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Effects of Mineral Supplements on the Length of the Tail and Wing Feathers in White Leghorns*
Effects of Mineral Supplements on the Length of the Tail and Wing Feathers in White Leghorns* W A L T E R A.
HENDRICKS
Beltsville Research Center, U. S. Department of Agriculture, Beltsville, Maryland (Received for Publication August 6, 1934)
T
HIS paper presents an analysis of a set of feather measurements obtained at the Beltsville Research Center during the course of an experiment designed to determine the effects of certain mineral supplements, particularly inorganic sulphur compounds, on the molt of White Leghorn hens. Since the molt is usually accompanied by a cessation of egg production, the importance of favorably influencing the growth of feathers, if it is possible to do so, is obvious. In view of the fact that feathers contain considerable amounts of sulphur, it was thought that the feeding of sulphur compounds might have some effect on the rate of growth and mature length of the new tail and wing feathers which were grown after the old feathers were dropped during the molt. The general plan of this experiment, in* This paper is the fifth prepared by the staff of the Poultry Nutrition Laboratory for the purpose of illustrating the application of the statistical methods, introduced by R. A. Fisher, to practical problems arising in experimental work in poultry nutrition. The sixth will appear in the next issue, and the seventh and last in the November issue. The entire series will be available in separates at a nominal cost.
eluding a description of the sulphur compounds fed, has been given in an earlier paper (Hendricks, 1933). Three groups of birds were used, each group representing a separate year's work. There were four lots of birds in the first group, three in the second, and four in the third. There were from fifteen to twenty-three birds in each lot at the beginning of each year's work. Six different mineral supplements were fed to these eleven lots of birds. The extent to which the feeding of each supplement was replicated from year to year is apparent from an examination of Table 1. The present paper, which is intended primarily for the purpose of illustrating a statistical procedure, is concerned only with a study of the data relating to feather length. The first three feathers, starting with the lowest, on the right half of the tail and the first three primary feathers, starting with the one next to the axial feather, on the left wing were measured on each bird in every lot once every week from the time the new feathers appeared until they stopped growing. The lengths were expressed in centimeters and the measurements were made to the nearest millimeter.
TABLE 1.—Birds receiving each mineral supplement
Group No.
Supplement A
Supplement B
Supplement C
Supplement D
1 2 3
Lotl
Lot 2 Lot 2a Lot 2b
Lot 3 Lot 3a Lot 3b
Lot 4 Lot 4a
[221]
Supplement E
Supplement F
Lot 5b
Lot 6b
222
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The resulting set of measurements was treated by analysis of variance. As a first step in analyzing the data, it was necessary to take stock of all known factors which may have contributed to the variability of the measurements and to determine for which of these factors the corresponding contributions to the variability could be estimated from the data. After this informa-
SCIENCE
total number of feathers measured. The correction is obviously nothing more than the quantity which is usually subtracted from the sum of the squares of a series of measurements in order to obtain the sum of the squares of their deviations from their arithmetic mean. After the total sum of squares was obtained, the magnitude of the contribution
'BETWEEN SUPPLEMENTS
TOTAL 1104
BETWEEN LOTS W I T H SAME SUPPLEMENT
S W I T H I N SUPPLEMENTS 1099
BETWEEN TAIL AND WING IN SAME LOT II WITHIN LOTS WITHIN T A I L s f B E T W E E N 1094 535 1 (.WITHIN
TYPES Or TAIL FEATHERS IN EACH LOT 2 2
TYPES OF TAIL FEATHERS IN EACH LOT 513 WITHIN WINGSj B E TWEEN TYPES OF WING FEATHERS IN EACH LOT 548 j , WITHIN TYPES OF WING FEATHERS IN EACH LOT
526
FIG. 1.—Sources of variability in the feather measurements and the number of degrees of freedom contributed by each.
tion was assembled, a plan of analysis was formulated. The author has found an outline, such as that presented in Figure 1, very helpful in keeping the general organization of the study clearly in mind. The numbers in the outline give the numbers of degrees of freedom contributed by each source of variability considered. For the study herein reported, suitable data were available for 1,105 feathers. Therefore, the total number of degrees of freedom was 1,104. The "total sum of squares," which, in the terminology of Fisher (1932), is a convenient manner of designating the sum of the squares of the deviations of all the measurements from their arithmetic mean, was calculated by squaring all of the measurements and subtracting from the sum of these squares a correction, obtained by dividing the square of the sum of all the measurements by the
of each of the known sources of variability to the total sum of squares was considered. Since the present study was made principally to determine the effects of the mineral supplements, the contribution of this particular source of variability was the first to be investigated. However, it should be borne in mind that the scheme of analysis adopted in the present study was only one of a number of possible methods of approaching the problem. Six different mineral supplements were fed. Therefore, this source of variability contributed 5 degrees of freedom. The magnitude of the contribution of this source of variability to the total sum of squares may be designated as the "sum of squares between supplements." It may be calculated by obtaining the value of the sum: ki(xi-x)2-\-kz(x2-x)2-\ fe,(i,-S)', in which ki, k2, k3, ks, ks, and ka denote the number of feathers, and Xi, x2, xs, xt, xs, and xe the average lengths of the feathers, measured on birds receiving supplements, A, B, C, D, E, and F respectively,
4
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x represents the average length of all the feathers measured. In actual practice, the value of the above sum was obtained by a method less laborious than carrying out the arithmetical processes indicated by the formula. The sum of the measurements in each of the 6 classes was first obtained. The 6 sums were squared and each square was divided by the' number of measurements, ki, in the corresponding class. The sum of squares between supplements was then obtained by subtracting from the sum of these quotients the same correction used in calculating the total sum of squares.
