SUBJECT-LEVEL TREND ANALYSIS IN CLINICAL TRIALS

SUBJECT-LEVEL TREND ANALYSIS IN CLINICAL TRIALS Alexandru-Ionut PETRISOR PhD, Assstant Professor, Sectons Urbansm and Landscape, School of Urbansm, "Ion Mncu" Unversty of Archtecture and Urbansm, Bucharest, Romana E-mal: alexandru_petrsor@yahoo.com Web-page: http://www.geoctes.com/a_petrsor Lvu DRAGOMIRESCU PhD, Assocate Professor, Department of Ecology, Faculy of Bology, Unversty of Bucharest, Romana E-mal: lvu_dragomrescu@yahoo.com Crstan PANAITE PhD Canddate, Intern, Dabetes Nutrton and Metabolc Dseases, N. Paulescu Insttute, Bucharest, Romana Romana E-mal: crstan.panate@gmal.com Alexandru SCAFA-UDRISTE PhD Canddate, Unversty Assstant, Carol Davla Unversty of Medcne and Pharmacy, Department of Internal Medcne and Cardology, Clnc Urgency Hosptal, Bucharest, Romana E-mal: Anamara BURG PhD, Assstant Professor, Unversty of South Carolna, Lancaster, S.C., USA E-mal: a_rusu@hotmal.com Abstract: In partcular stuatons, clncal trals researchers could have a potental nterest n assessng trends at the level of ndvdual subjects. Ths paper establshes a common approach and apples t n two dfferent stuatons, one from nutrtonal medcne and one from cardovascular medcne. The approach conssts of runnng as many regresson models as the number of subjects, lookng at the behavor of some parameter of nterest n tme. The regresson parameters, partcularly the slope of the regresson lne, offer the general sense of the trend and allow for testng ts statstcal sgnfcance. Extrapolaton at the level of the entre sample s possble usng some verson of the bnomal test. In both cases, sgnfcant results were obtaned despte of small sample szes. Key words: smple lnear regresson; slope; coeffcent of determnaton; bnomal test 5

. Introducton Clncal trals are used to evaluate new drugs or treatments, ncludng new technologes, assess new screenng programs, or ways to organze and delver health servces []. Epdemologsts dstngush three types of clncal trals: prophylactc - used to prevent dseases, therapeutc - used to treat dseases, and nterventonal - used to ntervene before the dsease s developed [],[3]. Whlst most often clncal trals, smlar to other epdemologc studes, analyze the mpact of nterventon on the development of a certan dsease, assessng the mpact of potental rsk factors, researchers mght be occasonally nterested to analyze trends at the level of the subject. Due to the fact that there are only several mportant moments when the parameters of nterest are checked (begnnng of the study, mdterm, endpont and eventually some other ntermedary moments), there are too few data to use a tme seres model. Nevertheless, the evoluton of the varable of nterest (weght n the frst case, n the second) n tme translates nto a smple lnear regresson model. For each regresson model, several parameters of nterest descrbe the trend. The sgn of the regresson slope, β, ndcates ether a decreasng or ncreasng trend. Whereas a test of sgnfcance for β could pnpont sgnfcant trends n some patents, the lmted number of tme mlestones results nto a general lack of sgnfcance. The very few sgnfcant trends cannot allow for further analyses. The same behavor characterzes the coeffcent of determnaton, R. To assess the overall trend, a sgn test could be used to assess whether the trend s decreasng (a percentage sgnfcantly larger than 50% or some other value of the regresson slopes are negatve) or postve. If suffcent data are avalable, the values of the regresson slopes could be used n conjuncton wth ther sgnfcance test and trends can be classfed as ether sgnfcantly