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Winter Term Practicum Presentation - Xiao Yan.html
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Winter Term Practicum Presentation - Xiao Yan.html
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<body class="quarto-light">
<div class="reveal">
<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Bayesian Marginal Structural Models</h1>
<p class="subtitle">An R Package Project</p>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
Xiao Yan | Supervisor: Kuan Liu
</div>
</div>
</div>
</section>
<section id="outline" class="slide level2" data-transition="fade">
<h2>Outline</h2>
<ul>
<li>Background</li>
<li>Motivation</li>
<li>Methods</li>
<li>Results</li>
<li>Summary</li>
</ul>
</section>
<section id="background" class="slide level2" data-transition="slide">
<h2>Background</h2>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Observational studies offer a viable, efficient, and low-cost design to readily gather evidence on exposure effects.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Although more practical, exposure mechanism is nonrandomized and <mark>causal inference methods</mark> are required to draw causal conclusions.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Popular approaches used in health research are predominantly <mark>frequentist</mark> methods.</li>
</ul>
</div>
</section>
<section id="motivation" class="slide level2" data-transition="slide">
<h2>Motivation</h2>
<div class="fragment">
<ul>
<li>A stream of Bayesian causal inference methods has been developed.</li>
</ul>
<div class="fragment">
<ul>
<li>Bayesian approaches have unique estimation features that are useful in many settings, however, there is a general lack of open-access software packages to carry out these analyses.</li>
</ul>
</div>
<div class="fragment">
<div class="fragment highlight-red">
<ul>
<li>Goal: build a user-friendly R package for Bayesian Marginal Structural Models (BMSMs).</li>
</ul>
</div>
</div>
</div>
</section>
<section id="methodology" class="slide level2" data-transition="slide">
<h2>Methodology</h2>
<p>2 estimation steps:</p>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Step 1. Bayesian treatment effect weight (similar to PS)</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Step 2. Bayesian non-parametric bootstrap to maximize the utility function with respect to the causal effect</li>
</ul>
</div>
<!-- ## Methodology {transition="slide"} -->
<!-- Next, we introduce the methodology used in the `bayesmsm` package to perform step 2 of Bayesian Marginal Structural Models (BMSMs) analysis. -->
<!-- ::: {.fragment .highlight-red} -->
<!-- - Main function: 'bayesmsm' -->
<!-- ::: -->
<!-- - Used to conduct Bayesian non-parametric bootstrap to calculate causal effect. -->
</section>
<section id="notation" class="slide level2" data-transition="fade">
<h2>Notation</h2>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>A longitudinal observational study with n subjects indexed by <span class="math inline">\(i\)</span>, <span class="math inline">\(i = 1, \ldots, n\)</span> and <span class="math inline">\(J\)</span> number of visits indexed by <span class="math inline">\(j\)</span>, <span class="math inline">\(j = 1, \ldots, J\)</span>.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li><span class="math inline">\(Y_{i}\)</span>, <span class="math inline">\(X_{ij}\)</span> and <span class="math inline">\(Z_{ij}\)</span> are random variables representing an end-of-study response, covariates and the treatment for individual <span class="math inline">\(i\)</span> at visit <span class="math inline">\(j\)</span>.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>History up to visit <span class="math inline">\(j\)</span> are denoted as <span class="math inline">\(\bar{X}_{ij}\)</span> and <span class="math inline">\(\bar{Z}_{ij}\)</span>.</li>
</ul>
</div>
</section>
<section id="methodology-bmsms" class="slide level2 smaller" data-transition="fade">
<h2>Methodology: BMSMs</h2>
<p>Using Bayesian decision theory and importance sampling technique, we maximize an expected utility function (a function involving only <span class="math inline">\(\theta\)</span>), <span class="math inline">\(\textbf{u}_{\mathcal{E}}(\Theta, \bar{v}_{i}^*)\)</span>, via posterior predictive inference,</p>
<p><span class="math display">\[ \hat{\Theta}
= argmax_{\theta} \int_{\bar{v}_{i}^*} u_{\mathcal{E}}(\Theta, \bar{v}_{i}^*)P_{\mathcal{E}}(\bar{v}_{i}^* \mid \textbf{V}_n) \ d\bar{v}_{i}^* \nonumber \\ \]</span></p>
<p><span class="math display">\[ = argmax_{\theta}\int_{\bar{v}_{i}^*} u_{\mathcal{E}}(\Theta, \bar{v}_{i}^*) \frac{P_{\mathcal{E}}(\bar{v}_{i}^* \mid \textbf{V}_n) }{P_{\mathcal{O}}(\bar{v}_{i}^* \mid \textbf{V}_n)}P_{\mathcal{O}}(\bar{v}_{i}^* \mid \textbf{V}_n) \ d\bar{v}_{i}^* \]</span></p>
<ul>
<li><p><span class="math inline">\(u(\Theta, \bar{v}_{i}^*)= log P_{\mathcal{E}}( Y_{i}^* \mid \bar{z}_{iJ}^*; \Theta)\)</span>, <mark>utility function</mark></p></li>
<li><p><span class="math inline">\(w_{i}^* = \frac{P_{\mathcal{E}}(\bar{v}_{i}^* \mid \textbf{v}_n)}{P_{\mathcal{O}}(\bar{v}_{i}^* \mid \textbf{v}_n)}\)</span> expanded to <mark>treatment assignment weight</mark></p></li>
</ul>
</section>
<section id="weighted-log-likelihood" class="slide level2" data-transition="fade">
<h2>Weighted log-likelihood</h2>
<div class="panel-tabset">
<ul id="tabset-1" class="panel-tabset-tabby"><li><a data-tabby-default="" href="#tabset-1-1">Normal Y</a></li><li><a href="#tabset-1-2">Binary Y</a></li></ul>
<div class="tab-content">
<div id="tabset-1-1">
<p><span class="math display">\[\begin{equation}
\mathcal{l}(\theta, \sigma^2 | Y, A) = \sum_{i=1}^{n} w_i \left( -\frac{1}{2} \log(\sigma^2) - \frac{1}{2\sigma^2} (y_i - A_i \theta)^2 \right)
