Predictive power of individual genetic and environmental factor scores

Twin Research (2000) 3, 99–108 © 2000 Macmillan Publishers Ltd All rights reserved 1369–0523/00 $15.00 www.nature.com/tr

Pr edi cti ve pow er of i ndi vi dual geneti c and envi r onmental factor scor es M arti ne A Thomi s1, Robert F Vl i eti nck 3, Hermi ne H M aes5, Cameron J Bl i mki e6, M arc van Leemputte2, A l brecht L Cl aessens1, Guy M archal 4 and Gaston P Beunen 1 1

Center for Physical Development Research, and 2Exercise Physiology and Biomechanics Laboratory, Faculty of Physical Education and Physiotherapy, Leuven 3 Center for Human Genetics, and 4Radiology Unit, Faculty of Medicine, Katholieke Universiteit Leuven, Belgium 5 Virginia Institute for Psychiatric and Behavioral Genetics, Department of Human Genetics, Virginia Commonwealth University Virginia, Richmond, VA, USA 6 Children’s Sport and Exercise Science, The New Children’s Hospital, Westmead, NSW, Australia Thi s study expl or es the use of an i ndi vi dual ’s geneti c (I GFS) and envi r onmental factor scor e (I EFS), constr ucted usi ng geneti c model fi tti ng of a mul ti var i ate str ength phenotype. M axi mal i sometr i c and dynami c str ength measur es, one maxi mal r epeti ti on l oad (1RM ) and muscl e cr osssecti onal ar ea (M CSA ) w er e measur ed i n 25 monozygoti c and 16 di zygoti c tw i n pai r s. The use of I GFS and I EFS i n pr edi cti ng the sensi ti vi ty to envi r onmental str ess w as eval uated by the associ ati on of the scor es w i th str ength tr ai ni ng gai ns after a 10-w eek hi gh r esi stance str ength tr ai ni ng pr ogr amme. Resul ts show a hi gh contr i buti on of geneti c factor s to the covar i ati on betw een maxi mal str ength and muscl e cr oss-secti onal ar ea (84–97%) at pr e-tr ai ni ng eval uati on. I ndi vi dual factor scor es expl ai ned the l ar gest par t of the var i ati on i n 1RM and other str ength measur es at pr etr ai ni ng and post-tr ai ni ng eval uati on r especti vel y. Genes that ar e sw i tched on due to tr ai ni ng str ess (gene–envi r onment i nter acti on) coul d expl ai n the decr ease i n expl ai ned var i ati on over ti me. A negati ve cor r el ati on w as found betw een I GFS and str ength tr ai ni ng gai ns (–0.24 to –0.51, P < 0.05); i ndi vi dual s w i th a hi gh I GFS tend to gai n l ess str ength than i ndi vi dual s w i th l ow I GFS. I ndi vi dual envi r onmental factor scor es have l ow er di ffer enti al pow er. The pr edi cti ve val ue of the I GFS has potenti al uti l i ty i n i denti fyi ng an i ndi vi dual ’s suscepti bi l i ty to envi r onmental str ess i n a var i ety of mul ti factor i al char acter i sti cs, eg di seases and i mpai r ments, and for sel ecti on of si b pai r s for QTL anal yses. Twin Research (2000) 3, 99–108. Keyw or ds: geneti c factor scores, tw i ns, predi cti on, envi ronmental stress, strength trai ni ng, si b pai r sel ecti on for QTLs

I ntr oducti on M ost common congeni tal mal formati ons (eg cl efti ng, spi na bi fi da, pyl ori c stenosi s, hi p di sl ocati on, cl ub feet), adul t di seases (eg i schemi c heart di sease, hypertensi on), and quanti fi abl e bi ol ogi cal trai ts (eg hei ght, w ei ght, bl ood pressure, i ntel l i gence) are mul ti factori al and are determi ned by both geneti c and envi ronmental factors. One of the major chal l enges i n the fi el d of human geneti cs i s to i denti fy i ndi vi dual s at hi gh ri sk for a gi ven di sease or a favourabl e phenotype, and to predi ct an i ndi vi dual ’s outcome i n an effi ci ent preventi on, trai ni ng or educati onal programme. The i mportance of geneti c and envi ronmental effects i n normal vari ati on can be studi ed usi ng data of geneti cal l y rel ated subjects.1,2 Correspondence: M Thomi s, Facul ty of Physi cal Educati on and Physi otherapy, Kathol i eke Uni versi tei t Leuven, Tervuursevest 101, B-3001 Leuven, Bel gi um. Tel : 32 16 32 90 86; Fax: 32 16 32 91 97; E-mai l : marti ne.thomi s@fl ok.kul euven.ac.be Recei ved 1 February 2000; accepted 9 February 2000

Recent devel opments use the technol ogi es of mol ecul ar bi ol ogy to map gene l oci expl ai ni ng vari ati on i n quanti tati ve trai ts (QTL).3 Heri tabi l i ty studi es esti mate the i mportance of geneti c factors at a popul ati on l evel . Stati sti cal procedures are now avai l abl e to esti mate i ndi vi dual l evel s of geneti c and envi ronmental determi nati on.4 In thi s paper w e use a pathanal yti c approach to construct i ndi vi dual geneti c and envi ronmental factor scores and to test w hether these scores predi ct the suscepti bi l i ty to envi ronmental changes. When mul ti vari ate observati ons are avai l abl e from geneti cal l y rel ated i ndi vi dual s, hypotheses can be tested about w hether the same envi ronmental and the same geneti c factors have pl ei otropi c i nfl uences on phenotypi cal l y correl ated measures.5,6 The parameters of such a geneti c factor anal ysi s can be esti mated, and i ndi vi dual genotypi c and envi ronmental factor scores (IGFS and IEFS) for each subject may be constructed by standard methods.4 The major questi on i n thi s study i s:

