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Bone Strength Statistical Distribution Functions for Broilers
RESEARCH NOTES Bone Strength Statistical Distribution Functions for Broilers JOSEPH P. HARNER, III Department of Agricultural Engineering, Kansas State University, Manhattan, Kansas 66506 JAMES H.WILSON Department of Agricultural Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (Received for publication June 11, 1984)
1985 Poultry Science 64:585-587 INTRODUCTION
The statistical analysis of data can involve parametric or nonparametric statistics. Parametric statistics involve a known distribution and the ability to define the continuous function. For nonparametric statistics, the underlying distribution of the data set is not easily specified. This form of analysis compares distribution rather than the parameters describing the continuous function. Nonparametric statistics have advantages: they are quick and easy to learn and apply, work of collecting data is reduced, sampling procedures may include several populations with very little assumed, and probability statements are not qualified as severely as with parametric procedures. However, if the form of the parent population is reasonably close to the distribution for normal theory then nonparametric procedures should not be used (Steel and Torrie, 1960). A review of the literature gave no indication as to the type of expected distribution of the mechanical properties of poultry bone. This study was undertaken to define the distribution patterns for the mechanical properies of bones from broilers.
MATERIALS AND METHODS
Forty-three femur and fifty tibia bones from Hubbard broilers from the same flock were tested to define the form of the distribution function. The two types of tests used to mea-
sure the breaking loads of the bones were the shear and three-point bending tests. The shear tests were performed on 43 femurs and 25 tibias using a double shear block apparatus (Wilson and Baker, 1979). The shear force was exerted over a 12.7-mm section located at the center of the shaft. The test position of each bone was such that the thinnest axis of the cross-section perpendicular to the diaphysis was parallel to the direction of loading. These tests resulted in the ultimate shear force and shear stress being evaluated for each bone. The three-point bending test was selected because fractures were related to this loading. Twenty-five tibias were tested. The ultimate bending force, bending stress, and flexural modulus of elasticity were determined using the bending test. The adjustable fulcra points of the three-point bending apparatus permitted different lengths of tibia bones to be tested in bending and maintained a length to diameter ratio near 10. Three-point bending tests were not performed on the femur because the length to diameter ratio (L/D) was less than five and a three point bending test gives erroneous results when L/D is less than 10. The normal, log-normal, and three parameter Weibull distributions (Ang and Tang, 1975; Hann, 1977) were fitted to each data set. The data was plotted and analyzed using the Graphic Data Analysis Computer Package developed by Admin and Woeste (1983).
585
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
ABSTRACT Shear and three-point bending tests were conducted on the femur and tibia bones from Hubbard broilers to establish the type of statistical distribution function that best fits bone strength data. In the past it was assumed that these types of data followed a normal distribution. The results of this investigation indicate that a log-normal function best describes the mechanical properties of the tibia and a normal function describes the properties of the femur. Overall the results of the distribution analysis indicate that the mechanical properties of bone tend to follow a log-normal distribution. (Key words: bone strength, normal, log-normal, broilers)
HARNER, III, AND WILSON
586
TABLE 1. Chi-square values for the distributions fitted to the data sets of the mechanical properties of the femur and tibia from broilers Distribution type
Mechanical property average value
Weibull
Critical chi-square value1
4.33
3.02
5.78
3.84
1.84
.94
1.29
3.84
6.38
3.77
4.13
3.84
11.78
4.92
5.20
3.84
2.29
.31
3.39
3.84
1.71
2.08
1.754
7.82
5.37
6.10
5.75
5.99
Normal
Modulus of elasticity 2 (8.4 GPa) Ultimate bending stress2 (62 MPa) Ultimate bending force2 (316 N) Ultimate shear stress2 (15 MPa) Ultimate shear force 2 (1010 N) Ultimate shear stress3 (5.4 MPa) Ultimate shear force 3 (1112N) 1
Statistical level P<.05.
2
Sample size is 25 tibiae.
3
Sample size is 43 femora.
"Critical value is 5.99.
The mechanical properties of the bones tested are summarized in Table 1. The observed and critical chi-square values for the seven sets of results are also given in Table 1. Figures 1 and 2 show the distribution curves for the
ultimate shear force and stress, respectively, of the femur. The normal functions described the data from the shear test of the femur better than the log normal or three parameter Weibull function. The distribution of ultimate shear stress of the femur could be modeled using any
Distribution Type Normal Log Noirmiil ,; WeibulT T '/
Distribution Tyje Log Normal Weibull
.08
0
5000
1000
1500
2000
Ultimate Shear Force (N)
FIG. 1. Distribution curves of the ultimate shear force of the femur.
5.0
Ultimate Shear Stress (MPa) FIG. 2. Distribution curves of the ultimate shear stress of the femur.
