USING DIFFERENTIAL GEOMETRIC LARS ALGORITHM TO STUDY THE EXPRESSION PROFILE OF A SAMPLE OF PATIENTS WITH LATEX-FRUIT SYNDROME

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1 Electronc Journal of Appled Statstcal Analyss EJASA (211), Electron. J. App. Stat. Anal., Vol. 4, Issue 2, e-issn , DOI /275948v4n2p Unverstà del Salento USING DIFFERENTIAL GEOMETRIC LARS ALGORITHM TO STUDY THE EXPRESSION PROFILE OF A SAMPLE OF PATIENTS WITH LATEX-FRUIT SYNDROME Lug Auguglaro*, Angelo Marcello Mneo Dpartmento d Scenze Statstche e Matematche, Unversty of Palermo, Italy Receved 29 September 21; Accepted 1 September 211 Avalable onlne 14 October 211 Abstract: Natural rubber latex IgE-medated hypersenstvty s one of the most mportant health problems n allergy durng recent years. The prevalence of ndvduals allergc to latex shows an assocated hypersenstvty to some plantderved foods, especally freshly consumed frut. Ths assocaton of latex allergy and allergy to plant-derved foods s called latex-frut syndrome. The am of ths study s to use the dfferental geometrc generalzaton of the LARS algorthm to dentfy canddate genes that may be assocated wth the pathogeness of allergy to latex or vegetable. Keywords: Latex-frut syndrome, varable selecton, penalzed regresson, hgh dmensonal, LARS 1. Introducton Natural rubber latex (NRL) s a product made prmarly from the rubber tree, Hevea braslenss. NRL s used n more than 4 medcal devces, such as gloves, catheters, face masks, stethoscope, et cetera. NRL s also used n other nonmedcal products, such as baloons, condoms, clothes, shoe soles, et cetera. IgE-medated allergy to NRL s a sgnfcant health problem n ndustralzed contres, especally among health care workers, patents wth congental malformatons, especally chdren wth spna bfda, and chldren wth a hstory of multple surgcal nterventons [5]. Accordng to some researchs, between 1% and 17% of medcal personnel n Europe and US are beleved to be senstve to NRL. The prevalence of NRL allergy n patents wth spna bfda s about 5% n ndustralzed countres, whle s close to n non * Correspondng author. E-mal: lug_auguglaro@unpa.t 227

2 Usng dfferental geometrc LARS algorthm to study the expresson profle of a sample of patents wth latex-frut syndrome ndustralzed countres. Moreover, senstvty was found n about 34% of chldren wth a hstory of 3 or more surgcal nterventons, that are a rsk factor for the development of NRL allergy n chldren but not n adults [5]. Intal symptoms for people allergc to NRL are localzed tchng, erythema, or contact urtcara after few mnutes from the NRL exposure. Progressve senstzaton can also lead to generalzed urtcara, angoedema, rhnts, conjunctvts, asthma, and anaphylactc shock mnutes after dermal or mucosal contact wth NRL protens. An ncreasng number of ndvduals allergc to NRL reports severe reactons to latex, ncludng generalzed urtcara, bronchospasm, and hypotenson. Approxmately 3-5% of ndvduals who are allergc to NRL show an assocated hypersenstvty to some plant-derved foods, especally fresh frut. Ths assocaton of allergy to latex and allergy to plant-derved foods s called latex-frut syndrome. An ncreasng number of plant sources, such as avocado, banana, chestnut, kw, peach, tomato, potato and bell pepper, has been assocated wth ths syndrome [15]. Am of ths paper s to dentfy a set of genes to screen the molecular profles of patents wth allergy to latex from patents wth allergy to frut. In lterature several methods have been proposed to dentfy genes that can be used to dscrmnate between two or more groups. For ths knd of problems the number of varables, say p, can be much larger than the sample sze n. In ths case, t s often assumed that only a small number of varables (genes) contrbutes