KINESIOLOGIA SLOVENICA 4 (1998)1 : 5-11 5 LINEAR AND NON-LINEAR MORPHOLOGY STRUCTURE MODEL$ Franci Ambrožič LINEARNI IN NELINEARNI MODELI STRUKTURE MORFOLOGIJE Abstract Ali studies tryingto find the structure of morphology till now used the linear correlation model. This work comparesthe classic linear approach w ith a non-lin- ear one. A sample of 686 adult males was measured with 23 anthropometric measures. The obtained da- ta was analysed with the SAS statistical package, us- ing Hotelling's principal component factor analysis method (procedure PRINCOMP) and the MTV method of data transformation (procedure PRIN- QUAL). The linear and non-linear latent factorsolu- tions (Kaiser-Guttman criterion was used for the number of factors) were rotated to an oblique solu- tion with the PROMAX method. Comparison of the two solutions showed unexpectedly small differ- ences. The latent structures are practically identical, the non-linear solution is somewhat cleaner and more in accord w ith the theoretical model. The question remains, however, if the same holds for younger or older males, females and other sub- spaces of the psychosomatic status. Keywords: morphology, mode/s, non-linearity, factor analysis, adults, males University of Ljubljana - Faculty of Sport, Gortanova 22, 51-1000 Ljubljana, Slovenia phone: ++38661140-1 0-77 fax: + +386 61 448-148 e-mail : Franci.Ambrozic@uni-lj .si Izvleček Vse dosedanje štud ije o strukturi morfologije so bile osnovane na linearnem korelacijskem modelu. V tem delu primerjamo klasični linearni pristop z ne- linearnim. Vzorec 686 odrasl ih moških je bil izmer- jen s 23. antropometričn i mi merami . Dobljene po- datke smo analizirali s statističnim paketom SAS, uporablj ajoč Hotellingovo metodo glavnih kompo- nent (procedura PRINCOMP) in MTV metodo trans- formacije podatkov (procedura PRINQUAL). Linearno in nelinearno latentno faktorsko strukturo (število faktorjev je bilo določeno s Kaiser-Gut- tmanovim kriterijem) smo zavrteli v poševnokotno rešitev s PROMAX metodo. Primerjava obeh rešitev je pokazala nepričakovano majhne razlike. Latentni strukturi sta praktično enaki, nelinearna je malce čis­ tejša in bolj v skladu s teoretičnim modelom. Ostaja pa vprašanje, če enako velja tudi za mlajše ali starejše moške, ženske in druge pod prostore psiho- somatičnega statusa. Ključne besede: morfologija, model, nelinearnost, faktorska analiza, odrasli, moški J 6 lntroduction The characteristics of the body in connection w ith sports have been the object of interest for a long ti rne. Research in the past went mainly in three di- rect ions: first, by comparing sportsmen of various sports w ith the general popu lati on and competitors of other sports; second, searching for distinct sub- groups - morpho logic types or somatotypes; and third, finding the latent structure of morphology. O ne the most comprehensive reviews of this re- search is given in the monograph »Struktura i razvoj morfoloških i motorički h d imenzija omlad ine« (Structure and Development of Morphologic and M otor Dimensions ofYouth) (12) . The theoretical model of morphology was concep- tualised w ith four latent dimensions - longitud inal dimensionality, transversal dimensionali ty, volumi- nosity and subcutaneous fat. Research on general sam ples of the male population did not always con- fi rm this model; sometimes the tra nsversal factor d id not appear (16) o r joined w ith voluminosity (17). Transformation into image metrics or part ialisation of social status did not give a d ifferent solution (1 O) . Female sam ples showed very differentsolutions (9); a study on seventeen-year-old girls gave even six la- tent dimensions (2). Di ffering solutions were ob- tained also with sam ples of PE students (13, 14), the exception was a study on several university centres (students) in Yugoslavia, where the theoretical mod- el was fully confirmed (11 ). The morphological struc- ture of children usually changes w ith age, the fou r latentd imensions merge and separate in various pe- riods, so that two-facto r, as well as four factor solu- tions are known (18). In light of this great instabili ty of structure some au- thors warned of the problems of classical approach- es and methods (3 ). Gredelj (7) for instance states that the obtained and theoretical structures differ too much for the measures to define human morpholo- gy well, he feels that the reason is the complexity of the anthropometric measures. A group of authors (6) criticises in their work the existent methods of com- puting »ideal weight« ind ices and proposes the quadratic polynomial regression model. This model gave better results on a sample of adult males than the classic approach. Po lynomial regression was used also by another group of authors (5) in analysing changes in morphological structure between sixteen and twenty years of age. AII this clearlyshows a need for veri fying one of the most basic suppositions in ki- nesiology t ill now - the l inearity of corre lat ion be- tween variables and the linear factor structu re mod- el based on them. =·a..-c :.~orožič LINEAR ANO NON-LINEAR MORPHOLOGY STRUC- \.;rl:: 'JODELS Figure 1: Linear and non-linear morphology structure mode/s Figure 1 clearly shows that the non-linear model re- sembles the human body much better, but finding that out from the gathered data is much more d iffi - cult. lnterested readers, w ho would l ike to learn more about the problem theoretically or at the level of variable pa irs, are referred to precedingstudies by this a uthor o r sources given therein (1 ). In this article we shall present on ly some methods dealing w ith analysing latent structures (both the linear and non- linear componentanalysis methods are described in some detai l in one of the reference books of the SAS statist ical package -1 5) . Classic principal componentanalysis is a well -known method (8), therefore we shall not present it here. The procedure PRINQUAL is another matter, so here is some basic in formation. This procedure is a data transformation method and comes from the works of Kruska l & Shepard, Young, Takane & Deleeuw , W insberg & Ramsay. It can also be used asa generalisation of the classic method of principal components to non-numerical variables or for find- ing non-linear relat ions between numeric and non- numeric variables. It contains three methods for da- ta transformation: MTV, M GV and MAC. Al i these at- tempt with certa in transformations to reduce the ran k of the covariance matrix of the transformed va riables. The MAC method can be used only if all the correlations between the variables are positive, in ou r case this is not so, therefore this method can- not be used. O f the remaining two we chose MTV beca use it is based on the principal components model, which was the one used as the reference (com parison) model. We are fully aware of the st ili present cont roversies and possible doubts on the choice ofthe principal components method and not one of the factor analysis methods (for a compre- hensive overview see Borg & M ohler - 4), it is a con- scious choice. It is a fact namely, that practically with- out exception all the stud ies of the latent structure of morphology have been made using the principal components method. Si nce the pu rpose of th is study isto compare the linear model w ith the non-linear Franci Ambrožič LINEAR ANO NON-LINEAR MORPHOLOGY STRUCTURE MODELS one, the choice of principal components method was completely logical . Al i the variables to be analysed are at least interva l (for nominal and ordinal variables other transforma- tions are used), therefore we can use linear or non- linear transformations; optimal, such as splines and monotonous splines; or non-optimal, such as expo- nential, power, logari thmic and other functions. Since we have no previous information about which function best linearises a certain variable and be- cause we are not sure that linear (splines with knots) or monotonous splines would lead to an optimal so- l uti on, we decided on non-monotonous splines without knots - polynom of order three (h igher order polynoms could of course also have been used), the method SPLINE. This procedure is also