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Relations of Weights and Volumes of Eggs to Measurements of Long and Short Axes*
Relations of Weights and Volumes of Eggs to Measurements of Long and Short Axes* WILLIAM DOWELL BATENI AND EARL W. HENDERSON:}:
Michigan Agricultural Experiment Station, East Lansing, Michigan (Received for publication April 23, 1941)
HE relationship between the volume, dimensions, and weight of eggs of domestic fowl seems to be a subject which has not been investigated extensively. Knowledge of such relationships should be useful in the general study of the inheritance of egg characteristics such as breaking strength, hatching power, and many other criteria. Several investigators have studied the relationship between dimensions and gross weight as well as the weight of parts of the egg but we have not found reports of quantitative determinations of the relation between volume and dimensions. From an interesting study of the progressive variation in the shape of the eggs of a pullet from the first to the fifty-fourth egg, Pearl (1909) proposed an "index" (breadth/length X 100), as a criterion. Asmundson and Baker (1940) postulated an equation for calculating volume from density and the length and breadth index but apparently they did not test the formula by physical determinations of volume. Estimations of volume from the specific gravity formula of Asmundson and Baker (1940) obviously are subject to error in direct proportion to the size of the air cells of the eggs. * Journal article No. 521 (n.s.) of the Michigan Agricultural Experiment Station. t Research associate in statistics. t Assistant professor of and research assistant in poultry husbandry.
PURPOSE
The purpose of this investigation was to determine the normal variations in volume, dimensions, relation of volume to dimensions, and weight of eggs of two classes; namely, white and brown colored shells. MATERIALS AND METHODS A random sample of approximately 104 eggs was selected from two breeds: 51 Single Comb White Leghorns and 53 Barred Plymouth Rocks. Each egg was weighed on the morning of the day following laying on a balance known to be accurate to one-tenth of a gram. Measurements of dimensions were taken with steel vernier calipers graduated to one-tenth of a millimeter. Photographs of each egg were taken to scale by E. J. Airola, graduate assistant in horticulture, for a permanent record, for the purpose of verifying the measurements, and for determining volume formulas. Volume was determined by immersing the eggs in a specially designed beaker (Fig. 1) filled with distilled water and measuring the water displaced in a burette. The special design of the beaker was necessary in order to permit a precise end point for measuring the overflow of water displaced and to save time. If water is allowed to flow from a tube attached to the upper portion of a beaker the amount which will flow as the result of displacement with a solid body is highly variable. The attach-
[S56J
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T
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
SS7
ment of the overflow tube to the bottom of the beaker reduces the variability. A slight hydraulic ram effect may be expected from the design but it was apparently uniform for a number of identical volume determinations were obtained with several eggs. Three volume measurements were taken of each egg and the average used in computations.
TABLE
Fio. 1. Beaker, specially constructed, for determining the volumes of eggs by the displacement of water.
slopes of the lines are also different. These differences may be attributed to the strains if not to breeds. RELATION OF WEIGHT AND SHORT AXIS Measurements were made of the short axes of the eggs for determining the relation between weight and short axis and for determing whether or not the long axis is better for predicting weight than the short axis. Table 2 contains predicting equations, standard errors of estimate, and correlation coefficients pertaining to predicting weights from the short axis; these were found by the method of least squares. The sizes of the standard errors of estimate in Table 2, 2.9S gm. and 3.25 gm., indicate that the predicting equation pertaining to Barred Rock eggs gives more
1 - -Data for predicting egg weight from long axis
Breed White Leghorn x = long axis, y' = predicted weight.
Predicting equation y ' = -14.86+12.59x y ' = -24.23+14.30x
Te(gm)
4.05 3.42
r
yx
.655 .787
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
RELATION OF WEIGHT AND LONG AXIS When the long axis measurements of eggs from the two breeds are plotted against weights the scatter diagrams fall along straight lines, suggesting a linear relation between these two variables. By the method of least squares, predicting equations shown in Table 1, together with the standard errors of estimate and correlation coefficients, were found for predicting the weight of an egg from the long axis. The symbol, y', means that the predicted weight is the value obtained on the average. The sizes of the standard errors of estimate, given in Table 1, indicate that the predicting equations give very good results. One can predict on the average, the weight of a Barred Rock egg to within 4.05 gm. about 68 times out of 100 times, and the weight of a White Leghorn egg to within 3.42 gm. Many of the predictions will be less. Values of the correlation coefficients listed in the last column of the above table also reveal a rather high degree of association between weights and long axis measurements of eggs from these breeds. The constant term in the predicting equation pertaining to Barred Rock eggs is different than the constant term in that pertaining to White Leghorn eggs; the
558
WILLIAM DOWELL BATEN AND EARL W. HENDERSON
BARRED BOCK
/
/
BAOREDROCK
o
v
Oft,
/ /
? ' (Je* 0.690
z u
5 so
-1
V=-Z.358+.97ZY
§ >
WHITE LEGHORN EGOS
j
L
SO «0 10 80 Y« WEIGHT IN GRAMS
WHITE LEGHQgN /.,
5» fit
(Wtf «-
1
Y<»-84.Z0+33.95z 35
50 60 10 80 r » WEIGHT IN GRAMS
40 4T «» Z - SHORT AXIS IN CM.
