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AIPTW tutorial
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‎3_Epidemiology_Analysis/c_causal_inference/1_time-fixed-treatments/5_AIPTW.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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"# Augmented Inverse Probability of Treatment Weights\n",
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"Augmented-IPTW (AIPTW) is a doubly robust estimator. Essentially, AIPTW combines the IPTW estimator and g-formula into a single estimate. Before continuing, I will briefly outline what a doubly-robust estimator is and why you would want to use one. In observational research with high-dimensional data, we (generally) are forced to use parametric models to adjust for many confounders. In this scenario, we assume that our parametric models are correctly specified. Our statistical model, $\\mathcal{M}$, must include the distribution that the data came from. \n",
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"\n",
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"With other estimators, like IPTW or g-formula, we have one chance to specify $\\mathcal{M}$ correctly. Doubly-robust estimators use a model to predict the treatment (like IPTW) and another model to predict the outcome (like g-formula). The estimator then combines the estimates, such that if either is correct, then our estimate will be consistent. Essentially, we get two chances to get the statistical model correct.\n",
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"\n",
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"A more in-depth description of doubly robust estimators is available in [this pre-print](https://statnav.files.wordpress.com/2017/10/doublerobustness-preprint.pdf)\n",
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"\n",
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"## AIPTW\n",
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"\n",
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"AIPTW takes the following form\n",
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"$$E[Y^a] = \\frac{1}{n} \\sum_i^n \\left(\\frac{Y \\times I(A=a)}{\\widehat{\\Pr}(A=a|L)} - \\frac{\\hat{E}[Y|A=a, L] \\times (I(A=a) - \\widehat{\\Pr}(A=a|L))}{1 - \\widehat{\\Pr}(A=a|L)}\\right)$$\n",
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"where $\\widehat{\\Pr}(A=a|L)$ comes from the IPTW model and $\\hat{E}[Y|A=a,L]$ comes from the g-formula. If we do some manipulations and assume an infinite sample size, we can end up with\n",
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"$$\\hat{E}^{IPW}[Y^a] \\times \\frac{\\Pr(A=a|L)}{\\widehat{\\Pr}(A=a|L, \\mathcal{M})} - \\hat{E}^{STD}[Y^a] \\times \\frac{\\Pr(A=a|L) - \\widehat{\\Pr}(A=a|L, \\mathcal{M})}{\\widehat{\\Pr}(A=a|L, \\mathcal{M})}$$\n",
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"from this form, we can see that if as long as one estimate is correct then AIPTW will be unbiased\n",
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"\n",
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"## An example\n",
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"To motivate our example, we will use a simulated data set included with *zEpid*. In the data set, we have a cohort of HIV-positive individuals. We are interested in the sample average treatment effect of antiretroviral therapy (ART) on all-cause mortality at 45-weeks. Based on substantive background knowledge, we believe that the treated and untreated population are exchangeable based gender, age, CD4 T-cell count, and detectable viral load. \n",
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"\n",
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"In this tutorial, we will focus on a complete case analysis. Therefore, we will drop the `cd4_wk45` column and all the missing data in `dead`. This will leave 517 observations with no 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": 1,
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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 12 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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"age_rs1 517 non-null float64\n",
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"age_rs2 517 non-null float64\n",
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"cd4_rs1 517 non-null float64\n",
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"cd4_rs2 517 non-null float64\n",
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"dtypes: float64(6), int64(6)\n",
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"memory usage: 52.5 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.doublyrobust import AIPTW\n",
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"\n",
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"df = load_sample_data(False)\n",
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"df[['age_rs1', 'age_rs2']] = spline(df, 'age0', n_knots=3, term=2, restricted=True)\n",
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"df[['cd4_rs1', 'cd4_rs2']] = spline(df, 'cd40', n_knots=3, term=2, restricted=True)\n",
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"\n",
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"dfcc = df.drop(columns=['cd4_wk45']).dropna()\n",
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"dfcc.info()"
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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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"Our data is now ready to conduct a complete case analysis using TMLE. First, we initialize TMLE with our complete-case data (dfcc), the treatment (art), and the outcome (dead)"
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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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"source": [
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"aipw = AIPTW(dfcc, exposure='art', outcome='dead')"
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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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"### Treatment Model\n",
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"First, we will specify our treatment model. We believe the sufficient set for the treatment model is gender (`male`), age (`age0`), CD4 T-cell (`cd40`) and detectable viral load (`dvl0`). To relax the functional for assumptions, we will model age and CD4 using restricted quadratic splines"
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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": 4,
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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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"\n",
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"----------------------------------------------------------------\n",
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"MODEL: art ~ male + age0 + age_rs1 + age_rs2 + cd40 + cd4_rs1 + cd4_rs2 + dvl0\n",
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"-----------------------------------------------------------------\n",
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" Generalized Linear Model Regression Results \n",
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"==============================================================================\n",
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"Dep. Variable: art No. Observations: 517\n",
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"Model: GLM Df Residuals: 508\n",
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"Model Family: Binomial Df Model: 8\n",
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"Link Function: logit Scale: 1.0000\n",
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"Method: IRLS Log-Likelihood: -206.06\n",
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"Date: Mon, 11 Mar 2019 Deviance: 412.12\n",
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"Time: 08:24:05 Pearson chi2: 510.\n",
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"No. Iterations: 5 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 1.4498 1.679 0.864 0.388 -1.841 4.741\n",
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"male -0.1159 0.321 -0.361 0.718 -0.745 0.513\n",
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"age0 -0.1026 0.059 -1.726 0.084 -0.219 0.014\n",
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"age_rs1 0.0048 0.003 1.706 0.088 -0.001 0.010\n",
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"age_rs2 -0.0077 0.006 -1.373 0.170 -0.019 0.003\n",
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"cd40 0.0041 0.004 0.964 0.335 -0.004 0.012\n",
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"cd4_rs1 -2.422e-05 1.2e-05 -2.014 0.044 -4.78e-05 -6.49e-07\n",
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"cd4_rs2 8.875e-05 4.55e-05 1.952 0.051 -3.81e-07 0.000\n",
