A New Proposed Statistically-Derived Compromise Cut-off CD4 Count Value for HIV Patients to Start using ARVs

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1 Amerca Joural of Appled Sceces Orgal Research Paper A New Proposed Statstcally-Derved Compromse Cut-off CD4 Cout Value for HIV Patets to Start usg ARVs Mara Mokgad Lekgayae ad Solly Matshosa Seeletse Departmet of Statstcs ad Operatos Research, Sefako Makgatho Health Sceces Uversty, PO Box 07, MEDUNSA, 0204, Gauteg Provce, South Afrca Artcle hstory Receved: Revsed: Accepted: Correspodg Author: Mara Mokgad Lekgayae Departmet of Statstcs ad Operatos Research, Sefako Makgatho Health Sceces Uversty, PO Box 07, MEDUNSA, 0204, Gauteg Provce, South Afrca Emal:mara.lekgayae@smu.ac.za Abstract: The study vestgated the relatoshp betwee statstcal outlers ad the cosstet values of the CD4 cout recommeded for startg the Atretrovral Therapy (ART) by HIV-postve patets. Low CD4 couts mply a low mmue system. It could be due to AIDS exstece or closeess to death. A effectve treatmet to curb HIV mpact s ART, whch s recommeded for low CD4 couts. However, coutres dffer the values used. Developed atos recommed start of ART whe the CD4 s stll hgh order to curb ts developmet. Poor coutres use very low CD4 couts. Some coutres keep chagg the CD4 value for ths purpose. The problem s that whe CD4 couts are too low, HIV may already be too advaced, makg t dffcult to save the patet from progresso to full AIDS. There should be CD4 values derved usg scetfc methods to assst stadardzg the CD4 cout value for ART commecemet. Usg a retrospectve sgle-ste cohort study desg, the study aalyses the CD4 couts usg robust statstcal methods together wth covetoal statstcal methods to study outlers ad the derve a compromse CD4 cout value that could serve as the stadard cut-off startg pot for usg ART. The CD4 cout cofdece lmts showed outlers to be above 300 cells/mm 3. The lower boud of 0 caot happe to ay lvg perso. The 300 upper boud s a value wth the maageable outlers that has ot reached crtcal HIV state. However, ths value s ear rsky CD4 cout values. The CD4 cout of 300cells/mm 3 dcates deteroratg HIV. If t s set as the startg pot of takg ARVs, the patets volved ca be saved from reachg paful states of lower CD4 couts.hiv patets mmue systems at ths level ca stll be boosted wthout them showg physcal weakess from eye specto. Keywords: ART, CD4, HIV, Outler Itroducto Curbg HIV/AIDS s udoubtedly desrable. ART was suggested for every HIV postve patet (Lchterfeld ad Roseberg, 20). CD4 elcts ART use because low values sgal HIV spread ad mmue system harm. ART are useful for HIV postve patets, especally for CD4 below 350 cells/mm 3. Poor coutres recommed startg ART at 200 cells/mm 3. These values have also chaged may tmes. Studyg CD4 outlers s vtal to maage HIV. Large outlers dcate HIV severty. Statstcal modellg ca help to detect severtes HIV, whle robust methods help to uderstad the bulk of the data such that outlers do ot dstort meags (Aderse, 2008). Robust statstcal methods are specalsed methods that ca resst flueces of outlers (Tofalls, 2008). CD4 Outlers A outler s a observato lyg very far from other values a radom sample, ether beg ureasoably low or hgh (Alqallaf et al., 2009). CD4 outlers are partcularly mportat detfyg severe HIV status. Robust methods eable uderstadg the data eve whe outlers fluece aalyses. Outlers are ot always clearly vsble. Whe usg tradtoal statstcal methods, outlers may be masked ad the estmates of resduals may be flated (Maroa et al., 2006). I CD4 couts, outlers ca dstort actualtes ad severtes may be obscured. Hece, the use of robust methods matas 206 Mara Mokgad Lekgayae ad Solly Matshosa Seeletse. Ths ope access artcle s dstrbuted uder a Creatve Commos Attrbuto (CC-BY) 3.0 lcese.

