Estimating Per Capita Rates Using Aggregate Measurements From Groups of Diverse Compositions

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1 Journal o Statstcal Theory and Applcatons, Vol. 4, No. (June 05, 9-03 Estatng Per Capta Rates Usng Aggregate Measureents Fro Groups o Dverse Copostons Donald N. Stengel Departent o Inoraton Systes and Decson Scences, Calorna State Unversty, Fresno 545 N. Backer Ave M/S PB07 Fresno, CA , USA Prsclla Chae-Stengel Departent o Inoraton Systes and Decson Scences, Calorna State Unversty, Fresno 545 N. Backer Ave M/S PB07 Fresno, CA , USA Abstract Ths paper consders the proble o estatng a varable ean or a populaton o eleents here data are only avalable as aggregate sus or groups o ultple eleents. The proposed odel addresses an addtonal coplcaton created hen the group easure ncludes the contrbuton o dverse eleents that ere only partally n operaton or present as part o the group durng the easureent perod. The odel also accounts or statstcal dependency beteen the contrbutons o ndvduals belongng to the sae group. The degree o statstcal dependency s relected n a correlaton coecent paraeter, hch, hle not observable, can be adusted to reduce heteroscedastcty n the group data. A sple exaple s provded to llustrate the odel. Keyords: Estaton, Saplng, Grouped data 00 Matheatcs Subect Classcaton: 6F0, 6D99, Introducton One o the authors as asked to desgn a study to estate the utlzaton rate per year o an te by ndvdual practtoners n a healthcare proesson, here easureents o utlzaton are generally avalable only on the bass o total cobned use by all partners n a practce because orders o supples and nventores are antaned on cobned bass. Whle unts o the varable beng easured are ultately assgnable to one ndvdual eber o the practce, the decoposton ro aggregate group-level easureent to ndvdual easureents s not observable. Slar crcustances o saple easureents beng avalable only or aggregates o ndvdual eleents occur n other settngs. Exaples are utlzaton o oce supples per orker here coon supply closets are used by all eployees n an adnstratve unt, consupton o ood by ndvduals n a lvng unt th a shared ktchen, and ater consupton per housng unt n neghborhood here sngle unts do not have ater eters. Publshed by Atlants Press Copyrght: the authors 9

2 A coon tool or addressng these stuatons s a rato estator. Cochran [] provdes a good presentaton o rato estators. The saplng n ths raeork easures to varables: the varable o prary nterest Y and an auxlary varable X that counts or easures the sze o the aggregated eleent. Rato estators ocus on the rato Y/X, hch s a proxy or the rate o varable Y per unt o varable X. The theory provdes equatons or pont and nterval estates o the rato. In cases here the total nuber o eleents s assued to be knon, the theory also provdes pont and nterval estates or the total o Y across the populaton. The assupton or the classcal case o rato estators s that Y s a lnear ultple o X, the varance n Y s a lnear ultple o X, and the resdual n Y ro the lnear relatonshp beteen Y and X s norally dstrbuted. For the proble studed here, the varable X ould be the nuber o unts ncluded n the aggregate group easureent o varable Y. I the varance n Y ncreases lnearly th the nuber o ndvdual eleents n the group beng easured, the varance o Y ould be equal to the varance or a group th a sngle eleent, ultpled by the nuber o unts n the group. Snce the varance o the su o a set o dentcal, statstcally ndependent rando varables s the varance o one o the varables ultpled by the nuber o rando varables, the assupton above ples that the contrbuton o each ndvdual eleent n a group aggregate s ndependent o other eleents n ther group. Ths assupton ay be too strong. In the case o the usage o a edcal supply by ndvdual healthcare practtoners, the usage by one proessonal n the group s hgher than the ean per capta rate n the populaton, the per capta rates o others n the sae group ay be ore lkely to be hgh relatve to the populaton. The varance thn a group ay be less than expected group ebers really dd peror ndependently o ther group ebershp. Ths ght occur because the ndvduals n a group have a partcular type o practce or ay nluence one another as to the anner n hch they execute ther practce. Although soe o ths dependency can be reoved by ncludng covarates n the analyss, the drvers o ntragroup dependency ay not be readly obvous. An alternatve approach that does not presue that varance aong eleents thn a group s equal to the varance aong eleents across the populaton s cluster saplng. Bhatt [] provdes a coprehensve treatent or analyzng the presence o reduced varaton th group or cluster eleents, called cluster eects. Typcally n cluster saplng, clusters are sapled randoly ro the populaton o clusters and then a rando saple o eleents thn each selected cluster s selected and easured. A an derence beteen the proble studed n ths paper and typcal cluster saplng applcatons s that the second level o cluster saplng provdes easureents or ndvdual eleents thn the saple, hle n ths paper t s assued there s a sngle easureent ro each group: the su o values across all eleents n the group or cluster. Another ssue that needs to be addressed n our ethodology s the pact o the coposton o the group on ntragroup varaton. A healthcare practce consstng o a sngle ull-te proessonal ay not have the sae varance as a healthcare practce o to hal-te proessonals. Although both practces have a slar sze n ters o contanng one ull-te equvalent proessonal, the latter case o to hal-te proessonals ay have a saller varance than the sngle ull-te proessonal. Ths paper presents a odel or estatng the rato o a varable per unt that addresses the possblty that thn-group varaton o ndvduals n a group s less than the varaton across all ndvduals n the populaton and consders the copostons o the sapled groups. Publshed by Atlants Press Copyrght: the authors 93

