Canonical joint distribution of Z-score and B-values under null and alternative hypothesis

Yujie Zhao, Yilong Zhang, and Keaven M. Anderson

When applying the AHR test in the time-to-event endpoint at the \(k\)-th analysis, we build the standardized treatment effect as the test statistics in Proschan, Lan, and Wittes (2006):

\[ Z_k = \frac{ \sum_{i=1}^{d_k} \Delta_i }{ \sqrt{\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0)} }, \]

where \(d_k\) is the total number of events at analysis \(k\). Here, \(\Delta_i = O_i - E_i\) with \(O_i\) is the indicator that the \(i\)-th event occurred in the experimental arm, and \(E_i = m_{i,1}/(m_{i,0} + m_{i,1})\) as the null expectation of \(O_i\) given \(m_{i,0}\) and \(m_{i,1}\). The numbers \(m_{i,0}\) and \(m_{i,1}\) refer to the number of at-risk subjects prior to the \(i\)-th death in the control and experimental group, respectively. When conditioning on \(m_{i,0}\) and \(m_{i,1}\), the \(O_i\) has a Bernoulli distribution with parameter \(E_i\). So the null conditional mean and variance of \(\Delta_i\) are 0 and \(V_i = E_i(1 - E_i)\), respectively. Unconditionally, \(\Delta_i\) are mean 0 random variables with variance \(E(V_i)\) under the null hypothesis. So, conditioned on \(d_k\), we have \(\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0)) = E(\sum_{i=1}^{d_k} V_i\)).

In this vignette, we discuss the statistical information. Mathematically, it is the inverse of the variance, depending on whether it is under null or alternative hypothesis:

\[ \begin{array}{c} \mathcal I_{k, H_0} = 1 /\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0), \\ \mathcal I_{k, H_1} = 1 /\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_1). \end{array} \]

The ratio between two statistical information under the null hypothesis is called information fraction:

\[ t_k = \frac{\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0)}{\text{Var}(\sum_{i=1}^{d_K} \Delta_i | H_0)} \approx \frac{d_k/4}{d_K / 4} = d_k /d_K, \]

In the rest of this vignette, we will discuss the value of \(\mathcal I_{k, H_0}, \mathcal I_{k, H_1}\). To derive these statistical information, we introduce the B-values:

\[ B_k = \frac{ \sum_{i=1}^{d_k} \Delta_i }{ \sqrt{\text{Var}(\sum_{i=1}^{d_K} \Delta_i | H_0)} } = \underbrace{ \frac{ \sum_{i=1}^{d_k} \Delta_i }{ \sqrt{\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0)} } }_{Z_k} \underbrace{ \frac{ \sqrt{\text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0)} }{ \sqrt{\text{Var}(\sum_{i=1}^{d_K} \Delta_i | H_0)} } }_{\sqrt{t_k}} = \sqrt{t_k} Z_k, \]

As depicted in Section 2.1.3 of Proschan, Lan, and Wittes (2006), the stochastic process formulation presented in Harrington, Fleming, and Green (1984) and Tsiatis (1982) provides insights that sequence of B-values, denoted as \({B_1, \ldots, B_K}\) with follows a multivariate normal distribution behaves asymptotically like Brownian motion.

For 2 B-values (\(B_i, B_j\) with \(i \leq j\)), we have

\[ \begin{eqnarray} \text{Cov}(B_i, B_j) & = & \text{Cov} \left( \frac{ \sum_{s=1}^{d_i} \Delta_s }{ \sqrt{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} }, \frac{ \sum_{s=1}^{d_j} \Delta_s }{ \sqrt{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} } \right) \\ & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Cov} \left( \sum_{s=1}^{d_i} \Delta_s, \sum_{s=1}^{d_j} \Delta_s \right) \\ & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Cov} \left( \sum_{s=1}^{d_i} \Delta_s, \sum_{s=1}^{d_i} \Delta_s + \sum_{s=1}^{d_j} \Delta_s - \sum_{s=1}^{d_i} \Delta_s \right) \\ & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s \right) + \text{Cov} \left( \sum_{s=1}^{d_i} \Delta_s, \sum_{s=1}^{d_j} \Delta_s - \sum_{s=1}^{d_i} \Delta_s \right) \\ & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s \right) \end{eqnarray} \]

Null hypothesis

The distribution of \(\{B_k\}_{k = 1, \ldots, K}\) has the following structure:

• \(B_1, B_2, \ldots, B_K\) have a multivariate normal distribution.
• \(E(B_k \;|\; H_0) = 0\) for any \(k = 1, \ldots, K\).
• \(\text{Var}(B_k \;|\; H_0) = t_k\).
• \(\text{Cov}(B_i, B_j \;|\; H_0) = t_i\) for any \(1 \leq i \leq j \leq K\).

The derivation of the last 2 statement is

\[ \begin{eqnarray} \text{Var}(B_k\;|\; H_0) & = & \frac{ \text{Var}(\sum_{i=1}^{d_k} \Delta_i | H_0) }{ \text{Var}(\sum_{i=1}^{d_K} \Delta_i | H_0) } = t_k\\ \text{Cov}(B_i, B_j \;|\; H_0) & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s\;|\; H_0)} \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s\;|\; H_0 \right) = t_i \end{eqnarray} \]

Accordingly, \(\{Z_k\}_{k = 1, \ldots, K}\) has the canonical joint distribution with the following properties:

• \(Z_1, Z_2, \ldots, Z_K\) have a multivariate normal distribution.
• \(E(Z_k \;|\; H_0) = 0\).
• \(\text{Var}(Z_k \;|\; H_0) = 1\).
• \(\text{Cov}(Z_i, Z_j \;|\; H_0) = \sqrt{t_i/t_j}\) for any \(1 \leq i \leq j \leq K\).

