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- Introduction
- Problem Setup
2.1. Causal Graph
2.2. Model With and Without Z
2.3. Strength of Z as a Confounder - Sensitivity Analysis
3.1. Goal
3.2. Robustness Value - PySensemakr
- Conclusion
- Acknowledgements
- References
The specter of unobserved confounding (aka omitted variable bias) is a infamous drawback in observational research. In most observational research, except we will moderately assume that therapy task is as-if random as in a pure experiment, we will by no means be really sure that we managed for all potential confounders in our mannequin. In consequence, our mannequin estimates may be severely biased if we fail to regulate for an necessary confounder–and we wouldn’t even understand it because the unobserved confounder is, nicely, unobserved!
Given this drawback, you will need to assess how delicate our estimates are to potential sources of unobserved confounding. In different phrases, it’s a useful train to ask ourselves: how a lot unobserved confounding would there need to be for our estimates to drastically change (e.g., therapy impact now not statistically important)? Sensitivity evaluation for unobserved confounding is an lively space of analysis, and there are a number of approaches to tackling this drawback. On this publish, I’ll cowl a easy linear technique [1] based mostly on the idea of partial R² that’s broadly relevant to a big spectrum of instances.
2.1. Causal Graph
Allow us to assume that now we have 4 variables:
- Y: end result
- D: therapy
- X: noticed confounder(s)
- Z: unobserved confounder(s)
It is a widespread setting in lots of observational research the place the researcher is desirous about realizing whether or not the therapy of curiosity has an impact on the end result after controlling for potential treatment-outcome confounders.
In our hypothetical setting, the connection between these variables are such that X and Z each have an effect on D and Y, however D has no impact on Y. In different phrases, we’re describing a state of affairs the place the true therapy impact is null. As will develop into clear within the subsequent part, the aim of sensitivity evaluation is with the ability to motive about this therapy impact when now we have no entry to Z, as we usually received’t because it’s unobserved. Determine 1 visualizes our setup.
Determine 1: Downside Setup
2.2. Mannequin With and With out Z
To reveal the issue that our unobserved Z could cause, I simulated some knowledge in step with the issue setup described above. You’ll be able to seek advice from this notebook for the small print of the simulation.
Since Z could be unobserved in actual life, the one mannequin we will usually match to knowledge is Y~D+X. Allow us to see what outcomes we get if we run that regression.
Primarily based on these outcomes, it looks like D has a statistically important impact of 0.2686 (p<0.001) per one unit change on Y, which we all know isn’t true based mostly on how we generated the information (no D impact).
Now, let’s see what occurs to our D estimate after we management for Z as nicely. (In actual life, we in fact received’t be capable of run this extra regression since Z is unobserved however our simulation setting permits us to peek backstage into the true knowledge technology course of.)
As anticipated, controlling for Z appropriately removes the D impact by shrinking the estimate in the direction of zero and giving us a p-value that’s now not statistically important on the 𝛼=0.05 threshold (p=0.059).
2.3. Power of Z as a Confounder
At this level, now we have established that Z is powerful sufficient of a confounder to remove the spurious D impact because the statistically important D impact disappears after we management for Z. What we haven’t mentioned but is precisely how robust Z is as a confounder. For this, we’ll make the most of a helpful statistical idea known as partial R², which quantifies the proportion of variation {that a} given variable of curiosity can clarify that may’t already be defined by the present variables in a mannequin. In different phrases, partial R² tells us the added explanatory energy of that variable of curiosity, above and past the opposite variables which might be already within the mannequin. Formally, it may be outlined as follows
the place RSS_reduced is the residual sum of squares from the mannequin that doesn’t embody the variable(s) of curiosity and RSS_full is the residual sum of squares from the mannequin that features the variable(s) of curiosity.
In our case, the variable of curiosity is Z, and we wish to know what quantity of the variation in Y and D that Z can clarify that may’t already be defined by the present variables. Extra exactly, we have an interest within the following two partial R² values
the place (1) quantifies the proportion of variance in Y that may be defined by Z that may’t already be defined by D and X (so the diminished mannequin is Y~D+X and the total mannequin is Y~D+X+Z), and (2) quantifies the proportion of variance in D that may be defined by Z that may’t already be defined by X (so the diminished mannequin is D~X and the total mannequin is D~X+Z).
