I/I BR. A FLY OF THE UNIVERSITY OF ILLINOIS 630.7 IJlGb cop AGRICULTURE NOTICE: Return or renew all Library Materials! The Minimum Fee for each Lost Book is $50.00. The person charging this material is responsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft mutilation, and underlining of books are reasons for discipli- nary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN L161 O-1096 . UNIVERSITY OF ILLINOIS Agricultural Experiment Station BULLETIN No. 288 PERSISTENCY OF LACTATION IN DAIRY COWS A Preliminary Study of Certain Guernsey and Holstein Records By W. L. GAINES URBANA, ILLINOIS, APRIL, 1927 CONTENTS PAGE INTRODUCTION. . 355 THE PROBLEM 356 THE METHOD 357 RESULTS FROM GUERNSEY RECORDS 367 Age and Persistency 368 Age and Initial Rate of Yield 369 Age and Yield 370 Persistency, Initial Rate, Age, and Yield 371 Correction Factors 375 Influence of Season 376 Variability in Persistency 377 Variability in Initial Rate of Yield 380 Increasing Rate of Yield with Advance in Lactation 381 Rate of Yield and Yield for the Year 382 Influence of Heredity and Environment 392 RESULTS FROM HOLSTEIN RECORDS 397 Age and Persistency , 397 Age and Initial Rate 398 Persistency and Initial Rate 398 Correction Factors 402 Variability in Persistency 402 Variability in Initial Rate of Yield 402 Rate of Yield and Yield for the Year 403 Influence of Heredity and Environment 405 DISCUSSION. 408 Selection of Records 408 Chemical Interpretation of the Lactation Curve 409 Breed Lactation Curves 411 Measures of Persistency 414 Correction Factors for Length of Record 414 Persistency as a Heritable Character 415 The Short-Time Test 416 SUMMARY AND CONCLUSIONS 422 LITERATURE CITED.. .424 PERSISTENCY OF LACTATION IN DAIRY COWS A Preliminary Study of Certain Guernsey and Holstein Records* By W. L. GAINES, Chief in Milk Production INTRODUCTION The term persistency of lactation is used to refer to the degree with which the rate of milk secretion is maintained as lactation advances. Cows ordinarily reach their highest rate of milk secretion in any one lactation period, shortly after calving. Following the flush of lactation, the rate of milk secretion declines more or less rapidly until the cow goes dry naturally or is dried up artificially by discontinuing milking. It is clear that, other things being equal, the more persistent a cow is (that is, the less rapidly she declines in rate of milk secretion), the more milk she will produce in a year 's time. It is commonly stated that the dairy breeds produce more milk per year than the beef breeds or unimproved cattle partly because they are more persistent milkers. As between the cows of the dairy breeds it is recognized, in turn, that some are more persistent than others. In the official testing of the dairy breeds, the short-time test, such as the 7-day test, has been very adversely criticised because it does not depend upon and does not measure (it is said) the quality of persistency. In general persistency of lactation is counted as a very important factor in the yearly yield and economical production of milk. As a matter of fact, however, our knowledge of persistency of lactation as a character of dairy cows is very limited so far as exact quantitative analysis is concerned. The American Guernsey Cattle Club has published about 15,000 yearly records of Guernsey cows in detail by calendar months. The Holstein-Friesian Association of America has published about 1,500 7-day records made at least eight months after calving, together with in each case a 7-day record made early in the same lactation. These breed records afford data of great value for quantitative study of persistency, and the present paper is the outcome of an attempt to analyze certain of them from the standpoint of the persistency of lactation shown by the individual records. Submitted for publication June 29, 1926. 355 356 BULLETIN No. 288 [April, The Guernsey records used (1,676) were limited to original entries published in Vols. 33 and 34 and No. 1 of Vol. 35 of the Herd Register. Only 365-day records which represented a single lactation and in which conception did not recur within 6 months after calving were used. This selection was intended to eliminate any disturbing influence of pregnancy on persistency. Also, only those records were used in which not more than 75 days elapsed from calving to the middle of the first full calendar month of the record, in order that the records should be fairly com- parable with respect to the time after calving at which they started. The Holstein records used (1,395) constitute all the eight-months- after-calving records published in Vols. 24 to 31 of the Advanced Regis- ter Year Book, with certain few exceptions noted later. The volumes named include nearly all the records of this class which have been published. THE PROBLEM The problem with respect to the Guernsey records may be pre- sented by considering the two records reproduced in Fig. 1. Mere in- spection of these records is sufficient to show that one of the cows (10233) was a very persistent milker. Ignoring the record for June be- cause it is for only part of the month, it is apparent that there is only a slight decrease in the monthly milk yields with advance in lactation. The corresponding fat yields show a tendency to increase rather than to decrease. The other cow (10372) shows by her record a tendency to decrease very rapidly in monthly, milk and fat yield with advance in lactation. She would be classed as a very non-persistent milker. It is not sufficient for quantitative study to say that one cow is persistent and another is not. The problem, first of all, is to derive a numerical value for persistency as shown by the individual record. This in itself is somewhat difficult and offers opportunity for difference of opinion as to what may be accepted as the best value for the purpose. The two records given in Fig. 1 represent opposite and rather ex- treme cases of persistency. What we should like to have is a completely representative picture of the breed as a whole with respect to the per- sistency character. Having given an acceptable numerical value of persistency for each record of a suitable group of cows, such a representa- tive picture may be obtained by the usual statistical methods. The character may then be studied after the same fashion as milk yield, fat yield, or fat percentage have already been studied by various investi- gators, notably at the Maine Station (cf. Gowen 8 ). Some of the questions relating to persistency are: the form of its frequency distribution curve; its relation to the age of the cow; its 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 357 relation to yearly yield; its relation to the rate of yield at the flush of lactation; its relation to genetic factors; its relation to environmental factors. In short, what are the attributes of persistency of lactation as a characteristic of dairy cows as shown by the advanced registry records? 10233 CHOICE OF ELMWOOD 51908 Official Year's Record, Class A Milk Butt er Fat Sire Sir Ponto of Elm wood 25492. Sire Katonah's Mack 17252. 1920 June 7, Iba. 919.2 % 4.28 Ibs. 39.34 Dam Faithful of Elm wood 39350. July. 1204.5 4.34 51.88 Dam Flossie of Elmwood 35949. Aug., 1181.9 4.30 51.53 Sire Rex of Eastside 12763 A. H. Sept.. 1179.6 4.40 51.90 Dam Dolly Vardftn of Eagtside 20756. Breeder M. E. Gifford, Sherman. N. Y. Oct., Nov., 1 230 . 1 1181.0 4.60 4.47 50.86 52.82 Owner George S. Love. Waukesha, Wig. Dec., 1212.0 4.51 54.06 Born Dec. 9, 1913. Coined June 4, 1920. Jan., 1921 120.">.4 4.75 57 . 2(i Sened Apr. 4, 1921. Feb.. 1077.8 5.05 54.43 Requirement for admission-. 360.00 Ibs. fat. Supervised by Wisconsin Station. 3 milkings daily. Mar.. Apr., May. 1115. 8 ! 109.1 1113.5 4.81 4.78 5.45 63.67 53.01 60.09 June 6, 223.8 5.45 12.20 Total 14820 3 4 68 653.25 1WZ DORA OF ELMENDORF 56322 Official Year's Record, Class A Milk Buttrr Fat Sire Imp. Lord Mar V. 1S901 A. H. 191 ft Ibs. % Ibs. Sire Imp. Lord Mar 14359 A. R. Dec. 22. 407.3 4 .56 22.67 Dam Imp. Count*** I. of Los Nour-ttr-s 36184. Jan., lf20 1009.4 4.56 73.39 Dam Imp. Dora of the Vranguo VI. 361!).'!. Feb., 12-14.1 4.96 01.71 Sire Imp. Galaxy's Sequel 16004 A. R. Mar., 1144.8 4.55 52.09 Dam Dora of the Yranguc R.G.A.S. 5572 P.S. Apr., 1022.9 4 . 52 46.24 Breeder J. B. Haggin, Lexington. Ky. May. 892.0 4 . 98 44.42 Owner R. M. Cooper, Jr.. Y\ wacky, H. C. June, 0*i6.8 4 . 57 30.03 Born Nov. 15. 1913. Calcetf. I>r. 1C, l'Jl':> July. 427.2 1 . 50 19.48 Serwl Oct. 8. 1020. Aug., 310.9 5.19 16.60 Requirement for admission: 300.00 ll>*. fut. Sept.. :>9 s 1 , 59 14.22 Supervised by South Carolina Station. Oct.. Sss . '> 5 . 40 15.60 3 milkings daily, Nov., 223,7 4 . 67 10.45 Hec. 20. I.V) 9 1 . <>3 7.22 Total 8792 4 71 414 11 FIG. 1. PHOTOGRAPHIC REPRODUCTION OF Two RECORDS FROM VOL. 35 OF THE GUERNSEY HERD REGISTER These two records are chosen as striking examples of the differences in persistency shown by the published records. They serve also to illus- trate the various data given in connection with each record. The leaves of the Herd Register were removed and backed by sheets of gummed paper board and then cut into cards, one to each individual record. Derived data for each record were recorded on its card. The records were thus brought into convenient form for manipulation for statistical purposes. As indicated, the first step is to derive a numerical expression for persistency, and a method of doing this we may consider in connection with the two records given in Fig. 1 as examples. THE METHOD The expression of persistency of lactation used in the present treat- ment is essentially an old one. Sturtevant 18 studying the average yield 358 BULLETIN No. 288 [April, of a herd of cows, used the method of expressing the decrease in yield of milk from month to month as a percentage of the yield of the previous month, and found that as thus expressed the decrease tended to be con- stant. This expression or its complement, the expression of the milk yield for any month as a percentage of the yield of the preceding month, has been used by various investigators, and gives a numerical measure of persistency, the meaning of which is readily grasped. Brody et al l have put the expression into the closely related form of an exponential equation. While this form may not be quite so easily understood as Sturtevant's it is on the whole a more reasonable ex- pression, giving concisely both the rate of decrease and the rate of yield. From experience 6 with the use of the exponential expression and its application to the group performance shown by Guernsey records, it seemed to the writer that it might be applied to individual records to give a reasonable numerical expression of persistency. In theory the idea is that the rate of milk secretion is continuously decreasing with advance in lactation in accordance with the equation: i/ in which y = yield in pounds; t = time in months from calving; -|- at is the rate of yield in pounds per month; e (= 2.71828) is the base of natural logarithms; a is the theoretical initial rate of yield; and k is the rate of change per month in the rate of yield a . In this equation k is the factor which is used as a measure of persistency. The minus sign means that the rate of yield is decreasing and k taken as a positive value is a measure of this decrease, differing from Sturtevant's expression of the percentage decrease per month in that it is reckoned as occuring con- tinuously instead of at monthly intervals. The rate of decrease is thus referred to the rate of yield at the immediate time rather than to the rate of yield a month preceding. We may consider the application of equation (1) to the records of Fig. 1. In the first place it is evident that the records are based on the calendar month, which varies from 28 to 31 days, and it is necessary to make allowance for this fact. This is conveniently done by reducing the monthly yields to an average yield per day. It is also apparent from Fig. 1 that there is a tendency for the percentage of fat in the milk to increase from month to month with advance in lactation. This is characteristic of the effect of advance in "The mathematical relations involved in the derivation and use of the equation are given in more detail in Bulletin 272". PERSISTENCY OF LACTATION IN DAIRY Cows 359 lactation, as has been shown in considerable detail by Turner. 19 Altho it happens in the two records shown in Fig. 1 that the average fat per- centage is very similar, as between a larger number of individuals there inevitably will be marked differences in this particular which should be taken into account. These individual and monthly differences in fat percentage and correlated percentage of other solids are taken into account by dealing with estimated energy yields in terms of 4-percent milk. The estimation is made by the formula 5 F.C.M. = AM + 15^, where F.C.M. is "fat-corrected milk," M is milk; and F is fat, all in pounds. One pound F.C.M. = one pound 4-percent milk = 331 large calories. Table 1 gives the average daily F.C.M. values for the eleven full calendar-month records of the two cows in Fig. 1. The record in this form shows more clearly than does Fig. 1 the marked difference in persistency of the two cows. The values are given graphically in Fig. 2. The next step in the work is to apply equation (1) to the data of Table 1. This cannot be done directly; the equation in form for application may be written, y d = Ae- k < (2) in which y d is the yield for a month expressed in terms of pounds of F.C.M. per day. Time, t, is reckoned to the middle of the month and 365A e' 5k e~ 5k =- = 'a j . Equation (2) may be converted to a linear I K form by taking logarithms on both sides, giving, logic?/ d = logio A kt Iogi e (3) By the use of equation (3), values for A and k are readily derived from the observed values of Table 1. Using the method of least squares, the equation for No. 10233 becomes y d = 41.854 , and for No. 10372, yd = 74.525 e~ 202893 '. The corresponding curves are given in di/ Fig. 2. If we wish to use equation (1) for cow No. 10372 we have -- = X 74.401e -*a3, that is, a = X .998344. In the case of No. I _ I 10233 the A constant is affected only slightly in the third decimal by the same transformation. The k constant is not affected in passing to the form of equation (1). For practical purposes, therefore, in the present data, we 365 . dy , 365 ^ may assume X yd = -jr and X A = a. We have thus expressed the rate of yield or lactation curve in terms of two constants, A and k, the values of which have been determined. 360 BULLETIN No. 288 lApril, TABLE 1. AVERAGE DAILY F.C.M. YIELDS OF Two Cows IN GUERNSEY ADVANCED REGISTER (Records for 11 full calendar months) Full calendar month Average daily F.C.M. yield (pounds) No. 10233 No. 10372 First 42.9 40.2 41.7 43.5 42.2 42.1 43.3 44.6 40.4 41.3 43.7 56.3 49.1 40.0 36.8 33.0 23.8 14.9 12.2 11.2 11.3 8.2 Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh 70 ?60 ^50 tJ b-1 ^0 Si _J 1 30 F "5 >20 \ N N 0.10 0.10 255 572 < o < e \ \ \ \ \ \ w , \, V \ \ \ \ \ \ b 1 - DC 10 \ X ^ f " v -_J ^ 3 2 4 (, & 10 12 Months after Calving (t) FIG. 2. EXTREMES IN PERSISTENCY (TABLE 1) AND FITTED CURVES Equations of curves, y d = Ae~ kt : No. 10233, y d = 41.9e' 002< ; No. 10372, y d = 74.5e~ 203 '. Similar curves have been derived, one for each of the 1,534 Guernsey records and 1,395 Holstein records studied. The individual curves are studied with reference to the A and k constants, particularly the k constant, which is used as a measure of persistency. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 361 While it is apparent from inspection of the records as first given in Fig. 1 that there is a great difference in the persistency of the two cows, we have now a numerical basis for the comparison of this difference in the factor k. No. 10372 is given a value of .203 and No. 10233, a value of -.002. Similar values may be determined for other cows from their records. It is convenient in the recording and in the subsequent handling of the values of k to multiply by 1000. This product, k X 10 3 , may be considered as expressing the rate of decrease in per imTe per month. In the case of No. 10233 the rate of energy yield with advance in lactation is not decreasing but increasing. Since we are treating k as a positive value representing the rate of decrease, we in this case must give the value of A; a minus sign. It would be absurd to suppose that this increasing rate of yield would continue indefinitely, especially in ac- cordance with the equation, but the fact of the record is that the rate of yield is increasing and this is simply recognized as a negative decrease. We shall see later that an appreciable proportion of the records studied show this same feature. While we are concerned primarily with persistency of lactation, that is, the k constant, we have to consider also the significance of the A constant. By the equation, t = at calving, and accordingly at that time Ae~ kt = A; that is, the initial rate of yield is the value of A. It will be apparent that A is thus a hypothetical quantity since the full rate of yield is not realized immediately at calving. In the case of No. 10233 the rate of yield, A, is actually realized at a time later than calving. In the case of No. 10372 the rate of yield, A, is never realized. No. 10372 is, of cour e, an extreme case. In general, the larger the value of k the more the value of A exceeds the maximum realized rate of yield. a But nevertheless it will be apparent that A is closely related to the maximum realized rate of milk secretion. Figs. 4 and 5 are intended to illustrate further the properties of the A and k constants of the equation. The determination of the A and k constants has been carried out by the use of graphic methods illustrated and described in Fig. 3. The alge- braic and geometric processes illustrated in Fig. 3 are all straightforward but there is an element of weakness, from the standpoint of mathematical precision, in fitting the straight line by the eye. In many cases the plotted observations are quite irregular and the fitting of the straight line re- quires the exercise of judgment. The determinations of A and k lack absolute precision on this account. Some records were so irregular that it was felt that extraneous factors must be so affecting the lactation "This fault of the expression, if it is a fault, might easily be remedied by dealing with the lactation curve starting some time after calving, for example, by taking the time origin at one month after calving. 