J. Anim Sci. 2007. 85:2031-2042. doi:10.2527/jas.2007-0016
© 2007 American Society of Animal Science
Evaluation of the impact of errors in the measurement of backfat depth on the prediction of fat-free lean mass
A. P. Schinckel*,1,
M. E. Einstein*,
K. Foster
and
B. A. Craig
* Department of Animal Sciences,
and
Department of Agricultural Economics, and
Department of Statistics, Purdue University, West Lafayette, IN 47907
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Abstract
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The development of regression equations to predict carcass composition typically assumes that the independent variables, such as backfat depth, are measured without error. However, technological and operator-specific types of measurement errors do exist. To evaluate the impact of measurement error for backfat depth, Monte Carlo simulation was used to model carcass fat-free lean mass (FFLM) in pigs. In the simulation, FFLM was a linear function of carcass weight and actual backfat depth (ABFD). Carcass weight was assumed to be measured without error, but measurement errors were generated such that the correlation (rBF) of the measured backfat depth (BFD) and ABFD ranged from 0.70 to 0.95. Two types of measurement errors were simulated: 1) constant variation that was additive to the variance of ABFD, and 2) variation proportional to the ABFD that was additive to the variance in ABFD. A total of 1,000 replications of 1,000 pigs were simulated. Within each type of measurement error, the absolute values of the regression coefficients and R2 values of the equations decreased as rBF decreased. The probability of the backfat depth squared (BFD2) being significant (P < 0.05) in the regression equation was increased when the measurement errors were proportional to ABFD. The occurrence of a significant BFD2 variable was 792 times out of 1,000 replications when rBF = 0.95 and increased to 996 times out of 1,000 when rBF = 0.85 for BFD with type 2 measurement errors. The inclusion of a CW x BFD variable in the regression equations (P < 0.05) increased (270 to 423 times out of 1,000) as rBF decreased from 0.85 to 0.70 for BFD with type 2 errors. Equations developed from BFD with measurement errors resulted in biased predictions of FFLM and changes in FFLM per unit change in BFD. The level and type of measurement errors that exist in the independent variables should be evaluated.
Key Words: carcass composition • measurement error • pig • prediction equation
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INTRODUCTION
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Pork carcass composition research has been conducted to evaluate the effects of experimental treatments, to model pig compositional growth, and to evaluate pork production systems (Schinckel and DeLange, 1996
; Schinckel et al., 2003). The prediction of carcass fat-free lean mass (FFLM) is of primary importance because FFLM is highly associated with the mass of trimmed lean cuts and carcass value (Akridge et al., 1992; Hicks et al., 1998; Berg et al., 1999). Also, fat-free lean growth rates are used to predict daily nutrient requirements (Schinckel and DeLange, 1996; NRC, 1998).
Currently, the majority of US and Canadian pork processors utilize carcass ultrasound or optical probe measurements to predict FFLM and estimate carcass value (Berg et al., 1999; Fortin et al., 2004). Carcass measurements collected by these technologies are subject to technology-specific and human measurement errors (McLaren et al., 1991; Boland et al., 1995; Moeller, 2002). The most common operator errors involve the incorrect placement or angle at which the optical or ultrasonic probe is placed into or on the carcass (S. D. M. Jones, W. M. Robertson, and T. Coupland, Agriculture and Agri-Ag Canada Research Station, Lacombe, Alberta, Canada, unpublished data; Boland et al., 1995; Moeller, 2002). Measurement errors due to incorrect placement angle would be expected to produce measurement errors for backfat depth (BFD) proportional to the actual backfat depth (ABFD; Moeller and Christian, 1998; Moeller, 2002).
The development of regression equations commonly assumes that the independent variables are measured without error (Neter et al., 1996). The impact of measurement errors, and specifically those proportional to the actual value of the carcass measurement, on the regression equations including such measurements has not been evaluated.
The objective of this research was to characterize and quantify the impact of different levels and types of measurement errors for BFD on the resulting regression equations developed to predict FFLM.
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MATERIALS AND METHODS
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The pigs in this trial were managed and harvested with approval of the Purdue Animal Care and Use Committee.