\end{equation}\]</span></p>
<ul>
<li><span class="math inline">\(\theta\)</span>: causal parameters on the mean</li>
<li><span class="math inline">\(\sigma\)</span>: causal parameter on the sd</li>
<li><span class="math inline">\(A\)</span>: design matrix of the causal outcome model</li>
</ul>
</div>
<div id="tabset-1-2">
<p><span class="math display">\[\begin{equation}
\mathcal{l}(\beta | Y, A) = \sum_{i=1}^{n} w_i \left( Y_i \eta_i - \log(1 + \exp(\eta_i)) \right)
\end{equation}\]</span></p>
<ul>
<li><span class="math inline">\(\beta\)</span>: causal parameters on the log-odds scale</li>
<li><span class="math inline">\(\eta\)</span>: <span class="math inline">\(A\)</span> * <span class="math inline">\(\beta\)</span>, the linear predictor</li>
</ul>
</div>
</div>
</div>
<div class="footer">
<p>Reference: <a href="https://www.stat.cmu.edu/~cshalizi/mreg/15/lectures/06/lecture-06.pdf">linearML</a> and <a href="https://www.ime.unicamp.br/~cnaber/optim_1.pdf">optim</a></p>
</div>
</section>
<section id="parallel-computing" class="slide level2">
<h2>Parallel computing</h2>
<p>Parallel computing for faster bootstrap calculation.</p>
<div class="sourceCode" id="cb1" data-code-line-numbers="|2|5-7"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href=""></a><span class="cf">if</span> (parallel <span class="sc">==</span> <span class="cn">TRUE</span>){</span>
<span id="cb1-2"><a href=""></a> numCores <span class="ot"><-</span> ncore</span>
<span id="cb1-3"><a href=""></a> <span class="fu">registerDoParallel</span>(<span class="at">cores =</span> numCores)</span>
<span id="cb1-4"><a href=""></a> </span>
<span id="cb1-5"><a href=""></a> results <span class="ot"><-</span> <span class="fu">foreach</span>(<span class="at">i=</span><span class="dv">1</span><span class="sc">:</span>nboot,</span>
<span id="cb1-6"><a href=""></a> <span class="at">.combine =</span> <span class="st">'rbind'</span>,</span>
<span id="cb1-7"><a href=""></a> <span class="at">.packages =</span> <span class="st">'MCMCpack'</span>) <span class="sc">%dopar%</span> {</span>
<span id="cb1-8"><a href=""></a> </span>
<span id="cb1-9"><a href=""></a> ... <span class="co"># Bootstrap calculation</span></span>
<span id="cb1-10"><a href=""></a> </span>
<span id="cb1-11"><a href=""></a> }</span>
<span id="cb1-12"><a href=""></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="putting-it-all-together" class="slide level2" data-transition="fade">
<h2>Putting it all together</h2>
<p>The complete function ‘bayesmsm’, at a glance:</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href=""></a>bayesmsm <span class="ot"><-</span> <span class="cf">function</span>(ymodel,</span>
<span id="cb2-2"><a href=""></a> nvisit,</span>
<span id="cb2-3"><a href=""></a> <span class="at">reference =</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">0</span>,nvisit)), <span class="co"># An example of never treated</span></span>
<span id="cb2-4"><a href=""></a> <span class="at">comparator =</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">1</span>,nvisit)),</span>
<span id="cb2-5"><a href=""></a> <span class="at">family =</span> <span class="st">"gaussian"</span>, <span class="co"># "gaussian" or "binomial"</span></span>
<span id="cb2-6"><a href=""></a> data,</span>
<span id="cb2-7"><a href=""></a> <span class="at">wmean =</span> <span class="fu">rep</span>(<span class="dv">1</span>, <span class="fu">nrow</span>(data)),</span>
<span id="cb2-8"><a href=""></a> <span class="at">nboot =</span> <span class="dv">1000</span>,</span>
<span id="cb2-9"><a href=""></a> <span class="at">optim_method =</span> <span class="st">'BFGS'</span>,</span>
<span id="cb2-10"><a href=""></a> <span class="at">estimand =</span> <span class="st">'RD'</span>,</span>
<span id="cb2-11"><a href=""></a> <span class="at">parallel =</span> <span class="cn">TRUE</span>,</span>
<span id="cb2-12"><a href=""></a> <span class="at">ncore =</span> <span class="dv">6</span>){</span>
<span id="cb2-13"><a href=""></a></span>
<span id="cb2-14"><a href=""></a> <span class="co"># load all the required R packages;</span></span>
<span id="cb2-15"><a href=""></a> <span class="cf">if</span> (<span class="sc">!</span><span class="fu">require</span>(foreach)){</span>
<span id="cb2-16"><a href=""></a> <span class="fu">install.packages</span>(<span class="st">"foreach"</span>,<span class="at">repos=</span><span class="st">"http://cran.r-project.org"</span>)</span>
<span id="cb2-17"><a href=""></a> <span class="fu">library</span>(foreach)</span>
<span id="cb2-18"><a href=""></a> }</span>
<span id="cb2-19"><a href=""></a> <span class="cf">if</span> (<span class="sc">!</span><span class="fu">require</span>(doParallel)){</span>
<span id="cb2-20"><a href=""></a> <span class="fu">install.packages</span>(<span class="st">"doParallel"</span>,<span class="at">repos=</span><span class="st">"http://cran.r-project.org"</span>)</span>
<span id="cb2-21"><a href=""></a> <span class="fu">library</span>(doParallel)</span>
<span id="cb2-22"><a href=""></a> }</span>
<span id="cb2-23"><a href=""></a> <span class="cf">if</span> (<span class="sc">!</span><span class="fu">require</span>(MCMCpack)){</span>
<span id="cb2-24"><a href=""></a> <span class="fu">install.packages</span>(<span class="st">"MCMCpack"</span>,<span class="at">repos=</span><span class="st">"http://cran.r-project.org"</span>)</span>
<span id="cb2-25"><a href=""></a> <span class="fu">library</span>(MCMCpack)</span>
<span id="cb2-26"><a href=""></a> }</span>
<span id="cb2-27"><a href=""></a></span>
<span id="cb2-28"><a href=""></a> <span class="co"># return error message if the input weight vector has different length comparing to the outcome Y;</span></span>
<span id="cb2-29"><a href=""></a> <span class="cf">if</span> (<span class="fu">length</span>(wmean) <span class="sc">!=</span> <span class="fu">nrow</span>(data)) {</span>