362 BULLETIN No. 288 [April, curve that for the purpose of the present study more trustworthy re- sults would be secured by not including such records. On this ground, out of 1,676 records considered, 142, or 8.5 percent, were excluded. In fitting the straight line it was attempted to make its slope correspond to the tendency of the plotted observations; and to adjust its level to the average level of the observations. As a check on the ac- Spring Clamp - / ' k Scale - o .oi .02 .03 .6U .05 .06 .07 Origin Scale Mo. 9498 A - 52.1 k .05S Rate of Yield fy, ) Log Scale FIG. 3. ILLUSTRATING THE METHOD OF DETERMINING THE A AND k CONSTANTS OF THE LACTATION CURVE The equation (2) of the curve is yd = Ae~ kl , in which yd is the yield for a month expressed in terms of F.C.M. per day; and t is time in months from calving as origin and reckoned to the middle of the month. Reference to Fig. 1 shows that there are 11 full calendar months in the cow's record and it is to these 11 observed values that the curve for each cow has to be fitted. The record (see Fig. 1) gives directly the pounds of milk and the percentage of fat for each month. The number of days in the month varies from 28 to 31. The average daily F.C.M. yield is computed by the use of a 500 mm. slide rule provided with a specially graduated slide. Consider a 30-day month and let M = recorded milk yield (Ibs.) and / the corresponding fat percentage, then the observed yd = ~30~~ = M (-01333 + .005/). A unity graduation (marked 30) is made on the slide, and with this coinciding with the unity graduation on scale D of the rule a graduation (marked 2.0) for 2.0 percent fat is made on the slide opposite 2333 1927] PERSISTENCY OP LACTATION IN DAIRY Cows 363 ( = .01333 + 2 x .005) of scale D. Similarly fat-percentage graduations are made by intervals of .1 to 9.0 percent. If the milk yield for a 30-day month is 1,071 pounds and the fat percentage 3.33, the slide is set as in the illustration with the "30 unity" opposite 1071 on scale D, and the runner is set at 3.33 on the fat-percentage scale of the slide. Under the runner the value 321 is read on scale D, and the observed yd = 32.1. To take care of months of 28, 29, or 31 days, it is clear that it is only necessary to provide appropriate unity graduations on the slide. Equation (2) is transformed to the linear logarithmic expression: logio yd = logio A - .4343fa (3) In fitting equation (3) graphically we have to plot yd on a log scale and time on an arithmetic scale. (The effect of the variation in the length of the calendar months is negligible here.) For the purpose of this plotting there are mounted on a drawing board two parallel guide bars, spaced to accommodate the length of the slide rule between them. These bars are each provided with 11 equally spaced notches in opposite pairs. The slide rule is provided at either end with a small lug to engage these notches. The arithmetic time spacing is thus provided for. The logarithmic yd spacing is provided for by the construction of the slide rule itself. The time origin (calving) is, of course, variable with respect to the month, being anywhere from 15 to 75 days preceding the middle of the first full calendar month. To locate this origin a scale is fixed alongside the extension of the guide bars, gradu- ated in days, 30.5 graduations equaling one space on the guide bars. Zero of this scale corresponds in position to the upper edge of the rule when the latter is placed in notch 1, for the first full calendar month. The description may be completed by following thru a determination. In oper- ation the plotting paper, of plain white letter size, is properly placed according to the level of production of the record under consideration, and held in position by the spring clamp shown at the top of the illustration. The rule is placed in notch 1 and the line ff is drawn along its upper edge. The value of yd for the first full calendar month is computed as above outlined and plotted by a small circle made thru the eye of the pointer on the runner. Graphically this is logio yd, unity of the scale D representing 10 pounds. (It is not necessary actually to read the value of yd.) The rule is then moved to notch 2 and yd for the second month plotted; and so on to the eleventh month. With the rule still in notch 11 the line gg is drawn along its upper edge. The rule is then removed, and with a celluloid triangle used as a straight edge the line hh is drawn to fit the plotted values by inspection. The line ii is then drawn at right angles to gg thru the point of intersection of lines gg and hh. This is accom- plished by adjusting the triangle against a guide strip (not shown) at the bottom of the apparatus. The rule is then adjusted on the time origin scale according to the number of days from calving to the middle of the first full month (32 days in the example), and the corresponding line jj drawn. Without moving the rule the runner is so set that the eye of the pointer coincides with the intersection of lines jj and hh. The corresponding reading on scale D gives the value of A, 32.1 in the example. The value of k is proportional to the distance between the points of intersec- tion of lines hh and ii with line ff. This distance corresponds to 10 units of time and therefore represents 4.343/c in terms of the spacing of scale D of the slide rule. This scale is based on 500 mm. between the unity graduations. Accordingly a distance of 2171 mm. (4.343 X 500) corresponds to a value of k = 1 and 217.1 mm. corre- sponds to a value of k = .1. A suitable k scale is easily prepared being simply a uni- formly graduated decimal scale with a length of 217.1 mm. between the zero and the . 1 graduations. The value of k is obtained by placing the k scale alongside of line ff with the zero on the line ii, and reading the scale at the point of intersection of lines f f and hh. In the example given k = .058. This completes the solution of the particular record. The device takes care of values of yd from 10 to 100. In the few cases where yd values of less than 10 are en- countered, they are handled by multiplying the milk yields by a suitable factor and correcting the reading for A accordingly. Where unusually large values of k are in- volved, as in the case of No. 10372, Fig. 1, the necessary plotting range is secured by the use of two regular sheets properly adjusted. Only the heavier solid lines are drawn, in practice, the lighter broken lines being shown for the sake of explanation. 364 BULLETIN No. 288 [April, FIG. 4. FORM OF THE THEORETICAL LACTATION CURVE WITH k CONSTANT (= 05) AND A VARIABLE Cows represented by these curves are given the same per- sistency value, and it will be clear that this does not mean they have the same rate of absolute decrease in rate of yield. If the ordinates of these curves were plotted on a logarithmic scale, the curves Q ^>^ I -py^ ^^ would appear as straight parallel lines, that is, all with the same slope. Constant persistency then means that the rate of decrease bears a constant ratio to the rate of yield; or the slope of the lac- tation curve expressed in logarithms is constant. It may be noted that the areas under the curves are pro- portional to the values of A, or any other ordinate. Curves of the same per- sistency result in yearly 4 a 12 yields proportional to their Time - Months A's. 40 fc u 2 t FIG. 5. FORM OF THE THEORETICAL LACTATION CURVE WITH A CONSTANT (= 40) AND k VARIABLE \ Time - Months If the ordinates of these curves were plotted on a logarithmic scale, the curves would appear as straight lines and with slopes proportional to their k's. The measure of per- sistency fc, is therefore based directly on the slope of the lactation curve when expressed in terms of logarithms. The areas under these curves would be inversely propor- tional to their k's (where k > 0) if the time considered were greatly extended. For the arbitrary period of a year the relation between k and yield is not simple, the yield 1 e~ 12fc being proportional to > K with A constant. However, within the range of the most frequent values of k, equal changes in k result in approximately equal inverse changes in yield. curacy of the fitting, a number of records (205) were selected at random and the theoretical twelve months' yields computed from the cor- responding equations. In determining the theoretical yield due regard 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 365 was had for the time after calving at which the record started in or- der that the computed yield should be comparable with the ob- served yield. The yield as thus computed was divided by the observed (F.C.M.) yield. The distribution of the resulting ratios is shown in Fig. 6. The mean ratio is 1.0001, indicating a very good aver- age agreement. The standard deviation of the ratios is .0194, and accordingly the computed yield equals the observed yield 1.3 percent of the observed yield. It appears, therefore, that the curves have been determined with a very fair degree of accuracy, as judged by the areas. (See also Table 17.) /n equency r s i ii_ n r __ r _T -Gb= 34 56 .98 1.00 102 1.04 1.06 Ratio FIG. 6. SHOWING THE RATIOS OF COMPUTED YIELDS TO ACTUAL YIELDS The data give an indication of the accuracy of the estimation of the constants of the lactation curve to the observed values. See also Table 17. The Holstein records studied consist, for each cow, of two 7-day records in the same lactation, the first of which was started not less than six days after calving and the second not less than eight months after calving. They afford thus only two observations for the determination of the lactation curve. In some cases more than one 7-day record eight months after calving is reported, and in such cases only the first in point of time is here considered. The same curve as above described has been fitted algebraically to these two observations. Obviously this does not give as satisfactory a basis as the eleven observations of the Guernsey records, but it is all the published data afford. Certain features of the derivation of the lactation 366 BULLETIN No. 288 [April, curve from the Holstein records are pointed out in connection with Fig. 7. The yields are considered, as in the Guernsey data, on an energy basis in terms of F.C.M. The constants of the lactation curve are com- puted thus: k = log e y l - log e -; and A = logr 1 (log e y t + where y\ = yield for the week of the first test in pounds F.C.M. ; y 2 = yield for the week of the second test in pounds F.C.M.; ti = time in 1 o I cc Time> *- FIG. 7. ILLUSTRATING CERTAIN CONDITIONS IN THE LACTATION CURVES DERIVED FROM HOLSTEIN RECORDS The solid line represents the curve of equation (1). The dotted lines represent deviations from (1) associated with the preceding and concurrent pregnancies (cf. Fig. 24 6 ). If the tests are conducted at 1 and 2 we may expect normal relations; if at 1 and 4, too high values for A; and A ; if at 3 and 2, too low values for k and A ; if at 3 and 4, too low a value for A. The Holstein records give no information as to occurrence of conception. The influence of pregnancy illustrated at the right of the diagram does not become appreciable before the fifth or sixth month of gestation. The effect on the lactation curve has been avoided in the Guernsey records by deal- ing only with those records in which gestation was not far enough advanced to be a material factor. months from calving to middle of first test; and fe = time in months from calving to middle of second test. As in the Guernsey data, k expresses the rate of decrease per month in the rate of yield ; but the rate of yield is expressed in pounds F.C.M. per week. The value of A in the Hol- stein lactation curves has therefore to be divided by 7 in order to be directly comparable with A in the Guernsey lactation curves. A table of natural logarithms and a 20-inch slide rule have been used freely in the computations. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 367 Records in which more than 139 days elapsed from calving to the start of the first test were excluded ; also records were excluded in which less than 130 days elapsed between the start of the two tests. Nineteen E55 35 55 75 95 Pays to First Test 135 155 FIG. 8. TIME OF CONDUCTING HOLSTEIN RECORDS The columns show the distribution of the records with respect to the time from calving to start of the first test. The curve shows the corresponding mean time between the two tests. The correlation between the two times is r = .498 .014. The mean time after calving for the second test is about the same (around 270 days) regardless of the time of conducting the first test. records, or 1.3 percent, were thus excluded. The distribution of the first tests with respect to the time after calving is given in Fig. 8. Some further details of the methods used are given in connection with the presentation of the results. RESULTS FROM GUERNSEY RECORDS Perhaps one of the first points of interest in connection with persis- tency values is the form of the distribution curve of these values for the breed as a whole. It develops, however, that it is necessary to make certain corrections to the values, and the several relations involved in these corrections will be considered first. 368 BULLETIN No. 288 [April, Age and Persistency. It is well known that milk yield is greatly affected by the age of the cow. It has been shown by Sanders 16 and by Gaines and Davidson 6 that persistency varies also with the age of the cow, older cows tending to be less persistent than younger cows. The mean persistency values for various age classes of the Guernsey records under study are given in Table 2 and shown graphically in Fig. 9. TABLE 2. VARIOUS AGE CLASSES AND CORRESPONDING MEAN VALUES FOR PERSISTENCY, THEORETICAL INITIAL RATE OF YIELD, AND YIELD FOR THE YEAR: GUERNSEY RECORDS Mean values Age in years (class mid-points) Number of records kX 10 3 Persistency A Initial rate of yield (pounds F.C.M. per day) Yield for year (pounds F.C.M.) 1.25.. 2 35.0 24.0 7 000 1.75 46 36.7 29.8 8 674 2.25 410 32.2 33.1 10 020 2.75 202 33 6 35 2 10 559 3.25 137 44.0 39.4 11 202 3.75 130 46.0 39.7 11 092 4.25 ... 117 51 2 43 3 11 756 4.75 70 59.3 45.1 11 657 5.25 81 56.7 45.9 12 043 5.75 51 47 2 42.9 11 833 6.25 62 62 4 46.8 11 871 6.75 32 60 6 44.8 11 688 7.5.. 76 53 7 46 8 12 500 8.5 44 55 7 47 5 12 454 9.5 38 62 1 46 1 11 974 10.5... . 16 66 9 45 3 11 188 11.5 10 62 43 6 11 100 12.5 4 52 5 47.5 12 750 13.5 3 35 34 10 500 14.5 2 75 49 11 000 15.5 16.5 1 65.0 40.6 9 500 It will be observed that the relation of the k values to age, while somewhat irregular, is certainly not linear. The Maine investigators have used an equation of the general form, y = a + bx + ex 2 + d log x, to express the relation between age and yield, and the precedent of their usage is followed in choosing an equation to describe the age-persistency data. The equation is a purely empirical one, and no particular biological significance can be attached to any of its constants. As a mathematical description, however, it seems well adapted to the data. The equation has been fitted to the age-persistency data by the "star-point method" of Smith. 17 This is essentially a "cut and try" 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 369 graphic method. The smooth curve thus derived has a close resemblance to the usual relation found between age and yield. Further information as to the relation between persistency and age is afforded by consider- ation of the several interrelations of the variables listed in Table 2. 00 70 ^_60 "b x ^50 .040 4 30 zo o o ,-*' o " -~, o V ^ ~~" \ ^ S o o / o / 'a o / 7 o o ' 3 5 7 9 II 13 15 17 Age -Years FIG. 9. ILLUSTRATING THE CHANGE IN PERSISTENCY WITH ADVANCING AGE, GUERNSEY RECORDS Equation of the curve: y = 21.8 + 3.00z - .2206z 2 + 31.78 log x, x being the age in years. According to the equation y reaches a maximum at x = 9.945 = 9 years, 11 months, 10 days, at which time y = 61.5. That is, the rate of decrease in the rate of yield per month at that age is 61.5 per mille per month. In the equation y a + bx + ex 2 + d Iogi z, the value of x at which y reaches either a maximum or a minimum is obtained by differentiating y with respect to x, setting the first derivative equal to zero, and solving for x. This gives ^ = b + 2 ex + (1 \ / 1 \ CJ^C 1 ( - ) - 0. and 2cx z + bx + .4343d = 0. from which we obtain loge 10/ \x/ -b (b* - 3.4744 cd)* x = -. 