Joint Distribution Under the Null and Alternatives

Why Compare the Joint Distributions?

In a group sequential design, the test statistics at successive interim looks are not separate, unrelated draws. Each cumulative statistic uses data observed earlier, so their joint behavior matters: a stopping rule applied across looks depends on the chance of particular sequences of statistics. The canonical joint model describes those sequences under the null hypothesis and under a specified, fixed alternative.

The key comparison is simple but consequential. In the canonical model, the alternative shifts the mean of each statistic according to the information available at that look. It does not change the covariance matrix. Thus, the null and fixed-alternative distributions have the same dependence structure but different locations. This statement relies on the canonical model’s assumptions; it is not a universal claim about every possible test statistic or data-generating process.

Definition: Let $K$ denote the number of interim looks, and let $0<I_1<\cdots<I_K$ be their cumulative information levels. Let $Z_i$ be the information-standardized cumulative test statistic at look $i$. In the canonical joint normal model, for a fixed standardized effect $\theta$, $$ (Z_1,\ldots,Z_K)^\mathsf{T}\sim N_K(\theta v,\Sigma), \qquad v_i=\sqrt{I_i}, \qquad \Sigma_{ij}=\sqrt{\frac{\min(I_i,I_j)}{\max(I_i,I_j)}}. $$ Under $H_0:\theta=0$, the mean vector is $0$. Under a fixed alternative $H_a:\theta=\theta_a$, the mean vector is $\theta_a v$. The covariance matrix $\Sigma$ is the same in both distributions.

Here, $N_K(m,\Sigma)$ denotes a $K$-variate normal distribution with mean vector $m$ and covariance matrix $\Sigma$. Each $Z_i$ has variance 1, so $\Sigma_{ii}=1$. For $i<j$, the correlation between $Z_i$ and $Z_j$ is $\sqrt{I_i/I_j}$. If information is expressed as fractions $t_i=I_i/I_K$, the same model has mean components $\theta\sqrt{I_Kt_i}$ and correlations $\sqrt{t_i/t_j}$ for $i<j$. The information scale and the scale of $\theta$ must be used consistently.

The fixed alternative means that $\theta$ is treated as a specified constant, not as a random quantity or a value re-estimated at each look. Its sign indicates the direction of the effect relative to the test’s chosen orientation. For a one-sided test aimed at positive effects, a positive $\theta_a$ shifts the means upward; a negative value shifts them downward. Under the null, the mean is zero at every look.

Why the Covariance Is Shared

The shared covariance follows from cumulative data and independent increments. A useful representation is $Z_i=(\theta I_i+W_i)/\sqrt{I_i}$, where the noise terms have mean zero, variance $I_i$, and covariance $\operatorname{Cov}(W_i,W_j)=\min(I_i,I_j)$. This representation expresses the model’s assumptions: information accumulates deterministically, the cumulative noise is Gaussian with independent increments, and the effect contributes a fixed drift proportional to information.

For $i\leq j$, the covariance calculation is

\operatorname{Cov}(Z_i,Z_j) =\frac{\operatorname{Cov}(W_i,W_j)}{\sqrt{I_iI_j}} =\frac{I_i}{\sqrt{I_iI_j}} =\sqrt{\frac{I_i}{I_j}}.

The drift terms $\theta I_i$ and $\theta I_j$ are constants when the alternative is fixed. Adding constants changes means but does not change covariance. Therefore the calculation is identical under $H_0$ and $H_a$. The entries of $\Sigma$ reflect how much data the looks share, not the value of the fixed effect.

The same point can be seen through increments. For adjacent looks, the newly accumulated noise between $I_{i-1}$ and $I_i$ is independent of the earlier noise, while the alternative adds a deterministic drift over that information interval. The observations $Z_i$ remain correlated because each includes earlier data. Independent increments therefore do not mean independent cumulative $Z$ statistics.

