Estimating Per Capita Rates Using Aggregate Measurements From Groups of Diverse Compositions
|
|
- Nicholas Alexander
- 5 years ago
- Views:
Transcription
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
PROBABILITY AND STATISTICS Vol. III - Analysis of Variance and Analysis of Covariance - V. Nollau ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE
ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE V. Nollau Insttute of Matheatcal Stochastcs, Techncal Unversty of Dresden, Gerany Keywords: Analyss of varance, least squares ethod, odels wth fxed effects,
More informationElastic Collisions. Definition: two point masses on which no external forces act collide without losing any energy.
Elastc Collsons Defnton: to pont asses on hch no external forces act collde thout losng any energy v Prerequstes: θ θ collsons n one denson conservaton of oentu and energy occurs frequently n everyday
More informationBAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS. Dariusz Biskup
BAYESIAN CURVE FITTING USING PIECEWISE POLYNOMIALS Darusz Bskup 1. Introducton The paper presents a nonparaetrc procedure for estaton of an unknown functon f n the regresson odel y = f x + ε = N. (1) (
More informationStatistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation
Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. ) with a symmetric Pcovariance matrix of the y( x ) measurements V
Fall Analyss o Experental Measureents B Esensten/rev S Errede General Least Squares wth General Constrants: Suppose we have easureents y( x ( y( x, y( x,, y( x wth a syetrc covarance atrx o the y( x easureents
More informationXII.3 The EM (Expectation-Maximization) Algorithm
XII.3 The EM (Expectaton-Maxzaton) Algorth Toshnor Munaata 3/7/06 The EM algorth s a technque to deal wth varous types of ncoplete data or hdden varables. It can be appled to a wde range of learnng probles
More informationExcess Error, Approximation Error, and Estimation Error
E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationCOMP th April, 2007 Clement Pang
COMP 540 12 th Aprl, 2007 Cleent Pang Boostng Cobnng weak classers Fts an Addtve Model Is essentally Forward Stagewse Addtve Modelng wth Exponental Loss Loss Functons Classcaton: Msclasscaton, Exponental,
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationEconomics 130. Lecture 4 Simple Linear Regression Continued
Economcs 130 Lecture 4 Contnued Readngs for Week 4 Text, Chapter and 3. We contnue wth addressng our second ssue + add n how we evaluate these relatonshps: Where do we get data to do ths analyss? How do
More informationx i1 =1 for all i (the constant ).
Chapter 5 The Multple Regresson Model Consder an economc model where the dependent varable s a functon of K explanatory varables. The economc model has the form: y = f ( x,x,..., ) xk Approxmate ths by
More information/ n ) are compared. The logic is: if the two
STAT C141, Sprng 2005 Lecture 13 Two sample tests One sample tests: examples of goodness of ft tests, where we are testng whether our data supports predctons. Two sample tests: called as tests of ndependence
More informationOn the number of regions in an m-dimensional space cut by n hyperplanes
6 On the nuber of regons n an -densonal space cut by n hyperplanes Chungwu Ho and Seth Zeran Abstract In ths note we provde a unfor approach for the nuber of bounded regons cut by n hyperplanes n general
More informationStatistics for Economics & Business
Statstcs for Economcs & Busness Smple Lnear Regresson Learnng Objectves In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable
More informationWhat is LP? LP is an optimization technique that allocates limited resources among competing activities in the best possible manner.
(C) 998 Gerald B Sheblé, all rghts reserved Lnear Prograng Introducton Contents I. What s LP? II. LP Theor III. The Splex Method IV. Refneents to the Splex Method What s LP? LP s an optzaton technque that
More informationtotal If no external forces act, the total linear momentum of the system is conserved. This occurs in collisions and explosions.
