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 possibly can’t randomise, you resort to observational information and Causal Inference strategies. 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 should you thought it couldn’t get extra thrilling than “vanilla” causal inference, learn on.
Personally, I’ve typically discovered myself in no less than two situations the place “simply doing causal inference” wasn’t simple. 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 area. Usually there isn’t a experimental information to be taught from, nor has there been any change (learn: “exogenous shock”), from the enterprise or product facet, 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 will not be possible—or not even wished. Even emulating a pure experiment is perhaps constrained legally, technically, or commercially (particularly when it’s about pricing), or when interference bias arises within the market.
These conditions open up the area for a “totally different” kind of causal inference. Though the strategy we’ll give attention to right here will not be the one one suited to the job, I’d love so that you can tag alongside on this deep dive into Regression Discontinuity Design (RDD).
On this publish, I’ll provide you with a crisp view of how and why RDD works. Inevitably, it will contain a little bit of math — a pleasing 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 issues 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 leap in!
Define
How and why RDD works
Regression Discontinuity Design exploits cutoffs — thresholds — to get well the impact of a therapy on an end result. Extra exactly, it seems to be for a pointy change within the likelihood of therapy project on a ‘operating’ variable. If therapy project relies upon solely on the operating variable, and the cutoff is unfair, i.e. exogenous, then we are able to deal with the models round it as randomly assigned. The distinction in outcomes simply above and beneath 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 check scores. That the cutoff is 90 is unfair — it might 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 usually are not actually totally different, are they? And those that scored 89.9 versus 90.1 — should you insist?
Making the cutoff might come all the way down to randomness, when it’s only a bout a number of factors. Perhaps the scholar drank an excessive amount of espresso proper earlier than the check — or too little. Perhaps they acquired dangerous information the evening earlier than, had been thrown off by the climate, or anxiousness hit on the worst potential second. It’s this randomness that makes the cutoff so instrumental in RDD.
And not using a cutoff, you don’t have an RDD — only a scatterplot and a dream. However, the cutoff by itself will not be geared up with all it takes to determine 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 approach, then its significance comes fully from the continuity assumption. The concept is an easy, counterfactual one: had there been no therapy, then there would’ve been no impact.
To floor the concept of continuity, let’s leap straight right into a traditional instance from public well being: does authorized alcohol entry enhance mortality?
Think about two worlds the place everybody and the whole lot is identical. Apart from one factor: a regulation that units the minimal authorized consuming age at 18 years (we’re in Europe, people).
On the earth with the regulation (the factual world), we’d count on alcohol consumption to leap proper after age 18. Alcohol-related deaths ought to leap too, if there’s a hyperlink.
Now, take the counterfactual world the place there isn’t a such regulation; there ought to be no such leap. Alcohol consumption and mortality would possible comply with a {smooth} development throughout age teams.
Now, that’s an excellent factor for figuring out the causal impact; the absence of a leap in deaths within the counterfactual world is the crucial situation to interpret a leap within the factual world because the impression of the regulation.
Put merely: if there isn’t a therapy, there shouldn’t be a leap in deaths. If there’s, then one thing apart from our therapy is inflicting it, and the RDD will not be legitimate.
The continuity assumption could 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 end result, say, danger of dying of topic (/mathbb{i}) below no therapy.
Discover that the right-hand facet 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 belief can’t be examined instantly. However, fortunately, we are able to proxy it. We are going to see placebo teams obtain this later within the publish. However first, we begin by figuring out what can break the belief:
- Confounders: one thing apart from the therapy occurs on the cutoff that additionally impacts the 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 regulation on the minimal age to devour alcohol (within the no-law world), however that does confound the impact we’re after, taking place on the identical age — the cutoff, that’s.
- Manipulating the operating variable:
When models can affect their place with regard to the cutoff, it might be that models 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 project 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 operating 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 check, nor by among the extra widespread causal inference strategies 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 operating variable too? These are questions that we deal with subsequent.