The sum of squares within supplements B and C and within supplements A, D, E, and F were calculated separately. Each of these two values was divided by the appropriate number of degrees of freedom to obtain the mean squares, or variances. These two estimates of variance were not significantly different. As a matter of fact, the latter was slightly greater than the former. Therefore, differences, if any, among the groups of birds had no significant effect on the lengths of the feathers, and the sum of squares between supplements previously obtained actually serves as a measure of the effect of differences in the mineral supplements.
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1935.
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The variability "within supplements" was investigated next. In general, the number of degrees of freedom contributed by the measurements within any class is one less than the number of measurements in the class. Therefore, in the present example, the number of degrees of freedom for the 6 classes was 6 less than the total number of measurements or 1,099. The sum of squares within supplements was obtained by calculating the sum of the squares of the deviations of the measurements from the arithmetic means for the classes in which they belonged. Both in the case of the degrees of freedom and the sum of squares, the sum of the values found for "between supplements" and "within supplements" must be equal to the value found for the "total." The validity of the method of analysis up to this point requires some justification. The 6 classes were treated as though the measurements in the respective classes had been obtained under the same essential conditions with but one exception, viz., the differences in the mineral supplements. An examination of Table 1 shows that this may not have been the case. Data were available for all 3 groups of birds only in the case of supplements B and C. If the lengths of the feathers depended, to any appreciable extent, upon the particular group of birds on which they were measured, the value found for the sum of squares between supplements could not be accepted as representing the effects of differences in the mineral supplements. It is fairly obvious that the mean for any one class would then depend, to some extent, upon the number of groups of birds represented in that class as well as upon the nature of the mineral supplement. There is a very simple test for determining whether the lengths of the feathers were influenced by differences among the groups of birds. If such differences were present, the variability of the measurements would be greater within those classes in which all 3 groups were represented than in the other classes.
The sum of squares within supplements may be separated into a component due to differences among the lots of birds receiving the same supplement and a component due to variability within the lots. There were eleven lots of birds, yielding 10 degrees of freedom "between lots." Of these 10 degrees of freedom, 5 were due to the mineral supplements. Therefore, the.number of degrees of freedom due to lots with the same supplement was (10 — 5) or 5. These 5 degrees of freedom may be identified specifically by an examination of Table 1. The number of degrees of freedom between lots receiving supplements A, B, C, D, E, and F are seen to be, respectively, 0, 2, 2, 1, 0, and 0, yielding a total of 5. The sum of squares between lots with the same supplement was calculated by treating the data for each of the supplements as a separate problem and adding the results. For any one supplement, the sum of the measurements was obtained for each lot of birds receiving that supplement. These sums were squared and each square divided by the number of measurements involved in the corresponding sum. The sum of squares between lots receiving that supplement was then calculated by subtracting from the sum of these quotients a correction, obtained by dividing the square of the sum of the measurements for the supplement by the number of measurements involved in that sum. After obtaining the sum of squares between lots for each of the six supplements, the six values were added to obtain the sum of squares between lots with the same supplement. From the above discussion, it is quite apparent that those supplements which contributed no degrees of freedom made no contribution to the sum of squares. The number of degrees of freedom within lots and the sum of squares within lots were obtained in a manner perfectly analogous to the corresponding calculations described under the discussion of the variability within supplements. Since there were
224 eleven lots of birds, the number of degrees of freedom within lots was (1,105 — 11) or 1,094. The calculation of the sum of squares requires some discussion. It will be recalled that for the study herein reported suitable data were available for 1,105 feathers. An examination of the data revealed that these feathers were measured on a total of 189 birds. Since 6 feathers were measured on every bird, measurements might have been available for 1,134 feathers. However, data for twenty-nine feathers were not available for various reasons. Some feathers were broken or lost before they stopped growing. In a few instances all three tail feathers or all three wing feathers of a bird were lost because of the bird going into a partial molt after the regular molt had been completed. Since each of the six feathers measured on a bird was of a distinct type, there was a possibility of an unequal distribution of these six types of feathers in some lots, which might invalidate the calculation of the sum of squares between lots by distorting the values of the means for some lots. For this reason, the sum of squares within lots was calculated separately