Satistical modelling of clinical trials

Transcription

1 Sept. 21st, /13

2 Patients recruitment in a multicentric clinical trial : Recruit N f patients via C centres. Typically, N f 300 and C 50. Trial can be very long (several years) and very expensive Previous models based on linear or exponential interpolation = Need for a probabilistic model that takes into account large variability of recruitment rates in different centres 2/13

3 Centre i opens at timeu i. Γ-model proposed by V. Anisimov (2; 1) : Recruitment processes of centres are independent Poisson processes Recruitment rate of centre i : λ i (unknown) Γ-Poisson model : (λ 1,...,λ C ) iid with Gamma distribution of parametersα and β. p α,β (x) = e βx x α 1 β α Γ(α) 1 3/13

4 On-going study at time t 1. Observed data is the number of patients recruited by each of C centres : (k 1,...,k C ). Independent but not identically distributed since λ L i Γ(α,β) and k L i P(λ i τ i ). Let τ i = t 1 u i (0 if negative). Estimation of (α,β) maximum likelihood technique : P[N i t 1 = k i ; 1 i C = E [ C (λ i τ i ) k i k i! log-likelihood (up to some constant) : e τ iλ i = C Γ(α+k i ) k i!γ(α) β α τ k i i (β +τ i ) α+k i L C (α,β) = α lnβ lnγ(α)+ 1 C C lnγ(α+k i ) (α+k i ) ln(β +τ i ) (ˆα C, ˆβ C ) = arg max (α,β) Θ L C (α,β) 4/13

5 As C +, the MLE is consistent and asymptotically normal. Approximated Fischer information matrix : if f(θ, k i,τ i ) = α lnβ lnγ(α)+lnγ(α+k i ) (α+k i ) ln(β +τ i ), then (θ = (α,β)) : and I(θ 0 ) 1 C C I i (θ 0 ) = 1 C C C(ˆθC θ 0 ) L N(0, I(θ 0 ) 1 ) E θ0 [ 2 f(θ 0, k i,τ i ). (I i ) 11 = E α0,β 0 [ψ (1) (α 0 + k i ) (I i ) 12 = (I i ) 21 = 1 β 2 0 ( 1 (I i ) 22 = α ( 0 1 β τ i /β τ i /β 0 ), ), +ψ (1) (α 0 ), 5/13

6 Bayesian re-estimation of the distribution of λ i : forward distribution of λ i conditionnaly of the information at t 1 : Γ(α+k i,β +τ i ) Overall recruitment rate : Λ = C λ i = C Γ(α+k i,β +τ i ), that can be approximated by a Γ(A, B) distribution by matching moments. In this framework, the (remaining) recruitment time ˆT t 1 has density : Then it is easy to get : P [ˆT Tf E[ˆT x N1 1 p T (x) = Γ(A+N 1) Γ(A)Γ(N 1 ) BA (x + B) N 1+A 6/13

7 If P [ˆT Tf is too small (say lower than p = 95%), we open M centres. C M Λ = λ i + λ i has the forward distribution Γ(α 0 + k i,β 0 +τ i ) λ i has a Γ(α 0,β 0 ) distribution Here, centres are supposed to open instantaneously, but it is possible to assume they open later in some interval [r i, s i. If P [ˆT Tf is too high, one can close centres to save money. λ i 7/13

8 Sensitivity parameters (see (3)) : P [ˆT Tf and E[ˆT are calculated with (ˆα C, ˆβ C ) estimated at t 1, instead of real parameters(α 0,β 0 ). The subsequent error is evaluated thanks to sensitivity parameters (e.g. αp [ˆT Tf ). When the overall rate is approximated by a Gamma distribution : althenors m = C α+k i β +τ i et v = C αe[ˆt N α(m v ) m (m v, m )2 β E[ˆT N β(m v ) m (m v. m )2 α+k i (β +τ i ) 2 8/13

9 Extension of the model Pareto distribution instead of Gamma distribution : p kp,x m (x) = k px kp m 1 x xm x 1 kp opening times u i of centres unkown, but we know the time of first recruitment v i. Then we assume u i uniformly distributed in [0, v i. 9/13

10 (centres opening times unkown) : N = 610, C = 77, T f = 3 years Study actually finished in 2.31 years. t Pareto - parameters (1.19, 1.39) (1.23, 1.30) (1.18, 1.22) Pareto - E[ˆT Gamma - parameters (1.17, 0.25) (1.08, 0.26) (1.31, 0.33) Gamma - E[ˆT At t 1 = 1, in the Γ-Poisson model, we get P [ˆT Tf Closing the smallest 10 centres steal leads to P [ˆT Tf /13

11 Model validation : ν i = number of centres having recruited exactly i patients FIGURE: Green : real data ; Bleu : Gamma model ; Rouge : Pareto model 11/13

12 Further research : simultaneaous modelling of screened, randomized and lost patients cost modelling exogenous variables? 12/13

13 [1 Vladimir V. Anisimov, Using mixed poisson models in patient recruit in multicentre clinical trials, Proceegings of th World Congress on Ingineering (London, United Kingdom), vol. II, [2 Vladimir V. Anisimov and Valerii V. Fedorov, Modelling, prediction and adaptive adjustment of recruitment in multicentre trials, Stat. Med. 26 (2007), no. 27, MR MR [3 Guillaume Mijoule, Stéphanie Savy, and Nicolas Savy, Models for patients recruitment in clinical trials and sensitivity analysis, Statistics in Medicine 31 (2012), no. 16, /13