Deriving the Model From Normal Observations

From Individual Observations to the Joint Model

At interim analyses, successive estimates are not based on separate samples: each later cumulative estimate reuses the observations already collected. The dependence between the estimates is therefore part of the model, not an adjustment added after the calculations. Starting from the observation-level model makes clear exactly where that dependence comes from.

Assume observations arrive in a fixed order, and let the analysis occur at prespecified cumulative sample sizes $n_1<n_2<\cdots<n_m$. For this tutorial, consider a one-sample normal-mean test with known standard deviation $\sigma>0$. Under the null hypothesis, the observation model is

H_0:\theta=\theta_0,\qquad X_i\ \text{independent and identically distributed as }N(\theta_0,\sigma^2).

The model assumes independent observations with the same mean and variance, exact normality, a known $\sigma$, and cumulative looks: the data at look $k$ are exactly $X_1,\ldots,X_{n_k}$. The look sizes are fixed rather than chosen in response to observed outcomes. These assumptions give an exact finite-sample result; they are not merely conditions for an approximate calculation.

Definition: At look $k$, let $\bar X_k=n_k^{-1}\sum_{i=1}^{n_k}X_i$ be the cumulative sample mean and define the standardized statistic $$ Z_k=\frac{\sqrt{n_k}(\bar X_k-\theta_0)}{\sigma}. $$ Under $H_0$, the canonical joint distribution is the joint distribution of the vector $(Z_1,\ldots,Z_m)$ at the specified cumulative looks.

Each $Z_k$ is standard normal under the null. The key question is how two such statistics covary when the later look includes all the earlier observations.

Deriving the Covariance From Shared Observations

Write $Y_i=(X_i-\theta_0)/\sigma$. Under $H_0$, the $Y_i$ are independent $N(0,1)$ variables, and $Z_k=n_k^{-1/2}\sum_{i=1}^{n_k}Y_i$. If $j\leq k$, the first $n_j$ terms occur in both sums; the terms from $n_j+1$ through $n_k$ occur only in the later one. Independence makes the covariance between a shared term and any other, distinct term zero. Thus

\begin{aligned} \operatorname{Cov}(Z_j,Z_k) &=\frac{1}{\sqrt{n_jn_k}} \operatorname{Cov}\left(\sum_{i=1}^{n_j}Y_i,\sum_{r=1}^{n_k}Y_r\right)\\ &=\frac{1}{\sqrt{n_jn_k}}\sum_{i=1}^{n_j}\operatorname{Var}(Y_i)\\ &=\frac{n_j}{\sqrt{n_jn_k}} =\sqrt{\frac{n_j}{n_k}}. \end{aligned}

The second equality follows because only the $n_j$ matching pairs of terms have nonzero covariance, and each has variance one. Since each $Z_k$ has variance one, this covariance is also the correlation. In terms of the original sample means, the same shared-data calculation gives $\operatorname{Cov}(\bar X_j,\bar X_k)=\sigma^2/n_k$ for $j\leq k$.

The vector of statistics is jointly normal, not just marginally normal: it is a linear transformation of the independent normal observations $X_1,\ldots,X_{n_m}$. Therefore, with $R$ denoting the correlation matrix, the exact null model is

(Z_1,\ldots,Z_m)^\mathsf{T}\sim N_m(\mathbf{0},R), \qquad R_{jk}=\sqrt{\frac{\min(n_j,n_k)}{\max(n_j,n_k)}}.

Here $N_m$ denotes an $m$-variate normal distribution with the displayed mean vector and covariance matrix. Because the looks are strictly increasing, each look adds at least one new independent observation; the resulting cumulative sums contain distinct increments, so $R$ is nonsingular. A repeated sample size would instead repeat a statistic and produce a singular matrix.

This derivation connects the observation-level model to the Brownian motion representation: cumulative sums of independent, mean-zero, variance-one observations have the same covariance pattern as Brownian motion observed at the corresponding sample sizes. The derivation here needs no continuous-time assumption; it follows directly from the finite collection of normal observations.

Worked Example: Covariance at Two Cumulative Looks

Suppose hypothetical observations follow the stated null model with known $\sigma=4$, and the two cumulative sample sizes are $n_1=6$ and $n_2=15$. First calculate the covariance of the sample means from the shared observations.

The first six observations occur in both means, so $\operatorname{Cov}(\bar X_1,\bar X_2)=\sigma^2/n_2=16/15=1.067$ square units of the measured outcome. The variances are $\operatorname{Var}(\bar X_1)=16/6=2.667$ and $\operatorname{Var}(\bar X_2)=16/15=1.067$, in the same squared units.

Consequently, the correlation of the two sample means, and also of their standardized statistics, is

\frac{1.067}{\sqrt{(2.667)(1.067)}}= \sqrt{\frac{6}{15}}=0.6325.

Thus under $H_0$, $(Z_1,Z_2)^\mathsf{T}$ has mean zero, unit variances, and correlation $0.6325$. The positive correlation reflects data reuse: the early observations contribute to both statistics. It does not mean that the two looks provide independent evidence.

Building a Multiple-Look Correlation Matrix

For any fixed schedule, apply the pairwise formula to every pair of looks. The entry $R_{jk}$ depends on the ratio of the earlier to the later sample size, not on the absolute size of either look. This gives the same correlation for schedules with the same relative spacing.