for those lots in which no measurements were missing and for those lots in which some measurements were missing. The corresponding variances were calculated by dividing each sum of squares by the appropriate number of degrees of freedom. These two estimates of variance were not significantly different. Therefore the method of analysis up to this point was considered to be quite satisfactory. The degrees of freedom and sum of squares within the various lots were then added to obtain the degrees of freedom and sum of squares "within lots." The sum of squares within lots may be separated into a component due to differences between the tail and wing in the same lot of birds and a component due to variability between tail feathers and variability between wing feathers in the same lot of birds. The latter sources of variability were designated as "within tail" and "within wing" in Figure 1 to be consistent with conventional terminology. They were listed separately because it was apparent from an inspection of the data that the variability between the tail feathers was considerably greater than the variability between the wing feathers. For any lot of birds, there was 1 degree of freedom between tail and wing. Since there were eleven lots of birds, the number of degrees of freedom between tail and wing for all lots was (11 X 1) or 11. The sum of squares between tail and wing for any lot of birds was calculated by treating the data for that lot as a separate problem. The respective
POULTRY
SCIENCE
sums of the measurements of the tail and wing feathers were obtained and squared. Each of the two squares was divided by the number of measurements involved in the corresponding sum. The sum of squares between tail and wing for the given lot of birds was then obtained by subtracting from the sum of the two quotients a correction, obtained by dividing the square of the sum of all the measurements for the lot by the number of measurements involved in that sum. The sum of squares "between tail and wing in the same lot" was obtained by calculating the sum of squares for each of the 11 lots and adding the results. The investigation of the variability within the tail and within the wing does not require a great deal of discussion. Of the 1,105 measurements used in the present study, 546 were of tail feathers and 559 were of wing feathers. Since there were eleven lots of birds, the number of degrees of freedom within the tail was (546 — 11) or 535 and the number of degrees of freedom within the wing was (559 — 11) or 548. The sum of squares within the tail and the sum of squares within the wing were calculated by methods identical with those used in the calculation of the sum of squares within supplements and within lots. As previously stated, the variability within the tail was greater than that within the wing. By applying the methods discussed earlier in this paper, it was easy to show that this difference in variability was not a spurious effect due to unequal distributions of feather types. Therefore, it seemed more desirable to permit the sum of squares for each of these two general classes of feathers to retain its identity than to pool the values into a single number expressing the sum of squares within tail and wing. For any lot of birds, the sum of squares within the tail may be separated into a component due to differences among the three types of tail feathers and a component due to variability among feathers of the same type. The former component may be designated as the sum of squares between types of tail feathers, and the latter as the sum of squares within types of tail feathers. The measurements of tail feathers for each lot of birds were treated separately and the results obtained for the 11 lots were added. There were 2 degrees of freedom between types of tail feathers for each lot of birds. Therefore, the number of degrees of freedom contributed by the 11 lots was (11 X 2) or 22. The calculation of the corresponding sum of squares requires no discussion. The method is obvious from a consideration of similar calculations described earlier in this paper.
JULY,
1935.
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The method of calculation of the degrees of freedom and sum of squares within types of tail feathers should also be quite clear from the illustrations of similar calculations already given. Since there were 546 measurements of tail feathers, eleven lots of birds, and three types of tail feathers in each lot of birds, the number of degrees of freedom within types of tail feathers was (546 —• 33) or 513. The corresponding sum of squares was calculated by methods identical with those described earlier in this paper. A similar analysis was made of the variability within the wing. The number of degrees of freedom between types of wing feathers was the same as the number of degrees of freedom between types of tail feathers, viz., 22. The number of degrees of freedom within types of wing feathers was (559 — 33) or 526. The calculation of the corresponding sums of squares requires no discussion. The results of the entire analysis up to this point are summarized in Table 2. By referring to the well-known tables of values of "z" given by Fisher (1932), it is apparent that the only sources of variability whose effects were significantly greater than the variability within types of tail feathers, or within types of wing feathers, in the same lot of birds were the differences between tail and wing, between types of tail feathers, and between types of wing feathers. The variance between types of tail feathers in the same lot was not significantly greater than the variance between types of wing feathers in the same lot. The variance within types of tail feathers in the same lot was significantly greater than that within types of wing feathers.
