ll B RAR.Y OF THE UNIVERSITY OF ILLINOIS 630.7 no. 95-312 co p. 2, AGRICW.TURE NON CIRCULATING CHECK FOR UNBOUND CIRCULATING COPY UNIVERSITY OF ILLINOIS Agricultural Experiment Station BULLETIN No.' 302 GROWTH AND SENESCENCE IN PUREBRED JERSEY COWS BY F. A. DAVIDSON URBANA, ILLINOIS, JANUARY, 1928 CONTENTS PAGE INTRODUCTION : 183 THE PROBLEM 184 SOURCE OF DATA 185 BIOMETRICAL ANALYSIS OF DATA 185 Course of Growth in Register-of -Merit Jersey Cows as Described by In- crease in Body Weight with Advancing Age 185 Truncation of Yearly Butterfat Frequency Distributions Due to Selective Effect of Register-of-Merit Requirement 199 Comparison of Means (V) and Standard Deviations (a) on Original Scale with Means (a) and Standard Deviations (s) on Logarithmic Scale . 209 Course of Growth and Senescence as Described by Rise and Fall in Yearly Butterfat Yields with Advancing Age 212 Influence of Level of Production Upon Age Curve of Milk Secretion 223 Nature of Selection of Register-of-Merit Production Requirement 224 SUMMARY 225 Course of Growth in Body Weight 226 Course of Growth and Senescence as Described by Rise and Fall in Yearly Butterfat Yields with Advancing Age 226 LITERATURE CITED 229 APPENDIX.. . 231 GROWTH AND SENESCENCE IN PUREBRED JERSEY COWS BY F. A. DAVIDSON* INTRODUCTION The American Jersey Cattle Club early established a special regis- ter known as the Register of Merit, in which all purebred Jersey cows are eligible to entry upon the fulfilment of a minimum milk-produc- tion requirement. This requirement is based upon the pounds of but- terfat produced during a single lactation and varies linearly with the age of the cows. Any purebred Jersey cow may have her production record entered in the Register of Merit as many times as she meets the requirement for the age at which her production is made. The cows that have only one record entered in the Register are spoken of as original-entry cows, those having more than one record entered are spoken of as reentry cows. Many thousands of entries have been made in the Register with the result that a large body of data on the milk production of Jersey cows has accumulated. These production records, however, are not quite representative of the production records of the purebred cows composing the breed as a whole owing to the selective and environmental influences imposed upon the cows entered in the Register. Gowen (1920) pointed out that the production requirement made by the Register of Merit eliminates many of the low-producing cows of the breed, which results in the truncation of the yearly butterfat- yield frequency distributions of the Register eows at successive ages. Gowen, altho recognizing this selective effect of the production require- ment, made no effort to correct for it and instead used the production records of a single herd of purebred Jersey cows as representative of the productions of the cows in the breed as a whole. It is a common practice among breeders to provide the best possi- ble environment, from the standpoint of both growth and production, for the cows they intend to submit for entry in the Register of Merit. Hence the reentry cows have a better chance to develop than the origi- nal-entry cows, since they are subjected for more than one lactation to This investigation was started by Mr. Davidson when a member of the Dairy Department of the University of Illinois and was completed by him under the direction of Professor Sewall Wright at the University of Chicago. The author wishes to express his appreciation to Professor Wright for his guidance and many valuable suggestions during the progress of the study. Submitted for publication September 14, 1927. 183 184 BULLETIN No. 302 [January, an environment highly stimulating to growth. Kildee and McCandlish (1916), Eckles (1918, 1920), and McCandlish (1920) all have demon- strated very clearly the influence of environment upon the rate of growth and milk secretion in dairy cows. Cows provided with the best possible environment show a distinctly superior rate of growth and milk production. There is also an inclination among breeders to select only the highest-producing cows in their herds to submit for reentry in the Register of Merit. Such a practice would have a tendency to bring about a genetic superiority of the reentry cows over the original- entry cows from the standpoint of milk production. This difference between the reentry and original-entry cows, altho never actually demonstrated by the breeders, has long been recognized by them, and as will be shown later, is not without justification. Since the reentry cows are kept under an environment which gives them a better chance to develop than the original-entry cows and since the former are also subject to selection by the breeders, it does not seem logical to lump the records of both the reentry and original-entry cows together when using them to make a study of the course of growth and senescence in purebred Jersey cows. Many investigators Pearl, Gowen, and Miner (1919), Hooper (1921), Brody, Ragsdale, and Turner (1923a), Turner, Ragsdale, and Brody (1924), and Graves and Fohrman (1925) have used these Register-of -Merit records for this purpose, but with the exception of Graves and Fohrman, all have lumped the original-entry and reentry records together in their studies. The latter have been the only investigators to separate the original- entry from the reentry records and study them separately. The results from this investigation, which will be discussed in more detail in another section, show that there is a marked difference between the original-entry and reentry cows, as measured by the rise and fall in their yearly butterfat productions with advancing age. THE PROBLEM In this study the Register-of-Merit records of the original-entry and reentry Jersey cows have been analyzed separately by means of biometrical methods from the following points of view: 1. The course of growth in Register-of-Merit Jersey cows as de- scribed by the increase in their body weights with advancing age. 2. The course of growth and senescence in Register-of-Merit Jer- sey cows as described by the rise and fall in their yearly butterfat yields with advancing age. This involved the preliminary problem of correcting for the truncation of the yearly butterfat frequency dis- 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 185 tributions at successive ages, due to the selective effect of the Regis- ter-of-Merit production requirement. 3. A comparison of the course -of growth and senescence in the original-entry and reentry cows. SOURCE OF DATA % The Register-of-Merit records involved in this study consist of all of the 365-day original-entry and reentry records as published in the yearly volumes of the Register of Merit up to and inclusive of the volume for 1920. These volumes contain 9,694 original-entry records and 2,628 reentry records, or a total of 12,322 records in all. Each record as published in the Register of Merit includes the following items of information concerning the cow. 1. Age at the beginning of the lactation period 2. Weight at the beginning of the lactation period 3. Length of the record in days 4. Total milk yield 5. Total butterfat yield 6. Percentage of butterfat in the milk 7. Designation of entry, whether first, second, etc. BIOMETRICAL ANALYSIS OF DATA COURSE OF GROWTH IN REGISTER-OF-MERIT JERSEY Cows AS DESCRIBED BY INCREASE IN BODY WEIGHT WITH ADVANCING AGE Dairy cows are peculiar in that they show very little, if any, tendency to fattening, this being especially true of Register-of-Merit Jersey cows. Hence the increase in the body weights of these cows with advancing age may be used as a fair measure of their rate of growth. The frequency distributions of the body weights for the Register- of-Merit Jersey cows at successive ages following 1.5 years of age for the original-entry cows and 2.5 years of age for the reentry cows, are reported in Tables 1 and 2 and Figs. 1 and 2 respectively. The histo- grams in Figs. 1 and 2 all tend towards the symmetrical or normal type. These histograms, however, are not representative of the true type of frequency distribution of body weight, owing to the fact that the body weights of the cows are in part estimated. The original-entry records include 1,096 records for which actual body weights of the cows are listed. These records were analyzed separately and the frequency distributions of body weight for these cows are reported in Table 3 and the last four distributions in Fig. 1. The fitted histograms in Fig. 1 may be assumed to represent the true type of distribution of body weight for Jersey cows. It will be noticed that these frequency dis- 186 BULLETIN No. 302 [January, 1 fc I-H O 5 o & o &. o OQ fc O I Q (M S5 . 1-1 i-c CO Tt< OS CO t- -IN MS O CN CO * i-l t>- Hostoo^^-i ;;;;;;: g liOO ^H N OS t>. CN >0 <-< CN OS (N t~ 00 00 CN 00 OS h- (N CO Op CO Q -^ O (N CO * O * ^ 00 O CO IO * -H i-c ^ t, OS rt< CO CN -! 1 IN o os os t^ g ^i t^ oo m -^ IN JL iH .1-1 IN t- IN IN -H O * to OS Tf CO . . . i-< IN t^ 00 p tO to 1-1 1 ... . Tj( rH 00 O t^ >O CO CO CO . O .... I-H -^H O O CO 00 to i-< 1 . . .i-icO5'-icOCOCOr~'3b-'COCO * ... ^tococo" O ... i-( CS *O I-H OS to 00 t to i^ "5 . . . - ;-c > co co t^ do to to co A"~ < rt CO ^< CO CO O COtOi-iOSOOOOXCOT(NCO5 i-ir-oo'nosiot^ia' g It jopoocociooooooccocicioiioo ) i.-; O >o p '" p o p 'C o c Q >_o o ^ O o o l ^ O to CD os ^ spnnod ui ?t{8i3^ 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 187 U5 oo Ci 4 CO 1 (Nl^t-CO o> 2 i - -CN^OO^OO^IN-H^ e s i . . i-l U5 U5 CO X5 CO IN N - 6 * 4 .*r.r.OO0- 00 c> ^ J, . -rH .CO^O^-OrHC, .rH . . . . X t- i .COO^-HOOOt^rH ^ 00 rt ^ t -HINtNINrHrH 2 i K< CN CO 00 OJ 00 IN CO IN IN rH rH . . . J, rHcooocN>acoo5t->ooco (J i CO CO CO CO CO rH rH 4 COCOcOOt*OCCOOOOOCOrH . . . . . ** r-tCOCOt^^^rH ...... i 4 CO CO rH f* CO tN Tjl t. CO N rH rH K CO 8 i rH rH CO X5 O Ol OS 00 O "5 ^" ,- CO rH If ^1 KJ CO CO N o >0 CO * rH UJ ^^ M QoioioQooooooooOQOo ' L T O L - O '-"^ O '-"^ O l ~ O '-^ O '" O O O >O O "o h H o CO o o i CN 5 4 ^C > ^ a i CO N 00 CO CN CO H o i (NrHrH^tUJrHrHrH 1 o h 01 i r-9 4 06 : : : :"S3 . . . 4 t- . rH (NrH ' '. '. '. CO I~ g s o 4 CO rH CO * OS 00 "5 OS "5 rH rH . . rH g 1 1 1 4 IO W ^l IO 00 CO CO 00 00 CO >O CO .1-1 3 o .3 II 4 Tl< : : :-" 2 : : S i ^Ot-OCO-HCOININ-H ^ * H o o 4 CO NrH ' '.'.'. -* UTIONS 4 CO IN d t~ CO t i lO 00 1~ CO .-1-1.. rHrHtNrH 8 1 4 o rHrHMrHrHrH '.'.'.'