From 5c938134b029e478fd02e63ae66428a052172329 Mon Sep 17 00:00:00 2001 From: Sophie Huebler Date: Thu, 17 Oct 2024 12:49:07 -0600 Subject: [PATCH] final commit for slides --- slides.html | 827 +++++++++++++++++++++++++++++++++++++++++----------- 1 file changed, 660 insertions(+), 167 deletions(-) diff --git a/slides.html b/slides.html index 6d89aa3..f469f24 100644 --- a/slides.html +++ b/slides.html @@ -1,12 +1,12 @@ +!function(t,e){"object"==typeof exports&&"object"==typeof module?module.exports=e():"function"==typeof define&&define.amd?define([],e):"object"==typeof exports?exports.ClipboardJS=e():t.ClipboardJS=e()}(this,function(){return n={686:function(t,e,n){"use strict";n.d(e,{default:function(){return o}});var e=n(279),i=n.n(e),e=n(370),u=n.n(e),e=n(817),c=n.n(e);function a(t){try{return document.execCommand(t)}catch(t){return}}var f=function(t){t=c()(t);return a("cut"),t};var l=function(t){var e,n,o,r=1 - - - + + - - My fancy presentation + + Meta-Analsis for the Binomial Model with Normal Random Effects @@ -485,10 +484,75 @@ ul.task-list{list-style: none;} ul.task-list li input[type="checkbox"] { width: 0.8em; -margin: 0 0.8em 0.2em -1em; vertical-align: middle; +margin: 0 0.8em 0.2em -1.6em; +vertical-align: middle; } +pre > code.sourceCode { white-space: pre; position: relative; } +pre > code.sourceCode > span { display: inline-block; line-height: 1.25; } +pre > code.sourceCode > span:empty { height: 1.2em; } +.sourceCode { overflow: visible; } +code.sourceCode > span { color: inherit; text-decoration: inherit; } +div.sourceCode { margin: 1em 0; } +pre.sourceCode { margin: 0; } +@media screen { +div.sourceCode { overflow: auto; } +} +@media print { +pre > code.sourceCode { white-space: pre-wrap; } +pre > code.sourceCode > span { text-indent: -5em; padding-left: 5em; } +} +pre.numberSource code +{ counter-reset: source-line 0; } +pre.numberSource code > span +{ position: relative; left: -4em; counter-increment: source-line; } +pre.numberSource code > span > a:first-child::before +{ content: counter(source-line); +position: relative; left: -1em; text-align: right; vertical-align: baseline; +border: none; display: inline-block; +-webkit-touch-callout: none; -webkit-user-select: none; +-khtml-user-select: none; -moz-user-select: none; +-ms-user-select: none; user-select: none; +padding: 0 4px; width: 4em; +color: #aaaaaa; +} +pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; } +div.sourceCode +{ color: #003b4f; background-color: #f1f3f5; } +@media screen { +pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; } +} +code span { color: #003b4f; } +code span.al { color: #ad0000; } +code span.an { color: #5e5e5e; } +code span.at { color: #657422; } +code span.bn { color: #ad0000; } +code span.bu { } +code span.cf { color: #003b4f; } +code span.ch { color: #20794d; } +code span.cn { color: #8f5902; } +code span.co { color: #5e5e5e; } +code span.cv { color: #5e5e5e; font-style: italic; } +code span.do { color: #5e5e5e; font-style: italic; } +code span.dt { color: #ad0000; } +code span.dv { color: #ad0000; } +code span.er { color: #ad0000; } +code span.ex { } +code span.fl { color: #ad0000; } +code span.fu { color: #4758ab; } +code span.im { color: #00769e; } +code span.in { color: #5e5e5e; } +code span.kw { color: #003b4f; } +code span.op { color: #5e5e5e; } +code span.ot { color: #003b4f; } +code span.pp { color: #ad0000; } +code span.sc { color: #5e5e5e; } +code span.ss { color: #20794d; } +code span.st { color: #20794d; } +code span.va { color: #111111; } +code span.vs { color: #20794d; } +code span.wa { color: #5e5e5e; font-style: italic; } - + @@ -1136,6 +1198,231 @@ display: block; margin-left: auto; margin-right: 0; } + + + + @@ -1143,29 +1430,343 @@
-

My fancy presentation

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For PHS 7045

+

Meta-Analsis for the Binomial Model with Normal Random Effects

+

Heirarchical and Empirical Bayesian Approaches

-John Doe +Sophie Huebler
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-

Section 1

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Section 1: Introduction to the Problem

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Slide 1

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The Goal

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  1. Create a computationally efficient implementation of the gibbs-metropolis-hastings hybrid algorithm for a heirarchical bayesian approach to meta analysis.

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  2. Create a computationally efficient function to compute mariginal likelihood and optimize it to empirically estimate parameters for random effects for an empirical bayesian approach to meta analysis.

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Meta-Analysis

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We will use a classic meta-analysis case to motivate this problem. Our treatment effect of interest is the odds ratio of an event occurring between treatment and control groups, and there are 7 studies which have estimated this effect by recording the number of events and sample size in a treatment sample and a control sample.

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Data
Trial R_it N_it R_ic N_ic
Balcon 14 56 15 58
Clausen 18 66 19 64
Multicentre 15 100 12 95
Barber 10 52 12 47
Norris 21 226 24 228
Kahler 3 38 6 31
Ledwich 2 20 3 20
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Meta-Analysis

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Each study therefore has a log odds ratio \(\theta_i\) which estimates the study specific treatment effect, and a standard error for that estimate. We assume that due to differing study protocol and random chance, each study is estimating a log odds ratio that is drawn from a normal distribution with center \(\mu\) and variance \(\tau^2\). We will call this the random effect population prior \(\pi(\theta_i | \mu, \tau)\).

