ers onal larce n y . T o ensu re the mo del is iden tifiable, w e imp ose a hierarch y on the r easons for not rep orting crimes. Namely , w e assume that the presence of a s p ouse dominates the use of a telephone int erview in determining wh ether or n ot a w oman rep orts suc h an in cident. S o if we could on ly observe it, the com- plete data und erlying T able 2 w ould tell us if a crime actually o ccurred and whether it wa s rep orted. If a crim e was n ot rep orted, the un ob s erv ed com- plete data could tell us if the crime wa s not rep orted b ecause of the presence of a sp ou s e, or was not rep orted b eca use the in terview w as conducted o ver the telephone. T he form of the complete (bu t unobs erv ed) data for r ep orting crimes is sho wn in T able 4 . Note that some ou tcomes are imp ossible, f or example, not rep orting b e- cause the sp ou s e is presen t wh en the w oman w as inte rview ed without the sp ouse’s presence. Suc h imp ossible outcomes are denoted by a dash in T a- ble 4 . W e no w present a mo del to analyze the probabilistic relationship b etw een the und erlying complete data and the observ ed data. The follo wing n otatio n is emp lo ye d: π = probabilit y of a telephone interview, 1 − τ = probabilit y of crimes not rep orted b ecause of telephone interview, 1 − ρ = probabilit y of rap e n ot rep orted b ecause sp ous e is presen t, 1 − δ = probabilit y of domestic violence not rep orted b ecause sp ouse is present, ω ij = p robabilit y of crime status i and interview status j , where j = 1 if sp ouse is present, 2 if sp ouse is not pr esen t; i = 1 if r ap e, 2 if domestic violence, 3 if other assault, 4 if p ersonal larceny , 5 if no crime. W e fit a m o del that assumes the indep endence b et w een crimes and sp ouse presence, s o that ω ij = c i · s j , wh ere c i denotes the probabilit y of eac h typ e of crime and s j denotes th e probabilit y of the sp ous e presen t d u ring the inter- view. Un d er the assum ptions describ ed ab o v e, the probabilities u nderlying the u nobserve d complete d ata are sho wn in T able 5 . 10 Q. Y U, E. A. ST A SNY AN D B. LI T able 4 F orm of unobserve d c omplete data Pers onal interview Sp ouse Present Not present Rap e Reported y 1111 y 1112 Not reported—sp ouse present y 1121 — Domestic Rep orted y 1211 y 1212 violence Not rep orted—sp ouse present y 1221 — Other assault Rep orted y 1311 y 1312 P ersonal larceny Rep orted y 1411 y 1412 No crime Rep orted y 1511 y 1512 T elep hone interview Sp ouse Present Not present Rap e Reported y 2111 y 2112 Not reported—sp ouse present y 2121 — Not reported—p hone interview y 2131 y 2132 Domestic Rep orted y 2211 y 2212 violence Not rep orted—sp ouse present y 2221 — Not reported—p hone interview y 2231 y 2232 Other Rep orted y 2311 y 2312 assault Not reported—p hone interview y 2331 y 2332 P ersonal larceny Rep orted y 2411 y 2412 No crime Rep orted y 2511 y 2512 In the observ ed data, s ome of the cells from the complete data are col- lapsed [see, e.g., Ch en and Fien b erg ( 1974 , 1976 )]. Hence, we observ e only sums of several cells rather than all 30 p ossible cells repr esen ted in the complete-data table. T able 6 presen ts the n otation for the observed data ta- ble and ind icates whic h cell counts from the complete data table are s ummed together to create the observ ed data. The p robabilities underlyin g the ob- serv ed data are similarly just the s ums of the probabilities underlying the unobserved complete data and are shown in T able 7 . 