Negative binomial endpoint • BATSS

Context

This case study is inspired by the CHANGE-MS study (Curtin et al. 2016), which was a multi-arm, phase II randomised controlled trial which aimed to study the efficacy and safety of temelimab for the treatment of relapsing-remitting multiple sclerosis (MS) and determine whether temelimab slowed down or stopped the progression of MS.

The primary endpoint is the cumulative number of new active lesions that were identified on brain MRI scans that were performed on 4 occasions (once a month) from weeks 12-24 post-randomisation. The data were overdispersed (Curtin et al. 2016; Hartung et al. 2022) and modelled using a negative binomial GLM; we will use the same model here.

Design

The chosen design has the following characteristics:

treatment arms: Our case study has 4 treatment arms: placebo (control), low, medium and high doses of temelimab (intervention arms referred to as arms “A”, “B” and “C”, respectively).
alternative hypotheses: This is a superiority trial where we are interested in demonstrating efficacy compared to the placebo arm and intervention arms will not be compared to one another.
interim analyses and maximum sample size: We will begin the interim analyses at 100 participants completing follow-up, and perform interim analyses every 40 participants completing follow-up thereafter. Our maximum sample size is 260 participants, which is the same as the original study (Curtin et al. 2016).
power: Similar to the original study, we aim to have 90% power to detect a 60% reduction (RR=0.4) in the mean cumulative number of lesions in the highest dose intervention arm (from a mean of 4 (control) to 1.6 lesions).
endpoint conditional distribution: the endpoint is assumed to follow a negative binomial distribution with nuisance parameter equal to \(\phi = 0.5\) when considering the parametrisation leading to the following variance: \[\text{Var}(Y) = \mu + \frac{\mu^2}{\phi}.\]
group allocation: we will consider equal allocation probabilities per group (no response-adaptive randomisation).
efficacy stopping rule: Early stopping of intervention arms for efficacy may occur if there is a high posterior probability of (any) benefit at look j, i.e., when \[\text{Prob}(\beta_k < \delta_{\epsilon}|y, X) > 1-b_{\epsilon}\left(\frac{\sum_{k=1}^{K} n_k}{N}\right)^{p_{\epsilon}}\] where
- \(\beta_k\) denotes the \(k\)th target parameter,
- \(\delta_{\epsilon} = 0\) denotes the (efficacy-related) clinically meaningful treatment effect value,
- \(N\) is the maximum sample size,
- \(n_k\) is the number of participants that have completed follow up in arm \(k=1,...,K\) at look \(j\) so that \((\sum_{k=1}^{K} n_k)/N\) corresponds to the information fraction,
- \(b_{\epsilon} = 0.009\) and \(p_{\epsilon}=3\) are tuning parameters that determine the shape of the efficacy function (Gotmaker et al. 2019).
futility stopping rule: Early stopping of intervention arms for futility may occur if there is a low posterior probability of seeing a minimum clinically important difference of a 20% reduction (i.e., RR=0.8), i.e., when \[ \text{Prob}(\beta_k < \delta_{f}|y, X) < b_f\] where
- \(\delta_{f} = log(0.8)\) denotes the (futility-related) clinically meaningful treatment effect values,
- \(b_{f} = 0.2025\) is a the cut-off value used to declare futility.
trial stopping rule: The trial will run until an efficacy or futility decision has been reached for each intervention arm or once the maximum sample size has been reached.

Note that, in the above,

the parameter values of \(b_{f}\), \(b_{\epsilon}\) and \(p_{\epsilon}\) have been optimised via a grid search to lead to suitable operating characteristics,
the parameters values of \(\delta_{\epsilon}\) and \(\delta_{f}\) have been defined by clinicians.

Self-defined R functions

In the following, we define the group allocation, futility and efficacy functions corresponding to the design described above

Group allocation

We need to generate a function that randomises the m participants of the next look according to the allocation ratios prob, where m and prob are ingredients described in the Ingredients section.

# function
treatalloc.fun = function(m,prob){
  prob = abs(prob)/sum(abs(prob)) 
  m0.g = floor(prob*m)
  m0   = sum(m0.g)
  factor(rep(names(prob),m0.g+rmultinom(1,m-m0,prob)),
         levels=names(prob))
}
# test on m = 40 patients and equal allocation per group
table(treatalloc.fun(m=40,prob=c(control=.25,A=.25,B=.25,C=.25)))
table(treatalloc.fun(m=40,prob=c(control=1,A=1,B=1,C=1)))

treatalloc.fun first allocates the largest possible number of units to the different groups given their exact target probabilities and then assigns randomly the remaining units to the different groups according to multinomial draws.

Note that

this corresponds to the function alloc.balanced available in the BATSS package.
when no RAR rule is used, the allocation probabilities used throughout the trial by batss.glm are the ones indicated under prob0.

