Monitoring of clinical trials with recurrent events Susanna Salem Georg-August University Goettingen Master thesis at the Department of Medical Statistics Supervisors: Prof. Tim Friede, Dr. Norbert Benda susanna.auberlen@stud.uni-goettingen.de July 4, 2018 Susanna Salem (GAUG) Master thesis July 4, 2018 1 / 36
Introduction 1 Method 2 Results 3 Discussion 4 Susanna Salem (GAUG) Master thesis July 4, 2018 2 / 36
Introduction Sample size calculation is an important step in any clinical trial. Inadequate sample sizes lead to underpowered studies or over-exposure of patients. A decision on the sample size has to be made at the planning stage and is based on α -level, power and assumptions about nuisance parameters (like baseline rates or overdispersion). But: specification of nuisance parameters before trial start is often difficult and imprecise. Susanna Salem (GAUG) Master thesis July 4, 2018 3 / 36
Introduction Blinded estimation of nuisance parameters at an interim stage of the study makes it possible to adapt the sample size or study length based on information from ongoing trials. Blinded = without knowledge about the allocation of patients to treatment groups. Susanna Salem (GAUG) Master thesis July 4, 2018 4 / 36
Introduction Different approaches: In Blinded sample size re-estimation (BSSR) one single re-estimation is done during the trial. In Blinded Continuous Monitoring (BCM) the information of the data is continuously analyzed (or e.g. monthly) and the study is stopped once sufficient information has been gathered, i.e. a stopping criterion is reached. Susanna Salem (GAUG) Master thesis July 4, 2018 5 / 36
Introduction Especially when patients with rare diseases or patients from vulnerable populations (e.g. children) are to be recruited, efficient designs are desirable. Example: The paediatric fingolimod trial 2-year study evaluating the efficacy and safety of fingolimod (oral) versus IFN beta-1a (injection) in children with multiple sclerosis (MS) aged 10 to 18 years Endpoint: annual relapse rate (ARR) No clinical trials on MS in children reported yet. High uncertainty about baseline rates and overdispersion at planning stage. Susanna Salem (GAUG) Master thesis July 4, 2018 6 / 36
Introduction - BCM for recurrent event data For recurrent event data BCM methods have been developed by Friede, H¨ aring & Schmidli (subm.). Friede, H¨ aring & Schmidli (subm.) showed in a simulation favorable results for the method: Scenarios based on paediatric fingolimod trial At trial start: two year recruitment phase and 2 year follow-up of 190 patients was planned Under planning assumptions: 44 months study length on average (60% of trials run till end) Power = 78.5 % With true baseline rates twice as high as expected: average study length reduces to 28.3 months Power = 85.3 % No inflation of type I error rate (type 1 error rates ranged from 2.25% to 2.5%) Susanna Salem (GAUG) Master thesis July 4, 2018 7 / 36
Introduction - Recurrent event data with time trends Declining rates (irrespective of group) have been reported for clinical trials on MS (Nicholas et al., 2012) Reasons for time trends: Regression to the mean effects (e.g. patients are recruited when the medical condition is most severe) How is type 1 error rate, power and study length affected when an actual negative time trend in the rates stays unconsidered in BCM? Susanna Salem (GAUG) Master thesis July 4, 2018 8 / 36
Aim 1 Investigate the robustness of BCM for clinical trials with event data with time trends For recurrent event data with time trends BSSR methods exist (Schneider, Schmidli & Friede, 2012). 2 Adapt BCM to data with time trends based on the methods already developed. 3 Evaluate in Monte Carlo trial simulations: Type 1 error rate Power Follow-up times and sample sizes Susanna Salem (GAUG) Master thesis July 4, 2018 9 / 36
Method - Model Non homogeneous Poisson Process: Non homogeneous Poisson process with negative time trend and gamma-distributed rates (analogous to Schneider, Schmidli and Friede, 2012) Non homogeneous Poisson process with rate λ ( t ): rates are a function of t numbers of events in any non-overlapping intervals are mutually independent random variables the number of events in any interval has a Poisson distribution Susanna Salem (GAUG) Master thesis July 4, 2018 10 / 36
