10  Conditional power and B-values

In some cases, rather than working with \(Z_{1}, Z_{2}, \ldots, Z_{k}\) as in Section 2.3, it is desirable to consider variables representing incremental sets of observations between analyses. We consider a discrete version of the Brownian motion formulation of Proschan, Lan and Wittes (Michael A. Proschan, Lan, and Wittes 2006) and apply the same principles to demonstrate how to compute conditional power and conditional error using the calculation tools we have been applying for group sequential designs. Those who prefer to skip the background materials can proceed directly to Section 10.4.1.

10.1 Z-values, B-values and S-values

We let \(X_{1}\), \(X_{2},\ldots\) be independent and identically distributed normal random variables with mean \(\delta\) and variance \(\sigma^2\), as before. We let \(0<n_1<n_2\ldots<n_k\) for some \(k>1\). For \(i=1,\) \(2,\ldots, k\) define \(\mathcal{I}_i=n_i/\sigma^2\) and \(t_i=n_i/n_k=\mathcal{I}_i/\mathcal{I}_k\). We next define treatment effect estimates, S-values, B-values, and Z-values as follows for \(i=1,2,\ldots,k\):

\[ \begin{array}{rclclcl} S_{i,j}&=&\sum_{m=n_i+1}^{n_j}X_m&\sim& N(n_{i,j}\delta,n_i\sigma^2),\\ \\ Z_{i,j}&=&S_{i,j}/(\sqrt{n_{i,j}}\sigma)&\sim& N(\sqrt{n_{i,j}}\theta,1)&\sim&N(\sqrt{\mathcal{I}_{i,j}}\delta,1), \\ \\ B_{i,j}&=&S_{i,j}/(\sqrt{n_k}\sigma)&\sim& N(\sqrt{n_k}t_{i,j}\theta,t_{i,j})&\sim&N(\sqrt{\mathcal{I}_k}t_{i,j}\delta,t_{i,j}). \end{array} \]

Note that \(B_i=\sqrt{t_i}Z_i\), \(1\le i\le k\). Letting \(\theta=\delta/\sigma\) we write a formulation of the above in terms of the standardized parameter \(\theta\):

\[ \begin{array}{rcl} \hat{\theta}_i&=&\hat{\delta}_i/\sigma\sim N(\theta,1/n_i),\\ \\ Z_{i}&=&\sqrt{n_i}\hat{\theta}_i\sim N(\sqrt{n_i}\theta, 1),\\ \\ B_{i}&=&\sqrt{t_i}Z_i\sim N(\sqrt{n_k} t_i\theta, t_i). \end{array} \]

These last forms for \(Z_i\) and \(B_i\) have been written in a general format that will apply to trials with many types of endpoints where \(\hat{\delta_i}\), \(\hat{\sigma_i}\) and \(\mathcal{I}_i\) take different forms. Scharfstein, Tsiatis and Robbins (Scharfstein, Tsiatis, and Robins 1997) have generalized work from many others by showing that if \(Z_i\) is an asymptotically efficient test for a single parameter in a parametric or semi-parametric model then the asymptotic theory holds. Proschan, Lan and Wittes (Michael A. Proschan, Lan, and Wittes 2006) provide extensive coverage of application of the Brownian motion version of B-values used here to group sequential trials with several different types of endpoints.

We now re-write bounds and boundary crossing probabilities in terms of B-values. For \(i=1,2,\ldots,k\) we define \(b_i=u_i\sqrt{t_i}\), \(a_i=l_i\sqrt{t_i}\) and note that

\[ \begin{array}{rcl} \alpha_i(\theta)&=&P_{\theta}\{\{Z_i \geq u_i\} \cap_{j=1}^{i-1}\{l_j\leqslant Z_j<u_j\}\}\\ \\ &=&P_{\theta}\{\{B_i \geq b_i\} \cap_{j=1}^{i-1}\{a_j\leqslant B_j<b_j\}\}.\\ \end{array} \tag{10.1}\]

Formulas for \(\beta_i(\theta)\) and \(\alpha_i^{+}(\theta)\) can be rewritten in an analogous fashion:

\[\begin{align} \alpha_i^{+}(\theta) & = & P_{\theta}\{\{B_i \geq b_i\} \cap_{j=1}^{i-1}\{B_j<b_j\}\}\\ \beta_i(\theta) & = & P_{\theta}\{\{B_i \leqslant a_i\} \cap_{j=1}^{i-1}\{a_j\leqslant B_j<b_j\}\} \end{align}\]