4. Ba yesian inference for the b ias-adjusting mo d el. 4.1. Why use a Bayesian mo del. O u r goal is to estimate the π , τ , ρ, δ and ω based on observ ations y in the m o del describ ed in Section 3 . W e adopt a Ba yesia n p ersp ectiv e that treats the parameters τ , ρ and δ as random v ariables to accoun t for the uncertain ties in τ , ρ and δ . W e incorp orate prior BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 11 T able 5 Pr ob abilities underlying unobserve d c omplete data Pers onal interview Sp ouse Present Not pr esent Rap e Rep orted (1 − π ) ρω 11 (1 − π ) ω 21 Not rep orted—sp ouse present (1 − π )(1 − ρ ) ω 11 — Domestic Rep orted (1 − π ) δ ω 12 (1 − π ) ω 22 violence Not reported—sp ouse present (1 − π )(1 − δ ) ω 12 — Other assault Rep orted (1 − π ) ω 13 (1 − π ) ω 23 P ersonal larceny Reported (1 − π ) ω 14 (1 − π ) ω 24 No crime Rep orted (1 − π ) ω 15 (1 − π ) ω 25 T elep hone interview Sp ouse Present Not pr esent Rap e Rep orted π τ ρω 11 π τ ω 21 Not rep orted—sp ouse present π (1 − ρ ) ω 11 — Not rep orted—ph one interview π ρ (1 − τ ) ω 11 π (1 − τ ) ω 21 Domestic Rep orted π τ δ ω 12 π τ ω 22 violence Not reported—sp ouse present π (1 − δ ) ω 12 — Not rep orted—ph one interview π δ (1 − τ ) ω 12 π (1 − τ ) ω 22 Other Rep orted π τ ω 13 π τ ω 23 assault Not rep orted—ph one interview π (1 − τ ) ω 13 π (1 − τ ) ω 23 P ersonal larceny Rep orted π ω 14 π ω 24 P ersonal larceny Rep orted π ω 15 π ω 25 information provided by the NCVS d ata from 1993 to 1997. Notice that we do not estimate π in this wa y since π , the prop ortion of telephone in terviews, is a fixed v alue and cannot b e influ enced by the prior surveys. W e would not use the prior in formation to estimate ω s in ce the crime rates change considerably b et w een the 93–97 an d the 98–04 time p eriods. T o demonstrate the difference, we bu ild t w o mo dels: Mo del A that assum es equal crime rates b et w een time p erio d s, and Mod el B that do es not mak e this assump tion. The lik eliho o d-ratio mo del comparison test sho ws an impr ov emen t of 740 in the G 2 with 4 degrees of f reedom of mo del B o ve r m o del A. The infl uence of the “gag” factors on crime rep orts migh t also change o ver the y ears, bu t th is change is not significan t as is sh o wn in the f ollo wing tests. This migh t result from the facts that features of the sur v ey instr u men t su c h as the question order, w ord ing of the questions and the gatew a y questions are consisten t o v er the y ears. T able 8 sho w s the num b er of rap es rep orted b y w omen when a sp ouse is or is not present d uring the int erview in th e 12 Q. Y U, E. A. ST A SNY AN D B. LI T able 6 F orm of observe d data Pers onal interview Sp ouse present N ot present Rap e rep orted x 111 = y 1111 x 112 = y 1112 Domestic violence rep orted x 121 = y 1211 x 122 = y 1212 Other assault rep orted x 131 = y 1311 x 132 = y 1312 P ersonal larceny rep orted x 141 = y 1411 x 142 = y 1412 No crime rep orted x 151 = y 1121 + y 1221 + y 1511 x 152 = y 1512 T elep hone interview Rap e rep orted x 21 = y 2111 + y 2112 Domestic violence rep orted x 22 = y 2211 + y 2212 Other assault rep orted x 23 = y 2311 + y 2312 P ersonal larceny rep orted x 24 = y 2411 + y 2412 No crime rep orted x 25 = y 2121 + y 2131 + y 2132 + y 2221 + y 2231 + y 2232 + y 2331 + y 2332 + y 2511 + y 2512 p erio d s 1993 to 1997 and 1998 to 2004. W e use the Breslo w–Da y metho d to test the h omogeneit y of the o d ds ratio in the continge ncy tables and T able 7 Pr ob abilities underlying the observe d data Pers onal interview Sp ouse Present Not present Rap e rep orted (1 − π ) ρω 11 (1 − π ) ω 21 Domestic violence rep orted (1 − π ) δ ω 12 (1 − π ) ω 22 Other assault rep orted (1 − π ) ω 13 (1 − π ) ω 23 P ersonal larceny rep orted (1 − π ) ω 14 (1 − π ) ω 24 No crime rep orted (1 − π )(1 − ρ ) ω 11 + (1 − π ) ω 25 (1 − π )(1 − δ ) ω 12 + (1 − π ) ω 15 T elep hone interview Rap e rep orted π τ ρω 11 + π τ ω 21 Domestic violence rep orted π τ δ ω 12 + π τ ω 22 Other assault rep orted π τ ω 13 + π τ ω 23 P ersonal larceny rep orted π τ ω 14 + π τ ω 24 No crime rep orted π (1 − ρ ) ω 11 + π ρ (1 − τ ) ω 11 + π (1 − τ ) ω 21 + π (1 − δ ) ω 12 + π δ (1 − τ ) ω 12 + π (1 − τ ) ω 22 + π (1 − τ ) ω 13 + π (1 − τ ) ω 23 + π ω 14 + π ω 24 BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 13 get a test statistic χ 2 = 0 . 