RDD and devices
In case you are already aware of instrumental variables (IV), you may even see the similarities: each RDD and IV leverage an exogenous variable that doesn’t trigger the result instantly, however does affect the therapy project, which in flip could affect the result. In IV this can be a third variable Z; in RDD it’s the operating variable that serves as an instrument.
Wait. A 3rd variable; possibly. However an exogenous one? That’s much less clear.
In our instance of alcohol consumption, it isn’t arduous to think about that age — the operating variable — is a confounder. As age will increase, so would possibly tolerance for alcohol, and with it the extent of consumption. That’s a stretch, possibly, however not implausible.
Since therapy (authorized minimal age) relies on age — solely models above 18 are handled — handled and untreated models are inherently totally different. If age additionally influences the result, by way of a mechanism just like the one sketched above, we acquired ourselves an apex confounder.
Nonetheless, the operating variable performs a key function. To know why, we have to have a look at how RDD and devices leverage the frontdoor criterion to determine causal results.
Backdoor vs. frontdoor
Maybe virtually instinctively, one could reply with controlling for the operating variable; that’s what stratification taught us. The operating variable is confounder, so we embody it in our regression, and shut the backdoor. However doing so would trigger some hassle.
Keep in mind, therapy project relies on the operating variable so that everybody above the cutoff is handled with all certainty, and actually not beneath it. So, if we management for the operating variable, we run into two very associated issues:
- Violation of the Positivity assumption: this assumption says that handled models ought to have a non-zero likelihood to obtain the alternative therapy, and vice versa. Intuitively, conditioning on the operating 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 operating variable, therapy is both all the time 1 or all the time 0. So, there’s no variation in therapy conditional on the operating variable to help such a query.
- Excellent collinearity on the cutoff: in estimating the therapy impact, the mannequin has no technique to separate the impact of crossing the cutoff from the impact of being at a selected worth of X. The outcome? 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 operating variable doesn’t make the operating variable the exogenous instrument that we’re after. As a substitute, the operating variable turns into exogenous by pushing it to the restrict—fairly actually. There the place the operating variable approaches the cutoff from both facet, the models are the identical with respect to the operating variable. But, falling simply above or beneath makes the distinction as for getting handled or not. This makes the operating variable a legitimate instrument, if therapy project 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 operating 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 operating variable acts as a confounder, working in opposition to the RDD as an alternative 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 can be 18 years and 1 month previous, on common. If something, a month or two distinction in age will not be 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 models 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 operating variable in figuring out the causal impact of a therapy on the result. Specifically, by letting the leap within the end result variable be fully 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 way of the sensible setup of a real-world RDD: we determine the important thing ideas; the operating variable and cutoff, therapy, end result, covariates, and at last, we estimate the RDD after discussing some essential modelling decisions, and finish the part with a placebo check.
RDD in Motion: Search Rating and itemizing efficiency Instance
In e-commerce and on-line marketplaces, the start line of the client expertise is trying to find an inventory. Consider the customer typing “Nikon F3 analogue digicam” within the search bar. Upon finishing up this motion, algorithms frantically type by way of the stock on the lookout for the perfect matching listings to populate the search outcomes web page.
Time and a focus are two scarce sources. So, it’s within the curiosity of everybody concerned — the client, the vendor and the platform — to order probably the most outstanding positions on the web page for the matches with the best anticipated likelihood to turn into profitable trades.
Moreover, place results in client behaviour recommend that customers infer greater 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 offered?
Speculation:
If an inventory is ranked greater on the search outcomes web page, then it’ll have a better likelihood of being offered, 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 aren’t 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 impact, anchoring bias and the useful resource principle of consideration. These are properly 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 manner. Personally, I additionally get excited when I’ve to brush up on some psychology.
However I’ve discovered by way of and thru {that a} principle is actually secondary to any initiative in my business (tech). Apart from a analysis crew and challenge, arguably. And it’s honest to say it helps us keep on-purpose: what we’re doing is to carry enterprise ahead, not mom science.
