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Information_Adaptive_Designs.html
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<div id="designing-precision-adaptive-randomized-trials" class="section level3">
<h3>Designing Precision Adaptive Randomized Trials</h3>
<p>Investigators are faced with many challenges in designing efficient,
ethical randomized trials due to many competing demands. A trial must
collect enough information to identify meaningful benefits or harms with
a desired probability, while also minimizing potential harm and
suboptimal treatment of participants. Satisfying these competing demands
is further complicated by the limited and imprecise information
available during the design of a study.</p>
<p>Rather than planning analyses around sample sizes, investigators can
plan analyses which occur when a specified level of precision is
reached: this is known as information monitoring <span class="citation">(Mehta and Tsiatis 2001)</span>. A precision-adaptive
design can reduce the risk of under- or overpowered trials by collecting
data until the precision is sufficient to conduct analyses. The
precision of an estimator is simply the reciprocal of the variance of
the estimator. Since this is unknown in practice, the variance is
estimated using the square of the standard error of the estimator.</p>
<p>We can estimate the precision required to achieve power <span class="math inline">\((1 - \beta)\)</span> to identify a treatment
effect <span class="math inline">\(\delta\)</span> with a <span class="math inline">\(s\)</span>-sided test with type I error rate <span class="math inline">\(\alpha\)</span> using:</p>
<p><span class="math display">\[\mathcal{I} = \left(\frac{Z_{\alpha/s} +
Z_{\beta}}{\delta}\right)^2 \approx
\frac{1}{\left(SE(\hat{\delta})\right)^2} =
\frac{1}{Var(\hat{\delta})}\]</span> This uses the square of the
empirical standard error (or the empirical variance estimate) to measure
the precision to which the treatment effect <span class="math inline">\(\delta\)</span> can be measured with the data in
hand. A precision-adaptive design can reduce the risk of under- or
overpowered trials by collecting data until the precision is sufficient
to conduct analyses.</p>
<hr />
</div>
<div id="approximate-precision-vs.-sample-size" class="section level3">
<h3>Approximate Precision vs. Sample Size</h3>
<p>Let <span class="math inline">\(T\)</span> denote treatment and <span class="math inline">\(C\)</span> denote control, and <span class="math inline">\(Y^{(A)}\)</span> denote the outcome of interest
under treatment assigment <span class="math inline">\(A\)</span>, where
<span class="math inline">\(A = 1\)</span> indicates assignment to the
treatment arm and <span class="math inline">\(A = 0\)</span> denotes
assignment to the control arm.</p>
<p>For a continuous outcome, the required information to estimate the
difference in means <span class="math inline">\(\delta_{DIM} =
E[Y^{(1)}] - E[Y^{(0)}]\)</span> depends on the sample size and
variances of outcomes in each treatment arm:</p>
<p><span class="math display">\[SE(\delta) =
\sqrt{\frac{\sigma^{2}_{T}}{n_{T}} +
\frac{\sigma^{2}_{C}}{n_{C}}}\]</span></p>
<p>For a binary outcome, the required information to estimate the risk
difference <span class="math inline">\(\delta_{RD} = E[Y^{(1)}] -
E[Y^{(0)}]\)</span> depends on the response rate in the control arm
<span class="math inline">\((\pi_{C} = \pi_{T} - \delta)\)</span>:</p>
<p><span class="math display">\[SE(\delta) = \sqrt{\frac{\pi_{T}(1 -
\pi_{T})}{n_{T}} + \frac{\pi_{C}(1 - \pi_{C})}{n_{C}}}\]</span></p>
<p>For an ordinal outcome with <span class="math inline">\(K\)</span>
categories, let <span class="math inline">\(\pi_{A}^{K} = Pr\{Y^{(A)} =
k\}\)</span> denote the probability of an outcome in category <span class="math inline">\(k\)</span> under treatment <span class="math inline">\(A\)</span>. The Mann-Whitney estimand <span class="math inline">\(\phi\)</span> is the probability of having an
outcome as good or better under the treatment arm relative to control
with an adjustment for ties:</p>
<p><span class="math display">\[\phi = Pr\{Y^{(T)} > Y^{(C)}\} +
\frac{1}{2} Pr\{Y^{(T)} = Y^{(C)}\}\]</span></p>
<p>This is also known as the competing probability. The
precision/information depends on <span class="math inline">\(\phi\)</span> <span class="citation">(Fay and
Malinovsky 2018)</span>:</p>
<p><span class="math display">\[SE(\delta) \approx \sqrt{\frac{\phi(1 -
\phi)}{n_{T}n_{C}}\left(1 + \left(\frac{n_{T} + n_{C} -
2}{2}\right)\left(\frac{\phi}{1 + \phi} + \frac{1 - \phi}{2 - \phi}
\right)\right)}\]</span></p>
<p>Alternatively, the precision/information can be obtained from the
distribution of outcomes under each treatment arm <span class="citation">(Zhao, Rahardja, and Qu 2008)</span>. Let <span class="math inline">\(N = n_{T} + n_{C}\)</span>:</p>
<p><span class="math display">\[SE(\delta) =
\sqrt{\frac{1}{12(n_{T}n_{C})}\left(N+1 - \frac{1}{N(N-1)}\right)\sum_{k
= 1}^{K}(\pi_{T}^{k}n_{T} + \pi_{C}^{k}n_{C})}\]</span></p>
<p>Expressions for other estimands can be obtained elsewhere <span class="citation">(Jennison and Turnbull 1999)</span>. In practice, the
parameters in these expressions are not precisely known a priori. The
advantage of an information-adaptive design is that the sample size is
not fixed a priori based on estimates of these parameters, but adapts
automatically to the precision of the accruing data.</p>
<hr />
</div>
<div id="covariate-adjustment-in-randomized-trials" class="section level3">
<h3>Covariate Adjustment in Randomized Trials</h3>
<p>Covariate adjusted analyses can also give greater precision than an
unadjusted analyses without introducing more stringent assumptions,
however the amount of precision gained in adjusted analyses are also not
precisely known a priori <span class="citation">(Benkeser et al.