2649 with 1 degree of fr eedom, and a p -v alue of 0.6068 , wh ic h means th at the in fluence of sp ous e-presence on rap e r ep orting is not significantly different for the tw o p erio ds. W e do th e same test on the rep orting of domestic violence and dra w the same conclusion (p -v alue = 0.3049). T o test the gag effect of telephone in terview on crime rep orts, w e cannot use the same metho d, b ecause the effect from “who was p resen t during the interview” is confounded in the d ata. W e use the data from yea r s 1993 to 1997 efficien tly through the u se of empir ical-Ba y esian estimates. The metho d is describ ed in the n ext section and in the r esu lts (Section 5 ), we clearly sho w that our metho d greatly impr ov es estimation without increasing computational load. 4.2. Inc orp or ating prior information. Since the parameters w e w an t to estimate are probabilities, we choose the Diric hlet distribution as the pr ior distribution since it is the n atur al conjugate family of prior distribu tions for the multinomia l distr ibution [Berger ( 1985 )]. W e now u se the data from 1993 to 1997 to form pr ior d istributions. Let X = ( x 1 , x 2 , . . . , x t ) b e m u ltinomial distributed with parameters n and P = ( p 1 , p 2 , . . . , p t ). Let the prior d istri- bution for P b e Diric hlet with parameters ( β 1 , β 2 , . . . , β t ). Then the p oste- rior distribu tion is also Diric h let with parameters B + X = ( β 1 + x 1 , β 2 + x 2 , . . . , β t + x t ). Under the square error loss fu nction, the Bay esian estima- tor of P is the p osterior mean. If w e set k = P t i =1 β i and λ i = β i /k , Λ = ( λ 1 , λ 2 , . . . , λ t ), which is a one-to-one tran s formation from the parameters ( β 1 , β 2 , . . . , β t ), then the estimate for P is E ( P | k , Λ , X ) = n n + k × X n + k n + k × Λ . W e wa nt to find prop er p arameters for the Diric hlet distr ibution to incor- p orate the in formation from the 1993 to 1997 data. W e know that the prior means of the p i are giv en b y E ( p i | k , λ i ) = λ i and w e use estimates for the probabilities from the 1993 to 1997 NCVS data as the λ i ’s. Using the mo del describ ed in S ection 3 and using the NC VS data f r om 1993 to 1997, we ob- tain estimates for τ , ρ, δ and ω easily [see, e.g., Stasny and Coke r ( 1997 )] by T able 8 Comp arison of the effe cts fr om sp ouse pr esenc e on r ap e r ep orts for the ye ars 1993–1997 and 1998–2004 1993–1 997 1998–2 004 Sp ouse present Not present Sp ouse present Not present Y es 9 211 Y es 9 152 No 22502 76513 N o 38376 1204 72 14 Q. Y U, E. A. ST A SNY AN D B. LI implemen tin g the EM algorithm. The parameter k can b e though t of as the prior sample size and it sp ecifies the exten t to wh ic h the estimator dep ends on th e prior information. Here w e prop ose an empirical-Ba yesian estimate of k . A detailed explanation and assessmen t of this m etho d can b e f ound in Bishop, Fien b erg and Holland ( 1975 ) and Carlin et al. ( 2000 ). Denote the Ba yes estimate for P to b e ˆ Q , w h ic h is the E ( P | k , Λ , X ). Then the risk function is R ( ˆ Q, P ) =  n n + k  2 (1 − k P k 2 ) +  k n + k  2 n k P − Λ k 2 , where k P k 2 = p 2 1 + p 2 2 + · · · + p 2 t . Differentia ting the risk f u nction with resp ect to k and setting th e resulting equation equal to 0 yields the estimate of k , ˆ k = (1 − k P k 2 ) / k P − Λ k 2 , that min imizes the r isk R ( ˆ Q, P ). The optimal v alue of k dep end s on the un kno w n v alue of P . If we use the MLE ˆ P = X/n to replace P , then the estimated optimal v alue of k is ˆ k = n 2 − t X i =1 x 2 i ! . t X i =1 x 2 i − 2 n t X i =1 x i λ i + n 2 t X i =1 λ 2 i ! . (4.1) The empirical-Ba yesia n estimat e of P is then P ∗ = n/ ( n + ˆ k )( X/n ) + ˆ k / ( n + ˆ k )Λ . (4.2) F rom equation ( 4.2 ), w e see that ˆ k determines how m uc h the estimator dep end s on the prior inf ormation. In equation ( 4.1 ) we can rewrite the de- nominator as P t i =1 ( x i − n λ i ) 2 . Ther efore, the m ore the pr ior information represent s the cur ren t d ata (i.e., the closer n λ i is to th e obs er ved mean of x i ), the more the estimators d ep end on the prior distribution. 