Non-normal endpoints
non-normal_endpoints.RmdIn previuos sections we have focused on normally distributed endpoints. Based on asymptotic theory, MAMS can also handle non-normal endpoints by exploiting the asymptotic properties of efficient score statistics (Jaki and Magirr 2013), as we will demonstrate for ordinal, binary, and time-to-event outcome data.
Ordinal and binary endpoints
Ordinal data consist of multiple different categories that have a natural order, which is common for quality-of-life scores, pain scores, and similar questionnaire-based outcomes. Our illustration here is motivated by the ASCLEPIOS study (Whitehead 1993) and its example analyses in (Whitehead and Jaki 2008) and (Jaki and Magirr 2012).
We design a MAMS trial with three experimental treatments and a control arm, one interim analysis after half the patients have provided an outcome measure, and triangular boundaries in a setting with an ordinal primary endpoint, under the assumption of proportional odds. We expect that under control conditions the probabilities of falling into each of six categories, ordered from best to worst, are 0.075, 0.182, 0.319, 0.243, 0.015, and 0.166. Suppose the interesting effect is a doubling in the probability of falling into one of the two best categories combined, from 25.7 to 51.4%, for any experimental arm. This corresponds to an odds ratio (OR) of 3.06 and a log-OR of 1.12. The uninteresting effect shall be one quarter of the interesting effect on the log-OR scale, i.e., a log-OR of 0.28 or an OR of 1.32.
To find the boundary values and sample sizes, we can use the function
ordinal.mams, which is a wrapper for mams with
additional inputs prob for the probabilities of falling
into each category (which must sum up to one), as well as
or and or0 for the interesting and
uninteresting treatment effects, respectively, on the OR scale:
prob <- c(0.075, 0.182, 0.319, 0.243, 0.015, 0.166)
mord <- ordinal.mams(prob = prob, or = 3.06, or0 = 1.32, K = 3, J = 2,
alpha = 0.05, power = 0.9, r = 1:2, r0 = 1:2, ushape = "triangular",
lshape = "triangular")
mord## Design parameters for a 2 stage trial with 3 treatments
##
## Stage 1 Stage 2
## Cumulative sample size per stage (control): 34 68
## Cumulative sample size per stage (active): 34 68
##
## Maximum total sample size: 272
##
## Stage 1 Stage 2
## Upper bound: 2.330 2.197
## Lower bound: 0.777 2.197The function ordinal.mams can also be used for binary
endpoints as they are a simple special case of ordinal data where \(prob\) has only two categories
(success/failure, yes/no, etc.) and the proportional odds assumption
becomes obsolete.
Time-to-event endpoints
Another useful extension of MAMS is to event-time outcomes, e.g., when the primary endpoint is survival. In that case the effect sizes \(δ\) and \(δ_0\) must be specified in terms of log-hazard ratios (log-HRs), which are assumed to be asymptotically normal, and the standard deviation is \(σ = 1\). Sample sizes are expressed in terms of events (e.g., deaths), \(e\), rather than numbers of patients, \(n\). As a consequence, we set \(r^{(1)}_k = 1\) and \(r^{(j)}_k = e^{(j)}_0 / e^{(1)}_0\). The underlying approximation should work well if the effect size is small, the number of allocated patients per arm is equal at each stage, and there are few ties in relation to the number of different event times.
Assume we want to design a MAMS trial with three experimental treatment arms and a control, using triangular boundaries. One interim analysis is to be conducted upon observing \(e^{(1)}_0\) events in the control arm, set to half of the total number of events in that arm. Our interesting effect size is a HR of 1.5, corresponding to a log-HR of 0.405, and the uninteresting effect size is a HR of 1.1, i.e., a log-HR of 0.095.
We can calculate the boundary values and sample sizes with the
function tite.mams, which is another wrapper for
mams() with additional inputs hr and
hr0 for the interesting and uninteresting treatment
effects, respectively, on the HR scale:
mtite <- tite.mams(hr = 1.5, hr0 = 1.1, K = 3, J = 2, alpha = 0.05,
power = 0.9, r = 1:2, r0 = 1:2, ushape = "triangular",
lshape = "triangular")
mtite## Design parameters for a 2 stage trial with 3 treatments
##
## Stage 1 Stage 2
## Cumulative number of events per stage (control): 81 162
## Cumulative number of events per stage (active): 81 162
##
## Maximum total number of events: 648
##
## Stage 1 Stage 2
## Upper bound: 2.330 2.197
## Lower bound: 0.777 2.197The sample size output here is given as the required number of events, which is obviously smaller than the required number of patients. We refer to (Whitehead 2001) for guidance how to estimate the maximum total number of patients to be recruited.