Derive the likelihood function based on the observed data d

Question

1. The continuous reassessment method (CRM) may take a different structure other than the power function, such as a logistic model, logistic model

logit(Π_j (α, β)) = ln ((Π_j (α, β)/(1-Π_j (α, β)) = α + βd_j, (1)

where d_j is the standardized dose at dose level j, j = 1, ... , J, α and β are unknown parameters.

(a) Derive the likelihood function based on the observed data D.

(b) Suppose that the trial has 5 dose levels, and let (d₁, d₂, . . . , d₅) = (0.1, 0.2, . . . , 0.5) denote the corresponding doses.

The observed data are

Dose level	1	2	3	4	5
# DLTs	0	0	2	3	0
# patients	3	6	12	7	0

The current dose is d₃, and the target toxicity probability is Φ = 0.3. If we take α ~ N(0, 10²) and β ~ Gamma(0.1, 0.1), determine the next dose level using the CRM dose-finding scheme.

(c) Let -y denote the dose of the MTD, i.e., Pr(DLT at γ) = Φ, and let θ denote the probability of DLT at the starting dose d1, i.e., Π₁ = θ. Show that the logistic model can be reparameterized by

α =  (d₁ logit (Φ) -γlogit(θ))/(d₁ - y)

β = (logit (θ) - logit(Φ))/(d₁ - y)

(d) As an alternative scheme to the CRM, escalation with overdose control (EWOC) can effectively control assigning patients to overly toxic doses. The next dose level j* is determined by

j* = arg_j=1,...,J min{|Pr(y < d_j| D) - λ|}

where Pr(γ, ≤ d_j | D) is the posterior probability that the MTD is not above the dose dj, and a is a small prespecified feasibility bound, e.g., λ = 0.25. Based on the prior distributions given in (b) and the repa¬rameterized model (2), determine the next dose level using the EWOC scheme.

(e) Other than assigning the prior distributions to α and β, we can take y ~ Unif(d₁, d_j) and θ Unif(0, Φ) for y and θ. Determine the next dose level using the EWOC scheme. Compare it with (d), and comment on your result.

2. The model-based designs are sensitive to the model specifications such as the model choice (e.g., the one-parameter power model or two-parameter logistic regression model), parametrization of the model, and the prior distribution of the unknown parameters. On the other hand, we cast dose finding in a Bayesian hypothesis testing problem. Specifically, we consider J hypotheses or nonparametric models,

H_j: |P_j - Φ|≤ ∈, j = 1,.....J,

where ∈ ≥ 0 is a small positive number, indicating the dose level j is the MTD when the model or hypothesis H_j is true. Under each H_j, we specify a prior distribution for p₁, ... ,p_j,

               Unif(P_k-i, P_k+i), k ≠ j;

Pk | H_j ~                                              k = 1,.....J

               Unif(Φ - ∈_, Φ + ∈), k = j;  

where Unif(a, b) represents the uniform distribution with a support of (a, b), and P₀ and p_J+1 are the lower and upper boundaries of the prior distribution, respectively.

(a) Let D_n = {(y₁, n₁), ... , (y_J, n_J)} denote the accumulated data up to the Nth patient, where yj is the number of DLTs and n_j is the number of patients at dose level j, and N = _J=1∑^J n_j., Let P(H_j) be the prior probability of Hi, derive the posterior probability that model H_j is true, i.e., P(H_j | D_n).

(b) Suppose that we treated N patients. Based on the posterior probabilities, the dose level, j* , for the (N + 1)th patient is determined according to a small prespecified feasibility bound A (0 < λ < 1),

j* = arg_j min|Σ_k=1^j P(H_k| D_n) - λ ID(H_k | D_n) -λ|,

where Σ_k=1^j P(H_k |D_n) is the posterior probability that the MTD is not above the dose level j. Consider three dose levels, i.e., J = 3, and suppose that 0 = 0.3, 6 = 0.05, P₀ = 0 and p_j+1 = 1. We specify a discrete uniform distribution for the prior model probability; that is, P(H_j) = 1/J, j = 1, . . . , J. The observed data are given by

Dose level	1	2	3
# DLTs	0	1	2
# patients	3	6	3

Determine the next dose level for the (N + 1)th patient according to λ = 0.35.