4.3. An EMB algorithm. The EMB algorithm is a v arian t of the EM algorithm [see, e.g., Dempster, Laird and Rubin ( 1977 )]. In the EM algo- rithm, the M-step inv olv es maximizing the complete data lik eliho o d func- tion to obtain the MLE for th e parameters. In EMB algorithm, we add a B-step in whic h the ˆ k in equation ( 4.1 ) is calculated based on the last M-step, and th en empirical-Ba yesia n estimates are ob tained by minim izing the risk fun ction. Th us, at the E-step, we fill in the u nobserved data with estimates b ased on the v alues from the last B-step (those p arameters us- ing Ba yesian estimates) or from th e last M-step (for those using classical estimates). W e illustrate th e us e of the EMB algorithm in the NCVS exam- ple. In the example the empirical-Ba y esian estimates are obtained for τ , ρ and δ . Using th e cell probabilities sho w n in T able 5 and the complete d ata f r om T able 4 , sub ject to the constrain t th at P i P j ω ij = 1, the lik eliho od function of the complete data h as a sim p le multiplicati v e form and can b e sp lit in to BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 15 fiv e factors, eac h a function of only the π , τ , ρ, δ and ω parameters. Th e lik e- liho o d function, w r itten so th at the fun ctions of the fiv e types of parameters are ob vious, is p rop ortional to the follo wing fu nction (a “+ ” in a subscript indicates summation o ver the corresp onding ind ex): π y 2+++ (1 − π ) y 1+++ × ρ y 1111 + y 2111 + y 2131 (1 − ρ ) y 1121 + y 2121 × δ y 1211 + y 2211 + y 2231 (1 − δ ) y 1221 + y 2221 × τ y 2111 + y 2112 + y 2211 + y 2212 + y 2311 + y 2312 × (1 − τ ) y 2131 + y 2132 + y 2231 + y 2232 + y 2331 + y 2332 (4.3) × ω y 1111 + y 1121 + y 2111 + y 2121 + y 2131 11 ω y 1211 + y 1221 + y 2211 + y 2221 + y 2231 12 × ω y 1311 + y 2311 + y 2331 13 ω y 1411 + y 2411 14 ω y 1511 + y 2511 15 ω y 1112 + y 2112 + y 2132 21 × ω y 1212 + y 2212 + y 2232 22 ω y 1312 + y 2312 + y 2332 23 ω y 1412 + y 2412 24 ω y 1512 + y 2512 25 ≡ (1 − π ) y 1+++ π y 2+++ ρ a 1 (1 − ρ ) a 2 × δ b 1 (1 − δ ) b 2 τ c 1 (1 − τ ) c 2 ( 5 Y i =1 2 Y j =1 ω y + i + j ij ) . Because of th e multiplicati v e form of this like lih o o d fu nction, we can accomplish maximization separately for the τ , ρ, δ and ω parameters. The closed form MLE s for these parameters are as follo ws: ˆ π = y 2+++ / ( y 1+++ + y 2+++ ) = (# ph one in terviews) / (# phone + p ersonal interviews), ˆ ρ = a 1 / ( a 1 + a 2 ) , ˆ δ = b 1 / ( b 1 + b 2 ) , ˆ τ = c 1 / ( c 1 + c 2 ) , ˆ ω ij = y + i + j /y ++++ , where a 1 , a 2 , b 1 , b 2 , c 1 and c 2 are as defined in equation ( 4.3 ). Based on the MLEs, w e determine the prior distribution for parameters b y calculating the k ’s through equation ( 4.1 ) th at minimize th e r isk functions. W e then obtain the empirical-Ba ye sian estimates τ ∗ , ρ ∗ and δ ∗ from equation ( 4.2 ). F or example, to estimate ρ , ˆ k ρ = 2 a 1 a 2 / ( a 2 1 + a 2 2 − 2 × ( a 1 + a 2 )( a 1 λ ρ + a 2 (1 − λ ρ )) + ( a 1 + a 2 ) 2 ( λ 2 ρ + (1 − λ ρ ) 2 )) 16 Q. Y U, E. A. ST A SNY AN D B. LI and ρ ∗ = ( a 1 + a 2 ) / ( a 1 + a 2 + ˆ k ρ ) × a 1 / ( a 1 + a 2 ) + ˆ k ρ / ( a 1 + a 2 + ˆ k ρ ) λ ρ , where λ ρ is the estimated v alue of ρ using the 1993 to 1997 NCVS d ata. Note that the B-step do es n ot add significan t computational load to the EM algorithm, sin ce eac h estimate has a closed form. The E-step of th e EMB algorithm is similar to that for th e EM algo- rithm, except that we use different estimates of p arameters to calculate the exp ectations of m iss ing cells. In our example, the E-step consists of obtain- ing th e exp ected cell coun