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Can i ndi vi dual geneti c or envi ronmental factor scores predi ct a subject’s suscepti bi l i ty to an envi ronmental stress factor, speci fi cal l y i n thi s study an arm fl exo trai ni ng programme? We hypothesi se that, dependi ng on the average heri tabi l i ty of the mul ti vari ate phenotype, the IGFS shoul d predi ct the val ue of the i ndi vi dual phenotype at l east at the pre-trai ni ng l evel . The hi gher the heri tabi l i ty of the trai ts, the smal l er the proporti on of the vari ance expl ai ned by non-shared envi ronmental factors, the greater the predi cti ve pow er of the IGFS. Large vari ati on i n envi ronmental scores, refl ecti ng a hi gh envi ronmental determi nati on, w i l l l ow er the predi cti ve pow er of the IGFS. The predi cti ve pow er of IGFS depends not onl y on the average heri tabl e and envi ronmental determi nati on of the trai ts but al so on the sw i tchi ng on of other genes i n response to the envi ronmental stress (genotype–envi ronment i nteracti on). The esti mati on of i ndi vi dual factor scores (IFS) i n the pre-stress condi ti on produces factor scores based on the geneti c and envi ronmental effects acti ng at thi s pre-stress l evel . If the stress acti vates new genes, then pre-stress IGFS w i l l predi ct the post-stress phenotypes and changes i n phenotypes l ess w el l . The mai n purpose of thi s study i s to test the predi cti ve pow er of i ndi vi dual geneti c and envi ronmental factor scores i n response to envi ronmental stress i n an empi ri cal trai ni ng study. A l though feasi bl e, i t i s ethi cal l y not desi rabl e to do i nterventi on studi es i n more rel evant mul ti factori al di seases such as hypertensi on, obesi ty, cancer of behavi oural di sorders etc, therefore w e deci ded to test the val i di ty of the i ndi vi dual factor scores on muscul ar strength. Speci fi c resi stance trai ni ng programmes are effecti ve i n i ncreasi ng muscl e strength and hypertrophy, and can, i n a standardi sed manner, be used as an envi ronmental stress factor i n untrai ned subjects. We constructed i ndi vi dual geneti c and envi ronmental factor scores for i sometri c arm strength and muscl e mass i n 25 monozygoti c and 16 di zygoti c mal e, adul t tw i n pai rs. The envi ronmental stress w as a 10-w eek heavy-resi stance trai ni ng programme for the el bow fl exors. Responses to thi s envi ronmental stress factor w ere measured as absol ute (post-trai ni ng mi nus pre-trai ni ng val ues) and rel ati ve i ncreases (gai n i n strength expressed as a percentage of i ni ti al strength) i n stati c and dynami c arm strength and muscl e hypertrophy after the trai ni ng programme. We hypothesi se that subjects w i th hi gher i ndi vi dual geneti c factor scores w i l l be l ess responsi ve to the envi ronmental stress (trai ni ng) than subjects w i th smal l er geneti c factor scores, and w i l l have correspondi ngl y smal l er strength gai ns fol l ow i ng trai ni ng. Twin Research

Subjects and methods Subjects The sampl e for thi s study consi sted of tw i n pai rs from Fl emi sh Brabant, Bel gi um. M al e vol unteer tw i ns 17–30 years of age w ere i ncl uded i f both members of a tw i n pai r had si mi l ar physi cal acti vi ty profi l es and had not started, nor recentl y stopped, strength trai ni ng duri ng the precedi ng year. Fortyone tw i n pai rs vol unteered, thei r mean age w as 22.4 ± 3.7 years. Subjects w ere ful l y i nformed of the measurement protocol before gi vi ng thei r w ri tten consent. The project w as approved by the l ocal medi cal ethi cs commi ttee. Zygosi ty w as determi ned by exami nati on of the fol l ow i ng geneti c markers: A BO, Rhesus (D, C, Cw, c, E, e), M NSs and Duffy(a,b). The pow er to detect di zygoti c (DZ) tw i ns w i th thi s set of geneti c markers w as 91% . Di fferences i n tw o geneti c markers w ere used to establ i sh di zygosi ty. The probabi l i ty of monozygosi ty (M Z) of pai rs w i th the same geneti c markers w as cal cul ated.7 A l l M Z pai rs had a probabi l i ty of monozygosi ty of at l east 95% . Tw enty-fi ve pai rs w ere cl assi fi ed as M Z and 16 as DZ. Training protocol Both members of the tw i n pai r parti ci pated i n a programme i n w hi ch mai nl y the el bow fl exors w ere trai ned. Duri ng 10 w eeks, fi ve sets of bi ceps curl s w ere performed, 3 ti mes a w eek on a trai ni ng apparatus (Kettl er Sport type 7408-150). Every w eek, the l oad of each set (w i th a preci si on of 0.5 kg) w as adjusted to each subject’s one repeti ti on maxi mal val ue (1RM ). The 1RM w as defi ned as the maxi mal resi stance that coul d be l i fted a si ngl e ti me through the ful l range of moti on. Duri ng each supervi sed trai ni ng sessi on, the fi rst set w as performed at 60% of 1RM w i th 14 repeti ti ons (reps), the second set at 75% of 1RM w i th 12 reps, the thi rd set at 80% of 1RM w i th 10 reps and 8 reps at 85% of 1RM for the fourth set. The fi fth set at 65% of 1RM w as performed unti l exhausti on. Measurement protocol and variables The esti mati on of IGFS and IEFS i s based on a mul ti vari ate phenotype. M easurements that eval uated maxi mal i sometri c strength i n the pre-trai ni ng condi ti on w ere chosen. These phenotypes consi sted of the maxi mal stati c vol untary contracti on at 140°, 110°, and 80° arm fl exi on (180° i s the arm i n ful l extensi on), and mean cross-secti onal arm muscl e area (cm 2). The eval uati on of maxi mal stati c vol untary contracti on w as done after one w eek of adaptati on to the trai ni ng apparatus usi ng l ow trai ni ng

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l oads (50–70% 1RM ) on a programmabl e dynamometer (Promett).8,9 Wi th thi s system, i sometri c, concentri c, and eccentri c contracti ons can be performed at di fferent speeds and ampl i tudes i mposed by the dynamometer. Subjects w ere asked to demonstrate maxi mal i sometri c strength and hol d i t for 3 seconds. The hi ghest regi stered moment duri ng thi s contracti on w as sel ected as the maxi mal i sometri c strength measurement expressed i n New ton meter (Nm). Test–retest correl ati ons ranged from 0.93 at the extreme angl es to 0.97 at the mi ddl e angl e (110°). The observer w as abl e to eval uate each subject’s maxi mal effort by vi sual i sed moment and el ectromyographi c si gnal s regi stered at M . bi ceps brachi i , M . brachi oradi al i s, M . brachi al i s, M . tri ceps brachi i and M . tri ceps brachi i . Computed tomography i magi ng scans w ere used to measure the mean crosssecti onal arm muscl e area.10 Techni cal error of measurement for muscl e area w as 0.16 cm 2 w i th a rel i abi l i ty of 0.99. The mean muscl e cross-secti onal area (M CSA ) of the four scans w as used i n the further anal yses. The dependent phenotypes to eval uate the strength gai n after trai ni ng w ere the absol ute and rel ati ve i ncreases i n 1RM , stati c strength at 110° fl exi on and strength at 140° fl exi on duri ng maxi mal concentri c contracti on at a speed of 60°/ s. Hypertrophi c adaptati ons to the heavy-resi stance strength programme w ere eval uated by absol ute and rel ati ve i ncreases i n mean muscl e cross-secti onal area of the arm, measured by CT-i magi ng.