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Data
Trial R_it N_it R_ic N_ic log(OR) Var(log(OR))
Balcon 14 56 15 58 -0.05 0.19
Clausen 18 66 19 64 -0.12 0.15
Multicentre 15 100 12 95 0.20 0.17
Barber 10 52 12 47 -0.36 0.24
Norris 21 226 24 228 -0.14 0.10
Kahler 3 38 6 31 -1.03 0.57
Ledwich 2 20 3 20 -0.46 0.95
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The Model

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\[ +\begin{aligned} +&i \in i,…,k \hspace{0.5cm} and \hspace{0.5cm} j \in \{C,T\}\\ +& Y_{ij}| \pi_{ij},n_{ij} \sim Bin(n_{ij}, \pi_{ij})\\ +&logit \pi_i^C | \gamma_i, \theta_i = \gamma_i - \theta_i/2\\ +&logit \pi_i^T | \gamma_i, \theta_i = \gamma_i + \theta_i/2\\ +& logit(\frac{\pi_{iT}}{\pi_{iC}}) = \theta_i | \mu, \tau^2 \sim N(\mu, \tau^2)\\ +& \mu \sim N(a_1, b_1 )\\ +& 1/\tau^2\sim gamma(a_2, b_2)\end{aligned} +\]

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Heirarichal Estimation

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From the model, we can easily write out the joint distribution and the full conditionals for \(\mu\), \(\tau^2\), \(\bf{\theta}\), and \(\bf{\gamma}\) (I leave this as an exercise to the listener). This allows us to use the gibbs sampling procedure to iteratively update posteriors. It is however important to note that the thetas and gammas do not have closed form full conditionals. Therefore we must also incorporate a metropolis hastings component wise updating procedure.

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Empirical Estimation

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Instead of iteratively updating the \(\mu\) and \(\tau^2\) variables, we can estimate them from the marginal likelihood using an MLE approach after integrating out all other parameters, thus reducing the problem to a single level model.

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Section 2: Solution Plan

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Heirarchical Bayesian Approach

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  • Full conditionals:

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    • Write in both R and c++
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  • Wrapper for the full algorithm:

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    • Rewrite with data table so that storage and manipulation is easier

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    • Pre-allocate memory

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  • Use microbenchmark to comapre efficiency to original code

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Empirical Bayesian Approach

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  • Marignal likelihood

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    • Use parallel programming for the R implementation that currently simulates average likelihood of the marginal likelihood

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    • Write the full likelihood function in C++ and then use GNU scientific library for numeric integration

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Section 3: Preliminary Results

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Heirarchical Bayesian

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Currently all posterior functions are written in modularized format in R and wrapped in a function that stores the results as a data table.

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  • Old function: 2 for loops, 4 lapply statements

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  • New function: 1 for loop

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              min   median  itr/sec mem_alloc  gc/sec n_itr n_gc total_time
+original 1.202172 1.202172 1.000000  1.582209 1.00000     1    6   3.135974
+updated  1.000000 1.000000 1.202172  1.000000 1.00181     1    5   2.608591
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Marginal Likelihood

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Original code:

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+
mlik_func <- function(mu,
+                      tau2,
+                      Y, 
+                      #simulate
+                      sim = 1000
+                      ){
+  
+#Set up
+  #number of studies
+  k = nrow(Y)
+  #initialize 
+  lik <- 0 
+  #test <-  0
+  
+#Outer loop !!! This is where parallelization can be implemented
+  for(j in 2:sim){
+    #Simulate theta and gamma
+    #based on mu and tau2 to optimize and take average lik
+    theta <- rnorm(k, mean = mu, sd = sqrt(tau2))
+    gamma <- rnorm(k, mean = 0, sd = 10)
+    
+    #Calculate odds (expit scale)
+    eta_C <- gamma - theta/2
+    eta_T <- gamma + theta/2
+    
+    #Transform odds to probabilities (p scale)
+    p_C <- 1/(1+exp(-eta_C))
+    p_T <- 1/(1+exp(-eta_T))
+    
+    #Construct priors (on theta and gamma only)
+    #Use log scale for easier computation so
+    #addition instead of multiplication for
+    #likelihood and priors
+    
+    pi_theta <- dnorm(theta, mean = mu, sd = sqrt(tau2), log = T)
+    pi_gamma <- dnorm(gamma, mean = 0, sd = 10, log = T)
+    
+    #Construct likelihood for both control
+    #and treatments, because the likelihood 
+    #depends on the data (do not construct
+    #likelihood for the parameters being estimated
+    #or integrated out)
+    
+    #Use probabilities found above
+    
+    lik_C <- dbinom(Y[,"ric"], Y[,"nic"], p_C, log = T)
+    lik_T <- dbinom(Y[,"rit"], Y[,"nit"], p_T, log = T)
+    
+    #Need to tranform back to exp scale
+    lik<- lik + exp(lik_C+lik_T+pi_theta+pi_gamma)
+    
+  }#End outer loop
+  
+  #Find average log likelihood
+  log_lik <- log(lik/sim)
+
+  #Return negative log likelihood
+  ret <- -sum(log_lik)
+  ret
+
+}
+
+
+
@@ -1223,7 +1824,7 @@

Slide 1

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Slide 1

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Slide 1

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Slide 1

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Slide 1

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