. o QUENCY 4 tN a8l5535Ra : : : : R 1 O o*Mdi t4dl Wctfit w P,und3 FIG. 1. FREJQUENCY DISTRIBUTIONS OF BODY WEIGHTS FOR ORIGINAL-ENTRY JERSEY Cows 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 189 tributions are not symmetrical but are skewed in the positive direction. A survey of the literature reveals the fact that the frequency distri- butions of the body weights and the weights of the parts of the bodies of animals, and even the weights of plant seeds, are of this type, that is, skewed in the positive direction. U gssisii i nl 3 IS >oV W/j*/ ;/ Pounds a i s 8 1 i i n * Body Weight in FbvHds FIG. 2. FREQUENCY DISTRIBUTIONS OF BODY WEIGHTS FOR REENTRY JERSEY Cows When the frequency distributions of the actual body-weight data are transformed to the logarithmic basis, that is, instead of the body weight the logarithm of the body weight is used as the variable, these distributions tend to become symmetrical and of the normal type. This peculiarity of these frequency distributions suggests that the 190 BULLETIN No. 302 [January, X 3 5 O M c K o 1-1 H ^ C s: - oo > S ll Z I z H i 3 a H t 3 ssss O si OS WOO 1 o 6 5 x "x g ai eom' oo fl> Cs O DO Cs "3 c * 3 o - t- CO CD 00 ^ o o 3.2 ^ H Jj H-H-H-H f oocoo 0000 a c cl^H iS H-H-H-H 00 00 00 OS 8OOO ooo -H-H-H-H < CO O CD OOOO ocooor~ 8888 C3 -H-H-H-H OCOf lOico OS OS OS OS IN CN IN IN 4 CNCOOOO -TO 'CCS COINi-HCO E c I 1/5 m iow - + a 3"o 3 2 1 CO CO O Ci O CO W O W CO ^^ Ci 03 1 -H-H -H-H-H-H -H-H -H -H -H -H-H O> Ol CN OS OS C< 00 1** f fH O COOS 09 -H CN CO -! -H -! CN CN >-l O O Ot>. 1 1 * * * * J 5 5 3 efficient c "3 3 o 09 5 < > O 1 H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H U e ^^rHddd-dddos'dosos'dos'dooeo a H H w S t D tOSOOOCfO>O 'CO -O 'CO COOO H 1 i e 3 u IO CN O U5 lO IO t^. IO *CO >k O -CD kOf II 11 II II II It _l_l 11 1 1 1 1 I 1 III! COiOOTjJcOOOh- .00 -CO -CO Of ~ -P H 'S 5 ^ leviatio * * * * J S < -2 2-S Standard 09 SSS'"28S3S*'?SS-2S8a3 try cows. Estimated an Estimated - ^i rtrtrt-HlNINCNININcNCNCOCOCOf "Of -H -H -H -H -H -H -H -H -H -H -H -H +HH -H -H -H -H -H 8.5 respectively of all original-en a O CN t CO i-i f CO I O O >O -f f -lOlO-HCOt^COOO f CN CO -OSO || io'S 3 5 Q 5 m 1 o "3 E oo f t^ t-~ oo t^ O t^ os b- OS .r^t-H -H-H-H-H-H-H+I-H -H -H -H :-H-H 00 -< CO tD OS OS CO CN f i-l CO -COf IO CO r~ s. x X X X cs o: Cs C! cs c: OS Cv c; * * * * cs are 5 . 5, 6 . 5, ghts include the < 3 5 Mean in i ctual** ro g3g-SgSSSoSg32 jse statist! actual wei 4 j ^ -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H -H 2-3 o "** H Estimat esansaassnausus ?1 c 8 I o . CO *The age **The es -.CNCNCOCOf f lOUSCOCCK.t-COCOOSO.-CN 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 191 factors affecting the body weights of these cows tend to have constant percentage effects, rather than constant absolute effects, thruout the entire scale. McAlister (1879) has shown that this type of frequency distribution can be fitted by the log-transformed equation of the normal frequency curve, the equation of the curve being y = = e in which W body weight, y = the ordinate, and a and s are the mean and standard deviation on the log scale. In order to make certain that the frequency distributions of the actual body-weight data were of the above-mentioned skewed type, they were fitted by both the normal frequency curve and the log-transformed frequency curve, the constants of which are reported in Table 4. In this table are also included the x 2 and probability values (P) measur- * fill orq/rratertfry ccwe -- Ory/'naffitfrf Ctwa *i/k actual Age /M Years (Jttwer clat) lifiifo) FIG. 3. MEAN BODY WEIGHTS OF ORIGINAL-ENTRY Cows ing the goodness of fit of these curves. In every case it will be found that the log-transformed frequency curve gives the better fit to the data. The fitted curves in Fig. 1 are the log-transformed curves. Altho the frequency distributions of the estimated plus the actual body-weight data do not represent the true type of frequency distri- bution of body weight, the means of these distributions show practi- cally the same trend with advancing age as do the means of the actual body-weight distributions (see Table 5 and Fig. 3). Gowen (1925) made a comparison between the estimated and actual weights of pure- bred Holstein cows and found that the means and standard deviations were practically the same in the two sets of data. Hence owing to the 192 BULLETIN No. 302 [January, 03 & O o fc B H tf I fc w 5 i i o i o P9 JH O c ffl fa o B! t CO w top^'^asoooocO'-KNb-mcO'-i CON -o i-c 00 O5 t- -H O> .-I .-< OS OS N CO Tf -J< CO * 00 -0) O IN IN IN CO P* CO CO CO * TJI in Tf U5 00 * -O :-H-H-H-H-H-H+l-H-H-H-H-H-H-H-H+l+l :- 5O (N Tf O 00 1-1 W 00 -i 00 O5 ^ ' -H -H -H -H -H -H -H -H -H-H -H +1 -H -H-H -H -H +I-H -H o >c ^< Tj< I-H o oo o b- n * 1-1 "5 o> oo "5 co >o o f- a> <-i oo oo 1-1 co * IN to t~ <* -o o * >n 'W 00 IN N (N C<5 IN IN CO OS C<3 "* * "O * K3 00 T l-H-H+i-H-H-H+l-H-H-H-H-H-H+l-H+l+l :+l OJ * 1C O O> CO N. O 1C I-H O >-i -- 1 00 i .'-i'* '00 rt i-c r-( i-H i-l rq (M 01 N O A. When integrated this equation assumes / the form of log - = KA(t ti) which when plotted gives an S curve with A X a point of inflection at its center. The formula embodied in this equation has been applied with more or less success by Robertson (1923) and Brody (1922) to the growth cycles in man, mice, rats, guinea pigs, rabbits, cattle, sheep, swine, and chickens. Brody and Ragsdale (1921) have shown that growth in the dairy cow con- sists of two extrauterine cycles and one intra uterine cycle. Data on purebred Holstein cows and purebred Jersey cows were presented to show the extrauterine cycles. The first extrauterine cycle commences slightly before birth, reaches its maximum at about 5 months of age and continues to the age of 15 months. The second cycle begins immediately after the first, reaches its maximum at about 20 months of age and continues to the age of maturity. Hence growth in dairy cows, after the age of 2 years, is non-cyclic in nature and follows an uninter- rupted progression to maturity. Brody next turned his attention to the growth of Register-of -Merit Jersey cows (Brody, Ragsdale, and Turner, 1923a), com- bining the weights of the original-entry and reentry cows. These data, as has been shown, represent the course of growth after the age of 2 years and are non- cyclic in nature. A quotation from Brody will best express his views in regard to these data. "The data show that after the age of 2 years the rate of growth de- clines in a non-cyclic manner. The course of decline in growth follows the course of decline of a monomolecular chemical reaction; that is, the percentage decline in growth with age is constant." The equation of the monomolecular chemical reaction used by Brody et al is W= A (1 e-*0 where A represents the weight of the animal at maturity, W the weight of the animal at any time t, and k the velocity constant of growth. This equation when applied to the data gives a fairly good fit. Later Brody and Ragsdale (1924) attempted to reach a growth 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 195 equation representing the whole course of development from birth to maturity. The Department of Animal Husbandry of the University of Missouri has for years been collecting a large number of linear measurements and body weights of purebred Jersey cows at intervals from birth to maturity. These data were used by Brody to represent the whole course of development in Jersey cows from birth to maturity. The cyclic fluctuations are obvious in these data, but in the present study they were not considered. The growth equation derived by Brody to fit the data is W = A Be~ kt where W is the weight or linear dimension at any time t , A is the limit reached at maturity, k is the velocity constant of growth, and Bis a constant locating the curve in point of time. This equation has the same form as the equation of a monomolecular chemical reaction with the exception that in a monomolecular reaction A and B have the same value and the curve begins at zero. Brody's interpretation of the curve is as follows: "Barring fluctuations due to the cyclic phenomena, the extrauterine course of growth in linear dimensions and in weight of the dairy cow follows an exponential law having the same form as the law representing the course of monomolecular change in chemistry. This suggests the interpretation that the general course of growth is limited by a monomolecular chemical process, and that the cyclic phenomena are due to sub- sidiary processes in the fundamentally exponential course of growth. . . . This is in accordance with expectations if it is assumed that each animal begins life with a definite endowment of limiting substance necessary for the process of growth, and that this endowment is used up at a constant rate (or percentage) of itself." In a later paper Brody (1926) gives a somewhat different interpretation. "One may, of course, with equally good logic, interpret this equation as indicating the production during the course of growth of a growth-retarding substance according to the monomolecular law." Altho there seems to be a striking similarity between the course of growth in animals and the course followed by a monomolecular chemical reaction, it seems doubtful whether such a complicated process as growth would follow so simple a chemical reaction. The same criticism holds true for the autocatalytic monomolecular theory of growth. The S curve of the autocatalytic monomole- cular reaction is a rather flexible curve and can be made to approximate closely a great many growth reactions in both animals and plants. However, the point of inflection of this curve is at its center, whereas most of the growth reactions show a point of inflection earlier in the reaction. Van de Sande-Bakhuyzen and Als- berg (1927) have given a very thoro criticism of the autocatalytic monomolecular chemical theory as applied to growth in animals and plants and have presented evidence to show that the reactions involved in growth cannot be represented by such a simple chemical theory. A growth curve similar to that derived by Brody (1924) but with a more general biological meaning may be derived in the following manner. Minot (1908) showed for a number of animals that the per- \y w centage increments in body weight. *-=-, , constantly decrease from "x birth to maturity. These percentage increments may be looked upon as measuring the average growth power of the body cells, if growth power may be defined as the percentage rate of increase in growth. 