<span id="cb2-30"><a href=""></a> <span class="fu">stop</span>(<span class="st">"The length of the weight vector does not match the length of Y."</span>)</span>
<span id="cb2-31"><a href=""></a> }</span>
<span id="cb2-32"><a href=""></a></span>
<span id="cb2-33"><a href=""></a> <span class="co"># load utility functions</span></span>
<span id="cb2-34"><a href=""></a> extract_variables <span class="ot"><-</span> <span class="cf">function</span>(formula) {</span>
<span id="cb2-35"><a href=""></a> <span class="co"># Get the terms of the formula</span></span>
<span id="cb2-36"><a href=""></a> formula_terms <span class="ot"><-</span> <span class="fu">terms</span>(formula)</span>
<span id="cb2-37"><a href=""></a></span>
<span id="cb2-38"><a href=""></a> <span class="co"># Extract the response variable name (if there is one)</span></span>
<span id="cb2-39"><a href=""></a> response_variable <span class="ot"><-</span> <span class="fu">attr</span>(formula_terms, <span class="st">"response"</span>)</span>
<span id="cb2-40"><a href=""></a> response_name <span class="ot"><-</span> <span class="cf">if</span> (response_variable <span class="sc">></span> <span class="dv">0</span>) {</span>
<span id="cb2-41"><a href=""></a> all_vars <span class="ot"><-</span> <span class="fu">all.vars</span>(formula)</span>
<span id="cb2-42"><a href=""></a> all_vars[response_variable]</span>
<span id="cb2-43"><a href=""></a> } <span class="cf">else</span> {<span class="cn">NA</span>}</span>
<span id="cb2-44"><a href=""></a></span>
<span id="cb2-45"><a href=""></a> <span class="co"># Extract predictor variable names</span></span>
<span id="cb2-46"><a href=""></a> predictor_names <span class="ot"><-</span> <span class="fu">attr</span>(formula_terms, <span class="st">"term.labels"</span>)</span>
<span id="cb2-47"><a href=""></a></span>
<span id="cb2-48"><a href=""></a> <span class="co"># Return a list of response and predictor variables</span></span>
<span id="cb2-49"><a href=""></a> <span class="fu">list</span>(<span class="at">response =</span> response_name, <span class="at">predictors =</span> predictor_names)</span>
<span id="cb2-50"><a href=""></a> }</span>
<span id="cb2-51"><a href=""></a></span>
<span id="cb2-52"><a href=""></a> variables <span class="ot"><-</span> <span class="fu">extract_variables</span>(ymodel) <span class="co"># Extract variable names from the formula</span></span>
<span id="cb2-53"><a href=""></a> Y_name <span class="ot"><-</span> variables<span class="sc">$</span>response</span>
<span id="cb2-54"><a href=""></a></span>
<span id="cb2-55"><a href=""></a> Y <span class="ot"><-</span> data[[Y_name]]</span>
<span id="cb2-56"><a href=""></a> A_base <span class="ot"><-</span> <span class="fu">data.frame</span>(<span class="fu">matrix</span>(<span class="at">data =</span> <span class="cn">NA</span>,</span>
<span id="cb2-57"><a href=""></a> <span class="at">nrow =</span> <span class="fu">nrow</span>(data),</span>
<span id="cb2-58"><a href=""></a> <span class="at">ncol =</span> <span class="fu">length</span>(variables<span class="sc">$</span>predictors)))</span>
<span id="cb2-59"><a href=""></a> <span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(variables<span class="sc">$</span>predictors)){</span>
<span id="cb2-60"><a href=""></a> initial_vector <span class="ot"><-</span> variables<span class="sc">$</span>predictors[i]</span>
<span id="cb2-61"><a href=""></a> split_vector <span class="ot"><-</span> <span class="fu">strsplit</span>(initial_vector, <span class="st">":"</span>)</span>
<span id="cb2-62"><a href=""></a> new_vector <span class="ot"><-</span> <span class="fu">unlist</span>(split_vector)</span>
<span id="cb2-63"><a href=""></a> <span class="cf">if</span> (<span class="fu">length</span>(new_vector)<span class="sc">==</span><span class="dv">1</span>){</span>
<span id="cb2-64"><a href=""></a> A_base[,i] <span class="ot"><-</span> data[, new_vector]</span>
<span id="cb2-65"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (<span class="fu">length</span>(new_vector)<span class="sc">></span><span class="dv">1</span>){</span>
<span id="cb2-66"><a href=""></a> A_base[,i] <span class="ot"><-</span> <span class="fu">apply</span>(data[, new_vector],<span class="dv">1</span>,prod)</span>
<span id="cb2-67"><a href=""></a> }</span>
<span id="cb2-68"><a href=""></a> }</span>
<span id="cb2-69"><a href=""></a></span>
<span id="cb2-70"><a href=""></a> A <span class="ot"><-</span> <span class="fu">cbind</span>(<span class="dv">1</span>, A_base)</span>
<span id="cb2-71"><a href=""></a> <span class="fu">colnames</span>(A)[<span class="dv">2</span><span class="sc">:</span><span class="fu">ncol</span>(A)]<span class="ot"><-</span> variables<span class="sc">$</span>predictors</span>
<span id="cb2-72"><a href=""></a></span>
<span id="cb2-73"><a href=""></a> wloglik_normal<span class="ot"><-</span><span class="cf">function</span>(param,</span>
<span id="cb2-74"><a href=""></a> Y,</span>
<span id="cb2-75"><a href=""></a> A,</span>
<span id="cb2-76"><a href=""></a> weight){</span>
<span id="cb2-77"><a href=""></a> <span class="co">#number of observations;</span></span>
<span id="cb2-78"><a href=""></a> n <span class="ot"><-</span> <span class="fu">length</span>(Y)</span>
<span id="cb2-79"><a href=""></a> theta <span class="ot"><-</span> param[<span class="dv">1</span><span class="sc">:</span><span class="fu">dim</span>(A)[<span class="dv">2</span>]] <span class="co">#causal parameters on the mean</span></span>
<span id="cb2-80"><a href=""></a> <span class="co">#number of parameter is determined by number of treatment variables, plus intercept;</span></span>