4c Age and Initial Rate of Yield. The mean initial rates of yield for several age classes are given in Table 2 and shown graphically in Fig. 10. The smooth curve is of the same type as above and has been fitted by the method of moments as developed by Miner 14 with some final arbitrary adjustments in the constants. Apparently the relation between age and theoretical initial rate of yield is more regular than is the relation be- tween age and persistency. A general parallelism between the two curves, Figs. 9 and 10, is evident. 370 BULLETIN No. 288 [April, Age and Yield. The change in yield with age has been extensively studied. Since we are dealing here with a comparatively small popula- tion, the age-yield data afford something of a check on the representa- tiveness of the selected records. The numerical data are given in Table 2, the graphic presentation in Fig. 11. It will be noticed that all three of the age curves are more or less similar, indicating that there is a positive correlation between A and A; of the lactation curve equation. The method of partial correlation would give an insight into the relation between any two of the variables ou " i_ 0. 242 o u_ 38 1 1 s 4 "o tx "o EE o ^ \, / / / / 1 I 1 1 5 9 7 9 11 15 n Age -Years FIG. 10. SHOWING THE CHANGE IN INITIAL RATE OF YIELD WITH ADVANCING AGE, GUERNSEY RECORDS Equation of the curve: y = 10.56 - .534x - .0132z 2 + 40.317 log x, where x is age in units of 6 months with origin at 1.5 months. This gives a maximum for y at x = 17.554 = 8 years, 10 months, 25 days of age. At this age the rate of yield is 47.3 pounds of F.C.M. per day by the equation. with other variables constant, if the regressions were linear. Plainly the condition of linear regression is not satisfied over the entire age range. From the data as plotted it is evident that linear regression may be approximated by dealing with ages under five years. These ages include 1,114 records, or 72.6 percent of the total, and are used in the correlation treatment following. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 371 Persistency, Initial Rate, Age, and Yield. The means, standard deviations, and coefficients of variability of the variables mentioned are given in Table 3. All the possible simple and partial correlations are given in Table 4. The simple correlations have been derived by the usual method, using class intervals of 10 in k X 10 3 ; 2, in A; six months, in age; and 1,000 pounds, in F.C.M. yield. The partial correlation coeffic- ients have been derived by the general formulae, J. fy. \/ J. \ J, (~, \y" "\ 7*12.3 = i- . -i,, and r 12.34 = r- ^ . -i,/ 120 ^J90 o do F 70 60 \ X 7 9 Age -Years 15 IS 17 FIG. 11. ILLUSTRATING THE CHANGE IN YIELD WITH ADVANCING AGE, GUERNSEY RECORDS Equation of the curve: y = 2393.8 - 434.14x + .2842x 2 + 13951.4 log x, x being the age in units of & months. Accordingly y reaches a maximum at x = 14.129 = 7 years, 24 days, at which age the yield for the year is 12,362 pounds F.C.M. The simple correlations of Table 4 show that there is a significant correlation between any two of the variables. The closest relationship is evident between the initial rate of yield and the yield for the year. The initial rate of yield is also closely related to age and to persistency. The relation of persistency to age and to yield for the year is less close than the other relations noted. 372 BULLETIN No. 288 [April, TABLE 3. STATISTICAL CONSTANTS OF THE VARIABLES INDICATED FOR AGES UNDER FIVE YEARS: GUERNSEY RECORDS Constants k X 10 3 Persistency A Initial rate (pounds F.C.M. per day) Age at calving (years) Yield for year (pounds F.C.M.) Mean 39.5 + .6 36.70 + .17 2.98 + .02 10612 + 41 Standard deviation 30.2 + .4 8.63 + .12 .85 + .01 2307 + 29 Coefficient of variability .... 76.44 1.09 23.51 .34 28.40 .25 21.74 .28 NOTE. The probable errors of the coefficients of variability thruout this paper are computed by the approximate formula, E CT = - . M TABLE 4. SIMPLE AND PARTIAL COEFFICIENTS OF CORRELATION BETWEEN PER- SISTENCY, INITIAL RATE OF YIELD, AGE, AND YIELD FOR THE YEAR: GUERNSEY RECORDS, UNDER FIVE YEARS Variables correlated Variables constant Coefficients Persistency and initial rate .480 + 016 Persistency and age .268 + .019 Persistency and yield .267 + .019 Initial rate and age .490 + .015 Initial rate and yield .691 + .011 Age and yield .308 + .019 Persistency and initial rate Age Yield .415 + .017 953 + 002 Age and yield .935 + .003 Persistency and age Initial rate 042 + 020 Yield .382 + .017 Initial rate and yield .009 + .020 Persistency and yield Initial rate. 943 + 002 Age 381 + .017 Initial rate and age -.933 + .003 Initial rate and age Persistency . . . 428 4- 017 Yield 403 + 017 Persistency and yield .141 + .020 Initial rate and yield Persistency . . 968 + 001 Age 645 + 012 Persistency and age .955 + .002 Age and yield Persistency 409 + 017 Initial rate 048 + 020 Persistency and initial rate . . -.025 .020 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 373 The question arises, to what extent is persistency related to age with initial rate of yield constant, and to initial rate of yield with age con- stant? The partial correlation coefficients give an index to these re- lations. Persistency is closely related to initial rate of yield with age constant, the partial correlation being .415 + .017. On the other hand, with initial rate of yield constant, persistency appears to be independ- ent of age, as shown by the coefficient .042 .020. This relationship is significant in considering the proper correction of the persistency values. Some of the partial correlations of Table 4 have no particular meaning except as a rough check on the computations. For example, if yield is constant, persistency and initial rate of yield bear to each other a definite relation and we should expect a correlation of 1. The correlation found is .953. Two reasons are apparent for the failure to realize the perfect correlation; first, the regressions are not exactly linear; and, second, the persistency and initial rate values of the lacta- tion curves have not been perfectly adjusted to the observed yield. In connection with the latter reason see Fig. 6. The low partial correlation between persistency and age, with initial rate of yield and total yield constant, is to be expected since the constant factors automatically re- quire that persistency also be constant, while age remains variable. Similar comment might be offered for others of the partial correlations, but in general it may be said that they show a fair degree of consistency in the data. Correction Factors. The evidence on the relations between persis- tency and age, and between persistency and initial rate of yield, show that a correction factor for persistency should be based on the initial rate of yield rather than on age. The correlations indicate quite clearly that such relation as exists between persistency and age is associated with the relation between age and initial rate of yield. The correlations, of course, say nothing as to whether persistency is due to initial rate, or whether initial rate is due to persistency. It seems reasonable to pre- sume that persistency is distinctly affected by the initial rate of yield, and that for purposes of comparison correction of the persistency values should be made accordingly. We shall return to the full number of records in deriving the per- sistency correction factors. The correlation surface is given in Table 5. The constants from the correlation surface are given in Table 6. The regression equation derived from the constants of Table 6 is k X 10s = 1.782 A 25.59. The curve of this equation and the mean persistency values of the several initial rate classes are shown in Fig. 12. The mean persistency values given in Fig. 12 show a quite regular agreement with the linear curve with a few exceptions at the extremes. 374 BULLETIN No. 288 [April, SB H H P o 1=1 2 3 2" " P QQ O u "a "o H i-nnsojoeq'-'r^toc^ot^-'i'iN-H i-i CN IN i-l <-i i-i I-" i e t~ CD ^ . H b- rH . . -&i n t~ IN rt rt t^ o M t- 00 CN -< m cS ^^ . ^ .^ . ^^ ^ ^rt-H . .rt -rt IQ CD i-HM'-H'-l -i-IN "-1 "-I o CO 2 rH CN CO I CN 5 O (N C^ -N-H .rH 9 Tl< <5 !< (N OC O5 35 . CO * Tjt ge - r-> C* r- s .rti-iincoo)coaOi-i'-i'Ot>.ocN-Hi-i - r-< IM i-l IN CN >-l <* . CO -H NN(I-(1-I M co o CNcNiCt~CN-<'Ni-lic*C CO CN CN i-l i-l i-l 2 CO N i i(Ni i K IN *" ^"COt^OCOO'H .!-! r ' ' rt N Ni * M --I CO i-t o N O .rtrtrt-VJlO rHrt 00 -IN N laieusioioioioioioinia^ifiiaifiu^teiaiaietoioioiQiaia COWi-i i-i IN CO T|t "3 CO I- 00 O5 O ^ CN CO ^< m CO t- 00 OS O i-i 3 "o r- : (tOI X 1) q^uoui aad aniui J3d es-eajoap jo yu ' PERSISTENCY OF LACTATION IN DAIRY Cows 375 Persistency ( k xio 3 ) 5 cv> ^ o K> < 3 O O O O O O O o { ^ff ^ -- ^ ^ o ^ ^ 5*^ o <: r^ k-lT ' 9-^ "*"' ^ * o "16 24 32. 40 48 56 64 7a &i Initial Rate of Yield -Lbs. F.C.M. per Pay (A) FIG. 12. SHOWING REGRESSION OF PERSISTENCY VALUES ON INITIAL RATE OF YIELD, GUERNSEY RECORDS Equation of the curve: y = 1.782x 25.59. The frequencies here are small, which may account for the variation in the mean values. It is possible that the entrance requirements of the advanced registry may have some effect at the lowest initial rate values. Thus a cow that starts with a rate of 18 pounds F.C.M. per day would need to have a persistency of about A; = in order to satisfy the mini- TABLE 6. STATISTICAL CONSTANTS FOR INITIAL RATE OF YIELD AND PERSISTENCY: GUERNSEY RECORDS Constants k X 10 3 Persistency A Initial rate of yield (pounds F.C.M. per day) Mean 44.25 +.55 39.19 +.17 Standard deviation 32.19 + .39 9.67 +.12 Coefficient of variability 72.75 + .89 24.67 + .30 Coefficient of correlation . . .535 - h.012 mum requirement of 250.5 pounds of fat at two years of age or under. Judging by the data of Fig. 12, no selective action is apparent except possibly in the case of the two cows of the 18-pound class. The correction factors for the persistency values are derived from the above equation k X 10 3 = 1.782A 25.59, to correct to the mean 376 BULLETIN No. 288 [April, value of A, 39.19. Correction to this value of A permits a minimum correction to a maximum number of the records. The correction formula is k c = k + 69.84 1.782 A, where k is the observed k X 10 3 , and k c is the corresponding corrected value. The age-correction factors to be applied to the yield for the year are derived from the equation given in Fig. 1 1 by the direct-ratio method used by Gowen 8 (chap. 4). The age of maximum yield is used as the base, or standard age. The age correction factors for the initial rate of yield are derived from the equation of Fig. 10 on the same basis as above. TABLE 7. MONTH OF CALVING AND MEAN VALUES FOR PERSISTENCY, INITIAL RATE OF YIELD, YIELD FOR YEAR, AND FAT PERCENTAGE: GUERNSEY RECORDS A fc\S -t f\9 Initial rate Yield for Month of calving Number of records X ID" Persistency corrected lor A. of yield corrected for age (pounds year cor- rected for age (pounds Fat percentage for year F.C.M. per F.C.M.) day) January .... 95 48.9 49.2 12 700 5.05 February . . . 128 49.9 49.0 12 297 5.15 March 166 50.0 52.2 12 855 5.05 April 158 50.8 48.4 12 411 5.06 May 126 49,8 45.6 12 040 5.12 June 96 37.3 44.0 12 646 5.01 July . 88 36.7 41.2 12 023 5.02 August. 1 . . . . 123 37.0 42.8 12 476 5 03 September. . 134 37.2 45.2 12 731 5.07 October 165 40.4 47.6 12 936 4.98 November . . 125 42.1 48.4 12 732 5.09 December. . . 130 52.5 50.4 12 508 5.00 It will be understood that the correction of milk yield for fat content (page 359) is really not a correction in the same sense as the age correction. It is rather a method of estimating and expressing the energy value of the milk. Influence of Season. The season of the year at which the cow calves may have a considerable influence on her production under ordi- nary conditions, largely because of the feed supplied from pastures. Sanders 16 has shown that the season of calving affects persistency under the usual practice of management. This seasonal effect is much less important in cows on official test because of the better and more uni- form care and feed which they receive. The data are tabulated in Table 7 according to the month in which the cow calved, and are also given in Fig. 13. It is apparent that the 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 377 total yield for the year is but little affected by the month of calving. The lowest yields occur with calving in May and July. The effect of the time of calving on initial rate of yield is somewhat more pronounced and regular, tending toward a low point for July calvers. The persistency values also show a pronounced drop in June, which continues for the sum- mer months. Apparently cows calving in the summer months hold up somewhat better than cows calving in the winter months when the initial rate of yield is allowed for. No direct corrections are made for the effect of month of calving on persistency or initial rate of yield. 100 I c V 60 0) 040 X 20 n 3 N W * (J< & fat Percentage ^ ^ ^ ^ -^ /*x . ^ ^*- . *-_ ^- .----. --^- r >- -=* ==S^ p. ^ A ~- ^-*-' --*-" h-~*~~~ , >e " V- "0~ --0-- - o-' -0 ~o-- M 0- kxi A 5 s , , . Vie fat d % Jan. Men. May July Sep. Nov. Jan. Month of Calving FIG. 13. EFFECT OF THE SEASON OF FRESHENING, GUERNSEY RECORDS The initial rate of yield (A) in pounds of F.C.M. per day is corrected for age. Persistency, (k X 10 3 ) rate of decrease per mille per month, is corrected for A. The yield for the year in cwt. F.C.M. (Yield) is corrected for age. The fat percentage for the year (Fat %) is taken directly from the records. Variability in Persistency. The distribution of the observed per- sistency values has been given in Table 5, and the constants derived from this array have been given in Table 6. The mean k value is .04425* ; "The same records under consideration here have been previously 8 dealt with in groups classified on the basis of the length of the gestation included in the record period. It may be noted that the k values previously determined from the group records varied from .03714 to .04361, and are thus lower than the mean here reported. The group records include contemporaneous reentries. How far the reentry records 378 BULLETIN No. 288 [April, standard deviation, .03219; and coefficient of variability, 72.75. Correction of these values on the basis presented above leads to the distribution given in Table 8. The two distributions are shown graphically on a percentage basis in Fig. 14. The corrected persistency TABLE 8. VARIABILITY IN CORRECTED PERSISTENCY VALUES AND ACCOMPANYING VARIABILITY IN CALCULATED YIELD: GUERNSEY RECORDS (The persistency values have been corrected for initial rate of yield to A = 39.19) Frequency k X 10 3 Persistency corrected for A (class mid- points) Yield for year A = 39.19 (pounds F.C.M.) 1 - 35 17 821 3 - 25 16 723 13 - 15 15 712 34 5 14 779 79 5 13 918 143 15 13 123 196 25 12 389 226 35 11 710 241 45 11 080 181 55 10 498 161 65 9 957 109 75 9 455 56 85 8 98& 43 95 8 556 25 105 8 153 10 115 7 777 8 125 7 427 135 2 145 6 795 155 1 165 6 242 2 175 5 992 Mean 44.81 11 260 Standard deviation. . . . Coefficient of variability 27.32 60.97 1 636 14.53 may be responsible for the lower k values of the group records in which they appear has not been determined. The primary purpose of this note, however, is to point out that it is not quite proper to assume that the average of the k's is the same as the k obtained from the average of the monthly data. As an extreme example, if we have two records with the same initial rate but in one record k = and in the other k = .2, the mean of the k's is .1; but if the two records are thrown together, the value of k determined from the group record would be, not .1, but about .06. Furthermore, this group lactation curve would not conform well to the curve of the exponential equation, having a considerably greater curvature than the equation provides. Now the observed group lactation curves previously presented show precisely this feature in a very small degree. But it would be straining the point perhaps to ex- plain the deviations of the observed curves on the basis of their being made up of records which individually correspond to the equation and therefore collectively differ from the equation in the direction found. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 379 values give a somewhat more regular distribution curve^than the raw values. The correction has, of course, caused a change of places among the individuals. The mean has been slightly increased and the standard deviation considerably lowered. The coefficient of variability is still very high as compared with the same constant for other characters. It is doubtful if this constant has its usual significance in the present case. 14 " 0) & 01 -40 \ Cc Tec ted 40 60 Persistency 120 160 200 FIG. 14. PERCENTAGE FREQUENCY DISTRIBUTIONS FOR PER- SISTENCY OF LACTATION, GUERNSEY RECORDS The corrected k's have been corrected for initial rate of yield as explained in text. Where the variability in persistency is based on a constant value of initial rate of yield, the yields for the year may be computed. The last column of Table 8 gives these computed yields on the basis of the mean initial rate. These values give 14.53 as the coefficient of variability. The persistency distribution of the two initial-rate arrays of Table 5 at A =38 and 40, by the same treatment give a coefficient of variability 380 BULLETIN No. 288 [April, in yield of 14.00. It appears, therefore, that the natural variability in persistency of Guernsey cows under the conditions and prescriptions of official test is responsible for a standard deviation in yearly yield equal to about 14.5 percent of the mean yield. Variability in Initial Rate of Yield. The distribution of the initial- rate-of-yield values has been given ;n Table 5. The coefficient of 16 o!4 i g- 1 * p uIlO 0)8 O I* u |4 5 ^ (BlttCKCiAS N AS 16 E4 5a 40 4S 56 64 7 60 S6 96 Initial Rate of Yield (A) FIG. 15. PERCENTAGE FREQUENCY DISTRIBUTIONS FOR INITIAL RATE OF YIELD, GUERNSEY RECORDS The corrected A's have been corrected for age of the cow at calving. TABLE 9. VARIABILITY IN INITIAL RATE OF YIELD CORRECTED FOR AGE: GUERNSEY RECORDS (Mean = 47.22; standard deviation = 9.62; coefficient of variability = 20.37) A fb 24 4 28 19 32 74 36 156 40 233 44 257 48 251 52 193 56 141 60 99 64 59 68 19 72 12 76 10 80 4 84 2 88 92 1 Class mid-points of initial rate of yield in pounds of F.C.M. per day. Frequency. variability (Table 6) is 24.67. Corrected for age, the values are given in Table 9. The distributions for the raw and corrected values are given graphically on a percentage basis in Fig. 15. The mean initial rate is of course increased by the age correction since the age of maximum initial rate has been used as a base. The mean of the age-corrected data is 47.22; the standard deviation is 9.62; the coefficient of variability, 20.37. PERSISTENCY OF LACTATION IN DAIRY Cows 381 This coefficient is quite directly comparable with the same measure of variability in yearly yield, since if persistency is constant the yield for the year is proportional to the initial rate and the coefficients of variability would therefore be the same. The constants of the age- and fat-corrected observed milk yields are: mean, 12,553; standard deviation, 2,503; coefficient of variability, 19.94. TABLE 10. AVERAGE DAILY YIELDS BY MONTHS FOR SPECIAL GROUP OF Cows: GUERNSEY RECORDS (Each record shows an increasing rate of yield with advance in lactation.) Months after calving Number of records Milk per day (pounds) Fat per day (pounds) Fat percentage F.C.M. per day (pounds) 1 72 30.0 1.33 4.44 32.0 2 83 28.6 1.31 4.59 31.2 3 83 27.5 1.31 4.77 30.6 4 83 26.9 1.32 4.92 30.6 5 83 26.7 1.36 5.09 31.1 6 83 26.7 1.38 5.17 31.4 7 83 27.1 1.42 5.22 32.1 8 83 27.1 1.43 5.27 32.3 9 83 27.6 1.46 5.29 32.9 10 83 27.5 1.46 5.29 32.8 11 83 26.8 1.44 5.37 32.3 30 29 Months after Calving (t) FIG. 16. SHOWING RATE OF YIELD FOR THE SPECIAL GROUP OF Cows, GUERNSEY RECORDS Each of the 83 records shows an increasing rate of yield with advance in lactation. The equation of the curve fitted by the method of least squares to the data of Table 10 is y d = 30.725e o 06439 '. Increasing Rate of Yield with Advance in Lactation, Advance in lactation usually implies a decrease in the rate of energy yield. A number of the records studied show an increase in the rate of energy yield with ad- vance in lactation. In Table 5 there are 83 such records, constituting 5.41 percent of the total. These records are of special interest because of their unusual, if not abnormal, nature. The results from them, 382 BULLETIN No. 288 [April, treated as a group, are given in Table 10 and Fig. 16. It is evident from Fig. 16 that these records give an average lactation curve which is not in very good conformity with the normal lactation curve. The first three observations show a normally decreasing rate of yield, after which there is a distinct trend toward an increasing rate, and finally, in the eleventh observation, there is evidence that a decreasing rate again has set in. It will be observed further that the initial rate of yield is much lower than the mean of all the records. This relation implies the presence of a dis- proportionate number of young cows. It is evident that the equation used does not afford a complete de- scription of the observed lactation curve for the records of Fig. 10 as a group. On the other hand, any simple expression of the observed lacta- tion curve must recognize a general tendency for the rate of yield to increase with advance in lactation, and this requirement the equation answers satisfactorily for the limited time range. Rate of Yield and Yield for the Year. It is intended under this head to consider the relation between the yearly yield and the rate of yield at various points in the lactation curve, for various age classes, and for all ages together. We have already seen (Table 4) that there is a marked correlation (r = .691) between initial rate and yearly yield, all records taken together. The question is, how does this correlation be- tween rate and yearly yield change as we consider the rate of yield at later times in the lactation; first, with age relatively constant; and second, without any age selection. Evidently a series of correlations may be used to indicate the common point in the lactation curves at which the rate of yield is most closely related to the year's yield. Such a point of closest relationship should be, theoretically, the ideal time to de- termine, by a short-time test, the probable year's production. Any differences due to, or associated with the age of the cow should be brought to light by such a series of correlations. The rates of yield, determined by the constants of the equations, have been computed for each lactation curve at ten additional points; namely, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 months after calving. The rate at ten months after calving was first computed algebraically by the use of a table of exponentials. The rates at the intermediate points were com- puted graphically by the use of an aritho-log chart improvised for the purpose. The and 10 logarithmic ordinates consisted of the scales of a 20-inch slide rule. Connecting these two logarithmic ordinates, ab- scissas were drawn for the 2-pound class intervals used. Nine inter- mediate ordinates were drawn to give equal spacing. A taut thread was properly adjusted to the end ordinates according to the two correspond- ing previously-determined values for a particular record. The points 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 383 t. <*N KNCC^^moOOlN ooooooooooo 5 00 -H-HHH-H-H-H-H-H-H-H-H mr--*osc'-it^io-}< . -H -H -H -H -H-H -H -H -H -H -H iOi-l^CVltDOO5'-O3O>coio>o5 M O t- 1 >, asotcxot^^Tfr^oiaoo fllTfHIMl-IOOOOrH'H'-l ooooooooooo 3 ffl | 1 H-H-H+I-H-H-H-H-H-H-H t-O5t^-00'1''^NCOO3'O COOOOOOTt.iO, 3 >o >. "o 3 -H-H-H+m-H+i-H-H-H-H mt^Ot^^MCOlNOSOJCO I'Jt-WOOM'f^t^tDCOOt^ >ocDt^ooO5O>O5a>O5O)ao 60 g 1 a o in 3 _ V E so .coeoroaot~oit^coco-^ t>.O5OO5Tft^t>-CC-0 IOCOXOOO1O5O3OOSO500 CO .s '8 coor~-*c^'- | ^ ( N | McO'* T-Hr- OS IN O500(N-*'<)"O2OOCO^ iOTtOOOOt^'COTt*'-HO-it^ IM>-iiNtDC^t>-t^^-*COOO Ot-OOOOO5O5OlO5O5OaO ler calving Months al o--io):oTi O Months after Calving FIG. 18. SHOWING THE CORRELATION BETWEEN RATE OF YIELD AT VARIOUS TIMES AFTER CALVING AND YIELD FOR THE YEAR, ALL AGES, GUERNSEY RECORDS The theoretical curve is based on the fitted lactation curves. The observed curve is based on the actual lactation curves. Data from Table 17. The smooth curves given in the graph are merely free-hand sketches. six months after calving. At this point on the lactation curve the two- year-old group and the seven-year-old group have reached practically the same high value, .980. All of the other age groups show the highest correlation at six months, except the six-year-old group, where the high- est coefficient is found at five months. The coefficients decrease in value after six months. These relations for the several age groups are shown graphically in Fig. 17. 386 BULLETIN No. 288 [April, Similar correlations have been computed for all ages lumped to- gether. They fall naturally in between the extremes given in Table 11 and Fig. 17. The correlation surface for the theoretical initial rate of yield and yield for the year is given in Table 12. Similar surfaces at three months, six months, and ten months after calving are given in Tables 13, 14, and 15 respectively. On account of the practical importance of the changes in the close- ness of the relation between the rate of yield and yearly yield with TABLE 12. CORRELATION "BETWEEN THEORETICAL INITIAL RATE OF YIELD AND TOTAL YIELD FOR THE YEAR: GUERNSEY RECORDS Yield for year, hundredweight F.C.M.: class mid-points 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 Total 18 2 2 20 2 7 2 2 13 22 1 4 7 5 1 1 1 20 24 1 15 17 4 37 26 1 4 17 20 6 6 54 28 4 14 31 20 6 2 77 30 1 16 25 }8 Ifi 7 2 105 32 5 16 20 38 23 10 7 119 34 3 32 24 36 20 8 5 1 129 36 6 13 25 34 99 22 q 1 139 38 2 11 21 28 20 28 14 3 127 40 1 6 19 21 31 28 14 2 ? 124 42 44 2 6 2 11 12 26 10 27 19 23 16 21 18 6 5 4 6 1 127 88 46 2 2 8 13 16 18 6 3 1 69 48 50 52 54 56 58 60 62 64 1 '2 2 4 2 1 2 1 1 'i 13 6 11 2 'i 1 11 11 4 4 5 1 1 1 1 12 7 7 5 4 5 3 1 8 6 10 5 5 7 1 1 2 12 9 5 4 3 3 4 1 1 2 5 8 2 5 '4 2 2 2 4 '4 2 2 3 1 1 2 '3 2 'i 2 1 '2 1 65 54 54 24 26 24 24 10 5 66 68 2 ? 'i 1 1 2 6 3 70 1 1 2 72 1 1 74 1 1 1 3 76 i 1 78 1 i 2 Total 5 25 105 196 236 254 220 195 145 68 43 17 14 4 6 1 1534 advance in lactation, a similar comparison has been made between the observed monthly yields and yield for the year. One correlation surface at six months after calving is given in Table 16; that is, the surface is based on the month the mid-point of which is closest to six-months after calving. The constants derived from Tables 12 to 16, and from similar tables for other points in the lactation curves, are given hi Table 17. Table 17 shows also the highest correlation between rate of yield and yearly yield at six months after calving. For the theoretical or smoothed lactation curve, the correlation at six months after calving PERSISTENCY OF LACTATION IN DAIRY Cows 387 is .980. The corresponding correlation for the actual yield of the sixth month and the yearly yield is .935. Both Comparisons show the maxi- mum correlation at six months after calving. The actual yield of the sixth TABLE 13. CORRELATION BETWEEN THEORETICAL RATE OF YIELD THREE MONTHS AFTER CALVING AND TOTAL YIELD FOR THE YEAR: GUERNSEY RECORDS Yield for year, hundredweight F.C.M.: class mid-points 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 Total 18 4 ? ? g 20 1 q 3 1 14 22 q ?5 7 41 24 5 41 38 5 1 90 26 23 e>?, ?fi 3 1 115 28 6 55 70 16 1 148 30 4 22 fiO 53 10 149 32 1 8 4?: 7? 30 9 162 34 ?, ?1 51 53 ?0 1 148 36 8 ?6 63 54 13 1 165 38 1 3 ?1 3fi 53 34 5 153 40 8 15 3? 45 7 1 108 42 1 2 7 13 20 13 8 64 44 2 12 15 19 6 1 55 46 2 2 13 12 11 2 1 43 48 1 2 7 11 R 3 30 50 2 2 4 2 4 14 52 1 2 1 9 54 5 1 6 56 1 1 3 2 7 58 2 2 60 1 1 62 64 1 1 2 Total 5 25 105 196 236 254 220 195 145 68 43 17 14 4 6 1 1534 TABLE 14. CORRELATION BETWEEN THEORETICAL RATE OF YIELD Six MONTHS AFTER CALVING AND TOTAL YIELD FOR THE YEAR: GUERNSEY RECORDS Yield for year, hundredweight F.C.M.: class mid-points 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 Total 16 5 3 1 1 10 18 ?1 q 30 20 1 54 3 58 22 39 86 4 129 24 2 103 49 1 155 26 4 155 27 1 187 28 ?6 149 q 184 30 1 75 96 172 32 1 105 31 1 138 34 1 9 134 g 152 36 ?q 77 1 107 38 1 55 30 86 40 5 32 9 46 42 4 23 1 28 44 11 9 20 46 6 fi 12 48 1 7 8 50 1 ? 3 52 2 1 3 54 ? 2 56 3 3 58 60 i 1 Total 5 25 105 196 236 254 220 195 145 68 43 17 14 4 6 1 1534 388 BULLETIN No. 288 [April, TABLE 15. CORRELATION BETWEEN THEORETICAL RATE OF YIELD TEN MONTHS AFTER CALVING AND TOTALYIELD FOR THE YEAR: GUERNSEY RECORDS Yield for year, hundredweight F.C.M.: class mid-pointa 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 Total g 1 1 g 2 2 10 1 2 1 1 5 j 3 5 4 3 1 17 14 3 9 18 g 3 3 1 45 16 3 28 26 13 7 1 78 18 7 34 52 19 12 4 128 20 1 16 46 49 32 7 2 153 22 2 47 75 49 25 6 1 205 24 8 46 53 33 13 1 1 155 26 4 21 61 57 22 11 176 28 4 28 44 29 10 3 1 119 30 1 7 30 42 28 11 119 32 1 12 47 32 10 5 107 34 6 ?<) 22 16 5 78 36 3 ?5 14 6 f, 50 38 ?! 11 7 10 3 1 34 40 3 4 10 2 1 20 42 1 ? 5 6 4 18 44 2 2 5 1 10 46 1 3 1 ? 7 48 f 2 50 3 3 52 54 1 1 56 1 1 Total 5 25 105 196 236 254 220 195 145 68 43 17 14 4 6 1 1534 TABLE 16. CORRELATION BETWEEN AVERAGE DAILY YIELD FOR SIXTH CALENDAR MONTH OF RECORD AND TOTAL YIELD FOR THE YEAR: GUERNSEY RECORDS Yield for year, hundredweight F.C.M.: class mid-points 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195 205 215 Total 14 1 1 16 3 3 4 10 18 1 13 13 4 31 20 8 38 ?3 ? 71 22 1 1 35 77 10 1 125 24 ^?, fi 55 6 139 26 ?, 18 103 57 5 3 188 28 6 50 74 ?6 ?, 1 159 30 ?, 11 73 R7 10 3 166 32 ft 35 61 47 4 1 153 34 8 46 fi? ?? ? 140 36 10 47 41 5 1 104 38 1 4 18 38 ?3 3 87 40 4 21 12 5 42 42 1 11 13 14 4 43 44 1 4 6 12 g 29 46 4 5 4 'g 'i 22 48 2 1 3 6 50 2 1 2 3 g 52 2 1 3 54 1 1 2 56 2 2 58 2 2 60 62 64 1 1 Total 5 25 105 196 236 254 220 195 145 68 43 17 14 4 6 1 1534 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 389 11 OS 8 f~00 88 U3CO CO-* iM CO i 1 CO .-HCO 88 ^H CO +1+1 lOt^ *o< 0000 +1+1 O-* ^H (^ OS 00 +1+1 10-* lO^H O3OS +1+1 00 OS 8? +1+1 Old oo co OS OS +1+1 t^^H COCO os os +1+1 ** * rt OS OS +1+1 ** rH 00 OS 00 +1+1 *-<00 00 "5 0000 O o (-! --H o CO oo os COCO r^ t^ OJiM cot^ OO coco ooo CO CO O^H coco COCO coco ** CD COCO CO ii ia .S *S h . *s -* CO coco coco CO CO co * CO CO 38 CD CO COCO t>-l> ^H COCO CO nn^ CO 1-1 Ot-i i ( i i SO I-H OSO5 00 oo os oo 00 OS 00 OS oo 00 OS oo 00 OS o o oo os 00 OS OS OS OO CO IIS 3*g -H i> CO +1+1 00 (N CO +1+1 t^Os t^OS +1+1 3S +1+1 coco oo co +1+1 co^ t^rH +1+) ** o +1+1 1O i-l COrH +1+1 *co l>r-H +1+1 ,-HCO OSO +1+1 +1 OSU5 f, OCO 00 GOT} O ^o. os 00 GO tt- t^t^ COt^. COI> COt^ cOt>- cOt^ COt t-t>. CO 3 l> ICIO CO-* O coco COTt< coco (N(N COCO 1 1 1 1 coco oso CO CO 00 OS coco t^t^. CO CO coco coco IOIO coco o .-H o TH I-l : '?. : TJ PH Theoretical . Theoretical. Observed. . . Theoretical . Observed. . . Theoretical. Observed. . . Theoretical . Observed. . . Theoretical . Observed. . . Theoretical. Observed.. . Theoretical . Observed. . . Theoretical. Observed. . . Theoretical. Observed. . . Theoretical. Observed. . . Observe 390 BULLETIN No. 288 [April, month is less closely related to the year's production, than is the cal- culated yield for that month. This is not surprising because of the considerable irregularities of the realized lactation curve. The con- dition is accentuated, furthermore, by the fact that the fat test for the month represents only two days, or one day in a minority of the records, thus increasing the variability of the F.C.M. observations. It is probable, in other words, that similar comparisons based on the raw milk yields would show higher correlations than those found for the F.C.M. values. One such correlation for raw milk yields has been computed for the 6th month and gives r = .938 + .002. The correlation coefficients for the theoretical and observed lac- tation curves are given graphically in Fig. 18. It is evident that the cor- relations change in a quite regular manner with advance in lactation, reaching a maximum during the fifth month of lactation. There is a reason for this in terms of the theoretical lactation curves. We have seen above (Fig. 4), with reference to the theoretical lactation curves, that the yields for the year are proportional to the ordinates of the curves at any specified time, provided k is constant. Theoretically, therefore, the only reason why there is not perfect cor- relation between the computed rate of yield and the observed yearly yield is because of variability in persistency. We must add to this reason the failure to adjust perfectly the theoretical lactation curves to the observed yields, and also some variability in the time after calving at which the record starts. The influence of variability in persistency, in affecting the correlation between the rate of yield at various times after calving and the total yield for twelve months, may be approached as follows: dij Consider the lactation curve as represented by equation (1), -f = at ae~ kt , the area under which, from t = to t = 12, representing the total yield for twelve months, is j- (1 e~ 12 *). It is clear from Fig. 5 K that if we have given a point on the lactation curve at the origin, t = 0, the area under the curve will vary inversely with the value of k. On the other hand, if we have given a point on the lactation curve at the opposite end, t = 12, the area will vary directly with k. At some point in the curve between t = and t = 12, the change in area with change in k passes from inverse to direct. Let this point be designated t' ; that is, t' is an assumed point in the lactation curve where a change in A; produces no change in area. Let Y = the area, or twelve months ' yield, = , (1 e~ 12A ); and let a' = the rate of yield at the time t', = ae~ kt '. 