ABSTRACT
When dealing with missing data in clinical trials, it is often convenient to work under simplifying assumptions, such as missing at random (MAR), and follow up with sensitivity analyses to address unverifiable missing data assumptions. One such sensitivity analysis, routinely requested by regulatory agencies, is the so-called tipping point analysis, in which the treatment effect is re-evaluated after adding a successively more extreme shift parameter to the predicted values among subjects with missing data. If the shift parameter needed to overturn the conclusion is so extreme that it is considered clinically implausible, then this indicates robustness to missing data assumptions. Tipping point analyses are frequently used in the context of continuous outcome data under multiple imputation. While simple to implement, computation can be cumbersome in the two-way setting where both comparator and active arms are shifted, essentially requiring the evaluation of a two-dimensional grid of models. We describe a computationally efficient approach to performing two-way tipping point analysis in the setting of continuous outcome data with multiple imputation. We show how geometric properties can lead to further simplification when exploring the impact of missing data. Lastly, we propose a novel extension to a multi-way setting which yields simple and general sufficient conditions for robustness to missing data assumptions.
Acknowledgments
The authors thank Søren Andersen for useful discussions.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Supplementary material
Supplemental data for this article can be accessed online at https://doi.org/10.1080/10543406.2022.2058525
ANCOVA-MI in a two-way tipping point setting: variance estimator
To verify (7), ignore first the MI step. When regressing an outcome of the form on
note that
by linearity and so
With MI and assuming that only is imputed so that imputations take the form
for
, we have
setting
Combining (4), (15), and (19), the within-imputations variance then is
Noting that cancels out in the between-imputations variance (6), and letting
and
, we arrive at (7).
ANCOVA-MI with subject-specific missing values: robustness arguments
We want to show that (14) is the solution to the maximization problem (13). Ignore first the MI step and consider a setting where we regress an outcome of the form on
. Write
. By direct calculation based on the expressions in (15), we get
Since the treatment is randomized (assuming no stratified randomization), converges to zero in probability when
and so
converges in probability to
or
, depending on whether
or 1. Hence, the first term on the left-hand side of (22) is of order
. The second term is of order
and so does not affect the sign of the partial derivative for large
. It follows that when
is not too small, the coordinate functions of
are strictly decreasing when
and strictly increasing when
. This means that the only feasible local (and so global) maximum in the set of extreme points of
is (14).
To translate this to an MI setting and the function (12), note that the partial derivative with respect to is the same as in (22) except for the denominator in the second term which will include a contribution from the between-imputations variance. Analogous arguments then apply to show that the maximum in the set of extreme points of
is given by (14).
It is useful to note that although the argument for optimality is inherently asymptotic, applicability in a specific setting can be verified directly by investigating the sign of (22) (or the equivalent expression in an MI setting).