You’re an avid information scientist and experimenter. You recognize that randomisation is the summit of Mount Proof Credibility, and also you additionally know that when you may’t randomise, you resort to observational information and Causal Inference methods. At your disposal are varied strategies for spinning up a management group — difference-in-differences, inverse propensity rating weighting, and others. With an assumption right here or there (some shakier than others), you estimate the causal impact and drive decision-making. However in the event you thought it couldn’t get extra thrilling than “vanilla” causal inference, learn on.
Personally, I’ve typically discovered myself in at the least two situations the place “simply doing causal inference” wasn’t easy. The widespread denominator in these two situations? A lacking management group — at first look, that’s.
First, the cold-start situation: the corporate needs to interrupt into an uncharted alternative house. Typically there is no such thing as a experimental information to study from, nor has there been any change (learn: “exogenous shock”), from the enterprise or product aspect, to leverage within the extra widespread causal inference frameworks like difference-in-differences (and different cousins within the pre-post paradigm).
Second, the unfeasible randomisation situation: the organisation is completely intentional about testing an concept, however randomisation is just not possible—or not even needed. Even emulating a pure experiment is likely to be constrained legally, technically, or commercially (particularly when it’s about pricing), or when interference bias arises within the market.
These conditions open up the house for a “totally different” kind of causal inference. Though the strategy we’ll give attention to right here is just not the one one fitted to the job, I’d love so that you can tag alongside on this deep dive into Regression Discontinuity Design (RDD).
On this put up, I’ll provide you with a crisp view of how and why RDD works. Inevitably, this may contain a little bit of math — a nice sight for some — however I’ll do my finest to maintain it accessible with traditional examples from the literature.
We’ll additionally see how RDD can deal with a thorny causal inference problem in e-commerce and on-line marketplaces: the impression of itemizing place on itemizing efficiency. On this sensible part we’ll cowl key modelling concerns that practitioners typically face: parametric versus non-parametric RDD, selecting the best bandwidth parameter, and extra. So, seize your self a cup of of espresso and let’s soar in!
Define
How and why RDD works
Regression Discontinuity Design exploits cutoffs — thresholds — to get well the impact of a therapy on an consequence. Extra exactly, it seems to be for a pointy change within the likelihood of therapy task on a ‘working’ variable. If therapy task relies upon solely on the working variable, and the cutoff is bigoted, i.e. exogenous, then we are able to deal with the items round it as randomly assigned. The distinction in outcomes simply above and under the cutoff offers us the causal impact.
For instance, a scholarship awarded solely to college students scoring above 90, creates a cutoff primarily based on take a look at scores. That the cutoff is 90 is bigoted — it may have been 80 for that matter; the road had simply to be drawn someplace. Furthermore, scoring 91 vs. 89 makes the entire distinction as for the therapy: both you get it or not. However relating to functionality, the 2 teams of scholars that scored 91 and 89 aren’t actually totally different, are they? And those that scored 89.9 versus 90.1 — in the event you insist?
Making the cutoff may come all the way down to randomness, when it’s only a bout a number of factors. Perhaps the coed drank an excessive amount of espresso proper earlier than the take a look at — or too little. Perhaps they bought unhealthy information the night time earlier than, have been thrown off by the climate, or nervousness hit on the worst attainable second. It’s this randomness that makes the cutoff so instrumental in RDD.
With out a cutoff, you don’t have an RDD — only a scatterplot and a dream. However, the cutoff by itself is just not geared up with all it takes to establish the causal impact. Why it really works hinges on one core identification assumption: continuity.
The continuity assumption, and parallel worlds
If the cutoff is the cornerstone of the method, then its significance comes completely from the continuity assumption. The thought is an easy, counterfactual one: had there been no therapy, then there would’ve been no impact.
To floor the thought of continuity, let’s soar straight right into a traditional instance from public well being: does authorized alcohol entry improve mortality?
Think about two worlds the place everybody and all the pieces is identical. Apart from one factor: a legislation that units the minimal authorized consuming age at 18 years (we’re in Europe, people).
On the planet with the legislation (the factual world), we’d anticipate alcohol consumption to leap proper after age 18. Alcohol-related deaths ought to soar too, if there’s a hyperlink.
Now, take the counterfactual world the place there is no such thing as a such legislation; there must be no such soar. Alcohol consumption and mortality would doubtless observe a {smooth} pattern throughout age teams.
Now, that’s a superb factor for figuring out the causal impact; the absence of a soar in deaths within the counterfactual world is the needed situation to interpret a soar within the factual world because the impression of the legislation.
Put merely: if there is no such thing as a therapy, there shouldn’t be a soar in deaths. If there may be, then one thing apart from our therapy is inflicting it, and the RDD is just not legitimate.