Now, allow us to see how strongly related Z is with D and Y in our knowledge by way of partial R².
It seems that Z explains 16% of the variation in Y that may’t already be defined by D and X (that is partial R² equation #1 above), and 20% of the variation in D that may’t already be defined by X (that is partial R² equation #2 above).
3.1. Objective
As we mentioned within the earlier part, unobserved confounding poses an issue in actual analysis settings exactly as a result of, in contrast to in our simulation setting, Z can’t be noticed. In different phrases, we’re caught with the mannequin Y~D+X, having no strategy to know what our outcomes would have been if we may run the mannequin Y~D+X+Z as an alternative. So, what can we do?
Intuitively, an inexpensive sensitivity evaluation method ought to be capable of inform us that if a Z such because the one now we have in our knowledge had been to exist, it will nullify our outcomes. Do not forget that our Z explains 16% of the variation in Y and 20% of the variation in D that may’t be defined by noticed variables. Subsequently, we count on sensitivity evaluation to inform us {that a} hypothetical Z-like confounder of comparable power could be sufficient to remove the statistically important D impact.
However how can we calculate that the unobserved confounder’s power ought to be on this 16–20% vary within the partial R² scale with out ever getting access to it? Enter robustness worth.
3.2. Robustness Worth
Robustness worth (RV) formalizes the thought we talked about above of figuring out the mandatory power of a hypothetical unobserved confounder that would nullify our outcomes. The usefulness of RV emanates from the truth that we solely want our observable mannequin Y~D+X and never the unobservable mannequin Y~D+X+Z to have the ability to calculate it.
Formally, we will write down as follows the RV that quantifies how robust unobserved confounding must be to vary our noticed statistical significance of the therapy impact (if the notation is an excessive amount of to comply with, simply bear in mind the important thing concept that the RV is a measure of the power of confounding wanted to vary our outcomes)
the place
- 𝛼 is our chosen significance degree (usually set to 0.05 or 5%),
- q determines the p.c discount q*100% in significance that we care about (usually set to 1, since we normally care about confounding that would scale back statistical significance by 1*100%=100% therefore rendering it not statistically important),
- t_betahat_treat is the noticed t-value of our therapy from the mannequin Y~D+X (which is 8.389 on this case as may be seen from the regression outcomes above),
- df is our levels of freedom (which is 1000–3=997 on this case since we simulated 1000 samples and are estimating 3 parameters together with the intercept), and
- t*_alpha,df-1 is the t-value threshold related to a given 𝛼 and df-1 (1.96 if 𝛼 is ready to 0.05).
We at the moment are able to calculate the RV in our personal knowledge utilizing solely the noticed mannequin Y~D+X (res_ydx).
It’s by no struck of luck that our RV (18%) falls proper within the vary of the partial R² values we calculated for Y~Z|D,X (16%) and D~Z|X (20%) above. What the RV is telling us right here is that, even with none specific information of Z, we will nonetheless motive that any unobserved confounder wants, on common, no less than 18% power within the partial R² scale vis-à-vis each the therapy and the end result to have the ability to nullify our statistically important outcome.
The rationale why the RV isn’t 16% or 20% however falls someplace in between (18%) is that it’s designed to be a single quantity that summarizes the mandatory power of the confounder with each the end result and the therapy, so 18% makes good sense given what we all know in regards to the knowledge. You’ll be able to give it some thought like this: because the technique doesn’t have entry to the precise numbers 16% and 20% when calculating the RV, it’s doing its finest to quantify the power of the confounder by assigning 18% to each partial R² values (Y~Z|D,X and D~Z|X), which isn’t too far off from the reality in any respect and truly does an excellent job summarizing the power of the confounder.
In fact, in actual life we received’t have the Z variable to double examine that our RV is right, however seeing how the 2 outcomes align right here ought to no less than provide you with some confidence within the technique. Lastly, as soon as we calculate the RV, we should always take into consideration whether or not an unobserved confounder of that power is believable. In our case, the reply is ‘sure’ as a result of now we have entry to the information technology course of, however on your particular real-life software, the existence of such a powerful confounder may be an unreasonable assumption. This might be excellent news for you since no real looking unobserved confounder may drastically change your outcomes.