3 . The Basc Model Suppose a populaton s coprsed o groups o one or ore ndvdual unts. There s a easurable characterstc varable Y or each ndvdual unt belongng to group. Hoever, only the su o the Y or a group s observable. A rando saple o n groups ll be dran ro the populaton and a easureent o Z Y or group s recorded. Indvdual unts ay only operate at a racton o a ull-te unt and the value o Y s scaled to relect the operatng level. A paraeter ndcates the operatng level, th or a unt that s operatng ull-te or 00%. Although the odel assues that ndvdual values o Y are not observable, the odel assues that the values o are observable or any group selected n the rando saple. The value o Y s presued to be norally dstrbuted, th a ean o µ and a varance o σ. The values o µ and σ are assued to be unknon. The expected covarance n Y or ndvdual unts ncluded n saple groups s presued to be ρ k σ or to ndvdual unts and k that belong to the sae group, here 0 ρ, and zero or to ndvdual unts belongng to derent groups. The paraeter ρ s the coecent o correlaton n varable Y or any par o ndvdual unts n the sae group. The assupton o coon ntragroup correlaton s eployed dely n cluster saplng odels, such as Scott and Holt [3]. In ths secton, e ll consder the paraeter ρ as a value set by the researcher; n a subsequent secton, e ll consder ho to select the value o ρ. Let represent the nuber o ndvdual unts n group. Based on knon results or sus o rando varables [4], e can conclude the expected value or the easured total Z or group s E( Z E( Y µ And the varance or the easured total Z or group s Var( Z Var( Y + Cov( Y, Y σ + ρσ < k Splttng the rst ter and reorganzng leads to Var k ( Z ( ρ σ + ρσ ( < k k Snce the nuber o ull-te equvalent unts n group s knon, the total observed value Z or group can be transored to a varable R or the per capta rate observed or the group, th the ollong expected value and varance: R Z / E( R E( Z / µ Var( R Var( Z / ( ( ρ σ /( + ρσ ( Publshed by Atlants Press Copyrght: the authors 94

4 For the case here a group conssts entrely o ully operatng ndvdual unts (.e., or all unts, Eq. ( sples to Var ( R ( ρ σ / + ρσ ( When ρ 0 and the value o each ndvdual unt s ndependent o the values o other ndvdual unts n ts group, the varance n Eq. ( s the alar expresson or the average o dentcal but ndependent rando varables. When ρ, the values o ndvdual unts are perectly correlated th other rs n the group, and the varance o the per capta rate s the sae regardless o the nuber o unts n the group. Pont estates and condence ntervals or µ can be obtaned by collectng the observed values o R ro the rando saple o groups and calculatng the nu varance estate. Hoever, snce each group has a derent varance or R due to derent group copostons, heteroscedastcty s generally present, and t s necessary to use a eghted estate o populaton paraeters. The eghts should be set such that the eghted varables have the sae expected varaton. Usng Eq. (, a sutable set o eghts ould be Var( R / σ ( ρ /( + ρ (3 The goal o the odel s to provde estates o ean µ and varance σ o the value o the varable beng studed as t apples to a ull-te sngle unt. Based on standard orulas used n statstcal sotare [5], e calculate the estates as ollos: The nu varance pont estate or µ s eghted ean o the R values. R n R / n Snce s set so that σ Var(R, the squared devaton o R ro the eghted ean s eectvely a sapled value ro a noral dstrbuton th ean zero and varance σ /. By averagng the eghted squared derences and correctng or saple varaton n the eghted ean, the unbased estate o varance σ correspondng to a sngle, ull-te unt s s n ( R R n here n s the nuber o observed groups. The standard error or the estate or µ s s e s n Publshed by Atlants Press Copyrght: the authors 95

