From Sample Size to Information
A group sequential design is planned at successive analyses, or looks. For a normal-mean design with known variance, each cumulative sample size determines the precision available at that look. Converting sample sizes to information puts the looks on the scale used by the canonical joint distribution: it also gives a direct way to construct the covariance matrix of the cumulative standardized statistics.
The mapping in this tutorial applies to a one-sample normal-mean model. Suppose observations \(X_1,X_2,\ldots\) are independent and identically distributed, with \(X_i\sim N(\mu,\sigma^2)\), where \(\sigma^2\) is known. At look \(j\), the first \(n_j\) observations have been collected, with \(1\le n_1<n_2<\cdots<n_K\). Thus, the sample at an earlier look is contained in the sample at every later look. The observation units and the meaning of \(\mu\) are the same across looks, and there is no unaccounted loss of observations or change in the variance model.
Information here is cumulative: \(I_j\) describes all observations available by look \(j\), not just those newly collected since the previous look. The increment describes only the new observations. This distinction matters when forming the joint distribution, because the cumulative statistics share observations while the increments do not.
Why the Mapping Is \(n_j/\sigma^2\)
For one observation from \(N(\mu,\sigma^2)\), the log likelihood as a function of \(\mu\), up to terms not involving \(\mu\), is \(-(X_i-\mu)^2/(2\sigma^2)\). Its score is \((X_i-\mu)/\sigma^2\), with variance \(1/\sigma^2\). Independent observations contribute additive information, so \(n_j\) observations contribute \(n_j/\sigma^2\). Equivalently, their total score has variance \(n_j/\sigma^2\). This justifies the information mapping under the stated model.
For testing \(H_0:\mu=\mu_0\), define the cumulative standardized statistic at look \(j\) by
Under the model, \(Z_j\) has variance 1. Under a fixed value of \(\mu\), its mean is \((\mu-\mu_0)\sqrt{I_j}\). For two looks \(i<j\), the first \(n_i\) observations are shared, and the remaining observations at look \(j\) are independent of them. Consequently, the covariance is determined by the shared observations:
The covariance matrix therefore has entries \(\Sigma_{ii}=1\), and, for \(i<j\), \(\Sigma_{ij}=\Sigma_{ji}=\sqrt{I_i/I_j}\). Under the normal model it is also the correlation matrix, since each statistic has variance 1. The full vector is multivariate normal: each statistic is a linear combination of the jointly normal observations. The covariance is the same under the null and fixed alternatives; changing \(\mu\) changes the means, not the covariance.
Worked Sample-Size Mappings
Worked Example: Three Looks With Known Variance
Suppose a hypothetical study measures an outcome in units and plans cumulative sample sizes \(n_1=25\), \(n_2=50\), and \(n_3=100\). Assume independent normal observations with known standard deviation \(\sigma=12\) units. Calculate the information at each look and the canonical covariance matrix.
The variance is \(12^2=144\) units squared. Applying \(I_j=n_j/\sigma^2\) gives
The looks are cumulative and ordered, as required: \(25<50<100\). The off-diagonal entries are
Thus the matrix, with rows and columns ordered by look, is
For example, the first and third standardized statistics have correlation 0.5000 because the first look's 25 observations make up one quarter of the 100 observations at look 3. The calculation relies on the same independent, known-variance normal model at every look; it would not be justified merely by knowing the three sample sizes.
Worked Example: Rounding Target Information to Feasible Sample Sizes
Consider a hypothetical normal outcome measured in units, with known \(\sigma=5\) units. A planner wants cumulative information targets of \(0.20\), \(0.50\), and \(0.80\) units\(^{-2}\). Find integer cumulative sample sizes and use the resulting actual information levels to form the covariance matrix.
Solving \(I=n/\sigma^2\) for \(n\) gives \(n=I\sigma^2=25I\). The target sample sizes are therefore \(5\), \(12.5\), and \(20\). A cumulative sample size must be an integer, so round the second target upward to 13 while retaining the first and third targets. The actual cumulative information levels are
The schedule is feasible and increasing: \(5<13<20\). Use the actual information values—not the original targets—to calculate the covariance entries:
The diagonal entries are 1, so the resulting matrix is
Rounding changed the second information level from \(0.50\) to \(0.52\) units\(^{-2}\), and that difference changes the correlations involving look 2. The variance and normal-model assumptions are unchanged; the calculation checks that the rounding produces an ordered, nested schedule.
