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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

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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

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1

Introduction

2

Method

3

Results

4

Discussion

Susanna Salem (GAUG) Master thesis July 4, 2018 2 / 36

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Introduction

Sample size calculation is an important step in any clinical trial. Inadequate sample sizes lead to underpowered studies or

  • ver-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

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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.

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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

  • nce sufficient information has been gathered, i.e. a stopping criterion

is reached.

Susanna Salem (GAUG) Master thesis July 4, 2018 5 / 36

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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.

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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

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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

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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

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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

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Method - Model

Intensity functions:

θ = (α0, α1) Log-linear baseline intensity function: λ0(t, θ) = exp(α0 + α1t)

exp(α0) is the baseline rate at t = 0 The time trend α1 is the same for both groups

Cumulative intensity function Λ0(t, θ) = t

0 λ0(u, θ)du

Susanna Salem (GAUG) Master thesis July 4, 2018 11 / 36

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Method - Model

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Intensity function

t in months λ(t, θ) = exp(α0 + α1t)

α1 −0.3 −0.5

Treatment group A B

β1 = log(0.5)

6 12 18 24 Susanna Salem (GAUG) Master thesis July 4, 2018 12 / 36

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Method - Model

Overdispersion:

Inter-individual variability was introduced through a gamma-distributed variable Uij ∼ Γ(ϕ−1, ϕ) with expectation 1 and variance ϕ

Intensity functions

Log-linear baseline rate: λi(t, α0, α1|Uij = uij) = uij exp(α0 + α1t) exp(xiβ1) Cumulative rates: Λij = uij 1

α1 exp(α0) (exp(α1T − 1) exp(xiβ1)

Treatment effect is parameterized as β1 = log

  • λ1

λ0

  • Susanna Salem (GAUG)

Master thesis July 4, 2018 13 / 36

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Method - Model

Likelihood:

Likelihood

L(ϕ, θ, β1) =

1

  • i=0

ni

  • j=1

yij

  • k=1

λ0(tijk ,θ) exp(xi β1) Λ0(Tij ,θ) exp(xi β1)

  • ×

Γ(ϕ−1+yij ) Γ(ϕ−1)yij ! (ϕΛ0(Tij ,θ) exp(xi β1))yij (1+ϕΛ0(Tij ,θ) exp(xi β1))yij +ϕ−1

Log-likelihood

l(β1, α0, α1, ϕ) =

1

  • i=0

ni

  • j=1

Yij

  • k=1

(α0 + α1tijk) +

1

  • i=0

ni

  • j=1

Yij (log(ϕ) + xiβ1) +

1

  • i=0

ni

  • j=1
  • log(Γ(Yij + ϕ−1)
  • −

log

  • Γ(ϕ−1)
  • −

1

  • i=0

ni

  • j=1
  • Yij + ϕ−1

log

  • 1 + ϕα−1

1

exp(α0) exp(xiβ1)(exp(α1Tij) − 1)

  • ni patients are recruited into two treatment groups i = 0, 1

(0 = control, 1 = treatment) tijk are the event times during the observation interval Tij from [8]

Susanna Salem (GAUG) Master thesis July 4, 2018 14 / 36

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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:

1

First, events with event times t∗

1 , ..., t∗ n are created from a

homogeneous Poisson process with constant rate λ∗ λ∗ > λ(t) for all t.

2

Second, events are deleted independently from each other with 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 [z1, z2, ...] which are uniformly distributed over the interval [0, 1]. An event t∗

k is deleted if zk > λ(t∗

k )

λ∗

(Stoyans, 1995, pp. 45-46).

