MODIFIED HALF SAMPLE VARIANCE ESTIMATION FOR MEDIAN SALES PRICES OF SOLD HOUSES: EFFECTS OF DATA GROUPING METHODS

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1 MODIFIED HALF SAMPLE VARIANCE ESTIMATION FOR MEDIAN SALES PRICES OF SOLD HOUSES: EFFECTS OF DATA GROUPING METHODS Katherne J. Thompson and Rchard S. Sgman Katherne J. Thompson, ESMPD, Room , U.S. Census Bureau, Washngton DC, Key Words: Varance of a Medan" Modfed BRR; Survey of Constructon I. Introducton The U.S. Census Bureau publshes estmates of medans for several characterstcs of new houses, wth a key estmate beng sales prce of sold houses. These estmates are calculated from data acqured from ntervews of home bulders by the Survey of Constructon (SOC). In the near future, the Survey of Constructon (SOC) wll move ts current varance estmaton system to the Census Bureau's re-engneered post-data-collecton processng system, the Standardzed Economc Processng System (STEPS). For sample desgns that do not use Posson samplng, the STEPS system uses replcaton methods to estmate standard errors. The SOC s a mult-stage probablty survey whose sample desgn s well suted to the modfed half sample (MHS) replcaton method ~ for reasons outlned n secton III.B. The lterature supports the use of Balanced Half-Sample Replcaton (e.g. Rao, Wu, and Yue (1992); Rao and Shao (1996); Kovacevc and Yung (1997)) and MHS replcaton (Judkns (1990)) for estmatng varances of medans from complex survey data. We consdered two methods of medan estmaton for varance estmaton purposes. The frst method uses the replcate weghts to estmate medans va replcated emprcal cumulatve-dstrbuton functons (.e., calculate the medan of each half-sample). The second method uses lnear nterpolaton of grouped contnuous data to approxmate the medan of each half-sample. The latter method s mplemented n VPLX (Varances from ComPLeX Survey, Fay (1995)), a varance estmaton software package developed at the Census Bureau. Drect calculaton of sample medans can be computatonally ntensve because t requres separate sorts for each value of a gven classfcaton varable. An alternatve estmaton method s to group the contnuous data nto dscrete ntervals (called bns) and use lnear nterpolaton over the nterval contanng the medan. Provded that the data are approxmately unformly dstrbuted over the nterval contanng the medan, nterpolaton yelds a good approxmaton. However, optmal bn wdths and locatons may change over tme, as the sample dstrbutons change. These consderatons motvated our research. In ths paper, we compare sx methods of medanestmaton for MHS replcaton: the sample medan and fve varatons usng lnear nterpolaton. Secton II provdes a bref overvew of the SOC desgn. Secton III presents general methodology. Secton IV descrbes the emprcal results from four months of SOC data that motvated the smulaton study presented n Secton V. Secton VI provdes our conclusons and recommendatons. II. SOC Sample Desgn The SOC unverse contans two sub-populatons: local areas that requre buldng permts and local areas that do not. The SOC sample unts selected from the frst sub- ~Balanced repeated replcaton wth replcate weghts of 1.5 and 0.5. populaton comprse the Survey of the Use of Permts (SUP), and those selected from the second sub-populaton, the Nonpermt Survey (NP). The SUP sample comprses the majorty of the SOC estmate. The two samples are multstage probablty samples stratfed by varables wth hgh expected correlaton wth the survey's key statstcs: housng starts, completons, and sales. The frst stage of the SUP and NP sample selecton s a subsample of Current Populaton Survey (CPS) Prmary Samplng Unts (PSUs), whch are contguous areas of land wth well-defned boundares. Thus, both surveys are conducted n the same PSUs but are otherwse ndependent samples. One PSU per stratum was selected. Selfrepresentng (SR) PSUs were ncluded n the sample wth certanty. Nonself-representng (NSR) PSUs were selected wth probablty proportonal to sze (PPS) from strata contanng more than one PSU. The second stage of SUP sample selecton s a stratfed systematc sample of permt-ssung places wthn sample PSUs (selected once a decade). In many cases, only one