ts for the complete data matrix (T able 5 ), giv en the observ ed data and the current estimates of the π , τ , ρ, δ and ω parame- ters. Th ese exp ectations are particularly simple in the case of discrete d ata [see L ittle and Ru bin ( 2002 )] and amount to prop ortionally allo cating the x ij k ’s of the observ ed d ata as sho wn in T able 6 to the y ij kl cells of T able 4 according to the curr en t parameter estimates. F or example, ˆ y 1121 = x 141 × (1 − π ∗ )(1 − ρ ∗ ) ω ∗ 11 (1 − π ∗ )(1 − ρ ∗ ) ω ∗ 11 + (1 − π ∗ )(1 − δ ∗ ) ω ∗ 12 + (1 − π ∗ ) ω ∗ 14 . Other exp ecte d cell counts can b e f ou n d similarly and, h ence, are not sh o wn here. The E-, M- and B-steps are rep eated until parameter estimates hav e con- v erged to the desired degree of accuracy , whic h in our case is when the sum of the r elativ e differences of all estimated probabilities b etw een t w o itera- tions is less than 0.0001. The co de for imp lemen ting the EMB algorithm on the NCVS d ata to adjust for resp ondent’s bias and estimate the rap e and domestic violence rates in th e y ears from 1998 to 2004 can b e found in the supplemental fi le [Y u, Stasny and Li ( 2008 )]. T o estimate the v ariances of the estimat ors, w e use the jac kknife. More sp ecifically , we lo ok on eac h quarter b et we en the y ears 1998 and 2004 as a sampling unit (S U). This results in 28 sampling units. Using all 28 SUs, w e obtain the b est estimate, sa y , m , for the parameter. By thro wing out the first SU, we use the jac kkn ife data s et of 27 “resampled” SUs to get another estimate, sa y , m 1 . In the next step a new reasmplin g is p erformed with the second SU b eing deleted, and a new estimate m 2 is obtained. The pro cess is rep eated for eac h samp le u nit, resulting in a set of estimates, m i , i = 1 , . . . , 28 . The error for estimation is giv en b y the formula σ 2 j = 27 P 28 i =1 ( m i − m ) 2 28 . T o calculate the m i , we alwa ys use the EMB algorithm and the same prior information, since different observ ations in the futur e w ould not c hange the prior kn o wledge. BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 17 T able 9 Observe d data (fr e quencies) for crimes fr om NCVS 1993 to 1997 Pers onal interview T elep hone Interview Sp ouse present Spouse n ot present Rap e 9 211 314 Domestic violence 12 581 652 Other assault 81 861 1903 P ersonal larceny 16 117 323 No crime 22393 74954 303610 EM estimates for crimes from 1993 to 1997 ˆ ω i j Sp ouse present Spouse n ot present Rap e 0.00057 7 0.00196 4 Domestic violence 0.00137 4 0.00467 6 Other assault 0.00247 5 0.00842 1 P ersonal larceny 0.00025 5 0.00086 8 No crime 0.22244 8 0.75694 2 ˆ π = 0 . 76 ˆ ρ = 0 . 14 ˆ δ = 0 . 07 ˆ τ = 0 . 53 5. Results. 5.1. Complexity of the c omputation. T o chec k the computational com- plexit y and the efficiency of the estimates, we use b oth the frequentist metho d and the Ba yesia n metho d to obtain the estimates. Sin ce eac h Bay esian estimate has a closed form, to get a Ba y esian estimate do es not add muc h computational load at eac h iteration. F rom the jac kknife analysis, the EM algorithm in the frequenti st m etho d iterates 77 . 57 times on a v erage to ob- tain conv erged estimates. F or the Ba yesian metho d, an a v erage of 76 . 43 iterations in the EMB alg orithm is requ ir ed for conv ergence. W e conclude that the Ba y esian metho d do es n ot add significan t computational load as compared w ith the traditional metho d. 5.2. Anal ysis with unweighte d data. W e fir st fit the mo del describ ed in Section 3 to the unw eigh ted data fr om 1993 to 1997 and form the prior distributions. The original d ata from 1993 to 1997 and the estimate s are sho w n in T able 9 . W e then fit the empirical-Ba y esian mo del to the data fr om T able 2 . The estimates obtained from the pro cedur e are sho wn in T able 10 . Notice th at the estimated probabilit y of rap e adjusting for the damp ening effect of the presence of a sp ouse and us in g a telephone inte r view is 0 . 