Susanna Salem (GAUG) Master thesis July 4, 2018 15 / 36

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Method - Example data

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

1 2 3 4

Year Patient

Example data: Treatment A

96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190

1 2 3 4

Year Patient

Example data: Treatment B

Parameters: α0 = log(1.338), α1 = -0.7, ϕ = 0.35 , β1 = log(0.5) , n = 190

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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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Method - Sample size

Patients are allocated to group 1 and group 0 with ratio k:1

BCM-CONST n0 =

(z1−α+z1−β)2 β⋆2

  • 1

λ0T

  • 1 +

1 k exp(β1⋆)

  • + ϕ⋆(1 + 1/k)
  • BCM-TREND

n0 =

(z1−α+z1−β)2 β⋆2

  • 1

Λ0(T,θ⋆)

  • 1 +

1 k exp(β1⋆)

  • + ϕ⋆(1 + 1/k)
  • Planning scenario for the simulations:

α = 0.025 (one-sided), Power = 80% (β = 0.2) Log-rate-ratio β1

⋆ = log(0.5),

  • Cum. baseline rates = 0.36*2 (T = 2 years follow-up),

Overdispersion ϕ⋆ = 0.82,

Under the planning scenario for the simulations: 95 patients per group

Susanna Salem (GAUG) Master thesis July 4, 2018 18 / 36

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Method - Example trial

5 10 15 5 10 15 20

Months after monitoring start I ^

CONST TREND blinded unblinded

I*

Susanna Salem (GAUG) Master thesis July 4, 2018 19 / 36

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Method - Stopping criterion

Effect is parameterized as β1 = log

  • λ1

λ0

  • Information about β1 is defined as I = var( ˆ

β1)−1 Wald-Tests were performed on the ML estimate ˆ β1 to test the hypotheses H0 : β1 ≥ 0 and H1 : β1 < 0 Critical information calculated before trial start: I⋆ = (z1−α+z1−β)2

β⋆

1 2 Susanna Salem (GAUG) Master thesis July 4, 2018 20 / 36

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Method - Parameter estimation

BCM-CONST BCM-TREND ML-estimation of parameters via fitting a negative binomial model to the data with R-function glm.nb() from the MASS package ML-estimation of parameter vector θ = (β1, α0, α1, ϕ) via maximisation of the log-likelihood with numerical optimization in R (R-function optim())

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Method - Parameter estimation

Variance of estimator in trend model was derived from Fisher information matrix I(θ) Distribution of ML estimator of parameter vector θ = (β1, α0, α1, ϕ) ˆ θML

a

∼ Np

  • θ, I(ˆ

θML)−1 (ˆ θML)i

a

∼ N

  • θi, [I(ˆ

θML)−1]ii

  • e.g. [2].

Susanna Salem (GAUG) Master thesis July 4, 2018 22 / 36

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Method - Information

BCM-CONST BCM-TREND Distribution of ML estimator ˆ β1

a

∼ N(β1, 1/I0 + 1/I1) IML =

1 1/I0+1/I1 = I0I1 I0+I1

with I0 =

n0

  • j=1

T0j λ0 1+ϕλ0

I1 =

n1

  • j=1

T1j λ1 1+ϕλ1

(Lawless, 1987) Distribution of ML estimator ˆ β1

a

∼ N(β1,

(I0+I1)[W ] I0I1[W ]−[Z]2 )

IML = I0I1[W ]−[Z]2

(I0+I1)[W ]

with I0 =

n0

  • j=1

Λ0(T0j ,θ) 1+ϕΛ0(T0j ,θ)

I1 =

n1

  • j=1

Λ1(T1j ,θ) 1+ϕΛ1(T1j ,θ) Hij = exp(α0) exp(xi β1) exp(α1Tij )Tij [W ] = (I0 + I1)

  • 1
  • i=0

ni

  • j=1

α−1

1

Hij Tij −

ϕ

  • H2

ij −(Λi (Tij ,θ))2 +2

  • Hij −Λi (Tij ,θ)
  • α2

1(1+ϕΛi (Tij ,θ))

  • −
  • 1
  • i=0

ni

  • j=1

Hij −Λi (Tij ,θ) α1(1+ϕΛi (Tij ,θ))

2 [Z] =

  • I0

n1

j=1 H1j α1(1+ϕΛ1(T1j ,θ)) − I1

n0

j=1 H0j α1(1+ϕΛ0(T0j ,θ))

  • Susanna Salem (GAUG)

Master thesis July 4, 2018 23 / 36

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Method - Blinded estimation

The estimated information ˆ IML is obtained by replacing unknown parameters with point estimates. Blinded estimation of rates with lumping approach:

Overall rate ˆ λ is estimated for the pooled data from both treatment groups Rates for groups are then obtained with the help of assumed treatment effect β⋆

1

ˆ λ0 =

2ˆ λ (1+exp(β⋆

1 )

ˆ λ1 = 2ˆ

λ exp(β⋆

1 )

(1+exp(β⋆

1 )

For model with time trend: blinded estimates are obtained the same way from cumulative pooled rate Λ(Tm, ˆ θ), with index m running over all observations from both treatment arms.