The continuity assumption may be written within the potential outcomes framework as:
start{equation}
lim_{x to c^-} mathbb{E}[Y_i(0) mid X_i = x] = lim_{x to c^+} mathbb{E}[Y_i(0) mid X_i = x]
label{eq: continuity_po}
finish{equation}
The place (Y_i(0)) is the potential consequence, say, threat of loss of life of topic (/mathbb{i}) below no therapy.
Discover that the right-hand aspect is a amount of the counterfactual world; not one that may be noticed within the factual world, the place topics are handled in the event that they fall above the cutoff.
Sadly for us, we solely have entry to the factual world, so the idea can’t be examined instantly. However, fortunately, we are able to proxy it. We’ll see placebo teams obtain this later within the put up. However first, we begin by figuring out what can break the idea:
- Confounders: one thing apart from the therapy occurs on the cutoff that additionally impacts the end result. As an example, adolescents resorting to alcohol to alleviate the crushing stress of being an grownup now — one thing that has nothing to do with the legislation on the minimal age to devour alcohol (within the no-law world), however that does confound the impact we’re after, occurring on the identical age — the cutoff, that’s.
- Manipulating the working variable:
When items can affect their place with regard to the cutoff, it might be that items who did so are inherently totally different from those that didn’t. Therefore, cutoff manipulation can lead to choice bias: a type of confounding. Particularly if therapy task is binding, topics could strive their finest to get one model of the therapy over the opposite.
Hopefully, it’s clear what constitutes a RDD: the working variable, the cutoff, and most significantly, affordable grounds to defend that continuity holds. With that, you’ve gotten your self a neat and efficient causal inference design for questions that may’t be answered by an A/B take a look at, nor by a few of the extra widespread causal inference methods like diff-in-diff, nor with stratification.
Within the subsequent part, we proceed shaping our understanding of how RDD works; how does RDD “management” confounding relationships? What precisely does it estimate? Can we not simply management for the working variable too? These are questions that we deal with subsequent.
RDD and devices
In case you are already aware of instrumental variables (IV), you might even see the similarities: each RDD and IV leverage an exogenous variable that doesn’t trigger the end result instantly, however does affect the therapy task, which in flip could affect the end result. In IV it is a third variable Z; in RDD it’s the working variable that serves as an instrument.
Wait. A 3rd variable; perhaps. However an exogenous one? That’s much less clear.
In our instance of alcohol consumption, it’s not arduous to think about that age — the working variable — is a confounder. As age will increase, so may tolerance for alcohol, and with it the extent of consumption. That’s a stretch, perhaps, however not implausible.
Since therapy (authorized minimal age) depends upon age — solely items above 18 are handled — handled and untreated items are inherently totally different. If age additionally influences the end result, by means of a mechanism just like the one sketched above, we bought ourselves an apex confounder.
Nonetheless, the working variable performs a key function. To know why, we have to take a look at how RDD and devices leverage the frontdoor criterion to establish causal results.
Backdoor vs. frontdoor
Maybe nearly instinctively, one could reply with controlling for the working variable; that’s what stratification taught us. The working variable is confounder, so we embrace it in our regression, and shut the backdoor. However doing so would trigger some hassle.
Bear in mind, therapy task depends upon the working variable so that everybody above the cutoff is handled with all certainty, and actually not under it. So, if we management for the working variable, we run into two very associated issues:
- Violation of the Positivity assumption: this assumption says that handled items ought to have a non-zero likelihood to obtain the other therapy, and vice versa. Intuitively, conditioning on the working variable is like saying: “Let’s estimate the impact of being above the minimal age for alcohol consumption, whereas holding age fastened at 14.” That doesn’t make sense. At any given worth of working variable, therapy is both at all times 1 or at all times 0. So, there’s no variation in therapy conditional on the working variable to help such a query.
- Good collinearity on the cutoff: in estimating the therapy impact, the mannequin has no solution to separate the impact of crossing the cutoff from the impact of being at a selected worth of X. The consequence? No estimate, or a forcefully dropped variable from the mannequin design matrix. Singular design matrix, doesn’t have full rank, these ought to sound acquainted to most practitioners.
So no — conditioning on the working variable doesn’t make the working variable the exogenous instrument that we’re after. As an alternative, the working variable turns into exogenous by pushing it to the restrict—fairly actually. There the place the working variable approaches the cutoff from both aspect, the items are the identical with respect to the working variable. But, falling simply above or under makes the distinction as for getting handled or not. This makes the working variable a sound instrument, if therapy task is the one factor that occurs on the cutoff. Judea Pearl refers to devices as assembly the front-door criterion.
LATE, not ATE
So, in essence, we’re controlling for the working variable — however solely close to the cutoff. That’s why RDD identifies the native common therapy impact (LATE), a particular flavour of the typical therapy impact (ATE). The LATE seems to be like:
$$delta_{SRD}=Ebig[Y^1_i – Y_i^0mid X_i=c_0]$$
The native bit refers back to the partial scope of the inhabitants we’re estimating the ATE for, which is the subpopulation across the cutoff. Actually, the additional away the information level is from the cutoff, the extra the working variable acts as a confounder, working in opposition to the RDD as a substitute of in its favour.
