Diagnostic Plots for Conditional Power

Why Plot Conditional Power?

A conditional-power value summarizes the chance of future success under a specified model and assumed effect, given the data observed so far. A single value can hide how strongly that chance depends on the interim Z-statistic, how much information remains, or which effect is assumed. Diagnostic plots make those dependencies visible. They help identify influential inputs and reveal whether a reported value is a robust description of the design or a consequence of one particular assumption.

The examples below use one remaining, upper-tail efficacy analysis. This setting gives a transparent formula for constructing the plots. With several future looks, conditional power is the probability of crossing at least one applicable future boundary, conditional on the interim data; it can be evaluated across a grid of inputs using the design’s joint distribution. The same plotting principles apply, but the single-look formula below must not be substituted for that multi-look calculation.

Model and Conditional-Power Formula

Suppose an interim analysis occurs at information \(I_k\), and the planned final analysis is at information \(I_K\), where \(0<I_k<I_K\). The information fraction is \(t=I_k/I_K\). Let \(Z_k=z\) be the observed interim Z-statistic. For this tutorial, \(\theta\) denotes the assumed standardized effect per square root of information: under that effect, the score process has drift \(\theta\) per unit information. Let \(c_K\) be the fixed upper-tail efficacy boundary at the final analysis, so success means \(Z_K\ge c_K\). Write \(\Phi\) for the standard normal cumulative distribution function.

The canonical model assumes normally distributed score increments with independent future increments, variance equal to the increment in information, and mean \(\theta\) times that increment. The planned final information and boundary are fixed, and the effect \(\theta\) is treated as specified rather than estimated as part of the conditional-power calculation. The interim result is below any rule that would already have stopped the trial; otherwise there is no future success probability to calculate under continued follow-up. These assumptions give the conditional distribution

The first argument of \(N\) is the conditional mean and the second is the conditional variance. Thus the conditional power (CP) is the probability that this future statistic reaches the final boundary:

This is a model-based probability conditional on the interim statistic, not a probability that the assumed effect is true. A different assumed effect, boundary, or continuation rule generally gives a different value.

Constructing the Three Diagnostic Views

For an interim-statistic plot, fix \(t\), \(\theta\), \(I_K\), and \(c_K\), then calculate conditional power over a range of plausible interim values \(z\). The formula shows that the conditional Z-score inside \(\Phi\) has slope \(\sqrt{t/(1-t)}\) in \(z\). Since \(\Phi\) is increasing, conditional power must increase with the observed interim Z-statistic when the other inputs are fixed.

For an assumed-effect plot, fix the observed \(z\), information fraction, and boundary, then vary \(\theta\). The slope of the conditional Z-score in \(\theta\) is \(\sqrt{I_K(1-t)}\), which is positive when information remains. Consequently, conditional power increases with the assumed effect in this upper-tail model. A plot of this kind displays dependence on an assumption; it does not determine which assumed effect is scientifically appropriate.

For an information-fraction plot, hold \(z\), \(\theta\), \(I_K\), and \(c_K\) fixed while varying \(t\) between zero and one. Unlike the other two views, this plot has no universal monotonic direction. Changing \(t\) changes both the information remaining and the conditional contribution of the observed Z-statistic. Its shape must be calculated for the inputs at hand and interpreted with care.

What the Curves Can—and Cannot—Establish

The information-fraction behavior can be examined by writing the argument of \(\Phi\) as \(q(t)=z\sqrt{t/(1-t)}+\theta\sqrt{I_K}\sqrt{1-t}-c_K/\sqrt{1-t}\). Differentiating for \(0<t<1\) gives

The denominator is positive, while the numerator can be positive or negative and can change sign as \(t\) changes. Since \(\Phi\) is increasing, CP has the same direction of change as \(q(t)\). For instance, use \(z=3\), \(\theta=0\), \(I_K=100\), and \(c_K=2\). At \(t=0.25\), \(q=(3(0.5)-2)/\sqrt{0.75}=-0.5774\), so CP \(\approx 0.282\). At \(t=0.75\), \(q=(3\sqrt{0.75}-2)/0.5=1.196\), so CP \(\approx 0.884\). Here CP increases over those points. A change in \(t\) is not merely “more data”: when \(z\) is held fixed, it compares hypothetical interim situations with the same standardized result at different information levels.

The effect plot is also useful for seeing the scale of dependence on the assumed effect. In the hypothetical setting \(I_K=100\), \(t=0.50\), \(z=1\), and \(c_K=2\), the conditional Z-score is \((0.7071+5\theta-2.000)/0.7071\). At \(\theta=0\), CP \(=\Phi(-1.828)\approx0.0337\); at \(\theta=0.20\), CP \(=\Phi(-0.4142)\approx0.339\); and at \(\theta=0.40\), CP \(=\Phi(1.000)\approx0.841\). The steep change illustrates that an assumed effect can be an influential input. It does not establish that any one value is credible.

These three views answer different questions. The interim-Z plot shows how the observed evidence enters the calculation. The information-fraction plot reveals the combined role of accrued and remaining information under a stated hypothetical comparison. The assumed-effect plot shows dependence on a model input. In contrast, predictive probability averages over uncertainty in the effect under a specified probability model; a conditional-power curve at a fixed \(\theta\) does not. Neither type of plot, by itself, evaluates the trial’s overall Type I error or replaces the design’s boundary-crossing calculations.

Plotting Checks and Failure Modes

A useful diagnostic display labels the varied input and its units, identifies the values held fixed, and states the boundary-crossing event. For example, label an assumed effect in the scale used to define \(\theta\), not merely “effect.” Mark the observed interim value on a Z-statistic plot when the purpose is to locate the current conditional-power estimate. If presenting several assumed effects, identify each curve rather than implying that the data selected one of them.

Check the plotted values against the formula at selected points. With fixed \(t\) and \(\theta\), increasing \(z\) must not lower CP. With fixed \(t\) and \(z\), increasing \(\theta\) must not lower CP under this upper-tail model. A violation suggests a calculation, coding, or labeling error. No corresponding monotonicity check should be imposed on the information-fraction plot: inspect its actual inputs and calculation instead.

Near \(t=1\), the conditional standard deviation \(\sqrt{1-t}\) approaches zero. The formula becomes numerically sensitive, and a plotted curve may change sharply. At \(t=1\), no future information remains; the conditional future statistic is already determined by the observed final statistic, so an ordinary future conditional-power calculation is no longer appropriate. Do not interpret a smoothed line through that endpoint as evidence of a gradual probability change.

The formula also has limits. It describes one final, upper-tail boundary with fixed information and the stated canonical normal model. A two-sided success event requires both tails to be included. Multiple future analyses require their joint crossing event and applicable boundaries, including any futility rule that defines continuation. If the statistic’s distribution, variance, information, or stopping rules differ from the assumptions, the plotted probabilities may not describe the design. In such cases use the appropriate design-specific conditional calculation rather than adjusting this formula informally.

A Practical Plotting Sequence