Adapting Altman's bankruptcy prediction model to the compositional data methodology

Adapti ng A ltman’ s bankruptcy pre dictio n model to the co mp os it i on a l d at a met h odo log y Fatemeh Keivani , Germ à C o enders * , Ge ò rgia Escaram ís Univer sitat de Gir ona, department of economics, Girona, Spa in March 2026 Abstract : Using standar d financial rati os as vari ab les in stati stical analy ses has been relat ed to several serious problems , suc h as extreme outliers, asymmetry , non - normali ty , and non- linearity . The compositional - data method ology has been successfully applied to solve thes e problems and has always yielded substantially different results when compar ed to standard finan cial rat ios. An under - researched area i s the use of fin ancial log- ratios computed with the compositional - data methodology to predict bankruptcy or the related terms o f bus iness de f ault, insolvenc y or failur e. Anothe r under - r esearched area is the use of machine learning methods i n combination with c ompositional log- ratios. The present art icle adapt s t he cla ssical Altma n bankruptcy pr ediction model and some of its extensions to the compos itional methodology with pairwis e log - ratios and thr ee common statis tical and machin e learning too ls : logistic r eg ression mode ls , k- nearest neighbours , and random forests , and compa res the results wit h standard financial ratios . Data f rom the sector in the Spanish e conomy with the larges t number of bankrupt fi rms according to the first two d igits of the NACE code (46XX “wholes ale tr ade, except of m otor vehicle s and mot orcycl es”) were o btained f ro m the I berian Balance sh eet Anal ysis System . The sample s ize (31,131 f irms, of w hich 9 7 were bankr upt) was d ivided into a training and a validation d atas et. The training data se t was downs ampled to one healthy fir m to each bankrupt fir m . No outliers were removed. Focusing on predictive performance, t he resu lt s show that compos itional methods are better than standard ratios in te rms of se nsitivity , with mixed res ul ts regarding specificity , compos itional random for ests a nd compositional logistic regression beha ving the best. Keyword s : Artificia l in telligenc e, da ta mining, big data, accounting ratios, log it mo dels, credit scoring, CoDa , SABI. JEL codes: C25, C45, C46, G32, G33, M41 . MSC cod es: 62H30, 62J12, 62P05 , 68 T 99. * Corresponding author: ge rma.coenders@udg.edu Keiv ani el al. ( 2026 ) , arX iv . 2 Intr oduction Using s tandard financial ratios as variables in statistic al models h as been relat ed to sever e violations of the models’ assumptions , such as extr em e outli ers , asymmet ry , connect ed to it sever e non -normality of the distributions, non- li nearity of the relationships, and even dependence of the results on the arbitrary decisio n regarding which accounting figure appears in the nu merator and which in the deno minator of the ratio ( Carre ras - Simó & Coenders, 2021; Coenders & Arimany - Se rrat, in press ; Durana et al . , 2025; I otti et al., 2024; L inar es - Mustarós e t al., 2018; 2022; Magrini , 2025 ) . The results of many sta tistical and machine learning analyses are in valid when all or some of these problems occur and the r esults and conclusions of said analyses can be affected to a great extent. T h e well - known Compositional Data ( CoDa) methodol ogy using log - ratio tr ans f ormations (Aitchison, 198 2 ; 1986; Coenders et al., 2023 a ; Gre en acre et al., 2023 ; Pawlowsky -Glahn et al., 2015 ) has been successf ul ly applied in connection with financial ratios over the l ast 10 years, to solve these problems ( Coenders, 2025; Coe nders & Arimany - Serrat, 2025; i n pr ess). This methodology was developed to deal with fixed - sum data in