A Method of Analyzing the Data of Chick Nutrition Experiments*

A Method of Analyzing the Data of Chick Nutrition Experiments* HARRY W. TITUS AND JOHN C. HAMMOND Beltsvitte Research Center of the U. S. Department of Agriculture, Beltsville, Maryland (Received for Publication July 28, 1934)

F

* This is the fourth of a series of papers on the application of statistical method to the planning and the interpreting of animal nutrition experiments.

diet. As a result, the usual method of variance analysis is not applicable, unless there is the same proportion of surviving males and females on the several diets, and this seldom happens. Brandt (1933) has published a method designed for the analysis of a "2 X s" classification but, as shown by Yates (1934), Brandt's method involves some assumptions which are not rigorously tenable. Yates (1933, 1934), however, has proposed a method of "weighted squares of means" which is reliable and easily applied. Another method which gives essentially the same results as Yates' method had been developed recently by Hendricks (1935). In the present paper Yates' method is used because it seems to offer some slight advantages in computation over the latter. The data used for demonstrating the application of Yates' method of analysis were obtained at the Beltsville Research Center in an experimental study of perosis (Titus, 1932). EXPERIMENTAL MATERIAL

One-day-old Rhode Island Red chicks were used. They were distributed at random among pens in a hot-water-heated brooder house. Electric brooders were used until the chicks were six weeks old. As a basal diet (see diet of pen 353 in Table 1), one previously demonstrated as being perosis-producing (Titus, 1932) was employed. The other diets were modifications of this basal. The four diets, for which data are reported in this paper may be characterized as follows:

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OR some time workers in poultry nutrition have keenly felt the lack of a suitable, accurate method of analyzing the data of chick nutrition experiments. Fortunately, recent developments in the application of variance analysis have supplied such a method. It is the purpose of this paper to demonstrate the application of this method. In the usual chick nutrition experiment a suitable number of chicks is equally distributed among the required number of pens and the chicks in these pens are fed the different diets, or are given the different treatments to be studied. Some mortality is almost inevitable and so it very rarely happens that there is the same number of chicks in each pen at the end of an experiment. Also, when an experiment is ended, it is a common experience to have unequal numbers of males and females in the same pen, as well as in the different pens. This would offer no serious difficulty in the application of variance analysis to the data, if the males and females grew at the same rate, for then the two sexes could be treated as a single group or class. There would thus be only one class for each diet and with but a simole, slight modification (Fisher, 1932, p. 44), the usual method of analysis would be applicable. -s* Since the two sexes ordinarily do not grow at the same rate, it is necessary to have two classes, i.e., males and females, for each

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XIV, No. 3

165

TABLE 1.—The diets fed to the four pens Ingredients* "Ani-

Pen Ground No. yellow corn

352 353 356 357

Percent 36.36 40.00 36.36 40.00

Special Dried Alfalfa Poul- Com- steamCorn ed mon butter- leaf gluten try meal scrap milk meal Yeast salt bone meal Foam" Per- Per- PerPercent cent cent cent 20.00 9.09 9.09 9.09 22.00 10.00 10.00 10.00 20.00 9.09 9.09 9.09 22.00 10.00 10.00 10.00

Percent 2.27 2.50 2.27 2.50

Percent 1.82 2.00 1.82 2.00

Percent 0.46 0.50 0.46 0.50

Percent 2.73 3.00 2.73 —

•

Crab Ground Total scrap crab meal shell Percent 9.09

Percent

—. —. —

9.09

— —.

—

Percent — 100.00 —. 100.00 —. 100.00 3.00 100.00

Percent

Pen 352: The basal diet plus 10 percent of rice bran. Pen 353 : The basal diet only. Pen 356: The basal diet plus 10 percent of crab scrap meal. Pen 357: The basal diet with 3 percent of ground crab shell replacing 3 percent of special steamed bone meal.