Arm efficacy stopping rule

We need to generate a function that leads to a logical output and takes as input

the ingredients
- posterior for the posterior probability of the target parameter being smaller than delta.eff = 0,
- n and N, respectively the sample size per arm at the look of interest and the max sample size to define the information fraction,
the additional parameters (to be added to eff.arm.control in batss.glm)
- \(b_{\epsilon}\) that we will name b.eff,
- \(p_{\epsilon}\) that we will name p.eff.

# function
efficacy.arm.fun = function(posterior,n,N,b.eff,p.eff){
  posterior > (1-(b.eff*(sum(n)/N)^p.eff))
}
# test at interim 1 for a parameter with a posterior = 0.999
efficacy.arm.fun(0.999, n=c(control=25,A=25,B=25,C=25),N=260,
                 b.eff = 0.009,p.eff=3)
# test at interim 1 for a parameter with a posterior = 0.9995
efficacy.arm.fun(0.9995, n=c(control=25,A=25,B=25,C=25),N=260,
                 b.eff = 0.009,p.eff=3)

Note that this correspond to the function eff.arm.infofract available in the BATSS package.

Trial efficacy stopping rule

We need to generate a function that, based on the ingredient eff.target (indicating if efficacy was reached for each target parameter at the stage of interest or before) leads to a logical output equal to TRUE if all target parameters reached efficacy and FALSE otherwise.

# function
efficacy.trial.fun = function(eff.target){
  all(eff.target)
}

# test 
efficacy.trial.fun(c(A=TRUE,B=TRUE,C=TRUE))
efficacy.trial.fun(c(A=TRUE,B=TRUE,C=FALSE))

Note that this corresponds to the function eff.trial.all available in the BATSS package and that this behaviour is the default one of the function batss.glm when eff.trial = NULL.

Arm futility stopping rule

We need to generate a function that leads to a logical output and takes as input

the ingredients posterior for the posterior probability of the target parameter being smaller than delta.fut = log(0.8),
the additional parameters \(b_{f}\) that we will name b.fut (and that needs to be added to fut.arm.control in batss.glm)

# function
futility.arm.fun = function(posterior,b.fut){
  posterior < b.fut
}
# test 
futility.arm.fun(0.9, b.fut=.2025)
futility.arm.fun(0.1, b.fut=.2025)

Note that this corresponds to the function fut.arm.simple available in the BATSS package.

Trial futility stopping rule

We need to generate a function that, based on the ingredient fut.target (indicating if futility was declared for each target parameter at the stage of interest or before) leads to a logical output equal to TRUE if all target parameters were declared futile and FALSE otherwise.

# function
futility.trial.fun = function(fut.target){
  all(fut.target)
}

# test 
futility.trial.fun(c(A=TRUE,B=TRUE,C=TRUE))
futility.trial.fun(c(A=TRUE,B=TRUE,C=FALSE))

Note that this corresponds to the function fut.trial.all available in the BATSS package and that this behaviour is the default one of the function batss.glm when fut.trial = NULL.

Monte Carlo Simulations

We consider three scenarios:

Scenario 1 = ‘global null’: each arm has a cumulative mean number of lesions equal to 4,
Scenario 2 = ‘one treatment works’: each arm has a cumulative mean number of lesions equal to 4 except the last one, “C”, that shows a 60% reduction in the cumulative mean number of lesions compared to the other groups (RR = 0.4),
Scenario 3 = ‘better, best’: the control arm has a cumulative mean number of lesions equal to 4 and the other arms respectively lead to a 20%, 40% and 60% reduction in the cumulative mean number of lesions compared to the reference group (i.e., RR = 0.8 for “A”, RR = 0.6 for “B” and RR = 0.4 for “C”).

Scenario 1

R = 25

scenario1 = batss.glm(   
  model           = y~treatment,
  var             = list(y = rnbinom,          
                         treatment = treatalloc.fun),
  var.control     = list(y = list(size = 1/2)), 
  family          = "nbinomial",
  link            = "log",
  beta            = c(log(4),log(1),log(1),log(1)),
  which           = c(2:4),
  R               = R,
  alternative     = c("less"),
  RAR             = NULL,
  prob0           = c(control = .25, A = .25, B = .25, C = .25),
  N               = 260,
  interim         = list(recruited=c(100,140,180,220)),
  eff.arm         = efficacy.arm.fun,
  delta.eff       = 0, 
  eff.arm.control = list("b.eff"=0.009, "p.eff"=3),
  delta.fut       = log(0.8), 
  fut.arm         = futility.arm.fun,
  fut.arm.control = list("b.fut"=0.2025),
  computation     = "parallel",
  mc.cores        = 10,
  H0              = TRUE,
  extended        = 1)

You can note that

y is generated via the function rnbinom with nuisance parameter equal to 0.5 (size = 1/2) and expected values on the log scale of respectively
- log(4) for the control group,
- log(4)+log(1) = log(4) for all other groups (as contrasts of type treatment are used for the factor treatment in the self-defined function treatalloc.fun),
the target parameters (corresponding to the shift in means of treatment arms “A”, “B” and “C” compared to the “control” on the log scale) are in position 2, 3 and 4 of the fitted coefficients obtained when using the formula y~treatment,
prob0 provides
- the (equal) allocation probabilities throughout the trial (as RAR = NULL),
- the names of the groups,
eff.arm and fut.arm are set to the functions defined above (i.e., efficacy.arm.fun and futility.arm.fun),
eff.trial and fut.trial are not specified and therefore equal to NULL (default) which leads to the behaviour wished in this case,
the values of the additional parameters of efficacy.arm.fun and futility.arm.fun (i.e., b.eff, p.eff and b.fut) are specified under eff.arm.control and fut.arm.control,
delta.eff and delta.fut are respectively set to 0 and log(0.8).