chemical and geological analys is, but the fixed sum is by no m eans ess ential. CoDa are contemp orar ily defined as arrays o f strictly positive numbers for which ratios between them a re considered to be relevant ( E gozcue & Pawlowsky - G lahn, 2019) , which pe r fectly fits the notion of f inancia l state ment analysis . Far from be ing a statistical re finement, the CoDa methodology leads to very subs tantial differ ences in the analysis resul ts whenever it has been compar ed wit h s t andard fin ancial ratios (Arimany - S errat et al., 2022; Carrer as - S im ó & Coenders, 2021; Coende rs & Arimany - Serrat, 2025 ; i n pre ss; Coenders et al ., 2023b; Creixans - T enas et al . , 2019; Dao et al., 2024; Escar amís & Arbussà , 2025; Her nandez - Romero & Coende r s, in press; Jo fre - Campuzano & Coenders, 2022; L inares - Must arós et al. , 2018; 2022; Magrini, 2025 ; Mola s - Colomer et al., 2024 ). An under - researched ar ea is that of the use of f inanci al - statement ratio s to predict outcom e variab les (Coenders & Ar imany - Serrat, 2 025 ; Molas - Colomer et al. , 2026 ) such as bankruptcy or the related terms of business defaul t , insolven cy or failur e . The only known ins tance of application o f the C oDa methodology here is that by Ma gr ini (2025 ) both using logistic r egr ession models , LASSO r egularization , an d random fo r ests . Magrini’ s work is also the first to c ombine a f inancial application of the CoDa methodology with machine learning . In bankruptc y prediction studies, p redictive accuracy bec omes a key pa rameter , beside s fulfilment or violation of the as sumptions. Magrini did not find subs tantial differences in a ccuracy betwee n standard and compositional methods. Since the seminal work by Altman (1968) , the proble m of using financial ratios to p redict bankruptcy has never ceased to attract interest ( Hanun & F erdiani, 2025; Insani et al. , 2024; Magrini, 2025; Martin - Melero et al ., 2025; Nugroho & Dewa yanto, 2025; Soukal et al., 2024 ; Staňková & Hampel, 2023; T aoushianis , 2025; V eganzones & S é verin , 2021 ; W iller do Prado et al. , 2016 ). The o riginal Altman’ s model and s ome popular vari ants Keiv ani el al. ( 2026 ) , arX iv . 3 continue to be used a nd a dapted to new machine le arning tools , a. k.a . statis tical learning , data mining , a n d artific ial intelligen ce tools (M arti n - Meler o et al., 202 5). T he literature on bankruptcy prediction with machine learning methods is growing apace ( Akusta & Gün, 2025; Antar et al., 2025; Anta r & T aya chi, 2025; Asif et al., 2025; Bal zano & Magrini, 2025; Chan et al . , 2025; Chen et a l., 20 25; Chohan et al., 2026; Cojocaru, & Ionescu, 2025; Dam & Nghiem, 2026; Dumitresc u et al., 2025; Gabrielli e t al., 2026; Gajdosikova et a l., 2026; Gnip et al., 2025; Gwalani et al . , 2025; Hussain et al., 2026; Kanojia & Arora , 2025; Khotimah & W anti W idodo, 2026; Kristanti et al., 2025; Lin et al., 2025; Liu , 2025 a ; L i u et al., 2025; L iu, 202 5b ; Martin - Melero et al., 2025 ; Mat suma ru & Katagiri, 2025; T anaka e t a l., 2025a; 2025b ; T hot a et al., 2026 ; V eganzones et al ., 2026; W ang et al., 2026; X u, 2026; Yu a n , 2025; Zhao, 2025. Zhu et al., 2026 ) . O f ten , eliminating a vast amount o f outliers ( Balza no & Magrini, 202 5 ; Magr i ni, 2025 ) , ev en in t he r ange o f one third o f the or iginal s ample (Martin - Melero et al., 2025) , is the p rice to be paid for using standa rd financial ratios , as some machine learning methods a r e relative ly immune to outliers, but no t all o f them . Machine lear ning t ools can be adapted to the CoDa methodology ( T olosana- Delgad o et al., 2019; T r an, 2025). The present ar ticle compares Altman’ s classical bankruptcy prediction model and some of its extensions with the