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10.21105.joss.06971.jats
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<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.2 20190208//EN"
"JATS-publishing1.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="1.2" article-type="other">
<front>
<journal-meta>
<journal-id></journal-id>
<journal-title-group>
<journal-title>Journal of Open Source Software</journal-title>
<abbrev-journal-title>JOSS</abbrev-journal-title>
</journal-title-group>
<issn publication-format="electronic">2475-9066</issn>
<publisher>
<publisher-name>Open Journals</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">6971</article-id>
<article-id pub-id-type="doi">10.21105/joss.06971</article-id>
<title-group>
<article-title>snSMART: An R Package for Small Sample, Sequential,
Multiple Assignment, Randomized Trial Data Analysis</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4838-0842</contrib-id>
<name>
<surname>Wang</surname>
<given-names>Sidi</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-7089-3591</contrib-id>
<name>
<surname>Fang</surname>
<given-names>Fang</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Tamura</surname>
<given-names>Roy</given-names>
</name>
<xref ref-type="aff" rid="aff-2"/>
</contrib>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-7113-6998</contrib-id>
<name>
<surname>Braun</surname>
<given-names>Thomas</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">https://orcid.org/0000-0002-1717-4483</contrib-id>
<name>
<surname>Kidwell</surname>
<given-names>Kelley M</given-names>
</name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<aff id="aff-1">
<institution-wrap>
<institution>University of Michigan</institution>
</institution-wrap>
</aff>
<aff id="aff-2">
<institution-wrap>
<institution>University of South Florida</institution>
</institution-wrap>
</aff>
</contrib-group>
<pub-date date-type="pub" publication-format="electronic" iso-8601-date="2024-05-22">
<day>22</day>
<month>5</month>
<year>2024</year>
</pub-date>
<volume>9</volume>
<issue>100</issue>
<fpage>6971</fpage>
<permissions>
<copyright-statement>Authors of papers retain copyright and release the
work under a Creative Commons Attribution 4.0 International License (CC
BY 4.0)</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>The article authors</copyright-holder>
<license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>Authors of papers retain copyright and release the work under
a Creative Commons Attribution 4.0 International License (CC BY
4.0)</license-p>
</license>
</permissions>
<kwd-group kwd-group-type="author">
<kwd>rare disease</kwd>
<kwd>clinical trial</kwd>
<kwd>snSMART</kwd>
<kwd>Bayesian</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="summary">
<title>Summary</title>
<p>Small sample, sequential, multiple assignment, randomized trials
(snSMARTs) are multistage trials designed to estimate first stage
treatment effects. By using data from all stages, snSMARTs provide
more precise estimates of these effects. Additionally, the design may
enhance participant recruitment and retention compared to standard
rare disease trials. To support the application of snSMART statistical
methods, we introduce the R package
<monospace>snSMART</monospace>.</p>
</sec>
<sec id="statement-of-need">
<title>Statement of need</title>
<p>The design and methods of snSMARTs are applicable to any disorder
or disease that affects a small group and remains stable over the
trial duration. Recent advances include methods for snSMARTs with
three active treatments
(<xref alt="Chao, Trachtman, et al., 2020" rid="ref-chao2020dynamic" ref-type="bibr">Chao,
Trachtman, et al., 2020</xref>;
<xref alt="Wei et al., 2018" rid="ref-wei2018bayesian" ref-type="bibr">Wei
et al., 2018</xref>,
<xref alt="2020" rid="ref-wei2020sample" ref-type="bibr">2020</xref>),
group sequential designs
(<xref alt="Chao, Braun, et al., 2020" rid="ref-chao2020bayesian" ref-type="bibr">Chao,
Braun, et al., 2020</xref>), placebo with two dose levels
(<xref alt="Fang et al., 2021" rid="ref-fang2021bayesian" ref-type="bibr">Fang