10.2 Incremental formulation

We now formulate a group sequential set of outcomes in terms of independent increments to simplify conditional power calculations. Define \(n_0=S_0=0\) and for \(0\leq i < j\leq k\), \(n_{i,j}=n_j-n_i\), \(t_{i,j}=n_{i,j}/n_k\), \(\mathcal{I}_{i,j}=n_{i,j}/\sigma^2\)

\[ \begin{array}{rclclcl} S_{i,j}&=&\sum_{m=n_i+1}^{n_j}X_m&\sim& N(n_{i,j}\delta,n_i\sigma^2),\\ \\ Z_{i,j}&=&S_{i,j}/(\sqrt{n_{i,j}}\sigma)&\sim& N(\sqrt{n_{i,j}}\theta,1)&\sim&N(\sqrt{\mathcal{I}_{i,j}}\delta,1), \\ \\ B_{i,j}&=&S_{i,j}/(\sqrt{n_k}\sigma)&\sim& N(\sqrt{n_k}t_{i,j}\theta,t_{i,j})&\sim&N(\sqrt{\mathcal{I}_k}t_{i,j}\delta,t_{i,j}). \end{array} \]

Analogous to before, \(B_{i,j}=\sqrt{t_{i,j}}Z_{i,j}\) and we can apply the ultimate conditional power and conditional error calculations below broadly to group sequential designs for which the general asymptotic theory is applicable. For \(0\le i < j\leq k\) we note that \(B_i=B_{0,i}\) and that \(B_j=B_i+B_{i,j}\) where \(B_i\) and \(B_{i,j}\) are independent and normally distributed. Thus, the conditional distribution of \(B_j\) assuming \(B_i=c\) for some \(1<i<j\leqslant k\) and constant \(c\) is normal with mean \(c+\sqrt{n_k}t_{i,j}\theta\) and variance \(t_{i,j}\).

10.3 Simple conditional power and conditional error

Proschan, Lan and Wittes (Michael A. Proschan, Lan, and Wittes 2006) and others compute a simple version of conditional boundary crossing probabilities that ignores interim stopping boundaries between an interim analysis that we condition on and some future analysis of interest. That is, assuming \(1\leq i < j\leq k\) we assume \(B_i=c\) and we compute conditional power as

\[ \begin{split} P\{B_j\geq b_j|B_i=c\}&=P\{B_{i,j}\geq b_j-c\}\\ &=1-\Phi\left(\frac{b_j-c-\theta t_{i,j}\sqrt{n_k}}{\sqrt{t_{i,j}}}\right)\\ &=1-\Phi\left(\frac{b_j-c-\delta t_{i,j}\sqrt{\mathcal{I}_k}}{\sqrt{t_{i,j}}} \right) \end{split} \tag{10.2}\]

10.3.1 Simple conditional power for the CAPTURE trial

We consider the interim data as previously displayed in Chapter 9, but restrict ourselves initially to the 2 interim analysis design. Recall that test statistics are in z and the design in CAPTURE. We apply Equation 10.2 to compute the simple conditional power and conditional error of crossing each future bound at the time of the first interim analysis. For each analysis, we place the information fractions (fractions of final sample size in this case) in ti and the B-value bounds for the design in b. We use the standardized parameter \(\theta\) formulation from equation Equation 10.2. The observed B-values corresponding to the first 3 analyses are placed in bobs. We then store the effect size powered for in theta1 and the observed interim effect size at the first analysis estimated by \(\hat{\theta}_1=Z_1/\sqrt{n_1}\) in thetahat. Finally, we compute the standard deviation for \(B_{1,j}\) for \(j=2,3,4\) as \(t_j-t_1\) and store it in sd1j.