A Monte Carlo simulation program was developed using parameters from the FFLM and carcass BFD data of Schinckel et al. (2003). The BFD measurements were taken 7.0 cm from the midline with an electronic probe (HGP4, Hennessy Grading Probe, Hennessy and Chong, Auckland, New Zealand) between the third and fourth ribs cranial to the last rib. The parameters of FFLM and 2 carcass measurements, carcass weight (CW, in kg) and ABFD (in mm), are presented in Table 1. The FFLM of each pig was calculated by the equation developed from the parameters. Specifically, for each pig, FFLM = 7.1605 + (0.59294 x CW) – (0.53846 x ABFD) + z, where the z is an independent, normal, random variable (mean = 0; SD = 2.8467). The equation had an expected R2 of 0.8346.
Two types of measurement errors were simulated, such that the correlation between the measured BFD and ABFD (rBF) ranged from 0.70 to 0.95. The first type of measurement error was modeled to be additive to ABFD. This BFD with additive measurement error (BF-AE) had substantially greater measurement error and total variance as rBF decreased. The equation used was BF-AE = ABFD + [(8/rBF)2 – 64]0.5 x z2, where rBF is the correlation between BF-AE and ABFD and z2 is a pig-specific value sampled from a standard normal distribution.
The second set of BFD measurements (BF-AEPE) were simulated to have increased additive measurement errors, for which the SD were proportional to ABFD. The equation used to simulate values of BF-AEPE was BF-AEPE = ABFD + {[(8/rBF)2 –64]0.5 x (ABFD/BFM) x z2}, where rBF is the correlation between BF-AEPE and ABFD and z2 is a pig-specific value sampled from a standard normal distribution.
A total of 1,000 pigs were simulated for each of 1,000 replications. Within each replicate (across levels of rBF and type of measurement error) the value of z2 was constant for each pig. The SAS software (SAS Inst. Inc., Cary, NC) was used to calculate the means, variances, and correlations among the BFD measurements and FFLM. The replicates of FFLM data for each type and level of measurement error were fitted to a series of regression equations (PROC GLM, SAS). These equations included independent variables, which included the quadratic and cubic effects of BFD and CW and linear cross-product variables. They were
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Equation [3] included the addition of cross-product variable (CW x BFD) to equation [1]. Equation [4] included the addition of a BFD3 variable to equation [2].
The number of times the regression coefficient being evaluated was significant at a probability of P < 0.05 was recorded. In addition, R2, RSD, and the estimated regression coefficients were calculated. The FFLM was predicted for each pig using the estimated prediction equation, and the slope of FFLM on BFD was calculated by taking the derivative of FFLM relative to the measured BFD. The mean predicted value and slope of FFLM relative to the measured BFD were summarized.
Variables produced from the regression analyses, including R2, RSD, and regression coefficient values, were fitted with a model including the effects of measurement error level, type of measurement error, and their interaction (PROC MIXED, SAS).
The predicted values for FFLM from the regression equations were calculated for each replicate for BFD of 12, 20, 28, 36, and 44 mm. The predicted slopes of FFLM per millimeter change in BFD were calculated for the regression equations including BFD and BFD2 for those same values of BFD. These variables were fitted to a model that included the effect of measurement error level, type of measurement error, BFD value, and their interactions. This model was used to evaluate the magnitude of biases in the prediction of FFLM or slope of FFLM on BFD for each factor and BFD level.
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RESULTS
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The simulated means for FFLM, CW, and ABFD were 45.01 kg, 94.01 kg, and 28.01 mm with SD of 6.999 kg, 10.99 kg, and 7.999 mm, respectively. The means and SD for the BFD measurements including measurement errors are in Table 2. As expected, the SD of the BFD measurements with additive errors increased as the accuracy of the measurement (rBF) decreased. The SD of the BFD measurements including errors proportional to ABFD have slightly greater total variance than the BFD measurements, which are independent to ABFD. This is due to the fact the sums of squares of the measurement errors were increased by squaring the errors that were proportional to the mean.
The correlations of the BFD measurements with ABFD and FFLM are in Table 3. The correlations of the BFD measurements with errors independent of ABFD were similar to the correlations of the BFD measurements with errors proportional to ABFD. The correlations of the BFD with errors proportional to ABFD with ABFD and FFLM were slightly lower than same correlations involving BFD measurements with errors independent of ABFD. This was caused by the slightly greater error variance and total variance of the BFD measurements with errors proportional to ABFD.
The probability of detecting additional independent variables beyond CW and the measured BFD was evaluated as the number of times the additional variable was significant (P < 0.05) out of the 1,000 replicated regression analyses (Table 4). The percentage of times that truly nonsignificant variable is expected to be found significant and included in the regression equation is the probability value used (P = 0.05). With no measurement error, ABFD2 was significant 51 times, CW x ABFD was significant 46 times, and ABFD3 was significant 54 times out of 1,000 regression analyses.
