For anima l s c l a s s i fi ed as Jerseys , and wi thi n years , the mean l i ve wei g hts at each age 0ere p l o tted aga i nst thei r standard devi ations . The pl ots for di ffere nt yea rs were very d i fferent . For animal s born i n 1966 the regressi on equati on descri b i ng the pl ot of the means and
standard deviati ons , for the u ntransformed data , from b i rth to 64 weeks of age wa s ,
Y = 0 . 06093 X - 6 . 6327
whi l e i n the animal s born in 1968 the rel ati ons h i p ' s equati on was ,
Y = 0 . 1 2640 X - 0 . 5868
So to obta i n a constant o ri g i n and to achi eve normal i ty , by a three parameter l ogari thmi c transformati on (see Chapter 2 ) , 110 woul d have had to have been added to the l i ve wei g hts from animal s born i n 1966 wh i l e the addi ti on of 4 . 5 woul d have g i ven a constant ori g i n to the pl ots for anima l s born i n 1968 . The l arge di fferences i n these rel a ti ons h i ps between means and s tandard devi ations , between years , di d not al l ow the ca l cu l ati on of a s i ng l e correcti on to a constant ori gi n for a l l years . As compari sons between years were of i nterest the data was ana l ysed usi ng a two parameter l og transformati on of l i ve wei ght
even though wi thi n years the appropriate three parameter l og transformati on
woul d have been super i or (Chapter 4 ) .
Hancock ( 1 951) wa s atl e to show tha t the average percentage
dev i ati ons from the mean set we i ghts (for 10 sets of monozygoti c twi ns)
decl i ned rapidly u p to 12 week s of age and then decrea sed sl owl y to 76 weeks df age when the stage of p regnancy cau sed a ri se i n the dev i a ti ons . The pl ots of means and standard d ev i ati ons (Fi g . 5 . 1 ; Tabl e 5 . 2 ) di ffer from the pl ots of the wi thi n set devi ati ons presented by Hancock a s
both wi thi n and between s e t vari a ti ons are i ncl uded i n the present pl ots ; of whi ch the between set vari ati on prov i des the greater proporti on
(Tabl e 5 . 3 ) . However the present mea sure of vari ation (i . e . for the Jerseys born in 1966 , Fi g 5 . 1 ) agai n decl i ned sharply to weani ng ( 12-16 weeks of age), bu t then i ncreased after wean i ng (presumabl y due to the between set vari ati on si nce Hancock showed the wi thi n set vari ati on d i d not i ncrease at thi s age) fol l owed by a stati sti cal l y s i gn i fi cant
69
s l ower decl i ne i n vari ation ( compared to that before 16 weeks of age ,
Tabl e 5 . 2) unti l the onse·t of pregnancy . The changes i n standard
dev i a ti on were simi l ar i n other yea�s al though not as cl early defi ned , partl y because the mea su rements i n other years were ta ken at monthly
i nterva l s from an earl i er age ( Ta bl e 5 . 1), and possi bly due to
d i fferi ng envi ronmental condi ti ons i n the di fferent years .
TABLE 5 . 2
Ma ssey Monozygotic Twi ns ( Jerseys born i n 1 966)
Regress i on equati ons and the corre l a t i ons between the mean l i ve wei g hts and the i r standard devi ati ons , fol l owi ng l i ve we i ght ' s transforma ti on to l ogari thms , over the per i ods i nd i cated ,
CORRELAT I ON
AGES n SLOPE ± S . E . M . CONSTANT ± S.E . M . r
(weeks ) COEFF I C I ENT
1) 1 - 16 16 -0 . 1 108 ± 0 . 0167 27� 923 ± 3 . 344 -0 . 8714
2) 17-64 18 - 0 . 0410 ± 0 . 0085 1 4 . 1 46 ± 2 . 086 -0 . 7696
3 ) 68-92 7 0 . 0058 ± 0 . 0649 1 . 604 ± 1 7 . 904 +0 . 0401
Hancoc k fou nd that di fferences between animal s in thei r s tage of pregnancy caused the wi th i n pa i r vari ation to i ncrea se fol l owi ng concepti on . The correl ati ons between means and standard devi ations from the present s tudy i ndi cate ·tha t fol l owi ng 64 weeks of age the vari ati on i n l i ve wei g ht did not conti nue to decrease , on a
l ogari thmi c bas i s , conf i rmi ng Hancock ' s fi ndi ngs . So for th i s reason , and because pregnancy al so a l tered the shape of the growth curves , the regres s i ons of l i ve wei ght on age were ca l cul ated from the l i ve wei ghts up to 64 weeks of age .