19S7J PERSISTENCY OF LACTATION IN DAIRY Cows 391 Then a = aV" and Y = a e' (1 e~ 12Ar ). We may determine the A/ value of t' by differentiating Y with respect to k and setting - = 0, ak whence dY ~dk > kt' a e + (1 - ka't'e k " - a'e k " 1 12e~ 12fc = 0, and t' = - - k I -*- e~ 12fc fc 1 - e- 12 *_ Solving for ' at the various class values of k found in the data, we have the results given in Table 18. It is evident that t' is not a fixed value but varies with the value of k. Applying the frequencies of Table 8 to the t' values of Table 18 leads to the constants: mean = 5.476 .006; standard deviation = .373 .005; coefficient of variability = 6.81 + .08. For the k's actually found, therefore, there is a point in the lacta- tion curves, t = 5.476, at which variation in k has little or no effect on TABLE 18. SHOWING THE POINTS, t', IN THE LACTATION CURVES AT WHICH THE YIELD FOR THE YEAR Is NOT AFFECTED BY CHANGES IN PERSISTENCY, k (The A; values are the various class values represented in the Guernsey records) k X 10 3 .. -35 -25 -15 -5 5 15 25 35 45 t' 6.42 6.30 6.18 6.06 5.94 5.82 5.70 5.58 5.46 A; X 10 3 .. 55 65 75 85 95 105 115 125 135 t' 5.35 5.23 5.11 5.00 4.88 4.77 4.66 4.55 4.45 k X 10 3 . . 145 155 165 175 185 195 205 215 t' 4.34 4.24 4.14 4.04 3.94 3.85 3.76 3.67 the area. The rate of yield at this point should be directly proportional to the yield for the year regardless of the persistency value. By inference we accordingly may expect to find the highest degree of correlation be- tween rate of yield and the yield for the year at or close to this point. This expectation is realized in the actual observations. Since the observed records start a few days after calving, there would be a tendency for the highest observed correlation to appear slightly later than the com- puted time. The constants of Table 17 afford the basis for estimating the yearly yield from the rate of yield per day. The equation from the smoothed lactation curves at six months after calving is y = (354.40z + 421.8) + 318; and from the actual yield of the sixth month y = (317.02x -f- 1431.3) 570. The probable errors are for a single estimate. Another point of interest in connection with Table 17 is in the mean values of the observed and calculated rates of yield. A fairly good 392 BULLETIN No. 288 [April, agreement is evident, but it appears that the k values have been slightly overestimated. That is, the calculated lactation curves average a slightly greater rate of decrease than is shown by the average actual data. Com- pare Fig. 6 in this connection. Influence of Heredity and Environment. It is well recognized by common observation that the yearly production of cows is influenced both by their ancestry and by the feed and care which they receive. Quantitative measures of the influence of heredity and environment on milk yield and fat percentage have been derived by statistical methods, and published (cf. particularly Gowen, 8 chap. 19). Similar methods are adopted here to study the relative effect of heredity and environment on persistency of lactation. Initial rate of yield, yield for the year, and fat percentage are also considered in the same connection as affording data of value and as a check on the results shown for persistency. The records were sorted into herd groups by the owner's name. Herds were selected which contained at least two unrelated cows, a and at least two cows by the same sire and from different dams. This condition was met by 72 herds containing 252 half-sisters by 97 sires, and 273 unrelated cows. The records of the cows with respect to persistency, initial rate of yield, yield for the year, and fat percentage have been correlated by the method of Harris. 12 The method is equiva- lent to using all possible combinations of the variables correlated, and gives a total number for half-sisters of 532; and for unrelated cows, 1,224. The derived constants are given in Table 19. Corresponding correlations have also been derived for half-sisters from a common dam and by different sires; for full sisters; and for cows related as dam and daughter. The three combinations just mentioned were limited tp the same herd, but on account of the limited number of such combinations all herds were included in which there were two or more cows of any of the relationships specified. The half-sister (common dam) combinations total 48; full-sister, 38; and dam-daughter, 54. The derived constants are given also in Table 19. Since we are dealing now with smaller groups, there is more chance that any group may not be entirely representative. The values of the means given in Table 19 afford some indication as to this. It will be seen that there is fairly good agreement between the means for the several groups except in the case of the full-sister group. In this group both the initial rate of yield (A) and decrease in rate of yield (ft) are high. Also the fat percentage of the full-sister group is distinctly above the average of all the records. There is consequently some question as to the repre- sentativeness of the full-sister group. 'By "unrelated" is meant that the cows were not related as half-sisters, full sisters or dams and daughters. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 393 The coefficients of correlation given in the last section of Table 19 afford an indication of the relative influence of heredity and en- vironment, within the range covered, on the performance of the cow. The coefficients for the unrelated herd mates show the degree of resem- blance in performance due to similarity of environmental factors within a single herd and to dissimilarity of such factors as between the several herds represented. The resemblance may be due in part, also, to blood relationship more remote than that of half-sister. Considering the data for fat percentage, which is well recognized as an individual and breed characteristic, it will be noted at once that daughters of a given sire within a particular herd tend to resemble each other more closely (r = .305) than do unrelated cows within a particular herd (r = .138). It will be recalled from the grouping used that each of the 72 herds is represented by at least two unrelated cows and by two half-sisters. It seems proper, therefore, to attribute the greater re- semblance of the half-sisters to the additional effect of the common par- entage on the sire's side. A more definite measure of the influence of the sire may be obtained by the path coefficient method of analysis of Wright. 21 It may be assumed that the correlation between a half-sister and an unrelated herd mate would be the same as between unrelated herd mates. If we let 1 and 2 stand for related cows and 3 and 4 for unrelated cows, then with re- spect to fat percentage from Table 18, n 2 = .305 and r 34 ( = r J3 = r 23 = TU = 7*24) = .138. The coefficient r u is determined by the action of two forces: the common herd, or environment, and the common parentage, or heredity. Representing these forces by E and //respectively, the re- sult is shown diagramatically in Fig. 19. The correlation between half- sisters is ri 2 = .3054 = h 2 -f e 2 ; the correlation between unrelated cows, r 34 = .1376 = e 2 . Hence h 2 = .3054 - .1376 = .1678. The correlation between half-sisters with the herd influence eliminated, r^.E becomes, 1^2 I ft7<2 - = = .1946. This is the correlation to be expected be- tween half-sisters on the sire 's side within a single large herd or where the herd conditions are the same. Additional similar coefficients are given in Table 20. It may be seen from Table 19 that half-sisters from a common dam show a similar degree of resemblance with respect to fat percentage as that shown by half-sisters by a common sire. The closer relationship, represented by full sisters or dam and daughter, shows a higher degree of resemblance. 394 BULLETIN No. 288 [April, li r-> OCOIMO"* (M C^ * lO * * OOOOOO 1C <* I-H !>. OO ^H I-H I-H co co C i-H rH (N (M (N CO _o 1 1 C<1 1 1 1-H +1+1 +1+1 +1+1 CO t> t i I ^ CO CO O5 * OO OO * c3 -a o s i S s o o Q r-J (V, i U) S3 & ^H t* *-< tt - Oi Oi T^ CO d OO O O5 id 03 13 _ oo PH li S-S-gg d t^ooc^Hcoi> O> O OO OOOOO i-H 1 g h, W h * COO 00 (M l^ CO CO1> CO ^H O ro o U a M i-H i-l (N CO O5 CO CO +1+1 +1+1 +1+1 TjH (N I> T-H TjH O mmon si 03 5 K oj PH O ^ co "^ "^ "^ co ^ lO -* 1C COCOOO o O K < ~ S3 H ERTAIN < IENTAL '. GO 43 O 5 ft. 03 > H 25 Q, :::: O O O5 t^OO t^ it II +1 +1 +1 +1 +1 + ^ OTt< OSr-ICO C^ CO COO5 1C (M o o oo o +1+1 +1+1 + '55 oj bO O (35 00 00 00 (35 T-I CO (N -^Tt* G $ , OO 00 (M COCOOO iO "7>CO CO CO CO Oi T^ OOO I-H *3 >> i-H 1 1 I 1 1 1 - ooooo G T3 O H . r-H o O 5T r ^ 0) t O vj 1 +1 +1 +1 +1 +1 +1 CO O5 CO Tj< CO O5 O5 rH i-H O 1C i-H o 1 +1 +1 +1 +1 +1 (35 (35 rH rH o *-< o o O r") s? co 4J ICCO COOO rH O5 00 e ^ fa !S s 00 I> CO 1C CO Ot^ CO co O 5 PH o -* C35 CO 00 CO CO T-H T-H IO O CO 6 jB . Oa to 1 tT a a co '53 1 if fe" Intragroupal Re. Unrelated herd mates* Half-sister herd mates (C-S) b Half-sister herd mates (C-D) C Full-sister herd mates *. . . . Dam-daughter herd mates. . . /dai Idai Unrelated herd mates Half-sister herd mates (C-S) Half-sister herd mates (C-D) Full-sister herd mates Dam-daughter herd mates S 1 oj T5 -a 1 a ^s "o 396 BULLETIN No. 288 [April, With respect to persistency of lactation it appears from Table 20 that there is no correlation between half-sisters by a common sire; but there is a considerable correlation between half-sisters from a common dam. There is also a significant correlation between dam and daughter. FIG. 19. ANALYSIS OF THE CORRELATION BETWEEN RELATED HERD MATES, AS DETERMINED BY HEREDITY AND ENVIRONMENT Unrelated herd mates are represented by 3, 4; related herd mates, by 1, 2. It is assumed that the correlation between unrelated herd mates (m) is the result of the common environment (E); the correlation between related herd mates (ri 2 ) is the re- sult of the common environment (E) plus the common parentage or heredity (H): rs4 = e 2 ; r i2 = relationship. e 2 + h 2 ; r J2 . E = the correlation associated with the blood TABLE 20. ESTIMATED COEFFICIENTS OF CORRELATION WITH RESPECT TO CERTAIN CHARACTERS, BETWEEN Cows OF VARIOUS BLOOD RELATIONSHIPS WITHIN AN INDEFINITELY LARGE HERD : GUERNSEY RECORDS ^) Relationship Characters Persistency of lactation corrected for A Initial rate of yield cor- rected for age Yield for year cor- rected for age Fat percentage for year Estimated coefficients of correlation Half-sister, common sire. . . Half-sister, common dam . . Full-sister -.001 +.029 .418 + .080 -.178 .106 .259 +.086 .186 + .028 .093 + .097 .387 .093 .458 + .073 .273 + .027 .424 + .080 .169 .106 .364 .080 .195 + .028 .156 + .095 .329 .098 .313 .083 Dam-daughter These relations suggest the interesting hypothesis that the persistency of a cow is affected by inheritance thru the dam but not thru the sire. The correlation between full sisters does not bear out such an hypothesis and in view of the large probable errors of the coefficients it may be questioned whether such an interpretation is justified. On the other hand, it has been noted above that the full-sister group does not conform closely to the average of all the records. Also, the correlation between full sisters with respect to yield for the year (r = .169) is less than is to 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 397 be expected from other results (Gowen, 8 p. 145). It seems clear that there is no resemblance between half-sisters on the sire's side with respect to persistency of lactation. This means either that the sire exerts no influence on his daughters or that all exert the same influence. RESULTS FROM HOLSTEIN RECORDS The Holstein data, in so far as they permit, will be presented in the same general fashion as has been used for the Guernsey records. The Holstein data do not include any record of the year's performance or date of freshening, so that relations involving these items cannot be considered. They do include one item not included in the Guernsey data, namely, the performance of the same cow in different lactations. There is, of course, a large amount of material of this kind available in the published Guernsey records, but the present study has been con- fined to original entries in the Guernsey records. Age and Persistency. The mean persistency values for the various age classes are given in Table 21 and Fig. 20. The mean values, like those for the Guernsey, are quite irregular, and a satisfactory repre- sentation by a smooth curve is difficult. It is clear, however, that there is a distinct rise in the curve at first, followed by a less rapid decline. TABLE 21. VARIOUS AGE CLASSES AND THE CORRESPONDING MEAN VALUES FOR PERSISTENCY AND THEORETICAL INITIAL RATE OF YIELD: HOLSTEIN RECORDS Age in years (class mid- points) Number of records k X 10 3 Persistency A Initial rate of yield (pounds F.C.M. per week) 1 75 ... 48 45.8 339.2 2.25 306 43.2 377.3 2 75 ... 193 47.1 415.8 3.25 136 56.6 461.8 3.75 119 58.9 491.1 4.25 84 62.4 528.1 4 75 ... 98 57.0 524.9 5.25 82 62.2 542.9 5.75 66 57.1 547.9 6.25 66 63.2 561.2 6.75 51 53.2 526.3 7.25 33 58.0 592.7 7.75 18 51.1 520.0 8.25 25 62.2 582.4 8.75 14 55.7 580.0 9.25 13 57.3 606.2 9.75 18 61.7 586.7 10.5 12 50.8 536.7 11.5 10 55.0 564.0 12.5 2 60.0 560.0 13.5 1 75.0 640.0 398 BULLETIN No. 288 [April, As compared with the Guernsey curve (Fig. 9), the changes with age are somewhat less pronounced. Age and Initial Rate. The mean values for initial rate of yield per week also are given in Table 21. Graphic presentation is given in Fig. 80 70 40 50 * a to Age -Years ia ILLUSTRATING THE CHANGE IN PERSISTENCY WITH ADVANCING AGE, HOLSTEIN RECORDS FIG. 20. Equation of the curve: y = 35.6 - 8.36x + . 88.73 log x, x being the age in years. According to the equa- tion, y reaches a maximum at x = 6.09 = 6 years, 1 month, 3 days, at which age y = 60.5. That is, the rate of decrease in the rate of yield per month at that age is 60.5 per mille per month. 21. The changes in rate of yield with age are quite regular up to years, after which there is considerable irregularity due in part to the small numbers represented. As compared with the similar data for Guernseys (Fig. 10), the changes with age are much more pronounced. It should be noted that the vertical scale of Fig. 21 is about half that of Fig. 10. Persistency and Initial Rate. The relation of persistency to initial rate of yield and to age may be shown approximately by dealing with the first part of the age curve in order to secure a reasonable approach to linear regression. For this purpose we may consider the records of those cows under 4*^ years of age at calving. There are 886 such records, making up 63.5 percent of the total, and the constants are given in Tables 22 and 23. Principal interest attaches to the correlation between persistency and age with initial rate constant, the partial coefficient of correlation PERSISTENCY OF LACTATION IN DAIRY Cows 399 being .006. This agrees with the Guernsey results in indicating that persistency is independent of age except as the initial rate of yield is associated with age. The correlation surface for persistency and initial rate of yield is given in Table 24 and the constants derived from this surface are pre- o ? 550 a. 2500 oo o o o o o ^ "^""^ -, o / / o X x 7 / / y / 7 " J I a 4 ^ e 10 la M Age -Years FIG. 21. SHOWING THE CHANGE IN INITIAL RATE OP YIELD WITH ADVANCING AGE, HOLSTEIN RECORDS Equation of the curve: y = 213.5 23.50a; .546s 2 + 645.78 log x, x being age in years. This gives y a maximum value at x 8.542 = 8 years, 6 months, 15 days, at which time y = 574.4, the maximum initial rate of yield in pounds F.C.M. per week. sented in Table 25. The correlation (r = .433) is not quite as high as that found for the Guernsey records (r = .535). The mean persistency value, k X 10 3 = 53.2, shows a considerably greater rate of decline than was shown by the Guernsey records. In this connection it is necessary to consider the initial rate of yield which for the Holstein records is 471 pounds of 4-percent milk per week, or 67.3 pounds per day as compared with 39.2 pounds of 4-percent milk per day for the Guernsey data. Thus while at first sight it might seem that the Hol- stein cows, as compared with the Guernsey, show a greater rate of de- 400 BULLETIN No. 288 [April, TABLE 22. STATISTICAL CONSTANTS OF PERSISTENCY AND INITIAL RATE OF YIELD FOR AGES UNDER FOUR AND ONE-HALF YEARS: HOLSTEIN RECORDS Constant k X 10 s Persistency A Initial rate (pounds F.C.M. per week) 50.2 + .7 426.1 + 2.3 Standard deviation 28.9 + .5 99.3 + 1.6 Coefficient of variability, 57.58 + .92 23.31 + .37 TABLE 23. SIMPLE AND PARTIAL COEFFICIENTS OF CORRELATION BETWEEN PER- SISTENCY, INITIAL RATE OF YIELD, AND AGE: HOLSTEIN RECORDS UNDER FOUR AND ONE-HALF YEARS Variables correlated Variables constant Coefficients Persistency and initial rate .407 + .019 Persistency and age .232 + 021 Initial rate and age .559 + 016 Persistency and initial rate Age .344 + .020 Persistency and age Initial rate .006 + .023 Initial rate and age Persistency .523 + .017 TABLE 24. CORRELATION SURFACE FOR INITIAL RATE OF YIELD AND PERSISTENCY: HOLSTEIN RECORDS Initial rate of yield, pounds F.C.M. per week (A): class mid-points 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800 840 880 Total -25 1 1 1 3 -15 ? 6 1 .. 8 - 5 7 3 3 ? 1 1 1 18 5 7 10 7 R 11 ? 