Fig 1.
Plots show the contour lines of two functions, chosen to illustrate identifiable and non-identifiable cases. Plot (A) is an identifiable case illustrated by Booth function lA(θ) = (θ1+2θ2−7)2+(2θ1+θ2−5)2, which has known minimum lA(1,3) = 0. Plot (B) illustrate non-identifiable case by Rosenbrock function with minimum lB(1,1) = 0. The star-shaped points mark the minima of the above functions. The bold contour represents the
for
. The dashed lines are profile paths projected on (θ1, θ2) Red circles mark the points where tangent hyperplanes correspond to parameters’ minimal or maximal values in CRα. Red circles are CI endpoints. The contours were calculated using marching squares algorithm implemented in Contour.jl package (https://github.com/JuliaGeometry/Contour.jl). They are provided for illustrative purposes only.
Table 1.
Comparison of CICO and stepwise profile likelihood methods for the cancer taxol treatment model.
Fig 2.
Search paths for the parameters’ CI endpoints of the cancer taxol treatment model.
The path of CI search for stepwise optimization-based algorithm (A, C) and CICO algorithm (B, D). Circles denote the points reached by the algorithm during the search and numbers above the circles indicate the number of likelihood function calls the algorithm makes to reach this point. The dotted line is the likelihood profile calculated separately for illustrative purposes. The dashed horizontal line marks the significance level α = 0.95. Red circles mark the estimated endpoints (if they exist) for CICO algorithm and black–the points, where the algorithm reaches the box constraints. It denotes non-identifiable case. (A) Estimation of lower and upper CI endpoints with the stepwise optimization-based method for a0 parameter. (B) Estimation of lower and upper CI endpoints with CICO method for a0 parameter. (C) Estimation of lower and upper CI endpoints with the stepwise optimization-based method for kd parameter. (D) Estimation of lower and upper CI endpoints with CICO method for kd parameter.
Table 2.
Comparison of LikelihoodProfiler and dMod for STAT5 dimerization model.