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Adding g-formula stochastic treatments
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‎3_Epidemiology_Analysis/c_causal_inference/1_time-fixed-treatments/2_gformula_stochastic.ipynb

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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Parametric g-formula: stochastic interventions\n",
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"In the previous tutorial we went over the basics of the parametric g-formula using `TimeFixedGFormula` for basic interventions. Additionally, we can use the g-formula to look at stochastic interventions. Stochastic interventions are treatment plans under which not necessarily everyone is treated, but some random percentage are treated.\n",
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"\n",
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"To estimate the g-formula for stochastic treatments, the process is fairly similar. However, instead of treating everyone, some percentage are treated. A random percentage are treated and then $\\hat{Y_i^a}$ are predicted and averaged. This process is repeated some number times and the average of the averaged potential outcomes is returned.\n",
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"\n",
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"For our example, we will return to the previous data set on ART among HIV-infected individuals and all-cause mortality. First, we will load the data (again ignoring missing data)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"<class 'pandas.core.frame.DataFrame'>\n",
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"Int64Index: 517 entries, 0 to 546\n",
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"Data columns (total 9 columns):\n",
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"id 517 non-null int64\n",
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"male 517 non-null int64\n",
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"age0 517 non-null int64\n",
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"cd40 517 non-null int64\n",
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"dvl0 517 non-null int64\n",
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"art 517 non-null int64\n",
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"dead 517 non-null float64\n",
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"t 517 non-null float64\n",
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"cd4_wk45 430 non-null float64\n",
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"dtypes: float64(3), int64(6)\n",
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"memory usage: 40.4 KB\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"import pandas as pd\n",
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"\n",
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"from zepid import load_sample_data, spline\n",
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"from zepid.causal.gformula import TimeFixedGFormula\n",
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"\n",
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"df = load_sample_data(timevary=False)\n",
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"dfs = df.dropna(subset=['dead']).copy()\n",
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"dfs.info()\n",
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"\n",
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"dfs[['cd4_rs1', 'cd4_rs2']] = spline(dfs, 'cd40', n_knots=3, term=2, restricted=True)\n",
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"dfs[['age_rs1', 'age_rs2']] = spline(dfs, 'age0', n_knots=3, term=2, restricted=True)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Similar to the previous tutorial, we initialize the `TimeFixedGFormula` with the data set (`dfs`), our treatment variable (`art`), and binary outcome (`dead`). Then we fit a regression model predicting all-cause mortality as a function of ART and our set of confounding variables (age, CD4 T-cell count, detectable viral load, gender)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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" Generalized Linear Model Regression Results \n",
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"==============================================================================\n",
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"Dep. Variable: dead No. Observations: 517\n",
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"Model: GLM Df Residuals: 507\n",
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"Model Family: Binomial Df Model: 9\n",
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"Link Function: logit Scale: 1.0000\n",
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"Method: IRLS Log-Likelihood: -202.83\n",
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"Date: Mon, 11 Mar 2019 Deviance: 405.67\n",
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"Time: 07:08:33 Pearson chi2: 534.\n",
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"No. Iterations: 6 Covariance Type: nonrobust\n",
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"==============================================================================\n",
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" coef std err z P>|z| [0.025 0.975]\n",
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"------------------------------------------------------------------------------\n",
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"Intercept -3.9822 2.621 -1.520 0.129 -9.119 1.154\n",
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"art -0.7278 0.393 -1.854 0.064 -1.497 0.042\n",
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"male -0.0773 0.334 -0.231 0.817 -0.732 0.578\n",
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"age0 0.1548 0.092 1.689 0.091 -0.025 0.334\n",
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"age_rs1 -0.0059 0.004 -1.493 0.135 -0.014 0.002\n",
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"age_rs2 0.0129 0.006 2.035 0.042 0.000 0.025\n",
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"cd40 -0.0121 0.004 -3.028 0.002 -0.020 -0.004\n",
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"cd4_rs1 1.887e-05 1.19e-05 1.581 0.114 -4.52e-06 4.23e-05\n",
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"cd4_rs2 -3.866e-05 4.57e-05 -0.846 0.398 -0.000 5.09e-05\n",
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"dvl0 -0.1254 0.398 -0.315 0.753 -0.905 0.654\n",