Genetic analyses The causes of vari ati on i n muscl e cross-secti onal area and maxi mal stati c strength at the di fferent el bow angl es w as fi rst studi ed i n a uni vari ate w ay.11 The si gni fi cance of addi ti ve geneti c vari ati on, speci fi c envi ronmental factors and common envi ronmental factors or domi nance geneti c vari ance w as tested w i th model fi tti ng.2 In order to construct the i ndi vi dual geneti c and envi ronmental factor scores, a common factor anal yti c model (Fi gure 1) w as appl i ed to the mul ti vari ate phenotype. The l oadi ng of common and vari abl e-speci fi c l atent factors on the phenotypes w as esti mated usi ng maxi mum l i kel i hood esti mati on i n M x.12 These l oadi ngs w ere then used i n the esti mati on procedure for the IGFS and IEFS. Thi s procedure i s a regressi on method that mi ni mi ses the di fferences betw een esti mated and true factor scores.13 It i s the preferred method w hen the pri mary i nterest i s the i ndi vi dual factor scores.14 The Thurstone regressi on method for the esti mati on of factor scores13 i s preferred above the Bartl ett esti mator 15 because the correl ati ons betw een true and esti mated factor scores are hi gher and di fferences betw een si mul ated and predi cted vari ances of the factor scores w ere somew hat smal l er for the regressi on than for the Bartl ett method.4 M ore detai l s on the model fi tti ng procedure and the constructi on of IGFS and IEFS are gi ven i n A ppendi x A .

Fi gur e 1 Path-di agram of the mul ti vari ate geneti c anal ysi s. Phenotypes are encl osed i n squares, and l atent factors are encl osed i n ci rcl es. A c and Ec are the addi ti ve geneti c and non-shared envi ronmental factors that are common to al l phenotypes. A 1–4 and E1–4 are addi ti ve geneti c and non-shared envi ronmental factors that are speci fi c to each phenotype. The numbers at each causal uni -di recti onal path i ndi cate path coeffi ci ents. Doubl e-headed arrow s i ndi cate correl ati ons betw een l atent factors (betw een addi ti ve geneti c factors, 1 for M Z tw i ns and 0.5 for DZ tw i ns) Twin Research

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Analysis of the predictive value of individual factor scores The predi cti ve val ue of the i ndi vi dual geneti c and envi ronmental factor scores based on pre-trai ni ng val ues w as tested by the associ ati on of these scores w i th pre- and post-strai ni ng and absol ute and rel ati ve trai ni ng responses i n 1RM , i sometri c strength at 110°, concentri c strength at 140° fl exi on at 60°/ sec and muscl e cross-secti onal area. These associ ati ons w ere tested by correl ati on coeffi ci ents. The di stri buti ons of al l vari abl es w ere tested for Gaussi an normal i ty usi ng the Shapi ro-Wi l k test. We furthermore tested for bi rth order effects and di fferences i n mean and vari ances betw een tw i n types w i th t tests and F tests, respecti vel y. In al l tests, the stati sti cal si gni fi cance l evel w as chosen at P < 0.05.

Resul ts Univariate and multivariate genetic analysis A l l data w ere normal l y di stri buted. Uni vari ate geneti c model fi tti ng on pre-trai ni ng phenotypes i ndi cated that a model w i th addi ti ve geneti c factors and uni que envi ronmental factors w as the most parsi moni ous. For the mean muscl e cross-secti onal area (M CSA ), there w as evi dence for a phenotypi c i nteracti on factor (one tw i n’s l arger M CSA goi ng together w i th a smal l er M CSA i n hi s co-tw i n); how ever, thi s w as due to a smal l er total vari ance i n M Z tw i ns than i n DZ tw i ns. Uni vari ate heri tabi l i ti es w ere 0.92, 0.75, 0.78 and 0.66 for M CSA , and stati c moments at 140°, 110° and 80°, respecti vel y.11 The geneti c contri buti ons i n thi s study correspond to

other tw i n studi es measuri ng maxi mal stati c or dynami c strength by arm pul l , hand gri p, pul l -ups or combi ned strength scores (h 2 = 60–83% ),16–20 and to studi es that esti mate the heri tabi l i ty i n regi onal arm muscul ature.21–23 The geneti c common factor model (Fi gure 1, Tabl e 1) fi tted the data w el l (χ2 = 61.04, df = 56, P = 0.30). The resi dual (co)vari ance matri x al so show ed smal l val ues. The common geneti c factor (A c) expl ai ned the l argest part of the vari ati on i n each phenotype (64–76% ), w hi l e phenotype-speci fi c geneti c and envi ronmental factors w ere l ess i mportant (0 to 20% ). The common envi ronmental factor onl y contri buted 1% to 16% of the vari ati on i n each phenotype (Tabl e 1A ). How ever, l eavi ng out thi s common factor w orsened the fi t of the model si gni fi cantl y. The hi gh geneti c correl ati ons among the four phenotypes ( > 0.85) i ndi cated that the same genes i nfl uenced strength at di fferent el bow angl es as w el l as the muscl e cross-secti onal area. Nonshared envi ronmental correl ati ons w ere hi ghest betw een the strength measures, but l ow betw een muscl e mass and i sometri c strength (Tabl e 1B). Construction of genetic and environmental factor scores The di stri buti on of IGFS and IEFS w as Gaussi an. The mean val ue of al l factor scores (Tabl e 2) w as not si gni fi cantl y di fferent from zero, but the vari ati on i n both geneti c and especi al l y envi ronmental factor scores w as si gni fi cantl y smal l er than the expected (1.0). Confi dence i nterval s around the IGFS w ere smal l er than those around the IEFS, and the confi dence i nterval s around the factor scores tended to be l arger i n DZ than i n M Z tw i ns.

Table 1 (A) Proportion of explained variance in each phenotype by genetic and environmental factors. Legend and abbreviations as in Fi gure 1 (numbers i n superscri pt gi ve path coeffi ci ents as i n Fi gure 1). (B) Bi -vari ate geneti c and envi ronmental correl ati ons (above diagonal) and percentage of explained covariation explained by genetic and environmental factors (below diagonal) A

MCSA 140°b 110°c 80°d

Proportion of explained variance by genetic and environmental factors Genetic variation Environmental variation Ac A1 A2 A3 A4 Ec E1 E2 a

B MCSA a 140°b 110°c 80°d a

(1)

0.64 0.76(2) 0.72(3) 0.64(4)

(5)

(9)

0.18

0.00(6) 0.00(7) 0.05(8)

0.01 0.07(10) 0.16(11) 0.11(12)

E3

E4

(13)

0.16

0.17(14) 0.13(15) 0.20(16)

Genetic factors MCSA a

140°b

110°c

80°d

Environmental factors MCSA a 140°b

110°c

80°d

– 97 95 95

0.88 – 88 89

0.88 0.97 – 84

0.85 0.96 0.96 –

– 3 5 4

0.17 0.40 – 16

0.13 0.31 0.44 –

0.12 – 12 11

M CSA : muscl e cross-secti onal area; b140°: maxi mal stati c moment at 140° of el bow fl exi on; c110°: maxi mal stati c moment at 110° of elbow flexion; d80°: maximal static moment at 80° of elbow flexion.

