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A Statistical Analysis of Some Experiments on Slipped Tendon*
A Statistical Analysis of Some Experiments on Slipped Tendon* T. T. MlLBY Iowa State College, Ames (Received for Publication December 23, 1932)
METHOD OF ANALYSIS
The data here analyzed are taken from the work of Hunter and Funk (1931), Payne, Hughes, and Leinhardt (1932), Titus (1932), and Mussehl, Ackerson, and Blish (1927). The work of the latter was not on slipped tendon, but a condition of "typical rickets" which developed itf one lot that was irradiated is described. The description of the malformation indicates that it might represent the condition known as slipped tendon. Four other lots in this experiment are also used in the analysis, in which no abnormal condition of this nature is reported. All experiments pertaining to slipped tendon were used in which sufficient data were given to enable the author to calculate the protein and mineral content of the rations and the percentages of slipped tendon. The percentage of protein, ash,
calcium, and phosphorus were calculated from average analyses given by Henry and Morrison (1923), Forbes, Beegle, and Mensching (1913), and Sherman (1928), when they were not given by the author. The numbers of slipped tendon in the various lots were those reported by the several investigators. It is possible that the criteria used by the various workers may not have included the actual slipping of the tendon in all cases. The material on which this analysis is based consists of 65 lots of chicks. The lots ranged in size from 20 to 40 chicks each. The amount of data is large enough to make practical the use of correlation in analysis. The analysis of the data was carried out according to the method of multiple correlation given by Wallace and Snedecor (1931). The variables studied were protein (A), ash (B), calcium (C), phosphorus(D), and the percentage of slipped tendon (X). The latter is the dependent variable. For reasons to be indicated later only the simple regression equation for estimating X from phosphorus was calculated, and the estimated value of X was calculated for each lot. The results of this study indicated that little additional information of value would be obtained from further analysis by other methods. RESULTS AND DISCUSSION
* Journal Paper No. 87 of the Iowa Agricultural Experiment Station. Ames, Iowa. Project No. S3. The author is indebted to Professors E. W. Henderson, G. W. Snedecor, and to A. E. Brandt for suggestions.
Table 1 shows means and standard deviations of the original items and the simple correlations between the variables. From these the multiple correlation coefficient is
[352]
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
ANY suggestions have been made by various workers concerning the cause of a malformation of the leg bones of chicks commonly known as slipped tendon. Several conflicting opinions have been presented. The writer has attempted a statistical analysis of the literature of the field, in order to ascertain whether or not there is any constant relationship between certain constituents of the ration and the incidence of slipped tendon.
M
NOVEMBER,
1933.
VOL. XII,
No.
6
353
TABLE 1.—Means and standard deviations of variables in percent and the simple correlations Protein-A Mean. 18.134 St. Dev. 1.452 Protein-A Ash-B Calcium-C Phosphorus-D
Ash-B
Calcium-C
Phosphorus-D
Slipped tendons-X
7.150 2.264 .2291
1.685 0.770 .2494 .8661f
0.964 0.280 -.0788 .4876f .3504*
33.080 29.064 -.0389 .0941 .0418 .6303t
* Significant. t Highly significant.
Lot
Actual % s . t.
Est. % s. t.
Errors of est.
Lot
Actual %s.t
50.0 0.0 0.0 45.5 0.0 95.8 81.5 90.0 60.8 28.6 25.0 17.4 0.0 81.8 73.9 0.0
30.0 30.0 30.0 30.0 30.0 55.1 39.8 55 1 55.1 24.5 23.1 17.6 11.6 47.8 34.9 0.0
+20.0 -30.0 -30.0 + 15.5 -30.0 +40.7 +41.7 +34.9 +5.1 +2.3 +1.9 -0.2 -11.6 +34.0 +39.0 -0.0
E xperiment 2—Payne et al. ( L932) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
60.0 30.0 0.0 10.0 16.7 0.0 10.0 3.3 63.3 10.0 3.3 26.7 86.7 43.3 76.7 0.0 56.7 . 10.0
30.8 48.9 27.6 16.5 40.6 3.9 22.3 3.9 53.1 25.6 20.4 37.0 69.4 42.6 72.7 37.4 53.1 25.0
Errors of est.
Experiment 2 (continued)
Experiment 1 Hunter & Funk (1931) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Est. %s.t.