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"==============================================================================\n"
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]
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}
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],
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"source": [
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"g = TimeFixedGFormula(dfs, exposure='art', outcome='dead')\n",
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"g.outcome_model(model='art + male + age0 + age_rs1 + age_rs2 + cd40 + cd4_rs1 + cd4_rs2 + dvl0')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"However, this time we do some backgound research and find that one potential intervention to increase ART prescriptions increases the probability of ART treatment to 80%. As a result, it is potentially misleading to compare to compare the treat-all vs treat-none scenarios. Instead, we will compare the stochastic treatment where 80% of individuals are treated with ART to the scenario where no one is treated.\n",
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"\n",
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"## Stochastic Treatment Plans\n",
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"To do this using `TimeFixedGFormula` we will instead call `fit_stochastic()` function instead of `fit()`. This function allows us to estimate a stochastic treatment. We specify `p=0.8` to have 80% of the population treated at random. By default, `fit_stochastic()` repeats this process 100 times and takes the average of these repeated random treatments. I will also use the `seed` argument to get replicable results. Let's look at the example"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"RD: -0.06041404870415\n"
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]
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}
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],
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"source": [
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"g.fit_stochastic(p=0.8, seed=1000191)\n",
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"r_80 = g.marginal_outcome\n",
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"\n",
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"g.fit(treatment='none')\n",
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"r_none = g.marginal_outcome\n",
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"\n",
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"print('RD:', r_80 - r_none)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Under the treatment plan where 80% of people are randomly treated, the risk of all-cause mortality would have been 6.0% points lower than if no one was treated. \n",
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"\n",
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"After reading some more articles, we find an alternative treatment plan. Under this plan, 75% of men and 90% of women start using HIV. For this plan, we are interested in a conditional stochastic treatment. Again, we want to compare this to the scenario where no one is treated\n",
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"\n",
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"## Conditional Stochastic Treatment Plans\n",
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"For conditionally stochastic treatments, we instead provide `p` a list of probabilities. Additionally, we specify the `conditional` argument with the group restrictions. Again, we will need to use the magic-g functionality. Below is the example of the stochastic plan where 75% of men are treated and 90% of women"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"RD: -0.058656195525173926\n"
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]
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}
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],
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"source": [
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"g.fit_stochastic(p=[0.75, 0.90], conditional=[\"g['male']==1\", \"g['male']==0\"], seed=518012)\n",
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"r_cs = g.marginal_outcome\n",
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"\n",
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"print('RD:', r_cs - r_none)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Under the treatment plan where 75% of men and 90% of women are randomly treated, the risk of all-cause mortality would have been 5.9% points lower than if no one was treated. This plan reduces the marginal mortality less than the previous stochastic plan because our HIV-infected population is predominantly men. \n",
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"\n",
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"# Conclusion\n",
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"In this tutorial, I detailed stochastic treatment plans using the g-formula. While presented for a binary outcome, the same procedure can also be used to estimate stochastic treatments for continuous outcomes. Please view other tutorials for information other functions in *zEpid*\n",
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"\n",
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"## Further Readings\n",
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"Ahern et al. (2016). Predicting the population health impacts of community interventions: the case of alcohol outlets and binge drinking. *AJPH*, 106(11), 1938-1943.\n",
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"\n",
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"Snowden et al. (2011) \"Implementation of G-computation on a simulated data set: demonstration of a causal inference technique.\" *AJE* 173.7: 731-738.\n",
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"\n",
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"Robins. (1986) \"A new approach to causal inference in mortality studies with a sustained exposure period—application to control of the healthy worker survivor effect.\" *Mathematical modelling* 7.9-12: 1393-1512"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.3"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}

‎3_Epidemiology_Analysis/c_causal_inference/README.md

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Throughout the following tutorials in this branch, we will make the following identifiability assumptions.