CoDa methodology an d three common s tatistic al and machine le arning tools : lo gistic reg ress ion models , random fores ts , and k- near est neighbours . Ra ther than focus ing on the violation of the ass umptions, we focus on comparing predictive performance. Our approach complements that of Magrini (2025) , firstly , by using the full s ample without tri mming outlie rs, which is poss i ble in the Co Da methodology , as i t tends t o produce few or no outliers ( L in ar e s - Mustarós et al., 2018 ) . Random forests are relative immune to outliers (Hia et al ., 2023), but logistic regression , being a parametric s tatistical mode l, and k - nearest neighbours , being a distance - based method, are no t . Secondly , by downsampling of healthy firm s , which is known to improve pr edicti ve performance when ther e are f ew bankrupt fir ms ( Antar & T ayachi, 2025; Cojoca ru & I onesc u, 2025; Gabrielli et al . , 2026; Gnip et al., 2025; Kanojia & Aror a, 2025; Liu et al., 2025; Ma rtin - Meler o et al., 2025; T anaka et al., 2025a ; 2025b; V eganzones et al., 2026). T o this end , we randomly select one healthy firm for each bankrupt fi r m in the training sample. T hird l y , by using an alternative log - ratio transformation te rme d pairwis e log -ratios (henceforth plr ; Greenacre, 201 9 ). Materi als and m ethods Standar d and CoDa r epr esentation of Alt man’ s mo del for bankruptcy pr ediction According to Altman (1968 ), the following five s tandard f inancial ratios are good predictors of bankruptcy: working capital ratio: (CA - CL)/ (N CA+CA) , retai ned earni ngs over assets: RE/( NCA+CA), re tur n o n assets: ( OR - OE)/( NCA+CA), (1) net worth ove r liab ilitie s: (NCA+CA - NCL - CL )/( NCL+CL ), turnover: O R /(NCA+CA), Keiv ani el al. ( 2026 ) , arX iv . 4 where NCA stands for non - cur ren t assets, CA f or cur rent assets, RE f or retained earnin gs, NCL for non - current liabilities , CL for current l iabilities, OR f or operating revenue and OE for operating expenses . In the origin al Altman’ s model, the c ompany marke t value is used inste ad of the book value of net worth ( NCA+CA - NCL - CL). The book value is used in this article as commonly d one, being available f or all c om panies, be they traded in the stock excha nge or not. Altman’ s model has been re for mulated m any times. T he mos t popul ar s uch reformul ations are arguably those by Amat el al. (2008) ; G r over and Lavin (2001); Springate (1978) ; and Zmijewski (1983). These formulations add or substitute five more standard finan cial ratios: p rofit ove r curren t liabili ties: (O R- OE) /CL , c urrent ratio: CA/CL, i nver ted lever age ratio : (NCA +CA - NCL - CL)/(NC A+CA), (2) return on equity : (O R- OE )/( NCA+CA - N CL - CL) , indebtedness : ( NCL +CL) /( NCA+CA ). The starting point of the CoDa methodology is not the ratios , but a composition of the D positive accountin g fig ures needed to compute them, which are call ed par ts or components. Positiveness is ess ential. For instanc e, one cannot use operating pro fit (O R- OE) which may be nega t ive, but the always positive operating revenue (O R ) and operat ing exp enses (OE) separatel y (Cr eixans - T enas et al., 2019). In o ur case the D =7 parts in the composition are NCA, CA, RE , NCL, CL , O R and OE. The D p art s ar e then subject to log - r atio transformatio ns that c ontain all the rel ative information among the m (L inar es - Mustarós e t al., 2018; Coenders & Arim any - Serr at, in pr ess). The simples t approach to the C oDa methodology in financial an alysis is transforming the accounting figures by means of plr ( Creix ans - T en as et al., 2019; Coenders & Arimany - Serr at, in press; Farre ras - Nogu er et al . , i n p r e ss ; Greenacre, 20 19 ; Mulet - Fort eza et al., in pr ess ) . This