The percentage of each ingredient in the basal diet and the three modifications, before the cod liver oil was added, is shown in Table 1. The chicks were weighed when placed in the pens and at age-intervals of two weeks thereafter. After the first symptoms of pero-

sis were observed, the chicks were carefully examined at regular intervals in order to ascertain what effect the several modifications of the perosis-producing diet had had on the development of the characteristic leg deformities. THE EXPERIMENTAL DATA Data on the mean live weights of the chicks at eight and twelve weeks are presented in Tables 2 and 3, respectively, and on the mean "degree" of perosis in Table 4. The following numerical notation was used to indicate the "degree" of perosis: (a) When a slight puffiness of the joints was

TABLE 2.—Mean live weights at eight weeks and their standard errors* Pen No.

Males

Females

Males and females (unweighted mean)

352 353 356 357

Grams 596.84 + 25.36 537.50 + 27.64 666.76 + 26.81 553.24 + 26.81

Grams 503.61 + 26.06 488.75 + 24.72 627.73 + 23.57 525.42 + 22.57

Grams 550.23 + 18.18 513.13 + 18.54 647.25 + 17.85 539.33 + 17.52

TABLE 3.—Mean live weights at twelve weeks and their standard errors* Pen No.

Males

Females

Males and females (unweighted mean)

352 353 356 357

Grams 1282.89 + 40.74 1047.81+44.40 1166.18+43.07 1130.29 + 43.07

Grams 1030.00+41.86 882.25 + 39.71 1083.18 + 37.86 1000.00 ±36.25

Grams 1156.45 + 29.20 965.03 + 29.78 1124.68 + 28.67 1065.15 + 28.15

* The probable error may be obtained by multiplying the standard error by 0.6745.

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* In each case, 2 parts of cod liver oil were mixed with 98 parts of the feed mixture just before it was fed

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TABLE 4.—Mean "degrees" of perosis and their standard errors* Pen No.

352 353 356 357

Males "Degree" of perosis 0.316 + 0.152 1.000 + 0.166 0.706 + 0.161 0.353 + 0.161

Females

Males and females (unweighted mean)

"Degree" of perosis 0.167 + 0.156 0.900 + 0.148 0.273 + 0.141 0.375 + 0.135

"Degree" of perosis 0.241 + 0.109 0.950 + 0.111 0.489 + 0.107 0.364 + 0.105

* The probable error may be obtained by multiplymg the standard error by 0.6745.

All birds in which no symptoms of perosis were detectable were assigned the value 0. ANALYSIS OF THE DATA In order to illustrate the application of the method of "weighted squares of means" the data on the live weights at eight weeks are used. A primary analysis of these data is made as follows: The first step is to tabulate the individual live weights of the males and females on each diet, as is shown in abbreviated form on work sheet No. 1. (In that which follows, the live weights of the males and females in each of the four pens will be referred to as subclasses. Thus, there are eight subclasses and these are the first eight numbered columns on work sheet No. 1). The total live weight of each subclass is then obtained, and the number of chicks in each subclass and the mean

live weight of each subclass inserted in the table. The total of each row is next obtained and these totals added to obtain the total live weight of all the chicks. This total must check with the total of the subclass totals. The second step is to tabulate the squares of all the individual live weights in a similar manner and obtain the subclass, as well as row, totals and check as before. The total live weight of each subclass (i.e., the total of each of the columns 1 to 8, inclusive) is then squared, divided by the number of live weights, and the results inserted in row 53. The figures in row 53 are then subtracted from the figures in row 52 to obtain the "sum of squares" for the eight subclasses. The third step is to calculate, the general correction term, the "sum of squares" due to subclasses, the error "sum of squares," and the total "sum of squares," as indicated on work sheet No. 1. The fourth step is to tabulate the above "sums •of squares" with their appropriate degrees of freedom, calculate the variances (or mean squares) and the standard deviations, insert the natural logarithms of the standard deviations and compute Fisher's z. This z is one-half of the difference between the natural logarithms of the variances which are being compared, or the difference between the natural logarithms of the corresponding standard deviations. Natural logarithms may be obtained from common logarithms by multiplying by 2.302S9. Since the 1 percent point (odds of 99 to 1) of z lies between 0.4846 and 0.5373, and is approxi-

TABLE 5.—Primary analysis of the live weights

Source of variation

d/f

Sums of squares

Variances (or mean squares)

Standard deviations

Subclasses Error (or residual)N^

7 145

516,387.36 1,772,420.81

73,769.62 12,223.59

271.61 110.56

152

2,288,808.17

—

Total

—

Nat. log. of standard deviations 5.6044 4.4778 z = 1.1266

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observed in the absence of any detectable bending or rotation of the leg bones, a numerical value of 1 was assigned. (b) When slight bending or rotation of the leg bones at the hock and/or slight bending or rotation of any leg bone above or below the hock was observed, a numerical value of 2 was assigned. (c) When pronounced deformity in one or both legs was observed, or when the chick became totally disabled, the numerical value of 3 was assigned.