196 BULLETIN No. 302 [January, Wright (1926) suggested briefly that the hypothesis that growth power falls off at a constant percentage rate leading to the curve / log log = a kt might often give a good fit to growth data. This curve may also be expressed in the form log W = A be ~ kt , curiously similar to Brody's formula. The derivation of this equation is as follows: dW Wdt =P (1) where W = body weight at any time t, and P = growth power of the body cells. Since the growth power is assumed to fall off at a constant percentage rate, - k Pdt ~ log P = C - kt P- e C ~ kt -- (3) Wdt logF = - \e C ~ kt + K e c - kt In equation (4) A is the logarithm of the weight of the animal at maturity; lOOfc is the constant percentage rate of decrease in growth power on the above interpretation, and b locates the curve in time; W is the weight at any time t. This equation differs from Brody's growth equation in that W is replaced by log W and A is the logarithm of weight instead of the actual weight at maturity. Also, it does not involve any simple chemical interpretation of growth. The curve for weight (W) is S-shaped with the point of inflection at - = 37 percent e of the final weight. We may now turn to the growth data of the original-entry and re- entry Register-of-Merit Jersey cows presented in Table 6. Equation (4) was applied to these data, and the values calculated from the fitted 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 197 equations are reported in Table 7. The fitted equations for original- entry and reentry cows are respectively : l oglo W = 2.9793 - .1273446-' 2763 ' lo glo W = 2.9930 - .134378e~ 2993 ' In these equations t = age and is measured in units of six months be- ginning with one year and three months as the origin. W = weight at any age t. TABLE 7. MEAN BODY WEIGHTS FOR ORIGINAL-ENTRY AND REENTRY JERSEY Cows WITH ADVANCING AGE Age in years Original-entry Reentry Mean observed Logio Mean observed Logio Mean calculated Mean ob- served Logio Mean observed Logio Mean calculated 1.5-2.0 766 2 . 88423 2 . 88270 2.0- 808 2.90741 2.90602 2.5- 836 2.92211 2.92371 870 2! 93952 2.93826 3.0- 867 2.93802 2.93713 911 2.95952 2.95242 3.5- 881 2.94498 2.94731 907 2.95761 2.96292 4.0- 906 2.95713 2.95503 940 2.97313 2.97070 4.5- 922 2.96473 2.96089 942 2.97405 2.97647 5.0- 925 2.96614 2.96534 949 2.97727 2.98074 5.5- 931 2.96895 2.96871 964 2.98408 2.98391 6.0- 937 2.97174 2.97126 973 2.98811 2.98626 6.5- 933 2.96988 2.97320 973 2.98811 2.98801 7.0- 934 2.97035 2.97468 982 2.99211 2.98930 7.5- 940 2.97313 2.97579 983 2.99255 2.99026 8.0- 944 2.97497 2.97664 982 2.99211 2.99097 8.5- 944 2.97497 2.97728 996 2.99826 2.99149 9. fl- 961 2.98272 2.97795 966 2.98498 2.99202 lC. 0- 964 2.98408 2.97852 949 2.97727 2.99246 11.0- 943 2.97451 2 . 97885 1007 3.00303 2.99271 12.0-13.0 960 2.98227 2.97904 954 2.97955 2.99283 14.1* 950 2.97772 2 . 97920 14.5* l6i2 3! 00518 2.99299 The equations of the curves fitted to observed values are: original-entry, logio W .127344e--z<; reentry, logio W = 2.9930 - . 134378e-<. *Average age of cows ranging from 13.0 to 18.5 years of age. 2.9793 - The smooth curves in Fig. 4 describing the course of growth in body weight of the original-entry and reentry Jersey cows are the fitted-growth curves represented by the above equations. It will be noted that the trends of these curves closely agree with the trends of the observed mean body weights with advancing age. Therefore it may be assumed that growth power in body weight of the original- entry and reentry Jersey cows after 2 years of age is falling off at a fairly constant percentage rate. It should be noted that this curve cannot be carried back to birth on this basis. Comparison of Course of Growth in Body Weight of Original-Entry and Reentry Cows The smooth curves in Fig. 4 describing the course of growth in the original-entry and reentry cows are plotted together for compar- 198 BULLETIN No. 302 [January, ison in Fig. 5. These curves, when compared, show that on the aver- age the reentry cows are distinctly larger and increase in weight more rapidly than do the original-entry cows. The same relation is indicated in the growth constants of the equations of these curves. The A constant, which is the logarithm of the body weight at ma- turity, is greater for the reentry cows than for the original-entry cows, the values of these constants being 2.9930 and 2.9793 respec- tively. The greater value of k for the reentry cows than for the original-entry cows indicates a more rapid rate of growth in the former, the values pf these constants being -- .2993 and -- .2763 respectively. 236 2.96 I u* Reentry Cows Original 'fntry Co ivs Fig. 5 ioy to of Calculated Mean Body Weighrs Equations of Curves Loa, K W= Z.9793-. 127344 e~ ' * Onyrfntry Loo, W = 2.993O-./34376e * 93t Reentry Fig. 6 Loo lo of Observed Mean Body Weights D o vOriginat~ Entry Cows without Reentries ........ ^Original Entries of Reentry Cows only Age in Years (Lower C/ass Limits) FIGS. 5 AND 6. COMPARISON OF BODY-WEIGHT CURVES OF ORIGINAL-ENTRY AND REENTRY Cows The graphs in Fig. 6 represent the mean body weights of the original-entry cows that do not have reentry records and the mean body weights of the original entries of the reentry cows only, that is, the mean body weights of the reentry cows when they made their first or original-entry record. The mean body weights of the original en- tries of the reentry cows are not significantly different from the mean body weights of the other original-entry cows. In view of this close agreement between the mean body weights, it is not likely that the reentry cows are selected for reentry on account of their superior body size. Hence it may be assumed that the greater size and more rapid rate of growth found in the reentry cows is due largely to the more favorable environment under which they are kept. This conclusion is in agreement with the experimental work of Eckles and Swett (1918) wherein they describe a difference in the course of growth between heavy-fed Jersey cows and light-fed Jersey cows, similar to that evi- denced between reentry and original-entry Jersey cows. 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 199 Since increase in weight with age may be due to an accumulation of inert substances within the body cells rather than to an increase in the mass of physiologically-active protoplasm within them, it is desirable to supplement the body-weight data with growth measure- ments that bear directly upon the increase in the mass of physiologi- cally, active tissue with advancing age. The primary function of a dairy cow is the secretion of milk, that is, all of her energy in excess of the requirements for maintenance, gestation excluded, is expended in the production of milk. Hence any change in the activity of the mammary gland with advancing age will reflect in a general way a similar change in the physiological activity of other organs of the body. The Register-of-Merit cows, as previously stated, must meet a minimum production requirement which varies with their age. This production requirement naturally eliminates many of the lower-pro- ducing cows of the breed and hence, if these Register-of-Merit pro- duction records are to be used to describe the course of growth and senescence in Jersey cows, some correction must be made for the records of the cows eliminated by the requirement. So far no one has attempted to estimate the number of cows eliminated by the require- ment. Most authors have considered it of minor importance, assum- ing that the requirement eliminated only a small percentage of all the cows making the Register of Merit. It will be remembered that this production requirement is levied upon the butterfat yields of the cows; therefore it was necessary to study the butterfat yields rather than the total milk yields of the cows. This, however, causes no serious disturbance in the interpretation, be- cause the butterfat content of Jersey milk contains in the neigh- borhood of 60 percent of the total energy of the yield. Tur: \~ CATION OF YEARLY BUTTERFAT FREQUENCY DISTRIBUTIONS DUE TO SELECTIVE EFFECT OF REGISTER-OF-MERIT REQUIREMENT The frequency distributions of the yearly butterfat yields at suc- cessive ages for the original-entry and reentry Register-of-Merit Jer- sey cows are reported in Tables 8 and 9 and Figs. 7 and 8 respectively. The histograms in Fig. 7 showing the frequency distributions of the yearly fat yields for the original-entry cows are severely truncated. The percentage of truncation, as will be shown later, ranges from 10 to 39. The corresponding histograms in Fig. 8 for the reentry cows, however, are only slightly truncated^ tis will likewise be shown later, ranging from 2 to 4 percent. This nearly complete lack of truncation of the reentry fat-yield distributions, as compared to original-entry 200 BULLETIN No. 302 [January, to I o O I 4 fe o S O i Q fe oo I H 00 . ro oo OP o -< -H CN n i -ot^ocororoi-c 9 3 4 o O f t- IN M CO <* "H . - . CD 4 a> t ; CN O> IN ^J< IN -iiO ro -CN -i-l . . W f) *!< IN i-H i-H c: I ' ' 00 05 CO IN -H t-. IN i-H go 00 . .(NrocOCNi-l Tf 00 : ;cN-*wiNi-t L N i t- ; 1 00 IN W (N CO i-l * t- i-l CM . . co m rf ro I-H 1-1 CO o IM 4 : : SMS8S ^e,- . . co ^ CO ; !i-lCNOClcOOOro^"5Cn ro 1-1 I-H ~ W "5 oo ro co IN "O T< i-i co ro I-H W 4 "5 CO'HOiMeOOOOO^NIN 00 CN O5 CO IN 1-1 ... c~. A :*gsn&S-"-- : | CO i :5?SSSSS"-" :- : : : 00 in CO 3S2S8833-" : : : : CO 4 ro co co ^* oo o oo *n co i*- CN i * I-H i-i 1-1 oi * -H CN >n CN I-H >n Ji CO 00 2 4 -H o IN in ro co oo t- 1^ * i-< 1-1 . . CO IN o ^ I-l NCN-l 00 >C O '*? O l -^ O >^ O 'C O >^ O '-^ O ^7 3 s H o IO oo s ci -,.' / '-t-:~ ___ . . . 