Worked Example: Three Unequally Spaced Looks

Consider hypothetical cumulative sample sizes $n_1=4$, $n_2=9$, and $n_3=16$, with known $\sigma=1.5$. The observation model under $H_0$ is independent normal sampling with mean $\theta_0=5.0$. Suppose the observed cumulative means are $5.6$, $5.4$, and $5.25$. The corresponding standardized statistics are

\begin{aligned} z_1&=\frac{\sqrt{4}(5.6-5.0)}{1.5}=0.8000,\\ z_2&=\frac{\sqrt{9}(5.4-5.0)}{1.5}=0.8000,\\ z_3&=\frac{\sqrt{16}(5.25-5.0)}{1.5}=0.6667. \end{aligned}

These are the observed values; the joint null distribution is determined by the sample sizes and assumptions, not by these observed values. Its off-diagonal correlations are $\sqrt{4/9}=0.6667$, $\sqrt{4/16}=0.5000$, and $\sqrt{9/16}=0.7500$. Hence

(Z_1,Z_2,Z_3)^\mathsf{T}\sim N_3\left( \begin{pmatrix}0\\0\\0\end{pmatrix}, \begin{pmatrix} 1&0.6667&0.5000\\ 0.6667&1&0.7500\\ 0.5000&0.7500&1 \end{pmatrix} \right).

The early and final statistics are less correlated than the middle and final statistics because the first look's four observations are a smaller fraction of the final sample than the second look's nine observations. The displayed $z$-values alone do not determine a joint probability or a sequential rejection decision; those require an appropriate decision rule.

When the Canonical Covariance Does Not Apply

The formula $\sqrt{n_j/n_k}$ is not a universal correlation for any two analyses performed at different sample sizes. It relies on cumulative nested data. A simple contrast shows why that condition matters.

Worked Example: Replacing the Earlier Sample

Suppose a hypothetical study observes eight independent normal outcomes in one cohort and eight new independent outcomes in a separate cohort, with known $\sigma=2$ and common null mean $\theta_0$. The first analysis uses the mean of the first cohort; the second analysis uses only the mean of the new cohort. Each standardized mean has variance one, but the two means use disjoint observations.

Independence gives $\operatorname{Cov}(\bar X_{\mathrm{first}},\bar X_{\mathrm{new}})=0$, so the standardized statistics also have correlation zero. If one incorrectly labels the second analysis a cumulative look at $n_2=16$, the nested-data formula would give $\sqrt{8/16}=0.7071$. That value is wrong for these data because the original eight observations were not retained in the second mean.

The lesson is to verify which observations contribute to each estimate before using the canonical correlation. Sample sizes alone are insufficient.

Assumptions, Diagnostics, and Alternatives

In a real analysis, the model should be checked against the data-collection process, not inferred from a plausible-looking correlation matrix. Confirm that each look uses the full cumulative sample, that new observations are independent of earlier ones, and that the variance is stable and known as assumed. Also confirm that the look sizes were fixed in advance for this exact derivation. If the number or timing of observations is changed in response to outcomes, the fixed-look argument does not by itself establish the same joint law.

Normality and a known variance are especially consequential for an exact result. Strong skewness, outliers, clustering, changing variances, or a variance estimated from the data can invalidate the stated finite-sample distribution or its correlation structure. A discrepancy between observed variability and the assumed $\sigma^2$ is a warning, though a small data set cannot reliably certify normality. The model is an exact consequence of its assumptions, not a guarantee that the assumptions hold.

The earlier material on the Brownian motion representation describes the same covariance pattern from a process perspective. The direct observation-level derivation is useful when the sampling design needs scrutiny: it identifies precisely which shared observations generate each covariance. A single-look normal test needs only one marginal standard normal statistic; repeated looks require the joint dependence as well. Neither approach repairs a design that is not cumulative or observations that are dependent.

A common mistake is to treat statistics at different looks as independent because each is individually standard normal under $H_0$. The correction is that marginal distributions do not determine a joint distribution: for nested samples, the shared data give $\operatorname{Corr}(Z_j,Z_k)=\sqrt{n_j/n_k}>0$. Ignoring this dependence can misstate the probability of crossing one or more repeated-testing thresholds. Conversely, imposing the nested-data formula on disjoint or replaced samples also misstates the joint model.

Key takeaway: Under independent, identically distributed normal observations with known variance and fixed cumulative looks, the standardized cumulative means are jointly normal under $H_0$, with unit variances and correlations $\sqrt{\min(n_j,n_k)/\max(n_j,n_k)}$. The nesting and observation-level assumptions are essential parts of this result.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

Under the model, what is $\operatorname{Corr}(Z_1,Z_2)$ when $n_1=10$ and $n_2=40$?
For cumulative looks at $n=(5,20,45)$, give the correlation between the first and third standardized statistics.
Why are the statistics jointly normal, rather than merely individually normal?
At the second look, a study discards the first cohort and analyzes only newly enrolled participants. Can the nested-sample correlation formula be used? Explain.

Solutions. (1) The correlation is $\sqrt{10/40}=\sqrt{0.25}=0.5000$. (2) The first and third looks share the first five observations, so the correlation is $\sqrt{5/45}=1/3=0.3333$. (3) Every $Z_k$ is a linear combination of the finite vector of independent normal observations; a vector of such linear combinations is multivariate normal. (4) No. The second analysis does not include the first cohort, so the statistics are not based on nested cumulative data; their covariance must be derived from the actual sampling design.