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Method - Scenarios for simulation

Parameter Value β1 log(0.375), log(0.5), log(1) Cumulative baseline rates 0.72, 1.44 α1 0, -0.3, -0.5, -0.7, -2.5, -3.5 ϕ 0.35, 0.82 Recruitment period 24 months Follow-up time 24 months Monitoring start time 13th month, 25th month Monitoring intervals monthly α − level 0.025 (one-sided) Power 0.8

2000 data sets per scenario (All simulations were done with R version 3.4.4)

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Method - Scenarios for simulation

Parameter Value β1 log(0.375), log(0.5), log(1) Cumulative baseline rates 0.72, 1.44 α1 0, -0.3, -0.5, -0.7, -2.5, -3.5 ϕ 0.35, 0.82 Recruitment period 24 months Follow-up time 24 months Monitoring start time 13th month, 25th month Monitoring intervals monthly α − level 0.025 (one-sided) Power 0.8

2000 data sets per scenario (All simulations were done with R version 3.4.4) Size of time trends of relapse rates in clinical MS trials corresponding to results from Nicholas et al., 2012 (from Schneider, Schmidli and Friede, 2013)

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Results - Type I error rate

Cum.Baserate: 0.72 phi: 0.35 Cum.Baserate: 0.72 phi: 0.82 Cum.Baserate: 1.44 phi: 0.35 Cum.Baserate: 1.44 phi: 0.82 monitoring.start: 13th month monitoring.start: 25th month monitoring.start: fixed design −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05

α1 Type I error rate BCM method

const trend

Monitoring start (month)

13th month 25th month fixed design

Type I error

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Results - Power

Cum.Baserate: 0.72 monitoring.start: 13th month Cum.Baserate: 0.72 monitoring.start: 25th month Cum.Baserate: 0.72 monitoring.start: fixed design Cum.Baserate: 1.44 monitoring.start: 13th month Cum.Baserate: 1.44 monitoring.start: 25th month Cum.Baserate: 1.44 monitoring.start: fixed design phi: 0.35 rate.ratio: 0.37 phi: 0.82 rate.ratio: 0.37 phi: 0.35 rate.ratio: 0.5 phi: 0.82 rate.ratio: 0.5 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00 0.80 0.85 0.90 0.95 1.00

α1 Power Monitoring start (month)

13th month 25th month fixed design

BCM

const trend Planning scenario

Power

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Results - Study length

Average study length in months for monitoring start after 13 months Time trend α1 BCM

  • 3.5
  • 2.5
  • 0.7
  • 0.5
  • 0.3

CONST 26.8 27.2 29.8 30.2 30.9 31.7 TREND 24.9 25.8 29.5 30.0 30.7 31.7 Average study length in months for monitoring start after 25 months Time trend α1 BCM

  • 3.5
  • 2.5
  • 0.7
  • 0.5
  • 0.3

CONST 30.4 30.4 31.5 31.7 32.1 32.6 TREND 29.6 29.8 31.3 31.6 32.0 32.6

Susanna Salem (GAUG) Master thesis July 4, 2018 29 / 36

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Results - Study length

Cum.Baserate: 0.72 monitoring.start: 13th month Cum.Baserate: 0.72 monitoring.start: 25th month Cum.Baserate: 0.72 monitoring.start: fixed design Cum.Baserate: 1.44 monitoring.start: 13th month Cum.Baserate: 1.44 monitoring.start: 25th month Cum.Baserate: 1.44 monitoring.start: fixed design phi: 0.35 rate.ratio: 0.37 phi: 0.82 rate.ratio: 0.37 phi: 0.35 rate.ratio: 0.5 phi: 0.82 rate.ratio: 0.5 phi: 0.35 rate.ratio: 1 phi: 0.82 rate.ratio: 1 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 20 30 40 20 30 40 20 30 40 20 30 40 20 30 40 20 30 40