involves d efining D - 1 l o g - ratios be tween only two accounting figures, so that each acc ount ing figure can be connected to any ot her (exhaustiveness) in only one way (non- r edundancy) thr ough the log - ratios. The D - 1=6 plr below follow both rules and can be interpreted according to c ommon financial concepts: ass et tangibility: log(NCA/C A), curr ent - asset turnover: log(O R /CA), margin: log(O R /OE) , current ratio: log(CA/CL), (3) debt maturity: log(NCL/CL), retained earnings over non - c ur rent liabilities: log (RE/NCL ). As recommended by Greena cre (2019), if we d raw a graph in which the accounting figures are the ve rtices (nodes) and the pl r a re the c onnections (edges), it is eas y to see that a n y accounting figure can be joined to any o ther through the plr in only one way , so that t he graph is connected and acyclic. For instance, NCA and O R are connected to each Keiv ani el al. ( 2026 ) , arX iv . 5 other only through the log ( NCA/CA) and log ( OR /CA ) p l r. Edges can be drawn as a rr ows without affecting the graph pr operties, the arrows pointing at the numerator of the plr for clarification purposes. Figure 1. Connected a cyclic graph among the D =7 acc ounting figures D - 1 plr can routinely be us ed as a se t of predicto rs in s tatistical model s , including logistic regression models (Coe nders & Greenacre, 2023; Coenders & Pawlows ky - Glahn, 2020). Furthermor e , the predictions are invar iant to the particu lar cho ice of D - 1 plr , as long as each acc ount ing figure can be c onnected to any other in only one way (Coenders & Pawlowsky - Glahn, 2020). More th an D -1 pl r would le ad to pe r fect collinearity . The distances among companies do depend on the plr choice. Rather th an selecting D - 1 plr , using all the possible D ( D -1)/2 plr is a better option for methods base d on di stance s, such as the k - nearest - neighbour method, because Euclidean distances computed on all the possible D ( D - 1)/2 plr ar e equivalent t o the stan dard d istance measur e for CoDa, term ed Aitchison’s distanc e ( Aitchison, 1983; Aitchis on et al., 2000) . M e thods which have the ability to s elect th e best predictors from a large ava ilable set , such as random forests , will also benefit from us ing all the pos sible D ( D - 1)/2 plr. Besi des, with random forests a nd k - nearest neighbours , t he predictions depend on the plr choice when using only of D - 1 of them , which makes results arbitrary and non - r eplicab le. Which accounting figure goes to the n umerator or the denominator of the plr does not change th e results. In our case, the set of D ( D -1)/2 =21 possible plr is: NCL OR CA NCA log( NCA/CA) RE OE CL log( RE / NCL ) log( NCL / CL ) log( CA / CL ) log(OR/ OE ) log( OR / CA ) Keiv ani el al. ( 2026 ) , arX iv . 6 log(NCA/CA), log (NCA /RE ), log(NCA/NCL ), log (NCA /CL ), log(NCA/O R ), log(NCA/OE ), log (CA/ RE) , log (CA/ NCL ), log (CA/ CL) , log(OR/ CA ), log(CA/OE ), (4) log (RE /NCL ), log( RE /CL ), log(RE/OR ), log(RE/OE), log (NCL /CL ), log(NCL/OR ), log( NCL/OE ), log(CL/OR ), log(CL/OE), log(OR /OE) . Thus, the 10 predictors in Equation s ( 1) and (2) w e re used for the standard - ratio approach under all three prediction methods. The predictors in Equation ( 3) w er e used for logistic regression under the compositional approac h, an d the predictors in Equation (4) were used for random fo r ests and k - nearest neighbours under the compositional approach. Statistic al analys is The logistic regres sion model was fi tted with the gl m function in t he basi s R sof twa re (R core team, 2025). The k - n earest - neighbour class ification was carried out with the knn function, which is part of the class R library ( V enables & Ripley , 2002) by tuning the best k value. T h e r andom- for