0003 26 + 0 . 001024 = 0 . 0013 50 . Thus, w e estimate ab out 1.35 r ap es p er 1,000 women. This com- pares with a rate of 0.79 p er 1,000 based on the r a w data (T able 1 ). Simi- 18 Q. Y U, E. A. ST A SNY AN D B. LI larly , the estimated probabilit y of domestic violence is 0 . 000702 + 0 . 002205 = 0 . 0029 07 . T his results in an estimate of 2.91 incidences of d omestic violence p er 1,000 w omen, whic h we compare to a rate of 1.66 p er 1,000 based on the r aw d ata (T able 10 ). W e estimate the probabilit y that a crime (except for p ersonal larcen y) is not r ep orted b ecause th e interview is conducted ov er the telephone is 1 − τ ∗ = 0 . 37. T h us, for interviews conducted ov er th e telephone w ith women who are victims of an y t yp e of p ersonal crime (except for p ersonal larceny), w e estimate that approximat ely 37% of the w omen did not rep ort the vic- timizatio n. F or this estimate, k / ( n + k ) = 21 / (21 + 3708 ) = 0 . 005 8 . Th at is, to get the estimate, w e dep end 0.58% on the prior information. Similarly , the probabilit y that a r ap e is not rep orted b ecause a s p ouse is present is ab out 1 − ρ ∗ = 0 . 86, with k / ( n + k ) = 1418 / (1 418 + 193) = 0 . 88. Th at is, for inte rviews with w omen who are victims of rap e and whose s p ouse was present d uring the in terview, w e estimate that 86 % of the women did not rep ort the victimization. T o get this estimate , we dep end 88% on the prior information. The p r obabilit y that d omestic violence is not r ep orted b ecause a sp ouse is p resen t is ab out 1 − δ ∗ = 0 . 92, and k / ( k + n ) = 227 / (227 + 404) is 0.36. That is, for the w omen who are victims of d omestic violence and whose sp ous e w as pr esen t durin g the in terview, w e estimate that 92% of the w omen did not r ep ort the victimization and the estimate dep ends 36% on the p rior information. T able 11 sho ws the estimated p arameters from the EM algorithm with- out using the p rior inf orm ation. Comparing the results in T able 10 and T a- ble 11 , we see the gains f r om the Ba y esian mod el: the v ariances (as sho wn in parentheses) of the estimat es from the Bay esian mod el are significantly lo we r than those from the classical metho d— V ar( ˆ ρ ) / V ar( ρ ∗ ) = 6 . 625 and V ar ( ˆ δ ) / V ar( δ ∗ ) = 1 . 86. Ba yesia n and frequ en tist metho ds estimate ab ou t the same effect of a telephone in terview on rep orting crime, and hav e simi- lar v ariances. This migh t b e due to the f act that the Bay esian estimate of τ dep end s little on pr ior information. On a verag e, the estimation of τ dep end s T able 10 EMB estimates f or crimes fr om 1998 to 2004 ˆ ω i j Sp ouse present Spouse n ot present Rap e 0.00032 6 0.00102 4 Domestic violence 0.000702 0.002 205 Other assault 0.00136 3 0.00428 1 P ersonal larceny 0.00014 4 0.00045 3 No crime 0.23895 4 0.75054 9 π ∗ = 0 . 72 ρ ∗ = 0 . 14(0 . 0 008) δ ∗ = 0 . 08(0 . 0 007) τ ∗ = 0 . 63(0 . 0008) Note . V ariances for the corresp onding estimates are shown in parentheses. BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 19 T able 11 EM estimates for crimes fr om 1998 to 2004 (not using the prior information) ˆ ω i j Sp ouse present Spouse n ot present Rap e 0.00036 7 0.00121 1 Domestic violence 0.000766 0.002 528 Other assault 0.00147 4 0.00486 3 P ersonal larceny 0.00013 5 0.00044 6 No crime 0.22986 0 0.75834 6 ˆ π = 0 . 72 ˆ ρ = 0 . 16(0 . 005 3) ˆ δ = 0 . 09(0 . 0013) ˆ τ = 0 . 61(0 . 00 08) 0 . 60% on p rior information, while the estimation of ρ dep ends 87 . 4 7% and the estimation of δ dep ends 38 . 87% on prior information. In all, w e conclude that to estimate r esp ondent bias, the Ba yesian mo del using pr ior information leads to more efficien t estimates. 