5 A 00(-α% condence nterval or µ s R t α /, nse µ R + t α /, n s e 3. Senstvty o Group Weghts to Intragroup Correlaton, Group Sze, and Group Coposton The group eghts used to deterne pont estates and condence ntervals on per capta levels are aected by the correlaton ρ beteen ndvdual unts n each group, the nuber o unts n each group, and the set o operatng levels o the unts n each group. As shon n the Appendx, the orula or the group eghts n Eq. (3 has the ollong equvalent expresson: + ( ρ CV / ( ρ( / (4 here the CV s the squared coecent o varaton n unt operatng levels : CV ( / and the average o the values s A careul exanaton o Eq. (4 ndcates that the eght ll ncrease as the nuber o unts ncreases or xed values o ρ and CV. Hoever, the rate o ncrease dnshes as gets large, asyptotcally approachng / ρ. The senstvty o group sze on the eght s greater hen the correlaton coecent s lo, as unt Y values are only eakly dependent on other unt values n ther group and the group per capta ean ll have less varance or larger groups. The exanaton o the eect o the coecent o varaton ter n Eq. (4 oers to nterestng nsghts. Frst, snce the value n the denonator ncreases CV ncreases, the eght or group gets saller as relatve varaton beteen the values becoes greater. Further, snce Eq. (4 ncludes no other ters related to the partcpaton levels o the unts, the average o the values has no eect on the eght the coecent o varaton reans the sae. So, or exaple, a group nvolvng to unts both operatng at 0.5 ll have the sae eght as a group th to unts operatng at.0. Stated derently, hen a group has unts th unor partcpaton levels, the eght assgned to the group s the sae or any average partcpaton level and no addtonal noraton about the ean per capta rate s provded erely by hgher average partcpaton rates. Ths concluson presues that unt perorance easureents are Publshed by Atlants Press Copyrght: the authors 96

6 dvsble, hch s a reasonable assupton the unt Y values are arly large or operatng levels o the unts are not ractons close to zero. The value o the denonator n Eq. (3 ll generally be less than one because or any group th ultple ndvdual unts operatng at nonzero levels < ( Thereore, or xed values o CV and, as the ntragroup correlaton coecent ncreases, the denonator ncreases and the eght or group decreases, approachng as ρ approaches one. Ths occurs because hen ρ, contrbutons to the group total ro ndvdual unts n the group are perectly correlated, so the group per capta rate eectvely vares lke a group coprsed o a sngle unt. On the other hand, as ρ approaches zero, the eght ncreases, approachng a value o + CV / ( / I CV 0, eght n Eq. (4 s, relectng the group per capta rate beng the result o unts operatng ndependently o other unts. 4. A Sple Exaple Suppose a researcher s nterested n the typcal nuber o syrnges used by a physcan n a partcular type o edcal practce over one year. A rando saple o sx practces that ocus on ths specalty has been selected and quered or ( total use o syrnges over a speced -onth perod, ( the nuber o physcans ho orked n the practce oce durng the perod, and (3 hat racton o a ull-te, ull year each ndvdual physcan orked durng the perod. The results ro the survey are n Table. Suppose e assue the ntragroup correlaton coecent ρ0.3. Usng the relatonshps ro the odel dened earler, the calculated eghts or the sx groups (rounded to three decal places appear n the nal colun o Table. Table. Saple results, use per ull-te equvalent physcan, and group eghts hen ρ0.3. Practce # Coposton # physcans, partcpaton levels Total syrnges used over year Syrnge use per FTE physcan, 00% physcans, each 50% physcans, each 00% physcans, one 00%, to 40% physcans, each 60% physcans, each 00% Weght or ρ 0.3 It s nterestng to note that hle Practce and Practce has one ull-te, ull year equvalent poston, Practce has a larger eght. Ths occurs because n the second case, there are to physcans contrbutng to the syrnge use, and hle there s only an expected correlaton o ρ0.3, the count ro Practce provdes ore noraton about the ean per capta use. Publshed by Atlants Press Copyrght: the authors 97

