A Worked Two-Look Normal-Mean Example

Two Cumulative Looks in a Normal-Mean Model

A group sequential design observes accumulating data at planned looks. At two looks, the later statistic reuses all the observations from the earlier look, so the two cumulative z statistics are correlated. That dependence matters when calculating their joint behavior and when describing what values at the second look are plausible after seeing the first.

This worked example uses independent normal observations and a known variance. The goal is to calculate the joint means, covariance and correlation of the two z statistics, then find the conditional distribution of the later statistic given the earlier one. These calculations describe the joint model; by themselves, they do not specify a stopping rule or adjust a test for repeated looks.

The observations added between looks are \(X_{n_1+1},\ldots,X_{n_2}\). Their independence from the first \(n_1\) observations is essential to the independent-increment representation. Define \(\Delta=\mu-\mu_0\) and \(I_k=n_k/\sigma^2\). Here \(I_k\) is the information for the normal mean at look \(k\), and the statistic’s mean under the fixed value \(\mu\) is \(\Delta\sqrt{I_k}\). The variance of each \(Z_k\) is 1.

The information fraction at the first look, relative to the second, is \(I_1/I_2=n_1/n_2\). Because the observations have the same known variance at both looks, this ratio also gives the squared correlation of the cumulative statistics.

Joint Mean, Covariance, and Correlation

Each \(Z_k\) is a linear combination of normal observations. Therefore \((Z_1,Z_2)\) has a bivariate normal distribution. Its mean vector follows from the mean of each sample mean:

The covariance comes from the shared observations. In particular, \(\bar X_2\) includes the first \(n_1\) observations that also determine \(\bar X_1\). Independence gives \(\operatorname{Cov}(\bar X_1,\bar X_2)=\sigma^2/n_2\), so scaling the two sample means into z statistics yields the following joint distribution:

The second argument of \(N\) here is the covariance matrix. Since both marginal variances are 1, the covariance equals the correlation. Write \(\rho=\sqrt{I_1/I_2}=\sqrt{n_1/n_2}\). The correlation is positive and less than 1: the statistics share data, but the later look also contains new observations.

Deriving the Conditional Transition

A conditional transition describes the distribution of \(Z_2\) after conditioning on \(Z_1=z_1\). For jointly normal variables, the conditional mean is the later marginal mean plus the correlation times the earlier statistic’s deviation from its mean. The conditional variance is the later variance multiplied by \(1-\rho^2\). These formulas can also be justified directly from the bivariate normal model.

Substituting \(m_1=\Delta\sqrt{n_1}/\sigma\), \(m_2=\Delta\sqrt{n_2}/\sigma\), and \(\rho=\sqrt{n_1/n_2}\) gives an equivalent form of the transition:

The second form shows the contribution of the new observations explicitly. Under \(H_0\), \(\Delta=0\), so the conditional mean is \(\rho z_1\). Under a fixed alternative, the extra term reflects the mean shift in the added observations. In both cases the conditional variance is positive when \(n_2>n_1\), and it does not depend on the observed value \(z_1\).

Interpretation, Checks, and Limitations

The joint distribution can be constructed either from the covariance of the nested sample means or from the first-look statistic and an independent added-batch statistic. The direct covariance calculation is compact. The batch representation is often a useful arithmetic check and makes clear why the cumulative z statistics themselves are not independent. Both approaches give the same transition under the stated model.

Several checks help identify mistakes. The look sizes must satisfy \(0<n_1<n_2\), so \(0<\rho<1\). Both marginal variances must equal 1, while the conditional variance \(1-n_1/n_2\) must be positive. The covariance matrix must be valid: its determinant is \(1-\rho^2=1-n_1/n_2>0\). If the computed correlation exceeds 1, the covariance or sample-size ratio has been mishandled.

A common mistake is to treat the independent observations added between looks as if the cumulative statistics \(Z_1\) and \(Z_2\) were independent. They are not: their correlation is \(\sqrt{n_1/n_2}\), since the second statistic includes the first look’s observations. Another mistake is to use the null conditional mean \(\rho z_1\) under a nonzero fixed alternative. Under that alternative, the correct mean includes the increment term \(\Delta(n_2-n_1)/(\sigma\sqrt{n_2})\).

These calculations rely on independent normal observations, a common known variance, fixed look sizes, and a fixed mean across the looks. If the variance is estimated, observations are dependent, or the mean changes during data collection, this exact finite-sample distribution need not apply. The result also describes the joint statistics without prescribing how to use them for a sequential decision; the design’s stopping boundaries and error-rate procedure must be specified separately.