Classical and Bayesian predictions applied to Bacillus toxin production

Bacillus thuringiensis is a bacterium with unusual properties that make it useful for pest control in ecoagriculture. It can form a parasporal crystal containing polypeptides (also called delta-endotoxins). These entomopathogenic toxins are made during the stationary phase of the bacterial growth cycle and were initially characterized as an insect pathogen. Nowadays, the use of saturated two-level designs is very popular. This method is especially used in industrial applications where the cost of experiments is expensive. Standard classical approaches are not appropriate to analyze data from saturated designs. It is due to the fact that they only allow to estimate the main factor effects and cannot assure enough freedom degrees to estimate the error variance. In this paper, we propose the use of empirical Bayesian procedures to get inferences for data obtained from saturated designs, inspired from Hadamard matrices. The proposed methodology is illustrated by assuming a dataset to prove the model robustness. The comparison between the two studied mathematical techniques, based on mean square error values (MSE), revealed that Bayesian method (BM) was more accurate than least square method (LSM): for example, the results showed that 2002 and 2000.7 mg/l for experimental and predicted (BM) data were obtained against 2002 and 1991 mg/l for experimental and predicted (LSM) data. This suggested method could be generalized in several application fields in biological sciences.