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Using a DAG to simulate data with the dagR library

Directed acyclic graphs (DAGs), and causal graphs in general, provide a framework for making assumptions explicit and identifying confounders or mediators of the relationship between the exposure of interest and outcome that need to be adjusted for in analysis. Recently, I ran into the need to generate data from a DAG for a paper I am writing with my peers Kevin McIntyre and Joshua Wiener. After a quick Google search, I was pleasantly surprised to see there were several options to do so. In particular, the dagR library provides “functions to draw, manipulate, [and] evaluate directed acyclic graphs and simulate corresponding data”.

Besides dagR‘s reference manual, a short letter published in Epidemiology, and a limited collection of examples, I couldn’t find too many resources regarding how to use the functionality provided by dagR. The goal of this blog post is to provide an expository example of how to create a DAG and generate data from it using the dagR library.

To simulate data from a DAG with dagR, we need to:

Create the DAG of interest using the dag.init function by specifying its nodes (exposure, outcome, and covariates) and their directed arcs (directed arrows to/from nodes).
Pass the DAG from (1) to the dag.sim function and specify the number of observations to be generated, arc coefficients, node types (binary or continuous), and parameters of the node distributions (Normal or Bernoulli).

For this tutorial, we are going to try to replicate the simple confounding/common cause DAG presented in Figure 1b as well as the more complex DAG in Figure 2a of Shier and Platt’s (2008) paper, Reducing bias through directed acyclic graphs.

library(dagR)
set.seed(12345)

Continue reading Using a DAG to simulate data with the dagR library

Parametric vs. Nonparametric Approach to Estimations

Parametric statistics assume that the unknown CDF $F$ belongs to a family of CDFs characterized by a parameter (vector) $\theta$ . As the form of $F$ is assumed, the target of estimation is its parameters $\theta$ . Thus, all uncertainty about $F$ is comprised of uncertainty about its parameters. Parameters are estimated by $\hat{\theta}$ , and estimates are be substituted into the assumed distribution to conduct inference for the quantities of interest. If the assumed distribution $F$ is incorrect, inference may also be inaccurate, or trends in the data may be missed.

To demonstrate the parametric approach, consider $n = 100$ independent and identically distributed random variables $X_1, …, X_n$ generated from an exponential distribution with rate $\lambda = 2$ . Investigators wish to estimate the 75 $^{th}$ percentile and erroneously assume that their data is normally distributed. Thus, $F$ is assumed to be the Normal CDF but $\mu$ and $\sigma^2$ are unknown. The parameters $\mu$ and $\sigma$ are estimated in their typical way by $\bar{x}$ and $\sigma^2$ , respectively. Since the normal distribution belongs to the location-scale family, an estimate of the $p^{th}$ percentile is provided by,

$x_p = \bar{x} + s\Phi^{-1}(p)$

where $\Phi^{-1}$ is the standard normal quantile function, also known as the probit.

set.seed(12345)
library(tidyverse, quietly = T)

# Generate data from Exp(2)
x <- rexp(n = 100, rate = 2)

# True value of 75th percentile with rate = 2
true <- qexp(p = 0.75, rate = 2) 
true

## [1] 0.6931472

# Estimate mu and sigma
xbar <- mean(x)
s    <- sd(x)

# Estimate 75th percentile assuming mu = xbar and sigma = s
param_est <- xbar + s * qnorm(p = 0.75)
param_est

## [1] 0.8792925

The true value of the 75 $^{th}$ percentile of $\text{Exp}(2)$ is 0.69 while the parametric estimate is 0.88.

Nonparametric statistics make fewer distributions about the unknown distribution $F$ , requiring only mild assumptions such as continuity or the existence of specific moments. Instead of estimating parameters of $F$ , $F$ itself is the target of estimation. $F$ is commonly estimated by the empirical cumulative distribution function (ECDF) $\hat{F}$ ,

$\hat{F}(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbb{I}(X_i \leq x).$

Any statistic that can be expressed as a function of the CDF, known as a statistical functional and denoted $\theta = T(F)$ , can be estimated by substituting $\hat{F}$ for $F$ . That is, plug-in estimators can be obtained as $\hat{\theta} = T(\hat{F})$ .

Continue reading Parametric vs. Nonparametric Approach to Estimations

The Probabilistic Index for Two Normally Distributed Outcomes

Consider a two-armed study comparing a placebo and treatment. In general, the probabilistic index (PI) is defined as,

$\text{PI} = P(X < Y) + \frac{1}{2} P(X = Y)$

and is interpreted as the probability that a subject in the treatment group will have an increased response compared to a subject in the placebo group. The probabilistic index is a particularly useful effect measure for ordinal data, where effects can be difficult to define and interpret owing to absence of a meaningful difference. However, it can also be used for continuous data, noting that when the outcome is continuous, $P(X = Y) = 0$ and the PI reduces to $P(X < Y)$ .