5.3. Anal ysis with weight-adjuste d data. W e then ap p ly our mo del to th e w eight- adjusted NC VS data to determine if the u s e of sample-based w eigh ts leads to conclusions differen t from those based on the ra w, unw eigh ted data. W e tak e the usual app roac h of standardizing the w eigh ts so that they sum to the actual sample size. Thus, a w oman’s standard ized weigh t is her original sample w eight divid ed by th e total of the sample-based wei gh ts for all w omen in the analysis. (Th e we igh ts used h ere are the cross-sectional w eigh ts deve l- op ed to m ak e the sample representa tiv e of the p opulation of in terest at the time of the survey . Because th er e are no longitudin al weigh ts a v ailable for th e NCVS, w e use the cross-sectional w eigh ts to reflect the so cio-economic and demographic mak eup of the p opulation while recognizing their limitations.) W e u se the weigh ted data from 1993 to 1997 to f orm the pr ior d istributions. These weig ht-adjusted data and the estimates are summ arized in T able 12 . W e again estimate τ , ρ and δ using the EMB-algorithm d escrib ed in Sec- tion 4 . The parameter estimates for crimes are sh own in T able 13 . T able 14 compares the original crime rates and our estimated crime rates for b oth w eighte d and unw eigh ted data. W e note that the crime rates are generally estimated at a h igher lev el when us ing the w eight ed data. 6. Conclusions and future researc h . W e ha v e sh o wn that estimated rates of rap e and domestic violence among w omen are increased un der a mo del that considers gag factor effects in rep orting suc h crimes based on the type of in terview an d w ho is presen t during the in terview. Also, w e used pr ior information to obtain more efficien t estimates. W e noticed that the t y p e of in terview and wh o is presen t dur ing the in terview may ha ve differen t influ- ences on differen t women. As rep orted by S tasn y and Coke r ( 1997 ), com- pared with wo men not rep orting rap e and domestic violence, those rep orting 20 Q. Y U, E. A. ST A SNY AN D B. LI T able 12 Weight-adjuste d data (fr e quencies) f or crimes f r om NCVS 1993 to 1997 Pers onal interview T elep hone interview Sp ouse present Spouse n ot present Rap e 8 . 96 224 . 36 337 . 35 Domestic violence 11 . 26 608 . 61 695 . 37 Other assault 84 . 99 937 . 59 1991 . 93 P ersonal larceny 15 . 10 121 . 16 323 . 33 No crime 22174 . 7 9 77079 . 1 8 301423 . 0 2 EM estimates for crimes from 1993 to 1997 ˆ ω i j Sp ouse present Spouse n ot present Rap e 0 . 00058 7 0 . 00207 9 Domestic violence 0 . 00138 3 0 . 00489 2 Other assault 0 . 00251 5 0 . 00889 5 P ersonal larceny 0 . 00024 9 0 . 00088 2 No crime 0 . 21568 8 0 . 76282 8 ˆ π = 0 . 75 ˆ ρ = 0 . 14 ˆ δ = 0 . 06 ˆ τ = 0 . 53 w ere y oun ger, had annual in comes of less than $15, 000, w ere unemplo y ed, ren ted rather than o wn their h omes, w ere not cur ren tly married, and had mo ved more than five times in the last three y ears. An imp ortan t area f or future researc h is to acc oun t f or some of these factors in the mo del. The problem with suc h analysis, of course, is that as other v ariables are used in creating cross-classified tables the d ata b eco mes very sparse, particularly in the cells in volving rep orting r ap e or domestic violence. Because rap e and domestic violence are r elativ ely rare eve n ts, we ha ve to com bine information from a num b er of y ears. T hus, we do not obtain enough rep eated m easures for a p anel analysis. The method s describ ed in T able 13 EMB estimates f or crimes fr om 1998 to 2004 (weight-adjuste d data) Estimates for crimes ˆ ω i j Sp ouse present Spouse n ot present Rap e 0.00037 2 0.00122 6 Domestic violence 0.000774 0.002 555 Other assault 0.00148 1 0.00488 7 P ersonal larceny 0.00013 5 0.00044 6 No crime 0.22983 9 0.75828 5 ˆ π = 0 . 72 ρ ∗ = 0 . 14(0 . 0 002) δ ∗ = 0 . 07(0 . 0 005) τ ∗ = 0 . 61(0 . 0025) BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 21 T able 14 Crime r ates (# of crimes p er 1000 p e ople) for ye ars 1998–2004 Fitted rates Original rates Un weighted data W eighted data Rap e 0 . 