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We additionally will assume no measurement error, no selection bias, and no interference.
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# Assumptions
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## Conditional Exchangeability
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Conditional exchangeability is the assumption that potential outcomes are independent of the treatment received
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conditional on some set of covariates. Using causal diagrams, this amounts to no open backdoor paths between the
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treatment and outcome. See the further reading list for publications on the assumption of conditional exchangeability
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and introductions to two different approaches to causal diagrams (directed acyclic graphs (DAG) and single-world
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intervention graphs (SWIG))
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### Further Reading
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Hernán MA, Robins JM. (2006). Estimating causal effects from epidemiological data. *Journal of Epidemiology
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& Community Health*, 60(7), 578-586.
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Greenland S, Pearl J, Robins JM. (1999). Causal diagrams for epidemiologic research. *Epidemiology*, 10, 37-48.
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Richardson TS, Robins JM. (2013). Single world intervention graphs: a primer. *In Second UAI workshop on
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causal structure learning*, Bellevue, Washington.
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Breskin A, Cole SR, Hudgens MG. (2018). A practical example demonstrating the utility of single-world
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intervention graphs. *Epidemiology*, 29(3), e20-e21.
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## Positivity
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The positivity assumption is that there are treated and untreated individuals at every combination of covariates. There
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are two potential positivity violations; deterministic or random. Deterministic positivity violations can never occur
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despite additional data collection. For an example of a deterministic positivity violation, consider the risk of death
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by hysterectomy. Since men lack a uterus, they are unable to receive a hysterectomy. Random positivity violations
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occur as a result of finite samples. In a small sample, it may just occur that we didn't observe anyone treated between
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ages 32-35. It isn't that no one could have been treated in that age group, we just didn't observe it in our sample.
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For these scenarios, we will assume that our statistical model correctly interpolates over these areas (often a
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strong assumption in small data sets)
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### Further Reading
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Westreich D, Cole SR. (2010). Invited commentary: positivity in practice. *American Journal of Epidemiology*,
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171(6), 674-677.
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Cole SR, Hernán MA. (2008). Constructing inverse probability weights for marginal structural models.
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*American Journal of Epidemiology*, 168(6), 656-664.
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## Causal Consistency
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Causal consistency is also referred to as treatment variation irrelevance. Under this assumption we assume that there
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is only one version of treatment (consistency) or that any differences remaining between treatments is irrelevant
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(treatment variation irrelevance). For example, consider a study on 200mg daily aspirin and all-cause mortality. In our
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study, we may be willing to assume that taking aspirin in the morning versus at night is irrelevant to all-cause
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mortality. This is an example of assuming treatment variation irrelevance. Generally, defining the treatment more
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precisely can get you out of this as an issue. There are also some additional approaches. I recommend reviewing the
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below readings for further discussions
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### Further Reading
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Cole SR, Frangakis CE. (2009). The consistency statement in causal inference: a definition or an assumption?.
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*Epidemiology*, 20(1), 3-5.
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VanderWeele TJ. (2009). Concerning the consistency assumption in causal inference. *Epidemiology*, 20(6), 880-883.
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VanderWeele TJ. (2018). On well-defined hypothetical interventions in the potential outcomes framework.
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*Epidemiology*, 29(4), e24-e25.
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## Correctly specified model
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Since we will be working with continuous and high-dimensional data, we will be using parametric regression models.
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We assume that these models are correctly specified. To make less restrictive assumptions regarding the functional
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forms of continuous variables, we will use splines throughout. Please refer to the Data Basics for an intro to
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using splines with *zEpid*
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Additionally, we will sometime uses machine learning approaches to relax this assumption further (see TMLE tutorials
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for some examples)

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