iterative and is supposed to converge to a global optimal solution. There were no missing data in our case, so we did not have to decide how the program me should treat them. Aim o f the study The principal purpose of this study was to find if the linear structure model (based on the Pearson corre- lation coefficient) describes sufficiently well the na- ture of the structure of morphology of adult males. METHODS Subject sample The subject sample comprised of 686 adult males between 18 and 2 7 years of age, taken frorn the pop- ulation of cl in ically healthy adult males, without manifest morphological or motor disorders - the base for samplingwere al l military draftees of the for- rner Yugoslavia, serving in 1973/74. The sample was a two-level group sample with optimal allocation, more deta ils are given in one of the articles of the re- search group (16). Variable sample The m9rphological sub-space is represented in this study by 23 anthropometric measures, chosen on the basis of the works of Pogačn ik and Momirovic and other authors. This sample includes all measures proposed in the lnternational Biologic Programme, w ith the addition of the va riable: hand length. A complete description of the measurement proce- dures is given in the previously cited work (16). Al i the variables are given bytheirorigina l name in order to make comparison with the ori ginal study easier. Longitudinal dimensionality: (VISINA) - body height, (DUZI RU) - arm length, (DUZISA) - hand length, (DUZINO) - leg length, (DUZIST) - foot length; Voluminosity: (TEZI NA) - body weight, (OPGRUD) - mean chest circumference, (OPNADL) - circumference of relaxed upper arm, (OPPODL) - lower arm circumference, (OPNA TK) - subgluteal thigh circumference, (OPPOTK) - calf circumference. Transversal dimensionality: (BIAKRO) - shoulderwidth, (DI LAKTI - elbow diameter, (DIRUZG) - wrist diameter, (SIRISA) - hand width, (BIKRIS) - pelvicwidth, (DIKOLJ) - knee diameter, (SISTOP) - foot width; Subcutaneous fat: (NAPAZU) - chest skin-fold, (NANALE) - back skin-fold, 7 (NA TRB U) - stomach skin-fold, (NANADL)- upper-arm skin-fold, (NABPOT) - thigh skin-fold; AI I the variables were measured three times, w ith the except ion of skin-folds and mean chest circumfer- ence, which were measured six times. Data analysis The data was pre-processed at the Faculty of Physical Culture in Zagreb, Croatia. The original measured items were condensed to the fi rst principal compo- nent, obtained from the covariance matrix of the original results, rescaled to anti image metrics. This procedure enhanced the reliability of the data and also gives no information on the real distributional parameters of the variables, which was in this case mandatory. Further analysis was performed atthe Faculty of Sport - University of Ljubljana, ona PC with the statistical package SAS. The procedures PRINCOMP and PRINQUAL were used. The number of factors in the linear case was determined w ith the Kaiser-Guttman criterion and the initial solu tion rotated with the PROMAX method toan obl ique solution. In the non- linear factor procedure (PRINQUAL) the data was transformed by the MTV method using non- monotonous spi i nes without knots (method SPLI NE) and the extracted number of factors fixed to the number of factors obtained in the linear solu tion to make comparison easier. RESU LTS Analysis of factor structures usually starts by taking a look at the correlation matrix and trying to see if the correlation coefficients between variables o f the same expected subspace are higher than their corre- lations with variables of other subspaces. Th is analy- sis was already made in a previous work by this au- thor (1) which led to this one, since the non-linear correlations differed sufficiently from the linear ones, 8 promising at least to some extent a somewhat changed structu re. Consequent ly, we shall bypass this analysis of the correlation matrix here, interest- ed readers are referred to the cited work. In order to make a comparison of the classic method w ith the non-linear one simpler, we executed both in the SAS