A more detailed analysis could be made if one wished to do so, but its value would be open to question. The sum of squares between tail and wing in the same lot, for example, could be separated into a component due to differences between the tail and wing
225
which remained constant from lot to lot and a component due to differential responses in the various lots. The former component contributed one degree of freedom and the latter contributed (11 — 1) or 10 degrees of freedom. The sum of squares due to the former component could be calculated by obtaining the sum of all the measurements of tail feathers and the sum of all the measurements of wing feathers, squaring each sum, dividing each of the two squares by the number of measurements involved in the corresponding sum, adding the two quotients, and subtracting from the sum of the quotients a correction, obtained by dividing the square of the sum of all the measurements by the total number of measurements. The correction used in this instance was the same as that used in calculating the total sum of squares. The sum of squares due to differential response would have to be obtained by difference, i.e., by subtracting the sum of squares between tail and wing just described from the sum of squares between tail and wing in the same lot given in Table 2. Several similar possibilities will doubtless suggest themselves to the minds of the readers of this paper without any further discussion on the part of the author. One additional aspect of the problem may be considered to advantage, viz., the question of how much of the variability within the types of tail and wing feathers for the same lot of birds was due to the individuality of
TABLE 2.—Analysis of variance of feather measurements , using all available measurements Variability
Between types of tail feathers in same lot Within types of wing feathers in same lot Total
Sum of squares
Mean square
5 5 11 22 22 513 526
2.4586 2.3320 925.8324 577.6303 327.5285 293.7468 158.8983
0.4917 0.4664 84.1666 26.2559 14.8877 0.5726 0.3021
1,104
2,288.4269
d/f
l
A IogeM. Squ.
-0.35494 -0.38135 +2.21640 + 1.63395 + 1.35027 -0.27879 -0.59850
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SCIENCE
TABLE 3.—Analysis of variance within tails, using only data for birds for which complete sets of three measurements were available d/f
Sum of squares
Mean square
^logeM. Squ.
Between birds in same lot Error
22 162 324
545.4125 238.3145 35.8900
24.7915 1.4711 0.1108
+ 1.60525 +0.19300 -1.10001
Total (within tail in same lot)
508
819.6170
Variability
the birds and how much was due to other causes, consisting for the most part of errors made in measuring the feathers. In this phase of the work it was desirable to use only data obtained for those birds for which complete sets of three measurements of tail feathers, or three measure-. ments of wing feathers, were available. Recent investigations by Yates (1933) indicate that such a selection of data might not have been necessary. However, a discussion of the methods presented by Yates is somewhat beyond the scope of this paper. The measurements of tail feathers for each lot of birds and the measurements of wing feathers for each lot of birds were treated separately, after which the results obtained for the various lots were combined in the case of each of these two general classes of feathers. For each lot of birds the necessary calculations involved the finding of new values of the degrees of freedom and sum of squares within tail feathers and within wing feathers and new values of the degrees of freedom and sum of squares between types of tail feathers and between types of wing
feathers. The degrees of freedom and sum of squares between birds in each case were found in a manner perfectly analogous to the method of calculating the degrees of freedom and sum of squares between types of feathers. The degrees of freedom and sum of squares due to error in each case were found by difference. The results of this phase of the work are summarized in Tables 3 and 4. It is quite apparent that very little of the variability within types of tail and wing feathers, as given in Table 2, was due to errors of measurement. The errors of measurement were quite negligible in comparison with the variability due to the individuality of the birds. The variability due to birds was significantly greater in the case of the tail feathers than in the case of the wing feathers. The same is true of the variability due to errors of measurement. The variability between supplements, as given in Table 2, was significantly greater than the variability due to errors of measurement, as given in Tables 3 and 4. This comparison was not strictly justifiable, for the variability between supplements, as
TABLE 4.—Analysis of variance within wings, using only data for birds for which complete sets of three measurements were available Variability
Between birds in same lot Total (within wing in same lot)
d/f
Sum of squares
Mean square
H loge M. Squ.
22 173 346
321.3491 144.2110 13.4655
14.6068 0.8336 0.0389
+ 1.34075 -0.09100 -1.62338
541
479.0256
JULY,
1935.
VOL.
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No.
given in Table 2 may have been affected slightly by some of the measurements which were discarded in estimating the error given in Tables 3 and 4. In order to be entirely consistent, the sum of squares between supplements would have to be calculated from exactly the same measurements used in calculating the sum of squares due to error. However, such a comparison is of very little practical significance, since it gives no information whatever regarding the results to be expected if the experiment were repeated with a different group of birds, unless those birds were carefully selected so that every bird used in such an experiment would be the exact counterpart of a corresponding bird used in the experiment discussed in this paper. The variability within types of tail feathers and within types of wing feathers, given in Table 2, is a more useful standard of comparison, since any conclusions based upon this variability as a measure of experimental error would be applicable to any group of birds of the
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same general breeding as those used in the experiment herein described. SUMMARY AND CONCLUSIONS Variability in the lengths of selected tail and wing feathers of White Leghorn hens, due to differences in the mineral supplements fed, was not significantly greater than the variability encountered between feathers of the same type in the same lot of birds. The variability due to differences in the mineral supplements seemed to be significantly greater than the variability due to actual errors of measurement made in measuring the feathers, but this comparison is of very little practical importance. REFERENCES
Fisher, R. A., 1932. Statistical methods for research workers, 4th edition. Edinburgh and London: Oliver and Boyd. Hendricks, W. A., 1933. A biometric study of molt in White Leghorn hens. Poul. Sci. 12:287-291. Yates, F., 1933. The principles of orthogonality and confounding in replicated experiments. Jour. Agr. Sci. 23 :108-145.
