Nonzero Null Margins And Shifted Boundary Coordinates

Why a Nonzero Null Requires Shifted Coordinates

Sequential boundaries are often first expressed for a statistic centered at zero. In clinical work, however, the value being tested may be a nonzero margin: for example, a specified difference in a clinical outcome. The statistic must then measure departure from that margin, not departure from zero. This changes the signed statistic coordinates and, if the same stopping decisions are to be retained, shifts the numerical boundary coordinates as well.

The key distinction is between changing coordinates and changing the design. A coordinate change leaves each stopping decision unchanged when the statistic and its boundaries are translated together. Keeping the same numerical limits after shifting the statistic generally changes the effect-scale thresholds and therefore defines a different rule.

Definition: Let $\theta$ be the parameter of interest, $\hat{\theta}_k$ its estimate at look $k$, and $\theta_0$ a prespecified null value, which may be nonzero. If $SE_k$ is the standard error at look $k$, the signed statistic centered at the null value is $$ Z_k(\theta_0)=\frac{\hat{\theta}_k-\theta_0}{SE_k}. $$ A positive value indicates an estimate above $\theta_0$, and a negative value indicates an estimate below $\theta_0$.

For a two-sided design, write $L_k$ and $U_k$ for the lower and upper efficacy limits at look $k$, with $L_k<U_k$. The rule stops for lower-tail evidence if $Z_k(\theta_0)\leq L_k$, stops for upper-tail evidence if $Z_k(\theta_0)\geq U_k$, and continues if $L_k<Z_k(\theta_0)<U_k$. These signed limits need not have equal magnitudes.

Model and Coordinate Transformation

For explicit derivations, suppose the sequential estimates follow the canonical normal information model. At look $k$, information is $I_k$, with $0<I_1<\cdots<I_K$, and $\hat{\theta}_k\sim N(\theta,1/I_k)$. The successive information increments are independent, so the joint distribution has the canonical covariance structure. Thus $SE_k=1/\sqrt{I_k}$, where the units of $I_k$ are inverse squared units of $\theta$. The model treats the information values as fixed by the design and assumes a common parameter $\theta$ across looks.

Define the statistic whose coordinates are centered at zero on the effect scale as $Z_k(0)=\sqrt{I_k}\hat{\theta}_k$. Then the nonzero-null statistic is obtained by subtracting the margin in standardized coordinates:

Z_k(\theta_0)=\sqrt{I_k}(\hat{\theta}_k-\theta_0) =Z_k(0)-\theta_0\sqrt{I_k}.

Consequently, if a rule is originally written in zero-origin coordinates with limits $L_k^{(0)}$ and $U_k^{(0)}$, the same rule in coordinates centered at $\theta_0$ has limits $L_k^{(0)}-\theta_0\sqrt{I_k}$ and $U_k^{(0)}-\theta_0\sqrt{I_k}$. Both limits shift by the same amount at a given look. When $\theta_0>0$, this translation is downward; when $\theta_0<0$, it is upward.

The equivalent effect-scale boundaries follow by solving the standardized inequalities for $\hat{\theta}_k$:

\hat{\theta}_k\leq \theta_0+\frac{L_k}{\sqrt{I_k}} \quad\text{(lower-tail stop)},\qquad \hat{\theta}_k\geq \theta_0+\frac{U_k}{\sqrt{I_k}} \quad\text{(upper-tail stop)}.

These are look-specific thresholds for the estimate, in the original units of $\theta$. They make explicit how a statistic-scale limit maps to an effect-scale decision. The null margin enters additively; it is not itself a standardized boundary.

Why the Canonical Null Calibration Is Preserved

Under the point null $H_0:\theta=\theta_0$, the centered statistic has mean zero and variance one at each look. In the canonical model, for $i<j$, its covariance is $\sqrt{I_i/I_j}$. This is the same joint null distribution as the canonical statistic centered at zero under a zero-valued null. Therefore a boundary design calibrated for that canonical null process can be used for a nonzero null value when the statistic is centered on $\theta_0$ and the boundary decisions are expressed in those centered coordinates.

A short derivation shows the covariance and the role of centering. Write $S_k=I_k(\hat{\theta}_k-\theta)$. Under the model, $S_k$ is a cumulative score with mean zero, variance $I_k$, and $\operatorname{Cov}(S_i,S_j)=I_i$ for $i<j$. Under $H_0:\theta=\theta_0$, $Z_k(\theta_0)=S_k/\sqrt{I_k}$. Hence:

E_{\theta_0}[Z_k(\theta_0)]=0,\qquad \operatorname{Var}_{\theta_0}[Z_k(\theta_0)]=1,\qquad \operatorname{Cov}_{\theta_0}[Z_i(\theta_0),Z_j(\theta_0)] =\frac{I_i}{\sqrt{I_iI_j}} =\sqrt{\frac{I_i}{I_j}}.

