Correction: Parameter estimation for multistage clonal expansion models from cancer incidence data: A practical identifiability analysis

Table 1 and Figs 4, 5, 7, 8 and 10 are incorrect. There is an error in the value of the threshold used to calculate the confidence intervals. The authors have provided the corrected versions here.

Table 1. Best-fit parameters and likelihood-based 95% confidence intervals.

Best-fit parameters and likelihood-based 95% confidence intervals for the fits of the two-, three-, and four-stage clonal expansion models (with parameterizations using only practically identifiable parameter combinations and given in Eqs. (2), (9), and (11) respectively) to age-specific incidence of pancreatic cancer.

Model	Parameter combination	Value	Likelihood-based 95% CI
Two-stage
	$p_2=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_1\right)-\sqrt{{\left(\alpha -\beta -{\mu }_1\right)}^2-4\alpha {\mu }_1}\right)$	-1.29E-1	(-1.32E-1, -1.26E-1)
	$q_2=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_1\right)+\sqrt{{\left(\alpha -\beta -{\mu }_1\right)}^2-4\alpha {\mu }_1}\right)$	6.21E-6	(5.58E-6,6.88E-6)
	r2 = vX/α	1.50E-2	(1.38E-2,1.65E-2)
Three-stage
	$p_3=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_2\right)-\sqrt{{\left(\alpha -\beta -{\mu }_2\right)}^2-4\alpha {\mu }_2}\right)$	-1.38E-1	(-1.42E-1,-1.34E-1)
	$q_3=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_2\right)+\sqrt{{\left(\alpha -\beta -{\mu }_2\right)}^2-4\alpha {\mu }_2}\right)$	1.57E-5	(1.38E-5,1.76E-5)
	$r_3=\sqrt{\nu X{\mu }_1/\alpha }$	2.35E-2	(2.23E-2,2.49E-2)
Four-stage
	$p_4=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_3\right)-\sqrt{{\left(\alpha -\beta -{\mu }_3\right)}^2-4\alpha {\mu }_3}\right)$	-1.50E-1	(-1.57E-1,-1.43E-1)
	$q_4=\frac{1}{2}\left(-\left(\alpha -\beta -{\mu }_3\right)+\sqrt{{\left(\alpha -\beta -{\mu }_3\right)}^2-4\alpha {\mu }_3}\right)$	4.59E-5	(3.90E-5,5.32E-5)
	r3 = (vXμ1μ2/α)1/3	2.66E-2	(2.54E-2,2.80E-2)

Fig 4. Two-stage model profile likelihoods.

Profiles of the relative negative log-likelihood (NLL) of the two-stage clonal expansion model as each of the parameter combinations p₂, q₂, and νX/α are varied while the remaining parameters are fit. The gray dotted line gives the α = 0.05 threshold for simultaneous confidence intervals based on the relative negative log-likelihood. All three parameters are identifiable.

Fig 5. Three-stage model profile likelihoods.

Profiles of the relative negative log-likelihood (NLL) of the three-stage clonal expansion model as each of parameter combinations p₃, q₃, νX, and μ₁/α are varied while the remaining parameters are fit. The gray dotted line gives the α = 0.05 threshold for simultaneous confidence intervals based on the relative negative log-likelihood. Parameter combinations p₃ and p₄ are identifiable, while νX and μ₁/α are practically unidentifiable.

Fig 7. Profile likelihood for the reparameterized combination of the three-stage model.

Profile of the relative negative log-likelihood (NLL) as the parameter $r_3=\sqrt{\nu X\mu 1/\alpha }$ varied while the remaining parameters are fit in the three-stage clonal expansion model. The gray dotted line gives the α = 0.05 threshold for simultaneous confidence intervals based on the relative negative log-likelihood. Parameter combination r₃ is identifiable.

Fig 8. Four-stage model profile likelihoods.

Profiles of the relative negative log-likelihood (NLL) of the four-stage clonal expansion model as each of parameter combinations p₄, q₄, νX, μ₁, and μ₂/α are varied while the remaining parameters are fit. The gray dotted line gives the α = 0.05 threshold for simultaneous confidence intervals based on the relative negative log-likelihood. Parameter combinations p₄ and q₄ are identifiable, while νX, μ₁, and μ₂/α are practically unidentifiable.

Fig 10. Profile likelihood for the reparameterized combination of the four-stage model.

Profile of the relative negative log-likelihood (NLL) as the parameter r₄ = (νXμ₁μ₂/α)^1/3 is varied while the remaining parameters are fit in the four stage clonal expansion model. The gray dotted line gives the α = 0.05 threshold for simultaneous confidence intervals based on the relative negative log-likelihood. Parameter combination r₄ is identifiable.