Worked Example: Comparing Three-Look Means and Covariances

Consider a hypothetical design with information levels $I_1=25$, $I_2=100$, and $I_3=225$. Suppose the fixed alternative is $\theta_a=0.20$ per square root of the information unit. The canonical model is appropriate here by assumption: the information levels are ordered and fixed, and the cumulative statistics follow the joint normal model with independent noise increments.

Under $H_0$, $\theta=0$, so the mean vector is $(0,0,0)^\mathsf{T}$. Under the fixed alternative, the components are $\theta_a\sqrt{I_i}$:

\theta_a v =0.20(\sqrt{25},\sqrt{100},\sqrt{225})^\mathsf{T} =0.20(5,10,15)^\mathsf{T} =(1,2,3)^\mathsf{T}.

The covariance matrix is calculated from the information ratios:

\Sigma= \begin{pmatrix} 1 & \sqrt{25/100} & \sqrt{25/225}\\ \sqrt{25/100} & 1 & \sqrt{100/225}\\ \sqrt{25/225} & \sqrt{100/225} & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0.5000 & 0.3333\\ 0.5000 & 1 & 0.6667\\ 0.3333 & 0.6667 & 1 \end{pmatrix}.

Thus the null distribution is $N_3((0,0,0)^\mathsf{T},\Sigma)$, while the fixed-alternative distribution is $N_3((1,2,3)^\mathsf{T},\Sigma)$. The later statistics have larger means under this positive alternative because more information has accumulated. Their variances remain 1, and their correlations remain the same. The statistics are positively correlated because the later cumulative statistics reuse earlier data.

Conditional Distributions Also Shift

The joint comparison implies a corresponding conditional comparison from one look to the next. For two adjacent looks $I_{i-1}<I_i$, the canonical model gives

Z_i\mid Z_{i-1}=z \sim N\left( \sqrt{\frac{I_{i-1}}{I_i}}\,z+ \theta\frac{I_i-I_{i-1}}{\sqrt{I_i}}, \frac{I_i-I_{i-1}}{I_i} \right).

The first term in the conditional mean carries forward the observed earlier statistic. The second is the drift contributed by the new information under the fixed alternative. Under $H_0$, that drift term is zero. The conditional variance depends on the information increment and is unchanged by $\theta$.

Worked Example: A Conditional Mean Under Each Hypothesis

Suppose two hypothetical looks occur at $I_1=40$ and $I_2=100$, and the first statistic is $Z_1=0.80$. Assume the canonical joint normal model with fixed information and independent Gaussian noise increments. Compare the conditional distribution at the second look under $H_0$ and under $\theta_a=0.15$.

The coefficient carrying forward $Z_1$ is $\sqrt{40/100}=\sqrt{0.40}\approx0.6325$. The conditional variance is $(100-40)/100=0.60$. Under the null, the conditional mean is

\sqrt{0.40}(0.80)+0 \approx 0.5060.

Under the alternative, the new-information drift is $0.15(100-40)/\sqrt{100}=0.15(60)/10=0.9000$, so the conditional mean is

\sqrt{0.40}(0.80)+0.15\frac{60}{10} \approx 0.5060+0.9000=1.406.

Consequently, $Z_2\mid Z_1=0.80$ is $N(0.5060,0.60)$ under the null and $N(1.406,0.60)$ under the fixed alternative. In each distribution the second argument is the variance, not the standard deviation. The alternative shifts the conditional mean; it does not alter the conditional variance.

A Marginal Probability Check

Since each marginal statistic has variance 1, the mean shift also gives a direct way to compare a single-look tail probability. This calculation concerns a marginal event at one look, not the probability of crossing a boundary at any of several looks. The latter depends on the joint distribution.