Lesson 0: Collsons, Rotatonal netc Energy, Torque, Center o Graty (Sectons 7.8 Last te we used ewton s second law to deelop the pulse-oentu theore. In words, the theore states that the change n lnear oentu
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationCollaborative Filtering Recommendation Algorithm
Vol.141 (GST 2016), pp.199-203 http://dx.do.org/10.14257/astl.2016.141.43 Collaboratve Flterng Recoendaton Algorth Dong Lang Qongta Teachers College, Haou 570100, Chna, 18689851015@163.co Abstract. Ths
More informationEcon107 Applied Econometrics Topic 9: Heteroskedasticity (Studenmund, Chapter 10)
I. Defnton and Problems Econ7 Appled Econometrcs Topc 9: Heteroskedastcty (Studenmund, Chapter ) We now relax another classcal assumpton. Ths s a problem that arses often wth cross sectons of ndvduals,
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2014. All Rghts Reserved. Created: July 15, 1999 Last Modfed: February 9, 2008 Contents 1 Lnear Fttng
More informationChapter 11: Simple Linear Regression and Correlation
Chapter 11: Smple Lnear Regresson and Correlaton 11-1 Emprcal Models 11-2 Smple Lnear Regresson 11-3 Propertes of the Least Squares Estmators 11-4 Hypothess Test n Smple Lnear Regresson 11-4.1 Use of t-tests
More informationSmall-Sample Equating With Prior Information
Research Report Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston Charles Lews June 009 ETS RR-09-5 Lstenng Learnng Leadng Sall-Saple Equatng Wth Pror Inforaton Sauel A Lvngston and Charles Lews ETS,
More informationHopfield Training Rules 1 N
Hopfeld Tranng Rules To memorse a sngle pattern Suppose e set the eghts thus - = p p here, s the eght beteen nodes & s the number of nodes n the netor p s the value requred for the -th node What ll the
More informationInternational Journal of Mathematical Archive-9(3), 2018, Available online through ISSN
Internatonal Journal of Matheatcal Archve-9(3), 208, 20-24 Avalable onlne through www.ja.nfo ISSN 2229 5046 CONSTRUCTION OF BALANCED INCOMPLETE BLOCK DESIGNS T. SHEKAR GOUD, JAGAN MOHAN RAO M AND N.CH.
More informationNegative Binomial Regression
STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...
More informationLecture 3 Stat102, Spring 2007
Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture
More informationA Particle Filter Algorithm based on Mixing of Prior probability density and UKF as Generate Importance Function
Advanced Scence and Technology Letters, pp.83-87 http://dx.do.org/10.14257/astl.2014.53.20 A Partcle Flter Algorthm based on Mxng of Pror probablty densty and UKF as Generate Importance Functon Lu Lu 1,1,
More informationFermi-Dirac statistics
UCC/Physcs/MK/EM/October 8, 205 Fer-Drac statstcs Fer-Drac dstrbuton Matter partcles that are eleentary ostly have a type of angular oentu called spn. hese partcles are known to have a agnetc oent whch
More informationUniversity of Washington Department of Chemistry Chemistry 452/456 Summer Quarter 2013
Lecture 8/8/3 Unversty o Washngton Departent o Chestry Chestry 45/456 Suer Quarter 3 A. The Gbbs-Duhe Equaton Fro Lecture 7 and ro the dscusson n sectons A and B o ths lecture, t s clear that the actvty
More informationStatistics for Business and Economics
Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear
More informationDepartment of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6
Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.
More informationDenote the function derivatives f(x) in given points. x a b. Using relationships (1.2), polynomials (1.1) are written in the form
SET OF METHODS FO SOUTION THE AUHY POBEM FO STIFF SYSTEMS OF ODINAY DIFFEENTIA EUATIONS AF atypov and YuV Nulchev Insttute of Theoretcal and Appled Mechancs SB AS 639 Novosbrs ussa Introducton A constructon
More informationChapter One Mixture of Ideal Gases
herodynacs II AA Chapter One Mxture of Ideal Gases. Coposton of a Gas Mxture: Mass and Mole Fractons o deterne the propertes of a xture, we need to now the coposton of the xture as well as the propertes
More informationBasic Business Statistics, 10/e
Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson
More informationComparison of Regression Lines
STATGRAPHICS Rev. 9/13/2013 Comparson of Regresson Lnes Summary... 1 Data Input... 3 Analyss Summary... 4 Plot of Ftted Model... 6 Condtonal Sums of Squares... 6 Analyss Optons... 7 Forecasts... 8 Confdence
More informationComputation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models
Computaton of Hgher Order Moments from Two Multnomal Overdsperson Lkelhood Models BY J. T. NEWCOMER, N. K. NEERCHAL Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore,
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationChapter 12 Lyes KADEM [Thermodynamics II] 2007
Chapter 2 Lyes KDEM [Therodynacs II] 2007 Gas Mxtures In ths chapter we wll develop ethods for deternng therodynac propertes of a xture n order to apply the frst law to systes nvolvng xtures. Ths wll be
More informationChapter 13. Gas Mixtures. Study Guide in PowerPoint. Thermodynamics: An Engineering Approach, 5th edition by Yunus A. Çengel and Michael A.