1 4 48 15 1 5 17 1?, 13 8 6 5 5 1 73 25 5 15 23 24 16 15 15 15 3 5 3 4 143 35 1 8 17 22 18 30 20 14 6 10 4 1 1 1 153 45 5 16 26 29 38 30 29 10 5 6 7 5 2 208 55 2 6 23 31 24 31 29 20 13 10 4 7 3 2 205 65 8 4 14 18 20 21 23 17 ?,?, 8 8 4 ?, 1 1 161 75 1 3 6 10 17 25 22 18 16 16 7 2 2 1 146 85 2 5 4 8 19 15 15 11 13 8 ?, ?, 8 1 109 95 3 6 9 12 10 6 8 3 3 1 55 105 1 1 2 5 7 3 4 2 1 2 i 29 115 2 1 1 2 3 2 1 2 1 15 125 1 3 2 2 1 1 1 11 135 1 3 4 145 1 1 2 155 165 1 1 2 175 1 1 2 Total 17 64 101 147 165 183 181 182 109 95 62 33 31 13 8 3 1 1395 crease in yield with advance in lactation, the very marked difference in level of production may require a modification of such a view. The Holstein lactation curves, it will be remembered, are based on two 7-day tests; the second test occurring not less than eight months after calving. It would seem that there might be a very natural tendency to conduct this second test only on those cows which were milking heav- 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 401 ily enough to make a creditable record, that is, to exclude cows of more rapid decrease in rate of yield. Such a practice would tend to make the TABLE 25. STATISTICAL CONSTANTS FOR INITIAL RATE OF YIELD AND PERSISTENCY : HOLSTEIN RECORDS Constant kX 10 3 Persistency A Initial rate of yield (pounds F.C.M. per week) Mean 53 2 + 5 471 +21 Standard deviation 28 1 + 4 116 +15 Coefficient of variability 52.85 + 67 24 63 + 31 Coefficient of correlation .433 - b .015 Persistency (kxio 8 ) r Jw t^ c o o o o o c ^ - ^ ^ e ^ >- ^ ^f o e o o ^ 0. >- ^_ o e 240 320 400 460 560 640 720 800 680 Initial Rate of Yield -Lbs. f.C.M. per Week (A) FIG. 22. SHOWING REGRESSION OF PERSISTENCY VALUES ON INITIAL RATE OF YIELD, HOLSTEIN RECORDS Equation of curve: y = 3.75 + .105x. k value of the records too low to be representative. On the other hand, since few, if any, of the cows concerned were on yearly test, there would not be the incentive to the owner to maintain milk flow at the highest possible level after the first test. This condition would tend to give a higher value to k than might prevail for the same cows on yearly official test. Other possible complicating factors have been mentioned above (Fig. 7). The regression of persistency on initial rate of yield is given in Fig. 22. The regression equation derived from the constants of Table 25 is k X 10 3 = 3.75 + .1054. It will be seen from Fig. 22 that the mean persistency values at either extreme of the initial-rate range do not 402 BULLETIN No. 288 [April, conform very well with the regression curve. On the whole, however, a linear description appears to be justified for the bulk of the records. A noticeable difference between the Holstein and Guernsey data is in the slopes of the curves, 1.782 for the Guernsey and .735 (= .105 X 7) for the Holstein. Correction Factors. The initial rate of yield is corrected for age to the age of maximum initial rate as per the relations shown in Fig. 21. A 20-inch slide .rule provided with a specially graduated slide has been used in making these age corrections. The application of the method will be readily apparent. The proper multiplication factors at certain specified ages are derived from the equation. Graduations are made on the slide corresponding to these factors, but labeled with the appropriate age value instead of the value of the factor itself. This method of computation is highly advantageous in point of time and convenience. It also permits rather fine age distinctions (.1 month) at those ages, say under three years, where the factors are changing most rapidly. This feature practically offsets the four-figure limitation of legibility inherent in the dimension of the rule. The persistency values are corrected for initial rate of yield to the mean initial rate, 741, according to Fig. 22, by the equation k c = k + 49.42 .105A; in which k is the observed value of A; X 10 3 , and k c the corresponding value corrected for initial rate of yield. Variability in Persistency. The distribution of the observed per- sistency values has been given in Table 24. Correction of these values for initial rate of yield leads to the distribution given in Table 26. The two distributions are presented graphically on a percentage basis in Fig. 23. The corrected persistency values give a more regular distribution, a slightly higher mean, and considerably lower standard deviation as compared with the raw values. The coefficient of variability is very high in either case. When expressed in terms of the theoretical yields, the coefficient of variability takes a much more usual value, namely 13.05 (see last column of Table 26). The corresponding figure for the Guernsey records is 14.53. Hence it appears that there is only slightly less variability in persistency for the Holstein records than that found above for the Guernsey records. Variability in Initial Rate of Yield. The distribution with respect to initial rate of yield has been given in Table 24. Correction of these values for age gives the distribution presented in Table 27. The coeffic- ient of variability is 18.84 and, as before mentioned, this value should be very closely related to the same constant for yield for the year. A note- worthy feature of these Holstein records is the high mean value of the 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 403 theoretical initial rate of yield, 571. This is equivalent to 81.6 pounds of 4-percent milk per day for the mature cow, or nearly double the comparable figure for the Guernsey records. \t> 14 Row Corrected k'a k'a -40 40 80 Persistency IEO (kxio 3 ) 160 200 FIG. 23. PERCENTAGE FREQUENCY DISTRIBUTIONS FOR PERSISTENCY OF LACTATION, HOLSTEIN RECORDS The corrected k's have been corrected for initial rate of yield, as explained in text. Rate of Yield and Yield for the Year. It is of interest to determine for the Holstein records the point in the lactation curve that affords the best index of yield for a year. The records do not give the actual year's production, so that a direct comparison such as that made for the Guernsey records is not possible. It is possible, however, to consider the problem on a theoretical basis. The points, t', on the lactation curves at which the area is unaffected by variation in k for the various persistency classes have been given in Table 18. The use of these values in conjunction with the persistency frequency distribution of Table 24 gives the following constants : mean 404 BULLETIN No. 288 [April, = 5.357 .005; standard deviation = .291 + .004; coefficient of vari- ability = 5.44 .07. From these results it may be assumed that the highest correlation between rate of yield and yield for the year would be found at about 5.36 months after calving. For any single random record the chances are even that the point i' would lie within about six TABLE 26. VARIABILITY IN CORRECTED PERSISTENCY VALUES AND CALCULATED YIELD FOR THE YEAR: HOLSTEIN RECORDS (The persistency values have been corrected for initial rate of yield to A =471.) Frequency k X 10 3 Persistency corrected for A (Class mid- points) Yield for year A =471 (pounds F.C.M.) 2 - 15 26 982 5 - 5 25 380 26 5 23 902 65 15 22 537 132 25 21 275 172 35 20 110 228 45 19 028 233 55 18 029 196 65 17 099 134 75 16 237 100 85 15 439 51 95 14 694 21 105 14 000 11 115 13 356 10 125 12 754 4 135 12 194 1 145 11 669 2 155 11 178 1 165 10 721 175 185 1 195 9 510 Mean 54.23 18 327 Standard deviation 25.08 2 392 Coefficient of variability .... 46.25 13.05 TABLE 27. VARIABILITY IN INITIAL RATE OP YIELD CORRECTED FOR AGE: HOLSTEIN RECORDS (Mean = 571; standard deviation = 107.6; coefficient of variability = 18.84) A... 308 336 364 392 420 448 476 504 532 f b 1 7 26 44 56 73 106 147 144 A.. 560 588 616 644 672 700 728 756 784 f 139 143 123 109 82 68 46 29 23 A.. 812 840 868 896 924 952 980 1008 1036 f 5 8 3 5 2 2 3 1 Class mid-points of initial rate of yield in pounds of F.C.M. per week. b Frequency. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 405 days of this mean value. It is to be expected on the basis of the above results that the rate of yield during the fifth month of lactation would be fully as highly correlated with the comparable yield for the year in the case of these Holstein cows as has been shown to be the case for the Guernsey cows. Influence of Heredity and Environment. The comparisons made here are similar to those made above for the Guernsey records, with respect to persistency and initial rate of yield. Data for yearly yield and fat percentage for the year being lacking, the comparisons cannot be made for these items. Those herds were selected in each of which there were at least two unrelated cows, as above defined, and also at least two half-sisters by a common sire and from different dams. The correlations with respect to persistency and initial rate of yield, between unrelated cows on the one hand and between half-sisters on the other hand, have been com- puted from the available records of the cows in these selected herds. Similar correlations have been determined for half-sisters from a common dam and by different sires; for full sisters; for dam and daughter; and for different lactation records of the same cow. All herds which pro- vided two or more records of any one of the relationships specified in this paragraph have been used in order to secure as large numbers as possible. It may be assumed that any correlation between unrelated cows is due to the environmental relationship of being in the same herd. For the other groups there is added to this environmental relationship the blood relationship specified by the basis of the selection and, for the one case, the identity of the animal. The number of herds used in the comparisons for unrelated cows and half-sisters by a common sire is 55; the number of combinations of unrelated cows, 1,620; the number of combinations of half-sisters by a common sire, 1,774; and the number of such common sires, 126. The number of combinations of half-sisters from a common dam is 78 and of full sisters, 64. There are 83 dam-daughter pairs, and 96 cows having two lactation records. The various statistical constants are given in Table 28. The mean values of Table 28 show a general agreement between the several groups, altho with some points of diversity. It will be noted that the half-sister (common sire) herd mates and the unrelated herd mates agree quite closely in age-corrected initial rate of yield and also in persistency corrected for initial rate of yield. The daughters of the dam-daughter group show an exceptionally high rate of decrease in yield. The constants of the lactation curves indicate that the daughters of the dam-daughter group would have a somewhat lower yearly yield than their dams. 406 BULLETIN No. 288 [April, The initial rate of yield of the second records of the 96 cows in the last group show an increase of 48.8 pounds F.C.M. per week over the first records. The persistency values are practically the same. Evi- dently the correction factors have served to correct the indirect influence TABLE 28. STATISTICAL CONSTANTS OF CERTAIN CHARACTERS WITHIN GROUPS SELECTED ON THE BASIS OF BLOOD AND ENVIRONMENTAL RELATIONSHIP: HOLSTEIN RECORDS Intragroupal relationship k X 10 3 Persistency corrected for A Theoretical initial rate of yield corrected for age (pounds F.C.M. per week) Means Unrelated herd mates 8 55.31 + .43 56.29 + .40 50.58 + 1.04 45.38 + 1.42 49.57 + 1.71 59.73 + 2.16 48.45 + 1.36 48.73 1.77 574.6 + 1.7 580.4 1.8 609.3 7.5 582.2 + 9.7 585.4 + 7.0 585.2 + 8.2 569.8 7.1 618.6 7.2 Half-sister herd mates (C-S) b Half-sister herd mates (C-D) C Full-sister herd mates T- t i , i_ j / dams. . Dam-daughter herd mates | daughterg i jj f first records. . Same cow, same herd*. . . . { second recordg Standard deviations Unrelated herd mates 25.74+ .31 25.22 .29 13.57 + .73 16.89 + 1.01 23.12 + 1.21 29.14 + 1.53 19.74+ .96 25.65 + 1.25 99.9 + 1.2 113.0 + 1.3 97.6 + 5.3 115.6 6.9 94.3 4.9 111.4 5.8 102.8 + 5.0 105.1 5.1 Half-sister herd mates (C-S) Half-sister herd mates (C-D) Full-sister herd mates T-I j ' i_ i_ j / dams. . Dam-daugher herd mates. ( daughters a i j 1 first records. . Same cow, same herd. . . . { gecond records Coefficients of variability Unrelated herd mates 46.53 .55 44.81 + .51 26.83 + 1.45 37.22 + 2.22 46.64 + 2.44 48.79 + 2.55 40.74 + 1.98 52.63 2.56 17.38+ .21 19.48 + .22 16.02 + .87 19.86 + 1.18 16.10 + .84 19.04 1.00 18.05 + .88 16.99 + .83 Half-sister herd mates (C-S) Half-sister herd mates (C-D) Full-sister herd mates Dam-daughter herd mates / dams \ daughters Same cow, same herd .... f first records 1 second records Coefficients of correlation Unrelated herd mates .144 + .016 .239 .015 .478 + .059 .067 + .084 .462 + .058 .461 + .054 .064 .017 .336 .014 .261 .071 .367 .073 .329 .066 .438 + .056 Half-sister herd mates (C-S) Half-sister herd mates (C-D) Full-sister herd mates Dam-daughter herd mates Same cow, same herd Not related as dam and daughter, full sister or half-sister. b Common sire, different dams. c Common dam, different sires. Successive lactation records of the same cow. 1927} PERSISTENCY OP LACTATION IN DAIRY Cows 407 of advance in age on persistency, but have not fully corrected the more direct influence of advance in age on initial rate of yield. Expressed in terms of yearly yield, the second record shows an increase in yield in excess of that expected on the basis of the average change in yield with age for the entire number (1,395) of cows. The excess increase amounts to about 8-percent of the first record. The most likely explanation for this excess increase is the artificial factor that there is no commercial object in conducting and reporting a second test on a cow unless that second test shows a relatively higher production than the first test. TABLE 29. ESTIMATED COEFFICIENTS OF CORRELATION WITH RESPECT TO PERSIS- TENCY AND INITIAL RATE OF YIELD, BETWEEN Cows OF VARIOUS BLOOD RELATIONSHIPS WITHIN AN INDEFINITELY LARGE HERD: HOLSTEIN RECORDS Relationship Characters Persistency of lactation corrected for A Initial rate of yield cor- rected for age Half -sister, common sire .111 + .016 .390 .065 - .090 + .084 .371 + .064 .370 + .059 .290 + .015 .211 + .073 .324 + .075 .284 + .068 .400 .058 Half-sister, common dam Full-sister Dam-daughter Same cow" Correlation between successive lactation records of the same cow. There is also the possibility of the excess increase being due to a physi- ological factor in the nature of an increased development of the mam- mary functions due to the exercise and training of the first test. The latter explanation has been noted by Gowen 9 and emphasized by Graves and Fohrman 10 . The coefficients of correlation show an appreciable resemblance between unrelated herd mates with respect to persistency and a smaller degree of resemblance with respect to initial rate of yield. The cor- relations for the various relationships within a single herd estimated by the method used above are given in Table 29. These correlations with environment constant indicate a very slight relation between half-sisters by a common sire with respect to per- sistency of lactation, namely, r = .111. But the relation between half- sisters from a common dam is much more marked, r = .390. The re- lation between dam and daughter is of the same order, r = .371. And both these are slightly more closely related than are the first and second records of the same cow, r = .370. 408 BULLETIN No. 288 [April, The resemblance in persistency of cows related thru the dam appears to be much greater than those related thru the sire. This is in agreement with the Guernsey data. Also, like the Guernsey, the Hoi- stein full sisters show no significant correlation in persistency. It is difficult to reconcile the results for full sisters with the results for other relationships, for full sisters have a common dam and it would seem should show as close a resemblance as half-sistefs from a common dam. While the small numbers reduce the significance of the correla- tions as a basis of generalization, the general agreement of the Guernsey and Holstein data make it appear that there is the possibility of a real difference in the correlation between the persistency values of cows according to their relationship thru the sire or thru the dam. DISCUSSION Selection of Records. Certain of the Guernsey records, taken in the order of publication, have been excluded in the results as presented above. Only original entries have been used. A further selection was made on the basis of time of breeding, so as to eliminate the complica- tion of the effect of advanced pregnancy on the lactation curve. The records showing very irregular lactation curves were also excluded. It is desirable to examine these records excluded in the above results. An indication of the effect of the selection on the basis of time of breeding is afforded by the graphic presentation of Fig. 24. The data on which this figure is based are taken from Bulletin 272 6 and include reentry records. There is considerable variability in the average initial rate of yield of the several groups separated according to the time of breeding (the farrow cows are shown as bred 12^ months after calving). There is a very slight tendency to a higher initial rate of yield in the groups in which breeding is delayed. On the other hand, there is quite a marked tendency to a slower rate of decrease in yield with the delayed breeding. The basis on which this rate of decrease has been determined practically eliminates the effect of pregnancy. Delayed breeding is sometimes practiced to secure as large a record for the year as possible. The relations shown in Fig. 24 may be taken to mean that there is no particular tendency to select inherently higher yielding cows to be held open, but that there is a tendency to so manage the groups in which breeding is delayed as to cause them to hold up in milk flow better than the average. On this account the exclusion of records with a service period of less than six months tends to disturb the persistency distribution curve and to give a yield for the year somewhat higher than would hold for all the original entries. 