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Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models

Fig 5

(A) Data obtained by solving Eqs (25) and (26), with θ = (α, β, x(0), y(0)) = (0.9, 1.1, 0.8, 0.3), at t = 0, 0.5, 1.0, …, 10 is corrupted with Gaussian noise with σ = 0.2. The MLE solution (solid curves) is superimposed, . (B)-(G) Bivariate profiles for (α, β), (α, x(0)), (α, y(0)), (β, x(0)), (β, y(0)) and (x(0), y(0)), respectively. In (B)-(G) the left-most panel shows 100 points along the contour (blue discs) and the MLE (pink disc). The middle- and right-most panels show 100 predictions of a(t) and b(t), respectively, associated with the 100 points along the contour, together with the profile-wise prediction interval (solid red). (H) Compares approximate prediction intervals obtained by computing the union of the profile-wise prediction intervals with the prediction intervals from the full likelihood function. Predictions (grey) are constructed by considering N = 10⁴ choices of θ and we plot solutions with only. These solutions define a gold-standard prediction interval (dashed gold) that we compare with the MLE solution (cyan) and with the approximate prediction interval formed by the union of the three bivariate trajectories (red).

doi: https://doi.org/10.1371/journal.pcbi.1011515.g005