Sensitivity of health-related scales is a non-decreasing function of

Transcription

1 Senstvty of health-related sales s a non-dereasng funton of ther lasses. Vasleos Maroulas and Demosthenes B. Panagotaos 2,* Insttute for Mathemats and ts Applatons, Unversty of Mnnesota, Mnneapols, USA. Emal: maroulas@ma.umn.edu 2 Offe of Bostatsts & Epdemology, Department of Nutrton Sene - Detets, Haroopo Unversty, 70 Eleftherou Venzelou Str., 76 7, Athens, Greee. Emal: dbpanag@hua.gr Tel., & Fax., *orrespondng author

2 Abstrat. In bomedal researh the use of dsrete sales whh desrbe haratersts of ndvduals are wdely appled for the evaluaton of lnal ondtons. However, the number of lasses (parttons) used n a dsrete sale has never been mathematally evaluated aganst the auray of a sale to predt the true ases. Ths wor, usng as auray marers the senstvty and spefty, revealed that the number of lasses of a dsrete sale affets ts estmatng ablty of orretly lassfyng the true dseased. In partular, t was proved that the senstvty of sales s a non-dereasng funton of the number of ther lasses. Ths result has partular nterest n lnal researh provdng a methodology for developng more aurate tools for dsease dagnoss. Key words: rs; health; senstvty. 2

3 . Introduton In bomedal lterature, dsrete varables, usually named sales or ndes, are wdely used for desrbng ndvduals haratersts, e.g. the level of depressve and anxety symptoms, the qualty of det, the sense of pan, for the evaluaton of varous lnal ondtons or behavors, le obesty, hypertenson, dabetes, ardovasular dsease rs (Kant 996, Kourlaba and Panagotaos 2009). Despte the fat that health-related sales are an mportant tool for the assessment of the relatonshps between ndvduals haratersts and ther health status, the methodology of ther onstruton s not utterly larfed. For example, t s unnown whh s the optmal number of values of the sale (.e., bnary, small range, suh as <5 lasses or large range), f ontnuous ranng (.e.,, 2, 3, 4, ) or dsontnuous (.e.,, 2, 5, 6, 7, 0, ) s better, whether speal weghts are needed for eah lass and other ssues that may mprove the dagnost auray of the sale n evaluatng a health outome. At ths pont t should be noted that ategorzaton of a ontnuous varable results n a loss of nformaton and onsequently, reduton n the ablty of the ndependent varable to aurately predt the dependent. At frst, the latter has not analytally been evaluated n terms of dagnost ablty of a sale; whle, seondly, n real prate use of ontnuous sales to measure ndvdual s behavoural attrbutes s dffult to apply n prate (e.g., t s dffult or even mpossble to aurately measure emotons, sense of pan, et usng ontnuous sales, beause of the nreased lelhood of mslassfaton). Thus, the use of dsrete, and partularly small-range (beause of potental mslassfaton) sales s usually the best approah used n prate. Reently, and based on both smulated and empral data t has been shown that nreasng the number of lasses, the senstvty (Se) of the sale s nreased, too (Kourlaba and Panagotaos 2009b). Spefally, usng omposte sales that are summatons of 3

4 dsrete random varables t was revealed that the dagnost auray of the sale nreases as the number of lasses of sale omponents nreases, as well. Moreover, t was observed that the use of ontnuous omponents s the optmal for ahevng the maxmum dagnost ablty. Therefore, the fndngs of ths wor are of major mportane for rs assessment tools n related researh sne they strongly suggest usng as many ategores as t an be used for a sale omponent. However, to the best of our nowledge, a mathematal proof of the aforementoned hypothess has never been presented.. Am The hypothess tested n ths wor was whether the number of lasses of an ordnal, dsrete varable, X, affets ts ablty for better predtng a response bnary varable Y that plays the role of a response outome. The predtng ablty was evaluated through the senstvty, Se, of the sale. The strategy followed was by onsderng two random dsrete varables X and X wth and lasses, respetvely; and one random varable Y from the Bernoull dstrbuton. Defnng (Se) P[X > Y] and (Se) P[X > Y] the senstvty of the X and X varables, respetvely ( and are the thresholds where the probablty of havng Y s maxmzed), we prove that (Se) (Se). 2. Proof Ths seton provdes a proof of our statement. A mathematal defnton of the senstvty s gven below. Defnton : Let Y be a random varable dstrbuted va Bernoull(p). For a random varable X, postvely assoated wth p P[Y], the senstvty of X, Se(X), n relaton to Y s defned as the ondtonal probablty, 4