2 meags from bulks of data whle detfyg aomales that may be revealg severtes. Robust Statstcal Methods Some effcet robust statstcal methods developed alog commo tradtoal methods are: Least Absolute Devato (LAD), Least Trmmed Squares (LTS), S- estmato ad M-estmators (Strutz, 200). Upgradg robust methods cludes MM-estmato whch combes the robustess of S-estmato wth the effcecy of M- estmato. Robust methods ca assst outler detfcato, whch s sometmes dffcult ad esurg that outlers do ot dstort the results (Dawso, 20). Outler Detecto Let Q ad Q 3 be the frst ad thrd quartles of a data set ad k a fece costat ormally chose to be ether.5 or 3. Accordg to Dovoedo (20), Tukey s boxplots boudares method defes outlers as observatos outsde the terval wth lower ad upper boudares: = Q k ( ) () L Q Q 3 = Q + k ( ) (2) U Q Q 3 3 HIV Treatmet ad Health The CD4 cout of a healthy perso rages from 500 to,200 cells/mm 3. Below these values for HIV postve patets are sgs for requrg ARVs. The U.S. Departmet of Health ad Huma Servces (HHS) provdes gudeles o usg ARVs ad other HIV medces to treat HIV fecto (Seleuou ad Souleymaou, 2009). These gudeles recommed that all HIV postve people should take ART ad emphasse that everyoe wth a CD4 cout below 350 cells/mm 3 should use ARVs. If a HIV-fected perso s CD4 cout drops rapdly or s below 200 cells/mm 3, startg ARV s mmet (Lchterfeld ad Roseberg, 20). A CD4 crease s a sg of mmue system recovery. Statstcal Techques Robust Statstcal Methods Least Absolute Devato Least Absolute Devato (LAD) s a robust statstcal optmzato procedure comparable to the geeral Ordary Least Squares (OLS) techque that approxmates a dataset. To desg the LAD problem, Wlcox (20) proposes a data set of pots (x, y ) wth =, 2,..., whch the problem s to fd a fucto f of a specfc form cotag parameters to be determed such that y f(x ). The approach s to search for estmated values of the ukow parameters that mmse the sum of the absolute values of the resduals: ( ) (3) S = y f x = M-Estmator M-estmators are a robust comprehesve class of estmators obtaed as the mma of sums of data fuctos (va de Geer, 2000). Cosderg a famly of probablty desty fuctos f parameterzed by θ, a MLE of θ s calculated for each data set by maxmsg the lkelhood fucto over the parameter space {θ}. Whe the observatos are depedet ad detcally dstrbuted, a MLEθ ˆ satsfes: ˆ θ = arg max, θ = f ( xθ ) (4) Least Trmmed Squares Least Trmmed Squares (LTS) s a robust statstcal method that fts a fucto to a data set (L, 2005). I OLS, the estmated parameter values β are those values that mmse the objectve fucto S(β) of squared resduals: S = r ( β ) 2 (5) = For a LTS aalyss, ths objectve fucto s replaced as follows: For a fxed value of β, let r (j) (β) deote the set of ordered absolute values of the resduals. It s sesble to detfy the outlers by usg Equato ad 2. LTS s obtaed from Equato 5 by removg the outlers. Measures of Accuracy Cosder the observatos y, y 2,...y ad the correspodg estmates yˆ ˆ ˆ, y2,... y geerated from a predcto model. The errors are: e = y ˆ y for =, 2,... (6) Thus, large devatos from the estmates wll gve large error values. Accuracy measures are defed usg these errors. Hece, large measure values mply less accuracy of the predcto model. The measures are the Cumulatve Forecast Error (CFE), Mea Absolute Devato (MAD), Mea Squared Error (MSE), Root Mea Squared (RMSE), stadard error (s e ), Mea Absolute Percetage Error (MAPE) (Elamr, 202) defed as: CFE = e (7) = 854