We chose here a low number of seeds/trials (R=25) to save time.

Scenario 2

R = 25

scenario2 = batss.glm(   
  model           = y~treatment,
  var             = list(y = rnbinom,          
                         treatment = treatalloc.fun),
  var.control     = list(y = list(size = 1/2)), 
  family          = "nbinomial",
  link            = "log",
  beta            = c(log(4),log(1),log(1),log(0.4)),
  which           = c(2:4),
  R               = R,
  alternative     = c("less"),
  RAR             = NULL,
  prob0           = c(control = .25, A = .25, B = .25, C = .25),
  N               = 260,
  interim         = list(recruited=c(100,140,180,220)),
  eff.arm         = efficacy.arm.fun,
  delta.eff       = 0, 
  eff.arm.control = list("b.eff"=0.009, "p.eff"=3),
  delta.fut       = log(0.8), 
  fut.arm         = futility.arm.fun,
  fut.arm.control = list("b.fut"=0.2025),
  computation     = "parallel",
  mc.cores        = 10,
  H0              = TRUE,
  extended        = 1)

Same as above except for beta which now leads to a mean of

exp(log(4)) = 4 for the control group,
exp(log(4) + log(1)) = 4 (no effect) for the groups “A” and “B”,
exp(log(4) + log(0.4)) = 1.6 (target effect) for group “C” which corresponds to a RR of 1.6/4 = 0.4.

Scenario 3

R = 25

scenario3 = batss.glm(   
  model           = y~treatment,
  var             = list(y = rnbinom,          
                         treatment = treatalloc.fun),
  var.control     = list(y = list(size = 1/2)), 
  family          = "nbinomial",
  link            = "log",
  beta            = c(log(4),log(0.8),log(0.6),log(0.4)),
  which           = c(2:4),
  R               = R,
  alternative     = c("less"),
  RAR             = NULL,
  prob0           = c(control = .25, A = .25, B = .25, C = .25),
  N               = 260,
  interim         = list(recruited=c(100,140,180,220)),
  eff.arm         = efficacy.arm.fun,
  delta.eff       = 0, 
  eff.arm.control = list("b.eff"=0.009, "p.eff"=3),
  delta.fut       = log(0.8), 
  fut.arm         = futility.arm.fun,
  fut.arm.control = list("b.fut"=0.2025),
  computation     = "parallel",
  mc.cores        = 10,
  H0              = TRUE,
  extended        = 1)

Same as above except for beta which now leads to a mean of

exp(log(4)) = 4 for the control group,
exp(log(4) + log(0.8)) = 3.2 (small effect) for group “A” which corresponds to a RR of 3.2/4 = 0.8,
exp(log(4) + log(0.6)) = 2.4 (medium effect) for group “B” which corresponds to a RR of 2.4/4 = 0.6,
exp(log(4) + log(0.4)) = 1.6 (target effect) for group “C” which corresponds to a RR of 1.6/4 = 0.4.

References

Curtin, Francois, Herve Porchet, Robert Glanzman, Hans Martin Schneble, Virginie Vidal, Marie-Laure Audoli-Inthavong, Estelle Lambert, and Hans Peter Hartung. 2016. “A Placebo Randomized Controlled Study to Test the Efficacy and Safety of GNbAC1, a Monoclonal Antibody for the Treatment of Multiple Sclerosis – Rationale and Design.” Multiple Sclerosis and Related Disorders 9: 95–100. https://doi.org/https://doi.org/10.1016/j.msard.2016.07.002.

Gotmaker, Robert, Michael J Barrington, John Reynolds, Lorenzo Trippa, and Stephane Heritier. 2019. “Bayesian Adaptive Design: The Future for Regional Anesthesia Trials?” Regional Anesthesia & Pain Medicine 44 (6): 617–22. https://doi.org/10.1136/rapm-2018-100248.

Hartung, Hans-Peter, Tobias Derfuss, Bruce AC Cree, Maria Pia Sormani, Krzysztof Selmaj, Jonathan Stutters, Ferran Prados, et al. 2022. “Efficacy and Safety of Temelimab in Multiple Sclerosis: Results of a Randomized Phase 2b and Extension Study.” Multiple Sclerosis Journal 28 (3): 429–40. https://doi.org/10.1177/13524585211024997.