et al., 2021</xref>), and continuous outcomes
(<xref alt="Hartman et al., 2021" rid="ref-hartman2021design" ref-type="bibr">Hartman
et al., 2021</xref>). Despite these developments, there is a lack of
software to implement these methods. The
<monospace>snSMART</monospace> R package addresses this need by
providing sample size calculations and trial data analysis using both
Bayesian and frequentist approaches. To our knowledge, no other R
packages offer similar functionalities.</p>
</sec>
<sec id="functionality-of-the-snsmart-package">
<title>Functionality of the snSMART package</title>
<p>We have summarized the functionality of all the
<monospace>snSMART</monospace> functions included in this package in
Table 1. The <monospace>BJSM_binary</monospace>,
<monospace>BJSM_c</monospace>, and <monospace>group_seq</monospace>
functions implement the Bayesian Joint Stage Modeling (BJSM) methods
to estimate treatment effects across all treatment arms in a snSMART
design with binary outcomes, continuous outcomes, and in a group
sequential trial design, respectively. The
<monospace>LPJSM_binary</monospace> function serves as the frequentist
equivalent to <monospace>BJSM_binary</monospace> and can be used for
sensitivity analysis. The <monospace>sample_size</monospace> function
performs Bayesian sample size calculations for a snSMART design with
binary outcomes, ensuring that the trial is scientifically valid,
ethically responsible, and resource-efficient.</p>
</sec>
<sec id="snsmart-comparing-two-dose-levels-with-placebo">
<title>snSMART comparing two dose levels with placebo</title>
<table-wrap>
<caption>
<p>Summary of the functionality of the snSMART package.</p>
</caption>
<table>
<colgroup>
<col width="38%" />
<col width="62%" />
</colgroup>
<thead>
<tr>
<th align="left">Function</th>
<th align="left">Description</th>
</tr>
</thead>
<tfoot>
<tr>
<td align="left"><italic>BJSM functions</italic></td>
<td align="left"></td>
</tr>
<tr>
<td align="left">BJSM_binary</td>
<td align="left">BJSM binary (3AT or P2D snSMART)</td>
</tr>
<tr>
<td align="left">BJSM_c</td>
<td align="left">BJSM (3AT snSMART with a mapping function and
continuous outcome)</td>
</tr>
<tr>
<td align="left">group_seq</td>
<td align="left">BJSM (interim analysis and final analysis of
group sequential 3AT snSMART)</td>
</tr>
<tr>
<td align="left"></td>
<td align="left"></td>
</tr>
<tr>
<td align="left"><italic>Frequentist functions</italic></td>
<td align="left"></td>
</tr>
<tr>
<td align="left">LPJSM_binary</td>
<td align="left">LPJSM (3AT or P2D snSMART)</td>
</tr>
<tr>
<td align="left"></td>
<td align="left"></td>
</tr>
<tr>
<td align="left"><italic>Sample size calculation</italic></td>
<td align="left"></td>
</tr>
<tr>
<td align="left">sample_size</td>
<td align="left">3AT snSMART sample size calculation</td>
</tr>
<tr>
<td align="left"></td>
<td align="left"></td>
</tr>
<tr>
<td align="left"><italic>S3 summary and print
methods</italic></td>
<td align="left"></td>
</tr>
<tr>
<td align="left">for class ’BJSM_binary’</td>
<td align="left">Summarize and print ’BJSM_binary’ object</td>
</tr>
<tr>
<td align="left">for class ’BJSM_binary_dose’</td>
<td align="left">Summarize and print ’BJSM_binary_dose’
object</td>
</tr>
<tr>
<td align="left">for class ’BJSM_c’</td>
<td align="left">Summarize and print ’BJSM_c’ object</td>
</tr>
<tr>
<td align="left">for class ’group_seq’</td>
<td align="left">Summarize and print ’group_seq’ object</td>
</tr>
<tr>
<td align="left">for class ’LPJSM_binary’</td>
<td align="left">Summarize and print ’LPJSM_binary’
object</td>
</tr>
<tr>
<td align="left">for class ’sim_group_seq’</td>
<td align="left">Summarize and print ’sim_group_seq’
object</td>
</tr>
</tfoot>
<tbody>
</tbody>
</table>
</table-wrap>
<p>This section details one of the snSMART designs, which investigates
the response rate of an experimental treatment at low and high doses
compared to placebo
(<xref alt="Fang et al., 2021" rid="ref-fang2021bayesian" ref-type="bibr">Fang
et al., 2021</xref>). In this design (Figure
<xref alt="[fig:snSMART-dose]" rid="figU003AsnSMART-dose">[fig:snSMART-dose]</xref>),