79 1 . 35 1 . 60 Domestic violence 1 . 66 2 . 91 3 . 33 Other assault 4 . 14 5 . 64 6 . 37 P ersonal larceny 0 . 60 0 . 60 0 . 58 No crime 992 . 82 989 . 50 988 . 1 2 this pap er are mainly us ed on con tingency table analysis. If su fficien t data are collected, we would lik e to implement our mo del in a panel analysis to disco ver how the crime rates c h ange ov er time. In our current analysis, we assume that if there is panel attrition in s u rv ey , the pattern of attrition is rand om. Brame and Piquero ( 2003 ) found that the pattern of panel attrition migh t b e r elated to the interview ee’s c h arac- teristics. In our f uture researc h , we w ant to explore the panel attrition. If it is nonr an d om panel attrition, we w ant to ad j ust our analysis accordingly so that the estimation of crime rates w ould b e more accurate. Our mo del w as d ev elop ed to adjus t for resp onse b iases caused by the mo de of int erviewing and the sensitiv e nature of questions in rep orting rap e and domestic violence. Th e mo dels describ ed in our pap er ma y b e useful in other s urve y s ampling settings where some kn o wn factors may result in resp onse bias. F or example, one can imagine that rep orting v arious sources of income could b e b iased b ecause of who is p resen t durin g the in terview. Moreo v er, if d ata are collected rep eatedly , the Ba y esian metho d w ould b e efficien t b ecause it incorp orate s the previously collected information into an imp ortant estimate. The EMB algorithm helps to mak e the compu tation easy . T o imp lemen t our metho d, first identify which part of the information w ould not c hange o v er time and then build th at part of the inform ation in to the p rior distrib utions. F ollo win g the EMB algorithm, we could easily obtain the estimates of p arameters that are of in terest. T he metho ds prop osed here, therefore, can b e easily applied in such cases. Ac kn o wledgment s. Th e authors th an k Donald Mercan te, the editor, as- so ciate ed itor and r eferees for constructiv e comments and suggestions that help ed to improv e the presenta tion of the pap er. SUPPLEMENT AR Y MA TER I AL R-cod e of EMB algorithm to adju st for resp onse b ias in NCVS data for es- timating rap e and d omestic violence rates (doi: 10.121 4/08-A OAS160 S UPP ; .txt). 22 Q. Y U, E. A. ST A SNY AN D B. LI REFERENCES Berger, J. O. (1985). Statistic al De cisi on The ory and Bayesian Analysis . Springer, New Y ork. MR0804611 Bishop, Y. M. M., Fi enberg, S. E. and Holland, P. W . (1975). Discr ete Multivariate Ana lysis : The ory and Pr actic e . MIT Press, Cambridge, MA. MR0381130 Brame, R. and Piquero, A. R. (2003). Selective attrition and the age-crime relationship. J. Quantitative Crimi nolo gy 19 107–12 7. Bureau of Justice St a tistics (1995). National Crime Victimi zation Survey R e design— F act She et . U .S. Dep artment of Justice, W ashington, DC. Bureau of Justice S t a ti stics (2001). National Crim e Victimization Survey , 1992- 1995[c omputer file] . U.S. Department of Commerce, Bureau of the Census. 9th ICPSR ed. 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T echnical Rep ort #592, Department of S t atistics, Oh io State Un iv. Q. Yu School of Pub lic Heal th Lousiana St a te Univ ersity 1615 Poydras Street Suite 140 0 New Orleans, Louisiana 70112 USA E-mail: qyu@lsuhsc.edu URL: http ://public health.lsuhsc.edu/F aculty pages/qyu/index.h tml E. A. S t asny Dep ar tment of St a tistics Ohio St a te University 404 Cockins Hall 1958 Neil A venue Columbus, Ohio 43210 USA E-mail: eas@stat.osu.edu BA YESI AN MODELS TO ADJUS T FOR RESPON SE BIAS IN S UR VEY DA T A 23 B. Li Dep ar tment of Experim ent al St a tistics Louisiana S t a te University Ba ton Rouge, Louisiana 70803 USA E-mail: bli@lsu.edu URL: http ://www.stat.lsu.edu/facult y/li /