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Table 2

Means, standard deviations and 95% confidence intervals (C.I.) of estimated factor scores

A

Total sample (n = 80) IGFS IEFS

Mean

0.006

0.015

SD

0.89

0.54b

MZ twins (n = 25) Twin 2

Twin 1

IGFS

IEFS

IGFS

IEFS

IGFS

IEFS

IGFS

0.000 0.93

0.000 0.61b

0.000 0.93

0.000 0.55b

0.027 0.90

0.086 0.37b

0.006 0.82

C.I.

± 0.60 (IGFS)

a

DZ twins (n = 16)

Twin 1

±1.5 (IEFS)

± 0.76 (IGFS)

Twin 2 IEFS –0.007 –0.58a

± 1.72 (IEFS)

b

P < 0.05; P < 0.01; significant difference from the expected standardised population parameters mean = 0 and SD = 1.

Strength training effects The effect of strength trai ni ng w as fi rst tested by anal ysi s of vari ance for repeated measurements. The one repeti ti on maxi mal strength i ncreased si gni fi cantl y by 45.4% on average, maxi mal i sometri c strength by 23.3% , and maxi mal concentri c strength by 24.9% . Hypertrophy of arm muscl e area w as smal l er but si gni fi cant (5.3% , P < 0.01). There w as a si gni fi cantl y l arger i ncrease i n M CSA i n DZ tw i ns than i n M Z tw i ns (F = 4.7, P < 0.05); how ever, no other zygosi ty i nteracti on effects w ere found, i ndi cati ng no di fference i n the muscul ar strength response to trai ni ng betw een M Z and DZ tw i ns. The vari ati on i n response to trai ni ng betw een i ndi vi dual s w as very l arge: the coeffi ci ent of vari ati on vari ed betw een 34% and 142% for the absol ute, and betw een 45% and 136% for rel ati ve changes i n 1RM and maxi mal concentri c strength respecti vel y. Changes i n 1RM scores, maxi mal i sometri c strength, and muscl e hypertrophy w ere comparabl e to trai ni ng effects in other strength trai ni ng programmes.24–28 Predictability and differential power of individual genetic and environmental factor scores Tabl e 3 show s the correl ati ons of IGFS and IEFS w i th the pre- and post-trai ni ng and absol ute and rel ati ve trai ni ng responses i n the four dependent phenotypes. IGFS w as hi ghl y posi ti vel y associ ated w i th both pre- and post-trai ni ng phenotypes (0.67–0.86). Subjects w i th a hi gh IGFS gai ned l ess absol ute and Table 3

rel ati ve strength as show n by the si gni fi cant negati ve correl ati ons w i th 1RM (–0.45––0.51), and rel ati ve change i n i sometri c and concentri c strength (–0.24). No si gni fi cant correl ati on w as found for the trai ni ng effects on muscl e mass (0.01, and –0.12). IEFS correl ated moderatel y w i th pre-trai ni ng i sometri c and concentri c strength (0.44 and 0.30, respecti vel y). No si gni fi cant associ ati on w as found betw een IEFS and trai ni ng effects i n the di fferent phenotypes, except for a si gni fi cant but l ow negati ve correl ati on w i th the i ncrease i n i sometri c strength (–0.24). Fi gure 2 show s the pow er of how w el l a subject’s basel i ne IFS predi cts hi s observed basel i ne 1RM strength and hi s future strength gai n after trai ni ng. In thi s fi gure, both IFS and 1RM scores w ere categori sed i nto quarti l e groups. Non-overl appi ng error bars i ndi cate si gni fi cant di fferences i n number of i ndi vi dual s posi ti oned i n the phenotypi cal quarti l e groups by contrasti ng the i ndi vi dual s accordi ng to tw o IFS quarti l e groups (A , B: IGFS < P25 agai nst IGFS * P75; and C, D: IEFS < P25 agai nst IEFS * P75). Before trai ni ng, subjects i n ei ther the l ow er or upper IGFS quarti l es al so had a hi gh or l ow 1RM strength score (Fi gure 2A ), w hi l e the extreme IEFS di d not di fferenti ate the subjects except somew hat i n the mi ddl e ( * P25– < P50) 1RM quarti l e (Fi gure 2C). For trai ni ng responses, an i nverse rel ati onshi p w as found. Subjects i n the hi gher IGFS quarti l e gai ned the l east strength (Fi gure 2B), w hereas subjects i n the l ow est IGFS quarti l e w ere i n the hi ghest quarti l es for thei r 1RM response. Contrasti ng the extremes of IEFS (Fi gure 2D) i ndi cated that i ndi vi dual s w i th a l ow IEFS at basel i ne tended

Correlations of IGFS and IEFS with the pre-, and post-training and training response phenotypes 1 RM a

Phenotypes isometric 110°

conc. 140° 60°/sec

MCSA b

IGFS with

pre-training post-training absolute change relative change

–0.79d –0.72d –0.45d –0.51d

–0.86d –0.71d –0.03 –0.24c

–0.78d –0.67d –0.15 –0.24c

–0.84d –0.83d –0.01 –0.12

IEFS with

pre-training post-training absolute change relative change

–0.05 –0.15 –0.09 –0.07

–0.44d –0.28c –0.14 –0.24c

–0.30c –0.22 –0.08 –0.18

–0.11 –0.10 –0.01 –0.03

a

1RM: one repetition maximal (kg); bMCSA: muscle cross-sectional area (cm 2); cP < 0.05; dP < 0.001. Twin Research

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Fi gur e 2 Di fferenti ati on i n 1RM phenotype by IGFS and IEFS. Both IFS and 1RM scores are categori zed i n four quarti l e groups. Nonoverl appi ng error bars i ndi cate si gni fi cant di fferences (* : P < 0.05) i n number of i ndi vi dual s posi ti oned i n the phenotypi cal quarti l e group by contrasti ng tw o IFS percenti l e groups. A and B compare the extreme quarti l es IGFS < P25 vs IGFS * P75, and C and D the extreme quarti l es IEFS < P25 vs IEFS * P75. The Y-axi s gi ves the number (n) of i ndi vi dual s i n a gi ven basel i ne IGFS or IEFS quarti l e w ho have 1RM strength scores i n a certai n quarti l e (X-axi s)

