Fast tipping point sensitivity analyses in clinical trials with missing continuous outcomes under multiple imputation

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.

KEYWORDS:

• Missing data
• multiple imputation
• replication

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 ${\tilde{Y}}_i=Y_i+\omega Z_i+\psi V_i$ on $A_i,{\mathbf{X}}_i$ note that $e_{{\tilde{Y}}_i}=e_{Y_i}+\omega e_{Z_i}+\psi e_{V_i}$ by linearity and so

(18) $\mathrm{R}\mathrm{S}{\mathrm{S}}_{\tilde{Y}}(\omega ,\psi )=\mathrm{R}\mathrm{S}{\mathrm{S}}_Y+{\omega }^2\mathrm{R}\mathrm{S}{\mathrm{S}}_Z+{\psi }^2\mathrm{R}\mathrm{S}{\mathrm{S}}_V+2\left(\omega \sum _{i=1}^n{e}_{Y_i}{e}_{Z_i}+\psi \sum _{i=1}^n{e}_{Y_i}{e}_{V_i}+\omega \psi \sum _{i=1}^n{e}_{Z_i}{e}_{V_i}\right).$(18)

With MI and assuming that only $Y_i$ is imputed so that imputations take the form ${\tilde{Y}}_i^{(j)}=Y_i^{(j)}+\omega Z_i+\psi V_i$ for $j=1,\dots ,m$, we have

(19) ${\overline{\mathrm{R}\mathrm{S}\mathrm{S}}}_{\tilde{Y}}(\omega ,\psi )={\overline{\mathrm{R}\mathrm{S}\mathrm{S}}}_Y[1+M(\omega ,\psi )/{\overline{\mathrm{R}\mathrm{S}\mathrm{S}}}_Y]$(19)

setting

(20) $M(\omega ,\psi )={\omega }^2\mathrm{R}\mathrm{S}{\mathrm{S}}_Z+{\psi }^2\mathrm{R}\mathrm{S}{\mathrm{S}}_V+2\left(\omega \sum _{i=1}^n{\bar{e}}_{Y_i}{e}_{Z_i}+\psi \sum _{i=1}^n{\bar{e}}_{Y_i}{e}_{V_i}+\omega \psi \sum _{i=1}^n{e}_{Z_i}{e}_{V_i}\right).$(20)

Combining (4), (15), and (19), the within-imputations variance then is

(21) $W[{\hat{\beta }}_A(\omega ,\psi )]=m^{-1}\sum _{j=1}^m\frac{{\mathrm{R}\mathrm{S}\mathrm{S}}_{\tilde{Y}}^{(j)}(\omega ,\psi )}{k(n-1)(n-p)}=W[{\hat{\beta }}_A][1+M(\omega ,\psi )/{\overline{\mathrm{R}\mathrm{S}\mathrm{S}}}_Y].$(21)

Noting that $\omega Z_i+\psi V_i$ cancels out in the between-imputations variance (6), and letting $\omega Z_i={\delta }_0{R}_i(1-A_i)$ and $\psi V_i={\delta }_1{R}_i{A}_i$, 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 $Y_i+w_i$ on $A_i,{\mathbf{X}}_i$. Write $D_i=A_i-\bar{A}-{\hat{\mathrm{\Sigma }}}_{A,\mathbf{X}}{\hat{\mathrm{\Sigma }}}_{\mathbf{X},\mathbf{X}}^{-1}({\mathbf{X}}_i-\bar{\mathbf{X}})$. By direct calculation based on the expressions in (15), we get

(22) $\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_i}\{{\hat{\beta }}_A(\mathbf{w})+z_{1-\alpha /2}\widehat{\mathrm{V}\mathrm{a}\mathrm{r}}[{\hat{\beta }}_A(\mathbf{w}){]}^{1/2}\}=\frac{D_i}{k(n-1)}+z_{1-\alpha /2}\frac{e_{Y_i+w_i}}{\sqrt{k(n-1)(n-p)\mathrm{R}\mathrm{S}{\mathrm{S}}_{Y+w}}}$(22)

Since the treatment is randomized (assuming no stratified randomization), ${\hat{\mathrm{\Sigma }}}_{A,\mathbf{X}}$ converges to zero in probability when $n\to \mathrm{\infty }$ and so $D_i$ converges in probability to $-\mathrm{P}(A_i=1)$ or $\mathrm{P}(A_i=0)$, depending on whether $A_i=0$ or 1. Hence, the first term on the left-hand side of (22) is of order $n^{-1}$. The second term is of order $n^{-3/2}$ and so does not affect the sign of the partial derivative for large $n$. It follows that when $n$ is not too small, the coordinate functions of $\mathbf{w}\mapsto {\hat{\beta }}_A(\mathbf{w})+z_{1-\alpha /2}\widehat{\mathrm{V}\mathrm{a}\mathrm{r}}[{\hat{\beta }}_A(\mathbf{w}){]}^{1/2}$ are strictly decreasing when $A_i=0$ and strictly increasing when $A_i=1$. This means that the only feasible local (and so global) maximum in the set of extreme points of $F$ is (14).

To translate this to an MI setting and the function (12), note that the partial derivative with respect to $w_i$ 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 $F$ 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).

Additional information

Funding

The author(s) reported that there is no funding associated with the work featured in this article.