5 Se(X) P[X > Y], () where s some onstant. Remar : The spefty of X, Sp(X), n relaton to Y, s another rteron for measurng the predtng ablty of a model. The spefty s defned as the ondtonal probablty, Sp(X) P[X< Y0]. The value, used n (), s a threshold that belongs to the doman of X. In prate, s the value of X that maxmzes the jont probabltes {Se(X), -Sp(X)}. Let (Ω, P) be a probablty spae, and X : Ω D be a dsrete random varable, where D {x, x 2,...,x }. Gven Y as n Defnton, the senstvty s gven n terms of the dseased populaton. Let denote the ondtonal probablty that the dsrete random varable X taes a fxed value x gven the dseased populaton,.e. P[X x Y ]. Hene aordng to () the senstvty wll be equal to, ( ) X Se (2) Proposton : For any N, the senstvty, Se(X ), s a non-dereasng funton as a funton of. Proof: It needs to be establshed that for any N Se(X ) Se(X ), (3) and aordng to (2) t s suffent to show that κ κ κ (4) The equalty n (3) holds when s hosen to be the medan value for any N. Consder a ()-partton, then - a, (5) for some onstant a,,...,. Furthermore one may argue wthout loss of generalty 5

6 that. Case :. Aordng to (5), ( ) a a κ κ And thus (4) has been establshed. Case 2: <. Assumpton : There exsts b R suh that b Hene from (5) and Assumpton, we have that ( ) a b (6), Furthermore, from (5) and (6) and onsderng a b for any,...,, one may observe that a (7) Hene (6) and (7) omplete the statement. Remar 2: Assumpton s requred as a ontrol property for the probabltes whh are exluded from the ()-partton, but nluded n the -partton. 3. Dsusson The am of ths wor was to nvestgate whether the number of lasses of a sale affets ts dagnost auray. It was proved that the senstvty of a sale, whh s a measure of dagnost auray that shows how orretly the sale lassfes the true dseased n medal researh, s a non-dereasng funton of the number of sale s lasses. Health measurement sales, and partularly the omposte ones, are useful 6

7 nstruments n bomedal researh, sne they an quantfy haratersts of the ndvduals that are dffult to measure, le behavors, atttudes and belefs (.e., psyhologal symptoms le depresson, severty of a dsease, health-related qualty of lfe, detary habts et). Despte the mportane of these tools no spef methodology has been proposed for ther development. One of the major unresolved ssues s the hoe of the number of lasses a sale should have. Untl now several sales have been developed usng dfferent methodologes, n order to measure the same haraterst, wth a varety of sorng systems, startng from a bnary odng to wder range. But, n all ases the hoe of the number of lasses was arbtrary. In a reent publaton by Kourlaba and Panagotaos based on four smulaton studes t was revealed that senstvty of a sale s a nreasng funton of sale lasses (Kourlaba and Panagotaos, 2009b). In partular, a ontnuous sale was ntally developed and afterwards thrteen other sales usng 800-tles, 500-tles, 200-tles, 00-tles, 50-tles, 5-tles, perentles, 8-tles, 6-tles, quntles, quartles, tertles and medan of the omponents of the ntal sale. All sales were tested aganst a bnary response outome. Results based on,000 smulated data sets revealed that the maxmum senstvty of the sale was obtaned when we use the maxmum number of lasses. However, smulatons studes, although they an gve an mpresson about the true relatonshps are always prone to bas. Thus, n ths wor, t was establshed that the senstvty of a sale s a nondereasng funton of the number of lasses used (or n other words the number of the values of the dsrete varable) under a tehnal assumpton (.e., Assumpton ). Ths assumpton provdes a ontrol on the mass of ponts whh are onsdered n the -partton, but exluded n the ()-partton. It s open f ths partton ould atually mprove the predtng ablty of a smulated model and of an empral paradgm. At ths pont t should be noted that sales usng small number of lasses may be 7

8 more omprehensble and easer appled n daly lnal prate; however, aordng to the smulated and proved results these sales result n low dagnost auray. The use of small-range sales ould also be an explanaton for the weaness of some studes to detet a sgnfant assoaton between a health measurement sale and a dsease outome, where the assoaton between what the sale ams to quantfy and the evaluated outome has long been understood (MCullough ML, 2000, Harna, 2002). In the prevously mentoned studes, small-sale ndes whh had been reated usng omponents wth only two lasses had been used. 4. Referenes. Kant, AK. (996). Indexes of overall det qualty: a revew. Journal Ameran Detet Assoaton 96: Kourlaba, G, Panagotaos, DB. (2009). Detary qualty ndes and human health: a revew. Maturtas 62: Kourlaba, G, Panagotaos, DB. (2009) The dagnost auray of a omposte ndex nreases as the number of parttons of the omponents nreases and when spef weghts are assgned to eah omponent. Journal of Appled Statsts 37: MCullough ML, Fesanh D, Stampfer MJ, Rosner BA, Hu FB, Hunter DJ, Varyam JN, Coldtz GA, Wllett WC. (2000) Adherene to the Detary Gudelnes for Amerans and rs of major hron dsease n women. Ameran Journal Clnal Nutrton 72: Harna L, Nodemus K, Jaobs DR, Jr., Folsom AR. (2002) An evaluaton of the Detary Gudelnes for Amerans n relaton to aner ourrene. Ameran Journal Clnal Nutrton 76,