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
Lognormal
RESEARCH NOTE
587
Distribution Type Normal Log Normal Weibull
D i s t r i b u t i o n Type Log-Normal Weibull
625
1250
1875
2500
Ultimate Shear Force (N)
FIG. 3. Distribution curves of t h e u l t i m a t e shear force of the tibia.
0
10
20
U l t i m a t e Shear S t r e s s
30
40
(MPa)
FIG. 4. Distribution curves of the u l t i m a t e shear stress of t h e tibia.
one of the three curves; however, the normal curve had a better fit. Figures 3 and 4 illustrate the density functions of the shear test results of the tibia. The ultimate shear force was described by the log-normal density function even though the other two distributions could not be rejected at p<.05 level (Table 1). None of the distributions fit the ultimate shear stress of the tibia. However, the chi-square of the log normal curves was closer to the critical chi-square value. The distribution of the ultimate bending force, bending stress, and modulus of elasticity were described by the log-normal distribution. The other two distributions were rejected at the P<.05 level. DISCUSSION The three-parameter Weibull distribution was not selected to model any of the distributions of the mechanical properties. Of the seven sets of data, the log normal distribution had the better fit in four of the sets of data. Another two sets of data could have been modeled using the log-normal distribution. The log-normal function described the properties of the tibia,
and the properties of the femur followed a normal distribution. The results of the distribution analysis indicated the mechanical properties of bone tend to follow a log-normal distribution rather than a normal distribution that has been assumed in past research. Because of the nonnormal distribution, the actual level of significance in an analysis of variance may increase from P<.05 to P<.07 or P<.08 (Page 129, Steel and Torrie, 1960).
REFERENCES A d m i n , L., and F. E. Woeste, 1 9 8 3 . Graphics D a t a Analysis C o m p u t e r Package. Agric. Eng. D e p t . Virginia Tech. Blacksburg, VA. Ang, A.H.S., and W. H. Tang, 1 9 7 5 . Probability Concepts in Engineering Planning and Design. J o h n Wiley and Sons, N e w York, NY. Hann, C. T., 1 9 7 7 . Statistics M e t h o d s in H y d r o l o g y . Iowa State Univ. Press, Ames, IA. Steel, R.G.D., and J. H. Torrie, 1 9 6 0 . Principles and Procedures of Statistics. McGraw-Hill B o o k C o m p a n y , N e w York, NY. Wilson, J. H., and J. L. Baker, 1 9 7 9 . D e t e r m i n a t i o n of t h e shear m o d u l u s in long shaft b o n e t o acc o m p a n y an elastic m o d u l u s . A m . Soc. Agric. Eng. Paper N o . 7 9 - 5 5 2 1 .
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
0
1985 Poultry Science 64:585-587 INTRODUCTION
The statistical analysis of data can involve parametric or nonparametric statistics. Parametric statistics involve a known distribution and the ability to define the continuous function. For nonparametric statistics, the underlying distribution of the data set is not easily specified. This form of analysis compares distribution rather than the parameters describing the continuous function. Nonparametric statistics have advantages: they are quick and easy to learn and apply, work of collecting data is reduced, sampling procedures may include several populations with very little assumed, and probability statements are not qualified as severely as with parametric procedures. However, if the form of the parent population is reasonably close to the distribution for normal theory then nonparametric procedures should not be used (Steel and Torrie, 1960). A review of the literature gave no indication as to the type of expected distribution of the mechanical properties of poultry bone. This study was undertaken to define the distribution patterns for the mechanical properies of bones from broilers.
MATERIALS AND METHODS
Forty-three femur and fifty tibia bones from Hubbard broilers from the same flock were tested to define the form of the distribution function. The two types of tests used to mea-
sure the breaking loads of the bones were the shear and three-point bending tests. The shear tests were performed on 43 femurs and 25 tibias using a double shear block apparatus (Wilson and Baker, 1979). The shear force was exerted over a 12.7-mm section located at the center of the shaft. The test position of each bone was such that the thinnest axis of the cross-section perpendicular to the diaphysis was parallel to the direction of loading. These tests resulted in the ultimate shear force and shear stress being evaluated for each bone. The three-point bending test was selected because fractures were related to this loading. Twenty-five tibias were tested. The ultimate bending force, bending stress, and flexural modulus of elasticity were determined using the bending test. The adjustable fulcra points of the three-point bending apparatus permitted different lengths of tibia bones to be tested in bending and maintained a length to diameter ratio near 10. Three-point bending tests were not performed on the femur because the length to diameter ratio (L/D) was less than five and a three point bending test gives erroneous results when L/D is less than 10. The normal, log-normal, and three parameter Weibull distributions (Ang and Tang, 1975; Hann, 1977) were fitted to each data set. The data was plotted and analyzed using the Graphic Data Analysis Computer Package developed by Admin and Woeste (1983).