HENDRICKS
Beltsville Research Center, U. S. Department of Agriculture, Beltsville, Maryland (Received for Publication August 6, 1934)
T
HIS paper presents an analysis of a set of feather measurements obtained at the Beltsville Research Center during the course of an experiment designed to determine the effects of certain mineral supplements, particularly inorganic sulphur compounds, on the molt of White Leghorn hens. Since the molt is usually accompanied by a cessation of egg production, the importance of favorably influencing the growth of feathers, if it is possible to do so, is obvious. In view of the fact that feathers contain considerable amounts of sulphur, it was thought that the feeding of sulphur compounds might have some effect on the rate of growth and mature length of the new tail and wing feathers which were grown after the old feathers were dropped during the molt. The general plan of this experiment, in* This paper is the fifth prepared by the staff of the Poultry Nutrition Laboratory for the purpose of illustrating the application of the statistical methods, introduced by R. A. Fisher, to practical problems arising in experimental work in poultry nutrition. The sixth will appear in the next issue, and the seventh and last in the November issue. The entire series will be available in separates at a nominal cost.
eluding a description of the sulphur compounds fed, has been given in an earlier paper (Hendricks, 1933). Three groups of birds were used, each group representing a separate year's work. There were four lots of birds in the first group, three in the second, and four in the third. There were from fifteen to twenty-three birds in each lot at the beginning of each year's work. Six different mineral supplements were fed to these eleven lots of birds. The extent to which the feeding of each supplement was replicated from year to year is apparent from an examination of Table 1. The present paper, which is intended primarily for the purpose of illustrating a statistical procedure, is concerned only with a study of the data relating to feather length. The first three feathers, starting with the lowest, on the right half of the tail and the first three primary feathers, starting with the one next to the axial feather, on the left wing were measured on each bird in every lot once every week from the time the new feathers appeared until they stopped growing. The lengths were expressed in centimeters and the measurements were made to the nearest millimeter.
TABLE 1.—Birds receiving each mineral supplement
Group No.
Supplement A
Supplement B
Supplement C
Supplement D
1 2 3
Lotl
Lot 2 Lot 2a Lot 2b
Lot 3 Lot 3a Lot 3b
Lot 4 Lot 4a
[221]
Supplement E
Supplement F
Lot 5b
Lot 6b
222
POULTRY
The resulting set of measurements was treated by analysis of variance. As a first step in analyzing the data, it was necessary to take stock of all known factors which may have contributed to the variability of the measurements and to determine for which of these factors the corresponding contributions to the variability could be estimated from the data. After this informa-
SCIENCE
total number of feathers measured. The correction is obviously nothing more than the quantity which is usually subtracted from the sum of the squares of a series of measurements in order to obtain the sum of the squares of their deviations from their arithmetic mean. After the total sum of squares was obtained, the magnitude of the contribution
'BETWEEN SUPPLEMENTS
TOTAL 1104
BETWEEN LOTS W I T H SAME SUPPLEMENT
S W I T H I N SUPPLEMENTS 1099
BETWEEN TAIL AND WING IN SAME LOT II WITHIN LOTS WITHIN T A I L s f B E T W E E N 1094 535 1 (.WITHIN
TYPES Or TAIL FEATHERS IN EACH LOT 2 2
TYPES OF TAIL FEATHERS IN EACH LOT 513 WITHIN WINGSj B E TWEEN TYPES OF WING FEATHERS IN EACH LOT 548 j , WITHIN TYPES OF WING FEATHERS IN EACH LOT
526
FIG. 1.—Sources of variability in the feather measurements and the number of degrees of freedom contributed by each.
tion was assembled, a plan of analysis was formulated. The author has found an outline, such as that presented in Figure 1, very helpful in keeping the general organization of the study clearly in mind. The numbers in the outline give the numbers of degrees of freedom contributed by each source of variability considered. For the study herein reported, suitable data were available for 1,105 feathers. Therefore, the total number of degrees of freedom was 1,104. The "total sum of squares," which, in the terminology of Fisher (1932), is a convenient manner of designating the sum of the squares of the deviations of all the measurements from their arithmetic mean, was calculated by squaring all of the measurements and subtracting from the sum of these squares a correction, obtained by dividing the square of the sum of all the measurements by the
of each of the known sources of variability to the total sum of squares was considered. Since the present study was made principally to determine the effects of the mineral supplements, the contribution of this particular source of variability was the first to be investigated. However, it should be borne in mind that the scheme of analysis adopted in the present study was only one of a number of possible methods of approaching the problem. Six different mineral supplements were fed. Therefore, this source of variability contributed 5 degrees of freedom. The magnitude of the contribution of this source of variability to the total sum of squares may be designated as the "sum of squares between supplements." It may be calculated by obtaining the value of the sum: ki(xi-x)2-\-kz(x2-x)2-\ fe,(i,-S)', in which ki, k2, k3, ks, ks, and ka denote the number of feathers, and Xi, x2, xs, xt, xs, and xe the average lengths of the feathers, measured on birds receiving supplements, A, B, C, D, E, and F respectively,
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x represents the average length of all the feathers measured. In actual practice, the value of the above sum was obtained by a method less laborious than carrying out the arithmetical processes indicated by the formula. The sum of the measurements in each of the 6 classes was first obtained. The 6 sums were squared and each square was divided by the' number of measurements, ki, in the corresponding class. The sum of squares between supplements was then obtained by subtracting from the sum of these quotients the same correction used in calculating the total sum of squares.