decreasng, not sgnfcant, or sgnfcantly ncreasng. Two examples of applyng the proposed approach are presented n ths paper. In the frst case, whle comparng the effcency of three weght loss programs, a queston of nterest s whether weght loss s consstent durng the perod when the treatment s admnstered. Weght s checked at some ntervals, and for each patent the effcency should translate nto a contnuous decrease of weght. In the second example, two echocardographc parameters (the ejecton fracton and the knetc score) were analyzed n relatonshp n a clncal study of the Acute Myocardal Reperfuson Syndrome lookng at rsk factors, predctors and crtera assessng the success of nterventons.. Methods.. Statstcal tests The steps taken n applyng the proposed methodology are:. For each subject, run smple lnear regressons accordng to the model Y=α+β TIME, where Y s the dependent varable montored n the study.. For each model, record the slope of the regresson lne, β, or the coeffcent of determnaton, R. Also, test for ther sgnfcance [6]: β t = 0 n t n ( Y ( α + β X ) n n, or = t X n X n R n =. 0 t n n = R = 53

3. Defne an ndcator varable to descrbe the trend as ether: I=-, f β<0;, f β>0; and no value otherwse, f β was found sgnfcant n very few cases, or I=-, f β<0 and p 0.05;, f β>0 and p 0.05; and 0 otherwse, f the number of subjects wth a sgnfcant trend s large enough to allow for further statstcal testng. 4. To analyze the overall trend, run the bnomal test to compare the proportons of subjects wth I= and, respectvely, - (derved from Pegorsch and Baler [7]): p 0.5 z = 0.5 n N ( 0,) p s the proporton of subjects wth I= or -. 0.5 s the proporton correspondng to the null hypothess H 0 : there s no overall trend (the proportons of subjects wth I= and, respectvely, - are equal, and each of them s 0.5). 5. The ndcator varable can be used n the Analyss of Co-Varance or logstc regresson, ether as dependent or ndependent varable, dependng upon the nterest of the researcher... Software mplementaton The steps descrbed above were mplemented n SAS. In order to use SAS, data were stored n an array wth the followng columns: TIME, the tme when each observaton was recorded; CLASS, any classfcaton varable (dentfyng the group to whch subjects belong, f any), and S0, S0,..., S0n, an dentfer of each subject (could be automatcally generated n Excel). Gven ths structure, the SAS code s provded below (comments are nserted between accolades {}). data name_of_dataset; nput tme class S0 S0... S0n; {Paste actual data here} ; proc reg; model S0 = tme / nfluence; {The nfluence opton was used n the second example to see f observatons recorded at some partcular tme are more relevant for dagnoss; f there s no nterest n testng t, do not use the / nfluence statement} by class; {Should analyses be run only for a partcular class, use where class = class_level nstead; f there are no classes, do not use any statement} {Repeat the statements startng wth proc reg; for S0,..., S0n} run; From the SAS output, retreve for each regresson model the values of ß and correspondng p-values for the test of H 0 : ß = 0 vs. H A : ß 0). Store all these n Excel n three columns and defne the followng based on Excel functons, replacng SIGN and BETA wth correspondng column names: The sgn of ß: SIGN= IF(BETA<0,"MINUS",IF(BETA=0,"ZERO","PLUS")) The sgnfcance of ß: SIGNIFICANCE=IF(p<0.05,"S","NS") To compute the proporton of subjects wth ß > 0 (respectvely ß < 0), use: X=COUNTIF(RANGE,"PLUS"), respectvely X=COUNTIF(RANGE,"MINUS"), where X s the poston of the cell where the result of the formula s computed and RANGE corresponds to FIRST CELL:LAST CELL of the column where the sgn of ß s stored. The sgn test can also be computed n Excel usng: Y=(X/N-0.5)/(0.5/SQRT(N)), where N s the total number of observatons. 54