to the response, whch leads to assume the sparsty of the model. For sparse model we mean a generc regresson model wth the coeffcent vector sparse,.e. wth many elements equal to zero. In ths feld, many varable selecton technques for hgh dmensonal statstcal models are based on the penalzed lkelhood approach. Some examples are the least absolute shrnkage and selecton operator (LASSO) estmator = argmn{ y-x 2-1 } proposed by Tbshran [14], where y-x 2 s the OLS loss functon, 1 s the L 1 -norm of the parameter vector and s a postve tunng parameter used to select the trade-off between sparsty of the estmated parameter vector and the predcton behavour of the model. Other mportant examples are the path followng algorthm proposed by Park and Haste [12] to estmate a generalzed lnear model wth L 1 -penalty functon and the smoothly clpped absolute devaton (SCAD) penalzed estmator proposed by Fan et al. [8], where the loss functon s y-x 2 and the penalty functon s a quadratc splne functon wth knots at and a, wth the tunng parameter and a a gven constant. In a recent paper, Efron et al. [7] ntroduced a new method to select mportant varables n a lnear regresson model called least angle regresson method (LARS). LARS algorthm can be descrbed as follows. Startng wth all coeffcents equal to, the LARS algorthm fnds the covarate that s most correlated wth the response varable and proceeds on ths drecton. Then, the algorthm takes the largest step possble n the drecton of ths covarate untl some other covarate has as much correlaton wth the current resdual. LARS algorthm proceeds n a drecton equangular between the two covarates untl a new covarate earns ts way nto the most correlated set (A) and then proceeds n the drecton that has an equal angle wth the three covarates untl a new covarate s ncluded n A, and so on. In recent years, there has been an enormous amount of research actvty devoted to automatc model-buldng algorthms: Yuan and Ln [16] have extended LARS algorthm to lnear regresson models wth grouped varables; Park and 228

3 Auguglaro, L., Mneo, A.M. (211). Electron. J. App. Stat. Anal., Vol. 4, Issue 2, Haste [12] have proposed a path followng algorthm for generalzed lnear models wth L 1 - penalty functon; Rosset and Zhu [13] studed the generc regularzed optmzaton problem for almost quadratc loss functons wth L 1 -penalty. These algorthms are based on the use of the L 1 - penalty functon to defne a path n the parameter space. Am of ths paper s to dentfy a set of dfferentally expressed genes usng the dfferental geometrc LARS (dglars)[3]. As we shall see n the followng of ths paper, the geometrcal theory underlyng the dglars can be used to extend the sure ndependence screenng method [9] used when we have an ultra-hgh dmensonal feature space (p>>n), as suggested by Auguglaro and Mneo [2]. 2. Dfferental Geometrc LARS (dglars) algorthm Let Y=(Y 1, Y 2,..., Y n ) T be a random varable vector havng a probablty densty functon p Y (y;, ) = a(y; ) exp{ (y T k ())}, yyr n, (1) wth respect to a σ-fnte measure on R n, where a(. ) and k(. ) are specfc gven functons, the canoncal parameter vares n the subset ΘR n and vares n a subset Λ of R +. The model (1) s called exponental dsperson model [1]. We have chosen to use ths model for ts hgh flexblty and for ts very general probablstc assumptons. We denote the mean value of Y by µ Y =µ(). Standard theory on the exponental dsperson model tells us that the mean value and the canoncal parameter are related to the gradent of the functon k(. ), namely, µ()= k()/ (see [1] for more detals), whch s called mean value mappng. Snce µ(.) s a one-to-one functon from ntθ onto =µ(ntθ), the exponental dsperson model may be parameterzed by (µ;σ 2 ), where σ 2 = -1 