second stage unt was selected. The thrd stage of SUP sample selecton s performed monthly: each month, Feld Representatves (FRs) select a systematc sample of buldng permts from the permt offces n each sampled permtssung place. The thrd-stage samples are ndependent by month; the frst and second stages are not. The second stage of NP sample selecton s a stratfed systematc sample of small land areas (1980 Census Enumeraton Dstrcts, or EDs). For the thrd stage of NP sample selecton, feld representatves completely canvass all of the roads n the sampled EDs (called segments). All new housng unts are ncluded n the NP sample wth certanty. Medan estmates are derved from the pooled SUP and NP samples and are calculated usng a post-stratfed weght for the SUP porton and an unbased weght for the NP porton. III. Methodology A. Medan-Estmaton Procedures 1. Sample Medan One procedure for estmatng the medan of a populaton s calculate the sample medan from ungrouped data, usng the sample weght to locate the medan as recommended n Kovar, Rao, and Wu (1988) and Rao and Shao (1996). 2. Lnear Interpolaton Another approach for estmatng the medan of a populaton s to group the sample data and nterpolate for the sample medan. Woodruff (1952) provdes the followng formula for lnear nterpolaton of a sample medan: 1 ^ --N - cf ffi= F-'(1)V) = ll + ( 2 f, ),() (2.1) where F = the cumulatve frequency of the characterstc usng sample weghts 11 = lower lmt of the bn contanng the medan 698

2 ~q - estmated total number of elements n the populaton cf = cumulatve frequency n all ntervals precedng the bn contanng the medan f~ = estmated total number of elements n the populaton of the nterval contanng the medan - wdth of the bn contanng the medan Ths s the method used by the current SOC producton varance estmaton system for monthly estmates and s also the lnear nterpolaton method employed by VPLX. We consdered two optons for settng the class sze (bn wdths) for the nterpolaton. The frst opton develops bns based on the specfc characterstc under consderaton usng the orgnal data. The second opton lnearly transforms the data to a standard scale and then uses a standard set of bns for every characterstc. We used the followng lnear transformaton: X / -- Xorgnal * (1,000/Q3) (2.2) where Q3 s the thrd quartle of the sample dstrbuton (estmated usng the sample weght). The nterpolated medan of the X / s multpled by (Q3/1000) to obtan an estmated medan of Xorgnal. Usng the orgnal data to develop medans has the advantage of producng producton ready estmates and SEs. Determnng the approprate bn wdth s dffcult, however. As the bn wdths get small, the varance estmates become more unstable. As the bn wdths ncrease, the bas of the estmate due to nterpolaton ncreases. The "optmal" bn sze balances estmate bas and varance-estmate stablty. Unfortunately, the optmal bn wdth may not reman constant between samples. Often, the dstrbutons change over tme, and the bns wdths/locatons n the sample should reflect ths change n scale. Moreover, the optmal bn wdth may be dfferent for dfferent values of a classfcaton varable: for example, the optmal bn wdth for the Mdwest's sales prce s probably dfferent from the optmal bn wdth for the South's sales prce. The desre to have the wdth of the bn depend on the sample motvated the lnear transformaton. Our procedure of lnearly transformng the data and then usng standard bn wdths s equvalent to smply dvdng the orgnal sample from 0 to Q3 nto x bns of equal wdth and placng the remander of the data nto one bn, whch, by desgn, s much larger than the others (contanng up to 25% of the sample). Indeed, the "standard" bn wdths used on the transformed data are not standard on the untransformed scale: they are data dependent. As the dstrbuton changes, the bn wdths on the untransformed (orgnal) scale also change. Usng the lnearly transformed data requres more bookkeepng n terms of scalng constants but easly allows for changes n the scale and shape of the dstrbuton. The procedure descrbed above was desgned for nonnegatve data. If the dstrbuton contans negatve values (e.g., a dstrbuton of net ncome), then a modfcaton of the lnear transformaton descrbed