Information and the canonical joint distribution

One statistical structure serves every endpoint: the sequence of Z statistics behaves like a Brownian motion observed at the looks.

Statistical information

The information about the treatment effect \(\theta\) at a look is the inverse of the variance of its estimate, \(I = 1/\operatorname{Var}(\hat\theta)\). It grows as data accumulate, and it — rather than calendar time or the number of patients — measures how far a study has progressed.

Endpoint Effect \(\theta\) Information
Continuous, two groups (\(n_1\), \(n_2\) patients, SD \(\sigma\)) difference in means \(\dfrac{1}{\sigma^2 (1/n_1 + 1/n_2)}\)
Binary, two groups risk difference \(\dfrac{1}{p_1(1-p_1)/n_1 + p_2(1-p_2)/n_2}\)
Binary, two groups log odds ratio \(\left(\sum 1/\text{cell counts}\right)^{-1}\) approximately
Time to event, allocation \(r:1\) log hazard ratio \(\approx d\, r/(1+r)^2\) with \(d\) events (\(d/4\) for 1:1)
Any estimate with standard error \(SE\) \(1/SE^2\)

The information fraction at look \(k\) is \(t_k = I_k / I_{\max}\). For a continuous endpoint it is simply the fraction of patients analysed; for a survival endpoint it is the fraction of the planned events.

The canonical joint distribution

Let \(\hat\theta_k\) be the estimate at look \(k\) and \(Z_k = \hat\theta_k \sqrt{I_k}\) the standardised statistic. For a very wide class of models,

\[ Z_k \sim N\!\left(\theta\sqrt{I_k},\, 1\right), \qquad \operatorname{Cov}(Z_j, Z_k) = \sqrt{I_j / I_k} \quad (j \le k). \]

Equivalently, the score statistics \(S_k = Z_k\sqrt{I_k}\) have independent increments: \(S_k - S_{k-1} \sim N\!\left(\theta (I_k - I_{k-1}),\; I_k - I_{k-1}\right)\). This holds exactly for normal data with known variance, and asymptotically for maximum-likelihood estimates in regular parametric models, generalised linear models, the log-rank test and Cox regression (Tsiatis 1982; Jennison and Turnbull 1997; Scharfstein et al. 1997).

The consequence is powerful: boundaries, power and expected sample size depend only on the information fractions, not on the endpoint. A design is solved once on the \(Z\) scale, and the endpoint enters only when information is converted into patients or events (Sample size).

Brownian motion and the B-value

Lan and Wittes (1988) rescaled the statistic as the B-value \(B(t) = Z(t)\sqrt{t}\). Under the canonical distribution, \(B(t)\) is a Brownian motion with drift \(\eta = \theta\sqrt{I_{\max}}\):

\[ B(t) \sim N(\eta\, t,\; t), \qquad \text{independent increments}. \]

Under \(H_0\) the drift is zero; under the design alternative it is \(\eta_1 = \theta_1\sqrt{I_{\max}}\). This picture makes several later results almost immediate, such as conditional power (Monitoring).

Figure 1: Simulated B-value paths under H0 (drift 0) and under the design alternative (drift 3.24, 90 % power for a one-sided 2.5 % test).

Fixed-sample information and the inflation factor

A fixed-sample, one-look study with one-sided level \(\alpha\) and power \(1-\beta\) at \(\theta_1\) needs

\[ I_{\text{fix}} = \left(\frac{z_{1-\alpha} + z_{1-\beta}}{\theta_1}\right)^2 , \]

that is, a drift \(\eta_{\text{fix}} = z_{1-\alpha} + z_{1-\beta}\) (use \(\alpha/2\) for a two-sided test). A sequential design with stricter boundaries needs a larger drift \(\eta\) for the same power, so its maximum information is \(I_{\max} = R \cdot I_{\text{fix}}\), where \(R = (\eta/\eta_{\text{fix}})^2\) is the inflation factor.

Table 1: Example: two-arm trial, effect 0.5 SD, two-sided alpha 0.05, power 90 %, 5 equally spaced looks.
Design Inflation factor Maximum n per arm
Fixed 1.000 85
O’Brien-Fleming 1.026 87
Pocock 1.207 102

With 1:1 allocation and \(\sigma = 1\), \(I = n/2\) for \(n\) patients per arm, so \(n = 2I\) per arm. The fixed design needs 85 patients per arm. The O’Brien-Fleming design needs only 2 more as a maximum, and most trials will stop earlier.

How jmsequential computes

Probabilities of crossing the boundaries are obtained by recursive numerical integration of the density of \(Z_k\) over the region where the study continues (Armitage et al. 1969; Jennison and Turnbull 2000, ch. 19). The density at look \(k\) is evaluated on a grid of points; Simpson’s rule integrates it forward to look \(k+1\). This needs only the normal density and distribution functions, is exact to about \(10^{-8}\), and takes about a millisecond per design. It is the same algorithm used by gsDesign and rpact, and jmsequential’s implementation agrees with both to within \(10^{-5}\).

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References

Armitage, P., C. K. McPherson, and B. C. Rowe. 1969. “Repeated Significance Tests on Accumulating Data.” Journal of the Royal Statistical Society. Series A (General) 132 (2): 235. https://doi.org/10.2307/2343787.
Jennison, Christopher, and Bruce Turnbull. 2000. Group Sequential Methods with Applications to Clinical Trials. In Chapman & Hall/CRC Interdisciplinary Statistics Series. Chapman & Hall/CRC. https://doi.org/10.1201/9781584888581.
Jennison, Christopher, and Bruce W. Turnbull. 1997. “Group-Sequential Analysis Incorporating Covariate Information.” Journal of the American Statistical Association 92 (440): 1330–41. https://doi.org/10.1080/01621459.1997.10473654.
Lan, K. K. Gordon, and Janet Wittes. 1988. “The b-Value: A Tool for Monitoring Data.” Biometrics 44 (2): 579. https://doi.org/10.2307/2531870.
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. https://doi.org/10.1080/01621459.1997.10473655.
Tsiatis, Anastasios A. 1982. “Repeated Significance Testing for a General Class of Statistics Used in Censored Survival Analysis.” Journal of the American Statistical Association 77 (380): 855–61. https://doi.org/10.1080/01621459.1982.10477898.