HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A Study of the Reproducblty of Measuremets wth HUR Leg Eteso/Curl Research Le A mportat property of measuremets s that the results should be the same uder equvalet codtos save for the uavodable statstcal fluctuatos. I physologcal measuremets we do however ever have detcal codtos. Yet, f a perso performs, let us say, a umber of MVC sometrc cotractos wth a short spa of tme (a day or so) uder smlar crcumstaces, such that we ca eglect trag effects, we should epect to obta readgs of the peak force whch do ot vary too much. We wll assume that the measuremet values fluctuate aroud a mea value ad the try to determe the se of the fluctuatos. Our prelmary result s that the measuremet values of peak torque (sometrc ad dyamc tests), ad mamum agular velocty (dyamc test) wll be wth about 0% of the mea value 95% of the cases. Ths s comparable wth results cted for soketc devces whch clam a varato of the measuremet results (peak torque, total work, average force) of the order of 9 4%. The data used the preset study s derved form a prevous study 3. I oe of the schemes employed that study they tested a group of fve wome practsers of aerobcs. Durg oe day of testg the wome performed fve sometrc eteso tests ad fve fleo tests, ad fve sets of dyamc eteso tests wth varyg resstace ad for both legs (left, rght), usg the stadard HUR test meu wth HUR Leg Eteso/Curl. Suppose we have obtaed a seres of values > 0 (,,..., ) by repeatg a measuremet tmes uder smlar codtos. Oe way to decrbe the varato of the result s to form the parameters 4 () ˆ ˆ where the wth a hat deotes the average Mamum volutary cotracto. Quoted (Mäkkö ad Martkae, 000) see ref. below. 3 N Mäkkö, V Martkae, Isometrset ja Dyaamset Mttaukset (Oulu dakossalatos, 000). Ms. 4 Dvdg wth the average value make the varatos for dfferet subjects comparable.
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) () ˆ As a eample of the data we gve the results of the sometrc fleo (curl) tests (left leg, mamum torque Nm) the followg table 5 : 86 77 99 0 5 8 86 0 3 84 8 0 3 4 85 80 89 3 90 79 89 7 Each colum gves the fve results of each subject (N 5). For each colum we caclulate the averages () ad the the 5 5 5 values for the -parameter (). Ths s repeated for rght leg data, ad eteso rght ad left leg data. It s foud that vares betwee about 0. ad 0.. We ca draw a correspodg hstogram for the total data 6 : Fgure Frequecy of the varato of sometrc data (totque) 5 The sometrc test cossts of two measuremets of MVC peak torque ad the better result s take as the represetatve value of the test. The jot agle fleo (curl) measuremets was set to 40 degrees ad the eteso measuremets to 0 degrees. 6 Data cossted of 90 ( 00 5) data pots because two complete tests were dropped. I the dagrams we have used all the -pots though for every test (cosstg of fve measuremets) oe -pot s determed by the four others sce ther sum accordg to the defto () wll be ero. Usg all the - pots gves us smoother curves ad forces the average to be eactly ero.
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) A clearer pcture of the statstcs of the data emerges f we draw the cumulatve dstrbuto of the varato: Fgure Dstrbuto of the varato of the sometrc data Ths shows e.g. that about 85 % of the measuremets the varato s less tha 0.05:.e. the measured value does ot eceed the true mea value by more tha 5 %. The cotous le the dagram s the curve of the ormal dstrbuto (Gauss) wth mea 0 ad stadard devato 0.045. The stadard devato was estmated from s usg (3) s As we ca see from the fgure the gaussa dstrbuto seems to descrbe the dstrbuto of the varato the sometrc measuremet data qute well. The stadard devato of the measuremet data ca be estmated usg 0.045 ad equ (6) the Apped: 0. 05 Thus, based o ths data we ca say that a measuremet of sometrc peak torque wll about 95 percet of the cases be wth ±0 % (0% /) 7 of the epected mea value of such measuremets. 7 The 95% cofdece terval of a ormal dstrbuto s gve by ( -.96, +.96 ). 3
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) That s, f we make two measuremets o a perso ad obta results that dffer by more tha about 4% (0 : 4) 8, the, wth a 95 percet certaty we may epect the dfferece to be sgfcat ad ot wth the bouds of statstcal varato. I the dyamc eteso tests the peak torque ad mamum agular velocty was measured durg MVC at resstaces steps betwee ad 8 bar. Ths was repeated fve tmes. The torque ad velocty data of the dyamc tests was processed the same way as the torque data of the sometrc tests. For peak torque data we used the measuremets at 4 ad 8 bar, ad for velocty data we used measuremets at ad 4 bar resstace (from both rght ad left leg). (Note: ad 4 bar data treated separately show practcally the same dstrbutos whece t makes sese to lump them together.) The dstrbuto of the varato for the peak torque s preseted the followg fgure: Fgure 3 Dstr. of vara. of peak torque of the dyamc tests The ormal dstrbuto draw the fgure has a stadard devato 0.037. For the varato of mamum agular veloctes we obta the dstrbuto: 8 The dfferece of two depedet radom varables wth ormal dstrbuto N(, ) wll have a ormal dstrbuto N(0, ). Formally the se of the sgfcat dfferece of two measuremets m ad m could be epressed as 0. m + m. 4