<span id="cb2-81"><a href=""></a> sigma <span class="ot"><-</span> param[(<span class="fu">dim</span>(A)[<span class="dv">2</span>]<span class="sc">+</span><span class="dv">1</span>)] <span class="co"># the remaining the parameter represent the standard deviation;</span></span>
<span id="cb2-82"><a href=""></a> mmat <span class="ot"><-</span> <span class="fu">as.matrix</span>(A) <span class="co">#design matrix of the causal outcome model, e.g., A = cbind(1, a_1, a_2);</span></span>
<span id="cb2-83"><a href=""></a> logl<span class="ot"><-</span> <span class="sc">-</span><span class="fl">0.5</span><span class="sc">*</span><span class="fu">log</span>(sigma<span class="sc">**</span><span class="dv">2</span>) <span class="sc">-</span> <span class="fl">0.5</span><span class="sc">*</span>((Y <span class="sc">-</span> mmat<span class="sc">%*%</span>theta)<span class="sc">**</span><span class="dv">2</span>)<span class="sc">/</span>(sigma<span class="sc">**</span><span class="dv">2</span>)</span>
<span id="cb2-84"><a href=""></a> wlogl<span class="ot"><-</span><span class="fu">sum</span>(weight<span class="sc">*</span>logl)</span>
<span id="cb2-85"><a href=""></a> <span class="fu">return</span>(wlogl)</span>
<span id="cb2-86"><a href=""></a> }</span>
<span id="cb2-87"><a href=""></a></span>
<span id="cb2-88"><a href=""></a> wloglik_binomial <span class="ot"><-</span> <span class="cf">function</span>(param,</span>
<span id="cb2-89"><a href=""></a> Y,</span>
<span id="cb2-90"><a href=""></a> A,</span>
<span id="cb2-91"><a href=""></a> weight){</span>
<span id="cb2-92"><a href=""></a> <span class="co"># number of observations;</span></span>
<span id="cb2-93"><a href=""></a> n <span class="ot"><-</span> <span class="fu">length</span>(Y)</span>
<span id="cb2-94"><a href=""></a> beta <span class="ot"><-</span> param[<span class="dv">1</span><span class="sc">:</span><span class="fu">dim</span>(A)[<span class="dv">2</span>]] <span class="co"># causal parameters on the log-odds scale (no sigma for binomial?)</span></span>
<span id="cb2-95"><a href=""></a> mmat <span class="ot"><-</span> <span class="fu">as.matrix</span>(A)</span>
<span id="cb2-96"><a href=""></a> eta<span class="ot"><-</span>mmat <span class="sc">%*%</span> beta <span class="co"># linear predictor</span></span>
<span id="cb2-97"><a href=""></a> logl <span class="ot"><-</span> Y<span class="sc">*</span>eta <span class="sc">-</span> <span class="fu">log</span>(<span class="dv">1</span><span class="sc">+</span><span class="fu">exp</span>(eta))</span>
<span id="cb2-98"><a href=""></a> wlogl<span class="ot"><-</span><span class="fu">sum</span>(weight<span class="sc">*</span>logl)</span>
<span id="cb2-99"><a href=""></a> <span class="fu">return</span>(wlogl)</span>
<span id="cb2-100"><a href=""></a> }</span>
<span id="cb2-101"><a href=""></a></span>
<span id="cb2-102"><a href=""></a> expit <span class="ot"><-</span> <span class="cf">function</span>(x){<span class="fu">exp</span>(x) <span class="sc">/</span> (<span class="dv">1</span><span class="sc">+</span><span class="fu">exp</span>(x))}</span>
<span id="cb2-103"><a href=""></a></span>
<span id="cb2-104"><a href=""></a> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"gaussian"</span>){</span>
<span id="cb2-105"><a href=""></a> wfn <span class="ot">=</span> wloglik_normal</span>
<span id="cb2-106"><a href=""></a> inits1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="fl">0.1</span>, <span class="fu">length</span>(A)), <span class="dv">4</span>) <span class="co"># Default initial values, 4 is for the SD;</span></span>
<span id="cb2-107"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"binomial"</span>){</span>
<span id="cb2-108"><a href=""></a> wfn <span class="ot">=</span> wloglik_binomial</span>
<span id="cb2-109"><a href=""></a> inits1 <span class="ot"><-</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="fl">0.1</span>, <span class="fu">length</span>(A)))</span>
<span id="cb2-110"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (<span class="sc">!</span>family <span class="sc">%in%</span> <span class="fu">c</span>(<span class="st">"gaussian"</span>,<span class="st">"binomial"</span>)){</span>
<span id="cb2-111"><a href=""></a> <span class="fu">stop</span>(<span class="st">"Current version only handles continuous (gaussian) and binary (binomial) outcomes."</span>)</span>
<span id="cb2-112"><a href=""></a> }</span>
<span id="cb2-113"><a href=""></a></span>
<span id="cb2-114"><a href=""></a></span>
<span id="cb2-115"><a href=""></a> <span class="co"># Parallel computing for bootstrapping</span></span>
<span id="cb2-116"><a href=""></a> <span class="cf">if</span> (parallel <span class="sc">==</span> <span class="cn">TRUE</span>){</span>
<span id="cb2-117"><a href=""></a> numCores <span class="ot"><-</span> ncore</span>
<span id="cb2-118"><a href=""></a> <span class="fu">registerDoParallel</span>(<span class="at">cores =</span> numCores)</span>
<span id="cb2-119"><a href=""></a></span>
<span id="cb2-120"><a href=""></a> results <span class="ot"><-</span> <span class="fu">foreach</span>(<span class="at">i=</span><span class="dv">1</span><span class="sc">:</span>nboot,</span>
<span id="cb2-121"><a href=""></a> <span class="at">.combine =</span> <span class="st">'rbind'</span>,</span>
<span id="cb2-122"><a href=""></a> <span class="at">.packages =</span> <span class="st">'MCMCpack'</span>) <span class="sc">%dopar%</span> {</span>
<span id="cb2-123"><a href=""></a></span>
<span id="cb2-124"><a href=""></a> calculate_effect <span class="ot"><-</span> <span class="cf">function</span>(intervention_levels, variables, param_estimates) {</span>
<span id="cb2-125"><a href=""></a> <span class="co"># Start with the intercept term</span></span>
<span id="cb2-126"><a href=""></a> effect<span class="ot"><-</span>effect_intercept<span class="ot"><-</span>param_estimates[<span class="dv">1</span>]</span>
<span id="cb2-127"><a href=""></a></span>