CHART I
CHART I I
RELATION OF WEIGHT AND SHORT AXIS OF EGOS
RELATION OF VOLUME AND WEIGHT of EGGS
accurate results than that pertaining to White Leghorn eggs. The values of the correlation coefficients between weight and short axis reveal a definite relationship between these two variables. The bands of "normality" shown in chart 1 are narrower than similar bands (not given) perTABLE 2 — Data for
Breed Barred Rock White Leghorn z=short axis, y'=predicted weight.
taining to long axes; this suggests that the short axis measurement might be better to use for predicting weight than the measurement of the long axis. The dots fall nearer the straight lines than in the charts (not given) pertaining to long axes. Since the standard errors of estimate
predicting egg weight from short axis Predicting equation
y'= -61.49+28.54z y ' = -84.20+33.95z
Te(gm)
Tyz
2.95 3.25
.835 .824
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
V
40 4,5 » Z - S H O R T AXIS IN CM.
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
-3)(l+r„) 2D
t = (ryz-ryx) y
where D is the third order determinant yx
D=
1
rx
r™
1
and where r^ is greater than ry*, and y, x, and z represent respectively the weight, long axis, and short axis of an egg and n the number of eggs measured. This test for determining whether or not the short axis is better to use for predicting the weight of a Barred Rock egg than the long axis, becomes t = (.835-.655) y
/^-3)(1+.163) = 2(.025)
6
14
where D=
1 .835 .655
.835 1 .163
.655 .163 = .025 1
On entering a t-table at 50 degrees of freedom it is seen that there is a significant difference between the two standard errors of estimate, 4.05 gm. and 2.95+ gm. This large value of t indicates that one can predict the weight of Barred Rock eggs more
accurately from the short axis than from the long axis. A difference of 4.05 gm. — 2.95 gm. = 1.10 gm. in the standard errors of estimate is significantly greater than zero. This test for the data pertaining to White Leghorn egjgs is as follows: t=(.824- - . 7 8 7 ) /
(51-3)(l+.377) 2(.048)
= .037X26.239=.971
where D=
1 824 787
.824 1 .377
.787 .377 1
.048
On entering a t-table at 48 degrees of freedom it is seen that there is no significant difference between these standard errors of estimate; that is, one can predict weights of White Leghorn egg from long axis about as accurately as from short axis. This test of Hotelling is very useful for it enables one to eliminate a great deal of work in many instances, since one measurement may be much easier to make than others. This test will prove a very important tool for research workers. The test can be made without finding the standard error of estimate for it involves only correlation coefficients and degrees of freedom. The constant terms in the equations in Table 2 are not the same; the slopes of the straight lines are quite different. These differences are attributed to differences between strains of stock used in this project. RELATION OF WEIGHT AND BOTH LONG AND SHORT AXES
Table 3 contains the equations for predicting weights of eggs from both long and short axis measurements. The values of the standard errors of estimate show that weights of eggs for these breeds can be predicted very accurately by using both long and short axes linearly. Better results are obtained when both axes
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were smaller when predicting weight from short axis than from long axis, it seems natural to test for a statistical significance between these standard errors of estimate; that is, to determine whether the short axis is better for predicting weight than long axis. Such a test could not be made prior to November of 1940 because of the mathematical difficulties involved in its development. Hotelling (1940) gave for the first time a test for determining which of two variables is better for predicting a third variable. Because of the importance of such a test it will be explained in detail. The test is as follows:
559
560
WILLIAM DOWELL BATEN AND EARL W. HENDERSON TABLE 3.—Data/ or predicting weights of eggs from long and short axes
Breed Barred Rock White Leghorn
Predicting equation*
o-e(gm)
i y xz
y ' = -109.48+10.25x+25.57z y ' = - l 0 6 . 8 2 + 10.22x+25.02z
.869 1.259
.987 .971
* x=long axis, z = short axis, y'=predicted weight.
RELATION OF LONG AXIS TO SHORT AXIS An interesting relation pertaining to the eggs of these breeds is the value of the average of the ratios of the short axis to the long axis. These averages for these breeds are AVERAGE OF RATIOS OF SHORT AXIS TO LONG AXIS
Arith. ave. Geometric ave.
Barred Rock 0.719 ± 0.00S 0.719
White Leghorn 0.722 ± 0.006 0.722
The difference of 0.003 between these values suggests that the ratio between the
two axes of eggs for several breeds might be the same. The standard deviations are very small, showing very little variation. The relationship between short and long axes of eggs is again revealed by the predicting equations shown in Table 4, which were found by the method of least squares; these do not differ a great deal. The slopes of the two lines, 0.716 and 0.717, are about the same when predicting equations without constant terms are used. The correlation coefficient between short axis and long axis is significantly greater than zero for the data pertaining to White Leghorns; this coefficient is not significantly different from zero for the Barred Rock data. EGGS AS CIRCULAR ELLIPSOID VOLUMES The volume of each egg was found by the displacement of water method after no little amount of effort to secure the volumes accurately. By using measurements as shown in Figure 2 made from photographs of these eggs it was found that an egg is approximately equal to the sum of the halves of two circular ellipsoids. In Figure 3 (k) is shown an outline of an egg where the curve EBDC is a circle and the solid A-EBDC is one-half of an ellipsoid with AF as one-half of its major axis and BC as equal to each of its other two axes, and the solid (F-EBDC) is (one-half of) a
TABLE 4.—Relation between long and short axes of eggs Breed
White Leghorn x=long axis; z'=short axis.
