Computing bounds under non-constant treatment effect
Keaven M. Anderson
Source:vignettes/articles/story-compute-npe-bound.Rmd
story-compute-npe-bound.RmdOverview
We consider group sequential designs with possibly non-constant
treatment effects over time. This can be useful for situations such as
an assumed non-proportional hazards model as laid out in
vignettes/articles/story-npe-background.Rmd. In general, we
assume \(K \geq 1\) analyses with
statistical information \(\mathcal{I}_k\) and information fraction
\(t_k=\mathcal{I}_k/\mathcal{I}_k\) at
analysis \(k\), \(1\leq k\leq K\). We denote the null
hypothesis \(H_{0}\): \(\theta(t)=0\) and an alternate hypothesis
\(H_1\): \(\theta(t)=\theta_1(t)\) for \(t> 0\) where \(t\) represents the information fraction for
a study. While a study is planned to stop at information fraction \(t=1\), we define \(\theta(t)\) for \(t>0\) since a trial can overrun its
planned statistical information at the final analysis. As before, we use
a shorthand notation in to have \(\theta\) represent \(\theta()\), \(\theta=0\) to represent \(\theta(t)\equiv 0\) for all \(t\) and \(\theta_1\) to represent \(\theta_i(t_k)\), the effect size at
analysis \(k\), \(1\leq k\leq K\).
For our purposes, \(H_0\) will represent no treatment difference, but it could represent a non-inferiority hypothesis. Recall that we assume \(K\) analyses and bounds \(-\infty \leq a_k< b_k<\leq \infty\) for \(1\leq k < K\) and \(-\infty \leq a_K\leq b_K<\infty\). We denote the probability of crossing the upper boundary at analysis \(k\) without previously crossing a bound by
\[\alpha_{k}(\theta)=P_{\theta}(\{Z_{k}\geq b_{k}\}\cap_{j=1}^{k-1}\{a_{j}\leq Z_{j}< b_{j}\}),\] \(k=1,2,\ldots,K.\) The total probability of crossing an upper bound prior to crossing a lower bound is denoted by
\[\alpha(\theta)\equiv\sum_{k=1}^K\alpha_k(\theta).\]
For non-binding bounds, we define the probability
\[\alpha_{k}^{+}(\theta)=P_{\theta}\{\{Z_{k}\geq b_{k}\}\cap_{j=1}^{k-1} \{Z_{j}< b_{j}\}\}\] which ignores the lower bounds when computing upper boundary crossing probabilities. The non-binding Type I error is the probability of ever crossing the upper bound when \(\theta=0\). The value \(\alpha^+_{k}(0)\) is commonly referred to as the amount of Type I error spent at analysis \(k\), \(1\leq k\leq K\). The total upper boundary crossing probability for a trial is denoted in this one-sided scenario by \[\alpha^+(\theta) \equiv\sum_{k=1}^{K}\alpha^+_{k}(\theta).\] We will primarily be interested in \(\alpha(\theta)\) to compute power when \(\theta > 0\). For Type I error, we may be interested in \(\alpha(0)\) for binding lower bounds, but more often we will consider non-binding Type I error calculations, \(\alpha^{+}(0)\).
We denote the probability of crossing a lower bound at analysis \(k\) without previously crossing any bound by
\[\beta_{k}(\theta)=P_{\theta}((Z_{k}< a_{k}\}\cap_{j=1}^{k-1}\{ a_{j}\leq Z_{j}< b_{j}\}).\]
Efficacy bounds \(b_k\), \(1\leq k\leq K\), for a group sequential design will be derived to control Type I at some level \(\alpha=\alpha(0)\).
Lower bounds \(a_k\), \(1\leq k\leq K\) may be used to control boundary crossing probabilities under either the null hypothesis (2-sided testing), the alternate hypothesis or some other hypothesis (futility testing).