statistical package. The purpose of this work was mainly to ascertain the appropriateness of the linear model and not the »real structure« of mor- phology, therefore we d id not attempt to analyse the data w ith various transformations, starting values or rotations - we just w ish to fi nd the concordance of the two models. To make comparison easier, we ex- tracted the same number of latent dimensions in both cases, w hich might not be the best idea, since it is possible t hat the non-linear procedure would ex- tract less latent d imensions- this has been left to fur- ther studies. =ranca Ambrožič LINEAR AND NON-LINEAR MORPHOLOGY STRUC-URE MODELS Table 1 : Comparison of t he linear and non-l inear so- lution - basic data PRINQ UAL MTV- iteration procedure lteration Average Largest Percentage Change change change of variance in variance 1 0.03703 2.46444 0.70288 0.00000 2 0.01152 0.63491 0.70669 0.00381 3 0.005F 0.39289 0 .70719 0.00050 4 0.00307 0.26-l65 o:T0732 0.00013 5 0.0021 5 0.1862.! o_-o-r O .0000 5 27 0.00001 0.00054 0_70744 0.00000 Factorl Factor2 Faaoc3 Fac:or-: Eigen value 9.2349 4.0255 1.7680 1 13,8 o/o variance 0.4015 0.1 750 0.0769 0.0.!95 Eigen value 9.2627 4.0859 1.7938 1.128- o/o variance 0.4027 0.1776 0.0780 0.0491 Legend: the upper two lines show the linear solution, lower two the non-linear one (first 4 factors) 1 We present the eigen values and percentage of ex- plained variance in the l inear and the non-linear so- lution (after transformation of variables) . Also the ini- tial solution, the factor pattern matrix and the corre- lations between the factors are shown. The fi rst information on the suitabili ty of the linear model in f inding the latentstructure of a space is the difference between the cornmon variance of th is space with the su pposition of li nearity of correlation between the variables and w ithout it. In table 1 we can notice that the final value of the iterative process (O . 707 44) is only slightly higher than the start ing val- ue (O. 70288) . This means thatthe use of a non-linear mode l d id not increase significant ly the comrnon variance. This does not necessarily mean that the la- tentstructu re w ill be the same, but it is a sign that the Table 2 : Comparison of linear and non-linear init ial solut ion F1 F2 FJ F4 N F1 N F2 NF3 NF4 VISINA 0.68712 -0.52125 0.31174 0.05422 0.68336 -0.52319 0.31930 0.06136 DUZIRU 0.62577 -0.55516 0.34379 0.03122 0.62365 -0.55234 0.35081 0.04416 DUZISA 0 .59277 -0.50170 -0.00051 0.34304 0 .59062 -0.50304 O 01676 0.33912 DUZINO 0.63335 -0.50751 0.42842 0.09333 0.63076 -0.50409 0.43229 0.10461 DUZIST 0.66918 -0.50830 0.19826 0.11354 0.66690 -0.50634 0.21032 0.10343 BIAKRO 0.58761 -0.21629 -0.09452 -0.04762 0.58909 -0.22572 -0.07600 -0.07867 DILAKT 0.6631 O -0.16182 -0.16027 0.05572 0.66545 -0.16387 -0.16919 0.07314 DIRUZC 0.39922 -0.27011 -0.36025 0.56516 0.36690 -0.27205 -0.41547 0.56863 SIRISA 0.60076 -0.22892 -0.33480 0.11295 0.59920 -0.23507 -0.32402 0.11428 BIKRIS 0.59875 -0.29955 0.19694 -0.3801 O 0.60119 -0.29406 0.20048 -0.3671 O DIKOLJ 0.52876 0.04067 0.41173 -0.35053 0.55289 0.04131 0.40191 -0.35753 SISTOP 0.5 3366 -0.31212 -0.26389 -0.16372 0.53757 -0.3 3562 -0.2 7144 -0.15232 NAPAZU 0.47677 0.67921 0.29984 0.19347 0.48262 0.68588 0.28012 0.1 91 27 NANALE 0.51494 0.68502 0.27903 0.09920 0.51699 0.68547 0.26521 0 .07646 NATRBU 0.44477 0.53199 -0.05464 0.40418 0.45261 0.56363 0.00517 0.38878 NANADL 0.42481 0.59177 0.45183 0.07328 0.43112 0.60333 0.43136 0.07192 NAPOTK \ 0.42866 0.55690 0.23204 0.15119 0.43458 0.55849 0.22066 0.18020 TEZINA 0.95928 0.03989 -0.09266 -0.03704 0.95969 0.02711 -0.0991 1 -0.03400 O PCRUD 0.79054 0.15169 -0.21205 -0.17921 0.79089 0.13880 -0.22201 -0.18068 OPNADL 0.73223 0.43243 -0.29194 -0.14049 0 .73560 0.42908 -0.29695 -0.14391 O PPODL 0.79005 0.21683 -0.29836 -0.21832 0.79030 0.21 123 -0.30626 -0.22126 O PNATK 0.78872 0.34728 -0.20341 -0.09385 0 .78848 0.34513 -0.2051 1 -0.08618 O PPOTK 0.74875 0.19414 -0.27829 -0.11998 0.74970 0.18405 -0.28849 -0.10249 Legend: F1 - F4 mark the factors o( the linear model, NF1 - NF4 o( the