Worked Example: Unequal Sample-Size Increments and the Alternative Mean
Suppose a hypothetical study has \(\sigma=10\) units and cumulative sample sizes \(n_1=30\), \(n_2=45\), and \(n_3=90\). Under a fixed alternative \(\mu=\mu_0+2\) units, calculate the information levels, the expected standardized statistics, and the covariance matrix.
The sample sizes are strictly increasing, and each look adds new independent observations, so the nested-sample assumptions hold. The information levels are
For each look, \(E(Z_j)=(\mu-\mu_0)\sqrt{I_j}\). Since \(\mu-\mu_0=2\) units, the means are
The covariance entries are unaffected by this alternative:
Thus the joint distribution has mean vector \((1.095,1.342,1.897)^\mathsf{T}\) and covariance matrix
The increments are 30, 15, and 45 observations, not three equal-sized groups. The result still follows because the cumulative samples are nested; equal increments are not needed for this covariance calculation.
Checks, Alternatives, and Limitations
Before using the matrix in a group sequential calculation, check that the sample sizes are cumulative, strictly increasing, and defined for the same outcome and population parameter. Information must also increase: with a fixed positive variance, \(n_i<n_j\) implies \(I_i<I_j\). The resulting covariance entries should be between 0 and 1, with later looks generally more correlated with nearby earlier looks than with much earlier ones.
This mapping uses a known variance. If \(\sigma^2\) is estimated from the data, substituting an estimate into \(n_j/\sigma^2\) does not by itself establish the same exact multivariate normal model for the standardized statistics. Likewise, correlated observations, changing variances, non-normal outcomes, or a change in the analysis population can invalidate the simple mapping or require a different information calculation. In those settings, do not use this matrix automatically; the model-specific information and joint distribution must be justified.
Calendar time and information are related but are not interchangeable. A planned calendar date does not determine information unless the amount and variance of data available by that date are known. Conversely, once the cumulative sample sizes and variance are specified under this model, the information levels follow directly. Equal sample-size increments are one possible schedule; unequal increments are also valid. The main requirement for this derivation is nested cumulative data under the stated model, not equal spacing between looks.
The result is an exact normal-mean calculation, whereas score-based canonical approximations in more general models may be asymptotic. Neither approach should be treated as a universal covariance formula without checking its assumptions. This tutorial establishes the mapping and covariance for the known-variance one-sample normal design; it does not determine stopping boundaries or error rates.
A common mistake is to use the increment \(n_j-n_{j-1}\) in place of the cumulative sample size when calculating \(I_j\), then treat that as the information at look \(j\). The correction is to calculate \(I_j=n_j/\sigma^2\) from all observations available at that look. For example, with \(n_1=25\), \(n_2=50\), and \(\sigma=12\), look 2 has information \(50/144\), not \(25/144\); the latter is only the information added after look 1.
Check Your Understanding
Answer each question using the known-variance normal-mean model described above. Solutions follow the questions.
- For \(\sigma=4\) units and cumulative sample sizes \(n_1=16\), \(n_2=64\), calculate \(I_1\), \(I_2\), and the covariance between the two standardized statistics.
- With \(\sigma=3\) units, what cumulative sample size corresponds exactly to information \(I=2\) units\(^{-2}\)?
- Suppose the new observations between two looks are independent, but the earlier observations are not included in the later cumulative statistic. Does the nested-sample covariance formula apply as written? Briefly explain.
- For two looks with information levels \(I_1=0.25\) and \(I_2=1\) units\(^{-2}\), state the covariance matrix and identify its off-diagonal entry.
Solutions. 1. \(I_1=16/16=1\) units\(^{-2}\), and \(I_2=64/16=4\) units\(^{-2}\). Their covariance is \(\sqrt{1/4}=0.5000\). 2. Solve \(n=I\sigma^2\): \(n=2(3^2)=18\) observations. 3. No. The derivation uses the first sample being contained in the later cumulative sample; without that nesting, the shared-data covariance need not be \(n_i\sigma^2\). 4. The off-diagonal entry is \(\sqrt{0.25/1}=0.5000\), so the matrix is \(\begin{pmatrix}1&0.5000\\0.5000&1\end{pmatrix}\).