Table 2.

Model selection using log marginal likelihood as the goodness of fit criteria. Including a gradient term (bulk milk PP coefficient) greatly improves the model fit. The inclusion of a random effect term is also strongly supported, however, the precise parameterization has little effect as does the choice of link function. Of the parameterizations explored, scaling the farm level random effect by the gradient parameter maximizes the marginal likelihood, as does the use of a logistic link function

Model	log (marginal likelihood)
	logistic	cloglog
θ	− 528.24	− 528.25
θ + βbulkmilki	− 281.05	− 282.15
θ + βbulkmilki + ϕi	− 273.58	− 274.20
θ + β(bulkmilki + ϕi)	− 272.98	− 273.11