s called dsperson parameter. In what follows we shall assume that the dsperson parameter s fxed. Under ths assumpton, followng Amar [1], the parameter space can be treated as a n-dmensonal Remannan manfold where µ Y plays the role of coordnate system and the Fsher nformaton matrx s a Remannan metrc. A generalzed lnear model s completely specfed by the followng assumptons: a) y=(y 1, y 2,..., y n ) T s a set of n ndependent observatons taken from (1); b) for each random varable Y we have a column of covarates x =(x 1, x 2,, x p ) T XR p, wth p<n. These covarates are related to the mean value of Y by a known functon such that µ =f(x T ), where BR p ; n order to smplfy our notaton, we denote µ()=(f(x 1 T ), f(x 2 T ),, f(x n T )) T. We assume that µ(. ) s an embeddng wth doman B; c) the dsperson parameter σ 2 does not depend on the vector of covarates. Gven the assumpton (b), µ(b)= B s a Remannan submanfold of, then we can generalze the noton of angle between two gven vectors. Let () be a dfferentable curve. Followng Kass and Vos [11], we have that the dervatve of the log-lkelhood functon L (()) wth respect to β s gven by L( μ( ; r( (2) μ( 229

4 Usng dfferental geometrc LARS algorthm to study the expresson profle of a sample of patents wth latex-frut syndrome where.;. γ μ( )) s the nner product between the current resdual vector r (()) and the -th base of the tangent space of B at µ(()). Usng expresson (2) we have the followng dfferental geometrc dentty L( cos( ( ) r( μ( μβ ( ( μβ ( ( cos( ( ) r( ( I ( ) μβ ( ( 1/ 2 (3) where ( s the local angle between r (()) and μ(, I (()) s the expected Fsher nformaton for () and s the norm defned on the tangent space of B at µ(()). μ( Condton (3) shows that the gradent of the log-lkelhood functon does not generalze the noton of equangular condton, snce we are not consderng the varaton related to (I (())) 1/2. To overcome ths problem, Auguglaro and Wt [3] propose a generalzaton of the LARS algorthm based on the followng condton: r u 1/ 2 ( )) ( I ( ))) L( )) cos( ( ) r( μ( A (4) u where r ( )) s the -th Rao score statstc evaluated n () and A s the actve set, namely, the set of ndces of covarates that are ncluded n the actual model. Condton (4) s called generalzed equangularty condton and t s used n [3] to defne a method that generalzes LARS to generalzed lnear models. Ths method s called dglars and the nterested reader can refer to [3] for more detals about t. When we work n a ultra-hgh dmensonal feature space, namely p>> n, the followng expresson u r ( )) r( )) 2 2 μ( )) = cos 2 (ρ (())) (5) can be used to defne a genune generalzaton of the sure ndependence screenng method [9] for generalzed lnear models. Usng expresson (5), Auguglaro and Mneo [2] have proposed the followng method to dentfy the relevant varables n a generalzed lnear model defned n a ultra-hgh dmensonal feature space. Followng Fan and Lv [9], we consder ω=(cos 2 (ρ 1 ( ˆ )), cos 2 (ρ 2 ( ˆ )),, cos 2 (ρ p ( ˆ ))) T, where ˆ s the maxmum lkelhood estmate of the parameter of a generalzed lnear model wth only the ntercept. For a gven value d<n, we sort the p component-wse magntudes of the vector ω n decreasng order and defne the submodel: M d ={{1, 2,, p} : cos 2 (ρ ( ˆ )) s among the frst d largest of all the correspondng values}. 23