to have a l ow er i mpact from the trai ni ng than i ndi vi dual s w i th hi gher basel i ne IEFS scores. Expressed as a rel ati ve ri sk, i ndi vi dual s havi ng a l ow IGFS, had a 4.1 ti mes i ncreased chance of havi ng a l ow basel i ne strength (RR = 4.1, CI 95: 1.9 to 8.79), w hi l e havi ng a hi gh IGFS i ncreased the chance of havi ng a hi gh 1RM performance seven-fol d (RR = 7.0, CI 95: 3.1 to 15.76). The rel ati ve ri sk i n i ndi vi dual s w i th a l ow IEFS of a l ow strength performance w as not si gni fi cantl y di fferent from one, but havi ng a hi gh IEFS gave a tw o-fol d i ncreased chance of havi ng a hi gh basel i ne strength score (RR = 2.0, CI 93: 1.006 to 4.0). Indi vi dual s w i th a hi gh IGFS had a si x-fol d si gni fi cantl y decreased chance (RR = 0.16, CI 93: 0.02 to 0.99) of a hi gh trai ni ng response; a si mi l ar rel ati ve ri sk w as observed for i ndi vi dual s w i th a l ow IGFS to have a l ow trai ni ng response. Rel ati ve ri sks to predi ct strength gai ns after trai ni ng based on the IEFS scores w ere not si gni fi cantl y di fferent from 1. Twin Research

Di scussi on To our know l edge, thi s i s the fi rst study that has i nvesti gated the use of i ndi vi dual geneti c and envi ronmental factor scores to predi ct the sensi ti vi ty of an i ndi vi dual to an envi ronmental stress. Indi vi dual geneti c and envi ronmental factor scores w ere constructed from a mul ti vari ate common geneti c factor model . Thi s model gave a good expl anati on of the observed covari ati on (l ow χ2); how ever, a more parsi moni ous model coul d be devel oped.29 Indi vi dual genotypi c and envi ronmental scores expl ai ned 60–74% of the vari ati on i n pre-trai ni ng phenotypes (r 2 from Tabl e 3). The proporti on of expl ai ned vari ance decreased w hen post-trai ni ng phenotypes and trai ni ng effects w ere predi cted. A fi rst possi bl e cause of thi s decrease coul d be envi ronmental factors, other than the trai ni ng programme, i nfl uenci ng the phenotypes duri ng trai ni ng. A l though subjects w ere asked to mai ntai n pre-

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trai ni ng physi cal acti vi ti es duri ng trai ni ng, w hi ch w as moni tored by a 7-day recal l questi onnai re every w eek,30 changes i n physi cal acti vi ty, di et and other envi ronmental factors coul d not be enti rel y control l ed duri ng the 10-w eek study peri od. A nother i mportant possi bi l i ty, how ever, i s that other, ‘new ’ geneti c factors, that do not contri bute to the geneti c vari ati on i n the pre-trai ni ng strength, may be sw i tched on duri ng trai ni ng (gene–envi ronment i nteracti on). The proporti on of expl ai ned geneti c vari ance by ‘new ’ geneti c factors i n the post-trai ni ng phenotype coul d not be expl ai ned by IGFS that are constructed from pre-trai ni ng phenotypes. The i mportance of ‘new ’ geneti c factors affecti ng the post-trai ni ng phenotype can be tested i n a l ongi tudi nal model i n w hi ch a speci fi c geneti c factor that onl y causes vari ati on i n the post-trai ni ng phenotype but not i n the pre-trai ni ng phenotype i s i ncl uded. Thi s speci fi c gene–envi ronment i nteracti on w as si gni fi cant for the 1RM and maxi mal stati c moment at 110° fl exi on, and expl ai ned 19–21% of the vari ati on i n post-trai ni ng strength but not for maxi mal torque i n eccentri c muscl e w ork or concentri c muscl e w ork at l ow er vel oci ti es (30° and 60°/ s).31 Usi ng a tw o-w ay anal ysi s of vari ance method Thi baul t et al 32 found no evi dence for a si gni fi cant genotype–trai ni ng i nteracti on i n peak torque output after 10 w eeks of i soki neti c knee fl exi on/ extensi on trai ni ng i n fi ve M Z tw i ns. There w as no evi dence for speci fi c geneti c factors i n our data to i nfl uence post-trai ni ng muscl e cross-secti onal area.31 We are not aw are of any method i n the trai ni ng l i terature that predi cts i ndi vi dual responses to strength trai ni ng. The predi cti on of i ndi vi dual trai ni ng effects, based on the i ndi vi dual geneti c factor scores i n thi s study w as more accurate for phenotypes w i th l arger average trai ni ng effects (1RM and rel ati ve changes). The observed negati ve rel ati onshi p betw een pre-trai ni ng genotype (IGFS) and strength i ncrease, someti mes referred to as the l aw of i ni ti al val ues,33 has al so been reported i n earl y strength trai ni ng studi es.34 Stronger subjects gai n l ess strength w i th a resi stance-trai ni ng programme than i ndi vi dual s w i th l ess strength. The resul ts i ndi cated that IGFS not onl y cl assi fi ed subjects i nto hi gh and l ow strength groups, but al so, i n spi te of the l arge vari abi l i ty i n trai ni ng responses, predi cted thei r l ow or hi gh strength gai n: of tw o i ndi vi dual s w i th si mi l ar strength at basel i ne, the one w i th the hi ghest IGFS w i l l gai n l ess from trai ni ng than the one w i th the l ow est IGFS. The i ndi vi dual envi ronmental factor scores, how ever, di d not have the same pow er to categori se subjects, or to predi ct thei r future trai ni ng gai ns. Envi ronmental factors uni que to i ndi vi dual s w i th a common effect on al l measured phenotypes (Ec), such as previ ous trai ni ng status or di et, onl y expl ai ned a smal l part of the