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Table 4. The number of times out of 1,000 replicates that a specific regression coefficient was significant, P < 0.051
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For the BFD measurements where measurement error was independent of ABFD, as rBF decreased and magnitude of the measurement errors increased, the probability of detecting additional independent variables as significant (P < 0.05) decreased and was less than 5%. This is likely caused by the fact that the error sums of squares used to test the significance of the partial sums of squares due to the additional variable increased as the measurements errors increased in magnitude.
The BFD measurements, which included measurement errors proportional to ABFD had identical and substantially increased probability of detecting the additional independent variables as being statistically significant. As expected with no measurement error, ABFD2 was significant in 5% of cases (51 out of 1,000 times). The inclusion of BFD2 increased to 792 times out of 1,000 regression analyses with a relatively low level of measurement error (rBF = 0.95). The number of times that BFD2 was significant was maximum at rBF = 0.85 with 996 out of 1,000 regression analyses.
It is important to note that the random measurement errors proportional to ABFD did not produce a quadratic relationship of the measured BFD to ABFD or FFLM. For example, the variance of BF-AEPE, with additive error proportional to ABFD, is the sum of the variance of ABFD and the variance due to measurement error. The variance due to measurement error equals (cSDe)2, where c is a constant and SDe is the SD of the measurement errors. If the SDe is proportional to ABFD, the variance of the measurement error and total variance of the BFD measurements increase as ABFD increases. The slope of FFLM per unit change in measured BFD equals the covariance (FFLM, BFD)/variance (BFD). If the measurement errors are proportional to ABFD, then the slope of FFLM to the BFD measurement decreases as BFD (measured or actual) increases. To account for this nearly linear decrease in the slope of FFLM to measured BFD (the derivative of FFLM on BFD), as BFD increases, the regression analyses have a substantially increased probability of detecting a significant positive regression coefficient for BFD2. This is because the first derivative of a function is linear, then the original function includes a quadratic variable.
The number of times that the product of CW times BFD was significant was greater than the 5% expected by chance when the BFD measurement included errors proportional to ABFD. The probability of detecting the product of CW times BFD as significant increased as the magnitude of the measurement increased and rBF decreased. This is likely caused by the fact that ABFD had a positive correlation with CW, and thus as CW increased, the ABFD values and SDe increased. This resulted in a tendency for the total variance in the measured backfat depths with errors proportional to ABFD to increase and caused the slopes of FFLM to BFD to decrease as CW increased. This is supported by the fact that the regression coefficients for the product of CW times BFD increased from 0.0007 to 0.0025 for BF-AEPE, as rBF decreased from 0.95 to 0.70.
The number of times that the BFD3 variable was significant was greatest when rBF ranged from 0.75 to 0.85. It is possible that at these levels of measurement errors proportional to ABFD, the change in the slope of FFLM relative to BFD [covariance (FFLM, BFD)]/[variance (BFD)] was close to quadratic in nature with respect to ABFD and thus increased the probability of detecting BFD3 as significant in the regression analyses. As the derivative of FFLM on BFD becomes closer to quadratic in nature, the greater the probability that the original equation includes a cubic variable.
The R2 and RSD values for the prediction equations including the linear effects CW and BFD variables, and equations including the additional BFD2 variable are shown in Table 5. The level of measurement errors, type of measurement errors, and their interaction had a significant (P < 0.001) impact on the RSD and R2 values of both equations. For the BFD measurements in which the magnitude of the measurement errors are independent of ABFD, the addition of the BFD2 variable added only 0.0002 to 0.0003 to the R2 and resulted in no change in the RSD. However, for regression equations including BFD measurements with measurement errors proportional to ABFD, the inclusion of the BFD2 variable increased the R2 from 0.0024 to 0.0047 and reduced the RSD of the regression equations from 0.013 to 0.035 kg. The impact of the addition of the BFD2 variable to equations of the same level of measurement error was small compared with the impact of the level of measurement error.
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Table 5. Coefficient of determination (R2) and residual SD (RSD) values for prediction equations only including the linear carcass weight and backfat depth and in equations including an additional backfat depth squared variable1
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The regression coefficients for equations including CW and BFD are shown in Table 6. Within each type of BFD measurement errors, the absolute values of the regression coefficients for BFD decreased as rBF decreased (P < 0.001). The absolute values of the regression coefficients for CW decreased as rBF decreased (P < 0.001), to a much lesser extent than the regression coefficients for BFD, and to a similar extent across the 2 types of measurement errors (measurement error level x measurement error type interaction, P = 0.78).