<span id="cb2-128"><a href=""></a> <span class="co"># Go through each predictor and add its contribution</span></span>
<span id="cb2-129"><a href=""></a> <span class="cf">for</span> (i <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span><span class="fu">length</span>(variables<span class="sc">$</span>predictors)) {</span>
<span id="cb2-130"><a href=""></a> term <span class="ot"><-</span> variables<span class="sc">$</span>predictors[i]</span>
<span id="cb2-131"><a href=""></a> term_variables <span class="ot"><-</span> <span class="fu">unlist</span>(<span class="fu">strsplit</span>(term, <span class="st">":"</span>))</span>
<span id="cb2-132"><a href=""></a> term_index <span class="ot"><-</span> <span class="fu">which</span>(<span class="fu">names</span>(param_estimates) <span class="sc">==</span> term)</span>
<span id="cb2-133"><a href=""></a></span>
<span id="cb2-134"><a href=""></a> <span class="co"># Calculate the product of intervention levels for the interaction term</span></span>
<span id="cb2-135"><a href=""></a> term_contribution <span class="ot"><-</span> param_estimates[term_index]</span>
<span id="cb2-136"><a href=""></a> <span class="cf">for</span> (term_variable <span class="cf">in</span> term_variables) {</span>
<span id="cb2-137"><a href=""></a> var_index <span class="ot"><-</span> <span class="fu">which</span>(variables<span class="sc">$</span>predictors <span class="sc">==</span> term_variable)</span>
<span id="cb2-138"><a href=""></a> term_contribution <span class="ot"><-</span> term_contribution <span class="sc">*</span> intervention_levels[var_index]</span>
<span id="cb2-139"><a href=""></a> }</span>
<span id="cb2-140"><a href=""></a></span>
<span id="cb2-141"><a href=""></a> <span class="co"># Add the term contribution to the effect</span></span>
<span id="cb2-142"><a href=""></a> effect <span class="ot"><-</span> effect <span class="sc">+</span> term_contribution</span>
<span id="cb2-143"><a href=""></a> }</span>
<span id="cb2-144"><a href=""></a></span>
<span id="cb2-145"><a href=""></a> <span class="fu">return</span>(effect)</span>
<span id="cb2-146"><a href=""></a> }</span>
<span id="cb2-147"><a href=""></a></span>
<span id="cb2-148"><a href=""></a> results.it <span class="ot"><-</span> <span class="fu">matrix</span>(<span class="cn">NA</span>, <span class="dv">1</span>, <span class="dv">3</span>) <span class="co">#result matrix, three columns for bootest, effect_ref, and effect_comp;</span></span>
<span id="cb2-149"><a href=""></a></span>
<span id="cb2-150"><a href=""></a> alpha <span class="ot"><-</span> <span class="fu">as.numeric</span>(<span class="fu">rdirichlet</span>(<span class="dv">1</span>, <span class="fu">rep</span>(<span class="fl">1.0</span>, <span class="fu">length</span>(Y))))</span>
<span id="cb2-151"><a href=""></a></span>
<span id="cb2-152"><a href=""></a> maxim <span class="ot"><-</span> <span class="fu">optim</span>(inits1,</span>
<span id="cb2-153"><a href=""></a> <span class="at">fn =</span> wfn,</span>
<span id="cb2-154"><a href=""></a> <span class="at">Y =</span> Y,</span>
<span id="cb2-155"><a href=""></a> <span class="at">A =</span> A,</span>
<span id="cb2-156"><a href=""></a> <span class="at">weight =</span> alpha <span class="sc">*</span> wmean,</span>
<span id="cb2-157"><a href=""></a> <span class="at">control =</span> <span class="fu">list</span>(<span class="at">fnscale =</span> <span class="sc">-</span><span class="dv">1</span>),</span>
<span id="cb2-158"><a href=""></a> <span class="at">method =</span> optim_method,</span>
<span id="cb2-159"><a href=""></a> <span class="at">hessian =</span> <span class="cn">FALSE</span>)</span>
<span id="cb2-160"><a href=""></a></span>
<span id="cb2-161"><a href=""></a> <span class="fu">names</span>(maxim<span class="sc">$</span>par) <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"(Intercept)"</span>, variables<span class="sc">$</span>predictors)</span>
<span id="cb2-162"><a href=""></a></span>
<span id="cb2-163"><a href=""></a> <span class="co"># Calculate the effects</span></span>
<span id="cb2-164"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">1</span>] <span class="ot"><-</span> <span class="fu">calculate_effect</span>(reference, variables, <span class="at">param_estimates=</span>maxim<span class="sc">$</span>par)</span>
<span id="cb2-165"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">2</span>] <span class="ot"><-</span> <span class="fu">calculate_effect</span>(comparator, variables, <span class="at">param_estimates=</span>maxim<span class="sc">$</span>par)</span>
<span id="cb2-166"><a href=""></a></span>
<span id="cb2-167"><a href=""></a> <span class="co"># Calculate the ATE</span></span>
<span id="cb2-168"><a href=""></a> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"binomial"</span>) { <span class="co"># binary outcomes</span></span>
<span id="cb2-169"><a href=""></a> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RD"</span>) { <span class="co"># Risk Difference</span></span>
<span id="cb2-170"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">3</span>] <span class="ot"><-</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">2</span>]) <span class="sc">-</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">1</span>])</span>
<span id="cb2-171"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RR"</span>) { <span class="co"># Relative Risk</span></span>
<span id="cb2-172"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">3</span>] <span class="ot"><-</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">2</span>]) <span class="sc">/</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">1</span>])</span>
<span id="cb2-173"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"OR"</span>) { <span class="co"># Odds Ratio</span></span>