+29.2 -18.9 -27.6 -6.5 -23.9 -3.9 -12.3 -0.6 + 10.2 -15.6 -17.1 -10.3 +17.3 +0.7 +4.0 -37.4 +3.6 -15.0
31 32 33 34 35
70.0 53.3 0.0 13.3 16.7
76.0 62.9 7.4 25.7 30.9
-6.0 -9.6 -7.4 -12.3 -14.2
Experiment 3—Mussehl et al. (1927) 2 3 4 5
0.0 90.5 0.0 0.0
46.9 92.6 38.8 38.8
-46.9 -2.1 -38.8 -38.8
Experiment 4—Titus (193 2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
50.0 40.0 12.5 0.0 62.5 70.0 30.0 20.0 25.0 17.5 25.0 12.5 46.9 12.5 46.9 56.2 25.0 37.5 0.0 53.1 50.0 56.2
30.2 2.8 34.6 9.9 30.2 31.5 32.3 2.8 32.8 33.5 34.2 34.6 30.2 34.6 30.2 30.2 34.2 2.8 9.9 30.2 39.4 30.2
+ 19.8 +37.2 -22.1 -9.9 +32.3 +38.5 -2.3 +17.2 -7.8 -16.0 -9.2 -22.1 + 16.7 -22.1 + 16.7 +26.0 -9.6 +34.7 -9.9 +22.9 +10.6 +26.0
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
TABLE 2.—The errors of estimate
3S4
POULTRY
SCIENCE
TABLE 3.—Analysis of variation by means of regression Degrees of freedom
Sum of squares
Mean square or variance
Due to regression Not accounted for
1 63
21479.49 32584.16
21479.49 517.21
22.742
Total
64
54063.65
844.74
29.064
Source of variation
of reduction of the variability of X by taking into account its correlation with D, the amount of phosphorus in the feed. This analysis is made according to the method of Fisher (1930). By taking into account the correlation between D and X, the standard deviation of X, 29.06 percent has been reduced to the standard error of estimate, 22.74. This is a reduction of 21.8 percent. By making allowance for the varying percentages of phosphorus we have accounted for a share of the variation in the incidence of slipped tendon but much of this variation is still unaccounted for. It is impossible to put a numerical value on the variation due to type of floor, or other factors, and it is studied only in a general way. No doubt much of this unaccounted-for variation is due to other factors in the ration or management about which we know little. Some of the possible factors which have been mentioned by various workers are brooding temperature, amount of exercise, and variations in susceptibility of the chicks. No doubt there may be others. Titus (1932) states that Rhode Island Reds are more susceptible than White Leghorns, and Payne, Hughes, and Leinhardt (1932) give data from their general flock which confirm this. SUMMARY
A statistical study has been made of 65 lots of chicks from various experiments in which slipped tendon has been reported. Correlations were calculated between protein, ash, calcium, phosphorus, and the percentage of slipped tendon. It has been
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
calculated to be .6803. This is a highly significant correlation, but it is only slightly larger than the simple correlation coefficient between D and X. Of the simple correlations between the independent variables and X, only that between D and X is significant. Taking into account all four of the independent variables does not improve our estimate of X to any extent over the estimate given by phosphorus alone. For this reason the multiple regression equation was not calculated. The simple regression equation of slipped tendon on phosphorus was found to be X (estimated value of X) equals 65.348 D minus 29.88. By substituting the values of percentages of phosphorus in the original data it is possible to calculate the. estimated percentages of slipped tendon and find the errors of estimate. A study of these errors of estimate is interesting. What has really been done is to remove the effect of phosphorus on percentages of slipped tendon. The errors of estimate are due to other factors which have not been taken into account. These errors of estimate are given in Table 2. Comparison of the rations and other variables in those lots in which the errors are large may give some hint as to the contributing causes of these errors. Without going into detail, it may be said that many of these errors confirm the author's conclusions concerning factors not included in this analysis. Some of the errors may possibly be traced to the size of the sample. Others cannot be explained at all in the light of our present knowledge. Results in Table 3 indicate the degree
Standard deviation
NOVEMBER,
1933.
VOL.
XII,
CONCLUSIONS
1. The correlations between protein, ash, calcium, and percentage of slipped tendon were not significant in the data studied. 2. The amount of phosphorus in the ration is positively correlated with the percentage of slipped tendon. The value of this correlation is .6303, which is highly significant. 3. The multiple correlation is not sufficiently larger than the simple correlation between phosphorus and slipped tendon to warrant its use in the place of the latter.