M A Y ,

1935.VOL.

XIV, No. 3 W O R K SHEET N O . 1 Live weights at eight

1

2

3

4

Pen 352

weeks

5

6

Pen 353

7

8

9

Pen 357

Pen 356

Total Females

Males

Females

Males

Females

Males

Females

570 410 725

650 480 450

395 555 580

485 475 415

670 615 380

565 570 530

635 695 590

485 700 520

4455 4500 4190

16 17 18 19 20 21 22 23 24

455 830 705 790

495 600 520

615

515 560 405 220 500

700 740

420 770 720 555 750 535 765

715 485

680 570 465 460 395 585 670 385 600

4595 4555 2815 2025 1645 1120 1435 385 6C0

11,340 19 596.84211

9,065 18 503.61111

8,600 16 537.50000

9,775 20 488.75000

11,335 17 666.76741

13,810 22 627.72727

9,405 17 553.23529

12,610 24 525.41667

85,940 153 561.69935

25 T o t a l 26Number 27 Mean

Squared 10

11 Pen 352

12

live weights at eight 13

Pen 353

weeks

14

15

16

17

18

Pen 357

Pen 356

Total

Males

Females

Males

Females

Males

Females

Males

Females

28 29 30

324,900 168,100 525,625

422,900 230,400 202,500

156,025 308,025 336,400

235,225 225,625 172,225

448,900 378,225 144,400

319,225 324,900 280,900

403,225 483,025 348,100

235,225 490,000 270,400

2,545,225 2,608,300 2,280,550

43 44 45 46 47 48 49 50 51

207^025 688,900 497,025 624,100

245^025 360,000 270,400

378^225

265^225 313,600 164,025 48,400 250,000

490',000 547,600

176,400 592,900 518,400 • 308,025 562,500 286,225 585,225

511^225 235,225

462^400 324,900 216,225 211,600 156,025 342,225 448,900 148,225 360,000

2,735,525 3,063,125 1,666,075 1,192,125 968,525 628,450 1,034,125 148,225 360,000

52 Total 7,026,600.00 4,710,175.00 4,696,850.00 4,983,175.00 7,744,625.00 8,946,900.00 5,448,825.00 7,004,100.00 50,561,250.00 53* 6,768,189.53 4,565,234.72 4,622,500.00 4,777,531.25 7,557,777.94 8,668,913.64 5,203,177.94 6,625,504.17 54 Sum of squares

258,410.47

144,940.28

74,350.00

205,643.75

186,847.06

277,986.36

245,647.06

378,595.83

•(Total weight)* Number of chicks

Sums of Squares for the Primary Analysis General correction term: Squared total weight of all the chicks (85,940)2 7,385,683,600 •— = = =48,272,441.83 Total number of chicks 153 153 Sum of Squares due to subclasses: Squared subclass total weight\ )-Generalcorrectionterm = 6,768,189.53+4,565,234.72+4,622,500.00+4,777,531.25 + Number of chicks in subclass/ 7,557,777.94+8,668,913.64+5,203,177.94+6,625,504.17-48,272,441.83=48,788,829.19-48,272,441.83=516,387.36

(

Error sum of squares: Sum of the "sums of squares" for all subclasses = 258,410.47 + 144,940.28+74,350.00+205,643.75+186,847.06+277,986.36+ 245,647.06+378,595.83 = 1,772,420.81 Total sum of squares: Sum of all the squared individual weights-General correction term =50,561,250.00-48,272,441.83 =2,288,808.17

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Males 1 2 3

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WORK SHEET N O . 2

Further analysis: The effect of diet and sex TABLE A.- -Mean live weights of chicks at eight weeks

Pen No.