The sum of squares within supplements B and C and within supplements A, D, E, and F were calculated separately. Each of these two values was divided by the appropriate number of degrees of freedom to obtain the mean squares, or variances. These two estimates of variance were not significantly different. As a matter of fact, the latter was slightly greater than the former. Therefore, differences, if any, among the groups of birds had no significant effect on the lengths of the feathers, and the sum of squares between supplements previously obtained actually serves as a measure of the effect of differences in the mineral supplements.
JULY,
1935.
VOL.
XIV,
No.
The variability "within supplements" was investigated next. In general, the number of degrees of freedom contributed by the measurements within any class is one less than the number of measurements in the class. Therefore, in the present example, the number of degrees of freedom for the 6 classes was 6 less than the total number of measurements or 1,099. The sum of squares within supplements was obtained by calculating the sum of the squares of the deviations of the measurements from the arithmetic means for the classes in which they belonged. Both in the case of the degrees of freedom and the sum of squares, the sum of the values found for "between supplements" and "within supplements" must be equal to the value found for the "total." The validity of the method of analysis up to this point requires some justification. The 6 classes were treated as though the measurements in the respective classes had been obtained under the same essential conditions with but one exception, viz., the differences in the mineral supplements. An examination of Table 1 shows that this may not have been the case. Data were available for all 3 groups of birds only in the case of supplements B and C. If the lengths of the feathers depended, to any appreciable extent, upon the particular group of birds on which they were measured, the value found for the sum of squares between supplements could not be accepted as representing the effects of differences in the mineral supplements. It is fairly obvious that the mean for any one class would then depend, to some extent, upon the number of groups of birds represented in that class as well as upon the nature of the mineral supplement. There is a very simple test for determining whether the lengths of the feathers were influenced by differences among the groups of birds. If such differences were present, the variability of the measurements would be greater within those classes in which all 3 groups were represented than in the other classes.
The sum of squares within supplements may be separated into a component due to differences among the lots of birds receiving the same supplement and a component due to variability within the lots. There were eleven lots of birds, yielding 10 degrees of freedom "between lots." Of these 10 degrees of freedom, 5 were due to the mineral supplements. Therefore, the.number of degrees of freedom due to lots with the same supplement was (10 — 5) or 5. These 5 degrees of freedom may be identified specifically by an examination of Table 1. The number of degrees of freedom between lots receiving supplements A, B, C, D, E, and F are seen to be, respectively, 0, 2, 2, 1, 0, and 0, yielding a total of 5. The sum of squares between lots with the same supplement was calculated by treating the data for each of the supplements as a separate problem and adding the results. For any one supplement, the sum of the measurements was obtained for each lot of birds receiving that supplement. These sums were squared and each square divided by the number of measurements involved in the corresponding sum. The sum of squares between lots receiving that supplement was then calculated by subtracting from the sum of these quotients a correction, obtained by dividing the square of the sum of the measurements for the supplement by the number of measurements involved in that sum. After obtaining the sum of squares between lots for each of the six supplements, the six values were added to obtain the sum of squares between lots with the same supplement. From the above discussion, it is quite apparent that those supplements which contributed no degrees of freedom made no contribution to the sum of squares. The number of degrees of freedom within lots and the sum of squares within lots were obtained in a manner perfectly analogous to the corresponding calculations described under the discussion of the variability within supplements. Since there were
224 eleven lots of birds, the number of degrees of freedom within lots was (1,105 — 11) or 1,094. The calculation of the sum of squares requires some discussion. It will be recalled that for the study herein reported suitable data were available for 1,105 feathers. An examination of the data revealed that these feathers were measured on a total of 189 birds. Since 6 feathers were measured on every bird, measurements might have been available for 1,134 feathers. However, data for twenty-nine feathers were not available for various reasons. Some feathers were broken or lost before they stopped growing. In a few instances all three tail feathers or all three wing feathers of a bird were lost because of the bird going into a partial molt after the regular molt had been completed. Since each of the six feathers measured on a bird was of a distinct type, there was a possibility of an unequal distribution of these six types of feathers in some lots, which might invalidate the calculation of the sum of squares between lots by distorting the values of the means for some lots. For this reason, the sum of squares within lots was calculated separately for those lots in which no measurements were missing and