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Fix TOC link for single RNA seq example notebook (#280)
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* fix toc link

* fixed toc and equation rendering issues

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Co-authored-by: Raj Agrawal <r.agrawal.mit@gmail.com>
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2 people authored and rfl-urbaniak committed Nov 29, 2023
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" - [Experimental design](#experimental-design)\n",
" - [Data ingestion](#data-ingestion)\n",
" - [Data preprocessing](#data-preprocessing)\n",
"- [Goal: Understand the effects that the cancer drug candidate Pracinostat SB939 has on other target genes](#goal-understand-the-effects-that-the-cancer-drug-candidate-pracinostat-sb939-has-on-other-target-genes)\n",
"- [Goal: Understand the effects that the cancer drug candidate Pracinostat SB939 has on other target genes](#goal:-understand-the-effects-that-the-cancer-drug-candidate-pracinostat-sb939-has-on-other-target-genes)\n",
" - [Exploratory data analysis](#exploratory-data-analysis)\n",
"- [Causal Probabilistic Program](#causal-probabilistic-program)\n",
" - [Model Description](#model-description)\n",
" - [Fitting observed data](#fitting-observed-data)\n",
" - [Predictions on unseen cells](#predictions-on-unseen-cells)\n",
"- [Computing causal queries](#computing-causal-queries)\n",
" - [Causal Query: Average treatment effect (ATE)](#causal-query-average-treatment-effect-ate)\n",
" - [Causal Query: Conditional average treatment effect (CATE)](#causal-query-conditional-average-treatment-effect-cate)\n",
" - [Causal Query: Average treatment effect (ATE)](#causal-query:-average-treatment-effect-(ate))\n",
" - [Causal Query: Conditional average treatment effect (CATE)](#causal-query:-conditional-average-treatment-effect-(cate))\n",
"- [References](#references)\n",
"\n",
"\n",
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"cell_type": "markdown",
"metadata": {},
"source": [
"#### Exploratory data analysis"
"### Exploratory data analysis"
]
},
{
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"In the plots above, we did not control for technical covariates (e.g., cell plate, replicate number, etc.). Here we fit a generalized linear model with a Poission likelihood to account for the heavy-tailed count nature of the data. We also include interactions between genes and confounders (e.g., cell type, read depth, etc.) to understand how treatment effects vary across cell types and experimental conditions, and model unobserved confounders. Our model is defined as follows:\n",
"\n",
"$$\n",
"\\begin{align*}\n",
"\\begin{split}\n",
" Z_{nk} &\\sim \\mathcal{N}(0, 1), \\quad{1 \\leq k \\leq K} \\quad (\\text{$K$ unobserved confounders for cell}) \\\\\n",
" \\tilde{X}_n &= (X_n, \\ Z_n) \\in \\mathbb{R}^{p + K} \\quad (\\text{observed and unobserved confounders}) \\\\\n",
" c_g &\\sim \\mathcal{N}(0, 1) \\quad (\\text{intercept}) \\\\\n",
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" \\eta_{gi} &\\sim \\mathcal{N}(0, 1 / \\sqrt{p + K}), \\quad{1 \\leq i \\leq p + K} \\quad (\\text{confounder-treatment interaction effect}) \\\\\n",
" \\log \\mu_{ng} &= c_g + \\tau_{g} T_n + \\theta_{g}^T \\tilde{X}_n + \\eta_{g}^T (\\tilde{X}_n \\odot T_n) \\quad (\\text{Poisson rate}) \\\\\n",
" Y_{ng} &\\sim \\text{Poisson}(\\mu_{ng}) \\quad (\\text{Poisson likelihood}) \\\\\n",
"\n",
"\\end{align*}\n",
"\\end{split}\n",
"$$\n",
"where $\\odot$ denotes elementwise multiplication, $p$ the number of confounders, $T_n$ treatment, $X_n$ observed confounders, and $Y_{ng}$ gene expression counts for cell $n$ and gene $g$.\n",
"\n",
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"\n",
"Guo Li, Yuan Tian, Wei-Guo Zhu. \"The Roles of Histone Deacetylases and Their Inhibitors in Cancer Therapy\". Frontiers in Cell and Developmental Biology, 2020. https://www.frontiersin.org/articles/10.3389/fcell.2020.576946/full\n",
"\n",
"HUGO Gene Nomenclature Committee. Histone deacetylase superfamily. HGNC, 2020. https://www.genenames.org/data/genegroup/#!/group/937\n",
"\n",
"Daryl C. Drummond et al. \"Clinical development of histone deacetylase inhibitors as anticancer agents\". Annual Review of Pharmacology and Toxicology, 2005. https://www.annualreviews.org/doi/10.1146/annurev.pharmtox.45.120403.095825.\n",
"\n",
"Xiang-Jiao Yang, Edward Seto. \"HATs and HDACs: from structure, function and regulation to novel strategies for therapy and prevention\". Oncogene, 2007. https://www.nature.com/articles/1210599.\n",
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