7 Practce 3 has a slar organzaton to Practce, but the to physcans orked at 00% rather than 50%. Yet, the eghts o the to practces are the sae, because the use by each hal-te physcan can be doubled and provde an equvalent estate o per capta use. Practces 4 and 5 each have three physcans and a total o.8 ull-te, ull year equvalent physcans, yet the eght or Practce 5 s larger. Ths happens because there s no varaton n the partcpaton ractons n Practce 5, hle Practce 4 has a postve coecent o varaton n partcpaton levels, so as explaned n the prevous secton, all other thngs equal, the eght s hgher or the group th a loer coecent o varaton. Practce 6 s at least ve tes as large as any o the other practces n the saple, yet the eght assgned to Practce 6 s less than tce as large as ost o the other practce groups. Ths happens because the correlaton coecent o ρ0.3 ndcates that syrnge use by one physcan nluences the use by other physcans n the group, thereby dnshng the value o the total syrnge use n the practce toard estatng the populaton ean. Usng these group eghts, the odel yelds the ollong saple statstcs or annual per capta syrnge use: Pont estate: 98 syrnges per year Saple standard devaton: syrnges per year Standard error o the estate o the populaton ean: 7. syrnges per year 95% condence nterval or the populaton ean: beteen 44 and 3540 syrnges/year Snce the value o the correlaton coecent s hypotheszed, the eect o the selecton can be exaned by recalculatng the eghts and statstcs or other values o ρ. Table shos the pact on group eghts or derent coecent values. When ρ0, practces are presued to be coprsed o statstcally ndependent ndvdual unts. When the unts n the group are unor n partcpaton level, the eght s equal to the nuber o unts. These eghts dnsh as ρ ncreases, relectng the hgher correlaton thn the group and correspondng dnshed value o each observaton n estatng µ relatve to the nuber o unts. When ρ, every practce has a eght o one regardless o group coposton, as perect correlaton beteen unts n a group eans that each group s per capta use relects the equvalent to noraton about µ provded by the value ro a practce th one ull-te physcan. Table. Group eghts or exaple usng derent values o correlaton coecent ρ. ρ Table 3 shos the eect o derent hypotheszed correlaton coecents on the saple statstcs. The pont estate s soehat senstve to the selecton, th greater senstvty hen ρ s closer to zero. The pont estate ncreases as ρ gets saller due to the act that the observed per capta levels or Practces 4, 5, and 6 ere hgher than or Practces,, and 3, and the eghts or the larger practces decrease ore as the coecent o correlaton ncreases. The 95% Publshed by Atlants Press Copyrght: the authors 98

8 condence nterval lts are tghter and ore senstve to changes n the coecent hen ρ s sall, snce the standard error o the estate s nversely related to the square root o the su o the group eghts, and that su ncreases at a grong rate as ρ decreases. Table 3. Saple statstcs or ndvdual per capta usage usng derent values o correlaton coecent ρ. ρ Pont Estate Saple Standard Dev Standard Error o Estate LCL (95% UCL(95% Estaton o the Intragroup Correlaton Coecent The odel presented n the second secton assues that a value or the paraeter ρ ll be speced beore beng appled to a set o observatons ro groups o unts. Ths paraeter ndcates the degree o correlaton aong pars o unts thn groups, allong dependency beteen the unts or the varable o nterest and hgher varaton n the aggregate group total than ould be expected ndvdual unts operated ully ndependently. Hoever, the degree o correlaton s not readly observable and thus s a practcal concern n applyng the odel. The ρ paraeter aects the varablty o the group aggregate varable and nluences the eghtng actors used to copensate or heteroscedastcty n saple per capta rates due to dverse group copostons. In turn, the choce o ρ aects the pont estate o the ean operatng level or a ull-te unt and the standard error o that estate. One can exane the eect o the choce o ρ on pont or nterval estates by dong senstvty analyss on the paraeter. The paraeter could be tested at derent levels, as as done n the exaple n the prevous secton. I the pont estate and condence nterval or µ do not change uch, the selecton o ρ s not a serous concern. I there are notceable derences, a conservatve approach ould be to den the condence nterval at the desred condence level to nclude the condence ntervals or the range o correlaton coecents tested. For exaple, th the syrnge use survey presented n the last secton, the researcher s uncertan o correlaton value, but beleves the correlaton coecent s no larger than 0.6, the nterval ro 38 to 3596 ould nclude the 95% condence ntervals or the per capta populaton ean or all values o ρ ro 0 to 0.6. I the researcher s uncertan about the degree o ntragroup correlaton, but has soe pror bele about value o ρ, a probablty dstrbuton could be dened on that paraeter and a Bayesan approach could be appled. Or, usng Monte Carlo sulaton and the odel o the pror secton, the sulaton or randoly generated values o ρ ould provde dstrbutons or the pont estate o µ and the upper and loer lts o the condence nterval or µ. When a arly substantal nuber o groups are ncluded n the saple, partcularly or a saple that has consderable varaton n the eghtng actors, the saple data can be used to nor the selecton o ρ. Snce the purpose o the eghtng actors s to antan equal varaton n the eghted resduals ro the populaton ean, scatterplots o the eghted resduals ro Publshed by Atlants Press Copyrght: the authors 99