Alternative hypothesis

Under the alternative hypothesis, for 2 B-values (\(B_i, B_j\) with \(i \leq j\)), the distribution of \(\{B_k\}_{k = 1, \ldots, K}\) has the following structure:

• \(B_1, B_2, \ldots, B_K\) have a multivariate normal distribution.
• \(E(B_k \;|\; H_1) = \theta_k t_k \sqrt{\mathcal I_{k, H_0}}\) for any \(k = 1, \ldots, K\).
• \(\text{Var}(B_k \;|\; H_1) = t_k \mathcal I_{k, H_0} / \mathcal I_{k, H_1}\).
• \(\text{Cov}(B_i, B_j \;|\; H_1) = t_i \; \mathcal I_{i, H_0}/\mathcal I_{i, H_1}\) for any \(1 \leq i \leq j \leq K\).

The last statement is derived as

\[ \begin{eqnarray} \text{Cov}(B_i, B_j \;|\; H_1) & = & \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s | H_1 \right) \\ & = & \underbrace{ \frac{1}{\text{Var}(\sum_{s=1}^{d_K} \Delta_s | H_0)} \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s | H_0 \right) }_{t_i} \underbrace{ \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s | H_1 \right) }_{1/\mathcal I_{i, H_1}} \bigg/ \underbrace{ \text{Var} \left( \sum_{s=1}^{d_i} \Delta_s | H_0 \right) }_{1/\mathcal I_{i, H_0}} \\ & = & t_i\; \mathcal I_{i, H_0}/\mathcal I_{i, H_1}. \end{eqnarray} \]

Accordingly, \(Z_k\) has the canonical joint distribution with the following properties:

• \(Z_1, Z_2, \ldots, Z_K\) have a multivariate normal distribution.
• \(E(Z_k \;|\; H_1) = \theta_k \sqrt{\mathcal I_{k, H_0}}\) with the treatment effect as \(\theta_k\) at the \(k\)-th analysis.
• \(\text{Var}(Z_k \;|\; H_1) = \mathcal I_{k, H_0} / \mathcal I_{k, H_1}\).
• \(\text{Cov}(Z_i, Z_j \;|\; H_1) = \sqrt{\frac{t_i}{t_j}} \frac{\mathcal I_{i, H_0}}{\mathcal I_{i, H_1}}\) for any \(1 \leq i \leq j \leq K\).

The last statement is because

\[ \begin{eqnarray} \text{Cov}(Z_i, Z_j \;|\; H_1) & = & \text{Cov}(B_i/\sqrt{t_i}, B_j/\sqrt{t_j}) \\ & = & \frac{1}{\sqrt{t_i t_j}} \text{Cov}(B_i, B_j) \\ & = & \frac{1}{\sqrt{t_i t_j}} \text{Var}(B_i) \\ & = & \sqrt{\frac{t_i}{t_j}} \frac{\mathcal I_{i, H_0}}{\mathcal I_{i, H_1}} \end{eqnarray} \]

When the local alternative assumption holds, we have \(\text{Cov}(Z_i, Z_j) \approx \sqrt{\frac{t_i}{t_j}}\), which is in the format of the canonical joint distribution introduced in Chapter 3 of Proschan, Lan, and Wittes (2006).

Summary

	B-value	Z-score
Expectation mean at the \(k\)-th analysis under \(H_0\)	0	0
Expectation mean at the \(k\)-th analysis under \(H_1\)	\(\theta_k t_k \sqrt{\mathcal I_{k, H_0}}\)	\(\theta_k \sqrt{\mathcal I_{k, H_0}}\)
Variance at the \(k\)-th analysis under \(H_0\)	\(t_k\)	1
Variance at the \(k\)-th analysis under \(H_1\)	\(t_k \mathcal I_{k, H_0} / \mathcal I_{k, H_1}\)	\(\mathcal I_{k, H_0} / \mathcal I_{k, H_1}\)
Covariance between the \(i\)-th and \(j\)th analysis under \(H_0\) (\(i\leq j\))	\(t_i\)	\(\sqrt{t_i/t_j}\)
Covariance between the \(i\)-th and \(j\)th analysis under \(H_1\) (\(i\leq j\))	\(t_i \; \mathcal I_{i, H_0}/\mathcal I_{i, H_1}\)	\(\sqrt{\frac{t_i}{t_j}} \frac{\mathcal I_{i, H_0}}{\mathcal I_{i, H_1}}\)

References

Harrington, DP, TR Fleming, and SJ Green. 1984. “Procedures for Serial Testing in Censored Survival Data.” In Strukturen Und Prozesse Neue Ansätze in Der Biometrie: 28. Biometrisches Kolloquium Der Biometrischen Gesellschaft Aachen, 16.–19. März 1982 Proceedings, 124–38. Springer.

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

Tsiatis, Anastasios A. 1982. “Group Sequential Methods for Survival Analysis with Staggered Entry.” Lecture Notes-Monograph Series 2: 257–68.