observed covari ati on. The envi ronmental vari ati on, uni que for each phenotype (E1–E4), w as more i mportant. The predi cti ve val ue of the IGFS and IEFS w as, how ever, not si gni fi cantl y i mproved compared w i th the predi cti ve val ues of the raw phenotypi c scores (eg pre-M CSA scores predi cti ng post-M CSA scores). Probabl y the hi gh heri tabi l i ti es of the phenotypes coul d expl ai n these observati ons. In phenotypes w i th l ow er heri tabi l i ti es l i ke compl ex di seases or behavi oural trai ts, the gai n i n predi cti ve val ue of the IFS coul d be l arger compared w i th the raw scores of the phenotypes. Thi s study demonstrated the feasi bi l i ty of usi ng mul ti vari ate geneti c model fi tti ng i n the constructi on of i ndi vi dual geneti c and envi ronmental factor scores, and the use of model fi tti ng i n predi cti ng the response to strength trai ni ng. The same approach, how ever, has potenti al for appl i cati ons i n mul ti factori al di seases, such as hypertensi on or obesi ty. Quanti fyi ng the IGFS and IEFS of an i ndi vi dual w i th a hi gh-ri sk phenotype (eg di astol i c bl ood pressure above 90 mmHg or a systol i c val ue above 140 mmHg) coul d i ndi cate w hether the cause of a hi gh ri sk phenotype i s mai nl y a geneti c predi sposi ti on (hi gh IGFS) or an envi ronmental devi ati on (hi gh IEFS).4 Consequentl y therapeuti c strategi es may more effi ci entl y concentrate on concrete acti ons on the regul atory mechani sms of hypertensi on i n the case of a hi gh geneti c predi sposi ti on, or di mi ni sh the negati ve envi ronmental stress factors, i f subjects w i th hypertensi on express hi gh envi ronmental factor scores. Besi des eti ol ogi cal cl assi fi cati on, thi s approach mi ght al so predi ct the therapeuti c outcome, and even gui de the progress by moni tori ng the evol uti on of the IEFS. The w ei ght matri x A (see A ppendi x 1, equati on 3) can be cal cul ated based on mul ti vari ate data from tw i ns, or an extended tw i n and fami l y desi gn. Thi s w ei ght matri x i s then mul ti pl i ed by i ndi vi dual screeni ng data to obtai n IGFS and IEFS for each i ndi vi dual . The resul ts of thi s study shoul d be i nterpreted i n the context of the fol l ow i ng l i mi tati ons. A l though the sampl e i s one of the l argest i n an experi mental strength-trai ni ng desi gn, geneti c anal yses usual l y requi re l arger sampl es. The pow er of thi s study i s suffi ci ent to test for the si gni fi cant contri buti on of geneti c factors agai nst a model w i th sol el y uni que envi ronmental contri buti ons to the observed vari ati on. The detecti on of a smal l proporti on of addi ti onal fami l i al envi ronmental factors or geneti c domi nance w oul d requi re much l arger sampl es. In the mul ti vari ate case, how ever, pow er i ncreases due to addi ti onal i nformati on, al though the pow er to di scri mi nate betw een di fferent hypothesi sed model s i s sti l l smal l . Further, resul ts onl y appl y to young adul t men, w ho may not be representati ve of the general popul ati on. Twin Research

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IGFS and IEFS al so i mprove the pow er of mappi ng Quanti tati ve Trai t Loci (QTL). Present strategi es are based on i denti fyi ng pol ymorphi c marker al l el es that are i nheri ted i denti cal l y by descent (IBD).35 To i ncrease the pow er of mappi ng QTLs, three strategi es are suggested by Lander and Botstei n:36

Leuven. A t the ti me of the study M AT w as supported by the Fund for Sci enti fi c Research, Fl anders (Bel gi um), as a Research A ssi stant, at present a Postdoctoral Fel l ow. Dr H M aes i s supported by the Carman Trust and grants HL48148 and M H45268.

1) genotypi ng of si bs w i th extreme phenotypes; 2) mul ti poi nt i nterval mappi ng; and 3) reduci ng envi ronmental vari ati on and geneti c vari ati on not associ ated w i th the QTL. Recent si mul ati on studi es have deal t w i th these i ssues37–42 or w i th actual data.3 Boomsma41 reports a tw o-fol d i ncrease i n pow er to detect l i nkage betw een a tw o-al l el e quanti tati ve trai t l ocus and a ful l y i nformati ve marker usi ng the Haseman-El ston regressi on approach w hen usi ng squared di fferences of i ndi vi dual geneti c factor scores (based on a mul ti vari ate M Z and DZ tw i n anal ysi s, i ncl udi ng geneti c vari ance that i s not accounted for by the QTL and envi ronmental vari ati on) compared w i th squared di fferences of phenotypi c scores betw een si bs. In a recent paper Boomsma and Dol an 42 performed pow er cal cul ati ons (number of si b pai rs to be studi ed to detect l i nkage) i n w hi ch both si b pai r sel ecti on and QTL anal ysi s w as based on an i ndi vi dual geneti c factor score approach. The use of factor scores w as show n to be uni versal l y more pow erful than the use of just a mul ti vari ate or mean phenotypi c data approach to detect l i nkage. The l oss i n pow er of usi ng the same sampl e to both cal cul ate the factor score regressi on matri x and to carry out the QTL anal ysi s, outw ei ghed the gai n i n pow er attri butabl e to the use of factor scores. In summary, thi s study expl ored the use of i ndi vi dual geneti c and envi ronmental factor scores i n predi cti ng an i ndi vi dual ’s suscepti bi l i ty to envi ronmental stress. The l arge proporti on of expl ai ned vari ance i n pre- and post-trai ni ng strength as w el l as i n strength i ncreases by IGFS and IEFS l eads to appl i cati on of these scores i n the devel opment of i ndi vi dual strength trai ni ng programmes, reval i dati on programmes and screeni ng for el i te athl etes. Furthermore, the i denti fi cati on of geneti c and envi ronmental sources of devi ati on i n i ndi vi dual s, has a major fi el d of appl i cati on i n di fferenti ati ng hi gh-ri sk phenotypes i n several mul ti factori al di seases. A l so, sel ecti on of i ndi vi dual s based on di scordant IGFS coul d i ncrease the pow er to map quanti tati ve trai t l oci .

A ck now l edgements Thi s study w as supported by grant OT/ 92/ 27 from the Research Fund of the Kathol i eke Uni versi tei t Twin Research

