How are group sequential analysis, random walks, and Brownian motion related?

Assume that I am planning a clinical trial comparing two groups using a binary outcome. I will do the $$\chi^2$$ test after 3 equal enrollment intervals: interim test #1 after $$m_1$$ enrollments in group 1 and $$m_2$$ enrollments in group 2; interim test #2 after $$2m_1$$ and $$2m_2$$ enrollments; and the final test after $$3m_1$$ and $$3m_2$$ enrollments. On the nth test (n = 1 ,2,3), I will compare the $$\chi^2$$ statistic to $$\frac{3}{n} \times c$$, where $$c$$ is the O'Brien Fleming constant. For $$N=3$$ and $$\alpha = 0.05$$, $$c = 3.940$$. (Note that this is not much different from 1.96^2 = 3.84.)

For each study interval, we see $$y_{1n}$$ outcomes in group 1 and $$y_{2n}$$ in group 2. Let $$\pi_1$$ and $$\pi_2$$ be the true outcome probabilities. The hypothesis being tested (H0) is that $$\pi_1 = \pi_2 = \pi$$. For each of the 3 intervals, we create a standard normal random variable $$U_n$$: $$U_n = \frac{y_{1n}/m_1 - y_{2n}/m_2}{\sqrt{\pi(1-\pi)(1/m_1+1/m_2)}}$$ Let $$S_n$$ be the sum of the $$U_n$$s, so \begin{align*} S_1 &= U_1 \quad \quad \quad \quad \sim N(0,1)\\ S_2 &= U_1 + U_2 \quad \quad \sim N(0,2)\\ S_3 &= U_1 + U_2 +U_3 \sim N(0,3) \end{align*} This is a 3-step Gaussian random walk starting at 0 with $$U_n$$ as the steps and $$S_n$$ as the location after $$n$$ steps. The steps $$U_1$$, $$U_2$$, and $$U_3$$ are independent but the locations $$S_1$$, $$S_2$$, and $$S_3$$ are not.

O'Brien and Fleming (1979) define $$Z_n^*$$ and $$T_n$$ as follows: $$Z_n^* = S_n/\sqrt{n} \sim N(0,1)$$ $$T_n = S_n^2/3 = (Z_n^*)^2\big(\frac{n}{3}\big)$$ $$(Z_n^*)^2$$ is the random variable corresponding to my $$\chi^2$$ statistics, which I am going to compare to $$\frac{3}{n} \times c$$. To get $$c$$, O'Brien and Fleming say we need to know the distribution of $$max\{T_n\}$$ (the maximum of $$T_1$$, $$T_2$$, and $$T_3$$) which they say has the same distribution as $$max\{[W(n/3)]^2\}$$ where $$\{W(t)|0 \le t \le1\}$$ "represents Brownian motion".

Q1: Are they correct that $$max\{T_n\}$$ for $$n = 1,2,3$$ has the same distribution as $$max\{[W(n/3)]^2\}$$ where $$\{W(t)|0 \le t \le1\}$$ "represents Brownian motion"?

They say that $$c = 3.940$$ is the 95th percentile value for $$max\{T_n\}$$ = $$max\{[W(n/3)]^2\}$$.

Q2: Is this correct? How does one get this value? Could I use this formula?

Since posting, I have partially answered my own questions.

Q1: Are O'Brien and Fleming correct when they say that the distribution of $$max\{T_n\}$$ has the same distribution as $$max \{ [W(n/3)]^2 \}$$ where $$\{ W(t)|0 \le t \le 1 \}$$ "represents Brownian motion"?

A1: Yes. Also, "Brownian motion" and "Wiener process" are used synonymously.

Q2: How does one get the value of $$c = 3.940$$ for $$N=3$$ "looks" and $$\alpha = 0.05$$?

A2: I was able to get close to this value using simulation in R. In fact, I was able to approximately reproduce their Table 1 of $$c$$ values for all their combinations of $$N$$ and $$\alpha$$. I was hoping that there was a better way than brute force simulation.

• One thing you didn't mention was what action you are going to take based on the results of the intermediate tests. If you are planning on stopping the enrollment depending on one of the intermediate results, then you might want to adjust your p-values to account for that. In other words, these don't seem like 3 independent tests at 0.05 with an overall experiment error rate of 0.05. Commented Feb 6, 2023 at 19:35
• On the nth test (n = 1 ,2,3), I will compare the $\chi^2$ statistic to $\frac{3}{𝑛}×𝑐$ , where $c$ is the O'Brien Fleming constant and will stop the study if $\chi^2 > \frac{3}{𝑛}×𝑐$. For $𝑁=3$ and $\alpha=0.05$, $𝑐=3.940$, so on the first interim look, I will stop the study if my $\chi^2$ value is greater than $3 \times 3.940 = 11.82$. This is obviously a stringent threshold, which is characteristic of O'Brien/Fleming. . Commented Feb 6, 2023 at 19:49

There are two other papers on the connection between Browian process and group-sequential designs:

1. Lan, K. K. G. and Wittes, J. (1988). The B-value: a tool for monitoring data. Biometrics 44, 579–585.
2. Lan, K. K. G. and Zucker, D. M. (1993). Sequential monitoring of clinical trials: the role of information and Brownian motion. Statistics in Medicine 12, 753–765