3. Results and Dscusson 3.. Example #: Weght loss Data had been produced by the study Effcency of the ntensve nutrtonal, pharmacologc and behavoral management of obesty correlaton of genetc, bomorphologcal and psychologcal factors (Natonal Unversty Research Councl grant #63 of 006). The am was to compare the effcency of three weght loss programs among 84 subjects: classcal nterventon (4 subjects), ntensve nterventon asssted by nutrtonsts (33 subjects), and ntensve nterventon asssted by psychologsts (7 subjects). The later two categores were joned n a group labeled ntensve nterventons (60 subjects). In ths case, the varable of nterest was actual weght, recorded n the begnnng of the study (0 months), durng the study, after and respectvely 4 months, and n the end of the study ( months). Its values were not recorded unformly, and the actual sample szes were dmnshed (Table ). Table. Subject level lnear regresson coeffcents of weght varaton n tme: Bucharest, 008 Classcal nterventon Intensve nterventon asssted by nutrtonsts Intensve nterventon asssted by psychologsts β t(β) p(t) β t(β) p(t) β t(β) p(t) -0.5 5.74 0.099-0.54 0.85 0.553-0.75.4 0.458 3.64-0.87 0.5456 -.09.48 0.378 3.8-0.83 0.5584 -.6.6 0.354-0.66 4.4 0.49 -.9 0.79 0.5739-7.0 3.56 0.743-0.89 6.4 0.0985 M -. 8.04 0.035 * -.00.. -.9.6 0.3537 5.64-0.7 0.6088 -.94.4 0.674-0.9.6 0.39 -. 3.74 0.664-3.39.3 0.46-0.79 5.9 0.068-0.7 9.63 0.0658 M -40.00.. -0.79.0 0.493 -.36 59.47 0.007 * -.8. 0.80 -.40.. -0.55.53 0.3686 3.00.. -0.54 6.3 0.000-0.00.. -0.00.. -8.00.. -0.07 0.03 0.9807-0.00.. -0.77.45 0.384-8.00.. -.63 49.65 0.08 * -.88.47 0.3798-3.95.58 0.3600.05-0. 0.968 -. 4.84 0.98-0.8.87 0.36-0.65.. -.69.67 0.3440-0.57 3.08 0.999-0.3 0.07 0.9534-0.54 5.94 0.06-0.97 3.05 0.07 -.64 0.4 0.03 * -.73 5.67 0. -3.47 3.99 0.564-0.00.. -.5.97 0.997-0.69 0.59 0.663 -.78.89 0.3.68-0.3 0.860-6.67.89 0.3 -.86.. -0.50 0.83 0.5573 -.3. 0.4380 0.00.. -..9 0.77-0.57 0.7 0.6047-3.47.85 0.354 -.09.7 0.3376-3.7 4. 0.53-3.56 0.30 0.843-5.69 4.3 0.5 -. 36.5 0.074 * 3.33.. -0.55 3.45 0.796 -.00 4.64 0.0434 * -0.69 3.7 0.674 -.78.. -.79 4.04 0.544 -.4 3.6 0.70-0.67 0.99 0.5030 -.0 0.96 0.5-0.47 3.09 0.993-0.40 0.33 0.7987-0.78 7.67 0.030 * -0.7 4.6 0.468-0.80 0.78 0.5769-0.76 60.6 0.005 * Notes: p-values use a modfed Mcheln scale, addng margnal sgnfcance to the uncertanty regon (0.05 p 0.): * ** M sgnfcant, hghly sgnfcant, margnally sgnfcant 55

The overall trend was very hghly sgnfcantly decreasng (p<0.00), wth slght varatons among subjects assgned to the classcal nterventon (p=0.05), ntensve nterventon asssted by nutrtonsts (p<0.00), and ntensve nterventon asssted by psychologsts (p<0.00). Comparsons among groups suggested that ntensve nterventon methods are more effcent wth respect to weght loss than the classcal ones (p=0.006 when jonng nterventons asssted by nutrtonsts and psychologsts, 0.07 otherwse), but no sgnfcant dfferences were detected between nterventons asssted by nutrtonsts and psychologsts (p=0.388). The results were checked for consstency wth fndngs usng a tradtonal approach, employng the Analyss of Varance (ANOVA) to test whether there are sgnfcant dfferences between the three groups. In ths case, we defned the weght loss as the dfference between the ntal (t=0) and fnal (t=) weght. The global F test was F=0.88 wth p<0.00, suggestng the exstence of sgnfcant dfferences between groups. No dfferences were found between nterventons asssted by nutrtonsts and psychologsts, but both of them dffered sgnfcantly from the classcal approach, beng more effcent. 