Refer ences 1 Eaves L, Last K, Young P, M arti n N. M odel -fi tti ng approaches to the anal ysi s of human behavi or. Heredity 1978; 41: 249–320. 2 Neal e M C, Cardon LR. Methodology for Genetic Studies of Twins and Families. Kl uw er A cademi c: Dordrecht, 1992. 3 Cardon LR, Smi th SD, Ful ker DW, Ki mberl i ng WJ, Penni ngton BF, DeFri es JC. Quanti tati ve trai t l ocus for readi ng di sabi l i ty on chromosome 6. Science 1994; 266: 276–279. 4 Boomsma DI, M ol enaar PCM , Orl ebeke JF. Esti mati on of i ndi vi dual geneti c and envi ronmental factor scores. Genet Epidemiol 1990; 7: 83–91. 5 M arti n N, Eaves L. The geneti cal anal ysi s of covari ance structure. Heredity 1977; 38: 79–95. 6 M cA rdl e JJ, Gol dsmi th HH. A l ternati ve common-factor model s for mul ti vari ate bi ometri c anal yses. Behav Genet 1990; 20: 569–608. 7 M eul epas, E, Vl i eti nck R, Van den Berghe H. The probabi l i ty of di zygosi ty of phenotypi cal l y concordant tw i ns. Am J Hum Genet 1988; 43: 817–826. 8 Van Leemputte M , Wi l l ems EJ. EM G quanti fi cati on and i ts appl i cati on to the anal ysi s of human movements. Med Sports Sci 1987; 25: 177–194. 9 Vande Broek G, Van Leemputte M , A ndri es R, Wi l l ems EJ. M echani cal muscl e properti es after tw o types of pl yometri c trai ni ng. In: Barabas A , Fabi an GY (eds). Biometric in Sports XII. Uni versi ty Press: Sj okof, ´ 1995, pp 98–101. 10 Thomi s M , Cl aessens A L, Vl i eti nck R, M archal G, Beunen G. A ccuracy of anthropometri c esti mati on of muscl e-cross-secti onal area of the arm i n mal es. Am J Hum Biol 1997; 9: 73–86. 11 Thomi s M , Beunen G, Van Leemputte M , M aes H, Bl i mki e CJ, Cl aessens A L, M archal G, Wi l l ems E, Vl i eti nck R. Inheri tance of stati c and dynami c arm strength and some of i ts determi nants. Acta Physiol Scand 1998; 163: 59–72. 12 Neal e M C. Mx: Statistical Modeling, 4th edn., Department of Psychi atry: Ri chmond, VA , 1997. 13 Thurstone LL. The Vectors of the Mind. Uni versi ty of Chi cago Press: Chi cago, 1935. 14 Sari s WE, De Pi jper M , M ul der J. Opti mal procedures for esti mati on of factor scores. Sociol Meth Res 1978; 7: 85–106. 15 Bartl ett M S. The stati sti cal concepti on of mental factors. Br J Psychol 1937; 28: 97–104. 16 Wei ss V. Di e Heri tabi l i t a¨ ten sportl i cher Tests, berechnet aus den Lei stungen zehnj a¨ hrl i chen Zw i l l i ngspaare. Leistungssport 1977; 9: 58–61. 17 Pi rnay F, Cri el aard JM . Infl uence de l ’h e´ r e´ di t e´ sur l es performances physi ques. M edic ´ du Sport 1983; 57: 221–225. 18 Jones B, Kl i ssouras V. Geneti c vari ati on i n the strength– vel oci ty rel ati on of human muscl e. In: M al i na RM , Bouchard C (eds). Sport and Human Genetics. Human Ki neti cs: Champai gn, IL, 1986, pp 155–163. 19 Reed T, Fabsi tz R, Sel by J, Carmel l i D. Geneti c i nfl uences and hand gri p strength norms i n the NLBI tw i n study mal e aged 59–69. Ann Hum Biol 1991; 18: 425–432. 20 M aes HHM , Beunen GP, Vl i eti nck RF, Neal e M C, Thomi s M , Vanden Eynde B, Lysens R, Si mons J, Derom C, Derom R. Inheri tance of physi cal fi tness i n 10-year-ol d tw i ns and thei r parents. Med Sci Sports Exerc 1996; 28: 1479–1491.

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21 Hoshi H, A shi zaw a K, Kouchi M , Koyama C. On the i ntrapai r si mi l ari ty of Japanese monozygoti c tw i ns i n some somatol ogi cal trai ts. Okajimas Folia Anatomica Japonica 1982; 58: 675–686. 22 M aes HH, Beunen G, Vl i eti nck R. Heri tabi l i ty of heal th- and performance-rel ated fi tness. Data from the Leuven Longi tudi nal Tw i n Study. In: Duquet W, Day J (eds). Kinanthropometry IV. E & FN Spon: London, 1993, pp 140–149. 23 Loos R, Thomi s M , M aes HH, Beunen G, Feys E, Derom C, Vl i eti nck R. Geneti c anal ysi s of muscul ari ty i n earl y adol escence: sex-speci fi c regi onal changes of the geneti c structure w i th age. J Appl Physiol 1997; 82: 1802–1810. 24 Davi es J, Parker DF, Rutherford OM , Jones DA . Changes i n strength and cross secti onal area of the el bow fl exors as a resul t of i sometri c strength trai ni ng. Eur J Appl Physiol 1988; 57: 667–670. 25 Narci M V, Roi GS, Landoni L, M i netti A E, Cerretel l i P. Changes i n force, cross-secti onal area and neural acti vati on duri ng strength trai ni ng and detrai ni ng of the human quadri ceps. Eur J Appl Physiol 1989; 59: 310–319. 26 Sal e DG. Neural adaptati on to strength trai ni ng. In: Komi (ed.). Strength and Power in Sport: The Encyclopaedia of Sports Medicine. Bl ackw el l Sci enti fi c Publ i cati ons: Oxford, 1992, pp 249–265. 27 Kraemer WJ, Fl eck SJ, Evans WJ. Strength and pow er trai ni ng: physi ol ogi cal mechani sms of adaptati on. In: Hol l oszy (ed.). Exercise and Sport Science Reviews, vol 24. Wi l l i ams and Wi l ki ns: Bal ti more, 1996, pp 363–397. 28 Chambers RL, M cDermott JC. M ol ecul ar basi s of skel etal muscl e regenerati on. Can J Appl Physiol 1996; 21: 155–184. 29 Thomi s M , Van Leemputte M , M aes H, Bl i mki e CJ, Cl aessens A L, M archal G, Wi l l ems E, Vl i eti nck R, Beunen G. M ul ti vari ate geneti c anal ysi s of maxi mal i sometri c muscl e force at di fferent angl es. J Appl Physiol 1997; 82: 959–967. 30 Bl ai r SN, Haskel l WL, Ho P et al. A ssessment of habi tual physi cal acti vi ty by a seven-day recal l i n a communi ty survey and control l ed experi ments. Am J Epidemiol 1985; 122: 794–804. 31 Thomi s M , Beunen G, M aes H, Bl i mki e CJ, Van Leemputte M , Cl aessens A L, M archal G, Wi l l ems E, Vl i eti nck R. Strength trai ni ng: i mportance of geneti c factors. Med Sci Sports Exerc 1998; 30: 724–731. 32 Thi baul t M C, Si moneau J-A , Cˆot e´ C, Boul ay M R, Lagass´e P, M arcotte M , Bouchard C. Inheri tance of human muscl e enzyme adaptati on to i soki neti c strength trai ni ng. Hum Hered 1986; 36: 341–347. 33 Schutz RW. A nal yzi ng change. In: Safri t M J, Wood TM (eds). Measurement Concepts in Physical Education and Exercise Science. Human Ki neti cs: Champai gn, IL, 1989; pp 207–228. 34 Hetti nger TL, M ul l er EA . M uscul ar performance and trai ni ng. In: Brow n RC, Kenyon GS (eds). Classical Studies on Physical Activity. Prenti ce-Hal l : New Jersey, 1968, pp 262–276. 35 Haseman JK, El son RC. The i nvesti gati on of l i nkage betw een a quanti tati ve trai t and a marker l ocus. Behav Genet 1972; 2: 3–19. 36 Lander ES, Botstei n D. M appi ng mendal i an factors underl yi ng quanti tati ve trai ts usi ng RFLP l i nkage maps. Genet 1989; 121: 185–199. 37 Eaves L, M eyer J. Locati ng human quanti tati ve trai t l oci : gui del i nes for the sel ecti on of si bl i ng pai rs for genotypi ng. Behav Genet 1994; 24: 443–455. 38 Ri sch N, Zhang H. Extreme di scordant si b pai rs for mappi ng quanti tati ve trai t l oci i n humans. Science 1995; 268: 1584–1589. 39 Ri sch N, Zhang H. M appi ng quanti tati ve trai t l oci w i th extreme di scordant si b pai rs: sampl i ng consi derati ons. Am J Hum Genet 1995; 58: 836–843. 40 Eaves L, Neal e M C, M aes H. M ul ti vari ate mul ti poi nt l i nkage anal ysi s of quanti tati ve trai t l oci . Behav Genet 1996; 26: 519–525.