for those lots in which some measurements were missing. The corresponding variances were calculated by dividing each sum of squares by the appropriate number of degrees of freedom. These two estimates of variance were not significantly different. Therefore the method of analysis up to this point was considered to be quite satisfactory. The degrees of freedom and sum of squares within the various lots were then added to obtain the degrees of freedom and sum of squares "within lots." The sum of squares within lots may be separated into a component due to differences between the tail and wing in the same lot of birds and a component due to variability between tail feathers and variability between wing feathers in the same lot of birds. The latter sources of variability were designated as "within tail" and "within wing" in Figure 1 to be consistent with conventional terminology. They were listed separately because it was apparent from an inspection of the data that the variability between the tail feathers was considerably greater than the variability between the wing feathers. For any lot of birds, there was 1 degree of freedom between tail and wing. Since there were eleven lots of birds, the number of degrees of freedom between tail and wing for all lots was (11 X 1) or 11. The sum of squares between tail and wing for any lot of birds was calculated by treating the data for that lot as a separate problem. The respective
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sums of the measurements of the tail and wing feathers were obtained and squared. Each of the two squares was divided by the number of measurements involved in the corresponding sum. The sum of squares between tail and wing for the given lot of birds was then obtained by subtracting from the sum of the two quotients a correction, obtained by dividing the square of the sum of all the measurements for the lot by the number of measurements involved in that sum. The sum of squares "between tail and wing in the same lot" was obtained by calculating the sum of squares for each of the 11 lots and adding the results. The investigation of the variability within the tail and within the wing does not require a great deal of discussion. Of the 1,105 measurements used in the present study, 546 were of tail feathers and 559 were of wing feathers. Since there were eleven lots of birds, the number of degrees of freedom within the tail was (546 — 11) or 535 and the number of degrees of freedom within the wing was (559 — 11) or 548. The sum of squares within the tail and the sum of squares within the wing were calculated by methods identical with those used in the calculation of the sum of squares within supplements and within lots. As previously stated, the variability within the tail was greater than that within the wing. By applying the methods discussed earlier in this paper, it was easy to show that this difference in variability was not a spurious effect due to unequal distributions of feather types. Therefore, it seemed more desirable to permit the sum of squares for each of these two general classes of feathers to retain its identity than to pool the values into a single number expressing the sum of squares within tail and wing. For any lot of birds, the sum of squares within the tail may be separated into a component due to differences among the three types of tail feathers and a component due to variability among feathers of the same type. The former component may be designated as the sum of squares between types of tail feathers, and the latter as the sum of squares within types of tail feathers. The measurements of tail feathers for each lot of birds were treated separately and the results obtained for the 11 lots were added. There were 2 degrees of freedom between types of tail feathers for each lot of birds. Therefore, the number of degrees of freedom contributed by the 11 lots was (11 X 2) or 22. The calculation of the corresponding sum of squares requires no discussion. The method is obvious from a consideration of similar calculations described earlier in this paper.
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The method of calculation of the degrees of freedom and sum of squares within types of tail feathers should also be quite clear from the illustrations of similar calculations already given. Since there were 546 measurements of tail feathers, eleven lots of birds, and three types of tail feathers in each lot of birds, the number of degrees of freedom within types of tail feathers was (546 —• 33) or 513. The corresponding sum of squares was calculated by methods identical with those described earlier in this paper. A similar analysis was made of the variability within the wing. The number of degrees of freedom between types of wing feathers was the same as the number of degrees of freedom between types of tail feathers, viz., 22. The number of degrees of freedom within types of wing feathers was (559 — 33) or 526. The calculation of the corresponding sums of squares requires no discussion. The results of the entire analysis up to this point are summarized in Table 2. By referring to the well-known tables of values of "z" given by Fisher (1932), it is apparent that the only sources of variability whose effects were significantly greater than the variability within types of tail feathers, or within types of wing feathers, in the same lot of birds were the differences between tail and wing, between types of tail feathers, and between types of wing feathers. The variance between types of tail feathers in the same lot was not significantly greater than the variance between types of wing feathers in the same lot. The variance within types of tail feathers in the same lot was significantly greater than that within types of wing feathers.