Chapter 3 Gas Mxtures Study Gude n PowerPont to accopany Therodynacs: An Engneerng Approach, 5th edton by Yunus A. Çengel and Mchael A. Boles The dscussons n ths chapter are restrcted to nonreactve deal-gas
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationProbability, Statistics, and Reliability for Engineers and Scientists SIMULATION
CHATER robablty, Statstcs, and Relablty or Engneers and Scentsts Second Edton SIULATIO A. J. Clark School o Engneerng Department o Cvl and Envronmental Engneerng 7b robablty and Statstcs or Cvl Engneers
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationF8: Heteroscedasticity
F8: Heteroscedastcty Feng L Department of Statstcs, Stockholm Unversty What s so-called heteroscedastcty In a lnear regresson model, we assume the error term has a normal dstrbuton wth mean zero and varance
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationECEN 5005 Crystals, Nanocrystals and Device Applications Class 19 Group Theory For Crystals
ECEN 5005 Crystals, Nanocrystals and Devce Applcatons Class 9 Group Theory For Crystals Dee Dagram Radatve Transton Probablty Wgner-Ecart Theorem Selecton Rule Dee Dagram Expermentally determned energy
More informationANALYSIS OF SIMULATION EXPERIMENTS BY BOOTSTRAP RESAMPLING. Russell C.H. Cheng
Proceedngs of the 00 Wnter Sulaton Conference B. A. Peters, J. S. Sth, D. J. Mederos, and M. W. Rohrer, eds. ANALYSIS OF SIMULATION EXPERIMENTS BY BOOTSTRAP RESAMPLING Russell C.H. Cheng Departent of Matheatcs
More informationarxiv: v2 [math.co] 3 Sep 2017
On the Approxate Asyptotc Statstcal Independence of the Peranents of 0- Matrces arxv:705.0868v2 ath.co 3 Sep 207 Paul Federbush Departent of Matheatcs Unversty of Mchgan Ann Arbor, MI, 4809-043 Septeber
More informationMultigradient for Neural Networks for Equalizers 1
Multgradent for Neural Netorks for Equalzers 1 Chulhee ee, Jnook Go and Heeyoung Km Department of Electrcal and Electronc Engneerng Yonse Unversty 134 Shnchon-Dong, Seodaemun-Ku, Seoul 1-749, Korea ABSTRACT
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More information1 Definition of Rademacher Complexity
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture #9 Scrbe: Josh Chen March 5, 2013 We ve spent the past few classes provng bounds on the generalzaton error of PAClearnng algorths for the
More informationMultipoint Analysis for Sibling Pairs. Biostatistics 666 Lecture 18
Multpont Analyss for Sblng ars Bostatstcs 666 Lecture 8 revously Lnkage analyss wth pars of ndvduals Non-paraetrc BS Methods Maxu Lkelhood BD Based Method ossble Trangle Constrant AS Methods Covered So
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationSignal-noise Ratio Recognition Algorithm Based on Singular Value Decomposition
4th Internatonal Conference on Machnery, Materals and Coputng Technology (ICMMCT 06) Sgnal-nose Rato Recognton Algorth Based on Sngular Value Decoposton Qao Y, a, Cu Qan, b, Zhang We, c and Lu Yan, d Bejng
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationLECTURE :FACTOR ANALYSIS
LCUR :FACOR ANALYSIS Rta Osadchy Based on Lecture Notes by A. Ng Motvaton Dstrbuton coes fro MoG Have suffcent aount of data: >>n denson Use M to ft Mture of Gaussans nu. of tranng ponts If
More informationOur focus will be on linear systems. A system is linear if it obeys the principle of superposition and homogenity, i.e.