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Table 6. Regression coefficients for equations predicting fat-free lean mass and having 2 types of measurement errors1
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The regression coefficients for the equations including CW, BFD, and BFD2 are shown in Table 7 and were not different at different levels of rBF. The mean values of the regression coefficients for BFD2 were –2.2 x 10–5 (SD = 0.0011) for BF-AE for rBF ranging from 0.70 to 0.95 and are not different from zero (P = 0.14). Overall, the regression coefficients for BFD and BFD2 in these equations were affected by the type and level of measurement errors and the interaction between the type and level of measurement error (P < 0.001). However, 97 to 98% of the total variance accounted by the model was due to the main effects of type and level of measurement errors. The regression coefficients for the BFD2 variable were positive (P < 0.001) for the BFD measurements with errors proportional to ABFD. The absolute value of the regression coefficient for BFD2 was greatest at rBF = 0.85. This may have been the level of measurement error that produced the decrease in the slope of FFLM relative to BFD that was most linear in nature. It should be noted that the slope of FFLM (kg) relative to BFD (mm) at constant CW is the derivative of the prediction equation FFLM = b0 + (b1 CW) + (b2 BFD) + (b3 BFD2), which is b2 + (2 b3 BFD).
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Table 7. Mean regression coefficients and SD for an equation (BF-AEPE) including carcass weight, measured backfat depth, and measured backfat depth squared1
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Although not shown, the regression coefficients for CW, BFD, and BFD x CW in equation [3] were affected (P < 0.001) by the type of measurement error and their interaction. The regression coefficient for the BFD x CW variable was not different from zero (P = 0.67 and 0.79) for prediction equations including BF-AE, whose measurement errors were independent of the value of ABFD. However, equations including BFD measurements with errors proportional to ABFD had positive regression coefficients (P < 0.001) for the CW x BFD variable.
The regression coefficients for equation [4], which included CW, BFD, BFD2, and BFD3, were affected (P < 0.001) by the type and level of measurement errors. The interaction of the level and type of measurement errors was significant (P < 0.001) for the BFD variables but not for CW (P = 0.70). Equations including BFD measurements with errors proportional to ABFD had negative regression coefficients for BFD3 (P < 0.001). The negative values of the regression coefficients for this type of measurement errors were greatest (P < 0.001) at rBF = 0.80. Equations that included BF-AE measurements had regression coefficients for BFD3 that did not differ from zero (P = 0.52 and 0.54, respectively).
The FFLM values predicted by the linear equations were analyzed by a model including the effects of level of measurement error, type of measurement error, and ABFD value (12, 20, 28, 36, and 44 mm), and their 2-and 3-variable interactions. The main effects of type and level of measurement error and the interaction of level and type of measurement error did not impact (P = 0.93, 0.99, and 1.00, respectively) the predicted FFLM values. This is due to fact that biases in the regression coefficient for BFD do not produce population-wide biases in the prediction of FFLM. The prediction of FFLM was affected (P < 0.0001) by the interactions of type of measurement error by BFD and level of measurement error by BFD. This indicates that for a sample of pigs with the population mean BFD, prediction equations including the linear variables were unbiased. But any specific subpopulation or sample of pigs whose BFD means were different from the population mean will have biased predictions of FFLM.
The FFLM values predicted by equation [2], which included CW, BFD, and BFD2, were impacted (P < 0.0001) by the BFD value, the level of measurement error, and type of measurement error, and all 2- and 3-variable interactions. After BFD value, the interaction of the level of measurement error by BFD value, the interaction of the type of measurement error and BFD, and the main effect of type of measurement error had the second, third, and fourth greatest impact on the FFLM values predicted. The main effect of level of measurement error and other interactions had substantially less impact than the other variables.