Predicting equation*
oj(cm.)
rXz
z'=3.301 + .163x z'= .716x
.168 .240
.163
z' = 3.701+.092x z'= .717x
.155 .207
.377
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are used than when either long axis or short axis is used. The multiple correlation coefficients, listed in the last column of Table 3, between weights and long and short axes are large, indicating a great deal of association between weights and the other variables. It is interesting to note that the coefficient of x and z in these predicting equations are about the same and that the constant terms —109.48 and —106.82 do not differ a great deal. These values suggest that one predicting equation might be found for predicting weights of eggs from long and short axes that will apply to many breeds. When only one axis was used the predicting equations pertaining to these two breeds differed widely, but when both axes were used the equations did not differ very much. The standard errors of estimate only differed by 0.39 gm.
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
561
circular ellipsoid with OF equal to one-half of its major axis and BC equal to each of its other axes. The volume of the egg shown is the sum of the volumes of these two solids, which are shown in Figure 3 (h) and Figure 3 (j). The volume of this egg is A-EBDC-F = A -
A-EBDC + F-EBDC
or Vol. of A-EBDC-F = y3 [4/3jta(z/2) 2 l + y2 [jif(z/2) 2 ] = 4/24;tz2(a + f) = 1/6JIZ2X = .S236z2x
where x = long axis and z = short axis. The volume of the egg A-EBDC-F is therefore equal to Vol. = V = 0.5236z2x
which is the volume of the circular ellipsoid A'-E'B'D'C'-F', shown in Figure 3 (1). This solid is a circular ellipsoid with major axis equal to the long axis of the solid in Figure 3 (k) and minor axes each equal
to its short axis. The volume of an egg is approximately equal to the volume of a circular ellipsoid with major axis equal to the long axis of the egg and with its minor axes each equal to the short axis of the egg. To determine how accurate this approximation is, it was assumed that the volume of an egg was linearly related to the product of the long axis and the square of the short axis, or that V = kxz2
where V = volume, x = long axis, z = short axis, and k is a constant to be found from the data. By the method of least squares the values of k were found for the data pertaining to the two breeds. These results are shown in Table 5, together with the standard errors of estimate and the correlation coefficients between volume and the product xz2.
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FIG. 2. Photographs of eggs from which measurements were taken.
WILLIAM DOWELL BATEN AND EARL W. HENDERSON
562
If each egg were a perfect circular ellipsoid its volume would be exactly equal to 0.S236 xz2. The coefficients of xz2 in the
equations in Table 5 are respectively 0.5212 and 0.5202; showing that the volumes of these eggs can be calculated approximately by calculating the volume of a circular ellipsoid as described. The
where x = long axis, z = short axis, V = volume, and a, b, and c are constants to be found from the data. The constants a, b, and c were found by the method of least squares after taking the logarithms of both numbers in the above equation. These equations, pertaining to both breeds, are listed in Table 6, together with the standard errors of estimate. The "standard errors" of estimate were found by finding the predicted values of log V, then the values of V, and finally the square root of the average of the squares
TABLE 5.—Relation of volume lo long axis times the square of the short axis Breeds Barred Rock. White Leghorn
Predicting equation* 2
V ' = .5212xz V'=.5202xz 2
* x=long axis, z=short axis, V'=predicted volume, f These were obtained by using V = a + b x z 2 .
rvk/Ot
1.22 0.93
976 988
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«) FIG. 3. Diagrams: showing the volume of an egg to be equivalent to the volume of ellipsoid, (h) Left part of the egg shown in (k). (j) Right part of the egg shown in (k). (k) Outline of an egg. (1) Ellipsoid with volume equal to the egg in (k).
volume of each Barred Rock egg was computed by using the equation V = 0.5212 xz2 and compared with the volume found by the displacement of water method. The average difference between observed (water displacement volume) and predicted values was 0.12 ex.; the maximum difference was 3.7 c.c. This was done for the White Leghorn eggs. The average difference was 0.02 c.c. and the maximum difference was 2.3 c.c. Seventy-five percent of the differences were less than or equal to 1 c.c. for each breed. These values show again that the predicting equations give on the average very good results. The sizes of the standard errors of estimate also reveal this. The values of the linear correlation coefficients between volume and (xz2), 0.976 and 0.988, indicate a very close association between volume and this product of the long axis and the square of the short axis. The volume of an egg was assumed to be related to the long and short axes by the following more complicated equation. V = a • xb • zc
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
563
of the differences between the observed and predicted volumes. The equations in Table 6 are a little better than those in Table 5 for predicting volume, but not enough to justify the extra
each egg was found and was considered to be the approximate density of the egg. The averages of these densities for both breeds are as follows:
TABLE 6.—Volume equations found from V = axaz°
The values of the above standard errors show that the densities of the individual eggs were scattered about the equal means with about the same variability. The above values suggest that the average density for all breeds might be 1.07 plus or minus a very small standard error.