non-linear model Franci Ambrožič LINEAR AND NON-LINEAR MORPHOLOGY STRUCTURE MODELS 9 Table 3: Comparison of the pattern matrices of both models F1 F2 F3 F4 NF1 NF2 NF3 NF4 VISINA 0.00163 0.91232 0.03618 0 .02656 -0.00329 0.91 784 0.03219 0.01470 DUZIRU -0.05299 0.92788 0.00258 -0.00853 -0.05909 0.9311 8 0.00305 -0.01501 DUZISA 0.03716 0.67592 0.00040 0.41712 0.03136 0.69251 0.02039 0.39728 DUZINO -0.1 5850 0.97457 0.12826 0.01014 -0.16183 0.97715 0 .12546 -0.00052 DUZIST 0.05552 0.82095 0 .00461 0 .1 2616 0.05928 0.82504 0 .00265 0.09881 BIAKRO 0.42513 0.33142 -0.08803 0.05758 0.43299 0.34130 -0.09978 0.02015 DILAKT 0.46786 0.29806 -0.00976 0 .1 8049 0.46845 0.29564 0.00617 0.19843 DIRU ZG 0.13583 0.20072 0.02028 0.73451 0.16833 0.15598 0 .01845 0.76070 SIRISA 0.53932 0.19511 -0.14935 0.30912 0.53130 0.20503 -0.12712 0.30953 BIKRIS 0.37875 0.55007 -0.1581 1 -0.36387 0.37498 0.54097 -0.16591 -0.35958 DIKOLJ 0 .1 7940 0.45339 0.21775 -0.46683 0.20686 0.44794 0.18565 -0.48183 SISTOP 0.60812 0.22248 -0.34514 0 .02538 0.61232 0.22731 -0.35326 0.04865 NAPAZU -0.01229 0.01192 0.90968 0 .01478 -0.00081 0.00683 0 .90448 -0.00107 NANALE 0.09260 -0.00883 0.85496 -0.06311 0 .1 0949 -0.01409 0.83198 -0.09847 NATRBU 0.12320 -0.1 2905 0.69891 0.37215 0.07922 -0.09041 0.77271 0.31544 NANADL -0.11 274 0.14046 0 .85824 -0.15413 -0.09862 0.12911 0.84483 -0.17126 NAPOTK 0.02931 0.01482 0. 74265 0.01335 0 .01393 0.02387 0.75863 0.02729 TEZI NA 0.67404 0.32583 0.19841 0.06325 0 .67611 0.32783 0.19437 O 06248 OPGRUD 0.79566 0.05570 0.07493 -0.04465 0 .80209 0.05052 0 .06490 -0.04098 OPNADL 0.85222 -0.21611 0.23210 -0.01055 0.85371 -0.22020 0.23758 -0.01104 OPPODL 0.90940 -0.06205 0 .04385 -0.05309 0 .91493 -0.0721 O 0 .04163 -O 04971 OPNATK 0.76093 -0.05636 0.26942 0.01001 0 .75146 -0.05680 0.28175 0 .01440 OPPOTK 0.79795 -0.03089 0 .08769 0.03059 0 .79185 -0.03221 0.09688 0.05297 Legend: values in bold show the factor-defining variables and the underlined values those where the greatest differences between the two mode/s exist Table 4: Comparison between the factor correlation matrices of both models F1 F2 F3 F4 NF1 NF2 NF3 NF4 1.00000 0.44133 0.42928 0.13677 1.00000 0.44727 0.42922 0.10263 0.44133 1.00000 0.04753 0.11 015 0.44727 1.00000 0.04883 0.1 0353 0.42928 0.04753 1.00000 -0.12478 0.42922 0.04883 1.00000 -0.15562 0.13677 0.11015 -0.12478 1.00000 0.1 0263 0.10353 -0.15562 1.00000 linear model is not too bad (in this case) . A similar sit- uation can be seen from the eigen va lues and the percentage of explained variance of the individual factors- in both cases we have the solution given by four latent dimensions, if we use the Kaiser-Guttman criterion (1 > 1 ). The eigen values and explained vari- ance in the non-linear model (after the optimal trans- formation of the original variables) are only slightly higherthan in the linear model. Let us see what hap- pened with the latent structure of morphology. A comparison of both orthogonal solutions (table 2) shows unexpectedly small d ifferences, the largest difference in the projections on the first principal component is only 0.03232 (wrist diameter) and in other factors 0.05522 (same va riable). The latent structures are practically identical, w ithout doubtthe same variables define the latent dimensions before and after transformation. An orthogonal solution is not the best one in our case, since we know that hu- man characteristics, properties and abi lities are in- ter-correlated, therefore we usual ly perform some kind of oblique rotation where the latent space al- lows also correlations between the factors. In our case we used the PROMAX rotation, on ly the pat- tern matrix is presented since the possible differ- ences between the linear and non-linear model wi ll be there more evident than in the structure matrix. It is quite obvious that no large differences exist be- tween the two models also in the obliquely rotated factor solution (table 3). The same va riables aga in define the factors, maybe we could say that the non- linear structure is somewhat»cleaner« since the pro- jections are a little