Males

352 353 356 357

Grams 596.84211 537.50000 666.76471 553.23529

Unweighted mean

558.58553

Difference: males— females

Unweighted mean of males and females

Grams 503.61111 488.75000 627.72727 525.41667

Grams 93.23100 48.75000 39.03744 27.81862

Grams 550.22661 513.12500 647.24599 539.32598

536.37626

52.20927

562.48089

Females

"Weight" of unweighted mean of males and females (4 X reciprocal of total)

Pen No.

Males

Females

Total: males and females

352 353 356 357

0.05263158 0.06250000 0.05882353 0.05882353

0.05555556 0.05000000 0.04S45455 0.04166667

0.10818714 0.11250000 0.10427808 0.10049020

9.243243 8.888889 9.589744 9.951219

36.972972 35.555556 38.358976 39.804876

Total

0.23277864

0.19267678

0.42545542

37.673095

150.692380

"Weight" of difference (reciprocal of total)

Weighted mean of means (males+females): Weighted mean of differences (males—females): Pen 352.—550.22661X36.972972 = 20,343.51305 Pen 352.—93.23100X9.243243= 861.75679 Pen 353—513.12500X35.555556=18,244.44467 Pen 353.—48.75000X8.888889= 433.33333 Pen 356—647.24599X38.358976 = 24,827.69340 Pen 356.—39.03744X9.5-9744= 374.35906 Pen 357—539.32598X39.804876 = 21,467.74120 Pen 357.—27.81862X9.951219= 276.82918 150.692380

37.673095

84,883.39232

51.66229

563.28921 Sums of squares for the further analysis: The effect of sex and diet. Sum of squares due to diet: (550.22661-563.28921) 2 X36.972972 = (-13.06260) 2 X36.972972 = 6,308.75 (513.12500-563.28921) 2 X35.555556=(-50.16421)2X35.555556= 89,473.71 (647.24599-563.28921)2X38.358976 = (+83.95678)2X38.358976=270,382.48 (539.32598-563.28921)2X39.804876= (-23.96323) 2 X39.804876= 22,857.41 389,022.35 Sum of squares due to sex: (52.20927) 2 X (number of diets) 2 0.42545542

1,946.27836

2

(52.20927) X 16 -=102,508.80 0.42545542

Sums of squares due to interaction between sex and diet: (93.23100-51.66229) 2 X9.243243 = (+41.56871)2X9.243243 = 15,971.93 (48.75000-51.66229) 2 X8.888889 = ( -2.91229) 2 X8.888889= 75.39 (39.03744-51.66229) 2 X9.589744=(-12.62485) 2 X9.589744= 1,528.48 2 2 (27.81862-51.66229) X9.951219 = (-23.84367) X9.951219= 5,657.47 23,233.27 Error sums of squares: The same as that obtained in making the primary analysis, i.e., 1,772,420.81.

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TABLE B.—Reciprocals of the numbers of chicks of each sex in each pen

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mately 0.4922, it is to be concluded that there are highly significant differences between the subclass mean live weights. For details regarding the z-test, see the second paper of this series (Titus and Harshaw, 1935). The number of degrees of freedom (d/f) for the subclasses is one less than the number of subclasses, i.e., (8-1) or 7; that for

several diets is used. This is the figure in the last row of the next to last column of Table A (work sheet No. 2). However, when there is no interaction, the corresponding weighted mean is used. This weighted mean of differences is the same as that used in calculating the "sum of squares" due to interaction in the present analysis.