n (2.2) s requred. To make all of the observatons n the sample non-negatve, replace Xorgnal wth X//= (Xorgnal -X(1)), where X(l) s the smallest observaton n the sample. Calculate Q3 from the dstrbuton of X//(usng the sample weght assocated wth Xong~n~), and apply (2.2) to the X//. To evaluate the frst opton, we used two dfferent sets of bn wdths (classfcaton szes): bns of sze $2000 (the same bn wdth used n the current producton varance estmaton system) and bns of sze $1000. [Note: The VPLX varance estmaton software would not allow any bn sze smaller than 1000 because the number of classes exceeded the allowable array range.] Based on our data analyss, we assumed that medan sales prce would always be larger than $36,000 and smaller than $550,000, so the frst orgnal-data classfcaton s always (low - 35,999) and the last orgnal-data classfcaton s always (550,000 - hgh): ths yelds 257 bns of sze $2000 or 514 bns of sze $1000, plus one bn of sze $36,000 and one bn whose wdth depends on the largest observaton n the sample. To evaluate the second opton, we used three dfferent sets of bn wdths: bns of sze 4, 25, and 50. The bns of sze 4 were chosen to be analogous to the bns of sze 2000 n terms of the number of bns: 251 bns total. The selecton of wdths 25 and 50 was somewhat arbtrary: we chose bn sze 50 to get a total of twenty bns for the data less than Q3; and we chose bn sze 25 to examne the effect of doublng the number of bns/halvng the wdth of the bns for data less than Q3. The transformed-data medan wll always be less than 1,000, so the last transformed-data classfcaton s always (1,000 - hgh). Ths procedure s desgned for symmetrc or postvely skewed dstrbutons. The data n the last bn s not used to estmate the medan because t s greater than Q3., whch s expected to be far from the medan. We guarantee that the frst and last bns are not mmedately below or above the bn contanng the medan by the standard bns szes: 6.7 bns per quartle for bn sze 50; 13.3 bns per quartle for bn sze 25, and 83.3 bns per quartle for bn sze 4. Consequently, there s no loss n precson n makng the last bn so much larger than the others. B. Varance Estmaton We used the Modfed Half Sample replcaton method (Fay, 1989 and Judkns, 1990) to estmate the varance of a medan. Modfed half-sample replcaton s a varaton of the "tradtonal" balanced half-sample (BRR) varance estmaton descrbed n Wolter (1985, Chapter 3), usng same replcate assgnment methodology as BRR (a Hadamard matrx) wth replcate weghts of 1.5 and 0.5 n place of the 2 and 0. The SE for a medan estmate usng MHS replcaton s gven by SE(Med) = ~ --~ 4,~,,:-', (Med r _ A/edo) 2 (2.3) where the r subscrpt refers to the replcate medan estmate (r = 1, 2... R) and the 0 subscrpt refers to the full sample medan estmate. Ths expresson contans a four (4) n the numerator because the MSE of the replcate estmates s too small by a factor of 1/(1-0.5) 2. See Judkns (1990). As stated n Secton II, nether the SUP nor the NP desgns are two-sample-unt-per-stratum desgns. To address the one sample unt per stratum problem, we "splt" the SR sample-unts nto two panels per sample unt usng the orgnal samplng methodology and form collapsed strata by parng two (or three) "smlar" NSR sample-unts. We then apply the half-sample approach n such a way that the elements contrbutng to the half samples are panels wthn sample unts for SR sample unts and are sample unts wthn collapsed strata for NSR sample unts. The current SOC producton varance system uses a Keyftz estmator (a pared dfference estmator) for NSR sample and a desgn-based estmator for SR sample to produce level estmate varances (Luery, 1990). Because SOC methodologsts had already collapsed NSR strata for ther pared dfference estmator, a B RR-lke applcaton was a logcal extenson of the pre-exstng varance estmaton 699