Again to the context of the minimal age for authorized alcohol consumption instance. Adolescents who’re 17 years and 11 months previous are actually not so totally different from these which are 18 years and 1 month previous, on common. If something, a month or two distinction in age is just not going to be what units them aside. Isn’t that the essence of conditioning on, or holding a variable fixed? What units them aside is that the latter group can devour alcohol legally for being above the cutoff, and never the previous.
This setup allows us to estimate the LATE for the items across the cutoff and with that, the impact of the minimal age coverage on alcohol-related deaths.
We’ve seen how the continuity assumption has to carry to make the cutoff an attention-grabbing level alongside the working variable in figuring out the causal impact of a therapy on the end result. Specifically, by letting the soar within the consequence variable be completely attributable to the therapy. If continuity holds, the therapy is as-good-as-random close to the cutoff, permitting us to estimate the native common therapy impact.
Within the subsequent part, we’ll stroll by means of the sensible setup of a real-world RDD: we establish the important thing ideas; the working variable and cutoff, therapy, consequence, covariates, and eventually, we estimate the RDD after discussing some essential modelling decisions, and finish the part with a placebo take a look at.
RDD in Motion: Search Rating and itemizing efficiency Instance
In e-commerce and on-line marketplaces, the start line of the customer expertise is trying to find an inventory. Consider the customer typing “Nikon F3 analogue digital camera” within the search bar. Upon finishing up this motion, algorithms frantically type by means of the stock in search of the very best matching listings to populate the search outcomes web page.
Time and a focus are two scarce assets. So, it’s within the curiosity of everybody concerned — the customer, the vendor and the platform — to order essentially the most distinguished positions on the web page for the matches with the best anticipated likelihood to develop into profitable trades.
Moreover, place results in client behaviour recommend that customers infer increased credibility and desirability from objects “ranked” on the prime. Take into consideration high-tier merchandise being positioned at eye-height or above in supermarkets, and highlighted objects on an e-commerce platform, on the prime of the homepage.
So, the query then turns into: how does positioning on the search outcomes web page affect an inventory’s probabilities to be bought?
Speculation:
If an inventory is ranked increased on the search outcomes web page, then it is going to have a better likelihood of being bought, as a result of higher-ranked listings get extra visibility and a focus from customers.
Intermezzo: enterprise or principle?
As with every good speculation, we’d like a little bit of principle to floor it. Good for us is that we’re not looking for the treatment for most cancers. Our principle is about well-understood psychological phenomena and behavioural patterns, to place it overly subtle.
Consider primacy effect, anchoring bias and the resource theory of attention. These are effectively concepts in behavioural and cognitive psychology that again up our plan right here.
Kicking off the dialog with a product supervisor shall be extra enjoyable this fashion. Personally, I additionally get excited when I’ve to brush up on some psychology.
However I’ve discovered by means of and thru {that a} principle is actually secondary to any initiative in my business (tech). Apart from a analysis staff and challenge, arguably. And it’s truthful to say it helps us keep on-purpose: what we’re doing is to carry enterprise ahead, not mom science.
Realizing the reply has actual enterprise worth. Product and industrial groups may use it to design new paid options that assist sellers get their listings on increased positions — a win for each the enterprise and the person. It may additionally make clear the worth of on-site actual property like banner positions and advert slots, serving to drive development in B2B promoting.
The query is about incrementality: would’ve itemizing (mathbb{j}) been bought, had it been ranked 1st on the outcomes web page, as a substitute of fifteenth. So, we need to make a causal assertion. That’s arduous for at the least two causes:
- A/B testing comes with a value, and;
- there are confounders we have to take care of if we resort to observational strategies.
Let’s increase on that.
The price of A/B testing
One experiment design may randomise the fetched listings throughout the web page slots, unbiased of the itemizing relevance. Breaking the inherent hyperlink between relevance and place, we’d study the impact of place on itemizing efficiency. It’s an attention-grabbing concept — however a pricey one.
Whereas it’s an affordable design for statistical inference, this setup is form of horrible for the person and enterprise. The person might need discovered what they wanted—perhaps even made a purchase order. However as a substitute, perhaps half of the stock they’d have seen was remotely a superb match due to our experiment. This suboptimal person expertise doubtless hurts engagement in each the brief and long run — particularly for brand new customers who’re nonetheless to see what worth the platform holds for them.
Can we consider a solution to mitigate this loss? Nonetheless dedicated to A/B testing, one may expose a smaller set of customers to the experiment. Whereas it is going to scale down the implications, it might additionally stand in the best way of reaching enough statistical energy by reducing the pattern measurement. Furthermore, even small audiences may be accountable for substantial income for some firms nonetheless — these with tens of millions of customers. So, chopping the uncovered viewers is just not a silver bullet both.