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TABLE 6.—Further analysis of live weights: the effect of diet and sex

Source of variation

Sums of squares

Variances (or mean squares)

Standard deviations

Nat. log. of standard deviations

3 1

389,022.35 102,508.80

129,674.12 102,508.80

360.10 320.17

5.8864" 5.7689 b

3 145

23,233.27 1,772,420.81

7,744.42 12,223.59

88.02 110.56

4.4776° 4.7056

» zfor diet=5.8864-4.7056= 1.1808. b zforsex = 5.7689-4.7056= 1.0633. 0 zforinteraction=4.4776-4.7056= -0.2280. error is the difference between the total number of individual observations and the number of subclasses, i.e. (153-8) or 145; and the total number of degrees of freedom is one less than the total number of observations, i.e. (153-1) or 152. The primary analysis having shown that there are significant differences among the subclass means, a further analysis is made for the purpose of determining the effect of diet and sex on live weight. The first step in making this analysis is to tabulate (Table A) the several sub-class mean live weights, the differences in mean live weight between the two sexes on each diet, and the unweighted mean live weight of the males and females on each diet. A tabulation (Table B) is also made of the reciprocals of the numbers of males and females on each diet, the sums of these reciprocals for each diet, and the corresponding "weights" to be assigned to the differences between the mean weights of the two sexes as well as the "weights" to be assigned to the unweighted mean live weights of the chicks on the several diets. See work sheet No. 2. The second step is to calculate the weighted mean of means (males -f- females) and the weighted mean of differences (males — females) and from these calculate the "sum of squares" due to diet, and the "sum of squares" due to interaction between sex and diet. The "sum of squares" due to sex is also calculated. In calculating the latter it is to be observed that when there is interaction between sex and diet the unweighted mean of the differences between the mean live weight of the males and the mean live weight of females on the

The expression, "interaction between sex and diet," has the same meaning as the expression, "differential response of sex to diet." Thus, if the difference in growth response of the males and females is not the same (within the limits of experimental error) on the several diets, there is said to be an interaction between sex and diet. The third step is to tabulate (see Table 6) the "sums of squares" with their appropriate degrees of freedom, calculate the mean squares (or variances) and the standard deviations, insert the natural logarithms of the standard deviations, and compute Fisher's z for diet, sex, and interaction.

The number of degrees of freedom for diet is one less than the number of diets, i.e., (4-1) or 3; for sex it is one less than the number of sexes, i.e., (2-1) or 1; and for interaction between sex and diet it is the product of the values for sex and diet, i.e., (1 x 3 ) or 3. Since the 1 percent point of z, for nt = 3 and n2 = 145, lies between 0.6651 and 0.7085 and is approximately 0.6831, it is concluded that there are highly significant differences between the mean weights of the chicks on the diets. A similar conclusion is reached regarding differences between the two sexes on the same diet, since the 1 percent point of z, for n1 = \ and n2 = 145, lies

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Diet Sex Interaction (sex and diet) Error

d/f

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between 0.9462 and 0.9784 and is approximately 0.9S95. The mean square for interaction is less than that for error and, hence, it is concluded that there is no interaction between sex and diet. Although there is no interaction, it is unnecessary, in this example, to recalculate the "sum of squares" due to sex by using the weighted mean of differences, because the weighted mean is nearly 99 percent of the unweighted mean and, thus, the recalculated standard deviation of the weighted mean live weights of the two sexes would be 99 percent of that found for the unweighted mean live weights. It may be noted that, when added together, the total of the "sums of squares" in Table 6 is not the same as that given in Table 5. This is because there is not the same number of chicks in each subclass. In the present example, the difference between the two is less than 0.08 percent of the total

given in Table 5. In general as the numbers of observations in the subclasses approach equality, the difference between the two totals approaches zero. The standard errors of the mean live weights may now be calculated as in the tables at the top of this page. Since 145 degrees of freedom were used in calculating the mean square (and standard deviation), a difference between any two means must be at least 1.976 times its standard error to be significant (odds of 19 to 1) and 2.608 times its standard error to be highly significant (odds of 99 to 1). See Wallace and Snedecor (1931) Table 16. If it is desired, the Mest for significance may be made, in which case, for the reason stated above, t must have a value of 1.976 for odds of 19 to 1 and a value of 2.608 for odds of 99 to 1. The method of making the f-test is illustrated below by four examples.