Predicting equation o-o*(cc.)
Breeds
Barred Rocks 1028 zi- 8 " White Leghorns.... V' = 0.646x
1.114 0.703
labor of computation in obtaining them. The eggs measured are nearly circular elipsoids; the volume can be computed very accurately by considering them as such. RELATION OF VOLUME AND LONG AND SHORT AXES Table 7 contains the predicting equations for predicting the volume of an egg from the long axis, the short axis, and from both long and short axes. By using Hotelling's test given above, it is found that the volume of an egg for these breeds can be predicted more accurately from short axis than from long axis. It is obvious on examining the standard errors of estimate that the volume can be predicted more accurately from both long and short axes than from either long or short axes. DENSITY OF EGGS The ratio of the weight and volume of
White Leghorn 1.07 ± .0016
RELATION OF VOLUME AND WEIGHT The values of the densities suggested that there might be a high degree of association between the volume of an egg and its weight. Table 8 contains the predicting equations for predicting volume from weight, together with the standard errors of estimate and the correlation coefficients between these variables. The volume of an egg can be predicted very accurately by multiplying the weight by a constant 0.933. The decrease in the error of estimate in predicting volume from weight by using an equation of the form V = a + by in preference to one of the form, V = hy, is negligible. The latter is much easier to employ. It appears as though the formula for predicting volume of these eggs is the same for both breeds, namely V = 0.933 y.
TABLE 7.—Relation of volume and long and short axes Breed
Predicting equation*
o-e(c.c)
Correlation coef.
Barred Rock. . White Leghorn
V'=V'=-
18.69+12.55x 22.28+13.28x
3.962 3.238
0.663 0.798
Barred Rock. . White Leghorn
V'=V'=-
61.47+27.S9z 81.97+32.46z
3.044 2.942
0.818 0.837
Barred Rock.. White Leghorn
V'= -109.72+10.30x-24.60z V ' = - 1 0 4 . 1 0 + 9.47x-24.44z
1.098 0.729
0.978f 0.986
* x=long axis, z=short aHs, V'=predicted volume, f Multiple correlation coefficient.
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* Obtained by finding the log V and then V.
Barred Rock 1.07 ± .0023
564
WILLIAM DOWELL BATEN AND EARL W. HENDERSON TABLE 8.—Volume predicted from weight
Breed
Predicting equation
"e
rv„
V'=-2.358+0.972y V'= 0.933y
0.890 0.905
0.986
V'= V'=
0.576 0.652
0.994
0.635+0.923y 0.933y
(2) Equation predicting weight (y') from short axis (z) A. Barred Rock y' = —61.49 + 28.S4z B. White Leghorn y' = —84.20 + 33.95z (3) Equations predicting weight y' from both axes A. Barred Rock y' = —109.48 + 10.25 X 25.57z B. White Leghorn y' = —106.82 + 10.22 X 25.02z (4) Relation of Long axis (x) to short axis (z) A. Barred Rock z = 0.716x B. White Leghorn z = 0.7l7x (5) Circular ellipsoid formulae for predicting volume ( V ) from long axis (x) and short axis (z). A. Barred Rocks V = 0.517 x10™ z IM2 . B. White Leghorn V = 0.646 x1028 z1'817.
The density of the Barred Rock eggs and White Leghorn eggs was the same; that is, 1.07. Volume (V) can be predicted from weight (y) of eggs of both breeds by the relation:V = 0.933y, the weight determined immediately after eggs were laid is a better index of volume than dimensions of long and short axes. Hotelling's method for evaluating criteria is demonstrated. A suggestion is made for determining the surface area of eggs.
SUMMARY
Relations of weights and volumes of eggs to dimensions of long and short axes were determined by physical measurements. Results generalized by mathematical formulae derived from measurements are as follows: (1) Equation predicting weight (y') from long axis (x) A. Barred Rock y' = 14.86 + 12.59x B. White Leghorn y' = -24.23 + 14.30x
REFERENCES
Asmundson, V. S., and G. A. Baker, 1940. Percentage shell as a function of shell thickness, egg volume, and egg shape. Poult. Sci. 19:227-232. Hotelling, Harold, 1940. The selection of variates for use in prediction with some comments on the general problem of nuisance parameter. The Annals of Math. Statistics, 11:271-283. Pearl, Raymond, 1909. Physiology of reproduction in domestic fowl. Journ. Exp. Zoo. 6:340-360.
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
Chart 2 shows graphically the relation between volumes and weights of eggs from these breeds. The sizes of the errors of estimate indicate that the weight of an egg is the best of the measurements made in this study for predicting the volume of an egg. After the eggs were three months old the weights were again found. In the meantime the eggs had been taken to several places on the campus (chemical laboratory, poultry offices, photographic room, and other places). Predicting equations were obtained for predicting volumes from these weight measurements. The errors of estimate were large showing that the eggs did not lose weight in the same manner. The scatter diagrams did not fall as near the straight lines as before. This verifies the common assumption that the volume of an egg can be calculated rather accurately from weight, provided of course that the weight is determined before moisture losses occur. Since the volume of an egg may be obtained rather accurately from the volume of an ellipsoid the surface area may also be computed from the surface of this ellipsoid.