Naturally, the best way to go is to go away the platform and its customers undisturbed — and nonetheless discover a solution to reply the query at hand. Causal inference is the fitting mindset for this, however the query is: how can we do this precisely?
Confounders
Listings don’t simply make it to the highest of the web page on a superb day; it’s their high quality, relevance, and the sellers’ repute that promote the rating of an inventory. Let’s name these three variables W.
What makes W difficult is that it influences each the rating of the itemizing and likewise the likelihood that the itemizing will get clicked, a proxy for efficiency.
In different phrases, W impacts each our therapy (place) and consequence (click on), serving to itself with the standing of confounder.
Due to this fact, our process is to discover a design that’s match for goal; one which successfully controls the confounding impact of W.
You don’t select regression discontinuity — it chooses you
Not all causal inference designs are simply sitting round ready to be picked. Generally they present up once you least want them, and generally you get fortunate once you want them most — like at the moment.
It seems to be like we are able to use the web page cutoff to establish the causal impression of place on clicks-through charge.
Abrupt cutoff in search outcomes pagination
Let’s unpack the itemizing suggestion mechanism to see precisely how. Right here’s what occurs below the hood when a outcomes web page is generated for a search:
- Fetch listings matching the question
A rough set of listings is pulled from the stock, primarily based on filters like location, radius, and class, and so on. - Rating listings on private relevance
This step makes use of person historical past and itemizing high quality proxies to foretell what the person is most definitely to click on. - Rank listings by rating
Greater scores get increased ranks. Enterprise guidelines combine in advertisements and industrial content material with natural outcomes. - Populate pages
Listings are slotted by absolute relevance rating. A outcomes web page ends on the okth itemizing, so the ok+1th itemizing seems on the prime of the subsequent web page. That is goes to be essential to our design. - Impressions and person interplay
Customers see the ends in order of relevance. If an inventory catches their eye, they could click on and look at extra particulars: one step nearer to the commerce.
Sensible setup and variables
So, what is precisely our design? Subsequent, we stroll by means of the reasoning and identification of the important thing substances of our design.
The working variable
In our setup, the working variable is the relevance rating (s_j) for itemizing j. This rating is a steady, complicated operate of each person and itemizing properties:
$$s_j = f(u_i, l_j)$$
The itemizing’s rank (r_j) is just a rank transformation of (s_j), outlined as:
$$r_i = sum_{j=1}^{n} mathbf{1}(s_j leq s_i)$$
Virtually talking, which means that for analytic functions—corresponding to becoming fashions, making native comparisons, or figuring out cutoff factors—figuring out an inventory’s rank conveys practically the identical info as figuring out its underlying relevance rating, and vice versa.
Particulars: Relevance rating vs. rank
The relevance rating (s_j) displays how effectively an inventory matches a selected person’s question, given parameters like location, value vary, and different filters. However this rating is relative—it solely has that means inside the context of the listings returned for that individual search.
In distinction, rank (or place) is absolute. It instantly determines an inventory’s visibility. I consider rank as a standardising transformation of (s_j). For instance, Itemizing A in search Z might need the best rating of 5.66, whereas Itemizing B in search Ok tops out at 0.99. These uncooked scores aren’t comparable throughout searches—however each listings are ranked first of their respective consequence units. That makes them equal when it comes to what actually issues right here: how seen they’re to customers.
The cutoff, and therapy
If an inventory simply misses the primary web page, it doesn’t fall to the underside of web page two — it’s artificially bumped to the highest. That’s a fortunate break. Usually, solely essentially the most related listings seem on the prime, however right here an inventory of merely average relevance results in a major slot —albeit on the second web page — purely because of the arbitrary place of the web page break. Formally, the therapy task (D_j) goes like:
$$D_j = start{instances} 1 & textual content{if } r_j > 30 0 & textual content{in any other case} finish{instances}$$
(Observe on world rank: Rank 31 isn’t simply the primary itemizing on web page two; it’s nonetheless the thirty first itemizing total)
The energy of this setup lies in what occurs close to the cutoff: an inventory ranked 30 could also be practically equivalent in relevance to 1 ranked 31. A small scoring fluctuation — or a high-ranking outlier — can push an inventory over the brink, flipping its therapy standing. This native randomness is what makes the setup legitimate for RDD.
The end result: Impression-to-click
Lastly, we operationalise the end result of curiosity because the click-though charge from impressions to clicks. Do not forget that all listings are ‘impressed’ when when the web page is populated. The clicking is the binary indicator of the specified person behaviour.
In abstract, that is our setup:
- Final result: impression-to-click conversion
- Remedy: Touchdown on the primary vs. second web page
- Operating variable: itemizing rank; web page cutoff at 30
Subsequent we stroll by means of the right way to estimate the RDD.