The least squares means for the predicted FFLM values for each combination of level of measurement error and BFD value are shown in Table 8. Equations with measurement errors overpredict (P < 0.001) the FFLM of pigs with greater than average (28 mm) ABFD and less than average FFLM, and underpredict (P < 0.001) the FFLM of pigs with less than average ABFD and greater than average FFLM. The magnitude to which the FFLM of pigs with less than 28 mm of BFD were underpredicted and pigs with more than 28 mm were overpredicted was impacted by the combination of the level of measurement error and BFD value. When the measurement errors were additive and improvement of the ABFD, the magnitude of the over- or under-prediction of FFLM was symmetric as the ABFD value increased or decreased from the mean value. The magnitude of the over- or underprediction of FFLM was not symmetric as the BFD value increased or decreased from the mean value when the SD of the measurement errors were proportional to ABFD. For example, for the BFD measurements with additive, proportional errors, the FFLM of pigs with 12 mm of BFD were underpredicted by 1.03, 2.67, and 4.39 kg and the FFLM of pigs with 44 mm BFD were overpredicted by 2.39, 3.70, and 4.78 kg at rBF values of 0.90, 0.80, and 0.70, respectively.
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Table 8. Least squares means for the fat-free lean mass (FFLM, kg) predicted from equations with backfat measurements collected with different levels of measurement errors1
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The equations predicted approximately 50% of the true differences between pigs at rBF = 0.70. Pigs with 12 mm of BFD and 53.62 kg of FFLM had 8.62 kg more lean than the average pig (28 mm of BFD, 45 kg of FFLM) but were predicted to have only 4.02 (BFAE) or 4.23 (BF-AEPE) kg more lean than the average pig. Pigs with 44 mm of BFD and 36.4 kg of FFLM were predicted to have 4.01 and 3.82 kg less FFLM than average pigs when the true difference was 8.6 kg. The prediction equations would be expected to rank different subpopulations of pigs correctly. However, a rBF = 0.70 the predicted variance among different subpopulations of pigs would be only 25% of the true variance among the subpopulations for FFLM. These statistics of the correlation between the actual and true subpopulation means and percentage of the true variance amongst the subpopulations predicted by the prediction equations have been evaluated by Gu et al., 1992; Hicks et al., 1998; Schinckel et al., 2001.
The mean FFLM predicted by the equations from the 1,000 replicates are shown in Figures 1 and 2 for rBF = 0.70 and rBF = 0.85 for the BFD measured with each type of measurement error. The magnitude of prediction biases increased (P < 0.001) as the level of measurement error increased.
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Figure 1. Predicted fat-free lean mass at different back-fat depths for equations with different types of measurement errors and with correlation of actual with measured backfat of 0.70. BF-AE = backfat depth measured with added random measurement error; BF-AEPE = backfat depth measured with random additive error and the SD of the errors is proportional to the actual backfat depth; and ABFD = actual backfat depth, in millimeters.
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Figure 2. Predicted fat-free lean mass at different back-fat depths for equations with different types of measurement errors and correlation of actual with measured back-fat of 0.85. BF-AE = backfat depth measured with added random measurement error; BF-AEPE = backfat depth measured with random additive error and the SD of the errors is proportional to the actual backfat depth; and ABFD = actual backfat depth, in millimeters.
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The changes in FFLM predicted per millimeter change in BFD are shown for each BFD measurement variable (Figures 3 and 4) from equation [2], which included CW, BFD, and BFD2. These figures show the impact of the quadratic terms on the predicted slope of FFLM on BFD. The slopes were affected by the type of measurement errors, level of measurement errors, the BFD value, and all 2- and 3-variable interactions (P < 0.0001). The primary factors producing the differences in the slopes were the 3 main effects and the interaction of the BFD value by the level of measurement error. The slopes for the BFD measurements with errors independent of ABFD are essentially constant over the range of ABFD. The absolute value of the predicted change in FFLM per millimeter change in BFD decreased (P < 0.001) as the measured BFD increased for equations that include BF-AEPE.
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Figure 3. Predicted change in fat-free lean mass, in kilograms, per millimeter of change in backfat depth for 6 types of measurement errors and correlation of actual with measured backfat of 0.70. The equation included carcass weight and linear and quadratic backfat depth. BF-AE = backfat depth measured with added random measurement error; BF-AEPE = backfat depth measured with random additive error and the SD of the errors is proportional to the actual backfat depth; and ABFD = actual backfat depth, in millimeters.
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Figure 4. Predicted change in fat-free lean mass, in kilograms, per millimeter of change in backfat depth for 6 types of measurement errors and correlation of actual with measured backfat of 0.85. The equation included carcass weight and linear and quadratic backfat depth. BF-AE = backfat depth measured with added random measurement error; BF-AEPE = backfat depth measured with random additive error and the SD of the errors is proportional to the actual backfat depth; and ABFD = actual backfat depth, in millimeters.