Michigan Agricultural Experiment Station, East Lansing, Michigan (Received for publication April 23, 1941)
HE relationship between the volume, dimensions, and weight of eggs of domestic fowl seems to be a subject which has not been investigated extensively. Knowledge of such relationships should be useful in the general study of the inheritance of egg characteristics such as breaking strength, hatching power, and many other criteria. Several investigators have studied the relationship between dimensions and gross weight as well as the weight of parts of the egg but we have not found reports of quantitative determinations of the relation between volume and dimensions. From an interesting study of the progressive variation in the shape of the eggs of a pullet from the first to the fifty-fourth egg, Pearl (1909) proposed an "index" (breadth/length X 100), as a criterion. Asmundson and Baker (1940) postulated an equation for calculating volume from density and the length and breadth index but apparently they did not test the formula by physical determinations of volume. Estimations of volume from the specific gravity formula of Asmundson and Baker (1940) obviously are subject to error in direct proportion to the size of the air cells of the eggs. * Journal article No. 521 (n.s.) of the Michigan Agricultural Experiment Station. t Research associate in statistics. t Assistant professor of and research assistant in poultry husbandry.
PURPOSE
The purpose of this investigation was to determine the normal variations in volume, dimensions, relation of volume to dimensions, and weight of eggs of two classes; namely, white and brown colored shells. MATERIALS AND METHODS A random sample of approximately 104 eggs was selected from two breeds: 51 Single Comb White Leghorns and 53 Barred Plymouth Rocks. Each egg was weighed on the morning of the day following laying on a balance known to be accurate to one-tenth of a gram. Measurements of dimensions were taken with steel vernier calipers graduated to one-tenth of a millimeter. Photographs of each egg were taken to scale by E. J. Airola, graduate assistant in horticulture, for a permanent record, for the purpose of verifying the measurements, and for determining volume formulas. Volume was determined by immersing the eggs in a specially designed beaker (Fig. 1) filled with distilled water and measuring the water displaced in a burette. The special design of the beaker was necessary in order to permit a precise end point for measuring the overflow of water displaced and to save time. If water is allowed to flow from a tube attached to the upper portion of a beaker the amount which will flow as the result of displacement with a solid body is highly variable. The attach-
[S56J
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
T
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
SS7
ment of the overflow tube to the bottom of the beaker reduces the variability. A slight hydraulic ram effect may be expected from the design but it was apparently uniform for a number of identical volume determinations were obtained with several eggs. Three volume measurements were taken of each egg and the average used in computations.
TABLE
Fio. 1. Beaker, specially constructed, for determining the volumes of eggs by the displacement of water.
slopes of the lines are also different. These differences may be attributed to the strains if not to breeds. RELATION OF WEIGHT AND SHORT AXIS Measurements were made of the short axes of the eggs for determining the relation between weight and short axis and for determing whether or not the long axis is better for predicting weight than the short axis. Table 2 contains predicting equations, standard errors of estimate, and correlation coefficients pertaining to predicting weights from the short axis; these were found by the method of least squares. The sizes of the standard errors of estimate in Table 2, 2.9S gm. and 3.25 gm., indicate that the predicting equation pertaining to Barred Rock eggs gives more
1 - -Data for predicting egg weight from long axis
Breed White Leghorn x = long axis, y' = predicted weight.
Predicting equation y ' = -14.86+12.59x y ' = -24.23+14.30x
Te(gm)
4.05 3.42
r
yx
.655 .787
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
RELATION OF WEIGHT AND LONG AXIS When the long axis measurements of eggs from the two breeds are plotted against weights the scatter diagrams fall along straight lines, suggesting a linear relation between these two variables. By the method of least squares, predicting equations shown in Table 1, together with the standard errors of estimate and correlation coefficients, were found for predicting the weight of an egg from the long axis. The symbol, y', means that the predicted weight is the value obtained on the average. The sizes of the standard errors of estimate, given in Table 1, indicate that the predicting equations give very good results. One can predict on the average, the weight of a Barred Rock egg to within 4.05 gm. about 68 times out of 100 times, and the weight of a White Leghorn egg to within 3.42 gm. Many of the predictions will be less. Values of the correlation coefficients listed in the last column of the above table also reveal a rather high degree of association between weights and long axis measurements of eggs from these breeds. The constant term in the predicting equation pertaining to Barred Rock eggs is different than the constant term in that pertaining to White Leghorn eggs; the
558
WILLIAM DOWELL BATEN AND EARL W. HENDERSON
BARRED BOCK
/
/
BAOREDROCK
o
v
Oft,
/ /
? ' (Je* 0.690
z u
5 so
-1
V=-Z.358+.97ZY
§ >
WHITE LEGHORN EGOS
j
L
SO «0 10 80 Y« WEIGHT IN GRAMS
WHITE LEGHQgN /.,
5» fit
(Wtf «-
1
Y<»-84.Z0+33.95z 35
50 60 10 80 r » WEIGHT IN GRAMS
40 4T «» Z - SHORT AXIS IN CM.