est classificat ion used t he randomForest function in the homonymous R l ibrary (Breiman, 2001), by gener ating 100 trees a nd randomly selecting 4 predictors in the standard - ratio approac h and 7 in the compositional approach, which is about one third of the 10 standa rd and 21 compos it ional ratios. T he sampl e was randomly divided into a training subset ( 70 % of the sample) and a validation subset to as sess predictive q uality (30 % of the sample) . The training subse t was downs ampled and one hea lthy firm wa s randomly selecte d for each bankrupt firm. The validation subset wa s kept entirel y . Thr ee commo n precision measur es computed from the validation subset are: 1. p r ed ictive accu racy (over all per centage o f correct predi ctions) , 2. sens itivity ( percent age of t r ue bankr upt companies predicted as so), 3. spec ificity ( percen tage of tr ue heal thy com panies predicted as so) , 4. balanced accur acy (arit hmeti c average of sensitiv ity and s pecificity) . Keiv ani el al. ( 2026 ) , arX iv . 7 Data collection and p re -pr ocessing W e def in ed a company as being bankrupt if under cr edit ors’ meeting ( “concurso de acreedor es” in Sp ain). To predict i f a company was bankr upt in the last av ailable y ear , the standard and compos itional ratio values of the year immediat ely before wer e u sed . Data gathering and p r e -processing were as follows : 1. Data fr om the sector in the Spanish economy w ith t he largest number of firms under creditors’ meetings in 2024 accordin g to t he first two digits o f the NACE code (4 6 XX “w holesale t rade, except of mot or vehicles and m otor cycles ” ) were selected. Filt ers incl uded having 5 or more empl oyees and data availabl e until at least 2023 . The data w ere obtained on 6/10/2025 fr o m t h e Iberian Balance sheet Ana lysi s Sy stem ( SA BI , access i ble at ht tps://sabi. bvdinf o. com/) database, developed by INFORMA D&B in collaboration wi th Bureau V an Dijk. 2. The sample size was 31,131 f irms, of which 97 we r e bankrupt. 3. No outliers w ere be remov ed, only inactiv e f ir ms having either zero total assets (NCA+CA) , zer o OR , an d zero OE duri ng the last available year or t he year befor e the last . 4. The r emaini ng zero s w ere imputed with the l og - ratio EM method ( Palar ea - Albalade jo & Martín - Fer nández, 2008) avai lable in the freewar e Co DaPack (Co mas - Cufí and Th ió - Henestrosa, 2 011; downloa dable at https://ima.udg. edu/codapac k/ ). The per centages of zero s were 1. 38 % for NCA , 0.01 % for CA , 5.54 % for RE, 22.58 % for NCL, and 0.05 % f or CL . 5. The sample was randomly divided into a training set (70 % of cases ) and a validation set (30 % of cas es). The tr aining set had 21,791 fi rms, of which 69 wer e bankrupt. T he validation s et had 9, 340 firms, of which 2 8 were bankrupt. T he downsampled training set thus contained 69 healthy and 69 bankrupt firms, and the validation set 9,312 healthy and 28 bankrupt fi r ms. Th e same s et s were used for all three methods bo t h for the stand ard and co mpositional approaches. T able 1 shows the sk ewness and ku rto sis statistics of the s tandard ratios in Equations (1) and (2) , and the plr in Equation (4) f or the full samp le. Sk ewness serves as a measure of asymmetry , and kurtosis as a mea sure of heavy distribution tails, in other words , out liers. A normal distribution uncontaminated by outliers has z ero skewnes s and kurtosis. All standard ratios but one have ve r y high s kewness and extr eme kurtosis , making them o n the paper unfit fo r statis tical modelling and non - robus t machine learning m ethods except after pruning outliers . Pr un ing outliers following the standard criterion ( e. g. , Ma rtin - Melero et al., 2025) of values above the thir d quar tile plus 1.5×( inter quartile range ) or below the firs t quartile minus 1.5×( inte rquartile r ange ) would r esult in dropping 9,544 cas es from the full s ample, which is 3 0. 