Abductive statistical methods improve the results of calibration curve bioassays: An example of determining zinc...

In this paper, we discuss the theory behind calibration curve experiments and their application to a zinc (Zn) bioavailability study with broiler chickens. Seven replicates of 16 male commercial broiler chicks were fed starter diets for 14 days. Six diets had different levels of a potential Zn source and one was a positive control with standard industry levels of Zn for comparison. Four commonly used methods of calculating bioavailability means and confidence intervals (CI) from a calibration curve (standard curve) experiment to estimate the bioavailability of a new zinc source in broiler chickens were compared. The methods compared were the following: 1) the Counter-Intuitive Method uses a multiple-range test to compare unknown test and standard samples; 2) the Intuitive Method uses standard linear regression and inverts the equation to predict Zn bioavailability for each replicate of test samples; 3) the Abductive Method uses Graybill's Equation, based on theory and observation, to estimate CI's; and 4) the Sophistic Method uses reverse regression, and calculates Zn bioavailability values directly from the equation. The Counter-Intuitive Method only gives information about which standards the test samples are, or are not, significantly different from respectively (average available Zn not predicted). The Intuitive Method ignores error about the standard curve and theoretically cannot estimate the CI directly ( $$  = 107.5 ± 15.8 mg Zn/kg). The Sophistic Method underestimates and overestimates the test sample mean values above and below the mean of the standards, respectively ( $$  = 96.6 mg Zn/kg). The Abductive Method has an advantage over the other methods: The mean prediction estimation is consistent with theory (107.5 ± 6.1 mg Zn/kg; $$ ). When test or "unknown" samples are near the mean of the standard samples, the CI is smaller than when near the extremes of the calibration curve. When calibration curve error is small (R ² > approximately 0.95), there is little advantage to using the Abductive Method, but when calibration curve error is larger, as in many bioassays with growing animals, the Abductive Method improves the accuracy of the CI calculations. The Abductive Method was used to demonstrate the influence of the number of replicate samples on experimental power and cost.