Thus, we may consider up to 3 values of \(\theta(t)\):
- under the null hypothesis \(\theta_0(t)=0\) for computing efficacy bounds,
- under a value \(\theta_1(t)\) for computing lower bounds, and
- under a value \(\theta_a(t)\) for computing sample size or power.
We refer to the information under these 3 assumptions as \(\mathcal{I}^{(0)}(t)\), \(\mathcal{I}^{(1)}(t)\), and \(\mathcal{I}^{(a)}(t)\), respectively. Often we will assume \(\mathcal{I}(t)=\mathcal{I}^{(0)}(t)=\mathcal{I}^{(1)}(t)=\mathcal{I}^{(a)}(t).\)
We note that information may differ under different values of \(\theta(t)\). For fixed designs, Lachin (2009) computes sample size based on different variances under the null and alternate hypothesis.
Spending bounds
We consider different boundary types in the gsDesign
package and simplify them into two types according to whether lower
bounds are binding or non-binding. The concept is to implicitly derive
Z-value bounds \(a_k, b_k,
k=1,\cdots,K\) based on probabilities specified in the following
table. We include the test.type argument from the
gsDesign::gsDesign() function for reference.
test.type |
Upper bound | Lower bound | Design type |
|---|---|---|---|
| 1 | \(\alpha_k^{+}(0)\) | None | One-sided efficacy |
| 2 | \(\alpha_k(0)\) | \(\alpha_k(0)\) | 2-sided symmetric |
| 3 | \(\alpha_k(0)\) | \(\beta_k(\theta_a)\) | \(\beta\)-spending with binding futility |
| 4 | \(\alpha_k^{+}(0)\) | \(\beta_k(\theta_a)\) | \(\beta\)-spending with non-binding futility |
| 5 | \(\alpha_k(0)\) | \(\beta_i(\theta_1)\) | \(\theta\)-spending with binding futility |
| 6 | \(\alpha^{+}(0)\) | \(\beta_i(\theta_1)\) | \(\theta\)-spending with non-binding futility |
This can be reduced to just two types distinguishing by whether or not lower bounds are binding or non-binding:
test.type |
Upper bound | Lower bound | Design type |
|---|---|---|---|
| 2, 3, 5 | \(\alpha_k(0)\) | \(\beta_k(\theta)\) | Binding lower bound |
| 1, 4, 6 | \(\alpha_k^{+}(0)\) | \(\beta_k(\theta)\) | Non-binding lower bound |
In this second table we have used \(\theta=0\) to derive the upper bound to
control Type I error in all cases. We have chosen some arbitrary \(\theta\) which could be 0 for any other
test.type, \(\theta_a\)
for \(\beta\)-spending or some
arbitrary \(\theta_1\) otherwise. We
note that for a one-sided design we let \(\beta_k(\theta)=0\) so that \(a_k=-\infty, k=1,\cdots,K\). For
test.type=3, 4 we let \(\theta=\theta_a\), while for
test.type=5, 6 \(\theta \geq
0\) is arbitrary. We note that asymmetric \(\alpha\)-spending bounds can be derived
using test.type > 2 and \(\theta=0.\)
Two-sided testing and design
We denote an alternative \(H_{a}\): \(\theta(t)=\theta_a(t)\); we will always assume \(H_a\) for power calculations; when using \(\beta\)-spending we will also use \(H_a\) for controlling lower boundary \(a_k\) crossing probabilities by letting \(\theta=\theta_a\) for lower bound spending. A value of \(\theta(t)>0\) will reflect a positive benefit. We will not restrict the alternate hypothesis to \(\theta_a(t)>0\) for all \(t\). The value of \(\theta(t)\) will be referred to as the (standardized) treatment effect at information fraction \(t\).
We assume there is interest in stopping early if there is good evidence to reject one hypothesis in favor of the other.