Thus the joint null crossing probability is determined by the centered limits and the information pattern, not by whether the clinical null value happens to be zero. This statement concerns the null calibration at the specified value and the stated model. It does not mean that one can retain arbitrary zero-origin numerical limits after changing the statistic's center.

Worked Example: Translating Existing Limits to a Positive Margin

Suppose $\theta$ is a treatment effect measured in clinical units, and the nonzero null value is $\theta_0=0.20$ units. At a particular look, let $I_k=25$ inverse-squared units. A stopping rule in zero-origin coordinates has limits $L_k^{(0)}=-2.4$ and $U_k^{(0)}=2.1$. Translate the limits while preserving the same effect-scale stopping decisions.

Here $\sqrt{I_k}=5$ per unit, so the null-margin shift is $\theta_0\sqrt{I_k}=0.20(5)=1.00$. The centered statistic is $Z_k(0)-1.00$, so subtract 1.00 from each zero-origin limit:

L_k^{(0.20)}=-2.4-1.00=-3.4,\qquad U_k^{(0.20)}=2.1-1.00=1.1.

The standard error is $1/5=0.20$ units. The original effect-scale thresholds were $0+(-2.4)(0.20)=-0.48$ units and $0+(2.1)(0.20)=0.42$ units. Using the translated limits gives $0.20+(-3.4)(0.20)=-0.48$ units and $0.20+(1.1)(0.20)=0.42$ units. The decisions are therefore unchanged; only the coordinate description has changed.

Worked Example: Applying a Centered Limit in Clinical Units

Suppose the outcome is measured in mmHg and the null margin is $\theta_0=5.00$ mmHg. At one look, $I_k=100$ inverse-mmHg-squared, so $SE_k=1/\sqrt{100}=0.100$ mmHg. A hypothetical estimate is $\hat{\theta}_k=5.23$ mmHg. The centered statistic is

Z_k(5.00)=\frac{5.23-5.00}{0.100}=2.30.

If the upper efficacy limit at this look is $U_k=2.00$, then $2.30\geq2.00$, so the rule stops at the upper boundary. The equivalent effect-scale threshold is $5.00+2.00(0.100)=5.20$ mmHg; the estimate of 5.23 mmHg exceeds it.

In zero-origin coordinates, $Z_k(0)=5.23/0.100=52.3$. The same upper boundary is $2.00+5.00/0.100=52.0$, and $52.3\geq52.0$ gives the same stop. Applying the number 2.00 directly to $Z_k(0)$ would not represent this rule.

Following the Shift Across Looks

Because the translation depends on $\sqrt{I_k}$, the same clinical margin produces a different numerical shift at each look. It is not generally correct to add or subtract one constant from the entire sequence of statistic-scale boundaries. The required translation is calculated separately using the information at each look.

Worked Example: Checking a Two-Look Stopping Path

Consider a hypothetical two-look design with margin $\theta_0=0.40$ units. At the first look, $I_1=25$, $\hat{\theta}_1=0.54$, and the limits in margin-centered coordinates are $-2.5$ and $2.5$. At the second look, $I_2=100$, $\hat{\theta}_2=0.48$, and the limits are $-2.0$ and $2.0$. Determine whether the path stops.

At look 1, $SE_1=1/5=0.20$ units and $Z_1(0.40)=(0.54-0.40)/0.20=0.70$. Since $-2.5<0.70<2.5$, the design continues. At look 2, $SE_2=1/10=0.10$ units and $Z_2(0.40)=(0.48-0.40)/0.10=0.80$. Since $-2.0<0.80<2.0$, it again remains inside the continuation region. There is no efficacy stop at either look.

Check the same path in zero-origin coordinates. The margin shifts are $0.40\sqrt{25}=2.0$ at look 1 and $0.40\sqrt{100}=4.0$ at look 2. Thus the zero-origin statistics are $0.70+2.0=2.70$ and $0.80+4.0=4.80$. The corresponding translated limits are $[-0.5,4.5]$ and $[2.0,6.0]$. The first statistic is between $-0.5$ and $4.5$, and the second is between $2.0$ and $6.0$, confirming the same continuation decisions at both looks.