CHART I
CHART I I
RELATION OF WEIGHT AND SHORT AXIS OF EGOS
RELATION OF VOLUME AND WEIGHT of EGGS
accurate results than that pertaining to White Leghorn eggs. The values of the correlation coefficients between weight and short axis reveal a definite relationship between these two variables. The bands of "normality" shown in chart 1 are narrower than similar bands (not given) perTABLE 2 — Data for
Breed Barred Rock White Leghorn z=short axis, y'=predicted weight.
taining to long axes; this suggests that the short axis measurement might be better to use for predicting weight than the measurement of the long axis. The dots fall nearer the straight lines than in the charts (not given) pertaining to long axes. Since the standard errors of estimate
predicting egg weight from short axis Predicting equation
y'= -61.49+28.54z y ' = -84.20+33.95z
Te(gm)
Tyz
2.95 3.25
.835 .824
Downloaded from http://ps.oxfordjournals.org/ at East Tennessee State University on May 30, 2015
V
40 4,5 » Z - S H O R T AXIS IN CM.
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
-3)(l+r„) 2D
t = (ryz-ryx) y
where D is the third order determinant yx
D=
1
rx
r™
1
and where r^ is greater than ry*, and y, x, and z represent respectively the weight, long axis, and short axis of an egg and n the number of eggs measured. This test for determining whether or not the short axis is better to use for predicting the weight of a Barred Rock egg than the long axis, becomes t = (.835-.655) y
/^-3)(1+.163) = 2(.025)
6
14
where D=
1 .835 .655
.835 1 .163
.655 .163 = .025 1
On entering a t-table at 50 degrees of freedom it is seen that there is a significant difference between the two standard errors of estimate, 4.05 gm. and 2.95+ gm. This large value of t indicates that one can predict the weight of Barred Rock eggs more
accurately from the short axis than from the long axis. A difference of 4.05 gm. — 2.95 gm. = 1.10 gm. in the standard errors of estimate is significantly greater than zero. This test for the data pertaining to White Leghorn egjgs is as follows: t=(.824- - . 7 8 7 ) /
(51-3)(l+.377) 2(.048)
= .037X26.239=.971
where D=
1 824 787
.824 1 .377
.787 .377 1
.048
On entering a t-table at 48 degrees of freedom it is seen that there is no significant difference between these standard errors of estimate; that is, one can predict weights of White Leghorn egg from long axis about as accurately as from short axis. This test of Hotelling is very useful for it enables one to eliminate a great deal of work in many instances, since one measurement may be much easier to make than others. This test will prove a very important tool for research workers. The test can be made without finding the standard error of estimate for it involves only correlation coefficients and degrees of freedom. The constant terms in the equations in Table 2 are not the same; the slopes of the straight lines are quite different. These differences are attributed to differences between strains of stock used in this project. RELATION OF WEIGHT AND BOTH LONG AND SHORT AXES
Table 3 contains the equations for predicting weights of eggs from both long and short axis measurements. The values of the standard errors of estimate show that weights of eggs for these breeds can be predicted very accurately by using both long and short axes linearly. Better results are obtained when both axes
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were smaller when predicting weight from short axis than from long axis, it seems natural to test for a statistical significance between these standard errors of estimate; that is, to determine whether the short axis is better for predicting weight than long axis. Such a test could not be made prior to November of 1940 because of the mathematical difficulties involved in its development. Hotelling (1940) gave for the first time a test for determining which of two variables is better for predicting a third variable. Because of the importance of such a test it will be explained in detail. The test is as follows:
559
560
WILLIAM DOWELL BATEN AND EARL W. HENDERSON TABLE 3.—Data/ or predicting weights of eggs from long and short axes
Breed Barred Rock White Leghorn
Predicting equation*
o-e(gm)
i y xz
y ' = -109.48+10.25x+25.57z y ' = - l 0 6 . 8 2 + 10.22x+25.02z
.869 1.259
.987 .971
* x=long axis, z = short axis, y'=predicted weight.
RELATION OF LONG AXIS TO SHORT AXIS An interesting relation pertaining to the eggs of these breeds is the value of the average of the ratios of the short axis to the long axis. These averages for these breeds are AVERAGE OF RATIOS OF SHORT AXIS TO LONG AXIS
Arith. ave. Geometric ave.
Barred Rock 0.719 ± 0.00S 0.719
White Leghorn 0.722 ± 0.006 0.722
The difference of 0.003 between these values suggests that the ratio between the
two axes of eggs for several breeds might be the same. The standard deviations are very small, showing very little variation. The relationship between short and long axes of eggs is again revealed by the predicting equations shown in Table 4, which were found by the method of least squares; these do not differ a great deal. The slopes of the two lines, 0.716 and 0.717, are about the same when predicting equations without constant terms are used. The correlation coefficient between short axis and long axis is significantly greater than zero for the data pertaining to White Leghorns; this coefficient is not significantly different from zero for the Barred Rock data. EGGS AS CIRCULAR ELLIPSOID VOLUMES The volume of each egg was found by the displacement of water method after no little amount of effort to secure the volumes accurately. By using measurements as shown in Figure 2 made from photographs of these eggs it was found that an egg is approximately equal to the sum of the halves of two circular ellipsoids. In Figure 3 (k) is shown an outline of an egg where the curve EBDC is a circle and the solid A-EBDC is one-half of an ellipsoid with AF as one-half of its major axis and BC as equal to each of its other two axes, and the solid (F-EBDC) is (one-half of) a
TABLE 4.—Relation between long and short axes of eggs Breed
White Leghorn x=long axis; z'=short axis.