355
6
The amount of phosphorus in the ration gives practically as good an estimate of the amount of slipped tendon as do protein, ash, calcium, and phosphorus combined. 4. Taking into account the variation in percentage of slipped tendon due to correlation lowers the standard deviation of the percentage of slipped tendon 29.06, to the standard error of estimate, 22.74, a reduction of 21.8 percent. 5. These results do not justify the conclusion that phosphorus is the chief causative factor of slipped tendon, but merely indicate that high percentages of slipped tendon are associated with high phosphorus in the ration. REFERENCES
Fisher, R. A., 1930. Statistical methods for research workers. London, Oliver and Boyd. Forbes, E. B., F. M. Beegle, and J. E. Mensching, 1913. Mineral and organic analyses of foods. Ohio Agr. Exp. Sta. Bui. 255:211-231. Henry, W. A. and F. B. Morrison, 1923. Feeds and Feeding. 18th ed. p. 709-721. Henry-Morrison Company. Hunter, J. E. and E. M. Funk, 1931. The production of slipped tendons in chicks by experimental feeding. Proc. 22nd Annual Meet Poul. Sci. Assoc. 45-47. Mussehl, F. E., C. W. Ackerson and M. J. Blish, 1927. The mineral metabolism of the growing chick. Poul. Sci. 6:239-242. Payne, L. F., J. S. Hughes, and H. F. Leinhardt, 1932. The etiological factors involved in the malformation of bones in young chickens. Poul. Sci., 11:158-165. Sherman, H. C. 1928. Chemistry of Food and Nutrition. 3rd ed. p. 587-593. The Macmillan Company. Titus, H. W., 1932. Perosis, or deforming leg weakness in the chicken. Poul. Sci. 11:117-125. Wallace, H. A. and G. W. Snedecor, 1931. Correlation and machine calculation. Iowa State College.
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
found that in the sample studied the only significant correlation is between phosphorus and slipped tendon. Further, it has been found that the multiple correlation is not sufficiently larger than the simple correlation between phosphorus and slipped tendon to warrant its use. The data indicate that phosphorus is the only factor of those studied which has any significant relationship with slipped tendon. A study of the errors of estimate was made in an attempt to explain the errors which were large. Most of these errors were due to special treatment to which the chicks were subjected by the experimenter. These special treatments were not capable of being subjected to statistical treatment by the method used. On the whole, these errors of estimate confirmed the conclusions of the authors. In several cases large errors could not be explained. Further study is needed before it can be definitely stated whether the relationship between phosphorus and slipped tendon is a causal one or not. However, it may be said that one cause of slipped tendon is either phosphorus or something in the feed which is very closely associated with phosphorus.
No.
METHOD OF ANALYSIS
The data here analyzed are taken from the work of Hunter and Funk (1931), Payne, Hughes, and Leinhardt (1932), Titus (1932), and Mussehl, Ackerson, and Blish (1927). The work of the latter was not on slipped tendon, but a condition of "typical rickets" which developed itf one lot that was irradiated is described. The description of the malformation indicates that it might represent the condition known as slipped tendon. Four other lots in this experiment are also used in the analysis, in which no abnormal condition of this nature is reported. All experiments pertaining to slipped tendon were used in which sufficient data were given to enable the author to calculate the protein and mineral content of the rations and the percentages of slipped tendon. The percentage of protein, ash,
calcium, and phosphorus were calculated from average analyses given by Henry and Morrison (1923), Forbes, Beegle, and Mensching (1913), and Sherman (1928), when they were not given by the author. The numbers of slipped tendon in the various lots were those reported by the several investigators. It is possible that the criteria used by the various workers may not have included the actual slipping of the tendon in all cases. The material on which this analysis is based consists of 65 lots of chicks. The lots ranged in size from 20 to 40 chicks each. The amount of data is large enough to make practical the use of correlation in analysis. The analysis of the data was carried out according to the method of multiple correlation given by Wallace and Snedecor (1931). The variables studied were protein (A), ash (B), calcium (C), phosphorus(D), and the percentage of slipped tendon (X). The latter is the dependent variable. For reasons to be indicated later only the simple regression equation for estimating X from phosphorus was calculated, and the estimated value of X was calculated for each lot. The results of this study indicated that little additional information of value would be obtained from further analysis by other methods. RESULTS AND DISCUSSION
* Journal Paper No. 87 of the Iowa Agricultural Experiment Station. Ames, Iowa. Project No. S3. The author is indebted to Professors E. W. Henderson, G. W. Snedecor, and to A. E. Brandt for suggestions.
Table 1 shows means and standard deviations of the original items and the simple correlations between the variables. From these the multiple correlation coefficient is
[352]
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
ANY suggestions have been made by various workers concerning the cause of a malformation of the leg bones of chicks commonly known as slipped tendon. Several conflicting opinions have been presented. The writer has attempted a statistical analysis of the literature of the field, in order to ascertain whether or not there is any constant relationship between certain constituents of the ration and the incidence of slipped tendon.