Estimating RDD
On this part, we’ll estimate the causal parameter, interpret it, and join them again to our core speculation: how place impacts itemizing visibility.
Right here’s what we’ll cowl:
- Meet the information: Intro to the dataset
- Covariates: Why and the right way to embrace them
- Modelling decisions: parametric RDD vs. not. Selecting the polynomial diploma and bandwidth.
- Placebo-testing
- Density continuity testing
Meet the information
We’re working with impressions information from one in all Adevinta’s (ex-eBay Classifieds Group) marketplaces. It’s actual information, which makes the entire train really feel grounded. That mentioned, values and relationships are censored and scrambled the place needed to guard its strategic worth.
An essential word to how we interpret the RDD estimates and drive selections, is how the information was collected: solely these searches the place the person noticed each the primary and second web page have been included.
This manner, we partial out the web page fastened impact if any, however the actuality is that many customers don’t make it to the second web page in any respect. So there’s a large quantity hole. We talk about the repercussion within the evaluation recap.
The dataset consists of those variables:
- Clicked: 1 if the itemizing was clicked, 0 in any other case – binary
- Place: the rank of the itemizing – numeric
- D: therapy indicator, 1 if place > 30, 0 in any other case – binary
- Class: product class of the itemizing – nominal
- Natural: 1 if natural, 0 if from an expert vendor – binary
- Boosted: 1 if was paid to be on the prime, 0 in any other case – binary
| click on | rel_position | D | class | natural | boosted |
| 1 | -3 | 0 | A | 1 | 0 |
| 1 | -14 | 0 | A | 1 | 0 |
| 0 | 3 | 1 | C | 1 | 0 |
| 0 | 10 | 1 | D | 0 | 0 |
| 1 | -1 | 0 | Ok | 1 | 1 |
Covariates: the right way to embrace them to extend accuracy?
The working variable, the cutoff, and the continuity assumption, provide you with all you have to establish the causal impact. However together with covariates can sharpen the estimator by lowering variance — if finished proper. And, oh is it straightforward to do it unsuitable.
The best factor to “break” concerning the RDD design, is the continuity assumption. Concurrently, that’s the final factor we need to break (I already rambled lengthy sufficient about this).
Due to this fact, the principle quest in including covariates is to it in such means that we scale back variance, whereas retaining the continuity assumption intact. One solution to formulate that, is to imagine continuity with out covariates and with covariates:
start{equation}
lim_{x to c^-} mathbb{E}[Y_i(0) mid X_i = x] = lim_{x to c^+} mathbb{E}[Y_i(0) mid X_i = x] textual content{(no covariates)}
finish{equation}
start{equation}
lim_{x to c^-} mathbb{E}[Y_i(0) mid X_i = x, Z_i] = lim_{x to c^+} mathbb{E}[Y_i(0) mid X_i = x, Z_i] textual content{(covariates)}
finish{equation}
The place (Z_i) is a vector of covariates, for topic i. Much less mathy, two issues ought to stay unchanged after including covariates:
- The practical type of the working variable, and;
- The (absence of the) soar in therapy task on the cutoff
I didn’t discover out the above myself; Calonico, Cattaneo, Farrell, and Titiunik (2018) did. They developed a proper framework for incorporating covariates into RDD. I’ll go away the main points to the paper. For now, some modelling pointers can hold us going:
- Mannequin covariates linearly in order that the therapy impact stays the identical with and with out covariates, because of a easy and {smooth} partial impact of the covariates;
- Hold the mannequin phrases additive, in order that the therapy impact stays the LATE, and doesn’t develop into conditional on covariates (CATE); and to keep away from including a soar on the cutoff.
- The above implies that there be no interactions with the therapy indicator, nor with the working variable. Doing any of those could break continuity and invalidate our RDD design.
Our goal mannequin could appear like this:
start{equation}
Y_i = alpha + tau D_i + f(X_i – c) + beta^prime Z_i + varepsilon_i
finish{equation}
For letting the covariates work together with the therapy indicator, the kind of mannequin we need to keep away from seems to be like this:
start{equation}
Y_i = alpha + tau D_i + f(X_i – c) + beta^prime (Z_i cdot D_i) + varepsilon_i
finish{equation}
Now, let’s distinguish between two methods of virtually together with covariates:
- Direct inclusion: Add them on to the end result mannequin alongside the therapy and working variable.
- Residualisation: First regress the end result on the covariates, then use the residuals within the RDD.
We’ll use residualisation in our case. It’s an efficient means scale back noise, produces cleaner visualisations, and protects the strategic worth of the information.
The snippet under defines the end result de-noising mannequin and computes the residualised consequence, click_res. The thought is easy: as soon as we strip out the variance defined by the covariates, what stays is a much less noisy model of our consequence variable—at the least in principle. Much less noise means extra accuracy.