3 structure. For the SR cases, we sort permts wthn predetermned sample-unt groups by geography and permt authorzaton date and systematcally splt the ordered sample nto two panels as suggested n Wolter (1985, p. 131). Ths method of assgnng unts to panels s referred to as the grouped balanced half sample (GBHS) method n Rao and Shao (1996) and s dscussed further n Secton V. For more detals on the replcate assgnments, see Thompson (1998). The SOC producton system uses the Woodruff method (Woodruff, 1952) to estmate the SE of a medan. Ths s not a replcate varance estmaton method. Ths methodology has had mxed success n the past accordng to survey analysts. IV. Emprcal Data Results Intally, we used four months of SOC sample data to examne the varances of the medan-estmaton methods for sales prce of sold houses: March 1997, May 1997, June 1997, and July We produced medans by regon and by type of fnancng. We used the same weght used by the SOC producton estmaton and varance systems (poststratfed for SUP sample and unbased for NP sample), poolng both surveys' data to obtan medans. Each set of varance estmates was produced usng 200 replcates. We found that the sx medan-estmaton methods produced three dstnct sets of SEs" one set for the sample medan, one set for the orgnal-data-nterpolated medans, and one set for the transformed-data-nterpolated medans. There was no clear relatonshp between bn wdth and SE estmates for the two sets of nterpolated medans. Indeed, wthn type of data (orgnal or transformed), the SEs were all very close. Clearly, there was a lnear transformaton and an nterpolaton effect. None of the medan-estmaton methods yelded SEs resemblng the publshed SEs, so there was no avalable argument for publcaton consstency. The emprcal results left us n a quandary. We had three dstnct sets of varance estmates, and no "gold standard" aganst whch to measure them. Because our emprcal results were nconclusve, we conducted a Monte Carlo smulaton study to evaluate the propertes of the MHS varance estmates produced from the dfferent medan estmators. V. Smulaton Study Comparson A. Procedure for Smulaton Study We created four fnte artfcal populatons based on a data analyss of four SOC sample populatons' one type-offnancng populaton (Conventonal Fnancng) and three regonal populatons (Mdwest (Regon 2), South (Regon 3), and West (Regon 4)). These populatons represented a varety of the types of SOC populatons from whch estmates are produced. Note that the SOC type-of-fnancng populaton s no.._!t ndependent of the SOC-regon populatons. To approxmate the fnte populaton of sales prce for houses sold, we generated w~ records for each sample unt, where w~ s the sample weght assocated wth unt. The dstrbutons of sales prce for sngle-unt sold houses could be approxmated by lognormal dstrbutons. The lognormal dstrbuton has the probablty densty functon fly) = 1 L.exp( 1 (log(y - 0)- ~) y - 0 ~/27to -2( o )2) for 0 < y < oo where 0 s the threshold parameter, ~ s the scale parameter, and o s the shape parameter. After performng ths data analyss, we generated four artfcal fnte populatons of bvarate random normal varables wth expected correlaton p=0.6 usng the method outlned n Naylor et al (1968). One of the two varables represented sales prce for houses sold and s generated usng the parameters determned above. Ths varable was exponentated and shfted by the approprate locaton parameters to obtan the sales prce varable. The second varable was dstrbuted as a standard normal and s used to form strata. Each populaton's sze was the estmated populaton total n the gven category rounded to the nearest 50. The sample sze s the orgnal sample sze rounded to the nearest 50. Model parameters and sample correlatons (between smulated sales prce and stratfyng varable) are reported n Table 1. We compared the percentles, sample skewness, and sample kurtoss of each smulated populaton to ts correspondng orgnal populaton, and they were qute close. To examne the effect of outlers n the orgnal populaton on the model, we removed outlers usng the resstant outer fences rule descrbed n Hoagln and Iglewcz (1987) and found that ths mproved agreement between the two populatons for the 90%, 95%, and 99% percentles. Table 1" Populaton Parameters and Sample Szes Populaton 0 o q p N Con. Fnancng Mdwest South West After generatng the fnte populatons, we sorted them by the stratfyng varable and formed 50 equal szed strata n each populaton. From these strata, we selected 5000 stratfed wthout-replacement