7 % of the companies , thus s er iously impai ring the sampl e repr esentativen ess. In t he case of p lr , only the margin r atio log(OR /O E) has high kurtosis, but to a much l esse r extent than the standard ratios. The rest of the plr behave w ell. In contrast wi th Magrini (2025), w e analyse the full s ample without pruning outliers. Keiv ani el al. ( 2026 ) , arX iv . 8 T able 1. Skewness and kurt osis of sta ndard ratios and plr in the full sample Sk ewne ss Kurtosis (CA - CL) /( NCA+CA ) -30.3 2220.3 RE/ (NCA +CA) 1.3 4.9 (O R- OE)/( NCA+CA) -23.9 1843.6 (NCA +CA - NCL - CL ) /(N CL+CL ) 129.9 19809.4 OR /(NCA+CA) 63.9 621 1.5 (O R- OE) /CL -167.6 29201.4 CA/CL 167.5 28935.9 (NCA +CA - NCL - CL ) /(N CA+CA) -44.7 3617.9 (O R- OE)/( NCA+CA - NCL - CL) -161.2 27301.5 (NCL +CL )/ (NCA +CA) 44.7 3617.9 log(NCA/CA) -0.9 2.8 log (NCA /RE ) 0.8 2.8 log(NCA/NCL ) 0.7 0.2 log (NCA /CL ) -0.9 2.9 log(NCA/O R) -0.9 3.0 log(NCA/OE ) -0.9 3.1 log( CA/ RE) 1.8 3.4 log (CA/ NCL ) 0.8 -0.4 log (CA/ CL) 0.9 6.3 log(OR/ CA ) -0.2 5.7 log(CA/OE ) 0.1 4.7 log (RE /NCL ) 0.3 0.1 log( RE /CL ) -1.2 2.0 log(RE/OR) -1.4 2.5 log(RE/OE) -1.4 2.5 log (NCL /CL ) -0.8 -0.3 log(NCL/OR) -0.8 -0.3 log(NCL /OE) -0.8 -0.3 log(CL/OR) -0.2 5.2 log(CL/OE) -0.3 6.1 log(OR /OE) -3.8 277.2 R esul ts According to the logistic r egression model with standard ratios , the relevan t predictor s (5% signi ficance) are ( NCA+CA - NCL - CL)/ (NCL +CL), (O R- OE) /C L , and CA/CL but, due to outliers , some p redicted probabilities are numer ically equal to 0 and 1, whic h af fects the r eliability o f pr ecisely the si gnif icance measures . W ith compositio nal ratios , no such problems are enc ount ered, and the rele vant predictors are l o g( OR/ CA) and log (RE /NCL ). T he k - nearest - neighbour method do es not produce any mea sure of variable impor tance. 5 nearest neighbours yie ld the highe st accuracy . The random - fo rest method can assess the im port ance of a predict or by th e mean decrease of Gini’ s impurity if the predictor is removed. The be st standard ratio s are (NCL+ CL)/(N CA+C A), (NCA+CA - NCL - CL) /( NCL +CL) , an d (NCA +CA - NCL - CL) /( NCA+CA), all connected to the br oad concept of long - ter m indeb tedness and lever age, and the best c ompositional ratios are Keiv ani el al. ( 2026 ) , arX iv . 9 log (RE /NCL ), log(CA/NCL ) , and log(NCL/O R) , a l so connec t ed to non - current liabilities and thus to long - term indebtedness. Figure 2. Te n most important predictor s for the random - forest method with standa r d (le ft) and compositional (right) r atios T able 2 . Pr ecision mea sures of the logistic regress i on model, 5- nearest ne ighbours, and the r andom forests with standard and compositional ratios Prediction method Accu r acy Sens itivi ty Spe c if ici ty Bal anced ac cura cy Logistic regressi on (sta ndard ) 66 % 82 % 66 % 74 % Logistic regressi on (c omposi ti o nal) 67 % 86 % 67 % 76 % k- nearest n eighbours (s ta n dard) 70 % 75 % 70 % 73 % k- nearest n eighbours ( c o mpositional ) 58 % 89 % 58 % 74 % Random fores t s (standard) 62 % 75 % 62 % 68 % Random fores t s (composit i on al) 65 % 89 % 65 % 77 % T able 2 contains the precision meas ur es for all app roaches computed from the va l idation set. Compositional methods a lways have highe r sensitivity that thei r standard counterpa rts. Random forests be at them also in spe cificity . Compos itional and standard logistic regression ha ve about the same specificity . Compos itional k - nearest neighbours perform worst in terms of s pecificity . T he overall a ss essment by the balanced accu racy i s that compositional random forests and compositional logistic regres sion perform b est. Disc ussi on Bankruptcy prediction is one of t he most commo n uses of financial ratios and yet a n under- researched area in the CoDa me thodology . This article adapt s Altman’ s prediction model and some of its adaptations to this methodology w ith t hr ee common pr ediction methods, one being a statis t ical model and the remaining two machine - learning method s, and compares the performance of standard and co mpositional financial ratios, computed a s p l r. Keiv ani el al. ( 2026 ) , arX iv . 