00 W /ANCING 4 CO IN CO CO t~ * CO CN -! TO i-l 00 ro o 3 a H |! i CO CO O3 00 00 CO t~ CO "r*T-l o * * o Q 4 O5 (35 * i 00 IN O CO i-H i-l r-l i-l . i-H CN 1 i-l -H i-H 1-H oe (x H 00 1 4 00 ro co o * oo oi IN o <*< -i-i -i-ii-< 00 t- H "TS h H 4 00 1-1 oo o t^ co t~ -H t co wmro H J, . r-< * CO * h- 00 'J 1 O 1< N CM 'N M 1 f~ M o s 4 t OS t~ Tf CO i-l ro CO CD >O TJH IN M N i-i IN IN IN IN --I s O a ^ i g J> CO O t~ 00 O CO t~ Tj< T*< O CO I-H CM 1-1 IN N ro -( CN 1-1 -H . . a CO H ' " 3? fe ^ 4 co 1-1 co t~ 1-1 *}< t~ co os oo ro 1-1 -i-c i-i CM CN ro ro IN 1-1 2 H 4 U5 i-l O5 t i-l r* 00 IN >/3 lO CO Tf i-l rH i-l -(NWCOMO2'HCO i-ioaxNmioooasoocj^i-iiN >-i N ro O O 'O O '^ O '-"^ O '-"^ O l -t C l: 7 O ^ o ocDcot-t~ooooo3OO 3 o H spunod ai pjoi/t ^c^ spunod u; ppi 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 201 Fat Yield in Pounds FIG. 7. FREQUENCY DISTRIBUTIONS OF FAT YIELDS FOB ORIGINAL-ENTRY JERSEY Cows 202 BULLETIN No. 302 [January, ^/ /rf in Pounds Fat Yteld i FIG. 8. FREXJUENCY DISTRIBUTIONS OF FAT YIELDS FOR REENTRY JERSEY Cows fat-yield distributions, may be attributed to the high level of produc- tion of the reentry cows, that is, their production is apparently beyond that specified by the Register-of-Merit production requirement. Owing GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 203 YaH in Cut. (liwer clan linns') FIG. 9. YEARLY MILK YIELDS OF REGISTER-OF-MERIT JERSEY Cows UNDER Six YEARS / 7 \ Orifirreffr/trf (IMS /fye t* te M a Yean _\ a = i. efffe / \ 3, ,Hlt3 \ P= .OtOT j i H J- rn^^ ~' =^, X S Reentry Cewt /foe totaif.e'Years i 7 ^ . S = .oiltl 4e }a tt n so w HO no ao ae He >s m fli/Mit/d trr C*t. (inter C/oss l/m'ts) FIG. 10. YEARLY MILK YIELDS OF REGISTER-OF-MERIT JERSEY Cows OVER Six YEARS to this practical completeness of the yearly fat-yield frequency dis- tributions of the reentry cows, they were considered as the type repre- sentative of yearly fat-yield frequency distributions for all purebred Jersey cows. A further analysis of these reentry fat-yield frequency distribu- tions revealed the fact that they, like the actual body-weight frequency distributions, are not symmetrical but are skewed in the positive direc- tion. Here again these fat-yield distributions when transformed to the logarithmic scale become symmetrical and can be described by the log-transformed equation of the normal frequency curve previously 204 BULLETIN No. 302 [January, cited. This peculiarity of these fat-yield frequency distributions sug- gests that the factors affecting yearly butterfat yield likewise tend to have a constant percentage effect rather than an absolute effect. This same peculiarity is also found in the frequency distributions of the yearly milk yields for the reentry cows reported in Figs. 9 and 10. Before going further into a discussion of these reentry fat-yield frequency distributions, it might be well to give a brief discussion of the significance of the constants of this log-transformed frequency curve. Heretofore it has been assumed that the arithmetic mean and standard deviation were appropriate statistics to measure the varia- bility in the milk yields of dairy cows. The use of the arithmetic mean and standard deviation presupposes that the frequency distri- bution of the data is symmetrical. This, however, as has been demon- strated, is not true in the case of milk and fat yield as well as body weight. Galton (1879) pointed out that the ordinary law of frequency of error based on the arithmetic mean demands that deviations in excess of the mean must be balanced by deviations of equal magnitude in deficiency. In other words, the frequency distributions must be sym- metrical. He also pointed out that in some cases where the deviations in excess of the mean are greater than those in deficiency, the geomet- ric mean and not the arithmetic mean best represents the mean of the data. McAlister (1879), who at Galton's suggestion gave a mathemat- ical proof of the applicability of the geometric mean to such data, essentially assumed the existence of this peculiarity of the factors affecting the data. When the body-weight, milk-yield, and fat-yield frequency distributions are transformed to the log scale, they become symmetrical, and hence the arithmetic mean and the standard devia- tion are appropriate statistics to apply on this scale. The arithmetic mean of the log scale is - , which equals log\/xi- x 2 x 3 x n , which in turn is the logarithm of the geometric mean on the original scale. The standard deviation on the log scale, when interpreted on the original scale, approximates the logarithm of (1 -f the coefficient of variation) and seems to be the best measure of the percentage varia- bility in data of this type. Application of Log -Trans formed Normal Frequency Curve to Yearly Fat-Yield Frequency Distributions of Reentry Cows The smooth curves describing the yearly fat-yield frequency dis- tributions for the reentry cows in Fig. 8 are fitted log-transformed GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 205 frequency curves, the general equation of these curves being 1 y = .s.r \/2i _ rlogz-012 where x = the fat yield and a and s are the mean and standard devia- tion on the log scale. The means (a) and the standard deviations (s) of these fitted curves, together with the probability values (P) measuring their goodness of fit, are reported in Table 10. It will be noted that the probability values of the goodness of fit of these curves, with few exceptions, indicate that there is a remarkably close agree- ment between the observed and calculated frequencies in the distri- butions. In view of this exceptionally good fit of the log-transformed normal frequency curve to the reentry fat-yield frequency distribu- tions, it was assumed that the frequency curve which best describes the yearly fat-yield frequency distributions characteristic of purebred Jersey cows is the log-transformed normal curve. TABLE 10. STATISTICS OF LOG-TRANSFORMED NORMAL CURVE FITTED TO FAT YIELDS OF REENTRY JERSEY Cows Age in years a s Fre- quency n* X 5 P** 2.5-3.0 2.6460 .0098 .08594 .0069 35 3 10.1639 .02 3.0- 2.6750 .0044 .10150 .0031 245 7 2.0768 .95 3.5- 2.6915 .0044 .10090 + .0031 238 7 7.4892 .38 4.0- 2.7075 .0037 .09346 .0026 285 7 6.3937 .50 4.5- 2.7220 .0038 .08783 .0027 247 6 2.3799 .88 5.0- 2.7175 .0036 .08645 .0026 260 6 10.4279 .11 5.5- 2.7340 .OO44 .10120 .0031 242 8 8.4483 .39 6.0- 2.7400 .0043 .08862 .0030 194 7 4.1042 .77 6.5- 2.7315 .0049 .09531 .0035 169 7 20.1871 .01 7.0- 2.7440 .0050 .09300 .0035 158 7 5.2092 .64 7.5- 2.7410 .0056 .09623 .0040 133 7 2.6176 .91 8.0- 2.7590 .0072 .10690 .0051 101 7 6.0326 .54 8.5- 2.7525 .0071 .09290 .0051 78 6 20.7688 .002 9.0- 2.7375 .0062 .09966 .0044 118 6 4.4881 .61 10.0- 2.7440 .0090 .09300 .0063 49 4 2.8702 .58 11.0- 2.7410 .0118 .10740 .0083 38 3 1 . 9832 .58 13.4*** 2.7250 .0094 .08621 + .0067 38 2 2.2453 .33 *Number of terms corrected for the loss of 3 degrees of freedom, viz., total frequency, a and s. **A value of P equal to . 10 or greater may be considered as representing a close agreement between the observed and fitted frequencies. The larger the value of P, of course the closer the agreement. ***Average age of cows ranging from 12.0 to 18.5 years of age. Correction for Truncation of Yearly Fat-Yield Frequency Distributions of Original-Entry Cows The original-entry yearly fat-yield frequency distributions in Fig. 7, being severely truncated, do not represent the yearly fat-yield frequency distributions of all original-entry Jersey cows. However, now that a frequency curve has been found that is representative of the true type of frequency distribution of the yearly fat yields of 206 BULLETIN No. 302 [January, CQ f> | ^CD^OOCO-OOWOO^O-O-N^O^ o u JERSEY *x IPIIIIIIIIPIPIII jNTRY >w *^ -w w ** - - o 5 II -^ MMeM o^^M^om^. o o to 1 c 2 22 --^^^-^^ ifl o 1 VMWH- ffl H o e _____,____ BITTED 1 ; S 1 O5 00 O ^ CO CO rooOCN(NOOOOCOTt<^^i-H INOCOO '^^IrHi-IlN -H+I-H+I-H+I-H-H-H-H-H-H-H+I-H-H -H o - .S Qt tO i iC^^i l*OCOOOOCONOOi 'CO O CO CO IN 1-1 Oi CO O i IN > '3 ^> coosOJNCft 1 , V W U "3 -H-H+l-H+l-H-H-H+l-H+l+l-H-H+l-H-H+l+l I - O --KNINi-KN a _ t>. co co co co co co co i" M 1 m to to 10 r>- p I-S l-H-H-H-H+l-H-H+l+l-H-H-H+l-H+m +1 ^ ' O5 W h- 00 "^ ^ Q ** r^ O tOC^ t> ^ O O & i -s IH |J f-i rt ^^i-HrtrHr-KNiNININCOCOeNCOCOiCCO < -I W 3 -H-H-H+l-H-H+l+m+l-H-H+l+l+l-H-H-H-H 1 '5 " S 2 o 3 S o a H 00 1-1 1~ 00 CO tO iO IN O CO CO t- CO U5 CO -H-H-H+I-H-H-H-H-H-H-H-H-H+HH-H C^ W^ ^l 4 00 ^ tO O tO tO Oi it Aje / Ufart fawer dost limts) FIG. 13. GEOMETRIC MEANS (a) AND STANDARD DEVI- ATIONS (s) OF YEARLY BUTTERFAT YIELDS FOR REENTRY Cows the age of maximum production may therefore be used as a measure of the influence of age on the physiological activity of the cows dur- ing senility. Brody et al (1923c) referred to this decline in the yearly fat yields during senility as a measure of senescence in the cows; in other words, the process of senescence was interpreted as apply- 214 BULLETIN No. 302 [January, ing only to the decline in the physiological activity of the cows after the age of maturity. Gowen (1924) also gives a similar inter- pretation to this decline in the rate of milk secretion after the age of maximum production. Such an interpretation of the process of senes- cence would lead one to believe that it is initiated by the onset of senility and is represented by the decline in the physiological activity of the cells during old age. Altho the evidence of senescence becomes very marked during senility, the process as a whole needs a broader and more general interpretation than that given by Brody and Gowen. Senescence the Result of Same Processes Which Determine Growth and Differentiation Review of Literature. Minot (1908) defined senescence as the process of growing old and associated it with the fundamental processes of growth and dif- ferentiation. He presented evidence based upon the growth increments of rabbits, guinea pigs, and chicks to show that the rate of growth is highest in the young organism and decreases as development proceeds and the rate of metabolism falls. He considers the rate of growth as a measure of the rate of senescence and therefore concludes that the rate of senescence is highest in youth and slowest in advanced life. Minot explains the phenomenon of senescence on the basis of the cytoplasmic changes taking place with age and presents evidence to show that growth and differentiation of the cytoplasm are the fundamental factors in senes- cence and death. He finds that in the young cell the amount of cytoplasm in relation to the amount of nuclear substance is least, but that during development the cytoplasm increases in proportion to the nuclear material, undergoes differen- tiation, and brings about senescence. Minot, however, did not explain how these changes in the cytoplasm bring about senescence. Child (1915) agrees with Minot in regarding the decrease in the growth power of the cells with advancing age as evidence of senescence, and also associates it with the cytoplasmic changes in the cells as age increases. Child, however, finds the cause of senescence in the ever- increasing mass of inactive protoplasm in the cells accompanying growth and dif- ferentiation. As the mass of inactive protoplasm increases, the mass of active pro- toplasm decreases, and hence the relative rate of metabolism decreases, which in turn brings about a decrease in the reproductive power of the cells. Child, there- fore, regards senescence as "primarily a decrease in the rate of the dynamic pro- cesses conditioned by the accumulation, differentiation and other associated changes of the material colloid