library(gsDesign)

n1 <- c(175, 353, 532, 635)
n2 <- c(175, 347, 518, 630)
x1 <- c(30, 55, 84, 101)
x2 <- c(14, 37, 55, 71)
z <- testBinomial(x1 = x1, x2 = x2, n1 = n1, n2 = n2)

n.fix <- nBinomial(p1 = 0.15, p2 = 0.1, beta = 0.2)
x <- gsDesign(
  k = 3,
  n.fix = n.fix,
  beta = 0.2,
  sfupar = -3
)
x <- gsDesign(
  k = 3,
  n.fix = n.fix,
  timing = c(350, 700) / x$n.I[3],
  beta = 0.2,
  sfupar = -3
)
x <- gsDesign(
  k = 3,
  n.fix = n.fix,
  timing = c(350, 700) / x$n.I[3],
  beta = 0.2,
  sfupar = -3
)
CAPTURE <- gsDesign(
  k = 3,
  n.fix = n.fix,
  timing = c(350, 700) / x$n.I[3],
  beta = 0.2,
  sfupar = -3,
  endpoint = "binomial",
  delta1 = 0.05
)

n <- (n1 + n2)[1:2]
ti <- c(n / CAPTURE$n.I[3], 1)
b <- CAPTURE$upper$bound * sqrt(ti)
bobs <- z[1:2] * sqrt(ti[1:2])
theta1 <- CAPTURE$delta
thetahat <- z[1:2] / sqrt(n)
t1j <- ti[2:3] - ti[1]
n1j <- c(n[2], CAPTURE$n.I[3]) - n[1]
sd1j <- sqrt(t1j)
b1j <- b[2:3] - bobs[1]

The simple conditional error after the first interim analysis is

simpCError <- pnorm(
  b1j,
  sd = sd1j,
  mean = 0,
  lower = FALSE
)
simpCError
#> [1] 0.1028575 0.2001852

Plugging the originally powered effect size produces one conditional power estimate.

simpCPwr1 <- pnorm(
  b1j,
  sd = sd1j,
  mean = theta1 * sqrt(CAPTURE$n.I[3]) * t1j,
  lower = FALSE
)
simpCPwr1
#> [1] 0.5596153 0.9523688

The conditional power assuming the observed interim effect size is

simpCPwrHat <- pnorm(
  b1j,
  sd = sd1j,
  mean = thetahat * sqrt(CAPTURE$n.I[3]) * t1j,
  lower = FALSE
)
simpCPwrHat
#> [1] 0.9056190 0.9421038

10.4 Conditional power and conditional error

As an alternative to futility bounds based on \(\beta\)-spending, stopping rules for futility are sometimes based on the conditional power of a positive trial given the value of a test statistic at an interim analysis. Thus, we consider the conditional probabilities of boundary crossing for a group sequential design given an interim result. Assume \(1\le i < j \leqslant k\) and \(B_i=c\). The conditional probabilities for first crossing an upper boundary at analysis \(j\) given \(B_i=c\) is

\[\begin{align} \alpha _{i,j}(\theta |B_i=c) & = & P_{\theta }\{\{B_{j}\geqslant b_j\}\cap_{m=i+1}^{j-1}\{a_m\leqslant B_{0,m}<b_m\}|B_{0i}=c\}\nonumber\\ & = & P_{\theta }\{\{B_{ij}\geqslant b_j-c\}\cap_{m=i+1}^{j-1}\{a_m-c\leqslant B_{im}<b_m-c\}\} \end{align}\]

Formulas for \(\alpha^{+}_{i,j}(\theta |B_i=c)\) and \(\beta_{i,j}(\theta |c)\) can be written analogously:

\[\begin{align} \alpha^{+}_{i,j}(\theta |B_i=c) & = & P_{\theta }\{\{B_{i,j}\geqslant b_j-c\}\cap_{m=i+1}^{j-1}\{ B_{im}<b_m-c\}\}\\ \beta _{i,j}(\theta |B_i=c) & = & P_{\theta }\{\{B_{i,j}\leqslant b_j-c\}\cap_{m=i+1}^{j-1}\{a_m-c\leqslant B_{i,m}<b_m-c\}\} \end{align}\]

Given the above and Equation 10.1, conditional boundary crossing probabilities can be computed using the same numerical tools as unconditional boundary crossing probabilities. The conditional design generated is that of Muller and Schafer (Müller and Schäfer 2001).