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The predicted change in FFLM per millimeter change in measured BFD is shown for regression equations for BF-AEPE at different levels of measurement errors (Figure 5). As the level of measurement errors increased, the overall absolute value of the predicted change in FFLM per millimeter change in BFD decreased (P < 0.001). The predicted change in FFLM per millimeter change in BFD was overestimated (P < 0.001) for pigs with less than 12 mm of BFD for BF-AEPE and rBF from 0.80 to 0.95.
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Figure 5. Predicted change in fat-free lean mass, in kilograms, per millimeter change in backfat depth for equations including backfat depth measurements with measurement errors that are proportional to the mean and whose variance is additive at 7 levels of measurement error (rBF = the correlation of the true and actual back-fat depth).
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DISCUSSION
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Currently, carcass ultrasound and optical probe measurements are used to predict FFLM and estimate carcass value by the majority of pork processors with carcass-value based marketing systems (Berg et al., 1999; Fortin et al., 2004). The optical probe and carcass ultrasound measurements are subject to operator error (Kempster et al., 1985; Sather et al., 1986; S. D. M. Jones, W. M. Robertson, and T. Coupland, Agriculture and Agri-Ag Canada Research Station, Lacombe, Alberta, Canada, unpublished data) associated with errors in the correct placement and angle in which the measurements are taken (Boland et al., 1995; Moeller, 2002). Such measurement errors would be expected to produce errors proportional to the ABFD.
Another potential source of errors due to technology is the correct measurement of all 3 layers of backfat. Some data suggests that A-mode ultrasound is less precise than B-mode ultrasound due to less consistent measurement of the third layer of backfat (Moeller, 2002). Backfat depths measured by A-mode ultrasound had smaller values, smaller SD, and correlations of lesser absolute value with ribbed carcass measurements than BFD measured with B-model ultrasound (Schinckel et al., 1994). Inconsistent measurement of the entire ABFD including the third layer of backfat could result in BFD measurements with decreased, similar, or increased total variance compared with ABFD depending on amount of variation produced by the measurement errors.
This paper examined the simpler cases of measurement errors that are independent of the actual backfat depths and whose variance were independent of or proportional to the actual backfat depths. It is possible with some technologies that do not measure the entire backfat thickness of fatter pigs such as the A-mode ultrasound that the measurement errors may be related to the actual backfat depths. It is also likely that measurement errors are a mixture of errors that are proportional to the actual values and errors independent of the actual values. Measurement errors due to incorrect angle or location of the measurement device would be expected to produce measurement errors proportional to the actual values. Still, some measurement errors may occur independent of the actual values. Examples of such errors would include rounding the backfat depth measurements to the nearest millimeter, differences in the pressure applied to achieve contact with ultrasonic measurements, or errors associated with the rate of withdrawal of the optical probe from the carcass during its measurement of the fat depth.
Few papers that have developed prediction equations for FFLM or other carcass endpoints have provided useful statistics to evaluate the magnitude of the measurement errors. Some have reported the correlations between the optical probe or carcass ultrasound measurements and ribbed carcass measurements. Under controlled laboratory conditions, the correlations of these BFD measurements have been at or above 0.90 with ribbed carcass BFD (Berg et al., 1994; Liu and Stouffer, 1995). This correlation is only a partially useful evaluation of the accuracy of the BFD measurement if the anatomical locations of the 2 measurements are identical. Other researchers have provided the correlations of the alternative BFD measurements with FFLM and carcass FFLM percentage. Under controlled conditions and with experienced trained operators, the carcass ultrasound and optical probe measurements have had correlations of similar magnitude with FFLM or carcass FFLM percentage as the ribbed carcass BFD measurements (Forrest et al., 1989; Gu et al., 1992; Gresham et al., 1994; Schinckel et al., 2001). The better means to quantify the magnitude of the measurement errors would be the variance between repeated measurements (Fortin, 1984; McClure et al., 2003; Olsen et al., 2007). In recently conducted research, the SD of the difference between repeated BFD and LM depth measurements for each pig increased as the values increased (A. Fortin, Agriculture and Agri-Ag Canada Research Station, Lacombe, Alberta, Canada, personal communication).
Backfat depth in pigs has been used as a measure of carcass FFLM percentage. Past trials have found correlations of –0.81 to –0.84 between BFD measurements measured off midline and FFLM percentage (Forrest et al., 1989; Liu and Stouffer, 1995; Schinckel et al., 2001). The measurement errors simulated in this paper reduce the correlation of the measured BFD and carcass FFLM percentage by the factor of rBF.