substratum." Consequently senescence is an in- evitable feature of growth and differentiation and is not limited to the senile stages in the life cycle. In the light of the more general interpretation of senescence, it appears logical to assume that the whole course of milk secretion in purebred Jersey cows from youth to old age is an expression of the senescent changes accompanying the growth and differentiation of their mammary glands. One of the outstanding features of senescence, 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 215 according to the above view, is the continuous decrease in the repro- ductive capacity (growth power) of the cells with advancing age. Such a process must necessarily be taking place in the mammary glands if their course of growth up to the age of maximum production may be described by the ever-decreasing rate of increase in the mean fat yields. Another outstanding feature of senescence, and according to Child the cause of this continuous decrease in the growth power of the cells, is the increase in the mass of inactive protoplasm in the cells accompanying growth and differentiation. As the mass of inactive pro- toplasm in the cells increases, the mass of active protoplasm decreases and likewise the relative rate of metabolism and functional activity of the cells. In other words, the relative functional activity of the cells in a gland constantly decreases as age increases. During the earlier stages of growth in a gland there is a rapid increase in the number of functional cells and the total functioning capacity of the gland increases regardless of the loss due to the decrease in the rela- tive functional activity of the component cells. However, a point is ultimately reached where the increase in the number of functional cells does not compensate for the decrease in the relative functional activity of the component cells and the total functioning of the gland begins to decrease. Such a process would bring about a change from a rising to a falling trend in the total functioning of the gland with advancing age. Considering the function of the mammary gland from this standpoint, a similar process must necessarily be taking place within it, if the rise and fall in the mean fat yields may be used as a measure of its functional activity with advancing age. Hence the process of senescence is indicated just as clearly in the ever-decreasing rate of increase in the mean fat yields up to the age of maximum pro- duction as in the ever-increasing rate of decrease in the mean fat yields beyond the age of maximum production. In other words, the process of senescence is an inevitable consequence of development and its evidences are ever present regardless of whether the organism is in the growing or senile phases of the life cycle. The course of growth and senescence in these purebred Register- of-Merit cows, as described by the rise and fall in their yearly fat yields with advancing age, may also be expressed in the form of a mathematical equation, providing an equation can be derived which not only represents the trend of the data but which also may be inter- preted upon the basis of the biological phenomena involved in the more general theory of growth and senescence. 216 BULLETIN No. 302 [January, Curves Describing Course of Growth and Senescence as Measured by Lactation in Dairy Cows Review of Literature. Pearl (1919) and Gowen (1924) have used the logar- ithmic equation, y = a + bx + ex 2 + d log x in which y = milk yield and x = age, to represent the rate of milk secretion with advancing age in all of the purebred breeds of dairy cattle. This equation was fitted with a great deal of ac- curacy to both the yearly milk yields and butterfat yields of Register-of-Merit Jersey cows. Altho this equation accurately represents the trend of the activity of the mammary gland as age increases, it is not based upon any general biologi- cal law, and hence may be considered as one of many empirical equations which change from an increasing to a decreasing trend with increasing values of the vari- ables. Brody (1923c) combined the yearly fat yields, at successive ages, of all the purebred breeds of dairy cattle, including the milking Shorthorns, and found that the equation of two simultaneous consecutive monomolecular chemical reactions could be fitted to the data with some degree of accuracy. This equation takes the form of M = A (ae~V) be~V) in which fc 2 and k\ are the velocity constants of growth and senescence respectively, M = the fat yield and t age. In order to give a biochemical interpretation to this equation, Brody and his coworkers assume that growth and senescence go on simultaneously from the beginning to the end of life, which assumption is not in harmony with their earlier statement. Their interpretation of this chemical equation is very clearly stated in the follow- ing quotation : "The whole course of milk secretion with age was therefore found to follow approximately the course of two simultaneous consecutive monomole- cular reactions. This is taken to mean that growth and senescence go on simul- taneously from the beginning to the end of life, and that each follows an ex- ponential law with age; and therefore perhaps that the course of the two pro- cesses are limited by two consecutive chemical reactions." Altho there is a sim- ilarity between the course of milk secretion with advancing age and the course of two consecutive monomolecular chemical reactions, it seems rather absurd to as- sume that such a complicated physiological function as milk secretion is due to two simple chemical reactions. Furthermore, this equation separates growth (fc 2 ) and senescence (/d) into two distinct processes, a separation that cannot be justi- fied according to the more generally accepted conception of senescence. If it may be assumed that the increasing trend in the yearly fat yields up to the age of maximum production, for both the original- entry and reentry cows, reflects in the main the growth of their mam- mary glands, then the growth equation of the type used describing the increase in body weight, logio-M" = A be~ kt , where M = fat yield, may be used to describe this trend in the yearly fat yields. Beyond the age of maximum productivity, however, the yearly fat yields of the cows change in trend and follow a decreasing course as age increases. This declining trend in the yearly fat yields after the age of maxi- mum production may be assumed to represent the effect of senility in the mammary glands of the cows. Hence if the above growth equa- tion is to be used to represent the whole course of milk secretion with advancing age, a corrective term must be added to it in order to ac- 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 217 count for this decline in the fat yields during senility. After the addi- tion of a corrective term, this growth equation takes the form of logio M = A be~ klt de kli in which M = the fat yield and t = age measured in units of 6 months, beginning with 1 year 3 months as the origin. The significance of the other constants will be brought out in the following discussion. The first part of the above equation logio M = A be~ hlt may be interpreted in a similar manner as pre- viously described under the section on growth in body weight, and broadly speaking represents the increase in fat yields due to the growth of the mammary gland. The function of the mammary gland, however, depends not only upon the number of cells composing it, but also upon the relative physiological activity of the cells. The physi- ological activity of the cells depends upon the amount of active proto- plasm within them, and since this constantly decreases with age and apparently at an ever-increasing rate, their relative physiological activity likewise decreases. Hence the corrective term de M that in- creases at an ever-increasing rate as age increases, must be subtracted from logio M = A be~ klt in order to account for this decrease in the relative physiological activity of the cells in the mammary gland ac- companying growth and senility. This equation may be derived as follows: letting M be the milk production, N the number of secreting cells, and A a measure of the physiological activity of the cells in producing milk, M = NA or log M = log N + log A (5) The rate of change in the milk production per unit of tissue would then be dM dN dA Mdt ~ Ndt Adt where t = time. The percentage change in the number of cells may be considered as a measure of the growth power of the cells (P) , that is, m - p < 7 > Ndt For convenience the percentage rate of decrease in the physiological activity per cell may be called S, that is, dA Adt Accordingly, then dM Mdt = S (8) = P - S (9) 218 BULLETIN No. 302 [January, If the growth power of the cells (P) falls off at a uniform percentage rate, then dP Pdt~ P = C ie - klt (10) If the percentage rate of loss in physiological activity in milk secre- tion per cell (S) is increasing at a uniform percentage rate, then dS Sdt ~ * S = C# k * (11) which after integration may be put in the form of log M = A - be~ klt - de klt (12) In this equation k^ and k 2 are both constants determining the velocity of the senescence process. k t performs this function by determining the percentage rate of decrease in the growth power of the cells and k z by determining the percentage rate of increase in the percentage rate of loss in the relative physiological activity of the cells. a This equation differs from Brody's equation in that it involves a growth equation of the type used to describe growth in body weight and fur- thermore is based upon a broad biological rather than a chemical theory of development. The above equation (12) representing the course of growth and senescence during development was applied to the geometric mean fat yields at successive ages for both the original-entry and reentry Jersey cows. The formulas of these fitted equations are respectively : log M = 2.67567 - .203800e~' 2349 ' - .0059618e' 1133 ' logio M = 2.7733 - .236561e~' 2303 ' - .0045624e - 0953 ' in which M = fat yields and t = age in units of six months, beginning with 1 year 3 months as the origin. The values calculated from these fitted equations are reported in Tables 15 and 16 respectively. The "The term deV is approximately d + k-it when h* is small. Thus dk-i is the approximate velocity constant of the percentage rate of decline in the physiologi- cal activity of the cells. 