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Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models
Fig 2
(A) Data obtained by solving Eq (22), with θ = (λ, K, C(0)) = (0.01, 100, 10), at t = 0, 100, 200, …, 1000 is corrupted with Gaussian noise with σ = 10. The MLE (cyan) solution is superimposed, 
. (B) Univariate profiles for λ, K and C(0), respectively. Each profile is superimposed with the MLE (vertical green) and the 95% threshold 
is shown (horizontal orange). Points of intersection of each profile and the threshold define asymptotic confidence intervals: 
; 
; and, 
. (C)-(E) C(t) trajectories associated with the λ, K and C(0) univariate profiles in (b), respectively. In each case we uniformly discretise the interest parameter using N = 100 points along the 95% confidence interval identified in (B), and solve the model forward with θ = (ψ, ω) and plot the family of solutions (grey), superimposing the MLE (cyan), and we identify the prediction interval defined by these solutions (solid red). (F) Compares approximate prediction intervals obtained by computing the union of the univariate profiles with the ‘exact’ prediction intervals computed from the full likelihood. Trajectories (grey) are constructed by considering N = 104 choices of θ and we plot solutions with 
only. These solutions define an ‘exact’ (or gold-standard) prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by taking the union of the three univariate intervals (red).
doi: https://doi.org/10.1371/journal.pcbi.1011515.g002