585
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
ABSTRACT Shear and three-point bending tests were conducted on the femur and tibia bones from Hubbard broilers to establish the type of statistical distribution function that best fits bone strength data. In the past it was assumed that these types of data followed a normal distribution. The results of this investigation indicate that a log-normal function best describes the mechanical properties of the tibia and a normal function describes the properties of the femur. Overall the results of the distribution analysis indicate that the mechanical properties of bone tend to follow a log-normal distribution. (Key words: bone strength, normal, log-normal, broilers)
HARNER, III, AND WILSON
586
TABLE 1. Chi-square values for the distributions fitted to the data sets of the mechanical properties of the femur and tibia from broilers Distribution type
Mechanical property average value
Weibull
Critical chi-square value1
4.33
3.02
5.78
3.84
1.84
.94
1.29
3.84
6.38
3.77
4.13
3.84
11.78
4.92
5.20
3.84
2.29
.31
3.39
3.84
1.71
2.08
1.754
7.82
5.37
6.10
5.75
5.99
Normal
Modulus of elasticity 2 (8.4 GPa) Ultimate bending stress2 (62 MPa) Ultimate bending force2 (316 N) Ultimate shear stress2 (15 MPa) Ultimate shear force 2 (1010 N) Ultimate shear stress3 (5.4 MPa) Ultimate shear force 3 (1112N) 1
Statistical level P<.05.
2
Sample size is 25 tibiae.
3
Sample size is 43 femora.
"Critical value is 5.99.
The mechanical properties of the bones tested are summarized in Table 1. The observed and critical chi-square values for the seven sets of results are also given in Table 1. Figures 1 and 2 show the distribution curves for the
ultimate shear force and stress, respectively, of the femur. The normal functions described the data from the shear test of the femur better than the log normal or three parameter Weibull function. The distribution of ultimate shear stress of the femur could be modeled using any
Distribution Type Normal Log Noirmiil ,; WeibulT T '/
Distribution Tyje Log Normal Weibull
.08
0
5000
1000
1500
2000
Ultimate Shear Force (N)
FIG. 1. Distribution curves of the ultimate shear force of the femur.
5.0
Ultimate Shear Stress (MPa) FIG. 2. Distribution curves of the ultimate shear stress of the femur.
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
Lognormal
RESEARCH NOTE
587
Distribution Type Normal Log Normal Weibull
D i s t r i b u t i o n Type Log-Normal Weibull
625
1250
1875
2500
Ultimate Shear Force (N)
FIG. 3. Distribution curves of t h e u l t i m a t e shear force of the tibia.
0
10
20
U l t i m a t e Shear S t r e s s
30
40
(MPa)
FIG. 4. Distribution curves of the u l t i m a t e shear stress of t h e tibia.
one of the three curves; however, the normal curve had a better fit. Figures 3 and 4 illustrate the density functions of the shear test results of the tibia. The ultimate shear force was described by the log-normal density function even though the other two distributions could not be rejected at p<.05 level (Table 1). None of the distributions fit the ultimate shear stress of the tibia. However, the chi-square of the log normal curves was closer to the critical chi-square value. The distribution of the ultimate bending force, bending stress, and modulus of elasticity were described by the log-normal distribution. The other two distributions were rejected at the P<.05 level. DISCUSSION The three-parameter Weibull distribution was not selected to model any of the distributions of the mechanical properties. Of the seven sets of data, the log normal distribution had the better fit in four of the sets of data. Another two sets of data could have been modeled using the log-normal distribution. The log-normal function described the properties of the tibia,
and the properties of the femur followed a normal distribution. The results of the distribution analysis indicated the mechanical properties of bone tend to follow a log-normal distribution rather than a normal distribution that has been assumed in past research. Because of the nonnormal distribution, the actual level of significance in an analysis of variance may increase from P<.05 to P<.07 or P<.08 (Page 129, Steel and Torrie, 1960).
REFERENCES A d m i n , L., and F. E. Woeste, 1 9 8 3 . Graphics D a t a Analysis C o m p u t e r Package. Agric. Eng. D e p t . Virginia Tech. Blacksburg, VA. Ang, A.H.S., and W. H. Tang, 1 9 7 5 . Probability Concepts in Engineering Planning and Design. J o h n Wiley and Sons, N e w York, NY. Hann, C. T., 1 9 7 7 . Statistics M e t h o d s in H y d r o l o g y . Iowa State Univ. Press, Ames, IA. Steel, R.G.D., and J. H. Torrie, 1 9 6 0 . Principles and Procedures of Statistics. McGraw-Hill B o o k C o m p a n y , N e w York, NY. Wilson, J. H., and J. L. Baker, 1 9 7 9 . D e t e r m i n a t i o n of t h e shear m o d u l u s in long shaft b o n e t o acc o m p a n y an elastic m o d u l u s . A m . Soc. Agric. Eng. Paper N o . 7 9 - 5 5 2 1 .
Downloaded from http://ps.oxfordjournals.org/ at New York University on April 27, 2015
0