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 409 The records which were excluded on account of the irregular nature of the lactation curve are treated in Fig. 25. The four individual lacta- tion curves given serve to illustrate the diversity found as between individual records. The curve for No. 6356 shows an increasing rate of yield for five months, followed by a decreasing rate of yield; the curve for No. 7405 is roughly the inverse of that for No. 6356. The curve for No. 8032 shows a high rate of decline in approximate conformity with the equation type for six months, after which the rate of yield holds nearly constant; the curve for No. 8777 on the other hand, shows an increasing rate of decline during the last six months. 57 55 53 51 49 29 Number of Cows 13 255 335 746 754575 572 Z, 235 172 196 55 Initial Rate Persistency 1200 1160 1120 HOG 1080 o 1060 1 a 3 4 5 fe 7 8 9 10 11' ia 13 Service Period -Months FIG. 24. SHOWING THE RELATION BETWEEN SERVICE PERIOD AND INITIAL RATE OP YIELD AND PERSISTENCY, GUERNSEY RECORDS Accepting the exponential curve as representing the normal course of affairs, it is apparent that the normal course is altered markedly in individual cases. When the individual curves of the irregular records (142) are thrown together and treated as a group, they fall in a fairly regular order (the circles in Fig. 25). Judged by the average curve alone, these irregular records conform very well to the exponential equation. The deviations of individuals are therefore of a compensating nature. Chemical Interpretation of the Lactation Curve* Brody et al, 1 on "Since this material was prepared, a somewhat more extensive treatment has been published by the author in an article, "Interpretation of the Lactation Curve," Jour. Gen. Physiol 10, No. 1, 27-31. 1926. 410 BULLETIN No. 288 [April, the basis of the equivalence of the equations of the lactation curve and that of a monomolecular reaction, suggest that the rate of milk secretion is limited by such a chemical reaction. The heterogeneous nature of the performance of individual cows naturally raises some question as to the validity of such an interpretation. 45 40 I cu n. 35 o III <0 20 15 468 Morrths after Calving 10 FIG. 25. IRREGULAR LACTATION CURVES, GUERNSEY RECORDS Four individual lactation curves (A.R. Nos. 6356, 7405, 8032, and 8777) are shown to illustrate the variation found in individual cases among the records ex- cluded on the ground of the irregular nature of the lactation curves. The circles repre- sent the average of the 142 records thus excluded. Equation of the corresponding smooth curve: y = 35.78e 03685( . On the basis of such a chemical interpretation the rate of decline in milk yield is governed by the velocity constant, k of the equation. Tables 8 and 26 show a very high degree of variability in the constant k. It may be questioned whether the velocity constant of a particular chemical reaction, under constant temperature conditions, would show such a high degree of variability. The conception of a limiting reaction 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 411 of this kind furthermore is complicated by the fact that 5 percent of the individual records show an increasing rate of yield with advance in lacta- tion for ten to twelve months. Apparently, if it is to be postulated that the rate of milk secretion is determined by a limiting substance, it must be further postulated that this limiting substance is not all present or active until some time (up to 12 months) after the beginning of lactation. Breed Lactation Curves. The initial-rate-of-yield distributions shown by the Guernsey records (Table 9) and by the Holstein records (Table 27) are brought together on a percentage basis in Fig. 26. There is apparent a very distinct difference in the two breeds with respect to the if 2 ih Guorns Hpfctelr -40 4O 8O I2O 160 Persistency (kxio s ) 200 FIG. 27. SHOWING PERCENTAGE FREQUENCY DIS- TRIBUTION CURVES OF PERSISTENCY OF LACTA- TION FOR GUERNSEY AND HOLSTEIN RECORDS The persistency values refer to the rate of decrease per mille per month in the rate of yield of 4 percent milk per month. The data of both curves have been corrected for initial rate of yield to the mean initial rate of yield: namely, 39.2 pounds F.C.M. per day for the Guernsey records and 67.3 pounds F.C.M. per day for the Holstein records. Hence it appears that while the Holstein records show a greater rate of decrease than the Guernsey records, yet when initial rate of yield is allowed for, the reverse is true. In view of the difference of the slope of the regression lines of persistency on initial rate of yield (Fig. 28), it is a question as to just how the two breed records may equitably be com- pared with respect to persistency, altho it would seem proper to say that the Holstein records show greater persistency (smaller value of k) than the Guernsey records. Ellinger 3 mentions that the Red Danish breed is more persistent 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 413 than the Jersey breed. His measure of persistency is the ratio of the milk yield of the second 10-week period of the lactation to the milk yield of the first 10-week period. Possibly the breed differences in persistency are associated with size (weight), the larger cow tending to be more per- sistent than the smaller cow. Carlyle and Woll 2 report results on persistency for cows in the experiment station herd classified in three groups: A, extreme dairy type (9 Jerseys, 4 Guernseys, 4 Holsteins) ; B, large dairy type (3 Jerseys, 5 Guernseys, 4 Holsteins); C, dual-purpose type (11 Shorthorns, 1 Red 120 40 Zi 48 4 80 96 |l Initial Rate of Yield -Lbs. f. C.M. per Pay 126 FIG. 28. SHOWING THE REGRESSION OF PERSISTENCY OF LAC- TATION ON INITIAL RATE OF YIELD FOR GUERNSEY AND HOLSTEIN RECORDS Polled). Their measure of persistency is the decrease in milk or fat yield of a late week in lactation as compared with an early week, the decrease being expressed as a percentage of the yield of the early week. Their results are somewhat variable, according to the particular two weeks of the lactation compared, but in general the extreme dairy type showed the smallest decrease, the large dairy type the largest decrease, and the dual-purpose type intermediate. These results are of interest in connection with the prevalent opinion as to the excellence of the dairy breeds in persistency of lactation, as compared with the dual-purpose breeds. It should be pointed out that the Wisconsin herd was probably composed of more or less select individuals and the same results might not hold for a larger and more representative population. The breeding experiments of the Iowa Station are of interest at this point. The experiments referred to are based on the mating of dairy bred bulls with scrub cows. McCandlish et al n have studied the persistency of lactation of scrub cows, of purebred dairy cows, and of crossbred cows by dairy bulls from the scrub cows. They have measured persistency by expressing each month 's milk yield as a percentage of the first month's milk yield. As thus expressed, the scrubs decrease most 414 BULLETIN No. 288 [April, rapidly, the dairy bred cows least rapidly, and the crossbreds are inter- mediate between the parent types. These results would indicate that there is a difference between dairybred cows and unimproved cows in respect to persistency of lactation. Correction of the persistency values for initial rate of yield would likely make the difference more pronounced. Apparently also the sire and dam influence the persistency of the off- spring about equally. Measures of Persistency? The measures of persistency mentioned above, it will be noted, are all based on a ratio of one sort or another. To total lactation yield these should be added the measure of Sanders/ 6 ; - ,' maximum day s yield The length of the lactation period, and consequently the lactation yield, are greatly affected by the length of the service period. Sanders has therefore applied a correction factor to the ratios to reduce them to a standard service period. If yield for a definite time period were sub- stituted for lactation yield in the ratio of Sanders, it is apparent the ratio would acquire a definite relation to the rate of decrease in yield. Turner 20 has presented a table for converting such ratios into the cor- responding percentage decrease per month. It would seem preferable, in the use of such a system, to substitute a smoothed value of an initial period yield in place of the maximum yield. It is evident that the system applied literally would not distinguish between the performance of those cows which show an increasing rate of yield and those which show an approximately equal decreasing rate of yield. Turner, in the paper mentioned, proposes also the method of dividing the yield for each calendar month by that of the preceding calen- dar month and using the arithmetic average of the ratios thus secured as a measure of persistency. It should be noted that such a method is mathematically unsound for the purpose in view, and where applied to the irregular data of individual records, tends to lead to too high results. Correction Factors for Length of Record. A set of correction factors for length of record from 200 to 365 days was presented in Bulletin 272, 6 based on an average persistency value for Guernsey records of k X 10 3 = 44.12. It was pointed out that the factors would vary according to the values of k. The k frequency distribution (Fig. 14 and Table 6) gives an indication of the variability of k. It is apparent from the relation between k and yield (r = .226) that the correction factors given are based on too high a value of k for low yields and too low a value of k for high yields. There is no simple way of taking account of "Since this material was prepared, a somewhat more extensive treatment has been given by the author in a paper submitted to the Journal of Agricultural Re- search entitled, "Measures of Persistency of Lactation." 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 415 this known relation, but as the factors stand the error to which they are subject is not very serious so far as application to the Guernsey Ad- vanced Registry records is concerned. It appears further that they should apply almost equally well to Holstein records, so far as may be judged by the present Holstein data. Persistency as a Heritable Character. Sanders, 16 from a study of the records of Shorthorn cows in an English milk recording society, came to the conclusion that the shape of the lactation curve, tho largely de- termined by environmental factors, is due partly to a genetic character- istic of the cow. His evidence for the genetic basis of persistency is the total relation between the standard deviation of the - - ratios of in- maximum dividual records as compared with the standard deviation of individual mean ratios. It is obvious that environment may be a large factor in determining persistency of lactation. Lactation may be terminated quickly at any stage by failure to remove the accumulated milk from the udder. It seems possible, therefore, that the character and frequency of milking may be factors in persistency. Obviously the feed supply is a very im- portant factor and probably largely determines the difference between advanced registry practice, which results in an average rate of decrease of about 5 percent per month, and commercial practice, which results in an average rate of decrease of about 10 percent per month. The data of McCandlish et al u presumably were obtained with en- vironment constant, or nearly so, that is, within the same herd. The results seem to show unequivocally that dairy-bred cows are superior to scrub cows in persistency of lactation, and that the character is inherited apparently in a blending fashion. It may be noted incidental- ly that the lactation curve of the dairy-bred cows shows a rate of decrease of about 10.6 percent per month. The results of the present paper indicate that persistency of lact- ation is nearly as definite an individual character as is initial rate of yield, judged by the fact that with the herd constant the correlation between two lactation records of the same cow is r = .370 for persistency and r = .400 for initial rate (Table 29). Neither of these coefficients is as high as similar comparisons reported by Gowen 8 for the Holstein breed in the case of yearly milk yield (r = .667) or fat percentage (r = .715). The effect of environment is not allowed for in the results of Gowen just quoted. It is very clear that there is considerable variability between the individual records of cows with respect to persistency and, admitting that this is in part due to genetic differences of the individual cows con- 416 BULLETIN No. 288 [April, cerned, it follows that either the sire or the dam, or both, must have an influence on the persistency of the daughter. In the case of the Guernsey records it seems clear that as between the 97 sires studied there was no difference in their effect on the persistency of their daughters. Either the sire has no influence on the persistency of the daughter, or the 97 sires were all genetically alike with respect to this character. The latter alternative is forced by the results of McCandlish referred to above. While the Holstein records show a statistically significant correlation be- tween half-sisters by a common sire and from different dams, the coeffic- ient itself (r = .111 .016) is too low to be of much practical importance. Obviously, if the differences in sires are not responsible for the inherent qualities of the daughters with respect to persistency, then differences in the dams must be responsible. Half-sisters from a common dam and by different sires, as well as daughter and dam, show a significant and material correlation, which is in accord with such an elimination con- clusion. The anomalous results for full sisters on the other hand raise the question as to whether there is really any genetic difference between individuals within either of the two breeds with respect to persistency of lactation. That is to say, it may be possible that there is very little variability of a genetic nature among our present dairy breeds with re- spect to persistency of lactation. The Short-Time Test. Gavin 7 was perhaps the first to show definite- ly, by statistical treatment, the correlation between milk yield for a short period at the start of lactation and milk yield for the entire lacta- tion. He found the correlation between lactation yield and yield for the 5th to 12th weeks to be r = .858 + .005, in the case of normal lactations 35 to 45 weeks in length. The corresponding relation for the maximum- days' yield was r = .839 + .006. The first correlation corresponds more or less closely in nature with the present correlation between the observed rate of yield two months after calving and yield for the year, shown in Table 17 to be r = .827 .006. Gavin's data were taken from the herd records of Lord Rayleigh in England. It is evident from the coefficients of correlation that the present Guernsey Advanced Regis- try data are very similar to these English herd records with respect to the closeness of the relation under consideration. Reference to Table 17 and Fig. 18 shows further that the correlation between rate of yield and yield for the year increases to r = .935 + .002 at six months after calving. This correlation would naturally be higher except for deviations of the observed lactation curve from the smoothed curve. The relation between the smoothed rate of yield and yearly yield is shown in Table 17 to be r = .980 .001. A question arises as to the records rejected because of unusual 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 417 irregularities. The correlation between the yield of the sixth month and the yearly yield for these 142 irregular records works out at r = .839 .017, or materially lower than that found for the bulk of the records. Inclusion of these 142 irregular records with the other 1,534 records gives a correlation of r = .928 .002. Since the rate of yield is so closely related to the yearly yield it is a question why, in the interest of economy, a short-time test should not be used. A short-time test conducted in the fifth month of lactation affords a well nigh perfect index of the cow 's 365-day energy yield under the conditions of the Guernsey Advanced Registry where pregnancy during the year period is eliminated as a factor. If the 365-day yield is the essential measure of a cow's performance, as seems to be assumed by the practice of the advanced registry system, why not use a six-months- after-calving short-time test to measure the cow 's performance? But the question should be asked, why select 365 days as a proper length of time to test a cow when that is