In apply, although, the residualisation barely moved the needle this time. We will see that by checking the change in customary deviation:
SD(click_res) / SD(click on) - 1 offers us about -3%, which is small virtually talking.
# denoising clicks
mod_outcome_model Despite the fact that the denoising didn’t have a lot impact, we’re nonetheless in a great spot. The unique consequence variable already has low conditional variance, and patterns across the cutoff are seen to the bare eye, as we are able to see under.
We transfer on to a couple different modelling selections that typically have a much bigger impression: selecting between parametric and non-parametric RDD, the polynomial diploma and the bandwidth parameter (h).
Modelling decisions in RDD
Parametric vs non-parametric RDD
You may surprise why we even have to decide on between parametric and non-parametric RDD. The reply lies in how every method trades off bias and variance in estimating the therapy impact.
Selecting parametric RDD is actually selecting to scale back variance. It assumes a selected practical kind for the connection between the end result and the working variable, (mathbb{E}[Y mid X]), and matches that mannequin throughout all the dataset. The therapy impact is captured as a discrete soar in an in any other case steady operate. The standard kind seems to be like this:
$$Y = beta_0 + beta_1 D + beta_2 X + beta_3 D cdot X + varepsilon$$
Non-parametric RDD, then again, is about lowering bias. It avoids robust assumptions concerning the world relationship between Y and X and as a substitute estimates the end result operate individually on both aspect of the cutoff. This flexibility permits the mannequin to extra precisely seize what’s occurring proper across the threshold. The non-parametric estimator is:
(tau = lim_{x downarrow c} mathbb{E}[Y mid X = x] – lim_{x uparrow c} mathbb{E}[Y mid X = x])
So, which must you select? Actually, it may well really feel arbitrary. And that’s okay. That is the primary in a collection of judgment calls that practitioners typically name the enjoyable a part of RDD. It’s the place modelling turns into as a lot an artwork as it’s a science.
I’ll stroll by means of how I method that alternative. However first, let’s take a look at two key tuning parameters (particularly for non-parametric RDD) that may information our ultimate choice: the polynomial diploma and the bandwidth, h.
Polynomial diploma
The connection between consequence and the working variable can take many types, and capturing its true form is essential for estimating the causal impact precisely. In case you’re fortunate, all the pieces is linear and there’s no want to consider polynomials — In case you’re a realist, then you definitely most likely need to find out how they will serve you within the course of.
In deciding on the fitting polynomial diploma, the objective is to scale back bias, with out inflating the variance of the estimator. So we need to enable for flexibility, however we don’t need to do it greater than needed. Take the examples within the picture under: with an consequence of low sufficient variance, the linear kind naturally invitations the eyes to estimate the end result on the cutoff. However the estimate turns into biased with solely a barely extra complicated kind, if we implement a linear form within the mannequin. Insisting on a linear kind in such a fancy case is like becoming your ft right into a glove: It form of works, however it’s very ugly.
As an alternative, we give the mannequin extra levels of freedom with a higher-degree polynomial, and estimate the anticipated (tau = lim_{x downarrow c} mathbb{E}[Y mid X = x] – lim_{x uparrow c} mathbb{E}[Y mid X = x]), with decrease bias.
The bandwidth parameter: h
Working with polynomials in the best way that’s described above doesn’t come freed from worries. Two issues are required and pose a problem on the identical time:
- we have to get the modelling proper for whole vary, and;
- all the vary must be related for the duty at hand, which is estimating (tau = lim_{x downarrow c} mathbb{E}[Y mid X = x] – lim_{x uparrow c} mathbb{E}[Y mid X = x])
Solely then we scale back bias as supposed; If one in all these two is just not the case, we threat including extra of it.
The factor is that modelling all the vary correctly is tougher than modelling a smaller vary, specifically if the shape is complicated. So, it’s simpler to make errors. Furthermore, all the vary is sort of sure to not be related to estimate the causal impact — the “native” in LATE offers it away. How can we work round this?
Enter the bandwidth parameter, h. The bandwidth parameters aids the mannequin in leveraging information that’s nearer to the cutoff, dropping the world information concept, and bringing it again to the native scope RDD estimates the impact for. It does so by weighting the information by some operate (mathbb{w}(X)) in order that extra weight is given to entries close to the cutoff, and fewer to the entries additional away.
For instance, with h = 10, the mannequin considers the vary of complete size 20; 10 on either side of the cutoff.
The efficient weight depends upon the operate, (mathbb{w}). A bandwidth operate that has a hard-boundary behaviour known as a sq., or uniform, kernel. Consider it as a operate that offers weights 1 when the information is inside bandwidth, and 0 in any other case. The gaussian and triangular kernels are two different regularly used kernels by practitioners. The important thing distinction is that these behave much less abruptly in weighting of the entries, in comparison with the sq. kernel. The picture under visualises the behaviour of the three kernels capabilities.