5 Auguglaro, L., Mneo, A.M. (211). Electron. J. App. Stat. Anal., Vol. 4, Issue 2, Ths s a straghtforward way to reduce the dmensonalty from p to d. Then, submodel M d s studed by usng the dglars algorthm. 3. Applcaton of dglars algorthm to a latex-frut allergy data set In oder to dentfy a set of genes to screen the molecular profles of patents wth allergy to latex from patents wth allergy to frut, a logstc regresson model estmated by usng the dglars algorthm was appled to a DNA mcroarray data set. From a sample of 6 patents allergc to latex and from a sample of 5 patents allergc to vegetable food, perpheral blood mononuclear cells were solated on fcoll gradent. Each sample was hybrdzed on Affymetrx human focus array and data were processed wth Affymetrx MAS 5. software. These data sets are avalable from the nternet ste of the Natonal Center for Botechnology Informaton (NCBI), n the Gene Expresson Omnbus (GEO) archves (URL: The data sets reference s GSE The two sample szes are very low, but usually ths s the dmenson of the sample sze of studes conducted n ths feld, snce n the World the people affected from the latex-frut syndrome are not so many and t s very dffcult to lead a medcal study wth a bg number of patents. Anyway, n spte of the sample sze, we thnk that the followng analyss s nterestng n ts own rght. The expresson profle of 8746 genes was used for the logstc regresson model estmated by usng the dglars algorthm. As suggested by Fan and Lv [9] a value of d = [n/log(n)] = 4 was used for the sure ndependence screenng step. Fgure 1. Panel (a) shows the densty of the Rao score test statstcs evaluated at the pont β ). Panel (b) shows the path of the Rao score test statstcs as functon of the number (k) of the algorthm Panel (a) n fgure 1 shows the densty of the Rao score test statstcs evaluated at the startng pont ( ), whle panel (b) shows the path of the Rao score test statstcs. The number of varables wth non-zero coeffcents was selected usng the Akake Informaton Crteron (AIC) and the Bayesan Informaton Crteron (BIC). ˆ( γ 231

6 Usng dfferental geometrc LARS algorthm to study the expresson profle of a sample of patents wth latex-frut syndrome Fgure 2. Panel (a) shows AIC and BIC used to select the number of predctors. Panel (b) shows the coeffcents path of the selected varables In ths settng, the Akake Informaton Crteron (AIC) s defned as AIC = - 2 L( A ()) + 2 A() where A() s the cardnalty of the actve set, whle the Bayesan Informaton Crteron (BIC) s defned as BIC = - 2 L( A ()) + log(n) A(). Panel (a) n fgure 2 shows AIC and BIC values as functon of the number of varables wth nonzero coeffcents. Usng these measures of goodness of ft only 3 varables were used for further analyss. Panel (b) shows the coeffcent paths of the selected varables. Ths set of genes was also dentfed usng the logstc regresson model wth L 1 -penalty functon. The dentfed genes are the nterleukn 13 (IL-13), the nterleukn 4 (IL-4) and chemokne (C-X-C motf) receptor 6 (CXCR6). In partcular, the nterleukn 13 (IL-13) and the nterleukn 4 (IL-4) are closely related wth the latex-frut syndrome, snce the products of these genes play a crtcal role n allergc reacton regardng antgen presentaton, IgE synthess, and actvaton of mast cell, respectvely (see [6] and [4] for further detals). The majorty of food allergen-specfc CD4 + T lymphocytes solated from food-allergc ndvduals was found to synthesze hgh levels of IL-4 and IL Conclusons In ths paper we have used the dfferental geometrc generalzaton of the LARS algorthm proposed by Auguglaro and Wt [3] and lnked to the sure ndependence screenng method by Auguglaro and Mneo [2] to select a set of genes to dscrmnate between patents wth latex allergy and wth frut allergy. Latex-frut syndrome s a well-defned dsorder affectng from 2% to 6% 232