Given the above and Equation 10.1, conditional boundary crossing probabilities can be computed using the same numerical tools as unconditional boundary crossing probabilities. We rewrite the incremental \(B\)-value bounds as incremental \(Z\)-value bounds as these are values produced in the function gsCP() provided below. Recalling that the incremental \(Z\)-value is denoted \(Z_{i,j}=B_{i,j}/\sqrt{t_{i,j}}\) for \(0\leq i<j\leq k\). If the interim B-value at analysis \(i\) is \(c\) as above, we define \(z_i=c/\sqrt{t_i}\). Then, crossing an upper bound with \(B_{i,j}\geq b_j-c\) later at analysis \(j>i\) is equivalent to

\[ Z_{i,j}=\frac{B_{i,j}}{\sqrt{t_{i,j}}} \geq \frac{b_j-c}{\sqrt{t_{i,j}}}= \frac{u_j\sqrt{t_j}-z_i\sqrt{t_i}}{\sqrt{t_{i,j}}}. \]

The incremental \(Z\)-value upper and lower bounds are defined respectively by

\[ u_{i,j}(z_{i})=\frac{u_{j}\sqrt{t_{j}}-z_{i}\sqrt{t_{i}}}{\sqrt{t_{i,j}}} \tag{10.3}\]

and

\[ l_{i,j}(z_{i})=\frac{l_{j}\sqrt{t_{j}}-z_{i}\sqrt{t_{i}}}{\sqrt{t_{i,j}}}. \tag{10.4}\]

10.4.1 Application to the CAPTURE trial

We now examine the observations after the initial interim that would be required to cross the bounds for the original design. The function gsCP() computes a new set of boundaries \(l_{i,j}\) and \(u_{i,j}\) from Equation 10.3 and Equation 10.4 based on results at interim \(i\). Values of \(\theta\) for which conditional power is computed are, by default,

• \(\theta = 0\), for conditional error.
• \(\theta = \hat{\theta}\) to compute conditional power at the interim trend as in, for example, Bauer and Köhne (Bauer and Kohne 1994), Proschan and Hunsberger (Michael A. Proschan and Hunsberger 1995) and Cui, Hung and Wang (Cui, Hung, and Wang 1999).
• \(\theta = \delta\), the treatment effect at which the trial was originally powered, as in Liu and Chi (Liu and Chi 2001).

xCP1 <- gsCP(x = CAPTURE, i = 1, zi = z[1])

We compute the conditional power after the first interim analysis based on the interim estimate of \(\theta\),

sum(xCP1$upper$prob[, 1])
#> [1] 0.9999172

the conditional error when \(\theta=0\),

sum(xCP1$upper$prob[, 2])
#> [1] 0.2449957

and the conditional power based on \(\theta=\delta\) where \(\delta\) is the value for which the trial was originally powered.

sum(xCP1$upper$prob[, 3])
#> [1] 0.9577563

The probability of crossing at the second interim analysis given the first is the same as previously computed in Section 10.3, interim treatment effect size, \(\theta = \theta_1\), the effect size originally powered for, and \(\theta = 0\), respectively.

xCP1$upper$prob[1, 1]
#> [1] 0.905619

simpCPwrHat[1]
#> [1] 0.905619

xCP1$upper$prob[1, 3]
#> [1] 0.5595968

simpCPwr1[1]
#> [1] 0.5596153

xCP1$upper$prob[1, 2]
#> [1] 0.1028575

simpCError[1]
#> [1] 0.1028575

As you can see, the conditional probability of success varies considerably depending on the value of \(\theta\). We will see later that there is substantial uncertainty concerning \(\theta\) at the first interim, thus calling into question use of any particular \(\theta\) value to compute conditional power.

Bauer, Peter, and F. Kohne. 1994. “Evaluation of Experiments with Adaptive Interim Analyses.” Biometrics 50 (4): 1029–41.

Cui, Lu, HM James Hung, and Sue-Jane Wang. 1999. “Modifications of Sample Size in Group Sequential Clinical Trials.” Biometrics 55 (3): 853–57.

Liu, Qing, and George YH Chi. 2001. “On Sample Size and Inference for Two-Stage Adaptive Designs.” Biometrics 57 (1): 172–77.

Müller, Hans-Helge, and Helmut Schäfer. 2001. “Adaptive Group Sequential Designs for Clinical Trials: Combining the Advantages of Adaptive and of Classical Group Sequential Approaches.” Biometrics 57 (3): 886–91.

Proschan, Michael A, and Sally A Hunsberger. 1995. “Designed Extension of Studies Based on Conditional Power.” Biometrics 51 (4): 1315–24.

Proschan, Michael A., K K Gordon Lan, and Janet Turk Wittes. 2006. Statistical Monitoring of Clinical Trials. A Unified Approach. New York, NY: Springer.

Scharfstein, Daniel O, Anastasios A Tsiatis, and James M Robins. 1997. “Semiparametric Efficiency and Its Implication on the Design and Analysis of Group-Sequential Studies.” Journal of the American Statistical Association 92 (440): 1342–50.