A SAS Macro for Finding Optimal k-means Clustering in One Dimension with Size Constraints

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1 Pape SD-02 A SAS Maco fo Fndng Optmal k-means Clusteng n One Dmenson wth Sze Constants Fengao Hu, Geoga Regents Unvesty; Robet E. Johnson, Vandeblt Unvesty ABSTRACT Wang and Song (2011) poposed a k-means clusteng algothm n one dmenson usng exact dynamc pogammng, whch guaantees optmalty. The algothm solved the clusteng poblem by beakng t nto smalle nested poblems. The one-dmensonal measue may, fo example, be baselne measues elated to a befoe-afte study and subects ae gouped (clusteed) on baselne befoe andomzaton. In ths pape we extend the wok by placng constants on the cluste sze, fo example, each cluste must be no less than the numbe of study ams. A SAS maco wll be pesented whch fnds the optmal clusteng gven the constant by mnmzng the wthn cluste oot mean squaed eo. An opton that andomly allocates subects to study ams s also ncluded. An example wll be gven whee a sample of pmay cae pactces ae to be allocated to teatment o contol. The study measues the degee to whch pmay cae physcans delve smokng cessaton counselng. Po to andomzaton, the pactces ae clusteed on the baselne measue. INTRODUCTION A pe-post desgn can help contol the wthn subect vaance to acheve the maxmum powe of analyss, snce each subect seves as ts own contol (Pak and Johnson, 2005). The vaance may be futhe contolled by statfcaton nto blocks po to andomzaton, whee the blocks ae elatvely homogeneous on the baselne pmay measues (Pak and Johnson, 2005). Although Pak and Johnson (2006) ponted out that teatng the baselne as a covaable povded maxmal contol of the vaance, wth o wthout pa-matchng, we ae nteested n undestandng the effect of baselne clusteng when consdeng an analyss of pe-post dffeences. In ode to statfy subects nto elatvely homogeneous blocks, a patton of the subects clusteed on the baselne values may be fomed so to mnmze the oot mean squaed eo (RMSE) of the baselne values. Snce subects wthn each block wll be andomzed to study ams, the subects baselne values must be clusteed such that not only the mnmal RMSE s acheved, but also each cluste must have at least the same numbe of subects as ams of the study. Heachcal clusteng methods, such as k-means (Lloyd, 1982), ae often used to dentfy clustes. The k-means pocedue does not always lead to the optmal clusteng snce the pocedue employs a andom path. These methods cannot easly constan the esult to have a mnmum numbe of subects fo each cluste. Unconstaned, the soluton may esult n one o moe clustes havng less than the equed numbe of subects. Fusng neghbong clustes can esolve ths, but may esult n a non-optmal set of clustes (Pak and Johnson, 2004). In ode to get an optmal set of clustes wth constants on numbe of subects, a modfed heachcal clusteng algothm s developed hee fo dentfyng clustes of unvaate baselne outcome data. It not only guaantees optmalty, but also places the desed constant on the mnmum numbe of subects n each cluste. The algothm s mplemented usng SAS/IML wth the maco ConClus (Constaned Un-dmensonal Clusteng). Fnally, we llustate how to use ths algothm by allocatng pmay cae pactces nto teatment and contol goups. The pactces ae clusteed on baselne physcan delvey of smokng cessaton counselng (Rothemch et. al, 2008). METHOD In ode to cluste on baselne to acheve mnmum RMSE wth the constant that evey cluste sze s no less than the numbe of study ams, we povde a new algothm that s an mpovement of the CKmeans.1d.dp algothm (Wang and Song, 2011). Snce, fo a fx set of clustes, mnmzng Eucldean sums of squaes (ESS) s equvalent to mnmzng RMSE when dealng wth unvaate data, we calculate RMSE by gettng the squae oot of the ato of wthnss and degees of feedom, whee wthnss epesents the Eucldean sums of squaes of wthn-cluste dstances fom each subect to ts coespondng cluste mean and the degees of feedom s the dffeence between total subects and numbe of clustes. Let x,, 1 n x be the non-descendng soted baselne values of n subects. We seek the cluste patton that mnmzes RMSE subect to the constant that evey cluste has no less than the numbe of study ams. Ou method extends that developed by Wang and Song (2011) and s equvalent to the algothm when 1. Suppose we have aanged subects nto m clustes wth mnmum wthnss. We ecod the coespondng mnmum wthnss 1