<span id="cb2-174"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">3</span>] <span class="ot"><-</span> (<span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">2</span>]) <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">2</span>]))) <span class="sc">/</span></span>
<span id="cb2-175"><a href=""></a> (<span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">1</span>]) <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> <span class="fu">expit</span>(results.it[<span class="dv">1</span>,<span class="dv">1</span>])))</span>
<span id="cb2-176"><a href=""></a> }</span>
<span id="cb2-177"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"gaussian"</span>){ <span class="co"># continuous outcomes</span></span>
<span id="cb2-178"><a href=""></a> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RD"</span>) { <span class="co"># Risk Difference</span></span>
<span id="cb2-179"><a href=""></a> results.it[<span class="dv">1</span>,<span class="dv">3</span>] <span class="ot"><-</span> results.it[<span class="dv">1</span>,<span class="dv">2</span>] <span class="sc">-</span> results.it[<span class="dv">1</span>,<span class="dv">1</span>]</span>
<span id="cb2-180"><a href=""></a> }</span>
<span id="cb2-181"><a href=""></a> }</span>
<span id="cb2-182"><a href=""></a></span>
<span id="cb2-183"><a href=""></a> <span class="co"># combining parallel results;</span></span>
<span id="cb2-184"><a href=""></a> <span class="fu">cbind</span>(i,results.it) <span class="co">#end of parallel;</span></span>
<span id="cb2-185"><a href=""></a> }</span>
<span id="cb2-186"><a href=""></a></span>
<span id="cb2-187"><a href=""></a> <span class="co">#saving output for the parallel setting;</span></span>
<span id="cb2-188"><a href=""></a> <span class="fu">return</span>(<span class="fu">list</span>(</span>
<span id="cb2-189"><a href=""></a> <span class="at">mean =</span> <span class="fu">mean</span>(results[,<span class="dv">4</span>]),</span>
<span id="cb2-190"><a href=""></a> <span class="at">sd =</span> <span class="fu">sqrt</span>(<span class="fu">var</span>(results[,<span class="dv">4</span>])),</span>
<span id="cb2-191"><a href=""></a> <span class="at">quantile =</span> <span class="fu">quantile</span>(results[,<span class="dv">4</span>], <span class="at">probs =</span> <span class="fu">c</span>(<span class="fl">0.025</span>, <span class="fl">0.975</span>)),</span>
<span id="cb2-192"><a href=""></a> <span class="at">bootdata =</span> <span class="fu">data.frame</span>(<span class="at">effect_reference =</span> results[,<span class="dv">2</span>],</span>
<span id="cb2-193"><a href=""></a> <span class="at">effect_comparator =</span> results[,<span class="dv">3</span>],</span>
<span id="cb2-194"><a href=""></a> <span class="at">ATE =</span> results[,<span class="dv">4</span>]),</span>
<span id="cb2-195"><a href=""></a> <span class="at">reference =</span> reference,</span>
<span id="cb2-196"><a href=""></a> <span class="at">comparator =</span> comparator</span>
<span id="cb2-197"><a href=""></a> ))</span>
<span id="cb2-198"><a href=""></a></span>
<span id="cb2-199"><a href=""></a> }</span>
<span id="cb2-200"><a href=""></a></span>
<span id="cb2-201"><a href=""></a> <span class="cf">else</span> <span class="cf">if</span> (parallel <span class="sc">==</span> <span class="cn">FALSE</span>) {</span>
<span id="cb2-202"><a href=""></a></span>
<span id="cb2-203"><a href=""></a> bootest <span class="ot"><-</span> <span class="fu">numeric</span>(nboot)</span>
<span id="cb2-204"><a href=""></a> effect_reference <span class="ot"><-</span> <span class="fu">numeric</span>(nboot)</span>
<span id="cb2-205"><a href=""></a> effect_comparator <span class="ot"><-</span> <span class="fu">numeric</span>(nboot)</span>
<span id="cb2-206"><a href=""></a></span>
<span id="cb2-207"><a href=""></a> <span class="cf">for</span> (j <span class="cf">in</span> <span class="dv">1</span><span class="sc">:</span>nboot) {</span>
<span id="cb2-208"><a href=""></a> alpha <span class="ot"><-</span> <span class="fu">as.numeric</span>(<span class="fu">rdirichlet</span>(<span class="dv">1</span>, <span class="fu">rep</span>(<span class="fl">1.0</span>, <span class="fu">length</span>(Y))))</span>
<span id="cb2-209"><a href=""></a></span>
<span id="cb2-210"><a href=""></a> maxim <span class="ot"><-</span> <span class="fu">optim</span>(inits1,</span>
<span id="cb2-211"><a href=""></a> <span class="at">fn =</span> wfn,</span>
<span id="cb2-212"><a href=""></a> <span class="at">Y =</span> Y,</span>
<span id="cb2-213"><a href=""></a> <span class="at">A =</span> A,</span>
<span id="cb2-214"><a href=""></a> <span class="at">weight =</span> alpha <span class="sc">*</span> wmean,</span>
<span id="cb2-215"><a href=""></a> <span class="at">control =</span> <span class="fu">list</span>(<span class="at">fnscale =</span> <span class="sc">-</span><span class="dv">1</span>),</span>
<span id="cb2-216"><a href=""></a> <span class="at">method =</span> optim_method,</span>
<span id="cb2-217"><a href=""></a> <span class="at">hessian =</span> <span class="cn">FALSE</span>)</span>
<span id="cb2-218"><a href=""></a></span>
<span id="cb2-219"><a href=""></a> <span class="fu">names</span>(maxim<span class="sc">$</span>par) <span class="ot"><-</span> <span class="fu">c</span>(<span class="st">"(Intercept)"</span>, variables<span class="sc">$</span>predictors)</span>
<span id="cb2-220"><a href=""></a></span>
<span id="cb2-221"><a href=""></a> <span class="co"># Calculate the effects</span></span>
<span id="cb2-222"><a href=""></a> effect_reference[j] <span class="ot"><-</span> <span class="fu">calculate_effect</span>(reference, variables, <span class="at">param_estimates=</span>maxim<span class="sc">$</span>par)</span>