Significance of difference between the mean live weights of the males and females on the same diet (for example, Pen 352): 596.84 — 503.61 t =

19 X 18

93.23

19 + 18

110.56

•

110.56

• X V9.243243 = 2.563*

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Pen 352: Standard error of mean live weight of males = ± V 12,223.59/19 = ±25.36 Standard error of mean live weight of females = ± V 12,223.59/18 = ±26.06 Standard error of unweighted mean live weight of males and females = ± V 12,223.59/36.972972 = ±18.18 Pen 353: Standard error of mean live weight of males = ±Vl2,223.59/16 = ±27.64 Standard error of mean live weight of females = ± V 12,223.59/20 = ±24.72 Standard error of unweighted mean live weight of males and females = ±Vl2,223.59/35.555556 ±18.54 Pen 356: Standard error of mean live weight of males = ± V 12,223.59/17 = ±26.81 Standard error of mean live weight of females : = ± V 12,223.59/22 = ±23.57 V 12,223.59/38.358976 = Standard error of unweighted mean live weight of males and females : ±17.85 Pen 357: Standard error of mean live weight of males = ±Vl2,223.59/17 = ±26.81 Standard error of mean live weight of females = ± Vl2,223.59/24 = ±22.57 Standard error of unweighted mean live weight of males and females = ±Vl2,223.59/39.804876 = ±17.52

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Significance of difference between the mean live weights of chicks of the same sex on any two different diets (for example, the males in pens 356 and 353): 666.76 — S37.S0 X16 129.26 t = • X 2.871 = 3.356t 110.56 + 16 110.56 Significance of difference between unweighted mean live weights of males and females on any two different diets (for example, the chicks in pens 356 and 353):

V17

647.25 — 513.13 *= •

/38.;

A/

= X 4.295 = 5.210t V 38.; 110.56 S.358976 + 35.555556 110.56 (males — females on any Significance of differences of the type (males — females on one diet) other diet) (for example, pens 352 and 357): 93.23 — 27.82

t

65.41

9.243243 + 9.951219

112.56

V-

X 2.189 = 1.295J

* Significant (odds of 19 to 1). t Highly significant (odds of 99 to 1). t Not significant (odds of about 4 to 1).

Analyses of the data on live weight at 12 weeks and the "degree" of perosis were made in exactly the same manner. The following differences attributable to sex and diet were found to be significant: The analysis of the live weight data Mean live weight at 8 weeks

it had no effect during the next four weeks. However, the advantage gained during the first eight weeks was maintained until the twelfth week. On the other hand, neither the addition of rice bran nor the substitution of ground crab shell for steamed bone meal

Mean live weight at 12 weeks

Mean "degree" of perosis

Comparisons of males and females M > F in Pen 352*

M > F in Pen 352t M > F in Pen 3S3t M > F in Pen 357*

M > F in Pen 356*

Comparisons of males on different diets Pen 356>Pen 353t Pen 356>Pen 357f

Pen 352>Pen 353f Pen352>Pen357*

Pen 353>Pen 352f Pen 353>Pen 357t

Comparisons of females on different diets Pen 356>Pen 352t Pen 356>Pen 353t Pen 356>Pen 3S7f

Pen 352>Pen 353* Pen 356>Pe'n 353f Pen 357>Pen353*

Pen 353>Pen 3S2f Pen 353>Pen 356t Pen 353>Pen 357f

Comparisons of chicks (males and females) on different diets Pen 356>Pen 352f Pen 356>Pen 353t Pen 356>Pen 357t

Pen 352>Pen 353t Pen 356>Pen 353f Pen 357>Pen 353* Pen352>Pen357f

Pen 353>Pen 352f Pen 353>Pen 356t Pen 353>Pen 357f

•Significant (odds of 19 to 1). t Highly significant (odds of 99 to 1)

shows that although the addition of crab scrap meal to the basal diet had a marked effect on growth during the first eight weeks,

had any statistically significant effect on growth during the first eight weeks, although both of these modifications of the