A more detailed analysis could be made if one wished to do so, but its value would be open to question. The sum of squares between tail and wing in the same lot, for example, could be separated into a component due to differences between the tail and wing
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which remained constant from lot to lot and a component due to differential responses in the various lots. The former component contributed one degree of freedom and the latter contributed (11 — 1) or 10 degrees of freedom. The sum of squares due to the former component could be calculated by obtaining the sum of all the measurements of tail feathers and the sum of all the measurements of wing feathers, squaring each sum, dividing each of the two squares by the number of measurements involved in the corresponding sum, adding the two quotients, and subtracting from the sum of the quotients a correction, obtained by dividing the square of the sum of all the measurements by the total number of measurements. The correction used in this instance was the same as that used in calculating the total sum of squares. The sum of squares due to differential response would have to be obtained by difference, i.e., by subtracting the sum of squares between tail and wing just described from the sum of squares between tail and wing in the same lot given in Table 2. Several similar possibilities will doubtless suggest themselves to the minds of the readers of this paper without any further discussion on the part of the author. One additional aspect of the problem may be considered to advantage, viz., the question of how much of the variability within the types of tail and wing feathers for the same lot of birds was due to the individuality of
TABLE 2.—Analysis of variance of feather measurements , using all available measurements Variability
Between types of tail feathers in same lot Within types of wing feathers in same lot Total
Sum of squares
Mean square
5 5 11 22 22 513 526
2.4586 2.3320 925.8324 577.6303 327.5285 293.7468 158.8983
0.4917 0.4664 84.1666 26.2559 14.8877 0.5726 0.3021
1,104
2,288.4269
d/f
l
A IogeM. Squ.
-0.35494 -0.38135 +2.21640 + 1.63395 + 1.35027 -0.27879 -0.59850
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TABLE 3.—Analysis of variance within tails, using only data for birds for which complete sets of three measurements were available d/f
Sum of squares
Mean square
^logeM. Squ.
Between birds in same lot Error
22 162 324
545.4125 238.3145 35.8900
24.7915 1.4711 0.1108
+ 1.60525 +0.19300 -1.10001
Total (within tail in same lot)
508
819.6170
Variability
the birds and how much was due to other causes, consisting for the most part of errors made in measuring the feathers. In this phase of the work it was desirable to use only data obtained for those birds for which complete sets of three measurements of tail feathers, or three measure-. ments of wing feathers, were available. Recent investigations by Yates (1933) indicate that such a selection of data might not have been necessary. However, a discussion of the methods presented by Yates is somewhat beyond the scope of this paper. The measurements of tail feathers for each lot of birds and the measurements of wing feathers for each lot of birds were treated separately, after which the results obtained for the various lots were combined in the case of each of these two general classes of feathers. For each lot of birds the necessary calculations involved the finding of new values of the degrees of freedom and sum of squares within tail feathers and within wing feathers and new values of the degrees of freedom and sum of squares between types of tail feathers and between types of wing
feathers. The degrees of freedom and sum of squares between birds in each case were found in a manner perfectly analogous to the method of calculating the degrees of freedom and sum of squares between types of feathers. The degrees of freedom and sum of squares due to error in each case were found by difference. The results of this phase of the work are summarized in Tables 3 and 4. It is quite apparent that very little of the variability within types of tail and wing feathers, as given in Table 2, was due to errors of measurement. The errors of measurement were quite negligible in comparison with the variability due to the individuality of the birds. The variability due to birds was significantly greater in the case of the tail feathers than in the case of the wing feathers. The same is true of the variability due to errors of measurement. The variability between supplements, as given in Table 2, was significantly greater than the variability due to errors of measurement, as given in Tables 3 and 4. This comparison was not strictly justifiable, for the variability between supplements, as
TABLE 4.—Analysis of variance within wings, using only data for birds for which complete sets of three measurements were available Variability
Between birds in same lot Total (within wing in same lot)
d/f
Sum of squares
Mean square
H loge M. Squ.
22 173 346
321.3491 144.2110 13.4655
14.6068 0.8336 0.0389
+ 1.34075 -0.09100 -1.62338
541
479.0256
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given in Table 2 may have been affected slightly by some of the measurements which were discarded in estimating the error given in Tables 3 and 4. In order to be entirely consistent, the sum of squares between supplements would have to be calculated from exactly the same measurements used in calculating the sum of squares due to error. However, such a comparison is of very little practical significance, since it gives no information whatever regarding the results to be expected if the experiment were repeated with a different group of birds, unless those birds were carefully selected so that every bird used in such an experiment would be the exact counterpart of a corresponding bird used in the experiment discussed in this paper. The variability within types of tail feathers and within types of wing feathers, given in Table 2, is a more useful standard of comparison, since any conclusions based upon this variability as a measure of experimental error would be applicable to any group of birds of the
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same general breeding as those used in the experiment herein described. SUMMARY AND CONCLUSIONS Variability in the lengths of selected tail and wing feathers of White Leghorn hens, due to differences in the mineral supplements fed, was not significantly greater than the variability encountered between feathers of the same type in the same lot of birds. The variability due to differences in the mineral supplements seemed to be significantly greater than the variability due to actual errors of measurement made in measuring the feathers, but this comparison is of very little practical importance. REFERENCES
Fisher, R. A., 1932. Statistical methods for research workers, 4th edition. Edinburgh and London: Oliver and Boyd. Hendricks, W. A., 1933. A biometric study of molt in White Leghorn hens. Poul. Sci. 12:287-291. Yates, F., 1933. The principles of orthogonality and confounding in replicated experiments. Jour. Agr. Sci. 23 :108-145.