2 n enty D[, m] of an n n matx D, whee n means the ntege pat of n. The second dmenson of D s n constaned snce n subects can be clusteed nto at most clustes. The last ow of the matx D ndcates clusteng all the subects, thus the mnmum value of the last ow n matx D coesponds to the numbe of clustes elated to the cluste patton wth the smallest wthnss value, the soluton to the ognal poblem. The matx D s bult dynamcally. Suppose we want to fnd the wthnss to place n D[, m ] and the coespondng th patton that leads to ths value. Let be the ndex of the smallest odeed data value n the last, m, cluste. The value of cannot be less than m 1because t must have at least subects n each cluste, that s at least th m ( 1) totally fo the fst m 1 clustes. Futhe, must be no geate than 1 snce othewse the m cluste would not meet the constant. Thus ( m 1) 1 1. It s evdent that D[ 1, m 1] must be the optmal wthnss fo the fst 1 ponts n m 1 clustes fo othewse one would have a bette soluton to D[, m ]. Ths establshes the optmal substuctue fo dynamc pogammng and leads to the ecuence equaton n D[, m] mn { D 1, m 1 d x,, x}, 1 n, 1 m ( m1) 1 1 whee d( x,, x ) s the sum of squaed dstances fom x,, x to the mean. The matx s ntalzed as D d x1 x [,1] (,, ), that s, wthnss fo clusteng subects nto one cluste, equals to the sum of squaed dstances fom all obsevatons to the mean. Usng the above ecuence, we can obtan D[ n, m] the mnmum wthnss f all n numbes ae clusteed nto m goups, wth mnmum RMSE n, m Dn, m. n m In ode to make the pogam moe effcent, as suggested n Wang and Song (2011), the values d( x,, x ) n the ecuence d( x,, x) can be computed pogessvely and stoed based on d( x,, x 1). Usng a geneal ndex fom to, we teatvely compute x 2 ( ) d( x,, x ) d( x,, x 1 ) ( x, 1), wth, 1 1, 1, whee, s the mean of ( x,, x ). To fnd a clusteng of data wth mnmum wthnss of D, an auxlay matx B s defned to ecod the ndex of the smallest numbe n cluste m n n n B[, m] agmn { D 1, m 1 d x,, x},1 n,1 m ( m1) 1 1 Then we backtack fom Bn, kto obtan the statng and endng ndces fo all clustes and geneate an optmal soluton to the k-means poblem. EXAMPLE Eghteen pmay cae pactces wee ecuted as eseach stes unde the auspces of the Vgna Ambulatoy Cae Outcomes Reseach Netwok (ACORN), a pactce-based eseach netwok. Befoe conductng the nteventon, each pactce s baselne ate of povdng cessaton counselng by suveyng a coss-sectonal sample of vstng smokes was detemned. The pactces wee then clusteed accodng to the baselne ate. Snce the study was a two-am study wth teatment and contol, the clusteng had the constant that evey cluste has at least two subects. A goup of 18 pactces can be goups n 1 to 9 clustes as pesented n Table 1. The wthn-cluste sum squaes and RMSE ae also pesented. The esults ndcate that clusteng pactces nto 8 clustes can acheve smallest RMSE, wth 3 pactces n fst cluste and sxth cluste, and two pactces n of the emanng clustes. Note that RMSE dops lttle fom 5 to 8 clustes. Fo ths study, 5 clustes wee used. Also note that the 9 th cluste s RMSE values ae geate than those fo the 8 th cluste. Such a skewed-u shape n the cluste RMSE plot s typcal when cluste sze s constaned (Fgue 1). 2

3 Numbe of Clustes Cluste Wthnss RMSE Cluste Sze Table 1: Wthnss, RMSE, and Cluste Sze fo Optmal Pattons of the Study Baselne Data Fgue 1: Example study: RMSE by the Numbe of Clustes. Numbe of Clustes Baselne ID Cluste Numbe Table 2: Optmal Pattonng of 18 Pmay Cae Pactces Into 1-9 Clustes The detals of how to assgn each pactce nto clustes ae pesented n Table 2. The pattonng fo a sngle cluste s obvous and s not pesented. Fo example, the column of Cluste 5 shows how to cluste 18 pactces nto 5 clustes wth the constant that each cluste has at least 2 subects: 5 pactces wth ID 7, 5, 4, 13 and 9 wee 3