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 219 TABLE 15. MEANS (a) AND STANDARD DEVIATIONS (s) OP YEARLY FAT YIELDS AT SUCCESSIVE AGES FOR ORIGINAL-ENTRY JERSEY Cows Age in years Logio geometric mean (a) Standard deviation on logio scale () Observed* Calculated** Observed* Calculated*** 1.5-2.0 2.51050 2.50784 .09895 .09982 2.0- 2 . 54222 2.54077 .10134 .100231 2.5- 2.56032 2.56653 .10405 .100643 3.0- 2.58806 2.58663 .09524 . 101056 3.5- 2 . 58795 2.60218 . 10866 . 101469 4.0- 2.62293 2.61410 .09369 . 101882 4.5- 2.62984 2.62311 .09790 . 102295 5.0- 2.62578 2.62978 .09964 . 102708 5.5- 2.64362 2.63452 .09668 .103120 6.0- 2.63695 2.63769 . 10570 . 103533 6.5- 2.62338 2.63954 .11602 . 103946 7.0- 2.63441 2.64027 .10954 . 104359 7.5- 2.62707 2.64003 .11994 . 104772 8.0- 2.64535 2.63892 .10467 . 105184 8.5- 2.61294 2.63701 .08708 . 105597 9. fl- 2.62339 2.63267 .11307 .106217 lC. 0- 2.63731 2.62440 . 10366 . 107042 11.0-12.0 2.61312 2.61301 .10627 . 107868 13.0**** 2 . 58522 2.58920 . 10875 . 108694 *The observed values of a and s are the constants of the log-transformed frequency curves fitted to the yearly fat-yield frequency distributions in Fig. 7. **The calculated values of a were derived from the equation logio M = 2.67567 . 2038e~- 1S4 " .00596 18e- 1133 ' where t = age in units of 6 months with the origin at 1 year 3 months and logio M = values of a. ***The calculated values of s were derived from the equation s = .099405 + .00041282J, where t = age in units of 6 months beginning with 1 year 3 months as the origin. ****Average age of cows ranging from 12.0 to 18. 5 years of age. TABLE 16. MEANS (a) AND STANDARD DEVIATIONS (s) OF YEARLY FAT YIELDS AT SUCCESSIVE AGES FOR REENTRY JERSEY Cows Age in years Logio geometric mean (a) Standard deviation on logio scale (s) Observed* Calculated** Observed* Calculated*** 2.5-3.0 2.6460 2.64865 .08594 .09386 3.0- 2.6750 2.67224 . 10150 .09400 3.5- 2.6915 2.69114 .10090 .09414 4.0- 2.7075 2.70580 .09346 .09429 4.5- 2.7220 2.71721 .08783 .09443 5.0- 2.7175 2.72603 . 08645 .09457 5.5- 2.7340 2.73276 . 10120 .09472 6.0- 2.7400 2.73780 .08860 .09486 6.5- 2.7315 2.74149 .09531 .09500 7.0- 2.7440 2.74405 .09300 .09515 7.5- 2.7410 2.74569 .09623 .09529 8.0- 2.7590 2.74656 .10690 .09544 8.5- 2.7525 2.74676 .09290 .09558 9. fl- 2.7375 2.74596 .09966 .09579 lC. 0- 2.7440 2.74330 .09300 .09608 11.0-12.0 2.7410 2.73895 . 10740 .09637 13.4**** 2.7250 2.72571 .08621 .09694 *The observed values of a and s are the constants of the log-transformed frequency curves fitted to the yearly fat-yield frequency distribution in Fig. 8. **The calculated values of a were derived from the equation logio M= 2.7733 .236561e-- ai .0045624e "', where t = age in units of 6 months, with the origin at 1 year 3 months and logio M = values of a. ***The calculated values of were derived from the equation s = .093714 + .000143418*, where t = age in units of 6 months, beginning with 2 years 9 months as the origin. ****Average age of cows ranging from 12 . to 18.5 years of age. smooth curves in Figs. 12 and 13 describing the rise and fall in the yearly fat yields of the original-entry and reentry cows are the fitted growth and senescence curves represented by the above equations. 220 BULLETIN No. 302 [January, It will be noted that the trends of the fitted curves are in fair agree- ment with the trends of the observed mean fat yields with advancing age. Therefore it may be assumed that the rise and fall of the yearly fat yields of the original-entry and reentry Jersey cows is in agree- ment with the biological theory of growth and senescence involved in the equation log M = A be~ klt de klt and hence is representative of the same processes in the development of the cow. Comparison of Course of Growth and Senescence in Original-Entry and Reentry Cows The smooth curves in Figs. 12 and 13 describing the course of milk secretion (yearly fat yields) with advancing age in the original- entry and reentry Jersey cows are plotted together for comparison in Fig. 14. A comparison of these curves shows that there is a marked f // JO J AjeutYeari (inner cJau bull FIG. 14. GEOMETRIC MEAN FAT YIELDS OF ORIGINAL- ENTRY AND REENTRY Cows difference between the trend in the yearly fat yields of the original- entry cows (whole population) and the trend in the yearly fat yields of the reentry cows as age advances. The yearly fat yields of the reentry cows besides being far superior to the yearly fat yields of the original-entry cows, increase at a greater rate during the period of growth and decrease at a slower rate during the period of senility. This same relationship may be deduced from a comparison of the velocity constants in the equations of these smooth curves in Fig. 14. The velocity constants (100/cJ, determining the percentage rate of decline in the growth power of the mammary gland cells, are practi- cally the same for both groups of cows, being 23.49 percent for the GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 221 original-entry cows and 23.03 percent for the reentry cows. On the other hand, in the last terms, de klt , which determine the rate of decline in the physiological activity of the mammary glands, a both d and k 2 are greater for the original-entry than for the reentry cows. In other words, the more rapid rate of increase as well as the less rapid rate of decrease in the mammary activity of the reentry cows is due almost entirely to the distinctly lower rate of decline in the physiological activity of the cells composing their mammary glands. Since the relative physiological activity of the mammary gland cells decreases at a greater rate in the original-entry than in the re- entry cows, it necessarily follows that the original-entry cows should reach the stage of maximum production sooner in life than the reentry cows. Such is the case, for the age of maximum production in the original-entry cows is 7 years 4.42 months, whereas in the reentry cows it is 8 years 9.22 months. Hence it is obvious that the original- entry and reentry Jersey cows are distinctly different in their courses of development as measured by the physiological activity of their mammary glands with advancing age. It will be interesting to note at this point that Robertson and Ray (1920) have found that over- growth in mice, due to heavy feeding, is correlated with late maturity and long life. The individuals showing overgrowth are very highly resistant to external disturbing factors and tend towards a relative paucity of tissue (inert tissue) accretion late in life. In view of these results of Robertson and Ray, the later maturity and slower rate of senescence in the reentry cows, as measured by the activity of their mammary glands, may be attributed in part to the more favorable environment under which the cows are kept. Difference in Genetic Constitution for Milk Production Between Original-Entry and Reentry Cows It was pointed out early in the discussion that there is a tendency on the part of breeders to select only the better-producing cows for reentry in the Register of Merit. This selection if practiced to an appreciable extent would bring about a genetic difference for milk production between the original-entry and reentry cows, and such seems to be the case. The present data and likewise the data of Graves and Fohrman (1925), who also made a separate study of the yearly fat yields of the original-entry and reentry Register-of-Merit Jersey cows, illustrates this genetic superiority of the reentry cows. The arithmetic mean fat yields at successive ages for reentry cows, "See footnote, page 218. 222 BULLETIN No. 302 [January, and the original entries of the reentry cows, as also the original en- tries of cows without reentry records, are shown in graphs I, II, and III respectively in Fig. 15. It will be noted that up to the age of maturity the original-entry fat yields of the reentry cows (II) lie somewhat above and tend to parallel the yearly fat yields of the other original-entry cows (III). Beyond maturity, however, there is very little difference between the yearly fat yields of the two groups of cows. Graves assumed that the difference between the original- Oriynaff/ftry fens nittlfat Reentry \Oryinal ' f/t tries ef /tecn/ry Cm fye i*Ytari (Jewer class l/nirs) FIG. 15. ARITHMETIC MEAN FAT YIELDS OF REGISTER- OF-MERIT JERSEY Cows (Data from Graves, U. S. D. A. Bui. No. 1352) entry yearly fat yields of the reentry cows (II) and their subsequent reentry fat yields (I) was due to the superior development of the cows brought about by their better environmental conditions. Such an assumption, however, is not entirely warranted as there is still an opportunity for further selection of the cows entered for the third, fourth, fifth, etc., times. Just as the cows to be entered for the sec- ond time are selected on their superior productive ability from all the original-entry cows, so may the cows entered for the third time be selected on their superior productive ability from all of the second- entry cows. In other words there may be a continual selection of the cows each time they are chosen for further entry in the Register of Merit. Such a selection, when coupled with the superior develop- ment of the cows resulting from their better environmental condi- tions, would naturally boost the reentry fat-yield curve higher and higher above the original-entry fat-yield curve with advancing age. Hence there is no doubt but that the superior productive ability of the reentry cows is due to the influence of both environmental and 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 223 hereditary factors, but owing to complexity of conditions it is well nigh impossible to determine the exact effect of either environment or heredity. INFLUENCE OF LEVEL OF PRODUCTION UPON AGE CURVE OF MILK SECRETION The percentiles of the yearly fat-yield frequency distributions for the original-entry and reentry Jersey cows are shown in Figs. 16 and 17 respectively. These percentiles were computed from the calculated (smoothed) means (a) and standard deviations (s) reported in Tables 15 and 16 respectively. In Figs. 16 and 17 the 50-percentiles are > GconctrK ffea. jfft /' feats (icmer e/ast t/fiit*) FIG. 16. PERCENTILES OF YEARLY FAT YIELDS FOR ORIGINAL-ENTRY JERSEY Cows represented by the geometric mean fat yields, above and below which lies 50 percent of the total area of the yearly fat-yield frequency curves. It will be noted that a given percentage of the area of the frequency curves above their means takes in a greater range in fat yield than the same percentage of the area of the curve below their means. This difference in the range of fat yield spanned by equal percentage areas of the frequency curves above and below their means is due, of course, to the skewness of the yearly fat-yield frequency distributions, which in turn has been interpreted as being the result of the constant percentage effects of the factors determining the rate of milk secretion. 