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"dvl0 -0.0158 0.399 -0.040 0.968 -0.797 0.765\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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"aipw.exposure_model('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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"`AIPTW` uses a logistic regression model to estimate the probabilities of treatment and the corresponding summary of the model fit are printed to the console. \n",
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"\n",
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"### Outcome Model\n",
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"Now, we will estimate the outcome model. We will model the outcomes as ART (`art`), gender (`male`), age (`age0`), CD4 T-cell (`cd40`) and detectable viral load (`dvl0`). Again, we will model age and CD4 using restricted quadratic splines "
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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": 5,
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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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"\n",
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"----------------------------------------------------------------\n",
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"MODEL: dead ~ art + male + age0 + age_rs1 + age_rs2 + cd40 + cd4_rs1 + cd4_rs2 + dvl0\n",
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"-----------------------------------------------------------------\n",
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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: -202.85\n",
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"Date: Mon, 11 Mar 2019 Deviance: 405.71\n",
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"Time: 08:25:18 Pearson chi2: 535.\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 -4.0961 2.713 -1.510 0.131 -9.413 1.220\n",
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"art -0.7274 0.392 -1.853 0.064 -1.497 0.042\n",
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"male -0.0774 0.334 -0.232 0.817 -0.732 0.577\n",
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"age0 0.1605 0.096 1.670 0.095 -0.028 0.349\n",
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"age_rs1 -0.0058 0.004 -1.481 0.139 -0.013 0.002\n",
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"age_rs2 0.0128 0.006 2.026 0.043 0.000 0.025\n",
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"cd40 -0.0123 0.004 -2.987 0.003 -0.020 -0.004\n",
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"cd4_rs1 1.872e-05 1.18e-05 1.584 0.113 -4.45e-06 4.19e-05\n",
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"cd4_rs2 -3.868e-05 4.59e-05 -0.842 0.400 -0.000 5.13e-05\n",
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"dvl0 -0.1261 0.398 -0.317 0.751 -0.906 0.653\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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"aipw.outcome_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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"Again, logistic regression is used to predict the outcome data\n",
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"\n",
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"### Estimation\n",
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"To estimate the risk difference and risk ratio, we will now call the `fit()` function. After this, `AIPTW` gains the attributes `risk_difference` and `risk_ratio`. Additionally, results can be printed to the console using the `summary()` function"
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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": 6,
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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: -0.06857139263314216\n",
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"RR: 0.5844630051369846\n",
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"----------------------------------------------------------------------\n",
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"Risk Difference: -0.0686\n",
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"Risk Ratio: 0.5845\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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"aipw.fit()\n",
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"\n",
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"print('RD:', aipw.risk_difference)\n",
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"print('RR:', aipw.risk_ratio)\n",
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"\n",
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"aipw.summary()"
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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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"Interpreting the risk difference, we would conclude that had everyone in our cohort been treated with ART, the risk of all-cause mortality would have been 6.9% points lower than had no one been treated.\n",
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"\n",
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"### Confidence Intervals\n",
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"To obtain correct confidence intervals, we need to use a bootstrap procedure. Influence curves are an alternative, but not currently available"
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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": 8,
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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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"95% LCL: -0.12324970574432371\n",
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"95% UCL -0.005298084807482341\n"
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]
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}
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],
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"source": [
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"rd_results = []\n",
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"for i in range(1000):\n",
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" dfs = dfcc.sample(n=dfcc.shape[0],replace=True)\n",
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" s = AIPTW(dfs,exposure='art',outcome='dead')\n",
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" s.exposure_model('male + age0 + age_rs1 + age_rs2 + cd40 + cd4_rs1 + cd4_rs2 + dvl0',\n",
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" print_results=False)\n",
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" s.outcome_model('art + male + age0 + age_rs1 + age_rs2 + cd40 + cd4_rs1 + cd4_rs2 + dvl0',\n",
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" print_results=False)\n",
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" s.fit()\n",
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" rd_results.append(s.risk_difference)\n",
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"\n",
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"\n",
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"print('95% LCL:', np.percentile(rd_results, q=2.5))\n",
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"print('95% UCL', np.percentile(rd_results, q=97.5))"
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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 counterfactual of everyone receiving treatment with ART, the risk of all-cause mortality was 6.9% points lower (95% CL: ) than the counterfactual where no one had been treated.\n",
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"\n",
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"# Conclusion\n",
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"In this tutorial, I introduced the concept of doubly-robust estimators and detailed augmented-IPTW. I demonstrated estimation with `AIPTW` using *zEpid* and how to obtain confidence intervals. Please view other tutorials for information on other functionality within *zEpid*\n",
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"\n",
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"## References\n",
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"Funk MJ, Westreich D, Wiesen C, Stürmer T, Brookhart MA, Davidian M. (2011). Doubly robust estimation of causal effects. *AJE*, 173(7), 761-767.\n",
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"\n",
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"Keil AP et al. (2018). Resolving an apparent paradox in doubly robust estimators. *AJE*, 187(4), 891-892.\n",
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"\n",
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"Robins JM, Rotnitzky A, Zhao LP. (1994). Estimation of regression coefficients when some regressors are not always observed. *JASA*, 89(427), 846-866."
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]
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}
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],
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