The whole lot put collectively: non- vs. parametric RDD, polynomial diploma and bandwidth
To me, selecting the ultimate mannequin boils all the way down to the query: what’s the easiest mannequin that does the nice job? Certainly — the precept of Occam’s razor by no means goes out of trend. In practise, this implies:
- Non- vs. Parametric: is the practical kind easy on each side of the cutoff? Then a single match, pooling information from each side will do. In any other case, nonparametric RDD provides the pliability that’s wanted to embrace two totally different dynamics on both aspect of the cutoff.
- Polynomial diploma: when the operate is complicated, I opt-in for increased levels to observe the pattern higher flexibly.
- Bandwidth: if simply picked a excessive polynomial diploma, then I’ll let h be bigger too. In any other case, decrease values for h typically go effectively with decrease levels of polynomials in my expertise*, **.
* This brings us to the commonly accepted suggestion within the literature: hold the polynomial diploma decrease than 3. In most use instances 2 works effectively sufficient. Simply be sure to decide mindfully.
** Additionally, word that h matches particularly effectively within the non-parametric mentality; I see these two decisions as co-dependent.
Again to the itemizing place situation. That is the ultimate mannequin to me:
# modelling the residuals of the end result (de-noised)
mod_rdd Decoding RDD outcomes
Let’s take a look at the mannequin output. The picture under exhibits us the mannequin abstract. In case you’re aware of that, all of it will come all the way down to deciphering the parameters.
The very first thing to have a look at is that handled listings have ~1% level increased likelihood of being clicked, than untreated listings. To place that in perspective, that’s a +20% change if the clicking charge of the management is 5%, and ~ +1% improve if the management is 80%. In the case of sensible significance of this causal impact, these two uplifts are day and night time. I’ll go away this open-ended with a number of inquiries to take residence: when would you and your staff label this impression as a chance to leap on? What different information/solutions do we have to declare this observe worthy of following?
The rest of the parameters don’t actually add a lot to the interpretation of the causal impact. However let’s go over them shortly, nonetheless. The second estimate (x) is that of the slope under cutoff slope; the third one (D x (mathbb(x))) is the extra [negative] factors added to the earlier slope to mirror the slope above the cutoff; Lastly, the intercept is the typical for the items proper under the cutoff. As a result of our consequence variable is residualised, the worth -0.012 is the demeaned consequence; it now not is on the size of the unique consequence.
Completely different decisions, totally different fashions
I’ve put this picture collectively to indicate a group of different attainable fashions, had we made totally different decisions in bandwidth, polynomial diploma, and parametric-versus-not. Though hardly any of those fashions would have put the choice maker on a very unsuitable path on this explicit dataset, every mannequin comes with its bias and variance properties. This does color our confidence of the estimate.
Placebo testing
In any causal inference technique, the identification assumption is all the pieces. One factor is off, and all the evaluation crumbles. We will fake all the pieces is alright, or we put our strategies to the take a look at ourselves (consider me, it’s higher once you break your personal evaluation earlier than it goes on the market)
Placebo testing is one solution to corroborate the outcomes. Placebo testing checks the validity of outcomes by utilizing a setup equivalent to the true one, minus the precise therapy. If we nonetheless see an impact, it indicators a flawed design — continuity can’t be assumed, and causal results can’t be recognized.
Good for us, we’ve a placebo group. The 30-listing web page minimize solely exists on the desktop model of the platform. On cell, infinite scroll makes it one lengthy web page; no pagination, no web page soar. So the impact of “going to the subsequent web page” shouldn’t seem, and it doesn’t.
I don’t suppose we have to do a lot inference. The graph under already tells us all the story: with out pages, going from the thirtieth place to the thirty first is just not totally different from going from another place to the subsequent. Extra importantly, the operate is {smooth} on the cutoff. This discovering provides a substantial amount of credibility to our evaluation by showcasing that continuity holds on this placebo group.
The placebo take a look at is without doubt one of the strongest checks in an RDD. It exams the continuity assumption nearly instantly, by treating the placebo group as a stand-in for the counterfactual.
After all, this depends on a brand new assumption: that the placebo group is legitimate; that it’s a sufficiently good counterfactual. So the take a look at is highly effective provided that that assumption is extra credible than assuming continuity with out proof.
Which implies that we must be open to the chance that there is no such thing as a correct placebo group. How can we stress-test our design then?
No-manipulation and the density continuity take a look at
Fast recap. There are two associated sources of confounding and therefore to violating the continuity assumption:
- direct confounding from a 3rd variable on the cutoff, and
- manipulation of the working variable.
The primary can’t be examined instantly (besides with a placebo take a look at). The second can.
If items can shift their working variable, they self-select into therapy. The comparability stops being truthful: we’re now evaluating manipulators to those that couldn’t or didn’t. That self-selection turns into a confounder, if it additionally impacts the end result.