M
NOVEMBER,
1933.
VOL. XII,
No.
6
353
TABLE 1.—Means and standard deviations of variables in percent and the simple correlations Protein-A Mean. 18.134 St. Dev. 1.452 Protein-A Ash-B Calcium-C Phosphorus-D
Ash-B
Calcium-C
Phosphorus-D
Slipped tendons-X
7.150 2.264 .2291
1.685 0.770 .2494 .8661f
0.964 0.280 -.0788 .4876f .3504*
33.080 29.064 -.0389 .0941 .0418 .6303t
* Significant. t Highly significant.
Lot
Actual % s . t.
Est. % s. t.
Errors of est.
Lot
Actual %s.t
50.0 0.0 0.0 45.5 0.0 95.8 81.5 90.0 60.8 28.6 25.0 17.4 0.0 81.8 73.9 0.0
30.0 30.0 30.0 30.0 30.0 55.1 39.8 55 1 55.1 24.5 23.1 17.6 11.6 47.8 34.9 0.0
+20.0 -30.0 -30.0 + 15.5 -30.0 +40.7 +41.7 +34.9 +5.1 +2.3 +1.9 -0.2 -11.6 +34.0 +39.0 -0.0
E xperiment 2—Payne et al. ( L932) 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
60.0 30.0 0.0 10.0 16.7 0.0 10.0 3.3 63.3 10.0 3.3 26.7 86.7 43.3 76.7 0.0 56.7 . 10.0
30.8 48.9 27.6 16.5 40.6 3.9 22.3 3.9 53.1 25.6 20.4 37.0 69.4 42.6 72.7 37.4 53.1 25.0
Errors of est.
Experiment 2 (continued)
Experiment 1 Hunter & Funk (1931) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Est. %s.t.
+29.2 -18.9 -27.6 -6.5 -23.9 -3.9 -12.3 -0.6 + 10.2 -15.6 -17.1 -10.3 +17.3 +0.7 +4.0 -37.4 +3.6 -15.0
31 32 33 34 35
70.0 53.3 0.0 13.3 16.7
76.0 62.9 7.4 25.7 30.9
-6.0 -9.6 -7.4 -12.3 -14.2
Experiment 3—Mussehl et al. (1927) 2 3 4 5
0.0 90.5 0.0 0.0
46.9 92.6 38.8 38.8
-46.9 -2.1 -38.8 -38.8
Experiment 4—Titus (193 2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
50.0 40.0 12.5 0.0 62.5 70.0 30.0 20.0 25.0 17.5 25.0 12.5 46.9 12.5 46.9 56.2 25.0 37.5 0.0 53.1 50.0 56.2
30.2 2.8 34.6 9.9 30.2 31.5 32.3 2.8 32.8 33.5 34.2 34.6 30.2 34.6 30.2 30.2 34.2 2.8 9.9 30.2 39.4 30.2
+ 19.8 +37.2 -22.1 -9.9 +32.3 +38.5 -2.3 +17.2 -7.8 -16.0 -9.2 -22.1 + 16.7 -22.1 + 16.7 +26.0 -9.6 +34.7 -9.9 +22.9 +10.6 +26.0
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
TABLE 2.—The errors of estimate
3S4
POULTRY
SCIENCE
TABLE 3.—Analysis of variation by means of regression Degrees of freedom
Sum of squares
Mean square or variance
Due to regression Not accounted for
1 63
21479.49 32584.16
21479.49 517.21
22.742
Total
64
54063.65
844.74
29.064
Source of variation
of reduction of the variability of X by taking into account its correlation with D, the amount of phosphorus in the feed. This analysis is made according to the method of Fisher (1930). By taking into account the correlation between D and X, the standard deviation of X, 29.06 percent has been reduced to the standard error of estimate, 22.74. This is a reduction of 21.8 percent. By making allowance for the varying percentages of phosphorus we have accounted for a share of the variation in the incidence of slipped tendon but much of this variation is still unaccounted for. It is impossible to put a numerical value on the variation due to type of floor, or other factors, and it is studied only in a general way. No doubt much of this unaccounted-for variation is due to other factors in the ration or management about which we know little. Some of the possible factors which have been mentioned by various workers are brooding temperature, amount of exercise, and variations in susceptibility of the chicks. No doubt there may be others. Titus (1932) states that Rhode Island Reds are more susceptible than White Leghorns, and Payne, Hughes, and Leinhardt (1932) give data from their general flock which confirm this. SUMMARY
A statistical study has been made of 65 lots of chicks from various experiments in which slipped tendon has been reported. Correlations were calculated between protein, ash, calcium, phosphorus, and the percentage of slipped tendon. It has been
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
calculated to be .6803. This is a highly significant correlation, but it is only slightly larger than the simple correlation coefficient between D and X. Of the simple correlations between the independent variables and X, only that between D and X is significant. Taking into account all four of the independent variables does not improve our estimate of X to any extent over the estimate given by phosphorus alone. For this reason the multiple regression equation was not calculated. The simple regression equation of slipped tendon on phosphorus was found to be X (estimated value of X) equals 65.348 D minus 29.88. By substituting the values of percentages