participants are equally assigned to receive either placebo, low dose,
or high dose in the first stage. They continue their initial treatment
for a pre-specified duration until their responses are measured at the
end of stage 1. In the second stage, all participants who initially
received placebo or low dose are re-randomized to either low or high
dose, regardless of their first stage response. Participants who
responded to the high dose are re-randomized between low and high
doses, while non-responders to the high dose continue with the high
dose in the second stage. The main goal of this snSMART is to estimate
and compare first stage response rates for low and high doses to
placebo by modeling the pooled data from both stages.</p>
<fig>
<caption><p>Study design of an snSMART with two dose levels and a
placebo. In stage 1, participants are randomized (R) to treatment P
(placebo), L (low dose), or H (high dose) with equal probability. At
time t, response to stage 1 treatment is assessed. Non-responders to
high dose stay on the same treatment in stage 2, while all the other
participants are equally re-randomized to either low or high dose in
stage 2. Interest is in the first stage response rate of placebo,
low and high
doses.<styled-content id="figU003AsnSMART-dose"></styled-content></p></caption>
<graphic mimetype="image" mime-subtype="png" xlink:href="dose_snSMART.png" />
</fig>
<p>Fang et al.
(<xref alt="2021" rid="ref-fang2021bayesian" ref-type="bibr">2021</xref>)
adapted the Bayesian joint stage model (BJSM) from Wei et al.
(<xref alt="2018" rid="ref-wei2018bayesian" ref-type="bibr">2018</xref>)
in Equations <xref alt="1" rid="eqU003A1">1</xref> and
<xref alt="2" rid="eqU003A2">2</xref>. For this design,
<inline-formula><alternatives>
<tex-math><![CDATA[m = P, L, H]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mo>,</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>
and <inline-formula><alternatives>
<tex-math><![CDATA[m' = L, H]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mi>′</mml:mi><mml:mo>=</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mi>H</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>,
where <inline-formula><alternatives>
<tex-math><![CDATA[P]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math></alternatives></inline-formula>
represents placebo, <inline-formula><alternatives>
<tex-math><![CDATA[L]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math></alternatives></inline-formula>
low dose, and <inline-formula><alternatives>
<tex-math><![CDATA[H]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>H</mml:mi></mml:math></alternatives></inline-formula>
high dose. The prior distribution for the response rate of placebo,
<inline-formula><alternatives>
<tex-math><![CDATA[\pi_P]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>π</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:math></alternatives></inline-formula>,
may be informed by natural history studies or previous trials and
specified as <inline-formula><alternatives>
<tex-math><![CDATA[\pi_P \sim Beta(\zeta_n, \eta_n)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>π</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo>∼</mml:mo><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>ζ</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>η</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>.
It is assumed that the drug doses have a weak tendency for higher
response rates than placebo, modeled as <inline-formula><alternatives>
<tex-math><![CDATA[\log(\pi_L/\pi_P) \sim N(\mu, \sigma^2)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mi>/</mml:mi><mml:msub><mml:mi>π</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>
and <inline-formula><alternatives>
<tex-math><![CDATA[\log(\pi_H/\pi_P) \sim N(\mu, \sigma^2)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>H</mml:mi></mml:msub><mml:mi>/</mml:mi><mml:msub><mml:mi>π</mml:mi><mml:mi>P</mml:mi></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>∼</mml:mo><mml:mi>N</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>μ</mml:mi><mml:mo>,</mml:mo><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>.