α1 Study length in months Monitoring start (month)

13th month 25th month fixed design

BCM

const trend

Study length

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Results - Number of patients

Average number of patients for monitoring start after 13 months Time trend α1 BCM

  • 3.5
  • 2.5
  • 0.7
  • 0.5
  • 0.3

CONST 166 167.9 179.1 180.8 182.6 185 TREND 158 162.6 178.4 180.3 182.3 185

Susanna Salem (GAUG) Master thesis July 4, 2018 31 / 36

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Results - Number of patients

Cum.Baserate: 0.72 monitoring.start: 13th month Cum.Baserate: 0.72 monitoring.start: 25th month Cum.Baserate: 0.72 monitoring.start: fixed design Cum.Baserate: 1.44 monitoring.start: 13th month Cum.Baserate: 1.44 monitoring.start: 25th month Cum.Baserate: 1.44 monitoring.start: fixed design phi: 0.35 rate.ratio: 0.37 phi: 0.82 rate.ratio: 0.37 phi: 0.35 rate.ratio: 0.5 phi: 0.82 rate.ratio: 0.5 phi: 0.35 rate.ratio: 1 phi: 0.82 rate.ratio: 1 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 −3.5 −2.5 −0.7 −0.5 −0.3 100 125 150 175 100 125 150 175 100 125 150 175 100 125 150 175 100 125 150 175 100 125 150 175

α1 Number of patients Monitoring start (month)

13th month 25th month fixed design

BCM

const trend

Number of patientes

Susanna Salem (GAUG) Master thesis July 4, 2018 32 / 36

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Summary

BCM-CONST seems to be robust against time trends of the size reported by [5] for MS trials. If negative time trends are higher, BCM-TREND leads to slightly more efficient trials.

Susanna Salem (GAUG) Master thesis July 4, 2018 33 / 36

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Discussion

Seasonal time trends in MS [9]. Possibility to plan a new study length (longer study) if rates were

  • ver-estimated at planning stage.

The true treatment effect has to be at least as large as the expected treatment effect at the planning stage of the study.

Susanna Salem (GAUG) Master thesis July 4, 2018 34 / 36

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Discussion

Thank you for your attention.

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References

Tim Friede and Frank Miller. Blinded continuous monitoring of nuisance parameters in clinical trials. Journal of the Royal Statistical Society: Series C (Applied Statistics), 61(4):601–618, 2012. Leonhard Held and D Saban´ es Bov´ e. Applied statistical inference, volume 10. Springer, 2014. Jerald F Lawless. Negative binomial and mixed poisson regression. Canadian Journal of Statistics, 15(3):209–225, 1987. Peter A Lewis and Gerald S Shedler. Simulation of nonhomogeneous poisson processes by thinning. Naval Research Logistics (NRL), 26(3):403–413, 1979. Richard Nicholas, Sebastian Straube, Heinz Schmidli, Sebastian Pfeiffer, and Tim Friede. Time-patterns of annualized relapse rates in randomized placebo-controlled clinical trials in relapsing multiple sclerosis: A systematic review and meta-analysis. Multiple Sclerosis Journal, 18(9):1290–1296, 2012. Raghu Pasupathy. Generating homogeneous poisson processes. Wiley Encyclopedia of Operations Research and Management Science, 2010. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2018. S Schneider, H Schmidli, and T Friede. Blinded sample size re-estimation for recurrent event data with time trends. Statistics in medicine, 32(30):5448–5457, 2013. Steve Simpson, Bruce Taylor, Leigh Blizzard, Anne-Louise Ponsonby, Fotini Pittas, Helen Tremlett, Terence Dwyer, Peter Susanna Salem (GAUG) Master thesis July 4, 2018 36 / 36