$PI = 0.5$ suggests an increased outcome is equally likely for subjects in the placebo and treatment group, while $PI > 0.5$ suggests an increased outcome is more likely for subjects in the treatment group compared to the placebo group, and the opposite is true when $PI < 0.5$ .

Simulation

Suppose $X \sim N(\mu_X, \sigma^{2}_{X})$ and $Y \sim N(\mu_Y, \sigma^{2}_{Y})$ represent the independent outcomes in the placebo and treatment groups, respectively and an increased value of the outcome is the desired response.

We simulate $n_X = n_Y = 50$ observations from each group such that treatment truly increases the outcome and the variances within each group are equal such that $\sigma^{2}_{X} = \sigma^{2}_{Y}$ .

# Loading required libraries
library(tidyverse)
library(gridExtra)

# Setting seed for reproducibility
set.seed(12345)

# Simulating data
n_X = n_Y = 50
sigma_X = sigma_Y = 1
mu_X = 5; mu_Y = 7

outcome_X = rnorm(n = n_X, mean = mu_X, sd = sigma_X)
outcome_Y = rnorm(n = n_Y, mean = mu_Y, sd = sigma_Y)

df <- data.frame(Group = c(rep('Placebo', n_X), rep('Treatment', n_Y)),
                 Outcome = c(outcome_X, outcome_Y))

Examining side-by-side histograms and boxplots of the outcomes within each group, there appears to be strong evidence that treatment increases the outcome as desired. Thus, we would expect a probabilistic index close to 1 as most outcomes in the treatment group appear larger than those of the placebo group.

# Histogram by group
hist_p <- df %>%
  ggplot(aes(x = Outcome, fill = Group)) +
    geom_histogram(position = 'identity', alpha = 0.75, bins = 10) + 
    theme_bw() +
    labs(y = 'Frequency')

# Boxplot by group
box_p <- df %>%
  ggplot(aes(x = Outcome, fill = Group)) +
    geom_boxplot() + 
    theme_bw() +
    labs(y = 'Frequency')

# Combine plots
grid.arrange(hist_p, box_p, nrow = 2)

plot of chunk unnamed-chunk-3

Continue reading The Probabilistic Index for Two Normally Distributed Outcomes

Kernel Regression

Motivation

For observed pairs $(x_i, y_i)$ , $i = 1, …, n$ , the relationship between $x$ and $y$ can be defined generally as

$y_i = m(x_i) + \varepsilon_i$

where $f(x_i) = E[y_i | x = x_i]$ and $\varepsilon_i \stackrel{iid}{\sim} (0, \sigma^2)$ . If we are unsure about the form of $m(\cdot)$ , our objective may be to estimate $m(\cdot)$ without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of $y = f(x) + \epsilon$ . Here, $x \sim Uniform(0, 10)$ and $\epsilon \sim N(0, 1)$ . The true function $f(x) = sin(x)$ is displayed in green.

Non-parametric approaches require only that $m(\cdot)$ be smooth and continuous. These assumptions are far less restrictive than alternative parametric approaches, thereby increasing the number of potential fits and providing additional flexibility. This makes non-parametric models particularly appealing when prior knowledge about $m(\cdot)$ ‘s functional form is limited.

Estimating the Regression Function

If multiple values of $y$ were observed at each $x$ , $f(x)$ could be estimated by averaging the value of the response at each $x$ . However, since $x$ is often continuous, it can take on a wide range of values making this quite rare. Instead, a neighbourhood of $x$ is considered.

Result of averaging $y_i$ at each $x_i$ . The fit is extremely rough due to gaps in $x$ and low $y$ frequency at each $x$ .

Define the neighbourhood around $x$ as $x \pm \lambda$ for some bandwidth $\lambda > 0$ . Then, a simple non-parametric estimate of $m(x)$ can be constructed as average of the $y_i$ ‘s corresponding to the $x_i$ within this neighbourhood. That is,

(1) $\begin{equation*} \hat{f}_{\lambda}(x) = \frac{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)~ y_i}{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)} = \frac{\sum_n K\left( \frac{x - x_i}{\lambda} \right) y_i}{\sum_n K\left( \frac{x - x_i}{\lambda} \right) } \end{equation*}$

where

$K(u) = \begin{cases} \frac{1}{2} & |u| \leq 1 \\ 0 & \text{o.w.} \end{cases}$

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function $K(u)$ (see my previous blog bost). It is, however, convention to use kernel functions of degree $\nu = 2$ (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating $f(x)$ with a Gaussian kernel and arbitrarily selected bandwidth of $\lambda = 1.25$ . The green line represents the true function $sin(x)$ .

Continue reading Kernel Regression