The prior distributions for the linkage parameters may vary, specified
as <inline-formula><alternatives>
<tex-math><![CDATA[\beta_{0m}, \beta_{1m} \sim Gamma(\omega, \psi)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:mi>G</mml:mi><mml:mi>a</mml:mi><mml:mi>m</mml:mi><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>ω</mml:mi><mml:mo>,</mml:mo><mml:mi>ψ</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>.</p>
<p>The BJSM is specified as follows:</p>
<p><named-content id="eqU003A1" content-type="equation"><disp-formula><alternatives>
<tex-math><![CDATA[Y_{i1m}|\pi_m \sim Bernoulli(\pi_m) ]]></tex-math>
<mml:math display="block" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false" form="prefix">|</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo>∼</mml:mo><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>n</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula></named-content>
<named-content id="eqU003A2" content-type="equation"><disp-formula><alternatives>
<tex-math><![CDATA[Y_{i2m'}|Y_{i1m}, \pi_{m'}, \beta_{1m}, \beta_{0m} \sim Bernoulli((\beta_{1m}\pi_{m})^{Y_{i1m}}(\beta_{0m}\pi_{m'})^{1-Y_{i1m}}) ]]></tex-math>
<mml:math display="block" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>2</mml:mn><mml:mi>m</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false" form="prefix">|</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>r</mml:mi><mml:mi>n</mml:mi><mml:mi>o</mml:mi><mml:mi>u</mml:mi><mml:mi>l</mml:mi><mml:mi>l</mml:mi><mml:mi>i</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>π</mml:mi><mml:mi>m</mml:mi></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:msup><mml:msup><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>π</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:msup><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula></named-content>
for <inline-formula><alternatives>
<tex-math><![CDATA[i = 1,...,N]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>.</mml:mi><mml:mi>.</mml:mi><mml:mi>.</mml:mi><mml:mo>,</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>;
and <inline-formula><alternatives>
<tex-math><![CDATA[j = 1, 2]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>;
where <inline-formula><alternatives>
<tex-math><![CDATA[Y_{ijm}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Y</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
is the outcome for participant <inline-formula><alternatives>
<tex-math><![CDATA[i]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>
at stage <inline-formula><alternatives>
<tex-math><![CDATA[j]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>j</mml:mi></mml:math></alternatives></inline-formula>
for treatment <inline-formula><alternatives>
<tex-math><![CDATA[m]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math></alternatives></inline-formula>
and takes the value 1 for response to treatment and 0 for no response;
<inline-formula><alternatives>
<tex-math><![CDATA[N]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi></mml:math></alternatives></inline-formula>
is the total sample size; <inline-formula><alternatives>
<tex-math><![CDATA[\beta_{0m}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
and <inline-formula><alternatives>
<tex-math><![CDATA[\beta_{1m}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
are the linkage parameters for non-responders and responders,
respectively; <inline-formula><alternatives>
<tex-math><![CDATA[\pi_m]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>π</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:math></alternatives></inline-formula>
is the first stage response rate for treatment
<inline-formula><alternatives>
<tex-math><![CDATA[m]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math></alternatives></inline-formula>;
<inline-formula><alternatives>
<tex-math><![CDATA[\beta_{1m}\pi_{m}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>π</mml:mi><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>
is the second stage response rate for first stage responders; and
<inline-formula><alternatives>
<tex-math><![CDATA[\beta_{0m}\pi_{m'}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mn>0</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>π</mml:mi><mml:mrow><mml:mi>m</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives></inline-formula>
is the second stage response rate for non-responders to treatment
<inline-formula><alternatives>
<tex-math><![CDATA[m]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math></alternatives></inline-formula>
in the first stage who receive treatment
<inline-formula><alternatives>
<tex-math><![CDATA[m']]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mi>′</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>
in the second stage.</p>
<p>To conduct the analysis in R, we can use the
<monospace>BJSM_binary</monospace> function. Users specify priors,
MCMC details, and BJSM model type (six beta or two beta). Here, we
assume the prior distribution of <inline-formula><alternatives>
<tex-math><![CDATA[\pi_P]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>π</mml:mi><mml:mi>P</mml:mi></mml:msub></mml:math></alternatives></inline-formula>
as <inline-formula><alternatives>
<tex-math><![CDATA[Beta(3, 17)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>B</mml:mi><mml:mi>e</mml:mi><mml:mi>t</mml:mi><mml:mi>a</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>17</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>,
<inline-formula><alternatives>
<tex-math><![CDATA[\beta_{jm}]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>β</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msub></mml:math></alternatives></inline-formula>
as <inline-formula><alternatives>
<tex-math><![CDATA[Gamma(2, 2)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mi>a</mml:mi><mml:mi>m</mml:mi><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>,
and the treatment effect ratio as <inline-formula><alternatives>
<tex-math><![CDATA[Normal(0.2, 100)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mi>o</mml:mi><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>l</mml:mi><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mn>0.2</mml:mn><mml:mo>,</mml:mo><mml:mn>100</mml:mn><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>.