random samples from each artfcal populaton usng the same samplng rate n each stratum (self-weghtng desgn). To perform the MHS replcaton, we sorted the sample wthn each stratum by stratfyng varable and then systematcally splt the sample nto two panels. Thus, the smulaton study captures some of the stratfcaton propertes of the SOC desgn and mmcs the panel assgnment for SR permt sample but does not take the multstage sample and PPS samplng nto account. We determned the medan of each fnte populaton (~p). Usng the 5000 samples, we estmated emprcal Mean Square Errors (MSE) and Mean Absolute Errors (MAE) for the followng sx medan-estmaton procedures: SM: the sample medan of each half-sample : nterpolated medans usng orgnal data, bns of sze 2000 (fxed bn wdth) IO1000: nterpolated medans usng orgnal data, bns of sze 1000 (fxed bn wdth) IT4: nterpolated medans usng lnearly transformed data, bns of sze 4 (data dependent bn wdth) IT25: nterpolated medans usng lnearly transformed IT50: data, bns of sze 25 (data dependent bn wdth) nterpolated medans usng lnearly transformed data, bns of sze 50 (data dependent bn wdth) The lnear transformaton was performed once for procedures IT4, IT25, and IT50. The orgnal data were transformed usng the full sample Q3, and these transformed data were assgned to the half-samples. Table 2 provdes the medan and thrd quartle of each fnte populaton, along wth the bn wdths on the orgnal scale for the transformed data. 700

4 Table 2: Medan, Thrd Quartle, and Bn Wdths on Orgnal Scale for Transformed Smulated Data Populaton Medan Q3 Bn Wdth Con. Fnancng Mdwest (Regon 2) South (Regon 3) West (Regon 4) To measure the precson of the sx medan-estmaton procedures over repeated samples, we calculated emprcal MSEs and Mean Absolute Errors (MAEs) for each procedure n each populaton. M(~), the emprcal MSE of medanestmaton procedure, was calculated r as M(~,) = 5ooo + (~' - ~p)2, where ~r s the estmated medan for sample r and estmator, ~ s the average of the ~r~, and ~p s the populaton medan. Ths s the emprcal MSE descrbed n Judkns (1990).The Mean Absolute Error (MAE) of each medan-estmaton procedure was calculated as MAE(~) = [~-~[~-~r - ~p[]/5000 as defned n DeGroot (1986). To compare the varance estmaton propertes of the dfferent medan-estmaton procedures, we calculated an MHS varance estmate (v~j) correspondng to each medanestmaton procedure from 1000 of the 5000 samples. These varance estmates were compared n terms of relatve bas [(~vj/1000)/m(~) - 1]; relatve stablty [[(2Vj- M(~))2/1000]'~/M(~)]; and error rate [(the number of samples where ~p< 0L or ~ > 0u)/1000 where 0L s the lower end of a 90% confdence nterval, and 0u~ s the upper end of a 90% confdence nterval]. These crteron are used n Kovar, Rao, and Wu (1988) and n Rao and Shao (1996). Wth an "optmal" varance estmator, both the relatve bas and relatve stablty wll be near zero, and the error rate wll be ten percent. B. Results Table 3 presents the emprcal root MSE, SE, the bas, and the MAE for each medan-estmaton procedure. Each of these statstcs was calculated from 5000 ndependent samples. The results from Table 3 can be summarzed as follows: The transformed-data-nterpolated medans wth bns of wdth 50 have the smallest root-mse n three of the four populatons (all but Regon 3), wth the transformed-datanterpolated medans wth bns of wdth 25 a close second. However, the root-mses of all sx procedures are very close n each populaton, so there s no dramatc loss n overall precson wth the choce of any partcular estmator. Smlarly, the transformed-data-nterpolated medans wth bns of wdth 50 have the smallest SE n each populaton, wth the transformed-data-nterpolated medans wth bns of wdth 25 a close second. Agan, the dfferences n SE are very close between all sx procedures (wthn approxmately 3% of each other n all populatons). The bas of the estmaton procedures does not have much nfluence on overall error. In all populatons, the bas as a percentage of the MSE s very small. The sx sets of MAEs n each populaton are very close, renforcng the concluson above regardng the equally-good performance of the dfferent medan-estmaton methods. Table 3" Precson of