4 gouped nto cluste 1, 2 pactces wth ID 6 and 17 wee gouped nto cluste 2, 3 pactces wth ID 12, 10 and 16 wee gouped nto cluste 3, 6 pactces wth ID 11, 2, 14, 18, 8 and 15 wee gouped nto cluste 4, and the emande, ID 3 and 1, wee gouped nto cluste 5. Ths was the patton used n the example study. THE MACRO OVERVIEW The algothm shown above s mplemented wth SAS/IML and povded as a maco, ConClus. The use supples the nput dataset name, vaable on whch clustes ae desed (should be an odnal numec vaable), vaable that can help connect cluste esult wth ognal dataset (e.g., ID o post-teatment measues), and the constant (e.g., numbe of study ams). Two datasets ae ceated. Cluste contans the optmal cluste pattons gven the constant fo all feasble numbe of clustes (example: Table 2). RMSE contans the wthnss, RMSE, and the cluste szes fo each of the pattons (example: Table 1). SYNTAX %ConClus(baselne, <optons>) Optons EXAMPLE baselne Name of numec vaable on whch the cluste patton s desed. Ths s equed and must be lsted fst. The vaable should contan values of an odnal measue. ID=< d-vaable> Name of vaable used to help connect cluste esult wth ognal dataset. DATA=<SAS-data-set> Name of nput data set. Defaults to the last data set, _last_. CONSTR=< > Postve ntege-valued constant. Defaults to 1 and must be a dvso of the numbe of obsevatons. data Example; nput Baselne datalnes; ; un; * Patton nto clustes wth two-am study; %ConClus(Baselne,d=ID,data=Example,const=2) CONCLUSION A modfed heachcal clusteng algothm was povded as a SAS maco fo dentfyng clustes of unvaate odnal data that not only guaantees optmalty but also places the desed constant on the mnmum numbe of subects n each cluste. The maco povdes two data sets: one descbes how many subects n each cluste and the mnmum wthn-cluste sum squaes and RMSE that can be acheved fo each numbe of clustes, and the othe one descbes how to goup the subects nto dffeent numbe of clustes to acheve mnmum RMSE. The algothm pesented hee can only be appled to one-dmensonal data. An algothm that eaches optmalty, s epeatable, and meets constants on cluste sze emans to be developed fo multdmensonal data. A futue veson of ths maco wll assst wth andomzaton of subects acoss study ams statfed by cluste. 4

5 REFERENCES Lloyd, S.P. (1982), Least Squaes Quantzaton n PCM. IEEE Tansactons on Infomaton Theoy, 28(2): Pak, M and Johnson, RE (2004), Methods fo Matchng Clustes on Baselne Outcome Measues Po to Randomzaton Poceedngs of Jounal of the Amecan Statstcal Assocaton, , Alexanda, VA: Amecan Statstcal Assocaton. Pak, M and Johnson, RE (2005), Statfcaton on Baselne Measue: The Vaance Effect Poceedngs of Jounal of the Amecan Statstcal Assocaton, Vol.4, Alexanda, VA: Amecan Statstcal Assocaton. Pak, M and Johnson, RE (2006), Desgn and Analyss Methods fo Cluste Randomzed Tals wth Pa-Matchng On Baselne Outcome: Reducton of Teatment Effect Vaance Poceedngs of Jounal of the Amecan Statstcal Assocaton, Alexanda, VA: Amecan Statstcal Assocaton. Rothemch, SF, Woolf, SH, Johnson RE et.al (2008) Effect on Cessaton Counselng of Documentng Smokng Status as a Routne Vtal Sgn: An ACORN Study. Annals of Famly Medcne, Vol. 6, No.1, Wang, H and Song, M (2011), CKmeans.1d.dp: Optmal k-means Clusteng n One Dmenson by Dynamc Pogammng. The R Jounal, Vol. 3/2, Decembe 2011 CONTACT INFORMATION You comments and questons ae valued and encouaged. Contact the autho at: Name: Fengao Hu Entepse: Geoga Regents Unvesty Addess: th Steet, Depatment of Bostatstcs Cty, State ZIP: Augusta, GA, Wok Phone: Fax: E-mal: fhu@gu.edu SAS and all othe SAS Insttute Inc. poduct o sevce names ae egsteed tademaks o tademaks of SAS Insttute Inc. n the USA and othe countes. ndcates USA egstaton. Othe band and poduct names ae tademaks of the espectve companes. 5