<span id="cb2-223"><a href=""></a> effect_comparator[j] <span class="ot"><-</span> <span class="fu">calculate_effect</span>(comparator, variables, <span class="at">param_estimates=</span>maxim<span class="sc">$</span>par)</span>
<span id="cb2-224"><a href=""></a></span>
<span id="cb2-225"><a href=""></a> <span class="co"># Calculate the ATE</span></span>
<span id="cb2-226"><a href=""></a> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"binomial"</span>) { <span class="co"># binary outcomes</span></span>
<span id="cb2-227"><a href=""></a> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RD"</span>) { <span class="co"># Risk Difference</span></span>
<span id="cb2-228"><a href=""></a> bootest[j] <span class="ot"><-</span> <span class="fu">expit</span>(effect_comparator[j]) <span class="sc">-</span> <span class="fu">expit</span>(effect_reference[j])</span>
<span id="cb2-229"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RR"</span>) { <span class="co"># Relative Risk</span></span>
<span id="cb2-230"><a href=""></a> bootest[j] <span class="ot"><-</span> <span class="fu">expit</span>(effect_comparator[j]) <span class="sc">/</span> <span class="fu">expit</span>(effect_reference[j])</span>
<span id="cb2-231"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"OR"</span>) { <span class="co"># Odds Ratio</span></span>
<span id="cb2-232"><a href=""></a> bootest[j] <span class="ot"><-</span> (<span class="fu">expit</span>(effect_comparator[j]) <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> <span class="fu">expit</span>(effect_comparator[j]))) <span class="sc">/</span></span>
<span id="cb2-233"><a href=""></a> (<span class="fu">expit</span>(effect_reference[j]) <span class="sc">/</span> (<span class="dv">1</span> <span class="sc">-</span> <span class="fu">expit</span>(effect_reference[j])))</span>
<span id="cb2-234"><a href=""></a> }</span>
<span id="cb2-235"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (family <span class="sc">==</span> <span class="st">"gaussian"</span>){ <span class="co"># continuous outcomes</span></span>
<span id="cb2-236"><a href=""></a> <span class="cf">if</span> (estimand <span class="sc">==</span> <span class="st">"RD"</span>) { <span class="co"># Risk Difference</span></span>
<span id="cb2-237"><a href=""></a> bootest[j] <span class="ot"><-</span> effect_comparator[j] <span class="sc">-</span> effect_reference[j]</span>
<span id="cb2-238"><a href=""></a> } <span class="cf">else</span> <span class="cf">if</span> (estimand <span class="sc">%in%</span> <span class="fu">c</span>(<span class="st">"RR"</span>,<span class="st">"OR"</span>)) {</span>
<span id="cb2-239"><a href=""></a> <span class="co"># print a warning message that say for continuous outcome, RR and OR specification are ignored. RD is the causal estimate;</span></span>
<span id="cb2-240"><a href=""></a> <span class="fu">warning</span>(<span class="st">"For continuous outcomes, RR and OR specifications are ignored. RD is the only applicable causal estimate."</span>)</span>
<span id="cb2-241"><a href=""></a> }</span>
<span id="cb2-242"><a href=""></a> }</span>
<span id="cb2-243"><a href=""></a></span>
<span id="cb2-244"><a href=""></a> }</span>
<span id="cb2-245"><a href=""></a></span>
<span id="cb2-246"><a href=""></a> <span class="co">#saving output for the non-parallel setting;</span></span>
<span id="cb2-247"><a href=""></a> <span class="fu">return</span>(<span class="fu">list</span>(</span>
<span id="cb2-248"><a href=""></a> <span class="at">mean =</span> <span class="fu">mean</span>(bootest),</span>
<span id="cb2-249"><a href=""></a> <span class="at">sd =</span> <span class="fu">sqrt</span>(<span class="fu">var</span>(bootest)),</span>
<span id="cb2-250"><a href=""></a> <span class="at">quantile =</span> <span class="fu">quantile</span>(bootest, <span class="at">probs =</span> <span class="fu">c</span>(<span class="fl">0.025</span>, <span class="fl">0.975</span>)),</span>
<span id="cb2-251"><a href=""></a> <span class="at">bootdata =</span> <span class="fu">data.frame</span>(effect_reference, effect_comparator, <span class="at">ATE=</span>bootest),</span>
<span id="cb2-252"><a href=""></a> <span class="at">reference =</span> reference,</span>
<span id="cb2-253"><a href=""></a> <span class="at">comparator =</span> comparator</span>
<span id="cb2-254"><a href=""></a> ))</span>
<span id="cb2-255"><a href=""></a></span>
<span id="cb2-256"><a href=""></a> }</span>
<span id="cb2-257"><a href=""></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="results-dag" class="slide level2" data-transition="slide">
<h2>Results (DAG)</h2>
<p>Figure: Longitudinal Directed Acyclic Graph (DAG) for 2 visits</p>
<div class="cell" data-reveal="true" data-layout-align="default">
<div class="cell-output-display">
<div>
<p></p><figure class=""><p></p>
<div>
<pre class="mermaid mermaid-js">flowchart TD
Cov1[Covariates 1] --> Cov2[Covariates 2]
Cov1 --> Treat1[Treatment 1]
Cov1 --> Treat2
Cov1 --> Outcome[Outcome]
Cov2 --> Treat2
Cov2 --> Outcome
Treat1 --> Treat2[Treatment 2]
Treat1 --> Cov2
Treat1 --> Outcome
Treat2 --> Outcome
</pre>
</div>
<p></p></figure><p></p>
</div>
</div>
</div>
</section>
<section id="results" class="slide level2" data-transition="fade">
<h2>Results</h2>
<p>Example usage of function <code>bayesmsm</code>:</p>
<div class="sourceCode" id="cb3" data-code-line-numbers="|1-2|3-4|5|7|10"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href=""></a>model <span class="ot"><-</span> <span class="fu">bayesmsm</span>(<span class="at">ymodel =</span> y <span class="sc">~</span> a_1<span class="sc">+</span>a_2,</span>
<span id="cb3-2"><a href=""></a> <span class="at">nvisit =</span> <span class="dv">2</span>,</span>
<span id="cb3-3"><a href=""></a> <span class="at">reference =</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">0</span>,<span class="dv">2</span>)),</span>