41 Boomsma DI. Usi ng mul ti vari ate geneti c model i ng to detect pl ei otropi c quanti tati ve trai t l oci . Behav Genet 1996; 26: 161–166. 42 Boomsma DI, Dol an CV. A compari son of pow er to detect a QTL i n si b-pai r data usi ng mul ti vari ate phenotypes, mean phenotypes, and factor scores. Behav Genet 1998; 28: 329–340. 43 Sage A P, M el sa JP. Estimation Theory with Application to Communications and Control. M cGraw -Hi l l : New York, 1971.

A ppendi x A The mul ti vari ate phenotype (P) (see Fi gure 1) can be expressed by the fol l ow i ng equati on: P(i j) = h c A c(j) + ec Ec(j) + h s A

(i j)

+ es E (i j)

(1),

w here i represents the four di fferent measured phenotypes (M CSA , STAT.M OM ENT 140°, STAT. M OM ENT 110°, STAT. M OM ENT 80°) and j the exami ned i ndi vi dual s (j = 1,…82). The phenotype P of each i ndi vi dual i s a functi on of hi s underl yi ng genotype (A c) that pl ei tropi cal l y i nfl uences al l four phenotypes, and envi ronmental factors (Ec) that are not shared i n fami l i es, and therefore uni que to each i ndi vi dual , but al so i nfl uence al l four phenotypes. The factor l oadi ngs of the measured phenotypes on the l atent factor A c and Ec are i ndi cated by h c and ec. The resi dual vari abl e-speci fi c vari ance i s al so parti ti oned i n geneti c factors (A 1 and A 4) and uni que envi ronmental factors (E1 to E4), path coeffi ci ents are i ndi cated by h s and es. The l oadi ng of each common and speci fi c factor i s esti mated by maxi mum l i kel i hood (M L) i n M x (Neal e12). The fol l ow i ng structural equati on i s sol ved: S ‰ Σ = Λ Ψ Λ' + Θ

(2),

w here S = observed 2p ⫻ 2p (p = number of phenotypes = 4) covari ance matri x of observati ons i n tw i n 1 and tw i n 2 (expressed i n devi ati ons from the group mean); Σ = predi cted 2p ⫻ 2p covari ance matri x of tw i n 1 and tw i n 2; Λ = 2p ⫻ 2m matri x, w here m = 2 i s the number of common l atent factors, contai ni ng the esti mated l oadi ngs of the common l atent factors on the four phenotypes of both tw i ns (path coeffi ci ents 1–4 and 9–12 i n Fi gure 1); the l oadi ngs are constrai ned to be equal for tw i n 1 and tw i n 2 and for M Z and DZ tw i ns; Ψ = 2m ⫻ 2m matri x of correl ati ons betw een the l atent factors; the correl ati on betw een A c i s 1 i n M Z tw i ns, 0.5 i n DZ tw i ns, the correl ati on Twin Research

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betw een the Ec factors i s 0 for both M Z and DZ tw i ns; Θ = 2p ⫻ 2p symmetri c matri x of esti mates of vari abl e-speci fi c uni que envi ronmental and geneti c vari ances that are equated for both members of the tw i n pai r and betw een M Z and DZ tw i ns (path coeffi ci ents 5–8 and 13–16 i n Fi gure 1). Wi thi n tw i n pai rs, the uni que geneti c factors are correl ated 1.0 i n M Z tw i ns and 0.5 i n DZ tw i ns. The constructi on of i ndi vi dual geneti c and envi ronmental factor scores i n the mul ti vari ate geneti c anal ysi s w as performed by the Thurstone regressi on techni que.4 The fol l ow i ng l i near expressi on w as used to obtai n the w ei ght matri x A , by mi ni mi si ng the sum of squares of the di fference betw een esti mated and true factor scores: A = Ψ Λ' (ΛΨΛ' + Θ)–1

(3),

w here Λ = matri x of l oadi ngs from mul ti vari ate phenotypes on common factor A c and Ec; Ψ = matri x of correl ati ons betw een l atent factors; Θ = di agonal matri x of uni que geneti c and envi ronmental vari ati on. Thi s w ei ght matri x w as used to compute factor scores for each subject by mul ti pl yi ng the w ei ght matri x by both the subject’s mul ti vari ate phenotypi c scores and hi s co-tw i n’s phenotypi c scores.

Twin Research

FSC = A P'

(4),

w here Fsc = [IGFS1, IEFS1, IGFS2, IEFS2], i s the vector of i ndi vi dual factor scores, IGFS = i ndi vi dual geneti c factor score, IEFS = i ndi vi dual envi ronmental factor score, subscri pts 1 and 2 i ndi cate tw i n 1 and tw i n 2; P = the measured mul ti vari ate phenotype of observati ons of tw i n 1 and tw i n 2, expressed i n Z-scores (n ⫻ 2p); A = (2m ⫻ 2p) w ei ght matri x, deri ved from equati on (3). Tw o-si ded, 95% confi dence i nterval s of the factor scores w ere cal cul ated as IGFS ± 1.96 ⫻ SEIGFS and IEFS ± 1.96 ⫻ SEIEFS for both M Z and DZ tw i ns. SEIGFS and SEIEFS are the square root of the di agonal el ements of the matri x from equati on 5, w hi ch i s a 2m ⫻ 2m matri x of the sampl i ng di stri buti on of constructed factor scores. M atri x V i s cal cul ated based on the factor l oadi ngs i n matri x Λ, the correl ati ons betw een the common factors i n matri x Ψ, and the esti mated covari ance matri x Σ of respecti vel y M Z and DZ tw i ns. Thi s fol l ow s from standard Kal man fi l teri ng techni ques:43 V = Ψ[Ψ–1 – Λ' Σ–1Λ]Ψ

(5).

The confi dence i nterval s onl y depend on the factor l oadi ngs and the amount of vari abl e-speci fi c uni que vari ance (i n Σ) and w i l l i ncrease i f the proporti on of uni que vari ance i ncreases.