only a part of her lactation period under the conditions that have developed in official testing? The selection of the 365-day period is undoubtedly connected with the economic significance of the year's yield under ordinary conditions of milk production, when the cows are bred to freshen once a year. The feed cost of milk production under such conditions bears a very definite relation to the yearly yield. The relation depends upon the two main physiological facts that a certain quantity of nutrients is required for body maintenance and a certain further quantity is required for the elaboration of the milk itself. For the sake of simplicity we may consider a cow of 1,100 pounds live weight. According to Haecker, 11 the maintenance of an 1100-pound cow for one year requires 3,182 pounds of digestible nutrients, and the elaboration of one pound F.C.M. (4-percent milk) requires .343 pounds of digestible nutrients (cf. Gaines 4 ). These quantitative relations lead 3182 to the equation y = .343 H , where y is the feed cost in pounds of x nutrients per pound F.C.M. and x is the yearly yield in pounds F.C.M. The curve of this equation is given in Fig. 29. Cost in nutrients may be translated directly into cost in dollars, as has been shown by Ross et al. lb Fig. 29 shows clearly how the yearly yield is related to the cost of production, and why high yearly production is so important from the standpoint of efficiency of production. It should be emphasized that the relation between yield and cost as a generalization is based on a calving interval (time from one calving to the next calving) of one year. The advanced registry record is often crowded past the point "X" of maximum efficiency of Fig. 29 and, also, usually fails to meet the calv- 418 BULLETIN No. 288 [April, ing interval condition mentioned above. While the selection of a year period as the time over which production is to be measured has a very fundamental basis in ordinary practice, the 365-day official record has 1.4 1.2 2 o ttl 1.0 <0 E 0) I .2 40 <40 60 100 F- C. M. per Year - Cwt. ISO 140 FIG. 29. THEORETICAL RELATION OF YEARLY YIELD TO EFFICIENCY OF PRODUCTION The solid curve represents the physiological limitation of efficiency of milk production by the 1100-pound cow, according to Haecker's feeding standard. The asterisk (*) on the curve shows the point at which is reached the often quoted efficiency of the cow, that is 18 percent of the feed nutrients recovered in the milk. The efficiency curve may be assumed to hold for any individual up to a certain point (X), beyond which some increase in yield is possible but at an extraordinary expense of nutrients and other costs, illustrated diagramatically by the broken curves. From the standpoint of economical milk production it is the point X on the yield curve that is of primary importance. The advanced registry record as a rule represents a point sqmewhere on the broken curve, often far past the point X. The broken curves are highly speculative. developed into a state where the significance of the record from the standpoint of efficiency of production is very uncertain. Fig. 30 presents a more or less diagramatic attempt to compare the advanced registry record with the ordinary record. The commercial 1927] PERSISTENCY OF LACTATION IN DAIRY Cows 419 dairy-man has found by experience that the best practice is to breed his cows to freshen about once a year, regardless of the value of the calf at birth. The reason for this is to be found in the great natural stimulus of milk secretion associated with the reproductive process, and which 4 6 & Months after Calving la FIG. 30. COMPARISON OP ADVANCED REGISTRY RECORD WITH ORDINARY RECORD The A.R. curve is that of the present Guernsey records. As compared with the A.R. record, the ordinary record starts at a lower initial rate and declines at a more rapid rate. Furthermore, the cow is bred to freshen about once a year and allowed a dry or rest period before freshening. Frequent reproduction is essential to economi- Pal milk production, regardless of the value of the calf. It is the area under the lower type of curve on which the comparison of Fig. 29 is based. Yield for a year obtained from the upper type of curve cannot be used in the comparison of Fig. 29. becomes apparent immediately following parturition. Practical ex- perience has shown also that it is desirable to allow one to two months dry period before calving. The importance of frequent reproduction may be demonstrated theoretically on the basis of the lactation curve. Assume* a dry period of about six weeks and a lactation yield in accordance with the general equation for a time two months less than the calving interval. Let c represent the calving interval in months, then the average yield per month for the interval is given by a It is the average yield over the calving interval that is economically important, and this may be computed in terms of a by assigning values to A; and c. Table 30 gives certain results computed by this formula. Graphic presentation is given in Fig. 31. It is apparent from Table 30 and Fig. 31 that, considering a per- sistency value of k X 10 3 = 100, which is about what may be expected "Since this paper was prepared the basis of the present assumption has been presented in more detail in a paper submitted by the author to the Journal of Dairy Science, entitled "Milk yield in relation to the recurrence of conception". 420 BULLETIN No. 288 [April, under ordinary conditions, the average yield per month or per year decreases quite rapidly as the calving interval increases. For example, consider a two-year period and an initial rate of yield of 1,000 pounds per month, cow A calving at the end of 12 months and again at 24 months, TABLE 30. AVERAGE YIELD PER MONTH FOR THE CALVING INTERVAL COMPUTED FROM THE LACTATION CURVE EQUATIONS (Explanation in text) Calving interval (months) Average yield per month expressed as a percentage of the initial rate of yield per month k X 10 3 = 25 A; X 10 3 = 50 k X 10 3 = 100 k X 10 3 = 150 10 72.51 73.27 73.73 73.98 74.05 73.83 73.26 72.47 71.54 70.51 65.94 65.89 65.58 65.08 64.46 62.93 61.19 59.34 57.47 55.59 55.07 53.95 52.68 51.32 49.91 47.09 44.34 41.74 39.30 37.05 46.59 44.89 43.16 41.43 39.75 36.56 33.68 31.09 28.79 26.75 11 12 13 14 16 18 20 22 24 70 40 i* o 50 8.2 20. 10 14 \{, 18 ZO Calving Interval - Months FIG. 31. SHOWING THE NECESSITY OF FREQUENT REPRODUCTION FOR HIGH AVERAGE YIELD It is assumed that milk is produced in accordance with the equation for a period two months less than the calving interval. The average yield, including both lactation and dry period, is greater the more frequent the breeding where k X 10 3 is not less than 50. 1927] PERSISTENCY OP LACTATION IN DAIRY Cows 421 cow B not freshening until 24 months. The two years' yield for A would be 12,643 [ = 2(.5268 X 1000 X 12)] pounds while for B it would be only 8,892 (= .3705 X 1000 X 24) pounds. Furthermore, until the age of maturity the initial rate of yield ordinarily increases with each calf, which would give a still further advantage to cow A, breeding every 12 months. The illustration serves to show the great influence of frequent freshening on the average yearly yield over a series of years. The glamour of a large year-record representing only the first 365 days of a much longer lactation period should not obscure the economic issue. In the economically practical type of lactation curve (ordinary record of Fig. 30), we have a lactation yield equal to, say, ten months' yield by the equation, and with a persistency value of about k X 10 3 = 100. It becomes of interest to determine the point in the lactation curve at which the ten months' yield is unaffected by variability in per- sistency. The solution is given by the value of i' in the equation t' = I I0e~ l0k -. For the value of k X 10 3 = 100, t' = 4.18. That is, k 1 e- lok under the conditions of commercial practice a short-time test should afford the best measure of the cow's production if conducted at 4.2 months after calving. The chief objection put forward to the short-time test is that it T does not measure persistency of lactation and consequently does not accurately indicate yearly production. As a matter of fact, a short-time test at the beginning of lactation is a better measure of persistency than is the yield for a year. This statement is warranted by the fact that the correlation between theoretical initial rate of yield and persistency (rate of decrease) for the present Guernsey records is r = .535, whereas the correlation between yield for the year and persistency is only r = -.226. Yield for the year of course embodies the result of persistency, but any accurate measure of this character must be based directly on the lactation curve. It is clear that if the short-time test is conducted at the stage of lactation where the yield for the portion of lactation under consider- ation is independent of variability in persistency, it becomes theoretically a perfect index of the yield for the longer period. For the purpose of representing the yearly production of the cow under ordinary condi- tions, bearing a calf yearly, it would seem that a short-time test con- ducted during the fourth month of lactation should afford a highly valuable measure of the cow's production from a practical standpoint. The yield for such a short-time test, like the yield for the year* would embody the result of persistency but would give no accurate measure of persistency. 422 BULLETIN No. 288 [April, As compared with initial rate of yield, or rate of yield at later stages of lactation, persistency is a much less important factor in determining the ordinary lactation yield. Differences in persistency are evidently very much subject to differences in factors of an environmental nature, and very little subject to differences in factors of a genetic nature, so far as indicated by the present records. From the inheritance stand- point, however, the present data are not extensive enough to establish satisfactorily the true condition of affairs, and judgment should be reserved pending investigation of a larger number of records. On the other hand, as compared with persistency, initial rate of yield is a much more powerful factor determining the ordinary lactation yield; it is less subject to environmental factors; and it seems to promise considerable opportunity of improvement by selective breeding. Cows with inherent capacity for high yearly yield are clearly necessary to the most efficient production of milk. High yield for the year is to be at- tained only thru high initial rate of yield. Note, in Table 12, how dis- tinctly the correlation surface of initial rate of yield and yield for the year is cut off at the upper right border. It is a question for serious consideration whether we may not progress as well or better in breeding and selecting high-yielding cows on the basis of a short-time yield soon after calving, as we may on the basis of a short-time test later in the lactation, or on the basis of the lactation yield itself. Given high initial rate of yield, and regular, fre- quent reproduction, persistency of lactation seems to be of minor im- portance in the problem of breeding and selecting efficient cows. SUMMARY AND CONCLUSIONS dy A curve of the type -j- = Ae~ kt has been fitted to each of 1,534 Guernsey records and 1,395 Holstein records. In the equation y is yield, dii -5- is the rate of yield, A is the theoretical initial rate of yield, t is time from calving, and k is the rate of decrease in the rate of yield. The value of k is used as a measure of persistency of lactation. The value of A is representative of the rate of yield shortly after calving. The A and k constants of the individual curves have been studied statistically, and in case of the Guernsey records, the yield for the year and fat percentage for the year, also. Yield has been measured on an energy basis in terms of 4-percent milk. It was found that A and k are quite closely related : for the Guernsey records r = .535; for the Holstein records r = .433. In both breeds A 1937] PERSISTENCY OF LACTATION IN DAIRY Cows 423 increases with age to about nine years and then declines while fc is independent of age with A constant. Yearly yield is more closely related to A than to k: r = .672 and r = -.226 respectively. The correlation between yearly yield and rate of yield increases with advance in lactation up to six months, after which it decreases. At six months r = .980 for the smoothed lactation curve and r = .935 for the raw lactation curve. From the mathematical properties of the lactation curve, it seems that for the ordinary lactation of ten months and k = .1, the best time of conducting a short-time test to estimate the lactation yield is 4.2 months after calving. The corrected persistency values show a high degree of variability, the coefficient of variability for the Guernsey and Holstein records being 60.97 and 46.25 respectively; and the corresponding coefficients for yearly yield as affected by persistency, 14.53 and 13.05. The mean initial rate corrected to age of maximum is, for the Guernsey records, 47.2; and for the Holstein records, 81.6 pounds 4-percent milk per day. Corrected to the same initial rate, the Holstein records show greater persistency (lower value of k) than the Guernsey records. Environment has a great effect on persistency. The evidence as to the influence of heredity on persistency with environment constant is inconclusive and conflicting : half-sisters by Guernsey sires show no correlation r = .001 + .029; half-sisters by Holstein sires, a feeble correlation, r = .111 .016. Half-sisters by a common dam show a material correlation, r = .418 .080 (Guernsey) and r = .390 .065 (Holstein). Dam-and-daughter relationships show r = .259 .086 (Guernsey) and r = .370 .059 (Holstein). Full sisters in both breeds show a non-significant negative correlation, r = .178 .106 (Guern- sey) and r = .090 + .084 (Holstein). Successive lactations of the same cow (Holstein) show a correlation of r = .370 + .059. All the correlations of this paragraph are for persistency corrected for initial rate of yield, and the correlations are corrected to eliminate the effect of common environment. In view of the uncertainty of effecting further genetic improvement in our dairy breeds with respect to persistency of lactation, the low cor- relation between persistency and yearly yield, and the high correlation between rate of yield and yearly yield, it must be granted that a properly conducted short-time test may serve as an excellent production record, from both economical and biological standpoints. 424 BULLETIN No. 288 1. BRODY, SAMUEL; RAGSDALB, ARTHUR C.; and TURNER, CHARLES W. The rate of decline of milk secretion with the advance of the period of lactation. Jour. Gen. Physiol. 5, 441-444. 1923. 2. CARLYLE, W. L., and WOLL, F. W. Studies in milk production. Wis. Agr. Exp. Sta. Bui. 102. 1903. 3. ELLINGER, TAGE. The variation and inheritance of milk characters. Proc. Nat, Acad. Sci, 9, 111-116. 1923. 4. GAINES, W. L. Feed cost of milk production as affected by the percentage fat content of the milk. Jour. Agr. Res. 29, 593-601. 1924. 5. GAINES, W. L., and DAVIDSON, F. A. Relation between percentage fat con- tent and yield of milk. 111. Agr. Exp. Sta. Bui. 245. 1923. 6. GAINES, W. L., and DAVIDSON, F. A. Rate of milk secretion as affected by advance in lactation and gestation. 111. Agr. Exp. Sta. Bui. 272. 1926. 7. GAVIN, WILLIAM. The interpretation of milk records. Jour. Roy. Agr. Soc. 73, 153-174. 1912. 8. GOWEN, JOHN W. Milk secretion. Williams and Wilkins. 1924. 9. GOWEN, JOHN W. Studies in milk secretion X. Relation between the milk yield of one lactation and the milk yield of a subsequent lactation in Guernsey advanced registry cattle. Jour. Dairy. Sci. 6, 102-121. 1923. 10. GRAVES, R. R., and FOHRMAN, M. H. Effect of age and development on butter- fat production of register-of-merit Jersey and advanced register Guernsey cattle. U. S. Dept. of Agr. Bui. 1352. 1925. 11. HAECKER, T. L. Investigations in milk production. Minn. Agr. Exp. Sta. Bui. 140. 1914. 12. HARRIS, J. A. On the calculation of intra-class and inter-class coefficients of correlation from class moments when the number of possible combinations is large. Biometrika 9, 446-472. 1913. 13. MCCANDLISH, A. C., GILLETTE, L. S., and KILDEE, H. H. Influence of en- vironment and breeding in increasing dairy production. II. Iowa Agr. Exp. Sta. Bui. 188. 1919. 14. MINER, JOHN RICE. Fitting logarithmic curves by the method of moments. Jour. Agr. Res. 3, 411-423. 1915. 15. Ross, H. A., HALL, H. F., and RHODE, C. S. The feed cost of milk and fat pro- duction as related to yields. 111. Agr. Exp. Sta. Bui. 244. 1923. 16. SANDERS, H. G. The shape of the lactation curve. Jour. Agr. Sci. 13, 169-179. 1923. 17. SMITH, J. W. Agricultural meterology. Macmillan. 1920. 18. STURTEVANT, E. L. Influence of distance from calving on milk yield. N. Y. (Geneva) Agr. Exp. Sta. 21-23. 1886. 19. TURNER, C. W. Factors affecting the percentage of fat in cow's milk. Mo. Agr. Exp. Sta. Bui. 222. 1924. 20. TURNER, C. W. A quantitative form of expressing persistency of milk or fat secretion. Jour. Dairy Sci. 9, 203-214. 1926. 21. WRIGHT, SEWALL. Correlation and causation. Jour. Agr. Res. 20. 557-585. 1921. UNIVERSITY OF ILLINOIS-URBANA