For i nd i vi dual hei fers i n the Ma ssey herd pl ots of ari thmeti c l i ve wei ght agai nst age showed an a l most l i nea r rel ati onshi p from bi rth to 64 weeks of age , I t was found that us i ng l ogari thmi c metameters the sl ope changed con stantl y , and i ncreased , g i v i ng a curvi l i near rel ati onsh i p . A sui tabl e me tameter for age st1ou l d have
7 0
compared to the l ogari thmi c , a contracted scal e at the i ni ti a l ages , but an expanded scal e at l ater ages , to reduce the curvature (a pos i ti ve quadrati c ) towards l i neari ty . The l og tangent transforma ti on possesses these prope rti es . A l i near rel a t i onshi p between the l og tangent of
age ( one week be i ng equated to hal f a degree ) and the l og of l i ve wei gh t was i ndeed found . Whether the l og tangent metameter wou l d prove of val ue i n other s i tuati ons , for other data , i s probl emati cal . For each data set va ri ous rel ati ons h i ps s hou l d be tested ; app l yi ng transformati ons a pri ori s hou l d be avoi ded .
The s l opes and constants for i nd i v i dual hei fers , cal cul ated fol l owi ng the transformation of l i ve wei ght to l ogari thms and age
(weeks/ 2 ) to i ts l og tangen t , are g i ven i n Appendi x Tabl es 5 . 10 (Jerseys ) and 5 . 1 1 ( non-Jerseys ) . S l opes and constants were then subjected to anal yses of vari ance . The s l opes were al so adju sted for di fferences i n the constant by ana l yses of covari ance . No compari sons of the wi thi n group regres s i ons are g i ven i n the Append i ces Tabl es as such compari sons , wi th only 2 a n i ma l s per grou p , are not poss i bl e mathema ti ea l ly .
A COMPAR I SON OF TRANS FORMED AND UNTRANSFORMED REGRESS I ONS
Average da i ly ga i n is sometimes cal cul ated by a regres s i on of ari thmeti c wei ght on ari thmeti c age ( Chap ter 10) . Thi s method of analys i s was compared wi th the transformed regressi ons u s i ng the
measurements from the Jerseys born i n 1966 . These an i ma l s were chosen as they were the mos t frequentl y wei g hed hei fers and the i r transformed regress i ons had yi el ded the uncorrected ' F ' val ues of the greatest s i ze .
A compari son of these ari thmeti c ( Appendix Tabl e 5 . 9 ) and l og ari thmic ( Append i x Tabl e 5 . 6 ) analyses reveal s the advantages o f a l ogari thmi c ana l ysi s . Si nce , for the ari thmeti c regre s s i ons , the correl ati ons between s l opes and cons tants , a l though negati ve , were non- s i gni fi cant , compensatory g rowth cou l d not be demons trated
stati s t i cal l y . Therefore i n the ana l ys i s of covari ance the ' F ' val ue for sl.ope was only s l i ghtly al tered by fi tti ng the regress i on on bi rth wei ght .
36 .00 z i ffi � 31 . 2 0 �
�
LU c.. 9 V) 28.80 • F IGURES 5 . 2A & 5 . 2sRELATIONSHIPS BETWEEN SLOPES AND CON STANTS. FROM THE LINEAR EQUATIONS RELATING LOG LIVE WEIGHT TO LOG TAN AGE (WEEKS I 2) , FOR MASSEY MONO ZVGOTJC TWIN HEIFERS. BORN IN 1966 • ... ... .... . . ' FIGURE 5.2A 1966
TWIN PAIRS JOINED BY LINES
- - - -• REGRESSION. IGNORI NG PAIRS
Y = -D.276X + 69.65 AND 1968 , ... ... ..,. ... ... ... ... ... .... ... ... 24 . 0 0 � ;!!'; er .., I- :c <.!> � u.. 0 � g LU c.. 9 <n 1 29 .oo 34 . 00 31 .60 29.20 26.80 24.40 · - · ... ... ... ... 1 �5. �0 1 42 .60 1 56 . 2 0
CONSTANT (P.OUNDS) LOGl� X 100
... ... ... .... ... ... ... ... ... ... ... FIGURE 5 . 2B 1968 ...
TWIN PAIRS JOINED BY LINES
- - - - • REGRESSION. IGNORING PAI RS
Y • -D . 255X + 64.51 •.
--���
.,
' ....-
' ' ·,> . .... • I • •\
• ' ' ' 1 6 3 . 0 0 . I • ' 'CONSTANT (P.OUNDsl LOG10 x 100
From the anal yses of var i a nce the ' F ' ra tios for s l ope and constant were , respecti vel y , 15 and 8 times greater usi ng the l ogari thmic �cal e . Al thoug h the ' F ' val ue for the corrected sl opes wa s on l y 30% greater for the transformed regressi ons the tra nsformati ons used cl early gi ve g reater twi n effi cienc i es tha n coul d be cal cul ated on the ori g i nal ari thmeti c sca l es .
RESULTS AND D I SCUSS ION