higher. The first latent dimension is a combination of voluminosity (circumferences and bodyweight) and transversal dimensionality (di- ameters), the second is longitudinal d imensionality and the third a very clear component of subcuta- neous fat. The existence of the fourth dimension is qu ite questionable, since it is actually defined by just one variable (wrist diameter) and could be pro- claimed a »single« factor even if itdoes contain some »add ition« of transversa l dimensional ity (knee di- ameter, pelvic width and hand width). Differences can be noted in justthree variables: w rist diameter, knee diameter and stomach skin fo ld · wh ich is not surprising since these variables are pre~ cisely the ones that had the most non-linear relations with the others (1). lf we take a closer look at w hat happened, we see that the variable wrist d iameter migrated from the second latent component (longi- tudinal dimensionality) to the first (voluminosity) and the fourth (transversal d imensionali ty), which is more logical since theory puts it there. The same thing happened with knee diameter and the variable stomach skin fold m igrated from al i the other com- ponents to the third (subcutaneous fat) where it be- longs. We can therefore say that the latent structure under the non-linear model is practically identical to that under the linear model, only that it is even more in accord with the theoretical model. Finally, let us look also at the correlations between the components (table 4), showing the association between the latent dimensions. The only difference we can see is t he correlation of the fourth compo- nent w ith the others. The strength of association with the first and second weakened, while it inc reased with the th ird. DISCUSSION Let us try to exp lain the obtained results. In studies where they used the linear model, the transversal di- mensionali ty component was obtained quite sel- dom, usually it merged with voluminosity or longi- tudinal dimensionality or the variables divided themselves between the two. Maybe the answer is actually in the non-linearity of some of the relations between the variables. In our (non-linear) example, the fourth component is defined by the variables wristdiameter, knee diameter and pelvic w idth. The other variables - foot w idth, hand width, elbow di- ameter and shoulder w idth, wh ich theory puts in the same component - correlate most w ith volum inosi- ty. Since we have al lowed also non-linear associa- tions the fourth component is cleaner and in conse- quence has a weaker correlation with voluminosity. ltwould be probablyworth th inkingabout strength- Franci Ambrožič LINEAR ANO NON-LINEAR MORPHOLOGY STRUCTURE MODELS en i ng th is su b-space w ith some add it ional variables to define it better. ' A com parison of both models of fi nd i ng latent struc- tu re showed a much greater congruence between the factors t han is shown by the pai rs of variables defining morphology. This is good, because it means that we do not need to revi se the already established latent structures and that the selection of variables re_presenting a particular sub-space is not problem- atI c. However, we feel that it is premature at this mo- ment to consider it a fact, since this is the find ing of one study, one sub-space of the psychosomatic sta- tus, one gender and one age category. It is namely quest1onable if these fi ndings w ill be confirmed also in other cases. It wil l be very interesting to see if the inclusion of a non-linear model will-at leastto some extent - abolish the great variability of the latent structu re in connection with gender and the age of the subJects in the samp le used. This study does, however, confirm the find ings of R.Joreskog (1967- 78, also Ba lderjahn (1989), Chou (1 991 ), Hu (1992), Muthen and Kaplan (1985, 1992), Tanaka (1984), Amemiya (1985), Browne (1985), Mooijaart and Ben ti er (1991 ), Satorra and Bentler (1990, 1991 ); ali in: Borg & Mohler - 4), w ho tested t he stability (ro- bustness) of factor analysis methods. 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