Worked Example: A Single-Look Upper-Tail Probability

Suppose a hypothetical final look has information $I=100$ and the fixed alternative is $\theta_a=0.12$. Assume the statistic at that look follows the canonical normal model, so it has variance 1 under both hypotheses. The alternative mean is $0.12\sqrt{100}=1.20$. For an upper-tail cutoff of 1.96, under $H_0$,

P(Z>1.96\mid H_0)=1-\Phi(1.96)\approx0.0250.

Under the fixed alternative, standardizing by its mean and standard deviation gives

P(Z>1.96\mid \theta=0.12) =1-\Phi(1.96-1.20) =1-\Phi(0.76) \approx0.2236.

Thus, under this specified alternative, the probability that the final-look statistic exceeds 1.96 is approximately 0.224. This does not give the probability of crossing the cutoff at any earlier or later look; such a calculation must account for the correlation among looks.

Assumptions, Diagnostics, and Alternatives

The shared-covariance comparison is useful only when its assumptions are reasonable. In exact normal models, it can follow directly from normal observations and fixed information. In other settings it may be an approximation. The main checks are whether the information levels are correctly ordered and treated as fixed, whether the statistics are standardized as specified, whether the effect is held fixed, and whether the joint normal independent-increment representation is plausible.

Warning signs include substantial non-normality in increments when a normal approximation is being relied on, information levels that depend on outcomes in a way not represented by the model, or statistics whose variance changes with the effect. If the model’s assumptions fail, one cannot simply assert that the alternative changes only the mean. The joint distribution may require a different covariance structure or a different approximation.

A common mistake is to reason, “The alternative changes the distribution, so it must change the covariance too.” That is not true in this canonical model: adding a deterministic drift changes the mean while leaving covariance unchanged. The opposite mistake is also possible: because covariance is shared, treating the cumulative statistics as independent under the alternative. The off-diagonal correlations are still present under both hypotheses.

The main comparison is between the canonical model and two tempting substitutes. First, treating each look’s $Z_i$ as an independent standard normal statistic loses the dependence caused by shared data. Second, specifying a different covariance matrix under each fixed effect contradicts the canonical construction unless a different data model justifies it. The Brownian motion representation is another way to express the same canonical dependence; it does not remove that dependence. These distinctions matter whenever probabilities involve more than one look.

The model has limits. It describes a fixed alternative, not an estimated or changing effect, and assumes a particular Gaussian cumulative structure. It does not by itself specify a stopping boundary, calibrate a repeated-testing error rate, or guarantee that an approximate model is accurate for small samples. Those tasks require additional design choices and, where needed, further modeling.

Key takeaway: Under the canonical joint normal model, $H_0$ and a fixed alternative share $\Sigma$. The null mean is zero, and the alternative mean at look $i$ is $\theta_a\sqrt{I_i}$. The shared covariance records overlap in cumulative data; the mean vector records the fixed drift.

Check Your Understanding

Answer each question using the canonical model. Solutions follow the questions.

For information levels $I_1=16$ and $I_2=64$, what are the correlation between the two statistics and the alternative mean vector when $\theta_a=0.25$?
Does a fixed positive alternative change the variance of an individual $Z_i$ in the canonical model? Explain.
For $I_1=25$, $I_2=100$, $Z_1=1$, and $\theta_a=0.10$, calculate the conditional mean and variance of $Z_2$ under the alternative.
Why is it incorrect to treat cumulative $Z$ statistics as independent merely because their increments are independent?

Solutions. (1) The correlation is $\sqrt{16/64}=0.5000$. The mean vector is $0.25(4,8)^\mathsf{T}=(1,2)^\mathsf{T}$. (2) No. Each statistic has variance $\Sigma_{ii}=1$ under both hypotheses; the fixed drift changes the mean, not the variance. (3) The conditional mean is $\sqrt{25/100}(1)+0.10(100-25)/\sqrt{100}=0.5000+0.7500=1.250$. The variance is $(100-25)/100=0.7500$. (4) Each later cumulative statistic includes data from earlier looks. Independent increments describe the newly added noise, but shared earlier data induce correlation between the cumulative statistics.