Why Calibrate a Spending Parameter?
A spending family describes how cumulative type I error is allocated as information accumulates. Its shape parameter controls the timing of that allocation: different parameter values can spend the same total significance level at different rates before the final analysis. Calibration chooses the parameter to match a design objective, such as a specified early-look expenditure or an efficacy boundary at a planned information fraction.
The objective must be expressed on the same scale as the spending function. A target cumulative expenditure is already on that scale. A target boundary is a test-statistic threshold, so it must first be translated into a null crossing probability. At the first look this translation is direct. At a later look, the relevant probability is cumulative crossing probability, accounting for the boundaries at all previous looks.
The parameter is calibrated for a specified overall significance level \(\alpha\), spending family, sidedness, and information fraction. It is not a replacement for boundary construction: after choosing the parameter, the sequential boundaries still need to be constructed under the group sequential null model.
Assumptions and the Calibration Target
For the usual canonical group sequential model under \(H_0\), the standardized statistics \(Z_i\) at information fractions \(0<t_1<\cdots<t_K=1\) have standard normal marginal distributions and joint normal dependence, with \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{t_i/t_j}\) for \(i<j\). The spending function specifies the desired cumulative probability of crossing an efficacy boundary under this null model. This calibration assumes the information fractions are correctly specified and that the chosen spending function is admissible: it starts at zero, is nondecreasing, and reaches \(\alpha\) at \(t=1\).
For a one-sided upper efficacy boundary \(c_1\) at the first look, \(Z_1\sim N(0,1)\) under \(H_0\), so its null crossing probability is \(P_0(Z_1\geq c_1)=1-\Phi(c_1)\). For a symmetric two-sided first-look boundary \(\pm c_1\), the total crossing probability is \(2[1-\Phi(c_1)]\). These are the cumulative expenditures at the first look, because there are no earlier looks to cross.
At a later look, the marginal tail probability \(P_0(Z_k\geq c_k)\) is generally not the new cumulative expenditure. Some paths have already crossed a previous boundary. The cumulative expenditure through look \(k\) is the probability of crossing at least one boundary by that look. Thus, a later-look target must be evaluated jointly with the preceding boundaries under the canonical joint distribution.
Calibrating the Power Spending Family
For the one-sided power spending family, the cumulative spending at information fraction \(t\) is \(A(t;\rho)=\alpha t^\rho\), where \(\rho>0\) is the shape parameter. The same form can represent total two-sided spending when the design allocates that spending symmetrically between the two tails. The parameter is determined directly when a target expenditure \(a^*\) is given at \(t^*\).
If \(0<a^*<\alpha\) and \(0<t^*<1\), both logarithms in the ratio are negative, so \(\rho>0\). The endpoint conditions are \(A(0;\rho)=0\) and \(A(1;\rho)=\alpha\). A target of zero at a positive information fraction or a target equal to \(\alpha\) strictly before the final look does not yield a finite positive \(\rho\) in this family.
Worked Example: Matching a Specified Early Expenditure
Suppose a one-sided design has \(\alpha=0.0500\), and the desired cumulative expenditure at \(t^*=0.400\) is \(a^*=0.0100\). Find the power parameter.
Substitute these values into the calibration equation:
The calibrated spending function is \(A(t)=0.0500t^{1.756}\). Checking the target, \(A(0.400)=0.0500(0.400)^{1.756}\approx0.0100\). Thus the design spends 0.0100 of cumulative one-sided type I error by information fraction 0.400. The parameter is positive, and the expenditure is between zero and \(\alpha\), as required.
The direct formula also gives a useful diagnostic: substitute the computed parameter back into \(A(t^*;\rho)\). A discrepancy beyond rounding usually signals an incorrect logarithm, a confusion between cumulative and incremental alpha, or use of a target expressed on the wrong sidedness scale.
Calibrating the Hwang-Shih-DeCani Family
The Hwang-Shih-DeCani family uses a parameter \(\gamma\) and is defined, for \(\gamma\ne0\), by
At \(\gamma=0\), the continuous limit is linear spending, \(A(t;0)=\alpha t\). Unlike the power family, the Hwang-Shih-DeCani calibration generally requires solving a nonlinear equation. For a target \(a^*\) at \(t^*\), solve
A numerical root finder can be used, with \(\gamma=0\) checked as the linear-spending case. For \(0<t^*<1\), the left side varies with \(\gamma\), so the selected root should be checked by substituting it back into the original spending equation. Large positive or negative values can also make the exponential calculation numerically unstable; stable numerical functions for evaluating \(1-\exp(-x)\) are preferable when \(x\) is close to zero.
Worked Example: Solving for a Hwang-Shih-DeCani Parameter
Take \(\alpha=0.0250\), \(t^*=0.500\), and target expenditure \(a^*=0.0150\). The required spending proportion is \(a^*/\alpha=0.6000\). Let \(x=\exp(-\gamma/2)\). Since \(t^*=0.5\), \(\exp(-\gamma)=x^2\), and the equation becomes
Therefore,
To verify, \(\exp(-\gamma/2)=2/3\) and \(\exp(-\gamma)=4/9\), so
The target is matched exactly apart from rounding. The positive parameter here produces the specified early expenditure; it does not change the final cumulative expenditure, which remains \(0.0250\).