The sensitivity evaluation method described above has already been carried out with all of its bells and whistles as a Python bundle beneath the title PySensemakr (R, Stata, and Shiny App variations exist as nicely). For instance, to get the very same outcome that we manually calculated within the earlier part, we will merely run the next code chunk.
Be aware that “Robustness Worth, q = 1 alpha = 0.05” is 0.184, which is precisely what we calculated above. Along with the RV for statistical significance, the bundle additionally supplies the RV that’s wanted for the coefficient estimate itself to shrink to 0. Not surprisingly, unobserved confounding must be even bigger for this to occur (0.233 vs 0.184).
The bundle additionally supplies contour plots for the 2 partial R² values, which permits for an intuitive visible show of sensitivity to potential ranges of confounding with the therapy and the end result (on this case, it shouldn’t be shocking to see that the x/y-axis worth pairs that meet the purple dotted line embody 0.18/0.18 in addition to 0.20/0.16).
One may even add benchmark values to the contour plot as proxies for potential quantities of confounding. In our case, since we solely have one noticed covariate X, we will set our benchmarks to be 0.25x, 0.5x and 1x as robust as that noticed covariate. The ensuing plot tells us {that a} confounder that’s half as robust as X ought to be sufficient to nullify our statistically important outcome (because the “0.5x X” worth falls proper on the purple dotted line).
Lastly, I wish to observe that whereas the simulated knowledge on this instance used a steady therapy variable, in follow the strategy works for any sort of therapy variable together with binary therapies. Alternatively, the end result variable technically must be a steady one since we’re working within the OLS framework. Nevertheless, the strategy can nonetheless be used even with a binary end result if we mannequin it utilizing OLS (that is known as a LPM [2]).
The likelihood that our impact estimate could also be biased on account of unobserved confounding is a standard hazard in observational research. Regardless of this potential hazard, observational research are an important device in knowledge science as a result of randomization merely isn’t possible in lots of instances. Subsequently, you will need to understand how we will deal with the difficulty of unobserved confounding by operating sensitivity analyses to see how strong our estimates are to potential such confounding.
The robustness worth technique by Cinelli and Hazlett mentioned on this publish is a straightforward and intuitive method to sensitivity evaluation formulated in a well-known linear mannequin framework. If you’re desirous about studying extra in regards to the technique, I extremely suggest having a look on the authentic paper and the package documentation the place you may study many extra fascinating purposes of the strategy similar to ‘excessive state of affairs’ evaluation.
There are additionally many different approaches to sensitivity evaluation for unobserved confounding, and I would love briefly point out a few of them right here for readers who wish to proceed studying extra on this matter. One versatile method is the E-value developed by VanderWeele and Ding that formulates the issue by way of threat ratios [3] (carried out in R here). One other method is the Austen plot developed by Veitch and Zaveri based mostly on the ideas of partial R² and propensity rating [4] (carried out in Python here), and one more latest method is by Chernozhukov et al [5] (carried out in Python here).
I wish to thank Chad Hazlett for answering my query associated to utilizing the strategy with binary outcomes and Xinyi Zhang for offering loads of helpful suggestions on the publish. Except in any other case famous, all pictures are by the writer.
[1] C. Cinelli and C. Hazlett, Making Sense of Sensitivity: Extending Omitted Variable Bias (2019), Journal of the Royal Statistical Society
[2] J. Murray, Linear Probability Model, Murray’s private web site
[3] T. VanderWeele and P. Ding, Sensitivity Analysis in Observational Research: Introducing the E-Value (2017), Annals of Inner Drugs
[4] V. Veitch and A. Zaveri, Sense and Sensitivity Analysis: Simple Post-Hoc Analysis of Bias Due to Unobserved Confounding (2020), NeurIPS
[5] V. Chernozhukov, C. Cinelli, W. Newey, A. Sharma, and V. Syrgkanis, Long Story Short: Omitted Variable Bias in Causal Machine Learning (2022), NBER
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