The other indicator of the accuracy of the BFD measurements is the RSD of the prediction of carcass lean percentage. As reviewed by Fortin et al., 2004, the RSD of predicting dissected lean yield or stat-standardized lean usually range from 1.7 to 2.7%. The prediction equation simulated had a RSD for FFLM percentage of 2.48 with a correlation of –0.84 between FFLM percentage and BFD. A more recent trial found RSD ranging from 3.22 to 4.32% for FFLM percentage (Johnson et al., 2004). The increased RSD for the prediction of FFLM percentage is produced by a weaker relationship between the BFD measurements and FFLM percentage likely produced by less accurate measures of backfat depth or FFLM. The European Community has a series of regulations (EC, 1994) outlining the evaluation of pig carcasses by different technologies. The prediction equations must be evaluated based on representative sampling and have RSD of 2.5% or less (EC, 1994). The impact of subpopulations on the RSD statistic has been evaluated in some detail (Engel et al., 2004). This requirement only authorizes the use of technologies with acceptable levels of measurement error.
It should be noted that the RSD is affected by measurement errors of the dependent variable and the independent variables. Increased measurement errors of FFLM and FFLM percentage would be expected when FFLM is determined based less on dissection and to a greater extent based on lipid analyses (Schinckel et al., 2001). Random measurement errors in FFLM would increase the RSD for the prediction of FFLM and FFLM percentage but would not be expected to bias the regression equations (Neter et al., 1996).
An assumption of conventional regression analyses is that the independent variables are measured without error. The results of this research indicate that both the magnitude of the measurement errors, as evaluated as the correlation between the measured BFD and ABFD, and the type of measurement errors affect the absolute value of the regression coefficients and degree of prediction bias. Prediction equations with BF-AE had similar R2 and RSD values at the same level of rBF, indicating that these statistics are not useful in evaluating prediction bias.
The simulation results indicate that if the random measurement errors and their SD are proportional to the true value of the independent variable, then the likelihood of incorrectly detecting additional variables, especially quadratic variables, substantially increases. It is possible that different prediction equations that include different sets of independent variables measured at identical anatomical locations but with different levels and types of measurement errors could result in drastically different regression equations with 1) different R2 and RSD values, 2) different sets of statistically significant independent variables including quadratic, cubic, and cross-product variables, 3) different predicted changes in FFLM per unit change in carcass measurement, and 4) different magnitude of prediction biases for specific subpopulations of pigs. In a recent trial, a regression equation predicting FFLM from an ultrafom BFD and CW had a greater RSD (3.62 vs. 3.46 kg) and substantially different independent variables and different predicted changes in FFLM per millimeter change in BFD than a prediction equation, which included the same measurements taken by an animal ultrasound system at the identical location (Johnson et al., 2004).
Canadian (Pomar et al., 2001; Fortin et al., 2004) and European researchers (Cook et al., 1989; Diestre et al., 1989; Engel et al., 2004) have concluded that linear regression models result in accurate prediction of pork carcass lean content. However, equations have been published that include cross product, quadratic, and cubic independent variables (Johnson et al., 2004). These prediction equations have greater RSD for FFLM and substantially greater RSD for FFLM percentage than those published by past researchers (Higbie et al., 2002; Fortin et al., 2004).
The simulation results indicate that prediction equations developed from BFD with measurement errors will produce biased equations with increased RSD. Equations developed from carcass measurements with errors proportional magnitude to the actual measurements will likely include extraneous variables. This result is supported by recent trials evaluating the Autofom 300 (UFOM, SFK Technology AI5, Herlev, Denmark). In Fortin et al., 2004, a prediction equation for lean percentage including UFOM BFD had a RSD of 1.71%. The inclusion of (BFD)2 was significant (P < 0.05). However, the addition of (BFD)2 had minimal impact on the RSD (1.70 kg). The prediction equation for FFLM percentage from UFOM BFD of Johnson et al. (2004) had an RSD of 4.23%. The prediction equation for FFLM had a RSD of 3.62 kg, a R2 of 0.395, and included a significant quadratic term [FFLM, kg = 20.76 – 1.04 (BFD, mm) + 0.015 (BFD)2 + 0.41 (CW, kg)]. The FFLM predicted by this equation decreases as BFD increases to 34.66 mm, and then the amount of lean increases as BFD increases. The equation predicts that pigs with 10 and 59.4 mm of BFD have the same FFLM (52.96 kg). Pigs with 20 and 49.4 mm of BFD have 47.0 kg of FFLM. Also, pigs with 28 and 41.5 mm of BFD have 44.4 kg of FFLM. It is possible that unrealistically large predicted quadratic relationship between FFLM and BFD, and the large RSD values for the UFOM, were caused by the existence of measurement errors proportional to the BFD.