3.. Example #: Clncal meanng of echocardographc parameters Data were generated wthn the Acute Myocardal Reperfuson Syndrome study BNP prognostc value of BNP correlated wth echocardographc ndces of systolc and dastolc functon n patents wth ST elevaton acute myocardal nfarcton wth ndcaton of reperfuson (Natonal Unversty Research Councl grant # of 006) lookng at rsk factors, predctors and crtera assessng the success of nterventons. In ths case, the varables of nterest were the ejecton fracton, defned as the dfference between end-dastolc and end-systolc volumes dvded by end-dastolc volume [4], and the knetc score, computed accordng to the gudelnes of the Amercan Socety of Echocardography as the rato between the sum of scores assgned to each segment of the left ventrcle ( = normal; = hypoknetc; 3 = aknetc; 4 = dysknetc) and ther number,.e. 6 [5]. Both values were recorded n the begnnng of the study (0 days), durng the study, after, 7 and respectvely 30 days, and n the end of the study (365 days). Even though 88 subjects were ncluded n the study, values of the ejecton fracton and knetc score were not always recorded, and the actual sample sze was dmnshed. The sgn test dd not detect any trend wth respect to the ejecton fracton (p=0.98), but detected a sgnfcantly decreasng trend of the knetc score (p=0.003). However, n ths partcular study further research questons that could be answered usng the proposed methodology. Among them, of partcular nterest was the predctve value for screenng purposes of recordng a value of ether or both the ejecton fracton and/or the knetc score at one of the partcular moments used (0,, 7, 30 or 365 days). In order to answer ths queston, we assgned ranks from to 5 to the par (ejecton fracton, tme) and (knetc score, tme) wth a maxmum mpact on the regresson lne, correspondng to ts poston on the tme scale:, for t=0;, for t=; 3, for t=7; 4, for t=30; and 5, for t=365. The magntude of the mpact was assessed based on the jackknfe resdual [6]: r ( ) = σ ( ) y + n ( α + β x ) ( μ ) x n x = x nμ x t n 3 56

In stuatons where the jackknfe resduals could not be computed, ranks were determned based on the value of the actual resduals [6]: e = Y ( α + β X ) The maxmum mpact corresponded to the maxmum absolute value of ether the jackknfe resdual or actual resdual, wthn each set of fve values computed for each patent. In order to test whether some moment has a hgher predctve value, we tested the statstcal sgnfcance of the dfference between the proporton of ts correspondng rank and 0. (proporton under the null hypothess), usng a modfed verson of the test proposed by Pagano and Gavreau [8]: z = p 0. p ( p) n N ( 0,) Results suggest that for both varables the frst moment (t=0) appears to be the most mportant, as ts correspondng p-values are the lowest (Table ). Nevertheless, for the ejecton fracton none of the moment was sgnfcant at 0.05. Due to the reduced sample szes, we accounted for addtonal margnal sgnfcance f p values were n the uncertanty area (0.05 p 0.). In the same order of mportance, the second moment s t=365 for the ejecton fracton and t=30 for the knetc score. The usage of two varables n the second study allowed for checkng the valdty of the proposed approach. Snce the ejecton fracton and the knetc score were very hghly sgnfcantly correlated (R =-0.74 overall, -0.6 at t=0, -0.7 at t=, -0.75 at t=7, -0.74 at t=30 and -0.84 at t=365, wth p<0.00 n all cases), we assessed