3 Ths measure adds the errors together ad ca be used to show f the model s over forecastg above the actual values (CFE<0) or uder forecasts (CFE>0): et = MAD = (8) The MAD measure s a average of the absolute errors. Smaller values of MAD show more accuracy of the predctor model whle larger oes mply less accuracy: 2 et = MSE = (9) The MSE measure s a average of the squared errors ad has useful mathematcal propertes related to the varace. Also, smaller MSE values show more accuracy of the predctor model: RMSE = MSE (0) The RMSE measure s the square root of the MSE ad the two measures are terpreted the same way: s = e e e ( ) 2 () = The s e measure s a stadard devato of the errors. Smaller values are also dcators of more model accuracy: 00 e MAPE = A (2) = The MAPE measure s a percetage of the absolute errors, wth smaller values also showg more predctor model accuracy. Statstcal Iferece Statstcal ferece refers to the process based o aalysed data to deduce propertes of a uderlyg dstrbuto (Held ad Bové, 203). It makes propostos about a populato usg data. Problems statstcal ferece are ofte related to statstcal modelg. Bascally, statstcal ferece coclusos are statstcal propostos such as pot estmate, terval estmate, rejecto of a hypothess ad clusterg of data pots to groups, amog others. I ths study, estmato s used ad the cut-off measure derved s a result of ths system. Methods Ths was a retrospectve cohort study o CD4 couts of HIV postve patets from Jauary 2006 to December 203 attedg treatmet at the Tshepag Clc of Dr. George Mukhar Academc Hosptal Gauteg Provce, South Afrca. Tshepag Clc receves patets from clcs surroudg towshps ad vllages. The clc ams to curb HIV advacemet ad mprove ART. The HIV patets records for the stated perod were about 350, whch made the study populato. The study used all the useful records of the etre populato. Thus the sample cossted of the remag records (38) after elmatg the usatsfactorly oes. Results CD4 Cout Statstcs Table shows abrdged CD4 couts. The bulk of the values rage from ad a otable amout s see from Few other values appear to be hgher tha 40. These may be cosdered caddates to be outlers. Fgure shows the spread the CD4 couts ad the umber of patets. The tal of values s show from 40 upwards. Ital Outler Idetfcato Equato ad 2 provde the lower ad upper lmts for outler detfcato. Let k =.5 to ehace stable cofdece tervals ad to avod L that s too deep the egatves. Usg the crude quartles from Table, the lower ad upper lmts are: ( ) = ( ) ( Q ) ( ) L = Q k Q3 Q = 00 U = Q + k Q = = There are o small outlers sce the L = 00 cells/mm 3 s below all the CD cout values Fg. ad Table. Actually, all CD4 couts 0. Sce L = 300, there are 28 CD4 couts detfed as (potetal) outlers class tervals above lower lmt 300 Table. Statstcs Groupg for ART Purpose Table 2 dcates 247 (77.7%) bulk of patets below 200 cells/mm 3, who should take ARVs. The ext larger group were betwee 200 ad 350 cells/mm 3, whch were also recommeded to take ARVs. Outlers ths case are oly 4, beg the healthy oes (>500 cells/mm 3 ) ad the less severe oe sat cells/mm 3. Table 3 dsplays the descrptve statstcs of the orgal CD4 cout values wth ther potetal outlers. Table 4 dsplays the orgal mea devato measures of the CD4 couts. Table 5 lsts the outlers detfed the box plot Fg. 2. Idetfyg Outlers Robust methods are used as there was evdece of outlers. The aalyss detfed 8 outlers whch were CD4 couts above 300 cells/mm 3. The box plot Fg. 2 detfes them. 855

4 Fg.. Shows the spread the CD4 couts ad the umber of patets. The tal of values s show from 40 upwards Table. CD4 couts CD4 Frequecy Percet Total Table 2. Grouped CD4 cout statstcs CD4 Frequecy 0 to to to or more 4 Total 38 Table 3. Descrptve CD4 cout statstcs Mea Stadard error Meda 7.5 Mode 30 Stadard devato Sample varace Kurtoss Skewess Rage 69 Mmum 2 Maxmum 62 Sum 4386 Cout 38 Cofdece level (95.0%) Table 4. Orgal mea devato measures of CD4 couts -.4E Fgure 2 dsplays the Box plot to detfy potetal CD4 outlers. The box plot cofrms the possblty of outlers. It also cofrms that there are o outlers below the lower lmt. However, stead of 8 outlers, 2 outlers detfed. Table 5. CD4 outlers #227 = 62 #309 = 539 #32 = 547 #268 = 509 #304 = 489 #222 = 479 #226 = 495 #29 = 439 #38 = 460 #303 = 430 #60 = 405 #253 = 408 Summary Statstcs The ext presetato cosders ew resampled data whe outlers have bee removed. Table 6 uses fewer resposes. Accordg to Equato 2, there are stll CD4 values above 300 cells/mm 3. Table 7 dsplays summary statstcs wthout outlers. Table 8 dsplays the orgal mea devato of CD4 couts. Table 9 dsplays the orgal mea devato of less CD4 couts. Table 0 dsplays the orgal mea devato of less CD4 couts ad VL outlers. Robust Statstcal Measures The robust measures assessed from the above aalyses are LAD, M-estmator ad LTS. Aalyss of Results Twelve (2) out of 4 measures were affected. Outlers therefore affected 85.7% (2/4) of the measures. Ths dcates the effect of outlers o the descrptve statstcs. Estmatg the Robust Measure Values Table 2 dsplays the orgal mea devato of CD4 couts less outlers ad leverage pots. 856

5 Fg. 2. Box plot of CD4 couts Table 6. Grouped CD4 couts: Outlers removed CD4 Frequecy 0 to to to Total 306 Table 7. Descrptve CD4 couts CD4 Mea Stadard error Meda Mode 30 Stadard devato Sample varace Kurtoss Skewess Rage 365 Mmum 2 Maxmum 367 Sum Cout 306 Cofdece level (95.0%) Table 8. Orgal mea devato measures of CD4 couts -.4E Table 9.Mea devato measures of CD4 couts less CD4 outlers -3E Table 0. Mea devato measures of CD4 couts less CD4 ad VL outlers Table. CD4 descrptve statstcs of dfferet codto Orgal No outlers Mea Std Error Meda Mode Std Dev Sample Var Kurtoss Skewess Rage Mmum 2 2 Maxmum Sum Cout % c. lev Table 2. Mea devato measures of CD4 couts less outlers ad leverage pots MAD MSE MAPE SE Data No outlers Dscusso Least Absolute Devatos LAD s the lowest sum of the absolute resdual values. From Equato 3, removg outlers leads to the lowest MAD Table 2. Thus, LAD = M-Estmators M-estmators are estmators obtaed as the mma of sums of fuctos of the data, obtaed usg Equato 4. These are equvalets of mmum MAPEs usg the 857