of phosphorus in the original data it is possible to calculate the. estimated percentages of slipped tendon and find the errors of estimate. A study of these errors of estimate is interesting. What has really been done is to remove the effect of phosphorus on percentages of slipped tendon. The errors of estimate are due to other factors which have not been taken into account. These errors of estimate are given in Table 2. Comparison of the rations and other variables in those lots in which the errors are large may give some hint as to the contributing causes of these errors. Without going into detail, it may be said that many of these errors confirm the author's conclusions concerning factors not included in this analysis. Some of the errors may possibly be traced to the size of the sample. Others cannot be explained at all in the light of our present knowledge. Results in Table 3 indicate the degree
Standard deviation
NOVEMBER,
1933.
VOL.
XII,
CONCLUSIONS
1. The correlations between protein, ash, calcium, and percentage of slipped tendon were not significant in the data studied. 2. The amount of phosphorus in the ration is positively correlated with the percentage of slipped tendon. The value of this correlation is .6303, which is highly significant. 3. The multiple correlation is not sufficiently larger than the simple correlation between phosphorus and slipped tendon to warrant its use in the place of the latter.
355
6
The amount of phosphorus in the ration gives practically as good an estimate of the amount of slipped tendon as do protein, ash, calcium, and phosphorus combined. 4. Taking into account the variation in percentage of slipped tendon due to correlation lowers the standard deviation of the percentage of slipped tendon 29.06, to the standard error of estimate, 22.74, a reduction of 21.8 percent. 5. These results do not justify the conclusion that phosphorus is the chief causative factor of slipped tendon, but merely indicate that high percentages of slipped tendon are associated with high phosphorus in the ration. REFERENCES
Fisher, R. A., 1930. Statistical methods for research workers. London, Oliver and Boyd. Forbes, E. B., F. M. Beegle, and J. E. Mensching, 1913. Mineral and organic analyses of foods. Ohio Agr. Exp. Sta. Bui. 255:211-231. Henry, W. A. and F. B. Morrison, 1923. Feeds and Feeding. 18th ed. p. 709-721. Henry-Morrison Company. Hunter, J. E. and E. M. Funk, 1931. The production of slipped tendons in chicks by experimental feeding. Proc. 22nd Annual Meet Poul. Sci. Assoc. 45-47. Mussehl, F. E., C. W. Ackerson and M. J. Blish, 1927. The mineral metabolism of the growing chick. Poul. Sci. 6:239-242. Payne, L. F., J. S. Hughes, and H. F. Leinhardt, 1932. The etiological factors involved in the malformation of bones in young chickens. Poul. Sci., 11:158-165. Sherman, H. C. 1928. Chemistry of Food and Nutrition. 3rd ed. p. 587-593. The Macmillan Company. Titus, H. W., 1932. Perosis, or deforming leg weakness in the chicken. Poul. Sci. 11:117-125. Wallace, H. A. and G. W. Snedecor, 1931. Correlation and machine calculation. Iowa State College.
Downloaded from http://ps.oxfordjournals.org/ at Florida Atlantic University on June 5, 2016
found that in the sample studied the only significant correlation is between phosphorus and slipped tendon. Further, it has been found that the multiple correlation is not sufficiently larger than the simple correlation between phosphorus and slipped tendon to warrant its use. The data indicate that phosphorus is the only factor of those studied which has any significant relationship with slipped tendon. A study of the errors of estimate was made in an attempt to explain the errors which were large. Most of these errors were due to special treatment to which the chicks were subjected by the experimenter. These special treatments were not capable of being subjected to statistical treatment by the method used. On the whole, these errors of estimate confirmed the conclusions of the authors. In several cases large errors could not be explained. Further study is needed before it can be definitely stated whether the relationship between phosphorus and slipped tendon is a causal one or not. However, it may be said that one cause of slipped tendon is either phosphorus or something in the feed which is very closely associated with phosphorus.
No.