2020)</span>. Instead of predicating the design on assumptions about the
potential gain in precision from covariate adjustment, a
precision-adaptive design automatically adjusts the sample size
accordingly.</p>
<p>The relative efficiency of a covariate adjusted estimator to an
unadjusted estimator is <span class="math inline">\(RE_{A/U} =
Var(\theta_{U})/Var(\theta_{A})\)</span>. The relative change in
variance of a covariate-adjusted analysis to an unadjusted analysis
is:</p>
<p><span class="math display">\[RCV_{A/U} = \frac{Var(\theta_{A}) -
Var(\theta_{U})}{Var(\theta_{U})} = \frac{1}{RE_{A/U}} - 1\]</span>
Alternatively, <span class="math inline">\(RE_{A/U} = 1/(1 +
RCV_{A/U})\)</span>: If a covariate adjusted analysis has a relative
efficiency of 1.25, the relative change in variance would be -0.2, or a
-20% change in variance. Since precision is the inverse of variance, the
relative change in precision of a covariate-adjusted analysis to an
unadjusted analysis is:</p>
<p><span class="math display">\[RCP_{A/U} = \frac{1/Var(\theta_{A}) -
1/Var(\theta_{U})}{1/Var(\theta_{U})} = Var(\theta_{U})/Var(\theta_{A})
- 1 = RE_{A/U} - 1\]</span> Alternatively, <span class="math inline">\(RE_{A/U} = 1 + RCP_{A/U}\)</span>: If a covariate
adjusted analysis has a relative efficiency of 1.25, the relative change
in precision would be 0.25, or a 25% change in precision.</p>
<hr />
</div>
<div id="sequential-analyses" class="section level3">
<h3>Sequential Analyses</h3>
<p>Pre-planned interim analyses allow investigators to stop a randomized
trial early for efficacy or futility <span class="citation">(Jennison
and Turnbull 1999)</span>. Precision-adaptive trials can integrate both
interim analyses and covariate adjustment, using a broad class of
methods <span class="citation">(Van Lancker, Betz, and Rosenblum
2022)</span>. <span class="citation">Mehta and Tsiatis (2001)</span>
illustrate information-adaptive designs in practice. For a tutorial on
implementing interim analyses, see <span class="citation">(<strong>Lakens2021?</strong>)</span>.</p>
</div>
<div id="references" class="section level3 unnumbered">
<h3 class="unnumbered">References</h3>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-Benkeser2020" class="csl-entry">
Benkeser, David, Iván Dı́az, Alex Luedtke, Jodi Segal, Daniel
Scharfstein, and Michael Rosenblum. 2020. <span>“Improving Precision and
Power in Randomized Trials for <span>COVID</span>-19 Treatments Using
Covariate Adjustment, for Binary, Ordinal, and Time-to-Event
Outcomes.”</span> <em>Biometrics</em> 77 (4): 1467–81. <a href="https://doi.org/10.1111/biom.13377">https://doi.org/10.1111/biom.13377</a>.
</div>
<div id="ref-Fay2018" class="csl-entry">
Fay, Michael P., and Yaakov Malinovsky. 2018. <span>“Confidence
Intervals of the Mann-Whitney Parameter That Are Compatible with the
Wilcoxon-Mann-Whitney Test.”</span> <em>Statistics in Medicine</em> 37
(27): 3991–4006. <a href="https://doi.org/10.1002/sim.7890">https://doi.org/10.1002/sim.7890</a>.
</div>
<div id="ref-Jennison1999" class="csl-entry">
Jennison, Christopher, and Bruce W. Turnbull. 1999. <em>Group Sequential
Methods with Applications to Clinical Trials</em>. Chapman;
Hall/<span>CRC</span>. <a href="https://doi.org/10.1201/9780367805326">https://doi.org/10.1201/9780367805326</a>.
</div>
<div id="ref-Mehta2001" class="csl-entry">
Mehta, Cyrus R., and Anastasios A. Tsiatis. 2001. <span>“Flexible Sample
Size Considerations Using Information-Based Interim Monitoring.”</span>
<em>Drug Information Journal</em> 35 (4): 1095–1112. <a href="https://doi.org/10.1177/009286150103500407">https://doi.org/10.1177/009286150103500407</a>.
</div>
<div id="ref-VanLancker2022" class="csl-entry">
Van Lancker, Kelly, Joshua Betz, and Michael Rosenblum. 2022.
<span>“Combining Covariate Adjustment with Group Sequential, Information
Adaptive Designs to Improve Randomized Trial Efficiency.”</span>
<em>arXiv Preprint arXiv:1409.0473</em>. <a href="https://doi.org/10.48550/ARXIV.2201.12921">https://doi.org/10.48550/ARXIV.2201.12921</a>.
</div>
<div id="ref-Zhao2008" class="csl-entry">
Zhao, Yan D., Dewi Rahardja, and Yongming Qu. 2008. <span>“Sample Size
Calculation for the Wilcoxon<span></span>mann<span></span>whitney Test
Adjusting for Ties.”</span> <em>Statistics in Medicine</em> 27 (3):
462–68. <a href="https://doi.org/10.1002/sim.2912">https://doi.org/10.1002/sim.2912</a>.
</div>
</div>
</div>