Medan-Estmaton Procedures Populaton Medan- Root SE Bas MAE Estmaton MSE Procedure Conventonal SM Fnancng O m :, )IO1000 ~ n m ~IT4 m 1, n n IT25 [ n IT m m m m Regon 2 SM m m m Mdwest IO IT n n IT IT Regon 3 SM South IO IT ~ IT l l 1 IT I I I Regon 4 SM West IO IT IT IT Table 4 summarzes the three dfferent comparson measures for the varance estmates n the four populatons. The numerators for the relatve bas and stablty and the coverage rates are based on 1000 samples. The denomnator for the relatve bas and stablty ("truth") are based on 5000 samples. An astersk (*) n the last column of Table 4 ndcates that the error rate s sgnfcantly dfferent from the nomnal error rate of 0.10 usng the normal approxmaton to the bnomal dstrbuton at the 90% confdence level. The varance estmates of the transformed-datanterpolated medans perform best n terms of relatve bas, stablty, and coverage (error rates). Specfcally, The varance estmates of the transformed-datanterpolated medans (IT4, IT25, IT50) have the smallest relatve bas. The dfference n estmaton method s qute pronounced n three of the four populatons, where the largest relatve bas of the transformed-data-nterpolated medans s less than one-half the sze of the smallest relatve bas of the orgnal-data-nterpolated and sample medans. In all four populatons, usng bns of wdth 50 on the transformed data yelded the smallest relatve bas; The varance estmates of the nterpolated medans had the best stablty. The sample medan had the poorest stablty n all four populatons. Ths result was expected due to the smoothng effect of nterpolaton. The transformed-data-nterpolated medans generally performed slghtly better than the orgnal-data-nterpolated medans; The confdence ntervals constructed from transformeddata-nterpolated medans and SEs have the best coverage: n each populaton, the data dependent bns (all wdths) yeld statstcally nomnal coverage [Note: there s no clear relatonshp between sze of bn wdth on the transformed scale and mproved/reduced error rates]. The coverage for the confdence ntervals constructed from orgnal-data- 701

5 nterpolated medans and SEs s very poor, yeldng very conservatve ntervals, and the coverage wth the sample medan s erratc. Table 4: Relatve Bas and Stablty for Varance Estmates and Error Rates and Coverage Error Rates Populaton Medan- Relatve Relatve Error Estmaton Bas Stablty Rate I Procedure Conventonal SM ! 11.0% Fnancng I %* IO %* IT %!IT % IT % Regon 2 I SM %* Mdwest 1O %* I %* IT : 10.1% IT ! 9.8% IT % Regon 3 SM %* South %* IO %* IT % IT % IT % Regon 4 SM % ~West IO %* IO %* IT % IT % IT % These tests of error rates have good power, as verfed through a smple power analyss. Let PA = bnomal error rate probablty under the alternatve hypotheses (PA ~ 0.10). Usng the normal approxmaton to the bnomal, for PA > 0.10, we have 90% confdence and x-percent power when the upper lmt of a 90% confdence nterval equals the x-percent lower lmt (one sded) under the alternatve hypothess. For PA < 0.10, we have 90% confdence andxpercent power when the lower lmt of a 90% confdence nterval equals the x-percent upper lmt under the alternatve hypothess. Solvng for P A, we fnd that we have 90% confdence and at least 70% power when PA -< or PA > (when IPA - Pol -> ). The power ncreases to 80% when PA -< or PA>_ To determne whether the dfferences n error rates between estmators was sgnfcant, we performed a one-way ANOVA n each populaton modellng each medan estmator as a treatment effect usng the varance stablzng arcsn-square root transformaton on the error rates. Because the error sums of squares for the transformed bnomal random varables s 821/n (Snedecor and Cochran, 1980), we tested for overall ft usng a ch-square(5) crtcal value. All tests are hghly sgnfcant: p-values of for Conventonal Fnancng; for Regon 2; for Regon 3; and for Regon 4. Thus, we can conclude that the sx treatments yeld dfferent results. Moreover, n all four populatons, all parwse dfferences between error rates greater than 0.10% are sgnfcant at the 95% jont confdence level (based on Scheff6 95% jont confdence ntervals for all parwse contrasts, usng the 95% confdence level due to the conservatve