Error and Bias in Using the Incomplete Gamma Function to Describe Lactation Curves

Abstract

Lactation curves can be generated and studied with mathematical models. The incomplete .gamma. function has the ability to generate curves of many shapes and can fit lactation curves affected by many different biological and environmental factors. The incomplete .gamma. function was used to fit lactation curves for 653 lactations at least 305 days long. For records of daily milk production for every day or every 30th day for the 1st 120, 210 or 305 days of the lactation, lactation curves were fitted by linear regression after a log transformation of the incomplete .gamma. function. Predictions of these curves for daily production and cumulative yield up to 305 days following freshening were compared to the records for daily production and cumulative yield for the same lactations. The bias and error for predicting daily milk production were high during the first week following freshening and then decreased. If the curve was fitted to production records for the first 120 or 210 days of the lactation, the bias and error of predicting daily production after day 120 or 210 of the lactation progressively increased until the end of the lactation. Fitting lactation curves to monthly observations of daily production increased the error in predicting 305-day cumulative yield from 247 to 300% compared to curves fitted from daily observations. When incomplete .gamma. curves were fitted to monthly observations of daily milk production over the entire 305-day lactation, the error in predicting 305-day cumulative yield (183.5 kg) was comparable to the prediction errors of centering date and test interval methods.