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) Fgure 4 Dstr. of vara. of ma. agular velocty of the dy. data I ths case the stadard devato of the supermposed ormal dstrbuto curve s 0.04. I fgure 3 ad 4 we see that the agreemet wth the ormal dstrbuto s ot perfect. Ths could suggest that the measuremets does ot eactly follow a ormal dstrbuto aroud a mea value but stead shows a hgher cocetrato aroud the ero varato pot 0. Oe mght speculate though ths data does ot warrat ay far reachg coclusos ths respect (the bumps are pretty much wth the statstcal fluctuatos) - e.g. that the dyamc tests the motorc program actvated also stables the results. Submamal efforts whch perhaps allow for greater cotrol of the moto could also stable the results. To test ths more data would be eeded. The esmato of of the stadard devato usg (3) s also fraught wth some ucertaty. Ideed, f were depedet ormal varables wth the dstrbuto N(0, ), the the sum χ N would obey the Ch-square dstrbuto. If the degrees of freedom N s greater tha 30 the Ch-square dstrbuto s well appromated by a ormal dstrbuto N( N, N ). I our case we obta from by droppg every ffth varable sce accordg to () the sum of the fve measuremets of every test s ero; thus, oly 4 5 are depedet varables. Wth these cosderatos we get N (4/5) 00 80 ad Prob 80 [.96 80 χ 80 +.96 80] 0. 95 Thus the 95% cofdece terval for the estmate of the stadard devato 5
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) s N N N χ wll be.96, 80 +.96 80 whch s (0.85,.5 );.e. the stadard varato s wth 5% of the estmate, where we for ca use the estmate s or (3) whch wll be of the same se (aroud 0.04). The pot of ths s oly to show that o bg varatos the estmate of the stadard devato are to be epected. I cocluso: Based o the data from the aerobc test group we may epect that oe ca make measuremets of mamum torque (sometrc test), peak torque ad mamum agular velocty (dyamc test) wth HUR Leg eteso/curl such that the varato s less tha about 0 % 95 percet of the cases. Ths s e.g. o the same level as has bee reported for measuremets wth soketc maches. Tests wth other groups of subjects wll most lkely show the same patter f the measuremets are doe properly 9. No bg chage the rage of varato s to be epected, but oly ew data wll tell more about ths. For stace, subjects wth poor motor cotrol may be epected to show larger varatos the results. Mathematcal Apped Suppose the depedet radom varables (,,..., ) have the ormal dstrbuto N(, ) wth > > 0, the the probablty desty fucto (pdf) for (4) ˆ ˆ s gve qute accurately by 9 The test subjects should e.g. have eough tme to famlare themselves wth the equpmet before the measuremets. 6
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) (5) f ( ) + e 3 π + The followg pcture (Fgure 5) shows a smulated test seres based o geeratg a seres of 5 ( 5) ormally dstrbuted umbers ( 00, 5) ad calculatg the - umbers (). Ths s repeated 00 tmes gvg 500 datapots whose dstrbuto s plotted the fgure. The cotous le the pcture s obtaed usg 5, 00, ad 5 (ths correspods to the case 0.045) by umercally computg the dstrbuto of (5). Apparetly t fts the smulated dstrbuto very well. Fgure 5 Dstrbuto of smulated data. Cotous le s the dstrbuto computed from (5). For small (that s, for << ) (5) approaches a ormal dstrbuto wth the stadard devato gve by (6) It follows that (5) s well appromated by a ormal dstrbuto f we have 0 0 The codto says that the mea value must be cosderable larger tha the stadard devato of the average value. Ths s also a obvous requremet for (4) to be a meagful descrpto of varato. 7
HUR Techcal Report 000--9 verso.05 / Frak Borg (borgbros@ett.f) 8 (7) >> I the preset case (wth about 0.05 ad 5) we get for the left had sde (7) a umber aroud 000 so ths codto s well satsfed. Ths supports the use of (4) as a measure of varato ad the use of the ormal dstrbuto characterg the dstrbuto of the varato. A curous property of the dstrbuto (5) s that t lacks secod momet;.e, the stadard devato s ot defed for t (the tegral dverges). Ths s of o practcal cosequece here because very bg -values ( the tal of (5)) are physcally mpossble/rrelevat. Mathematcally oe may also adopt a alteratve defto of varato gve by (7) ˆ whch may be epected to be mathematcally better behaved certa respects. The varable (7) has geeral a complcated formula for the probabllty desty fucto. E.g. the case the varable (the square of that (7)) (8) ( ) y y + takes values the rage 0 to. If ad y are depedet radom varables wth the same N(,)-dstrbuto, the probablty desty fucto for (8) becomes (9) ( ) ( ) ( ) + 0 4!! 4 ) ( u u e du u e e f π π Ths dstrbuto s of terest oly for small values of /, cotrary to (5).