SSTEM MODELLIN In order to solve a control syste proble, the descrptons of the syste and ts coponents ust be put nto a for sutable for analyss and evaluaton. The followng ethods can be used to odel physcal
More informationApplied Mathematics Letters
Appled Matheatcs Letters 2 (2) 46 5 Contents lsts avalable at ScenceDrect Appled Matheatcs Letters journal hoepage: wwwelseverco/locate/al Calculaton of coeffcents of a cardnal B-splne Gradr V Mlovanovć
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationLeast Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2015. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationECONOMICS 351*-A Mid-Term Exam -- Fall Term 2000 Page 1 of 13 pages. QUEEN'S UNIVERSITY AT KINGSTON Department of Economics
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott
More informationReliability estimation in Pareto-I distribution based on progressively type II censored sample with binomial removals
Journal of Scentfc esearch Developent (): 08-3 05 Avalable onlne at wwwjsradorg ISSN 5-7569 05 JSAD elablty estaton n Pareto-I dstrbuton based on progressvely type II censored saple wth bnoal reovals Ilhan
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More information1 Review From Last Time
COS 5: Foundatons of Machne Learnng Rob Schapre Lecture #8 Scrbe: Monrul I Sharf Aprl 0, 2003 Revew Fro Last Te Last te, we were talkng about how to odel dstrbutons, and we had ths setup: Gven - exaples
More informationSeveral generation methods of multinomial distributed random number Tian Lei 1, a,linxihe 1,b,Zhigang Zhang 1,c
Internatonal Conference on Appled Scence and Engneerng Innovaton (ASEI 205) Several generaton ethods of ultnoal dstrbuted rando nuber Tan Le, a,lnhe,b,zhgang Zhang,c School of Matheatcs and Physcs, USTB,
More informationLecture 6: Introduction to Linear Regression
Lecture 6: Introducton to Lnear Regresson An Manchakul amancha@jhsph.edu 24 Aprl 27 Lnear regresson: man dea Lnear regresson can be used to study an outcome as a lnear functon of a predctor Example: 6
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationDetermination of the Confidence Level of PSD Estimation with Given D.O.F. Based on WELCH Algorithm
Internatonal Conference on Inforaton Technology and Manageent Innovaton (ICITMI 05) Deternaton of the Confdence Level of PSD Estaton wth Gven D.O.F. Based on WELCH Algorth Xue-wang Zhu, *, S-jan Zhang
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationCHAPT II : Prob-stats, estimation
CHAPT II : Prob-stats, estaton Randoness, probablty Probablty densty functons and cuulatve densty functons. Jont, argnal and condtonal dstrbutons. The Bayes forula. Saplng and statstcs Descrptve and nferental
More informationChapter 6. Supplemental Text Material
Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.
More informationChapter 5 Multilevel Models
Chapter 5 Multlevel Models 5.1 Cross-sectonal multlevel models 5.1.1 Two-level models 5.1.2 Multple level models 5.1.3 Multple level modelng n other felds 5.2 Longtudnal multlevel models 5.2.1 Two-level
More information, are assumed to fluctuate around zero, with E( i) 0. Now imagine that this overall random effect, , is composed of many independent factors,
Part II. Contnuous Spatal Data Analyss 3. Spatally-Dependent Rando Effects Observe that all regressons n the llustratons above [startng wth expresson (..3) n the Sudan ranfall exaple] have reled on an
More informationStatistics II Final Exam 26/6/18
Statstcs II Fnal Exam 26/6/18 Academc Year 2017/18 Solutons Exam duraton: 2 h 30 mn 1. (3 ponts) A town hall s conductng a study to determne the amount of leftover food produced by the restaurants n the
More informationChat eld, C. and A.J.Collins, Introduction to multivariate analysis. Chapman & Hall, 1980
MT07: Multvarate Statstcal Methods Mke Tso: emal mke.tso@manchester.ac.uk Webpage for notes: http://www.maths.manchester.ac.uk/~mkt/new_teachng.htm. Introducton to multvarate data. Books Chat eld, C. and
More informationUsing the estimated penetrances to determine the range of the underlying genetic model in casecontrol
Georgetown Unversty From the SelectedWorks of Mark J Meyer 8 Usng the estmated penetrances to determne the range of the underlyng genetc model n casecontrol desgn Mark J Meyer Neal Jeffres Gang Zheng Avalable
More information7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA
Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More informationCentral tendency. mean for metric data. The mean. "I say what I means and I means what I say!."