nature of the procedure). Absolute dfferences between two error rates s greater than or equal to are sgnfcant. Consequently, error rate comparsons between medan-estmaton-method varances are statstcally meanngful. C. Valdaton of Smulaton Results Usng Randomly Grouped Balanced Half Sample Replcaton Inferences from ths smulaton study are as vald as the varance estmates used. Rao and Shao (1996) establsh the asymptotc nconsstency of the grouped balanced half sample (GBHS) estmator for estmatng the SE of quantles from samples wth a fxed number of strata as the strata sample szes nh--'oo. Instead, they recommend a repeatedly grouped balanced half sample (RGBHS) estmator,.e. repeatng the random panel assgnment T tmes and usng the average of the T GBHS estmators. Because we used the MHS varance estmator n all our applcatons, so GMHS refers to GBHS wth replcate weghts of 1.5 and 0.5, and RGMHS refers to RGBHS wth replcate weghts of 1.5 and 0.5. We performed a small smulaton study (300 samples per populaton) comparng GMHS and RGMHS (T = 15) varance estmaton for the sx medan-estmaton procedures. Because the RGMHS estmator requres a great deal of computer overhead (4,500 runs per procedure for T-15 ), we restrcted our comparsons to two of the four sample populatons (the largest and smallest). Table 5 presents the relatve bas, stablty, and error rates for 90% confdence ntervals calculated from the frst 300 samples for each medan-estmaton procedure for the GMHS and RGMHS n the Conventonal Fnancng and n the Regon 2 (Mdwest) populatons. An astersk ndcates that an error rate s sgnfcantly dfferent from the nomnal error rate of 10%. The results n Table 5 can be summarzed as follows: The relatve bases are generally the same usng GMHS and RGMHS for each treatment, although the RGMHS varance estmate does reduce the relatve bas for the SM procedure by twenty-fve percent n the Regon 2 populaton; As expected, the RGMHS procedure yelds more stable varance estmates. In the Conventonal Fnancng populaton, the reducton s as great as thrty-fve percent for three of the sx medan-estmaton procedures. However, the mprovements n stablty for all medan-estmaton procedures are less pronounced n the Regon 2 populaton, and nether the RGMHS and GHMS varances have good stablty; In both populatons, the error rates for the SM confdence ntervals constructed from the GMHS and RGMHS SEs are the same and are ndeed nomnal. Error rates constructed from the GMHS and RGMHS SEs for other treatments are close, and for most treatments these error rates are not sgnfcantly dfferent from 10%. The error rates for RGMHS orgnal-data-nterpolated medans ( and IO1000) n the Conventonal Fnancng populaton are sgnfcantly less than 10%, provdng more evdence that the orgnal-data-nterpolaton procedures are too conservatve (although ths pattern s not seen n the Regon 2 populaton). In Regon 2, the error rates for both the GMHS and RGMHS transformed-data-nterpolated medans wth bns of wdth 4 and the error rates for the GMHS transformed-data-nterpolated medans wth bns of wdth 25 are sgnfcantly hgher than nomnal. We beleve that the conflctng results between Tables 4 and 5 for confdence nterval coverage for the dfferent medan-estmaton 702

6 procedures n Regon 2 s caused by the nherent nstablty n the varance estmates due to small sample sze n that populaton, snce the Table 5 error rates wthn medanestmaton procedure are very smlar for the GMHS and RGMHS SEs. In terms of relatve bas and error rates, the results n Table 5 are farly consstent for the two varance estmates for each medan-estmaton procedure. The RGMHS estmator does mprove the stablty, but mproved stablty does not appear to be reflected n confdence nterval coverage (at least for these samples). The consstency between the GMHS and RGMHS results renforces our earler conclusons vs-h-vs the dfferent estmaton procedures. Moreover, t supports the varance estmaton methodology used n the larger smulaton study and n SOC producton" comparable results are acheved wth 1/15 the replcate estmates. Table 5" Relatve Bas, Stablty, and Error Rates Usng GMHS and RGMHS Varance Estmaton?opulaton dedan- Relatve Estmaton Bas Procedure Son. gm ~'nancng [02000 [O1000 [T4 [T25 [T50 ~egon 2 ~M Mdwest) [02000 [O1000 IT4 [T25 IT50 GMHS RGMHS Stablty GMHS RGMHS Error Rate GMHS RGMHS * * " 14.0" 13.0' VI. Concluson We explored the effect of usng varatons of two dfferent methods of estmatng the medan of contnuous data on MHS varance estmaton" drect estmaton versus lnear nterpolaton. Lnear nterpolaton requres classfyng contnuous data nto bns of standard wdth. Ths wdth can be arbtrary, and "optmal" wdths may change as the sample dstrbuton changes over tme. The lnear transformaton based on the thrd quartle appeared to correct ths problem. Wth the transformed data, the bns' locatons change dependng on the data. Our emprcal results ndcated that the choce of method has a pronounced mpact on the varance estmates gven modfed half sample replcaton. Our smulaton study results examned the propertes of the dfferent medan-estmaton procedures on the varance estmates, usng the grouped MHS varance estmator. In all four smulated populatons, the transformed-data-nterpolated medans performed the best, usually by a wde margn. Snce all three bns wdths consdered wth transformed data appeared to have the same varance estmaton propertes, we recommend usng the fewest number of bns examned,.e. use twenty-one bns (bns of sze 50 on the transformed scale). The recommended method has several advantages. Frst, t takes the scale of the dfferent dstrbutons nto account through the lnear rescalng. Second, the larger bn sze should amelorate some of the samplng effects. Fnally, usng lnear nterpolaton saves computng resources by avodng sortng each half-sample. Acknowledgments The authors would lke to thank Elzabeth Huang and James Fagan of the U.S. Census Bureau for ther helpful comments on earler versons of ths manuscrpt, and J.N.K Rao for hs useful comments on the orgnal smulaton study. Ths paper reports the results of research and analyss undertaken by Census Bureau staff. It has undergone a more lmted revew than offcal Census Bureau publcatons. Ths report s released to nform nterested partes of research and to nform dscusson. References DeGroot, Morrs (1986). Probablty and Statstcs. Readng, MA: Addson-Wesley Publshng, Inc. Fay, Robert E. (1989). Theory and Applcaton of Replcate Weghtng for Varance Calculatons. Proceedngs of the Secton on Survey Research Methods, Amercan Statstcal Assocaton. Fay, Robert E. (1995), "VPLX: Varance Estmaton for Complex Surveys, Program Documentaton," unpublshed Bureau of the Census Report. Hoagln, D.C. and Iglewcz, B. (1987). Fne-tunng Some Resstant Rules for Outler Labelng. Journal of the Amercan Statstcal Assocaton, 83, pp Judkns, Davd R. (1990). Fay's Method for Varance Estmaton. Journal of Offcal Statstcs, 6, pp Kovar, J.G, Rao, J.N.K, and Wu, C.F.J. (1988). Bootstrap and Other Methods to Measure Errors n Survey Estmates. The Canadan Journal of Statstcs, 16, pp Kovacevc, Mlorad and Yung, Wesley (1997). Varance Estmaton for Measures of Income Inequalty and Polarzaton -- An Emprcal Study. Survey Methodology, 23, pp Luery, Donald M (1990). Survey of Constructon Techncal Paper. Unpublshed draft Bureau of the Census nternal documentaton. Naylor, Thomas H., Balnt~, Joseph L., Burdck, Donald S., and Chu, Kong (1968). Computer Smulaton Technques. New York: John Wley and Sons, Inc. Rao, J.N.K., Wu, C.F.J., and Yue, K. (1992). Some Recent Work on Resamplng Methods for Complex Surveys. Survey Methodology, 18, pp Rao, J.N.K. and Shao, J. (1996). On Balanced Half- Sample Varance Estmaton n Stratfed Random Samplng. Journal of the Amercan Statstcal Assocaton, 91, pp Snedecor, George W. and Cochran, Wllam G. (1980). Statstcal Methods. Iowa: The Iowa State Unversty Press. Thompson, Katherne J. (forthcomng n 1998). Evaluaton of Modfed Half-Sample Replcaton for Estmatng Varances for the Survey of Constructon (SOC). Washngton, DC: U.S. Bureau of the Census. (Techncal Report #ESM-9801, avalable from the Economc Statstcal Methods and Programmng Dvson). Wolter, Krk M. (1985). Introducton to Varance Estmaton. New York: Sprnger-Verlag, Inc. Woodruff, Ralph S. (1952). Confdence Intervals for Medans and Other Poston Measures. Journal of the Amercan Statstcal Assocaton, 47, pp