A Simple Diagnostic for the Positivity Assumption for Continuous Exposures

Abstract

The positivity or experimental treatment assignment assumption is a fundamental requirement in causal analyses, invoked to ensure that identifiability holds without extrapolating beyond what the observed data can reveal. Positivity is well understood in the context of binary and categorical treatments, and has been thoroughly discussed-from how the assumption can be assessed to approaches that may be used when the assumption is suspected not to hold. Positivity extends to the context of continuous exposures, such as doses, however it has been given very little formal consideration. In this manuscript, we propose a method for assessing whether the positivity assumption is violated in a given dataset, relying on a principled concept in regression analysis. We demonstrate the diagnostic tool in various simulated settings, as well as in an application involving warfarin dosing.

Keywords: causal inference; dose; experimental treatment assignment assumption; generalized propensity score; hat‐value.

FIGURE 1

Depictions of linear regression for an outcome $y$ with (a) one variable, (b) two variables, one of which is binary, and (c) and (d) two continuous variables. In each plot, the pink square indicates a point where prediction could be made without extrapolation (although may require some interpolation), whereas the blue and cyan triangles are points where extrapolation may occur. (a) A single predictor, $x$. (b) Two predictors, $x$ (continuous) and $z$ (binary). (c) Two continuous, correlated predictors. (d) Bivariate plot of predictors in (c).

FIGURE 2

An illustration of two scenarios with a continuous dose $d$ and two confounders $x_1$ and $x_2$, each of which following a uniform distribution. The plots in the two panels depict the relations in a setting (a) without and (b) with positivity violations. The bottom row displays scatter plots between $d$ and each respective confounder in the two settings. The top row illustrates the joint behavior of the variables, with the upper left graph showing $d$ versus $(x_1,x_2)$, and the upper right quadrant, the joint behavior of the confounders $(x_1,x_2)$. Note that here, positivity requires that the space defined by the jointly observable confounders $(x_1,x_2)$ be filled over the range of the exposure $d$; the joint distribution of only the confounders need not cover the square defined by the product of the range of $x_1$ and $x_2$. (a) No positivity violation. (b) Positivity violation.

FIGURE 3

Left column: scatter plots of $d$ versus $x$, with original data points depicted by grey circles and the candidate values as cyan triangles. Right column: density plot of ${\widehat{\varphi }}_0$ computed under a null scenario assuming $d$ follows a uniform (solid grey) or normal (solid black) distribution, or by randomly sampling with (dashed black) or without (dotted black) replacement from the empirical dose distribution; the observed $\widehat{\varphi }$ is marked by a cyan X. The first row displays the results from a setting with no positivity violations, whereas the second row provides results from a setting where positivity violations are present. (a) No positivity violation. (b) No positivity violation. (c) Positivity violation. (d) Positivity violation.

FIGURE 4

Left column: scatter plots of $d$ versus $x$, with original data points depicted by grey circles and the candidate values as cyan triangles. Middle column: results when candidate points are taken at the $5^{th}$, $50^{th}$, and $95^{th}$ percentiles of the dose distribution; the density plot of ${\widehat{\varphi }}_0$ is computed under a null assuming $d$ follows a uniform (solid grey) or normal (solid black) distribution, or by randomly sampling with (dashed black) or without (dotted black) replacement from the empirical dose distribution, with the observed $\widehat{\varphi }$ marked by a cyan X. Right column: same as the middle, but candidate points restricted to only the $50^{th}$ percentile of the dose distribution (in this setting, all density plots coincide and thus cannot be distinguished). The first row displays the results from a setting with no positivity violations, whereas the second row provides results from a setting where positivity violations are present. (a) No positivity violation. (b) No positivity violation. (c) No positivity violation. (d) Positivity violation. (e) Positivity violation. (f) Positivity violation.

FIGURE 5

Scatter plots of $d$ versus $x$: original data points (left), weighted by normalized inverse density of exposure weights (middle), and weighted by normalized inverse probability of exposure weights estimated from a multinomial logistic regression fit to quintiles of the exposure distribution (right). (a) Original data, (b) weighting by inverse density, and (c) weighting by inverse probability.

FIGURE 6

The International Warfarin Pharmacogenetics Consortium data: (a) a histogram of doses, (b) a normal QQ plot of doses, and (c) the hat‐value diagnostic, with $\widehat{\varphi }$ marked by a cyan X and the density of ${\widehat{\varphi }}_0$ under the assumption of a normally distributed dose (solid line) with no positivity violations or under random sampling from the empirical exposure distribution with (dashed line) or without (dotted line) replacement.