<span id="cb3-4"><a href=""></a> <span class="at">comparator =</span> <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">1</span>,<span class="dv">2</span>)),</span>
<span id="cb3-5"><a href=""></a> <span class="at">family =</span> <span class="st">"gaussian"</span>,</span>
<span id="cb3-6"><a href=""></a> <span class="at">data =</span> testdata,</span>
<span id="cb3-7"><a href=""></a> <span class="at">wmean =</span> Wmsm<span class="sc">$</span>weights,</span>
<span id="cb3-8"><a href=""></a> <span class="at">nboot =</span> <span class="dv">1000</span>,</span>
<span id="cb3-9"><a href=""></a> <span class="at">optim_method =</span> <span class="st">"BFGS"</span>,</span>
<span id="cb3-10"><a href=""></a> <span class="at">estimand =</span> <span class="st">"RD"</span>,</span>
<span id="cb3-11"><a href=""></a> <span class="at">parallel =</span> <span class="cn">TRUE</span>,</span>
<span id="cb3-12"><a href=""></a> <span class="at">ncore =</span> <span class="dv">6</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="footer">
<p>Inspired by <code>gfoRmula</code> and <code>ltmle</code>, see: <a href="https://kuan-liu.github.io/causal_Quarto/section3.html#implementing-targeted-maximum-likelihood-estimation">Causal analysis with time-varying treatment</a></p>
</div>
</section>
<section id="bootstrap-results" class="slide level2" data-transition="fade">
<h2>Bootstrap Results</h2>
<div class="sourceCode" id="cb4"><pre class="sourceCode numberSource r number-lines code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href=""></a>model<span class="sc">$</span>bootdata</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="columns">
<div class="column" style="width:100%;">
<div class="cell">
<div class="cell-output cell-output-stdout">
<pre><code> effect_reference effect_comparator ATE
1 2.342642 -0.7230695 -3.065711
2 2.279753 -0.8778490 -3.157602
3 2.243634 -0.8274983 -3.071133
4 2.320007 -0.8651029 -3.185109
5 2.273139 -0.8059713 -3.079110
6 2.342425 -0.8598928 -3.202318</code></pre>
</div>
</div>
</div>
</div>
<div class="fragment">
<p>This model output allows users to plot and summarize the bootstrap results.</p>
</div>
</section>
<section id="results-1" class="slide level2" data-transition="fade">
<h2>Results</h2>
<p>There are also other functions in this package available to visualize and interpret the results:</p>
<ul>
<li>‘plot_ATE’</li>
<li>‘plot_APO’</li>
<li>‘plot_est_box’</li>
</ul>
</section>
<section id="other-functions-in-the-package" class="slide level2 smaller scrollable" data-transition="fade">
<h2>Other functions in the package</h2>
<div class="panel-tabset">
<ul id="tabset-2" class="panel-tabset-tabby"><li><a data-tabby-default="" href="#tabset-2-1">plot_ATE</a></li><li><a href="#tabset-2-2">plot_est_box</a></li></ul>
<div class="tab-content">
<div id="tabset-2-1">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Winter-Term-Practicum-Presentation---Xiao-Yan_files/figure-revealjs/unnamed-chunk-6-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div>
<div id="tabset-2-2">
<div class="cell">
<div class="cell-output-display">
<div>
<figure>
<p><img data-src="Winter-Term-Practicum-Presentation---Xiao-Yan_files/figure-revealjs/unnamed-chunk-7-1.png" width="960"></p>
</figure>
</div>
</div>
</div>
</div>
</div>
</div>
</section>
<section id="summary" class="slide level2" data-transition="slide">
<h2>Summary</h2>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Suitable for both continuous and binary Y.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Parallel computing option is provided in <code>bayesmsm</code> for faster calculation.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Simplifies complex BMSM analysis for users.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Our package can be downloaded on GitHub: <a href="https://kuan-liu-lab.github.io/bayesmsm/">Kuan-Liu-Lab/bayesmsm</a></li>
</ul>
</div>
</section>
<section id="future-work" class="slide level2" data-transition="slide">
<h2>Future work</h2>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Bayesian parametric estimation of treatment assignment weights (step 1 of BMSM).</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Improve computational efficiency for larger datasets.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Write documentation (vignette) for this package.</li>
</ul>
</div>
<div class="fragment fade-in-then-semi-out">
<ul>
<li>Possibly extend this package for survival outcomes.</li>
</ul>
</div>
</section>
<section id="references" class="slide level2" data-transition="fade">
<h2>References</h2>
<ul>
<li>Liu, K. (2021). Bayesian causal inference with longitudinal data. Tspace.library.utoronto.ca. https://tspace.library.utoronto.ca/handle/1807/109330</li>
<li>Saarela, O., Stephens, D. A., Moodie, E. E. M., & Klein, M. B. (2015). On Bayesian estimation of marginal structural models. Biometrics, 71(2), 279–288. https://doi.org/10.1111/biom.12269</li>
<li>Robins, J. M., Hernán, M. A., & Brumback, B. (2000). Marginal structural models and causal inference in epidemiology. Epidemiology, 11(5), 550–560. https://doi.org/10.1097/00001648-200009000-00011</li>
</ul>
</section>
<section id="references-1" class="slide level2">
<h2>References</h2>
<ul>
<li>Liu, K., Saarela, O., Feldman, B. M., & Pullenayegum, E. (2020). Estimation of causal effects with repeatedly measured outcomes in a Bayesian framework. Statistical Methods in Medical Research, 29(9), 2507–2519. https://doi.org/10.1177/0962280219900362</li>
</ul>
</section>
<section id="questions" class="slide level2" data-transition="fade">
<h2>Questions?</h2>
<p>Thank you for your attention ;)</p>
<div class="footer">
<p>Xiao Yan (Supervisor: Kuan Liu)</p>
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<p>Practicum Term 2 Presentation</p>
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