If \(a_k= -\infty\) at analysis \(k\) for some \(1\leq k\leq K\) then the alternate hypothesis cannot be rejected at analysis \(k\); i.e., there is no futility bound at analysis \(k\). For \(k=1,2,\ldots,K\), the trial is stopped at analysis \(k\) to reject \(H_0\) if \(a_j<Z_j< b_j\), \(j=1,2,\dots,i-1\) and \(Z_k\geq b_k\). If the trial continues until stage \(k\) without crossing a bound and \(Z_k\leq a_k\) then \(H_1\) is rejected in favor of \(H_0\), \(k=1,2,\ldots,K\). Note that if \(a_K< b_K\) there is the possibility of completing the trial without rejecting \(H_0\) or \(H_1\) unless \(a_K=b_K.\)
Haybittle-Peto and spending bounds
The recursive algorithm of the previous section allows computation of both spending bounds and Haybittle-Peto bounds. For a Haybittle-Peto efficacy bound, one would normally set \(b_k=\Phi^{-1}(1-\epsilon)\) for \(k=1,2,\ldots,K-1\) and some small \(\epsilon>0\) such as \(\epsilon= 0.001\) which yields \(b_k=3.09\). While the original proposal was to use \(b_K=\Phi^{-1}(1-\alpha)\) at the final analysis, to fully control one-sided Type I error at level \(\alpha\) we suggest computing the final bound \(b_K\) using the above algorithm so that \(\alpha(0)=\alpha\).
Bounds computed with spending \(\alpha_k(0)\) at analysis \(k\) can be computed by using equation (9) for \(b_1\). Then for \(k=2,\ldots,K\) the algorithm of the previous section is used. As noted by Jennison and Turnbull (1999), \(b_1,\ldots,b_K\) if determined under the null hypothesis depend only on \(t_k\) and \(\alpha_k(0)\) with no dependence on \(\mathcal{I}_k\), \(k=1,\ldots,K\). When computing bounds based on \(\beta_k(\theta)\), \(k=1,\ldots,K\), where some \(\theta(t_k)\neq 0\) we have an additional dependency with \(a_k\) depending not only on \(t_k\) and \(b_k\), \(k=1,\ldots,K\), but also on the final total information \(\mathcal{I}_K\). Thus, a spending bound under something other than the null hypothesis needs to be recomputed each time \(\mathcal{I}_K\) changes, whereas it only needs to be computed once when \(\theta(t_k)=0\), \(k=1,\ldots,K\).
Bounds based on boundary families
Assume constants \(b_1^*,\ldots,b_K^*\) and a total targeted one-sided Type I error \(\alpha\). We wish to find \(C_u\) as a function of \(t_1,\ldots t_K\) such that if \(b_k=C_ub_k^*\) then \(\alpha(0)=\alpha.\) Thus, the problem is to solve for \(C_u\). If \(a_k\), \(k=1,2,\ldots,K\) are fixed then this is a simple root finding problem. Since one normally normally uses non-binding efficacy bounds, it will normally be the case that \(a_k=-\infty\), \(k=1,\ldots,K\) for this problem.
Now we assume constants \(a_k^*\) and wish to find \(C_l\) such that if \(a_k=C_la_k^*+\theta(t_k)\sqrt{\mathcal{I}_k}\) for \(k=1,\ldots,K\) then \(\beta(\theta)=\beta\). If we use the constant upper bounds from the previous paragraph, finding \(C_l\) is a simple root-finding problem.
For 2-sided symmetric bounds with \(a_k=-b_k\), \(k=1,\ldots,K\), we only need to solve for \(C_u\) and again use simple root finding.
At this point, we do not solve for this type of bound for asymmetric upper and lower bounds.
Sample size
For sample size, we assume \(t_k\), and \(\theta(t_k)\) \(1,\ldots,K\) are fixed. We assume \(\beta(\theta)\) is decreasing as \(\mathcal{I}\) is decreasing. This will automatically be the case when \(\theta(t_k)>0\), \(k=1,\ldots,K\) and for many other cases. Thus, the information required is done by a search for \(\mathcal{I_K}\) that yields \(\alpha(\theta)\) yields the targeted power.