Based on previous in-sample and out-of-sample simulation (Schinckel et al., 2000a,b), prediction equations developed using carcass measurements with the least measurement errors resulted in the smallest genetic population and sex biases. The simulation results indicated that regardless of the level of measurement error in future carcass measurements, the prediction equations should be developed under conditions and by personnel that result in the smallest magnitude of measurement errors. This will result in the highest correlations of the carcass measurements with FFLM, and the most accurate equations in terms of their ability to estimate true genetic population and sex differences.
This paper examined the simpler cases of measurement errors that are independent of the actual backfat depths and whose variance were independent of or proportional to the actual backfat depths. It is possible with some technologies that do not measure the entire backfat thickness of fatter pigs such as the A-mode ultrasound, that the measurement errors may be related to the actual backfat depths. It is also likely that measurement errors are a mixture of errors whose SD are proportional to the actual values and errors whose SD are independent of the actual values. Measurement errors due to incorrect angle or location of the measurement device would be expected to produce measurement errors proportional to the actual values. Still, some measurement errors may occur that are independent of the actual values. Examples of such errors would include rounding the backfat depth measurements to the nearest millimeter, and differences in the pressure applied to achieve contact with ultrasonic measurements or the rate of withdrawal of the optical probe during its measurement of the fat depth.
Several additional types and combinations of types of measurement errors were simulated but not reported in this paper. Each type of measurement error produced different regression equations with respect to the variables included in the regression equation, the absolute value of the regression coefficients, and magnitude of prediction bias at the same level of rBF. Thus, the type and magnitude of measurement errors in the independent variables must be accounted for to reduce their impact on the resulting regression equations.
It is possible that the impact of measurement errors on the regression analyses could be accounted for by utilizing prior information obtained from repeated carcass measurements. The procedure would need to account for the variance and type of measurement errors. The method would become increasingly complex as the number and type of carcass measurements increased. Regression equations including 2 or more carcass measurements could include numerous quadratic, cubic, and cross product variables (linear by linear). Also, the measurement errors for the carcass measurements such as BFD and loin muscle depth (S. D. M. Jones, W. M. Robertson, and T. Coupland, Agriculture Canada, Agriculture and Agri-Ag Canada Research Station, Lacombe, Alberta, Canada, unpublished data; Boland et al., 1995) may be related, and those relationships may need to be accounted for. The procedure should result in prediction equations that are more precise (decreased true SE of the regression coefficients) and have a low probability (significance level set by the investigator) of including extraneous independent variables.
Schmidt (1976) presents a very concise and informative treatment of errors in measurement in regression and the use of prior information to obtain maximum likelihood estimates in the case of additive error; however, proportional measurement error does not appear to have been treated in the literature. When the measurement error is proportional to the level of some regressor, Xi, then the relationship between the observed and latent values is amended to the special case where the measurement error µi = 1/Øi Xi with Øi defined as a random variable that is directly related to the deviation of the angle of probe entry from the perpendicular. Thus, the relationship between the measurement error and the observed value of X*i is µi = 
iX*i where i =1/(1 + Øi) and has expected value of . Forming a first order Taylor approximation for the measurement error yields µi 
X* + (i – )X* + (X*i – X*). Subtracting the leading right hand side term from both sides, squaring both sides, and taking expectations leads to the following approximation of the variance of the measurement error, 
X*2Var(i) + 2Var(X*) + 2Cov(i, X*). Clearly, this term introduces quadratic effects. One could even anticipate cubic effects if the true form of the variance is of higher order than quadratic in nature.
In conclusion, equations used to predict carcass composition of pigs have been developed in the past with little evaluation of the level of measurement errors in the carcass measurements used as independent variables. The level and type of errors associated with the carcass measurements substantially impacts the accuracy and prediction biases of the regression equations developed. The magnitude and type of measurement errors associated with the independent variables should be evaluated before their inclusion in any prediction equation of scientific and economic importance. Prediction equations developed from inaccurate carcass measurements will produce biased estimates of carcass component mass and percentage.
1 Corresponding author: aschinck{at}purdue.edu
Received for publication January 8, 2007.
Accepted for publication April 2, 2007.
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Abstract
INTRODUCTION