10 As reported in the literature, the C oDa methodology improves on the skewness and kurtosis of the analysis var iables and outlier pruning, which can improve the fulfilment of statis tical ass umptions bu t can al so affect th e sample rep resentati vity , i s avoided . Unlik e research d one so far with compositional bankruptcy prediction ( Magrini , 2025) , we have us ed the full s ample . The results show that be yond the fu lfilment o r not of sta tistical ass umptions, the comparison between the standard and compositional approach is also a matter of predi ctive per forman ce. C ompos itional methods a r e better in terms of sens itivity , with mixed results regarding specificity , compositiona l random forests and compositional logistic regression beha ving the best. Magrini ( 2025) did not f ind any substa ntial difference betwe en standard and compos itional rat ios for logistic r egression and random fo rest s, but this a uthor pruned the extreme outliers. In spite of their known sens i tivity to outli er s, when it comes to predi ctive per formance, log istic r egression and k nearest neighbours are just mar gi nally wor se fo r the outlier - r ich standard r atios. In spite of their known robustness t o outliers, when it comes to predictive performance, random fo r ests be have much worse for standard ratios. W e ar e of course awar e that th ese findi ngs are no t generalizable beyond this pa r ticular sector of the Spanis h economy an d the par ticul ar accounting figures and r atios chos en . The possibility o f using other log - r atio trans formations than plr deserves a comment . Centred log - ratios a.k.a. clr (Aitchison, 1983) a re coherent with Aitchison’ s distances and therefore can be used f or t h e k - near est - neighbour method with identica l results as the D ( D - 1)/2 plr . The family o f isometric log - rati o transfor matio ns a.k. a. ilr (Egozcue et al., 2003) has the same property : a ny ilr trans formation can be used for the k - n earest - neighbour method with identical res ults as the clr and the D ( D - 1)/2 plr . Any ilr transformation can also be used as p redictor in a lo gistic regres sion model with iden tical for ecasts as the D - 1 plr ( Coenders & Pawlows ky - Glahn, 2020). Tr e e - based methods in general and r andom forests in particula r cons tit ute another story . While t h e i lr family of transformations is a respect able choice (Magrini, 2025) , it m ust be borne in mind that the forecasts depe nd on the particular ilr selected within the fam ily ( T empl & Gonzalez - Rodriguez, 2024 ) , and that any of these forecasts als o differ from those obtained with the D ( D - 1)/2 plr . T he main advantage of plr i s th at t her e is only one way o f selecting all possible D ( D - 1)/2 plr , which makes t he r esults le ss a rbitrar y and always repl icable ( E ngle & Chaput, 2021 ; T olosana- Delgado et al., 2019 ) . O n the contrary , using all possible ilr is generally u nfeasib le. W ith ju st D =7 accou nting fig ures ther e are 1 . 