Figuring out the reply has actual enterprise worth. Product and business groups might use it to design new paid options that assist sellers get their listings on greater positions — a win for each the enterprise and the consumer. It might additionally make clear the worth of on-site actual property like banner positions and advert slots, serving to drive progress in B2B promoting.
The query is about incrementality: would’ve itemizing (mathbb{j}) been offered, had it been ranked 1st on the outcomes web page, as an alternative of fifteenth. So, we need to make a causal assertion. That’s arduous for no less than two causes:
- A/B testing comes with a worth, and;
- there are confounders we have to cope with if we resort to observational strategies.
Let’s broaden on that.
The price of A/B testing
One experiment design might randomise the fetched listings throughout the web page slots, unbiased of the itemizing relevance. Breaking the inherent hyperlink between relevance and place, we might be taught the impact of place on itemizing efficiency. It’s an attention-grabbing concept — however a expensive one.
Whereas it’s an inexpensive design for statistical inference, this setup is sort of horrible for the consumer and enterprise. The consumer might need discovered what they wanted—possibly even made a purchase order. However as an alternative, possibly half of the stock they’d have seen was remotely an excellent match due to our experiment. This suboptimal consumer expertise possible hurts engagement in each the quick and long run — particularly for brand spanking new customers who’re nonetheless to see what worth the platform holds for them.
Can we consider a technique to mitigate this loss? Nonetheless dedicated to A/B testing, one might expose a smaller set of customers to the experiment. Whereas it’ll scale down the implications, it might additionally stand in the best way of reaching ample statistical energy by reducing the pattern measurement. Furthermore, even small audiences could be liable for substantial income for some firms nonetheless — these with tens of millions of customers. So, reducing the uncovered viewers will not be a silver bullet both.
Naturally, the best way to go is to go away the platform and its customers undisturbed — and nonetheless discover a technique to reply the query at hand. Causal inference is the best 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 an excellent day; it’s their high quality, relevance, and the sellers’ status 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 in addition the likelihood that the itemizing will get clicked, a proxy for efficiency.
In different phrases, W impacts each our therapy (place) and end result (click on), serving to itself with the standing of confounder.
Subsequently, 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. Typically they present up while you least want them, and generally you get fortunate while you want them most — like in the present day.
It seems to be like we are able to use the web page cutoff to determine the causal impression of place on clicks-through price.
Abrupt cutoff in search outcomes pagination
Let’s unpack the itemizing advice 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 consumer historical past and itemizing high quality proxies to foretell what the consumer is most definitely to click on. - Rank listings by rating
Greater scores get greater ranks. Enterprise guidelines combine in adverts and business content material with natural outcomes. - Populate pages
Listings are slotted by absolute relevance rating. A outcomes web page ends on the okayth itemizing, so the okay+1th itemizing seems on the prime of the subsequent web page. That is goes to be essential to our design. - Impressions and consumer interplay
Customers see the leads to 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 strictly our design? Subsequent, we stroll by way of the reasoning and identification of the important thing substances of our design.
The operating variable
In our setup, the operating variable is the relevance rating (s_j) for itemizing j. This rating is a steady, advanced operate of each consumer and itemizing properties:
$$s_j = f(u_i, l_j)$$
The itemizing’s rank (r_j) is solely 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—akin to becoming fashions, making native comparisons, or figuring out cutoff factors—realizing an inventory’s rank conveys almost the identical data as realizing its underlying relevance rating, and vice versa.
Particulars: Relevance rating vs. rank
The relevance rating (s_j) displays how properly an inventory matches a selected consumer’s question, given parameters like location, worth vary, and different filters. However this rating is relative—it solely has that means inside the context of the listings returned for that specific 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 outcome units. That makes them equal by way of 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 probably the most related listings seem on the prime, however right here an inventory of merely average relevance leads to a primary slot —albeit on the second web page — purely as a result of arbitrary place of the web page break. Formally, the therapy project (D_j) goes like:
$$D_j = start{circumstances} 1 & textual content{if } r_j > 30 0 & textual content{in any other case} finish{circumstances}$$
(Observe on world rank: Rank 31 isn’t simply the primary itemizing on web page two; it’s nonetheless the thirty first itemizing general)
The power of this setup lies in what occurs close to the cutoff: an inventory ranked 30 could also be almost equivalent in relevance to at least one 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 result: Impression-to-click
Lastly, we operationalise the result of curiosity because the click-though price from impressions to clicks. Keep in mind that all listings are ‘impressed’ when when the web page is populated. The press is the binary indicator of the specified consumer behaviour.