Method - Model Intensity functions: θ = ( α 0 , α 1 ) Log-linear baseline intensity function: λ 0 ( t , θ ) = exp( α 0 + α 1 t ) exp( α 0 ) is the baseline rate at t = 0 The time trend α 1 is the same for both groups � t Cumulative intensity function Λ 0 ( t , θ ) = 0 λ 0 ( u , θ ) du Susanna Salem (GAUG) Master thesis July 4, 2018 11 / 36
Method - Model Intensity function 0.7 Treatment group A 0.6 B 0.5 λ ( t , θ ) = exp ( α 0 + α 1 t ) 0.4 0.3 0.2 α 1 β 1 = log(0.5) 0.1 0 −0.3 −0.5 0.0 0 6 12 18 24 t in months Susanna Salem (GAUG) Master thesis July 4, 2018 12 / 36
Method - Model Overdispersion: Inter-individual variability was introduced through a gamma-distributed variable U ij ∼ Γ( ϕ − 1 , ϕ ) with expectation 1 and variance ϕ Intensity functions Log-linear baseline rate: λ i ( t , α 0 , α 1 | U ij = u ij ) = u ij exp( α 0 + α 1 t ) exp( x i β 1 ) Cumulative rates: Λ ij = u ij 1 α 1 exp( α 0 ) (exp( α 1 T − 1) exp( x i β 1 ) � � λ 1 Treatment effect is parameterized as β 1 = log λ 0 Susanna Salem (GAUG) Master thesis July 4, 2018 13 / 36
Method - Model Likelihood: Likelihood � y ij n i 1 � ( ϕ Λ 0 ( T ij ,θ ) exp( x i β 1 )) yij Γ( ϕ − 1 + y ij ) λ 0 ( t ijk ,θ ) exp( x i β 1 ) L ( ϕ, θ, β 1 ) = � � � × (1+ ϕ Λ 0 ( T ij ,θ ) exp( x i β 1 )) yij + ϕ − 1 Λ 0 ( T ij ,θ ) exp( x i β 1 ) Γ( ϕ − 1 ) y ij ! i =0 j =1 k =1 Log-likelihood l ( β 1 , α 0 , α 1 , ϕ ) = Y ij n i n i n i 1 1 1 log(Γ( Y ij + ϕ − 1 ) � � � � � � � � � ( α 0 + α 1 t ijk ) + Y ij (log( ϕ ) + x i β 1 ) + − i =0 j =1 k =1 i =0 j =1 i =0 j =1 n i 1 � Γ( ϕ − 1 ) � � Y ij + ϕ − 1 � � 1 + ϕα − 1 � log � � log exp( α 0 ) exp( x i β 1 )(exp( α 1 T ij ) − 1) − 1 i =0 j =1 n i patients are recruited into two treatment groups i = 0 , 1 (0 = control, 1 = treatment) t ijk are the event times during the observation interval T ij from [8] Susanna Salem (GAUG) Master thesis July 4, 2018 14 / 36
Method - Data simulation Different algorithms exist to simulate non homogeneous Poisson processes (see e.g. Pasupathy, 2010) . To events with negative time trend the thinning method was used: First, events with event times t ∗ n are created from a 1 , ..., t ∗ 1 homogeneous Poisson process with constant rate λ ∗ λ ∗ > λ ( t ) for all t. Second, events are deleted independently from each other with 2 probability 1 − λ ( t ) λ ∗ . A proof that this results in an non homogeneous Poisson process with rate λ ( t ) can be found in Lewis & Shedler (1979). Practically, events can be deleted with the help of random variables [ z 1 , z 2 , ... ] which are uniformly distributed over the interval [0 , 1]. An k is deleted if z k > λ ( t ∗ k ) event t ∗ (Stoyans, 1995, pp. 45-46). λ ∗ Susanna Salem (GAUG) Master thesis July 4, 2018 15 / 36
Method - Example data Example data: Example data: Treatment A Treatment B 95 190 94 189 93 188 92 187 91 186 90 185 89 184 88 183 87 182 86 181 85 180 84 179 83 178 82 177 81 176 80 175 79 174 78 173 77 172 76 171 75 170 74 169 73 168 72 167 71 166 70 165 69 164 68 163 67 162 66 161 65 160 64 159 63 158 62 157 61 156 60 155 59 154 58 153 57 152 56 151 55 150 54 149 53 148 52 147 51 146 Patient Patient 50 145 49 144 48 143 47 142 46 141 45 140 44 139 43 138 42 137 41 136 40 135 39 134 38 133 37 132 36 131 35 130 34 129 33 128 32 127 31 126 30 125 29 124 28 123 27 122 26 121 25 120 24 119 23 118 22 117 21 116 20 115 19 114 18 113 17 112 16 111 15 110 14 109 13 108 12 107 11 106 10 105 9 104 8 103 7 102 6 101 5 100 4 99 3 98 2 97 1 96 0 1 2 3 4 0 1 2 3 4 Year Year Parameters: α 0 = log(1 . 338), α 1 = -0.7, ϕ = 0.35 , β 1 = log(0 . 5) , n = 190 Susanna Salem (GAUG) Master thesis July 4, 2018 16 / 36
Method - BCM General steps for both methods: 1 Plan fixed sample size and calculate stopping criterion I ⋆ . 2 Start monitoring: Calculate estimate of information ˆ I from the data in a blinded fashion. 3 Monthly data looks were implemented 4 Stop trial if enough data is gathered, i.e the observed information in the data reaches the critical value (ˆ I ≥ I ⋆ ) or all patients reach maximal follow-up time Susanna Salem (GAUG) Master thesis July 4, 2018 17 / 36
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