224 BULLETIN No. 302 [January, Another point of interest which may be deduced from these yearly fat-yield percentiles is the fact that at the higher levels of production there is a somewhat greater percentage increase in the milk secretion up to the age of maximum production than at the FIG. 17. PERCENTILES OF YEARLY FAT YIELDS FOR REENTRY JERSEY Cows lower levels of production, and a correspondingly smaller decrease thereafter. Another consequence is that at the higher levels the age of maximum production comes somewhat later in life than at the lower levels. These result from the slight increase in the values of s with age. NATURE OF SELECTION OF REGISTER-OF-MERIT PRODUCTION REQUIREMENT Review of Literature. The nature of the selection of the Register-of-Merit production requirement has been a matter of much concern ever since the estab- lishment of the Register of Merit by the American Jersey Cattle Club. Almost every investigator who has studied the production records of the Register-of- Merit cows has in some way or other referred to the selective effects of the re- quirement, but none have actually estimated the percentage of cows eliminated. Gowen (1921) made a study of the nature of the selection of the Register-of- Merit production requirement and concluded that the cows under 2 years of age and over 11.8 years of age were handicapped more by the requirement than the cows of the intervening ages. He also recognized that the cows in the neighbor- hood of 5 years of age were handicapped more than the cows of other intervening ages. Gowen, however, made no estimate of the percentage of all the cows elim- inated by the requirement at the various ages. Hooper (1921) also made a study 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 225 of the nature of the selection of the Register-of-Merit production requirement and concluded that 2-year-old Jersey cows were handicapped the least and 5-year- old cows the most by the selective influence of the requirement. Here again no estimate was made concerning the percentage of all the cows eliminated at the various ages. The area of the yearly fat-yield frequency distributions at successive ages truncated by the Register-of-Merit production requirement for both the original-entry and reentry cows is shown in Figs. 16 and 17 respectively. The percentiles indi- cated by the broken lines under the Register-of-Merit production- requirement curves in Figs. 16 and 17 may be assumed to represent the percentage of the cows at each age eliminated by the requirement. It will be noted, for the original-entry cows, that the production- requirement curve just includes the 10-percentile for cows 2 years of age and then increases linearly with age, cutting the 15- and 20-per- centiles, up to the 23-percentile for cows 5 years of age. Above the age of 5 years the production requirement remains constant at 360 pounds, but decreases in its selective effect as it nears the age of max- imum production, where it just touches the 20-percentile. Beyond the age of maximum production the requirement again begins to increase in its selective effect and continues to increase thruout the remaining life of the cows, cutting the 39-percentile at the age of 13 years. Hence it may be assumed that the percentage of all the original-entry cows at each age eliminated by the requirement in- creases from 10 percent of the 2-year-old cows to 23 percent of the 5-year-old cows and then decreases to 20 percent of the 7.5-year-old cows, following which it continually increases, eliminating 39 percent of the 13-year-old cows. The production-requirement curve in Fig. 17 follows a similar trend in its selective influence as age advances, but eliminates only a very small percentage of the reentry cows. Less than 2 percent of the 2-year-old reentry cows, 4 percent of the 5- year-old cows, approximately 2 percent of the 8.5-year-old cows and only 4 percent of the 13-year-old cows are eliminated by the require- ment. Therefore it is obvious that the high level of production of the reentry cows almost entirely excludes them from the selective influence of the Register-of-Merit requirement. 226 BULLETIN No. 302 [January, SUMMARY In this study the Register-of-Merit records of 9,694 original-entry and 2,628 reentry cows were analyzed separately by means of bio- metrical methods in an effort to determine the course of growth and senescence in Jersey cows. Course of Growth in Body Weight. The increase in body weight of the cows with advancing age may be expressed by the growth equation \og w W- A be~ kt in which W is the weight at any age t, and A is the logarithm of the body weight at maturity. 100/c is the con- stant percentage rate of decrease in the growth power per unit weight, e is the base of the natural logarithms, and b is a constant locating the curve in point of time. The formulas of the equations repre- senting the growth data for the original-entry and reentry cows are respectively : logio W = 2.9793 - .12736-- 2763 ' logio W = 2.9930 - .13446- 2993 ' A comparison of the constants in these equations shows that there is a distinct difference between the course of growth in body weight in the original-entry and reentry cows. The reentry cows attain a greater weight at maturity and increase in weight more rapidly than do the original-entry cows. Both groups reach their maximum weight at approximately 8 years of age. It was found that there is no genetic difference between the original-entry and reentry cows for body size; hence it may be assumed that the greater size and the more rapid rate of growth of the reentry cows is due largely to the more favorable environment under which they are kept. Course of Growth and Senescence as Described by Rise and Fall in Yearly Butterfat Yields with Advancing Age. It was necessary first to correct for the truncation of the yearly butterfat frequency distributions of the cows, due to the selective effect of the production requirement of the Register of Merit. The frequency distributions of the reentry cows at successive ages are only slightly truncated at the lower levels, whereas the distributions of the original-entry cows are severely truncated. It was found that the frequency curve best adapted to these yearly fat-yield frequency distributions of both the original-entry and reentry cows is the log-transformed equation of the normal curve 1 [logo; a] 2 = ~ L 1928] GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 227 in which a and s are the mean and standard deviation respectively on the log scale. Owing to the severe truncation of the original-entry distributions, it was necessary to fit them in a peculiar manner in order to determine the curves of the whole populations of which the original-entry cows are a truncated sample. The percentage of the cows eliminated by the production requirement was estimated from the fitted frequency curves. Only 2 percent to 4 percent of the re- entry cows are eliminated, whereas 10 percent to 39 percent of the original-entry cows are eliminated. The true means of the yearly fat yields at successive ages for both the original-entry and reentry cows were determined by the fitted log-transformed frequency curves. These means increase with ad- vancing age up to a maximum (age of maximum production) but at an ever-decreasing rate, and then decrease at an ever-increasing rate. This rise and fall in the yearly fat yields of the cows may be ex- pressed in part by an equation of the same type as the growth equa- tion representing the increase in body weight with age. However, a cor- rective factor must be added to this equation in order to take into ac- count the decrease in the fat yields after the age of maximum produc- tion. The corrected equation takes the form of logic M = A be~ klt de klt in which M = fat yield at any age t. The first part of this equation, logio M = A be~ klt broadly speaking, may be interpreted as repre- senting the increase in the fat yield with advancing age due to the growth of the mammary gland, 100/c 1 being the constant percentage rate of decrease in the growth power per unit volume of the gland. The total function of the mammary gland, however, depends not only upon the number of cells composing it, but also upon the relative physiological activity of the cells. The physiological activity of the cells depends upon the amount of active protoplasm within them, and since this constantly decreases with advancing age and apparently at an ever-increasing rate, their relative physiological activity likewise decreases. Hence the corrective term de ktt may be said to represent the decline in the relative physiological activity of the cells in the mammary gland accompanying growth and senility. The theory of senescence involved in the above interpretation may be said to be in accordance with the general theory of senescence advanced by Child (1915). Consequently the process of senescence is indicated just as clearly in the ever-decreasing rate of increase in the mean fat yields up to the age of maximum production as in the ever-increasing rate of decrease in the mean fat yields beyond the age of maximum production. 228 BULLETIN No. 302 [January, The formulas of the equations representing the mean fat yields with advancing age for the original-entry and reentry cows are respec- tively : logic M = 2.6757 -- .2038e - 2349< - .00596e 1133 ' logio M = 2.7733 - .2366e~' 2303 ' - .00456e' 0953< These equations when graphed show that there is a distinct dif- ference between the rise and fall in the yearly fat yields of the orig- inal-entry and the reentry cows. The fat yields of the reentry cows are far superior to the fat yields of the original-entry cows and in- crease at a greater rate with advancing age. After the age of maxi- mum production (during senility) the fat yields of the reentry cows do not decline so rapidly as do the fat yields of the original-entry cows. The age of maximum production in the reentry cows is 8 years 9.22 months and in the original-entry cows 7 years 4.42 months. It was found that the genetic difference for milk production be- tween the original-entry and reentry cows, altho significant, was not great enough to account entirely for the superior productive ability of the latter. Hence it may be assumed that both heredity and en- vironment play an important part in bringing about the distinctly superior rate of milk secretion in the reentry cows. 