9 the procedure can be exaned to assess the eectveness o the eghtng actors. The squared resduals can be plotted aganst the values o eghts. I the spread o eghted resduals s arly unor across the eghts, the selected correlaton coecent s probably sound. On the other hand, the spread appears to ether ncrease or decrease as the eghts ncrease, ths suggests that relatve szes o the eghts are not approprate or reducng heteroscedastcty and a derent value or the correlaton coecent ght be approprate. Snce, as noted earler, or any group th ultple ndvdual unts th a nonzero operatng level, < ( there s a negatve relatonshp beteen the selecton o the correlaton coecent ρ and the eghts as calculated n Eq. (3. As such, the larger eghts need to be ncreased to balance the eghted squared resduals, decreasng the value o ρ ould reduce the balance, hereas the larger eghts need to be reduced n relatve agntude, ncreasng the value o ρ should result n an proveent. A narrong spread o the eghted squared resduals as the eghts ncrease ndcates that the eghts are too aggressve and a larger value o ρ should be tred. Alternatvely, a denng spread o eghted squared resduals suggests the hgher eghted groups are relatvely undereghted and ρ should be decreased. Another approach to settng the correlaton coecent s to start th the assupton o no ntragroup correlaton eect (ρ 0 and test the level o heteroscedastcty s sgncant. One opton s to place the sapled groups nto categores based on the eghts and apply the Levene test [6] to the eghted resduals to see the derences n category varaton are sgncant. Another opton s to run a regresson on the eghted squared resduals aganst the eght values (or group szes to see there s a sgncant relatonshp. The Whte test [7] and Breusch-Pagan test [8] take ths approach. I the test concludes that hooscedastcty ust be reected, the value o the correlaton coecent could be reset and retested or heteroscedastcty. It should be noted that henever a derent correlaton coecent ρ s used, not only are the eghts aected, but n turn, the estate o the ean per capta estate ll change as ell. As a consequence, the resduals beteen the observed per capta rate and estated populaton per capta ean need to recalculated and not erely reeghted. There s no guarantee that heteroscedastcty can be reoved sucently th any correlaton coecent ρ beteen zero and one. Ths stuaton ay occur the basc assupton o the odel that groups have a slar degree o ntragroup correlaton s strongly volated. 6. Dscusson Most statstcal sotare packages allo the entry o a eghtng varable or calculatng saple statstcs or a sngle saple ean. To apply the ethods n ths paper, the eghts ould need to be calculated separately or the sotare ould need to be augented th a scrpt n a language lke R or Python. Whle the orula or calculatng the eghted ean s standard across statstcal sotare tools, there are derences n the calculatons o the eghted saple varance, standard error o the ean estate, and condence ntervals. One source o the derence s the base used to calculate the saple varance. In ths paper and n SAS [5], the base used to calculate the Publshed by Atlants Press Copyrght: the authors 00