Practical Checks, Alternatives, and Limitations

The direct approach is to subtract the null margin before standardizing: calculate $(\hat{\theta}_k-\theta_0)/SE_k$, then compare with limits defined for that centered statistic. An equivalent approach is to retain a zero-origin statistic and translate every boundary by the appropriate $\theta_0/SE_k$. A third representation is to transform the parameter to $\theta^*=\theta-\theta_0$; its null value is zero, and its estimate is $\hat{\theta}_k-\theta_0$. These are algebraically equivalent when applied consistently. The centered-statistic form is usually easiest to interpret because zero directly denotes the null value.

Representation	Statistic	Limits for an equivalent rule
Centered at the null margin	$(\hat{\theta}_k-\theta_0)/SE_k$	$L_k,\ U_k$
Centered at zero on the effect scale	$\hat{\theta}_k/SE_k$	$L_k+\theta_0/SE_k,\ U_k+\theta_0/SE_k$
Shifted parameter	$(\hat{\theta}_k-\theta_0)/SE_k$	$L_k,\ U_k$

Several checks help catch implementation errors:

At every look, verify the identity $Z_k(0)=Z_k(\theta_0)+\theta_0/SE_k$. The shift should vary with the standard error.
Translate both lower and upper limits by the same signed amount. Reversing the sign of the shift moves the limits the wrong way.
Convert each limit back to an effect-scale threshold. Equivalent coordinates must produce identical thresholds in the units of $\theta$.
Under $H_0:\theta=\theta_0$, the centered statistic should have mean zero. A nonzero null mean is a warning that the margin was not subtracted correctly.

A common mistake is to calculate the centered statistic but compare it with the old zero-origin limits without translating or reinterpreting them. The mistaken reasoning is that “the limits are already calibrated, so their numbers should stay the same.” The correction is to identify what coordinate system those limits belong to. If they were calibrated for $Z_k(0)$, either retain $Z_k(0)$ and its original limits, or use $Z_k(\theta_0)$ and subtract $\theta_0/SE_k$ from both limits. Using the centered statistic with unchanged zero-origin limits generally changes the effect-scale thresholds and the sequential error probability.

The canonical covariance result relies on the stated normal information model. If standard errors are estimated, information is random, the estimator is biased, or the effect is changing across looks, the exact canonical null distribution may not apply; the centering identity remains algebraically true, but calibration may need a model appropriate to those features. The margin should also be specified on the same effect scale as $\hat{\theta}_k$. For nonlinear transformations, shifting a parameter on one scale is not automatically equivalent to shifting it on another.

Key takeaway: A nonzero null value shifts the signed statistic by $\theta_0/SE_k$ at each look. To preserve a sequential rule, shift its lower and upper limits consistently, or keep the original statistic and limits together. Always verify equivalence on the effect scale.

Check Your Understanding

Answer each question, then compare with the short solutions below.

At a look with $SE_k=0.25$ units and $\theta_0=0.50$ units, how far apart are $Z_k(0)$ and $Z_k(\theta_0)$?
A centered upper limit is $1.8$, with $\theta_0=3.0$ mg and $SE_k=0.20$ mg. What is the equivalent upper threshold for $\hat{\theta}_k$?
In a canonical model under $H_0:\theta=\theta_0$, what is the mean of $Z_k(\theta_0)$, and why?
At two looks with standard errors $0.20$ and $0.10$, should a fixed nonzero margin shift the zero-origin statistic by the same amount at both looks? Explain.

Solutions. (1) The difference is $\theta_0/SE_k=0.50/0.25=2.00$, so $Z_k(0)=Z_k(\theta_0)+2.00$. (2) The threshold is $3.0+1.8(0.20)=3.36$ mg. (3) Its mean is zero because under the null $E[\hat{\theta}_k]=\theta_0$, so $E[(\hat{\theta}_k-\theta_0)/SE_k]=0$. (4) No. The shifts are $0.20/SE_k$, giving 1.00 and 2.00 respectively; the shift changes as the standard error changes.

Representation	Statistic	Limits for an equivalent rule
Centered at the null margin	\((\hat{\theta}_k-\theta_0)/SE_k\)	\(L_k,\ U_k\)
Centered at zero on the effect scale	\(\hat{\theta}_k/SE_k\)	\(L_k+\theta_0/SE_k,\ U_k+\theta_0/SE_k\)
Shifted parameter	\((\hat{\theta}_k-\theta_0)/SE_k\)	\(L_k,\ U_k\)