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110.56

9.243243 X 9.951219

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DISCUSSION The method of "weighted squares of means" is of wide application and may be used for analyzing data on growth, bone ash, perosis, egg weight, and pellagra, and by suitable application of the principle of regression (covariance analysis) may be used in the case of data on egg production and hatchability. The data presented in the present paper clearly show the inadequacy of methods of "correcting" the average growth rate of chicks on a given series of diets by multiplying the weight of the females by some previously determined factor (Ackerson and Mussehl, 1930). In the present case, at eight weeks, the ratios of the average live weights of the males to those of the females were 1.19, 1.10, 1.06, and 1.0S, respectively, for the chicks in pens 352, 353, 356, and 357. The ratio of the unweighted mean live weight of the males to that of the females, in all four pens, was 1.10. Had this factor been used, it is clear that some of the differences in growth response among the pens would have appeared to be either greater or less than they really were. In any case, a certain amount of artificiality would have been introduced into the analysis with the result that the usual tests of significance would not have been applicable. The method of variance analysis, in addition to giving a numerical measure of the reliance that may be placed in the observed differences in growth response on a series of diets, enables one to determine whether or not there was a differential response of sex to diet. In the present example, although there were both large and small differences between the average live weights (at both eight and twelve weeks) of the males and females, the statistical analysis clearly showed that there was no significant

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basal diet had a distinctly beneficial effect during the whole of the twelve-week period studied. When differences in growth are reported on different diets it is necessary to state the age at which the differences become apparent, since a diet which stimulates early growth (i.e., growth between hatching and the point of inflection of the growth curve) may not continue to have a stimulating effect. This was demonstrated several years ago by Titus and Jull (1928). From the analysis of the data on perosis it is to be concluded that all the modifications of the perosis-producing diet had a beneficial effect in that they tended to reduce the "degree" of perosis. The effect of the addition of 10 percent of rice bran was the most pronounced. In the case of the addition of the same quantity of crab scrap meal there was only a slight reduction of perosis among the males although the reduction among the females was very marked. In spite of the fact that sex seemed to have some effect on the degree of perosis, this effect was not consistent on all four diets. Furthermore, the interaction between sex and diet was found not to be statistically significant. The substitution of ground crab shell also reduced the "degree" of perosis to a marked extent. These results are in agreement with those previously reported by Titus (1932). In view of some of the current theories regarding the etiology of perosis, it may be well to call attention to the fact that although rice bran is rich in phosphorus, the addition of 10 percent was just as effective, if not more effective, in reducing the "degree" of perosis than was the reduction of the phosphorus content of the diet by the substitution of ground crab shell (largely calcium carbonate) for the special steamed bone meal.

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differential response of sex to diet (i.e., no interaction between sex and diet).

Fisher, F. E., 1932. Statistical Methods for Research Workers, 4th ed. Oliver and Boyd, Edinburgh and London. Hendricks, W. A., 1935. Analysis of variance considered as an application of simple error theory. (In press.) Ann. Math. Stat. Titus, H. W., 1932. Perosis, or deforming leg weakness, in the chicken. Poul. Sci. 11:117-125. , and M. A. Jull, 1928. The growth of Rhode Island Reds and the effect of feeding skimmilk on the constants of their growth curves. Jour. Agr. Res. 36:515-540. , and H. M. Harshaw, 1935. An adaptation of the "randomized block" to nutrition experiments. Poul. Sci. 14:3-15. Wallace, H. A., and G. W. Snedecor, 1931. Correlation and machine calculation. Iowa State College of Agr. and Mech. Arts Official Pub. 30 (4), pp. 71. Yates, F., 1933. The principles of orthognality and confounding in replicated experiments. Jour. Agr. Sci. 23:108-145. , 1934. The analysis of multiple classifications with unequal numbers in the different classes. Jour. Amer. Stat. Assn. 29 :51-66.

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SUMMARY A method of analyzing the data of chick nutrition experiments, known as the method of "weighted squares of means," is presented. By means of this method it is possible to obtain accurate measures of the reliance that may be placed in observed experimental differences in growth, bone ash, perosis, egg weight, pellagra and other factors. By suitable application of the principle of regression this method may be applied to data on egg production and hatchability. REFERENCES Ackerson, C. W., and F. E. Mussehl, 1930. Sex differences in the normal growth rate of chicks. Jour. Agr. Res. 40:863-866. Brandt, A. E., 1933. The analysis of variance in a "2 x s" table with disproportionate frequencies. Jour. Amer. Stat. Assn. 28:164-173.

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XIV, No. 3

MAY,