Predicting equation*
oj(cm.)
rXz
z'=3.301 + .163x z'= .716x
.168 .240
.163
z' = 3.701+.092x z'= .717x
.155 .207
.377
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are used than when either long axis or short axis is used. The multiple correlation coefficients, listed in the last column of Table 3, between weights and long and short axes are large, indicating a great deal of association between weights and the other variables. It is interesting to note that the coefficient of x and z in these predicting equations are about the same and that the constant terms —109.48 and —106.82 do not differ a great deal. These values suggest that one predicting equation might be found for predicting weights of eggs from long and short axes that will apply to many breeds. When only one axis was used the predicting equations pertaining to these two breeds differed widely, but when both axes were used the equations did not differ very much. The standard errors of estimate only differed by 0.39 gm.
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
561
circular ellipsoid with OF equal to one-half of its major axis and BC equal to each of its other axes. The volume of the egg shown is the sum of the volumes of these two solids, which are shown in Figure 3 (h) and Figure 3 (j). The volume of this egg is A-EBDC-F = A -
A-EBDC + F-EBDC
or Vol. of A-EBDC-F = y3 [4/3jta(z/2) 2 l + y2 [jif(z/2) 2 ] = 4/24;tz2(a + f) = 1/6JIZ2X = .S236z2x
where x = long axis and z = short axis. The volume of the egg A-EBDC-F is therefore equal to Vol. = V = 0.5236z2x
which is the volume of the circular ellipsoid A'-E'B'D'C'-F', shown in Figure 3 (1). This solid is a circular ellipsoid with major axis equal to the long axis of the solid in Figure 3 (k) and minor axes each equal
to its short axis. The volume of an egg is approximately equal to the volume of a circular ellipsoid with major axis equal to the long axis of the egg and with its minor axes each equal to the short axis of the egg. To determine how accurate this approximation is, it was assumed that the volume of an egg was linearly related to the product of the long axis and the square of the short axis, or that V = kxz2
where V = volume, x = long axis, z = short axis, and k is a constant to be found from the data. By the method of least squares the values of k were found for the data pertaining to the two breeds. These results are shown in Table 5, together with the standard errors of estimate and the correlation coefficients between volume and the product xz2.
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FIG. 2. Photographs of eggs from which measurements were taken.
WILLIAM DOWELL BATEN AND EARL W. HENDERSON
562
If each egg were a perfect circular ellipsoid its volume would be exactly equal to 0.S236 xz2. The coefficients of xz2 in the
equations in Table 5 are respectively 0.5212 and 0.5202; showing that the volumes of these eggs can be calculated approximately by calculating the volume of a circular ellipsoid as described. The
where x = long axis, z = short axis, V = volume, and a, b, and c are constants to be found from the data. The constants a, b, and c were found by the method of least squares after taking the logarithms of both numbers in the above equation. These equations, pertaining to both breeds, are listed in Table 6, together with the standard errors of estimate. The "standard errors" of estimate were found by finding the predicted values of log V, then the values of V, and finally the square root of the average of the squares
TABLE 5.—Relation of volume lo long axis times the square of the short axis Breeds Barred Rock. White Leghorn
Predicting equation* 2
V ' = .5212xz V'=.5202xz 2
* x=long axis, z=short axis, V'=predicted volume, f These were obtained by using V = a + b x z 2 .
rvk/Ot
1.22 0.93
976 988
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«) FIG. 3. Diagrams: showing the volume of an egg to be equivalent to the volume of ellipsoid, (h) Left part of the egg shown in (k). (j) Right part of the egg shown in (k). (k) Outline of an egg. (1) Ellipsoid with volume equal to the egg in (k).
volume of each Barred Rock egg was computed by using the equation V = 0.5212 xz2 and compared with the volume found by the displacement of water method. The average difference between observed (water displacement volume) and predicted values was 0.12 ex.; the maximum difference was 3.7 c.c. This was done for the White Leghorn eggs. The average difference was 0.02 c.c. and the maximum difference was 2.3 c.c. Seventy-five percent of the differences were less than or equal to 1 c.c. for each breed. These values show again that the predicting equations give on the average very good results. The sizes of the standard errors of estimate also reveal this. The values of the linear correlation coefficients between volume and (xz2), 0.976 and 0.988, indicate a very close association between volume and this product of the long axis and the square of the short axis. The volume of an egg was assumed to be related to the long and short axes by the following more complicated equation. V = a • xb • zc
RELATIONS OF WEIGHTS AND VOLUMES OF EGGS TO AXES MEASUREMENTS
563
of the differences between the observed and predicted volumes. The equations in Table 6 are a little better than those in Table 5 for predicting volume, but not enough to justify the extra
each egg was found and was considered to be the approximate density of the egg. The averages of these densities for both breeds are as follows:
TABLE 6.—Volume equations found from V = axaz°
The values of the above standard errors show that the densities of the individual eggs were scattered about the equal means with about the same variability. The above values suggest that the average density for all breeds might be 1.07 plus or minus a very small standard error.