Calibrating from a First-Look Boundary
A boundary can be used as the calibration target by converting it to the corresponding first-look null crossing probability. This conversion depends on whether the design is one-sided or two-sided, so the sidedness must be stated before solving for the spending parameter.
Worked Example: Matching a Two-Sided First-Look Boundary
Consider a two-sided design with total \(\alpha=0.0500\). At \(t^*=0.500\), the desired symmetric first-look boundary is approximately \(\pm2.576\). Under \(H_0\), each tail has probability approximately \(0.0050\), so the total first-look expenditure is \(a^*\approx2(0.0050)=0.0100\). Calibrate the power family to this target.
Using total two-sided spending in the power formula gives
The resulting function is \(A(t)=0.0500t^{2.322}\), where \(A(t)\) is total spending across both tails. At \(t=0.500\), \(A(t)\approx0.0100\), or approximately \(0.0050\) in each tail under symmetric allocation. The implied first-look cutoff is therefore approximately \(2.576\) in absolute value. This checks the parameter against the stated boundary and confirms that the target was treated as total, not per-tail, expenditure.
Later-Look Targets and Numerical Calibration
A specified boundary at a later look cannot usually be converted to a spending target using only a standard normal tail area. For example, in a one-sided design with two looks, the cumulative null crossing probability through look 2 is
where \(\operatorname{Corr}(Z_1,Z_2)=\sqrt{t_1/t_2}\). This probability depends on both boundaries and on their dependence. The spending equation at look 2 must match this cumulative probability, not merely \(1-\Phi(c_2)\).
A practical calibration proceeds by choosing a trial value of the family parameter, constructing boundaries sequentially so that each cumulative expenditure matches the spending function at its look, and evaluating the boundary at the target look. A root finder then changes the parameter until the target boundary is reached. At every trial value, the earlier boundaries must be recalculated, because changing the parameter changes the preceding spending and hence those boundaries.
Record the significance level, one-sided or two-sided allocation, target information fraction, and target expenditure or boundary.
At the first look, use the standard normal tail probability. At a later look, compute the cumulative null crossing probability using all boundaries through that look and their canonical joint distribution.
Use the direct logarithmic formula for power spending. For Hwang-Shih-DeCani or a later-look boundary target, use a numerical root finder on the relevant calibration equation.
Recompute the spending and boundaries using the fitted parameter. Check the target, the final expenditure \(\alpha\), monotonicity, and the intended sidedness.
Diagnostics, Alternatives, and Limitations
Calibration is only as reliable as its target and model inputs. Check that information fractions are in \((0,1]\), that a proposed early cumulative expenditure lies between zero and \(\alpha\), and that a two-sided target distinguishes total expenditure from expenditure in each tail. For a later boundary, verify that the numerical calculation uses the canonical correlation structure and accounts for all previous crossings. A marginal tail calculation at a later look is not a substitute.
Power-family calibration has a closed-form solution and is easy to verify, but its available spending shapes are restricted to that family. Hwang-Shih-DeCani calibration is also a one-parameter choice, but usually requires numerical solution. Alternatively, a designer can specify boundary targets or expenditures at several looks and solve a more general design problem; that approach is not the same as fitting one shape parameter to one target. Choosing a familiar parameter without checking its implied expenditure is simpler, but may fail to meet the design objective.
A single target does not generally determine all desirable properties of a design. Many spending curves can agree at one information fraction while differing elsewhere. Calibration also does not guarantee adequate power, appropriate expected sample size, or robustness to misspecified information timing; these require separate design assessments. Treat the fitted parameter as a way to meet a stated spending or boundary objective, not as a complete design criterion.
- Wrong reasoning: “The desired boundary is at the second look, so its one-sided normal tail probability is the cumulative alpha to use.”
- Correction: At a later look, cumulative expenditure is the probability of crossing any boundary up to that look. Calculate this joint crossing probability before matching the spending function.
- Wrong reasoning: “A two-sided boundary of \(c\) implies expenditure \(1-\Phi(c)\).”
- Correction: For a symmetric two-sided first-look boundary, total expenditure is \(2[1-\Phi(c)]\); \(1-\Phi(c)\) is the expenditure in one tail.
Check Your Understanding
For each question, identify the relevant calibration equation or interpretation.
- For power spending with \(\alpha=0.0400\), \(t^*=0.250\), and \(a^*=0.0100\), find \(\rho\).
- A one-sided first-look boundary is \(c_1=2.054\). What cumulative expenditure does it imply approximately?
- Why is \(1-\Phi(c_2)\) generally not the cumulative expenditure through a second look?
- For a symmetric two-sided first-look boundary \(c_1=2.326\), what is the approximate total expenditure?
Solutions. (1) \(\rho=\log(0.0100/0.0400)/\log(0.250)=\log(0.25)/\log(0.25)=1.000\). (2) \(1-\Phi(2.054)\approx0.0200\), so the one-sided cumulative first-look expenditure is approximately 0.0200. (3) It omits paths that crossed the first-look boundary and ignores the dependence between statistics; cumulative expenditure is the probability of crossing at least one boundary by look 2. (4) Each tail is approximately 0.0100, so total expenditure is approximately \(2(0.0100)=0.0200\).