Central tendency "I say hat I means and I means hat I say!." Popeye Normal dstrbuton vdeo clp To ve an unedted verson vst: http://.learner.org/resources/seres65.html# mean for metrc data mportant propertes
More informationAN ANALYSIS OF A FRACTAL KINETICS CURVE OF SAVAGEAU
AN ANALYI OF A FRACTAL KINETIC CURE OF AAGEAU by John Maloney and Jack Hedel Departent of Matheatcs Unversty of Nebraska at Oaha Oaha, Nebraska 688 Eal addresses: aloney@unoaha.edu, jhedel@unoaha.edu Runnng
More informationSee Book Chapter 11 2 nd Edition (Chapter 10 1 st Edition)
Count Data Models See Book Chapter 11 2 nd Edton (Chapter 10 1 st Edton) Count data consst of non-negatve nteger values Examples: number of drver route changes per week, the number of trp departure changes
More informationOn Pfaff s solution of the Pfaff problem
Zur Pfaff scen Lösung des Pfaff scen Probles Mat. Ann. 7 (880) 53-530. On Pfaff s soluton of te Pfaff proble By A. MAYER n Lepzg Translated by D. H. Delpenc Te way tat Pfaff adopted for te ntegraton of
More informationA Comparative Study for Estimation Parameters in Panel Data Model
A Comparatve Study for Estmaton Parameters n Panel Data Model Ahmed H. Youssef and Mohamed R. Abonazel hs paper examnes the panel data models when the regresson coeffcents are fxed random and mxed and
More informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
More informationUsing Multivariate Rank Sum Tests to Evaluate Effectiveness of Computer Applications in Teaching Business Statistics
Usng Multvarate Rank Sum Tests to Evaluate Effectveness of Computer Applcatons n Teachng Busness Statstcs by Yeong-Tzay Su, Professor Department of Mathematcs Kaohsung Normal Unversty Kaohsung, TAIWAN
More informationOutline. Prior Information and Subjective Probability. Subjective Probability. The Histogram Approach. Subjective Determination of the Prior Density
Outlne Pror Inforaton and Subjectve Probablty u89603 1 Subjectve Probablty Subjectve Deternaton of the Pror Densty Nonnforatve Prors Maxu Entropy Prors Usng the Margnal Dstrbuton to Deterne the Pror Herarchcal
More informationHere is the rationale: If X and y have a strong positive relationship to one another, then ( x x) will tend to be positive when ( y y)
Secton 1.5 Correlaton In the prevous sectons, we looked at regresson and the value r was a measurement of how much of the varaton n y can be attrbuted to the lnear relatonshp between y and x. In ths secton,
More informationDepartment of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution
Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More informationEstimation of Reliability in Multicomponent Stress-Strength Based on Generalized Rayleigh Distribution
Journal of Modern Appled Statstcal Methods Volue 13 Issue 1 Artcle 4 5-1-014 Estaton of Relablty n Multcoponent Stress-Strength Based on Generalzed Raylegh Dstrbuton Gadde Srnvasa Rao Unversty of Dodoa,
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationEFFECTS OF MAGNITUDE UNCERTAINTIES ON SEISMIC HAZARD ESTIMATES
79 EFFECTS OF MAGNITUDE UNCERTAINTIES ON SEISMIC HAZARD ESTIMATES Davd A RHOADES And Davd J DOWRICK SUMMARY Magntude uncertantes affect dfferent coponents of an estate of sesc hazard n a varety of ways,
More informationDurban Watson for Testing the Lack-of-Fit of Polynomial Regression Models without Replications
Durban Watson for Testng the Lack-of-Ft of Polynomal Regresson Models wthout Replcatons Ruba A. Alyaf, Maha A. Omar, Abdullah A. Al-Shha ralyaf@ksu.edu.sa, maomar@ksu.edu.sa, aalshha@ksu.edu.sa Department
More informationSolving Fuzzy Linear Programming Problem With Fuzzy Relational Equation Constraint
Intern. J. Fuzz Maeatcal Archve Vol., 0, -0 ISSN: 0 (P, 0 0 (onlne Publshed on 0 Septeber 0 www.researchasc.org Internatonal Journal of Solvng Fuzz Lnear Prograng Proble W Fuzz Relatonal Equaton Constrant
More informationDepartment of Economics, Niigata Sangyo University, Niigata, Japan
Appled Matheatcs, 0, 5, 777-78 Publshed Onlne March 0 n ScRes. http://www.scrp.org/journal/a http://d.do.org/0.6/a.0.507 On Relatons between the General Recurrence Forula o the Etenson o Murase-Newton
More informationON WEIGHTED ESTIMATION IN LINEAR REGRESSION IN THE PRESENCE OF PARAMETER UNCERTAINTY
Econoetrcs orkng Paper EP7 ISSN 485-644 Departent of Econocs ON EIGTED ESTIMATION IN LINEAR REGRESSION IN TE PRESENCE OF PARAMETER UNCERTAINTY udth A Clarke Departent of Econocs, Unversty of Vctora Vctora,
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More information