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 229 LITERATURE CITED BRODY, SAMUEL 1926 Time relations of growth. I. Genetic growth constants of animals. Jour. Gen. Physiol. 8, 233. BRODY, SAMUEL, AND RAGSDALE, ARTHUR C. 1921 The rate of growth in the dairy cow. Jour. Gen. Physiol. 3, 623. 1922 The equivalence of age in animals. Jour. Gen. Physiol. 5, 205. 1924 Rate of growth of the dairy cow. V. Extrauterine growth in linear di- mensions. Jour. Gen. Physiol. 6, 329. BRODY, SAMUEL, RAGSDALE, ARTHUR C., AND TURNER, CHARLES W. 1923a Rate of growth of the dairy cow. II. Growth in weight after the age of two years. Jour. Gen. Physiol. 5, 445. 1923b Relation between growth in weight and increase of milk secretion with age. Jour. Gen. Physiol. 6, 21. 1923c Growth and senescence as measured by the rise and fall of milk se- cretion with age. Jour. Gen. Physiol. 6, 31. CHILD, CHARLES MANNING 1915 Senescence and rejuvenescence. University of Chicago Press. Chicago. DONALDSON, HENRY HERBERT 1915 The rat. Wistar Inst. Mem. 6. 1915. ECKLES, C. H., AND SwETT, W. W. 1918 Some factors influencing the growth of dairy heifers. Mo. Agr. Exp. Sta. Res. Bui. 31. ECKLES, C. H. 1920 The normal growth of dairy heifers. Mo. Agr. Exp. Sta. Res. Bui. 36. FISHER, RONALD AYLMEB 1925 Statistical methods for research workers. Oliver and Boyd. Edinburgh, London. GALTON, FRANCIS 1879 The geometric mean in vital and social statistics. Roy. Soc. (London) Proc/29, 365. GOWEN, JOHN W. 1919a Report of progress on animal husbandry investigations in 1919. Ann. Rpt. Maine Agr. Exp. Sta. 1919. 1919b Variations and mode of secretion of milk solids. Jour. Agr. Res. 16, 79. 1920 Studies in milk secretion. V. On the variations and correlations of milk secretion with age. Genetics 5, 111. 1921 Mean butterfat yield of the different breeds and the advanced registry requirements. Ann. Rpt. Maine Agr. Exp. Sta. 1920. 1924 Milk secretion. Williams & Wilkins Co. Baltimore, Md. 1925 Studies on conformation in relation to milk-producing capacity in dairy cows. Jour. Agr. Res. 30, 865. GRAVES, R. R., AND FOHRMAN, M. H. 1925 Effect of age and development on butterfat production of register-of- merit cattle. U. S. Dept. Agr. Bui. 1352. HOOPER, J. J. 1921 Studies of dairy cattle. Ky. Agr. Exp. Sta. Res. Bui. 234. KlLDEE, H. H., AND McCANDLISH, A. C. 1916 Influence of environment and breeding in increasing dairy production. Iowa Agr. Exp. Sta. Bui. 165. LOEB, JACQUES 1906 The dynamics of living matter. Columbia University Press. New York. 230 BULLETIN No. 302 McAiJSTER, DONALD 1879 The law of the geometric mean. Roy. Soc. (London) Proc. 29, 367. McCANDLiSH, ANDREW C. 1920 Environment and breeding as factors influencing milk production. Jour. Heredity 11, 204. MINOT, CHARLES S. 1891 Senescence and rejuvenescence. Jour. Physiol. 12, 97. 1908 The problem of age, growth and death. Putnam's Sons, New York. OSTWALD, WlLHELM 1908 Vortrage und Aufsatze iiber Entwicklungsmechanik der Organismen. Leipzig. 1908 Uber die zeitlichen Eigenschaften der Entwicklungsvorgange. Vortrage und Aufsatze iiber Entwicklungsmech., herausgeg von Wilh. Roux, Heft 5. PEARL, RAYMOND, GOWEN, JOHN W., AND MINER, JOHN RICE 1919 Studies in milk secretion. VII. Transmitting qualities of Jersey sires for milk yield, butterfat percentage, and butterfat. Ann. Rpt. Maine Agr. Exp. Sta. PEARSON, KARL 1914 Tables for statisticians and biometricians. London. 1900 On the criterion that a given system of deviations, etc. Phil. Mag. and Jour. Sci. 50, 157. 1902 On the systematic fitting of frequency curves. Biometrika 2, 1. READ, J. MARION 1912 Intrauterine growth cycles of the guinea pig. Arch. Entwickl. Mech. Organ. 35, 708. ROBERTSON, T. BRAILSFORD 1916 The normal growth of the white mouse. Jour. Biol. Chem. 24, 363. 1923 The chemical basis of growth and senescence. Lippincott. Philadelphia and London. ROBERTSON, T. BRAILSFORD, AND RAY, L. A. 1920 Experimental studies of growth. XV and XVI. Jour. Biol. Chem. 42 and 44. TURNER, C. W., RAGSDALE, A. C., AND BRODY, SAMUEL 1924 The relation between age, weight and fat production in dairy cows. Mo. Agr. Exp. Sta. Bui. 221. VAN DE SANDE-BAKHUYZEN, H. L., AND ALSBERG, CARL L. 1927 The growth curve in annual plants. Physiol. Rev. 7, 151. WRIGHT, SEWALL 1926 A frequency curve adapted to variation in percentage occurrence. Jour. Amer. Statis. Assoc. 21, 162. 1926 Reviews. Jour. Amer. Statis. Assoc. 21, 493. APPENDIX Fitting Log-Transformed Normal Frequency Curve by Method of Least Squares to Truncated Yearly Butterfat Frequency Distributions of Original-Entry Regis- ter-of-Merit Jersey Cows (/. /)* In fitting frequency curves Pearson (1900) has shown that x 1 = 7 , where /. = observed frequency and f c = calculated frequency, should be minimum. Thus an ordinary least-square fit, (/. / e ) 2 = minimum, gives too much weight to the high values of / at the expense of the low values. The proper weighting may be secured by multiplying the squared residuals by j For example, the following dif- ferences contribute equally to x 2 : /. /. A/ (A/) 7. (A/)> 1.10 1 .10 .01 .01 10.31 10 .31 .10 .01 101.00 100 1.00 1.00 .01 In actual calculation, however, j must be used as an approximation to j This, of course, introduces an error, but one of no practical importance. In fitting the log of frequency curves, a residual A/ in the primary curve is represented by -4- in the log curve (since d log/ = -~ J and the squared residual by ,, Therefore in fitting the log frequencies by least squares, each squared residual should be weighted by / in order to make 4 minimum, since ~ f = For example, /. /. log/ log/, Alog/ (Alog/)> / t (Alog/) 1.10 1 .041 .041 .0016 .0018 10.31 10 1.013 1 .013 .00016 .0016 101.00 100 2.004 2 .004 .000016 .0016 Here again f t instead of f t must be used in weighting in the actual calculation. The equation of the log-transformed normal curve fitted to the truncated yearly fat-yield frequency distributions of the original-entry cows is N' flog x- a\ 2 7 J in which N' = the total frequency under the curve, including the unknown por- tion below the Register-of-Merit production requirement; a and s are the mean and the standard deviation on the log scale. It is convenient in fitting to trans- form to logic z instead of log e x and to measure x in 100-pound units. As the class ranges are in 50-pound units (except for the first class) the observed frequencies 231 232 BULLETIN No. 302 [January, give an approximation to -\y, ~?QQ~ V, in general. Letting y = f a = | for complete classes _ t pogio x al 2 N' logic 6 J L g J 2/ = " K This equation cannot be fitted by direct methods because of the truncation of the data. It can be fitted, however, by taking the logarithms of the frequencies, which throws it into the form of a parabola, a convenient form for fitting by least squares. Letting y' = logio y o and x' = logio x This is of the general form y' = A + Bx'-\- C(x'Y. Where _ a logio e _ _ _ _ logio e The constants A, B, and C were determined by the method of least squares as follows: The general least-square normal equations of the parabola y' = A + Ex + C (x') 2 , using the weight f o , are ABC f a + fa*' + /o(*') 2 = ./> (*') + /(*'>* + / (*') 3 = f,x'y In these formulae y' = log /o except in the first class of each distribution, which is of varying range. The effect of the truncation is illustrated in Fig. 18. Taking, for example, cows 3.5 to 4.0 years of age, the first recorded butterfat class is that for 300 to 350 pounds. This class is truncated by the requirement to varying ex- tents indicated by the shading. An average of the truncation is at 314.4 pounds, leaving a class range of 35.6 pounds instead of the usual 50 pounds, and a class mid-point of 3322 pounds. The observed frequency of 104 must be rated up by the ra ti o (giving 146.1) to obtain the magnitude of the ordinate on a scale com- 35.6 parable to those for the complete classes. In this case then y' logio ( )/<> \range/ The whole procedure for fitting this age class is given in Table 17. The method of determining the theoretical frequencies (/) is reported in Table 18 and is the same as that described by Wright (1926). N" in Table 18 GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 233 S w >3 > c* r^(NO>oor^wowioiN ^fOcpTj-CiCCCiO ONOoo>ot^ocf~iot~ NT(ttO^l (ONt^fO-f-HOSOSOUi O)Oh-t tOt-COO <>> t^cot^-iot^-^^ccc^cocc MCOCOIN'NCO^frXO c t- i p cr O5 $ I^tCON^IN(O5'H ~-> n a> IN o n *-> rt(N-Hl-l I t- 3S \ oo-oc<5e<5Tj-H COOOMOOOOO5?D l ^^^C^ !NIN S'HtOOOOOO'^'CO'HO >CC5Xt~O5>OOMCO !O^t--o>rotc<3t o *iN"OW!NOO'-ir^^ oaon H H,J, O H C<5t-MHIX>lNWOcO coosr*' 'oc^ococccoo OOCOC5O>1 < i-TMOfiO .^i"Wcooo "j<~*ooot~'*r-r-'- l o tONiO e S C-l e| v? OOON"5tOIMt^C^'H (N !O 1O * IN it rH I N h * *? - >< ~H , cOh-COCsO^fl^WOCD O^C5OCCO' W C^' '00 NO>O"OOCOC. (N5INOOi-i'H'Ot-- ^^ H *OONtOIN'OOIN'-l ift -H 00 W CO IN i-c n FOLLOW] FREQI "H O- H C3'fC300O'i'3O^l Mr-?5i^csi-'5o^)S30 USiOtOCOt^t^t^WOOOO -PROCEDUR *: Scoc<5iNoaio'^Me'i * o> S5 N -H i-" S c 1 r^ H F&t yield class mid-point X ~'"i-i~'~'"'": ccr^Mi^^t^Mt^Mt^ COW^"^"'3>fflOr^h. 3 H S 3 "e **> s N O II II +++ ' + ' gag- n n n j o ?5 ^ Q ^ ss CON"-" +++ 234 BULLETIN No. 302 [January, w ,i g 3 5 CO M a 3 5 H t 3 !Z &: j < o cj CM 02 2 Hg O O I a T H i -; . . . .10 . 5 00 00 b- * O5 IN "5 CO * I ^ 00 Ui CO 1-1 1^ O C55 IN fi rt CO O5 >-l IN i-l CO 00 i-" ^ x co n * oo ro >r^^(N^ e "H o>05in^raioiNTj<^ioc5050iC O "-^ |! ^ O 'O O l -^ O '^ '-T O ^C O 7 i ? ? ? T T T *? *? *f T T *T T T i 1 1928} GROWTH AND SENESCENCE IN PUREBRED JERSEY Cows 235 FIG. 18. REGISTER-OF-MERIT PRODUCTION REQUIREMENT CURVE was calculated from the total frequency of the truncated data, 2/ = 636. As 79.8 percent of the area of the theoretical curve is above the point of truncation (314.4 pounds) and 636 cows are recorded above this point, the percentage fre- G"yc quencies must be multiplied by = 7.974 to obtain the theoretical frequen- / y.o cies which will give minimum x 2 - This gives a total theoretical frequency, N" = 797.4, slightly different from the figure N' = 800.8, obtained from the solution of the normal equations. The latter is used only as a rough check to N". In calcu- lating the probability from x 2 , it must be noted that 3 degrees of freedom are lost in the calculation of N' a, and s from the data (see Fisher 1925). The 9 contri- butions to x 2 thus yield 6 degrees of freedom and in Elderton's table are entered under ri = 7 (n r being one greater than the number of degrees of freedom) . UNIVERSITY OF ILLINOIS-URBANA