Label placebo as 1, low dose as 2, and high dose as 3 in the dataset.
The output is a <monospace>BJSM_dose_binary</monospace> object with
posterior samples and estimates of linkage parameters, treatment
response rates, and pairwise response rate differences.</p>
<code language="r script">BJSM_dose_result <- BJSM_binary(
data = data_dose, prior_dist = c("beta", "gamma"),
pi_prior = c(3, 17), normal.par = c(0.2, 100), beta_prior = c(2, 2),
n_MCMC_chain = 2, n.adapt = 1000, BURN.IN = 10000,
MCMC_SAMPLE = 60000, ci = 0.95
)
summary(BJSM_dose_result)</code>
<code language="r script">Treatment Effects Estimate:
Estimate Std. Error C.I. CI low CI high
trtP 0.08606853 0.04004852 0.95 0.01694565 0.1618828
trtL 0.39969511 0.06130935 0.95 0.28185110 0.5202667
trtH 0.73414788 0.07501235 0.95 0.58710144 0.8763916
Differences between Treatments:
Estimate Std.Error C.I. CI low CI high
diffPL -0.3136266 0.07345504 0.95 -0.4577648 -0.1696336
diffLH -0.3344528 0.07967433 0.95 -0.4895912 -0.1785552
diffPH -0.6480794 0.08559511 0.95 -0.8071207 -0.4772492
Linkage Parameter Estimate:
Estimate Std. Error C.I. CI low CI high
beta[1,1] 0.9763364 0.1640819 0.95 0.65222142 1.2973089
beta[2,1] 0.8560191 0.3257939 0.95 0.23204941 1.4280772
beta[1,2] 1.0749284 0.1869756 0.95 0.70435649 1.4426901
beta[2,2] 0.9872268 0.2503916 0.95 0.48669193 1.4458416
beta[1,3] 0.3824723 0.1899827 0.95 0.05813823 0.7529239
beta[2,3] 1.0703154 0.1657493 0.95 0.74952233 1.4055420</code>
<p>The response rates for placebo, low dose and high dose are
estimated to be <monospace>trtP</monospace> 0.09 (95% credible
interval (CI): 0.02 - 0.16), <monospace>trtL</monospace> 0.40 (95% CI:
0.28 - 0.52), and <monospace>trtH</monospace> 0.73 (95% CI: 0.59 -
0.88) respectively. Other estimated outcomes are also clearly
presented in the R output above.</p>
</sec>
<sec id="discussion">
<title>Discussion</title>
<p>We introduced and demonstrated the <monospace>snSMART</monospace>
package for analyzing snSMART data using Bayesian methods. BJSM is
often more efficient, but frequentist methods are recommended for
sensitivity analysis. The package will be updated with new designs and
methods to aid in finding effective treatments for rare diseases.</p>
</sec>
<sec id="acknowledgments">
<title>Acknowledgments</title>
<p>This work was supported by a Patient-Centered Outcomes Research
Institute (PCORI) award (ME-1507-31108). We thank Boxian Wei,
Yan-Cheng Chao, and Holly Hartman for contributing their original R
code to the creation of this package. We also thank Mike Kleinsasser
for assisting with the publication of the R package on CRAN.</p>
</sec>
</body>
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