10 (unbased saple varance s the su o classes nus one. Hoever, n SPSS [9], the base used s the su o the group eghts nus one. The ustcaton or usng the nuber o groups nus one s that hle groups vary n sze and coposton, each group n the saple provdes a sngle per capta estate, and, hen eghted to relect group coposton, provdes a slar contrbuton to other groups n estatng the populaton paraeters. Lkese, the desgnaton o the nuber o degrees o reedo or the t-statstc used to create condence ntervals or the ean ay der based on the sae ssue. Snce the saple varance n the odel n ths paper s an estate based on the nuber o class observatons, the condence ntervals on the estate o the populaton ean eploy a t-dstrbuton usng the nuber o groups nus one as the nuber o degrees o reedo. For algorths here the saple varance s calculated usng the total o the group eghts n the denonator, the degrees o reedo ll be the su o the group eghts nus one. Obvously, varables other than ntragroup correlaton and group coposton ay be eectve n explanng varaton n per capta levels across groups. For the exaple n ths paper, perhaps subspecaltes o a group practce or geographcal locaton ould be sgncant. Weghted analyss o varance or eghted least-squares regresson ould allo the ncorporaton o the group eghts based on hypotheszed ntragroup correlaton and coposton, as ell at the consderaton o other varables. Hoever, as th the coputaton o eghted saple varance and condence ntervals, hen usng eghted algorths n statstcal sotare packages, care should be taken to see ho the eghts are used and degrees o reedo are deterned. In addton to provdng nerences about the populaton ro hch the saple s dran, the statstcs generated ro a saple can be used to deterne hether the per capta use or a specc group s hgh or lo relatve to the populaton, gven ts coposton. The group s per capta value can be evaluated based on ts poston th respect to the probablty dstrbuton created by a lnear transoraton o a t-dstrbuton havng ( a ean equal to the eghted ean saple per capta rate, ( a standard devaton equal to the saple standard devaton n per capta aount dvded by the square root o the eght assgned to the group, and (3 the nuber o degrees o reedo based on the nuber o groups used n the calculaton o the saple varance nus one. Wth coputer tools, the per capta aount or the group n queston can be converted to a quantle th respect to that dstrbuton. Ths evaluaton could be appled to groups hether or not they ere not part o the saple used to generate the saple statstcs. Appendx: Dervaton o the Alternate Forula or the Group Weghts In the presentaton o the odel n the second secton, the orula Eq. (3 or the approprate eghts or each observed class as derved usng the ntragroup correlaton coecent ρ and partcpaton levels o the ndvdual unts coprsng the group. In the thrd secton o the paper, alternatve orula Eq. (4 as cted that relates the approprate eght to correlaton coecent, the nuber o unts n the group, and the relatve varaton n partcpaton levels th the group. The alternatve orula s derved here. The rst orula or eghts (Eq. (3 derved th the basc odel as ( ρ /( + ρ Publshed by Atlants Press Copyrght: the authors 0

11 Invertng the equaton and algebrac anpulaton results n the ollong equaton: ( ( ( ( ( / + ρ Snce the denonator s the square o the su o unt ractons, and the su o the unt ractons s equal to the nuber o unts tes the average unt racton, usng ths substtuton n the equaton yelds ( ( / + ρ Usng the ollong ( + the equaton becoes ( ( ( ( / + ρ ρ Applyng the coputatonal orula or the varance o a set o data F Var ( then extractng the squared coecent o varaton n the values usng ( F Var CV olloed by cancellaton o ters and re-nverson o the equaton, results n Eq. (4: CV / ( ( / ( + ρ ρ Publshed by Atlants Press Copyrght: the authors 0

12 Reerences [] Cochran, W. G., Saplng Technques (3rd ed. (John Wley & Sons, Ne York, NY, 977. [] Bhatt, M., Cluster Eects n Mnng Coplex Data. (Nova Scence Publsher's, Ne York, NY, 0. [3] Scott, A. J., and Holt, D., The Eect o To-Stage Saplng on Ordnary Least Squares Methods, Journal o the Aercan Statstcal Assocaton, 77(380, ( [4] Feller, W., An Introducton to Probablty Theory and ts Applcatons (Vol., 3rd ed. (John Wley & Sons, Ne York, NY, 967. [5] SAS Insttute, Inc., SAS/STAT 9. User s Gude, Chapter 9 (SAS Insttute, Cary, NC, 009. [6] Levene, H., Contrbutons to Probablty and Statstcs: Essays n Honor o Harold Hotellng, eds. Olkn, Hotellng, et al (Stanord, CA: Stanord Unversty Press, Stanord, CA, 960, pp [7] Whte, H., A Heteroskedastcty-Consstent Covarance Matrx Estator and a Drect Test or Heteroskedastcty, Econoetrca, 48(4, ( [8] Breusch, T. S., and Pagan, A. R., A Sple Test or Heteroscedastcty and Rando Coecent Varaton, Econoetrca, 47(5, ( [9] IBM Corporaton, IBM SPSS Algorths, t Test Algorths, One Saple t Test, [onlne]. Avalable at: Publshed by Atlants Press Copyrght: the authors 03