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Find out how to know the unknowable in observational research
- Introduction
- Drawback Setup
2.1. Causal Graph
2.2. Mannequin With and With out Z
2.3. Energy of Z as a Confounder - Sensitivity Evaluation
3.1. Aim
3.2. Robustness Worth - 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 are able to moderately assume that remedy task is as-if random as in a pure experiment, we are able to by no means be really sure that we managed for all potential confounders in our mannequin. Because of this, our mannequin estimates may be severely biased if we fail to manage for an vital confounder–and we wouldn’t even realize it because the unobserved confounder is, effectively, 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., remedy impact not statistically important)? Sensitivity evaluation for unobserved confounding is an energetic space of analysis, and there are a number of approaches to tackling this drawback. On this submit, I’ll cowl a easy linear technique [1] primarily based 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 we now have 4 variables:
- Y: consequence
- D: remedy
- X: noticed confounder(s)
- Z: unobserved confounder(s)
This can be a frequent setting in lots of observational research the place the researcher is fascinated by understanding whether or not the remedy 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 remedy impact is null. As will turn out to be clear within the subsequent part, the aim of sensitivity evaluation is with the ability to motive about this remedy impact when we now have no entry to Z, as we usually gained’t because it’s unobserved. Determine 1 visualizes our setup.
Determine 1: Drawback Setup
2.2. Mannequin With and With out Z
To exhibit the issue that our unobserved Z could cause, I simulated some knowledge in keeping with the issue setup described above. You’ll be able to confer with this pocket book for the small print of the simulation.
Since Z could be unobserved in actual life, the one mannequin we are able to usually match to knowledge is Y~D+X. Allow us to see what outcomes we get if we run that regression.
Based mostly 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 primarily based on how we generated the info (no D impact).
Now, let’s see what occurs to our D estimate once we management for Z as effectively. (In actual life, we after all gained’t be capable of run this extra regression since Z is unobserved however our simulation setting permits us to peek behind the scenes into the true knowledge era course of.)
As anticipated, controlling for Z accurately removes the D impact by shrinking the estimate in direction of zero and giving us a p-value that’s not statistically important on the =0.05 threshold (p=0.059).
2.3. Energy of Z as a Confounder
At this level, we now have established that Z is powerful sufficient of a confounder to eradicate the spurious D impact because the statistically important D impact disappears once we management for Z. What we haven’t mentioned but is precisely how robust Z is as a confounder. For this, we are going to 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 prevailing 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 can 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 embrace 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 prevailing 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 complete 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 complete 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. Aim
As we mentioned within the earlier part, unobserved confounding poses an issue in actual analysis settings exactly as a result of, not like in our simulation setting, Z can’t be noticed. In different phrases, we’re caught with the mannequin Y~D+X, having no option 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 strategy ought to be capable of inform us that if a Z such because the one we now have in our knowledge have 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. Due to this fact, we anticipate sensitivity evaluation to inform us {that a} hypothetical Z-like confounder of comparable energy could be sufficient to eradicate the statistically important D impact.
However how can we calculate that the unobserved confounder’s energy must be on this 16–20% vary within the partial R² scale with out ever accessing it? Enter robustness worth.
3.2. Robustness Worth
Robustness worth (RV) formalizes the thought we talked about above of figuring out the required energy 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 are able to write down as follows the RV that quantifies how robust unobserved confounding must be to vary our noticed statistical significance of the remedy impact (if the notation is an excessive amount of to comply with, simply keep in mind the important thing concept that the RV is a measure of the energy of confounding wanted to vary our outcomes)
the place
- is our chosen significance degree (usually set to 0.05 or 5%),
- q determines the % discount q*100% in significance that we care about (usually set to 1, since we normally care about confounding that would cut back statistical significance by 1*100%=100% therefore rendering it not statistically important),
- t_betahat_treat is the noticed t-value of our remedy 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 and df-1 (1.96 if is ready to 0.05).
We are actually 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 express information of Z, we are able to nonetheless motive that any unobserved confounder wants, on common, at the least 18% energy within the partial R² scale vis-à-vis each the remedy and the end result to have the ability to nullify our statistically important consequence.
The explanation 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 required energy of the confounder with each the end result and the remedy, so 18% makes good sense given what we all know concerning 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 energy 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 really does an awesome job summarizing the energy of the confounder.
After all, in actual life we gained’t have the Z variable to double verify that our RV is right, however seeing how the 2 outcomes align right here ought to at the least offer you some confidence within the technique. Lastly, as soon as we calculate the RV, we must always take into consideration whether or not an unobserved confounder of that energy is believable. In our case, the reply is ‘sure’ as a result of we now have entry to the info era course of, however in your particular real-life software, the existence of such a powerful confounder is likely to be an unreasonable assumption. This is able to be excellent news for you since no reasonable unobserved confounder may drastically change your outcomes.
The sensitivity evaluation method described above has already been applied with all of its bells and whistles as a Python package deal beneath the title PySensemakr (R, Stata, and Shiny App variations exist as effectively). For instance, to get the very same consequence that we manually calculated within the earlier part, we are able to merely run the next code chunk.
Observe 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 package deal 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 package deal 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 remedy and the end result (on this case, it shouldn’t be shocking to see that the x/y-axis worth pairs that meet the pink dotted line embrace 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 are able to 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 must be sufficient to nullify our statistically important consequence (because the “0.5x X” worth falls proper on the pink dotted line).
Lastly, I wish to observe that whereas the simulated knowledge on this instance used a steady remedy variable, in apply the tactic works for any type of remedy variable together with binary remedies. However, the end result variable technically must be a steady one since we’re working within the OLS framework. Nevertheless, the tactic can nonetheless be used even with a binary consequence if we mannequin it utilizing OLS (that is known as a LPM [2]).
The chance that our impact estimate could also be biased attributable to unobserved confounding is a typical hazard in observational research. Regardless of this potential hazard, observational research are a significant device in knowledge science as a result of randomization merely isn’t possible in lots of instances. Due to this fact, you will need to know the way we are able to 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 submit is a straightforward and intuitive strategy to sensitivity evaluation formulated in a well-known linear mannequin framework. In case you are fascinated by studying extra concerning the technique, I extremely advocate looking on the authentic paper and the package deal documentation the place you may study many extra attention-grabbing functions of the tactic resembling ‘excessive state of affairs’ evaluation.
There are additionally many different approaches to sensitivity evaluation for unobserved confounding, and I would really like briefly point out a few of them right here for readers who wish to proceed studying extra on this subject. One versatile method is the E-value developed by VanderWeele and Ding that formulates the issue by way of threat ratios [3] (applied in R right here). One other method is the Austen plot developed by Veitch and Zaveri primarily based on the ideas of partial R² and propensity rating [4] (applied in Python right here), and yet one more current strategy is by Chernozhukov et al [5] (applied in Python right here).
I wish to thank Chad Hazlett for answering my query associated to utilizing the tactic with binary outcomes and Xinyi Zhang for offering numerous invaluable suggestions on the submit. Except in any other case famous, all photos 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 Chance Mannequin, Murray’s private web site
[3] T. VanderWeele and P. Ding, Sensitivity Evaluation in Observational Analysis: Introducing the E-Worth (2017), Annals of Inside Medication
[4] V. Veitch and A. Zaveri, Sense and Sensitivity Evaluation: Easy Put up-Hoc Evaluation of Bias Attributable to Unobserved Confounding (2020), NeurIPS
[5] V. Chernozhukov, C. Cinelli, W. Newey, A. Sharma, and V. Syrgkanis, Lengthy Story Brief: Omitted Variable Bias in Causal Machine Studying (2022), NBER
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