In abstract, that is our setup:
- Final result: impression-to-click conversion
- Therapy: Touchdown on the primary vs. second web page
- Working variable: itemizing rank; web page cutoff at 30
Subsequent we stroll by way of the 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 way to embody 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 considered one of Adevinta’s (ex-eBay Classifieds Group) marketplaces. It’s actual information, which makes the entire train really feel grounded. That stated, values and relationships are censored and scrambled the place crucial 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 consumer noticed each the primary and second web page had 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 focus on 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 way to embody them to extend accuracy?
The operating variable, the cutoff, and the continuity assumption, provide you with all it’s worthwhile to determine the causal impact. However together with covariates can sharpen the estimator by decreasing variance — if executed proper. And, oh is it simple to do it mistaken.
The best factor to “break” in regards to the RDD design, is the continuity assumption. Concurrently, that’s the final factor we need to break (I already rambled lengthy sufficient about this).
Subsequently, the primary quest in including covariates is to it in such approach that we cut back variance, whereas maintaining the continuity assumption intact. One technique 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 operating variable, and;
- The (absence of the) leap in therapy project 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;
- Preserve the mannequin phrases additive, in order that the therapy impact stays the LATE, and doesn’t turn into conditional on covariates (CATE); and to keep away from including a leap on the cutoff.
- The above implies that there be no interactions with the therapy indicator, nor with the operating variable. Doing any of those could break continuity and invalidate our RDD design.
Our goal mannequin could seem 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 form 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 result mannequin alongside the therapy and operating variable.
- Residualisation: First regress the result on the covariates, then use the residuals within the RDD.
We’ll use residualisation in our case. It’s an efficient approach cut back noise, produces cleaner visualisations, and protects the strategic worth of the information.
The snippet beneath defines the result de-noising mannequin and computes the residualised end result, click_res. The concept is easy: as soon as we strip out the variance defined by the covariates, what stays is a much less noisy model of our end result variable—no less than in principle. Much less noise means extra accuracy.
In apply, although, the residualisation barely moved the needle this time. We are able to see that by checking the change in commonplace deviation:
SD(click_res) / SD(click on) - 1 offers us about -3%, which is small virtually talking.
# denoising clicks
mod_outcome_model <- lm(click on ~ l1 + natural + boosted,
information = df_listing_level)
df_listing_level$click_res <- residuals(mod_outcome_model)
# the impression on variance is proscribed: ~ -3%
sd(df_listing_level$click_res) / sd(df_listing_level$click on) - 1Although the denoising didn’t have a lot impact, we’re nonetheless in a great spot. The unique end result variable already has low conditional variance, and patterns across the cutoff are seen to the bare eye, as we are able to see beneath.
We transfer on to a couple different modelling selections that usually have an even 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 would possibly 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 cut back variance. It assumes a selected practical type for the connection between the result and the operating variable, (mathbb{E}[Y mid X]), and matches that mannequin throughout the whole dataset. The therapy impact is captured as a discrete leap in an in any other case steady operate. The standard type seems to be like this:
$$Y = beta_0 + beta_1 D + beta_2 X + beta_3 D cdot X + varepsilon$$
Non-parametric RDD, alternatively, is about decreasing bias. It avoids robust assumptions in regards to the world relationship between Y and X and as an alternative estimates the result operate individually on both facet of the cutoff. This flexibility permits the mannequin to extra precisely seize what’s taking place 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 might 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 way of how I method that alternative. However first, let’s have a look at two key tuning parameters (particularly for non-parametric RDD) that may information our remaining resolution: the polynomial diploma and the bandwidth, h.
Polynomial diploma
The connection between end result and the operating variable can take many kinds, and capturing its true form is essential for estimating the causal impact precisely. Should you’re fortunate, the whole lot is linear and there’s no want to consider polynomials — Should you’re a realist, then you definitely in all probability need to learn the way they will serve you within the course of.
In deciding on the best polynomial diploma, the aim is to cut back bias, with out inflating the variance of the estimator. So we need to permit for flexibility, however we don’t need to do it greater than crucial. Take the examples within the picture beneath: with an end result of low sufficient variance, the linear type naturally invitations the eyes to estimate the result on the cutoff. However the estimate turns into biased with solely a barely extra advanced type, if we implement a linear form within the mannequin. Insisting on a linear type in such a fancy case is like becoming your toes right into a glove: It sort of works, but it surely’s very ugly.
As a substitute, 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 total vary, and;
- the whole vary ought to 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 cut back bias as supposed; If considered one of these two will not be the case, we danger including extra of it.
The factor is that modelling the whole vary correctly is harder than modelling a smaller vary, specifically if the shape is advanced. So, it’s simpler to make errors. Furthermore, the whole 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 both sides of the cutoff.
The efficient weight relies on the operate, (mathbb{w}). A bandwidth operate that has a hard-boundary behaviour is named a sq., or uniform, kernel. Consider it as a operate that provides 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 beneath visualises the behaviour of the three kernels capabilities.
Every part 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 type easy on either side of the cutoff? Then a single match, pooling information from either side will do. In any other case, nonparametric RDD provides the flexibleness that’s wanted to embrace two totally different dynamics on both facet of the cutoff.
- Polynomial diploma: when the operate is advanced, I opt-in for greater levels to comply with the development 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 properly with decrease levels of polynomials in my expertise*, **.
* This brings us to the commonly accepted advice within the literature: hold the polynomial diploma decrease than 3. In most use circumstances 2 works properly sufficient. Simply be sure to decide mindfully.
** Additionally, word that h matches particularly properly 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 result (de-noised)
mod_rdd <- lm(click_res ~ D + ad_position_idx,
weight = triangular_kernel(x = ad_position_idx, c = 0, h = 10), # that is h
information = df_listing_level)Decoding RDD outcomes
Let’s have a look at the mannequin output. The picture beneath exhibits us the mannequin abstract. Should you’re aware of that, all of it will come all the way down to decoding the parameters.
The very first thing to take a look at is that handled listings have ~1% level greater likelihood of being clicked, than untreated listings. To place that in perspective, that’s a +20% change if the clicking price of the management is 5%, and ~ +1% enhance if the management is 80%. On the subject of sensible significance of this causal impact, these two uplifts are day and evening. I’ll go away this open-ended with a number of inquiries to take house: when would you and your crew label this impression as a chance to leap on? What different information/solutions do we have to declare this monitor 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 rapidly, nonetheless. The second estimate (x) is that of the slope beneath cutoff slope; the third one (D x (mathbb(x))) is the extra [negative] factors added to the earlier slope to replicate the slope above the cutoff; Lastly, the intercept is the typical for the models proper beneath the cutoff. As a result of our end result variable is residualised, the worth -0.012 is the demeaned end result; it not is on the dimensions of the unique end result.
Completely different decisions, totally different fashions
I’ve put this picture collectively to point out a group of different potential 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 mistaken path on this specific dataset, every mannequin comes with its bias and variance properties. This does color our confidence of the estimate.
Placebo testing
In any causal inference methodology, the identification assumption is the whole lot. One factor is off, and the whole evaluation crumbles. We are able to faux the whole lot is alright, or we put our strategies to the check ourselves (imagine me, it’s higher while you break your individual evaluation earlier than it goes on the market)
Placebo testing is one technique to corroborate the outcomes. Placebo testing checks the validity of outcomes through the use of 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 have now a placebo group. The 30-listing web page reduce solely exists on the desktop model of the platform. On cell, infinite scroll makes it one lengthy web page; no pagination, no web page leap. So the impact of “going to the following web page” shouldn’t seem, and it doesn’t.
I don’t suppose we have to do a lot inference. The graph beneath already tells us the whole story: with out pages, going from the thirtieth place to the thirty first will not be totally different from going from some other place to the following. Extra importantly, the operate is {smooth} on the cutoff. This discovering provides quite a lot of credibility to our evaluation by showcasing that continuity holds on this placebo group.
The placebo check is likely one of the strongest checks in an RDD. It assessments the continuity assumption virtually 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 check is highly effective provided that that assumption is extra credible than assuming continuity with out proof.
Which implies that we have to be open to the chance that there isn’t a correct placebo group. How can we stress-test our design then?
No-manipulation and the density continuity check
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 operating variable.
The primary can’t be examined instantly (besides with a placebo check). The second can.
If models can shift their operating variable, they self-select into therapy. The comparability stops being honest: 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 result.
As an example, college students who didn’t make the reduce for a scholarship, however go on to successfully smooth-talk their establishment into letting them cross with a better rating. That silver tongue may assist them getting higher salaries, and act as confounder after we examine the impact of scholarships on future revenue.
So, what are the indicators that we’re in such situation? An unexpectedly excessive variety of models simply above the cutoff, and a dip just under (or vice versa). We are able to see this as one other continuity query, however this time by way of the density of the samples.
Whereas we are able to’t check the continuity of the potential outcomes instantly, we are able to check the continuity of the density of the operating variable on the cutoff. The McCrary check is the usual instrument 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 operating variable. If (f(x)) jumps at x = c, it means that models have sorted themselves simply above or beneath the cutoff — violating the belief that the operating variable was not manipulable at that margin.
The internals of this check is one thing for a distinct publish, as a result of fortunately we are able to rely rdrobust::rddensity to run this check, off-the-shelf.
require(rddensity)
density_check_obj <- rddensity(X = df_listing_level$ad_position_idx,
c = 0)
abstract(density_check_obj)
# for the plot beneath
rdplotdensity(density_check_obj, X = df_listing_level$ad_position_idx)
The check exhibits marginal proof of a discontinuity within the density of the operating variable (T = 1.77, p = 0.077). Binomial counts are unbalanced throughout the cutoff, suggesting fewer observations just under the brink.
Often, this can be a crimson flag as it might pose a thread to the continuity assumption. This time nonetheless, we all know that continuity really holds (see placebo check).
Furthermore, rating is completed by the algorithm: sellers haven’t 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 usually are not 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 end result on the cutoff, that not is actually at the cutoff after we take samples additional away from it, and a pair of) the bandwidth-induced uncertainty that’s overlooked 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 can be 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 acquired extra clicks than these caught on the backside of web page one. A placebo check backed this up—on cell, the place there’s infinite scroll and no actual “pages,” the impact disappears. That offers us extra confidence within the outcome. Backside line: the place an inventory exhibits up issues, and prioritising prime positions might enhance engagement and create new business 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 in all probability indicators much more worth to customers. So this is perhaps 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 would possibly nonetheless win general.
Conclusion
Regression Discontinuity Design will not be your on a regular basis causal inference methodology — it’s a nuanced method finest saved for when the celebs align, and randomisation isn’t doable. Just be sure you have an excellent grip on the design, and be thorough in regards to the core assumptions: attempt to break them, after which strive tougher. When you have got what you want, it’s an extremely satisfying design. I hope this studying serves you properly the following time you get a chance to use this methodology.
It’s nice seeing that you just acquired this far into this publish. If you wish to learn extra, it’s potential; simply not right here. So, I compiled a small record of sources for you:
Additionally take a look at the reference part beneath for some deep-reads.
Pleased to attach on LinkedIn, the place I focus on extra matters just like the one right here. Additionally, be happy to bookmark my private web site that’s a lot cosier than right here.
All photographs on this publish are my very own. The dataset that I used is actual, and it isn’t publicly out there. Furthermore, the values extracted from it are anonymised; modified or omitted, to keep away from revealing strategic insights in regards to 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