the correlaton at the subject level lookng at the correlaton of the ranks descrbed above. Spearman's coeffcent of correlaton was 0.89 wth p=0.086, fallng nto the uncertanty regon. Ths could be due to reducng the overall sample as the computaton of ranks was not always possble snce fve values were not always recorded for each varable for ndvdual subjects. Table. Tests of the predctve value of the moment when the ejecton fracton and knetc score are recorded: Bucharest, 008 Ejecton fracton (n=8) Knetc score (n=86) Rank Frequency z p Rank Frequency z p 0.8.63 0.056 M 0.3.33 0.0099 ** 0.0 0.05 0.480 0.5.3 0.095 M 3 0.7 0.63 0.643 3 0..55 0.0054 * 4 0. 0. 0.468 4 0..09 0.083 * 5 0.4.6 0.056 M 5 0.30.90 0.087 * Notes: p-values use a modfed Mcheln scale, addng margnal sgnfcance to the uncertanty regon (0.05 p 0.): * sgnfcant, ** hghly sgnfcant, M margnally sgnfcant 4. Concluson Both examples ndcate that the proposed approach was able to produce sgnfcant results despte of the reduced sample szes. Comparsons wth classcal approaches, answerng partally the same research queston, suggest that the proposed methodology s vald and could be used to answer the partcular queston of assessng subject-level trends s 57

clncal trals. The only dsadvantage s that t employs a large number of analyses, problem resolved by employng approprate software. References. Bernard, Y. L échocardogramme de stress, Revue Médcale Susse, no. 58, June, 999, onlne source: http://revue.medhyg.ch/prnt.php3?sd=9884, accessed on March 7, 009. Gords, L. Epdemology, Phladelpha, PA: W. B. Saunders Company, 996, pp. 90 3. Habash-Bseso, D. E., Rokey, R., Berger, C. J., Weer, A. W. and Chyou, P.-H. Accuracy of Nonnvasve Ejecton Fracton Measurement n a Large Communty-Based Clnc, Clncal Medcne and Research 3, no., 005, pp. 7-8 4. Hussey, J. R. BIOS 757: Intermedate Bometrcs. Course notes/sprng 998, Columba, SC: School of Publc Health, Unversty of South Carolna, 998, pp. 9,5, 9, 30 5. Llenfeld, A. M. and Llenfeld. D. E. Foundatons of Epdemology, nd ed., New York, NY: Oxford Unversty Press, 980, pp. 57-58 6. Mausner, J. S. and Kramer, S. Mausner & Bahn Epdemology An Introductory Text, Phladelpha, PA: W. B. Saunders Company, 985, pp.95 7. Pagano, M. and Gauvreau, K. Prncples of Bostatstcs, st ed., Belmont, CA: Duxbury Press, 993, pp. 96 8. Pegorsch, Walter W. and John A. Baler Statstcs for Envronmental Bology and Toxcology, st ed., London, UK: Chapman and Hall, 997, pp. Codfcaton of references: [] Gords, L. Epdemology, Phladelpha, PA: W. B. Saunders Company, 996, pp. 90 [] Llenfeld, A. M. and Llenfeld. D. E. Foundatons of Epdemology, nd ed., New York, NY: Oxford Unversty Press, 980, pp. 57-58 [3] Mausner, J. S. and Kramer, S. Mausner & Bahn Epdemology An Introductory Text, Phladelpha, PA: W. B. Saunders Company, 985, pp.95 Habash-Bseso, D. E., Rokey, R., Berger, C. J., Weer, A. W. and Chyou, P.-H. Accuracy of Nonnvasve [4] Ejecton Fracton Measurement n a Large Communty-Based Clnc, Clncal Medcne and Research 3, no., 005, pp. 7-8 [5] Bernard, Y. L échocardogramme de stress, Revue Médcale Susse, no. 58, June, 999, onlne source: http://revue.medhyg.ch/prnt.php3?sd=9884, accessed on March 7, 009 [6] Hussey, J. R. BIOS 757: Intermedate Bometrcs. Course notes/sprng 998, Columba, SC: School of Publc Health, Unversty of South Carolna, 998, pp. 9,5, 9, 30 [7] Pegorsch, Walter W. and John A. Baler Statstcs for Envronmental Bology and Toxcology, st ed., London, UK: Chapman and Hall, 997, pp. [8] Pagano, M. and Gauvreau, K. Prncples of Bostatstcs, st ed., Belmont, CA: Duxbury Press, 993, pp. 96 58