7 Auguglaro, L., Mneo, A.M. (211). Electron. J. App. Stat. Anal., Vol. 4, Issue 2, of patents wth latex allergy. A number of envronmental and genetc factors seems to contrbute to the latex-senstvty phenotype. Although exposure to natural rubber latex products s necessary for senstzaton, t s not suffcent. A number of other envronmental and genetc factors seems to contrbute to the latex-senstve phenotype. Known rsk factors for latex allergy nclude an atopc hstory, concomtant food allerges, and delayed skn reactons to NRLcontanng products. Although there s overwhelmng support for a genetc component for allergc dsease, the multgenc nature of the phenotype has made the dentfcaton of susceptblty genes a dffcult task. A large number of studes have explored general rsk factors for allergc dsease by means of the canddate gene approach or genome-wde analyses whch are based on the penalzaton methods, such as the generalzed lnear model wth the L 1 -penalty functon or the rdge regresson. In ths paper, we have seen that the dglars method can also be used wth good results. In spte of the sample sze very low, our analyss dentfes two of the well known genes that are related wth the latex allergy, namely the nterleukn 4 (IL-4) and the nterleukn 13 (IL-13), that several studes have dentfed. Products of these genes play a crtcal role n allergc reacton regardng antgen presentaton, IgE synthess, and actvaton of mast cell, respectvely. Acknowledgement We want to thank the Unversty of Palermo for partally supportng ths research. We want also to thank an anonymous referee for hs/her valuable comments. References [1]. Amar, S.I. (1985). Dfferental-Geometrcal Methods n Statstcs. New York: Sprnger-Verlag. [2]. Auguglaro, L., Mneo, A.M. (29). Applyng Dfferental Geometrc LARS Algorthm to Ultrahgh Dmensonal Feature Space. In Actes de XVIemes Rencontres de la Socété Francophone de Classfcaton, Grenoble, 2-4 September 29, [3]. Auguglaro, L., Wt, E.C. (29). Generalzng LARS Algorthm Usng Dfferental Geometry. In Book of Short Papers of the 7 Meetng of the Classfcaton and Data Analyss Group of the Italan Statstcal Socety, Catana, 9-11 September 29, [4]. Blanco, C., Sanchez-Garca, F., Torres-Galvan, M.J., Dumperrez, A.G., Almeda, L., Fgueroa, J., Ortega, N., Castllo, R., Gallego, M.D., Carrllo, T. (24). Genetc bass of the latex-frut syndrome: Assocaton wth HLA class II alleles n a Spansh populaton. Journal of allergy and clncal mmunology, 114, [5]. Brehler, R., Kuttng, B. (21). Natural rubber latex allergy: a problem of nterdscplnary concern n medcne. Archves of nternal medcne, 161, [6]. Brown, R.H., Hamlton, R.G., Mntz, M., Jedlcka, A.E., Scott, A.L., Kleeberger, S.R. (25). Genetc predsposton to latex allergy: role of nterleukn 13 and nterleukn 18. Anesthesology, 12, [7]. Efron, B., Haste, T., Johnstone, I., Tbshran, R. (24). Least angle regresson. The Annals of Statstcs, 32,

8 Usng dfferental geometrc LARS algorthm to study the expresson profle of a sample of patents wth latex-frut syndrome [8]. Fan, J., L, R. (21). SCAD, Penalzed lkelhood, Oracle estmator. Journal of the Amercan Statstcal Assocaton, 96, [9]. Fan, J., Lv, J. (28). Sure ndependent screenng for ultrahgh dmensonal feature space. Journal of the Royal Statstcal Socety seres B, 7, [1]. Jørgensen, B. (1987). Exponental dsperson models. Journal of the Royal Statstcal Socety seres B, 49, [11]. Kass, R.E., Vos, P.W. (1997). Geometrcal Foundaton of Asymptotc Inference. New York: John Wley. [12]. Park, M.Y., Haste, T. (27). L 1 -regularzaton path algorthm for generalzed lnear models. Journal of the Royal Statstcal Socety seres B, 69, [13]. Rosset, S., Zhu, J. (27). Pecewse lnear regularzated soluton paths. The Annals of Statstcs, 35, [14]. Tbshran, R. (1996). Regresson Shrnkage and Selecton va the Lasso. Journal of the Royal Statstcal Socety seres B, 58, [15]. Wagnerand, S., Breteneder, H. (22). The latex-frut syndrome. Bochemcal Socety Transactons, 3, [16]. Yuan, M., Ln, Y. (26). Model selecton and estmaton n regresson wth grouped varables. Journal of the Royal Statstcal Socety seres B, 68,