As an example, college students who didn’t make the minimize for a scholarship, however go on to successfully smooth-talk their establishment into letting them move with a better rating. That silver tongue also can assist them getting higher salaries, and act as confounder after we research the impact of scholarships on future earnings.
So, what are the indicators that we’re in such situation? An unexpectedly excessive variety of items simply above the cutoff, and a dip slightly below (or vice versa). We will see this as one other continuity query, however this time when it comes to the density of the samples.
Whereas we are able to’t take a look at the continuity of the potential outcomes instantly, we are able to take a look at the continuity of the density of the working variable on the cutoff. The McCrary take a look at is the usual software for this, precisely testing:
(H_0: lim_{x to c^-} f(x) = lim_{x to c^+} f(x) quad textual content{(No manipulation)})
(H_A: lim_{x to c^-} f(x) neq lim_{x to c^+} f(x) quad textual content{(Manipulation)})
the place (f(x)) is the density operate of the working variable. If (f(x)) jumps at x = c, it means that items have sorted themselves simply above or under the cutoff — violating the idea that the working variable was not manipulable at that margin.
The internals of this take a look at is one thing for a unique put up, as a result of fortunately we are able to rely rdrobust::rddensity to run this take a look at, off-the-shelf.
require(rddensity)
density_check_obj 
The take a look at exhibits marginal proof of a discontinuity within the density of the working variable (T = 1.77, p = 0.077). Binomial counts are unbalanced throughout the cutoff, suggesting fewer observations slightly below the brink.
Normally, it is a pink flag as it might pose a thread to the continuity assumption. This time nevertheless, we all know that continuity truly holds (see placebo take a look at).
Furthermore, rating is completed by the algorithm: sellers don’t have any means to govern the rank of their listings in any respect. That’s one thing we all know by design.
Therefore, a extra believable clarification is that the discontinuity within the density is pushed by platform-side impression logging (not rating), or my very own filtering within the SQL question (which is elaborate, and lacking values on the filter variables aren’t unusual).
Inference
The outcomes will do that time round. However Calonico, Cattaneo, and Titiunik (2014) spotlight a number of points with OLS RDD estimates like ours. Particularly, about 1) the bias in estimating the anticipated consequence on the cutoff, that now not is actually at the cutoff after we take samples additional away from it, and a pair of) the bandwidth-induced uncertainty that’s ignored of the mannequin (as h is handled as a hyperparameter, not a mannequin parameter).
Their strategies are applied in rdrobust, an R and Stata package deal. I like to recommend utilizing that software program in analyses which are about driving real-life selections.
Evaluation recap
We checked out how an inventory’s spot within the search outcomes impacts how typically it will get clicked. By specializing in the cutoff between the primary and second web page, we discovered a transparent (although modest) causal impact: listings on the prime of web page two bought extra clicks than these caught on the backside of web page one. A placebo take a look at backed this up—on cell, the place there’s infinite scroll and no actual “pages,” the impact disappears. That provides us extra confidence within the consequence. Backside line: the place an inventory exhibits up issues, and prioritising prime positions may increase engagement and create new industrial prospects.
However earlier than we run with it, a few essential caveats.
First, our result’s native—it solely tells us what occurs close to the page-two cutoff. We don’t know if the identical impact holds on the prime of web page one, which most likely indicators much more worth to customers. So this is likely to be a lower-bound estimate.
Second, quantity issues. The primary web page will get much more eyeballs. So even when a prime slot on web page two will get extra clicks per view, a decrease spot on web page one may nonetheless win total.
Conclusion
Regression Discontinuity Design is just not your on a regular basis causal inference technique — it’s a nuanced method finest saved for when the celebs align, and randomisation isn’t doable. Just remember to have a superb grip on the design, and be thorough concerning the core assumptions: attempt to break them, after which strive tougher. When you could have what you want, it’s an extremely satisfying design. I hope this studying serves you effectively the subsequent time you get a chance to use this technique.
It’s nice seeing that you just bought this far into this put up. If you wish to learn extra, it’s attainable; simply not right here. So, I compiled a small listing of assets for you:
Additionally try the reference part under for some deep-reads.
Completely happy to attach on LinkedIn, the place I talk about extra subjects just like the one right here. Additionally, be at liberty to bookmark my private website that’s a lot cosier than right here.
All pictures on this put up are my very own. The dataset that I used is actual, and it’s not publicly accessible. Furthermore, the values extracted from it are anonymised; modified or omitted, to keep away from revealing strategic insights concerning the firm.
References
Calonico, S., Cattaneo, M. D., Farrell, M. H., & Titiunik, R. (2018). Regression Discontinuity Designs Utilizing Covariates. Retrieved from http://arxiv.org/abs/1809.03904v1
Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). Strong nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6), 2295–2326. https://doi.org/10.3982/ECTA11757