141 distinct balanc es for the ilr transformation and it is easy to imagine what can happen when D is lar ge. clr have also be en used for random forests ( Nathwani et al., 2022), being als o r eplicable, but they yield different r esults from both ilr and plr . T olos ana- Delgado et al. ( 2019) reported su peri or perfo rmanc e of p lr ove r ilr o r clr . Fur ther r esearch can include adapting other machine - learning methods li ke neural networks and deep learning , naï ve B ayes classifi er , partial- lea st - squar es disc riminant analysis (PLS - DA) , s upport vector machines (S VM) , natural gr adi ent boosting (NGBoost), adaptative boosting ( AdaBoos t), cate gorical boosting (CatB oostc lassifier), Keiv ani el al. ( 2026 ) , arX iv . 11 light gradient boosting (lightGBM ) , and extreme gradient boosting (XGBoost) . Th e so - called hybrid models , ens emble methods , or m eta models combine more than one machine - learning approach and are also worth explori ng (Akusta & Gün , 2025; A si f et al., 2025 ; Choha n et al . , 2026; Huss ain et al., 2026 ; Liu , 202 5b ; V eganzones e t al., 2026 ). Each m achine lear ning method can have a differ ent set of compa ti bl e log - ratio transformations within the CoDa methodology an d t his has to be e xpl o r ed in the first place. Besides , each machine lea rning method may be robust to outliers to a dif f erent extent, if at all . Additional metrics to compare and int erpret the re sults of the methods can also be used (Antar et al., 2025; Chan et al., 2025 ; Thota et al., 2026 ) , like the positive and negative pr edicti ve values , Cohen’ s kappa , th e F 1 s c o re , and the ar ea under the r eceiver operating ch ara cteri stic (RO C) curve . The ch oice of the bes t method may be based not only on predictive quality but also on convenience and interpretability ( Antar et al., 20 25; Chan et al., 2025; Gwalani et al., 2025 ; Hussain et al., 2026; L in e t al., 2025; Molnar , 2020; 2025 ) . Some methods provide an explic it an d interpr etable decision rule and s ome do not, some methods make it possible to s elect the be st predictor subset a nd some do not, some are we ll known to a wide pr actiti oner au dience and som e are not , or not to th e same extent. When machin e learning methods are not inte r pretable ante - hoc, i nterpreta bility of machine learning result s can be enhanced post-hoc by means of the so - call ed explaina ble artific ial intelligenc e (X AI) methods, Shapley additive expl anations - SHAP - ( Antar et al., 2025; Chan et al. , 2026; Hus sain et al., 2026 ; Lin e t al. , 2025 ) an d l ocal in t erpr etable model - agnostic explanations - LI ME - (Antar et al., 2025; Lin et al., 2025) being the most popular . A strong limitation of our s tudy is that o ur sample had a very small number of bankrupt firms. Using a larger portion of the country’ s economy than the 2 - digit NACE clas sificatio n is a po ssibility , although predictio n s m ay be industry - specific ( Balzano & Magrini, 2025). Acknow ledgments This work was supported by the Spanish Ministry o f Science, Innovation a nd Universities MCIN/AEI/10.13039/501 10001 1033 and ERDF - a wa y of making Europe (grant number PID2021-123833OB- I 00); the Spanish Ministry of Health ( grant number CIBERCB06/02/1002); and AGAUR and the Dep ar tment of C limate Action, Food a nd Rural Agenda of Generalitat de Catalunya (grant number 2023 - CLI MA - 00037). Conf lict s of interest The au thors decl are no confl ict of interest. Data availability The cu rated dat a are available fro m the correspond ing author upon reas onable request. Keiv ani el al. ( 2026 ) , arX iv . 12 References Aitchison, J. ( 1982). T he statistica l analysis of compos itional data (with discuss ion). Journal of the R oyal Sta tistical Society Ser i es B ( Statistical Methodology ), 4 4(2 ), 139-177. https ://doi.org/10.1 1 1 1/j. 2517 -6161.1982.tb01 195. x Aitchison, J. (1983). Principal component analysis of compositional data. 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