Predicting equation o-o*(cc.)
Breeds
Barred Rocks 1028 zi- 8 " White Leghorns.... V' = 0.646x
1.114 0.703
labor of computation in obtaining them. The eggs measured are nearly circular elipsoids; the volume can be computed very accurately by considering them as such. RELATION OF VOLUME AND LONG AND SHORT AXES Table 7 contains the predicting equations for predicting the volume of an egg from the long axis, the short axis, and from both long and short axes. By using Hotelling's test given above, it is found that the volume of an egg for these breeds can be predicted more accurately from short axis than from long axis. It is obvious on examining the standard errors of estimate that the volume can be predicted more accurately from both long and short axes than from either long or short axes. DENSITY OF EGGS The ratio of the weight and volume of
White Leghorn 1.07 ± .0016
RELATION OF VOLUME AND WEIGHT The values of the densities suggested that there might be a high degree of association between the volume of an egg and its weight. Table 8 contains the predicting equations for predicting volume from weight, together with the standard errors of estimate and the correlation coefficients between these variables. The volume of an egg can be predicted very accurately by multiplying the weight by a constant 0.933. The decrease in the error of estimate in predicting volume from weight by using an equation of the form V = a + by in preference to one of the form, V = hy, is negligible. The latter is much easier to employ. It appears as though the formula for predicting volume of these eggs is the same for both breeds, namely V = 0.933 y.
TABLE 7.—Relation of volume and long and short axes Breed
Predicting equation*
o-e(c.c)
Correlation coef.
Barred Rock. . White Leghorn
V'=V'=-
18.69+12.55x 22.28+13.28x
3.962 3.238
0.663 0.798
Barred Rock. . White Leghorn
V'=V'=-
61.47+27.S9z 81.97+32.46z
3.044 2.942
0.818 0.837
Barred Rock.. White Leghorn
V'= -109.72+10.30x-24.60z V ' = - 1 0 4 . 1 0 + 9.47x-24.44z
1.098 0.729
0.978f 0.986
* x=long axis, z=short aHs, V'=predicted volume, f Multiple correlation coefficient.
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* Obtained by finding the log V and then V.
Barred Rock 1.07 ± .0023
564
WILLIAM DOWELL BATEN AND EARL W. HENDERSON TABLE 8.—Volume predicted from weight
Breed
Predicting equation
"e
rv„
V'=-2.358+0.972y V'= 0.933y
0.890 0.905
0.986
V'= V'=
0.576 0.652
0.994
0.635+0.923y 0.933y
(2) Equation predicting weight (y') from short axis (z) A. Barred Rock y' = —61.49 + 28.S4z B. White Leghorn y' = —84.20 + 33.95z (3) Equations predicting weight y' from both axes A. Barred Rock y' = —109.48 + 10.25 X 25.57z B. White Leghorn y' = —106.82 + 10.22 X 25.02z (4) Relation of Long axis (x) to short axis (z) A. Barred Rock z = 0.716x B. White Leghorn z = 0.7l7x (5) Circular ellipsoid formulae for predicting volume ( V ) from long axis (x) and short axis (z). A. Barred Rocks V = 0.517 x10™ z IM2 . B. White Leghorn V = 0.646 x1028 z1'817.
The density of the Barred Rock eggs and White Leghorn eggs was the same; that is, 1.07. Volume (V) can be predicted from weight (y) of eggs of both breeds by the relation:V = 0.933y, the weight determined immediately after eggs were laid is a better index of volume than dimensions of long and short axes. Hotelling's method for evaluating criteria is demonstrated. A suggestion is made for determining the surface area of eggs.
SUMMARY
Relations of weights and volumes of eggs to dimensions of long and short axes were determined by physical measurements. Results generalized by mathematical formulae derived from measurements are as follows: (1) Equation predicting weight (y') from long axis (x) A. Barred Rock y' = 14.86 + 12.59x B. White Leghorn y' = -24.23 + 14.30x
REFERENCES
Asmundson, V. S., and G. A. Baker, 1940. Percentage shell as a function of shell thickness, egg volume, and egg shape. Poult. Sci. 19:227-232. Hotelling, Harold, 1940. The selection of variates for use in prediction with some comments on the general problem of nuisance parameter. The Annals of Math. Statistics, 11:271-283. Pearl, Raymond, 1909. Physiology of reproduction in domestic fowl. Journ. Exp. Zoo. 6:340-360.
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Chart 2 shows graphically the relation between volumes and weights of eggs from these breeds. The sizes of the errors of estimate indicate that the weight of an egg is the best of the measurements made in this study for predicting the volume of an egg. After the eggs were three months old the weights were again found. In the meantime the eggs had been taken to several places on the campus (chemical laboratory, poultry offices, photographic room, and other places). Predicting equations were obtained for predicting volumes from these weight measurements. The errors of estimate were large showing that the eggs did not lose weight in the same manner. The scatter diagrams did not fall as near the straight lines as before. This verifies the common assumption that the volume of an egg can be calculated rather accurately from weight, provided of course that the weight is determined before moisture losses occur. Since the volume of an egg may be obtained rather accurately from the volume of an ellipsoid the surface area may also be computed from the surface of this ellipsoid.