6 observed values as the dvsors, detfed Table 2. Hece, s M-est = Least Trmmed Squares LTS s a robust statstcal method that fts a fucto to a data set of data, whch curtals the sum of squared resduals Equato 5, also Table 2. The, LTS = The stadard error s smaller for No outlers row Table, whch seems to cofrm that the dea of removg outlers leads to uaffected statstcal results. Dscusso ad Cocluso The values obtaed for robust measures are LAD = ; LTS = as well as M est = The statstcal outlers obtaed from robust methods ad extreme values that are prescrbed the startg cut-off for startg to use ARVs dd ot dcate to have ay relatoshp. However, the adjustmet based o statstcal gudeles s possble ad ca be a useful adjustmet creasg ARV usage by HIV patets order to maage HIV. More specfcally, the value of 300 cells/mm 3 was obtaed for a upper boud. Recommedatos The study recommeds that: Health Authortes should revse the startg level of usg ARVs to 300 cells/mm 3 The study also recommeds that a study: Should be coducted determe the CD4 couts that relate to death stage of a HIV postve patet Smlar to ths oe should be udertake o a bvarate sample of CD4 couts ad vral load ad compare the results Ackowledgemet The authors would lke to ackowledge Dr M.M Motshwae for recrutg the frst author to postgraduate studes the Departmet. They would also lke to express ther apprecatos to the ukow referees for ther commets ad suggestos whch mproved the presetato of the paper. Fudg Iformato Ths research was supported ad fuded by Sefako Makgatho Health Sceces Uversty. Author s Cotrbutos Mara Mokgad Lekgayae: Coducted the study, cotrbuted to the study methodology, data aalyss ad terpretato of results. Wrote the paper, revewed, revsed edted ad falsed the mauscrpt. Solly Matshosa Seeletse: Supervsed the study, collected ad developed the tal mauscrpt. Desged the research pla ad orgased the study. Ethcs Ths artcle s orgal ad cotas upublshed materal. The authors have read ad approved the mauscrpt ad o ethcal ssues are volved. Refereces Alqallaf, F., S. Va Aelst, V.J. Yohaad R.H. Zamar, Propagato of outlers multvarate data. A Stat., 37: DOI: 0.24/07-AOS588 Aderse, R., Moder Methods for Robust Regresso. st Ed., SAGE Publcatos, Lodo, ISBN-0: , pp: 07. Dawso, R., 20. How sgfcat s a boxplot outler. J. Stat. Educ., 9: -2. Dovoedo, Y.H., 20. Cotrbutos to outler detecto methods: Some theory ad applcatos. Ph.D. Thess, Uversty of Alabama, Tuscaloosa, USA. Elamr, E.A.H., 202. O uses of mea absolute devato: Decomposto, skewess ad correlato coeffcets. METRON, 70: DOI: 0.007/BF Held, L. ad D.S. Bové, 203. Appled Statstcal Iferece: Lkelhood ad Bayes. st Ed., Sprger Scece ad Busess Meda, Berl, ISBN-0: , pp: 376. L, L.M., A algorthm for computg exact least-trmmed squares estmate of smple lear regresso wth costrats. Comput. Stat. Data Aal., 48: DOI: 0.06/j.csda Lchterfeld, M. ad E.S. Roseberg, 20.Atretrovral combato therapy markedly reduces rsk of heterosexual HIV- trasmsso. Evdece-Based Med., 7: PMID: Maroa, R.A., R.D. Mart ad V.J. Yoha, Robust Statstcs: Theory ad Methods. st Ed., Wley, Chchester, ISBN-0: , pp: 436. Seleuou, I. ad M. Souleymaou, Determats of survval AIDS patets o atretrovral therapy a rural cetre the Far-North Provce, Cameroo. Tropcal Med. It. Health, 4: DOI: 0./j x Strutz, T., 200. Data Fttg ad Ucertaty: A Practcal Itroducto to Weghted Least Squares ad Beyod. st Ed., Veweg + Teuber Verlag, Wesbade, ISBN-0: , pp: 244. Tofalls, C., Least squares percetage regresso. J. Moder Appled Stat. Meth., 7:

7 Va de Geer, S.A., Emprcal processes M- estmato: Applcatos of emprcal process theory. Cambrdge Uversty Press. Wlcox, C.W., 20. Bas: The Ucoscous Decever. st Ed., Xlbrs Corporato, ISBN-0: , pp: