fuzzy regression discontinuity when treatment is not binary

by Lenna Price Published 3 years ago Updated 2 years ago

When the probability of treatment changes at the threshold, but not from 0 to 1, this is known as “fuzzy regression discontinuity (FRD)”. 5, 6

Full Answer

What is “fuzzy regression discontinuity?

What I have described thus far is referred to as a “Sharp Regression Discontinuity” There is also something called a “Fuzzy Regression Discontinuity” This occurs when rules are not strictly enforced Examples Birth date to start school Eligibility for a program has other criterion Whether punishment kicks in (might be an appeal process)

Can regression discontinuity designs be used to re-analyze data from a trial?

This study demonstrates that regression discontinuity designs can be used to re-analyze data from a trial to address concerns about non-compliance in the trial. 11.•.

Does regression discontinuity still work for incumbency?

where Wiis chosen by someone, but ˘iis random and unknown when Wiis chosen Lee shows that regression discontinuity approaches still work in this case Incumbency

What is regression discontinuity analysis in nursing?

Abstract Regression discontinuity (RD) analysis is a rigorous nonexperimental1approach that can be used to estimate program impacts in situations in which candidates are selected for treatment based on whether their value for a numeric rating exceeds a designated threshold or cut-point.

What is fuzzy regression discontinuity?

In the Fuzzy Regression Discontinuity (FRD) design, the probability of receiving the. treatment needs not change from zero to one at the threshold. Instead, the design allows. for a smaller jump in the probability of assignment to the treatment at the threshold: lim.

What is the difference between sharp and fuzzy regression discontinuity?

If deterministic, the regression discontinuity takes a sharp design; if probabilistic, the regression discontinuity takes a fuzzy design. In sharp designs, the probability of treatment changes from 0 to 1 at the cutoff. There are no cross-overs and no no-shows.

What are the assumptions of regression discontinuity?

Required assumptions. Regression discontinuity design requires that all potentially relevant variables besides the treatment variable and outcome variable be continuous at the point where the treatment and outcome discontinuities occur.

What is bandwidth regression discontinuity?

Regression Discontinuity Design (RDD) is a quasi-experimental impact evaluation method used to evaluate programs that have a cutoff point determining who is eligible to participate.

What is regression kink design?

ABSTRACT. A regression kink design (RKD or RK design) can be used to identify casual effects in settings where the regressor of interest is a kinked function of an assignment variable.

What is McCrary test?

– McCrary (2008) provides a formal test for manipulation of the assignment variable in an RD. The idea is that the marginal density of X should be continuous without manipulation and hence we look for discontinuities in the density around the threshold.

What is the distinguishing feature of regression discontinuity designs?

The unique characteristic which sets RD designs apart from other pre-post group designs is the method by which research participants are assigned to conditions. In RD designs, participants are assigned to program or comparison groups solely on the basis of a cutoff score on a pre-program measure.

What is the difference between a sharp Rd design and a fuzzy Rd design?

In addition to these two characterizations, the existing literature typically distinguishes two types of RD designs: the sharp design, in which all subjects receive their assigned treatment or control condition, and the fuzzy design, in which some subjects do not.

What is spatial regression discontinuity?

Spatial regression discontinuity. • Spatial regression discontinuity is a special case that recognizes geographic borders as sharp cutoff points.

What is optimal bandwidth?

We derive the asymptotically optimal bandwidth under squared error loss. This optimal bandwidth depends on unknown functionals of the distribution of the data and we propose simple and consistent estimators for these functionals to obtain a fully data-driven bandwidth algorithm.

How do you do a regression discontinuity in Python?

1:026:3613.2) Regression Discontinuity in Python - YouTubeYouTubeStart of suggested clipEnd of suggested clipRate now let's create the variable. Group if the age is above 21 years it's a treatment. Group ifMoreRate now let's create the variable. Group if the age is above 21 years it's a treatment. Group if the age is below 21 it's a control group let's create the variable. Age that is h cell minus 21.

What is a forcing variable?

Under an RD design, the effect on an intervention can be estimated as the difference in mean outcomes between treatment and comparison group units, adjusting statistically for the relationship between the outcomes and the variable used to assign units to the intervention, typically referred to as the “forcing” or “ ...

How many regression functions are there in FRD?

In the FRD design, there are four regression functions that need to be estimated: the expected outcomegiven the forcing variable, both on the left and right of the cutoff point, and the expected value of thetreatment variable, again on the left and right of the cutoff point. In principle, we can use different binwidthsfor each of the four nonparametric regressions.

Why does the overlap assumption hold in a trivial manner?

This assumption holds in a trivial manner, because conditional on the covariates there is no variation in thetreatment. However, this assumption cannot be exploited directly. The problem is that the second assumptionthat is typically used for matching-type approaches, the overlap assumption which requires that for all valuesof the covariates there are both treated and control units, or

What is the third set of tests?

third set of tests involves estimating jumps at points where there should be no jumps. As in the treatmenteffect literature (e.g.,Imbens, 2004), the approach used here consists of testing for a zero effect in settingswhere it is known that the effect should be 0.

What is regression discontinuity design?

In statistics, econometrics, political science, epidemiology, and related disciplines, a regression discontinuity design (RDD) is a quasi-experimental pretest-posttest design that aims to determine the causal effects of interventions by assigning a cutoff or threshold above or below which an intervention is assigned . By comparing observations lying closely on either side of the threshold, it is possible to estimate the average treatment effect in environments in which randomisation is unfeasible. However, it remains impossible to make true causal inference with this method alone, as it does not automatically reject causal effects by any potential confounding variable. First applied by Donald Thistlethwaite and Donald Campbell to the evaluation of scholarship programs, the RDD has become increasingly popular in recent years. Recent study comparisons of randomised controlled trials (RCTs) and RDDs have empirically demonstrated the internal validity of the design.

Is estimated effect unbiased?

The estimated effects are only unbiased if the functional form of the relationship between the treatment and outcome is correctly modelled. The most popular caveats are non-linear relationships that are mistaken as a discontinuity.

Can an RDD exploit exogenous characteristics of the intervention to elicit causal effects?

Despite the absence of an experimental design, an RDD can exploit exogenous characteristics of the intervention to elicit causal effects. If all students above a given grade — for example 80% — are given the scholarship, it is possible to elicit the local treatment effect by comparing students around the 80% cut-off.

Can you test for validity of a regression?

Testing the validity of the assumptions. It is impossible to definitively test for validity if agents are able to determine their treatment status perfectly. However, some tests can provide evidence that either supports or discounts the validity of the regression discontinuity design.

Is treatment assignment at the threshold good?

Treatment assignment at the threshold can be "as good as random" if there is randomness in the assignment variable and the agents considered (individuals, firms, etc.) cannot perfectly manipulate their treatment status. For example, suppose the treatment is passing an exam, where a grade of 50% is required.

Is RDD unbiased?

When properly implemented and analysed, the RDD yields an unbiased estimate of the local treatment effect. The RDD can be almost as good as a randomised experiment in measuring a treatment effect.

What is regression discontinuity analysis?

Regression discontinuity (RD) analysis is a rigorous nonexperimental1 approach that can be used to estimate program impacts in situations in which candidates are selected for treatment based on whether their value for a numeric rating exceeds a designated threshold or cut-point. Over the last two decades, the regression discontinuity approach has been used to evaluate the impact of a wide variety of social programs (DiNardo and Lee, 2004; Hahn, Todd, and van der Klaauw, 1999; Lemieux and Milligan, 2004; van der Klaauw, 2002; Angrist and Lavy, 1999; Jacob and Lefgren, 2006; McEwan and Shapiro, 2008; Black, Galdo, and Smith, 2007; Gamse, Bloom, Kemple, and Jacob, 2008). Yet, despite the growing popularity of the approach, there is only a limited amount of accessible information to guide researchers in the implementation of an RD design. While the approach is intuitively appealing, the statistical details regarding the implementation of an RD design are more complicated than they might first appear. Most of the guidance that currently exists appears in technical journals that require a high degree of technical sophistication to read. Furthermore, the terminology that is used is not well defined and is often used inconsistently. Finally, while a number of different approaches to the implementation of an RD design are proposed in the literature, they each differ slightly in their details. As such, even researchers with a fairly sophisticated statistical background can find it difficult to access practical guidance for the implementation of an RD design.

What is nonparametric regression?

In the broadest sense, nonparametric regression is a form of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. In other words, instead of estimating the parameters of a specific func-

What is internal validity of RD design?

RD design is considered to be internally valid if a valid causal interference can be made for the sample that is being observed, as opposed to the population to which these findings will be generalized. (Shadish, Cook, and Campbell, 2002). Without establishing the internal validity of the RD design, no causal interpretation can be made. While a valid RD design can identify a treatment effect in much the same way a randomized trial does, in order for an RD design to be valid, a clear discontinuity in the probability of receiving treatment must exist at the cut-point, and candidates’ ratings and the cut-point must be determined independently of each other. This condition can be ensured if the cut-point is determined without knowledge of candidates’ rat-ings and if candidates’ ratings are determined without knowledge of the cut-point.34 If not, the internal validity of the RD design is called into question.

What is the first formal way of choosing the optimal bandwidth?

The first formal way of choosing the optimal bandwidth, which is used widely in the literature, is called the “leave-one-out” cross-validation procedure. Recently, Ludwig and Miller (2005) and Imbens and Lemieux (2008) have proposed a version of the “leave-one-out” cross-validation procedure that is tailored for the RD design. This cross-validation procedure can be carried out as follows (a visual depiction of this procedure is shown in Figure 6):

What is the purpose of graphical analysis in RD?

As mentioned before, the main purpose of the graphical analysis in an RD design is to provide a simple way to visualize the relationship between an outcome variable and a rating variable as well as to indicate the magnitude of the discontinuity at the cut-point. For these purposes, we recommend that researchers follow three steps in selecting a bin width for a graphical RD presentation:

What is RD approach?

The RD approach is appealing from a variety of perspectives. Situations that lend themselves to an RD approach occur frequently in practice, and one can often obtain existing data and use it post hoc to conduct analyses of program impact — at significantly lower cost than conducting a random assignment study. Even in prospective studies, the RD approach can avoid many of the pitfalls of a random assignment design, since it works with the selection process that is already in place for program participation rather than requiring a random selection of participants.3 However, because it is a nonexperimental approach, it must meet a variety of conditions to pro-vide unbiased impact estimates and to approach the rigor of a randomized experiment (for ex-ample, Hahn, Todd, and van der Klaauw, 2001; Shadish, Cook, and Campbell, 2002). Specifi-cally:

What is generalizability in RD?

Another issue to consider when planning and implementing an RD study is generalizability. Much of the current literature notes that even for an internally valid, adequately powered RD study with a correctly specified functional form, the comparison of mean outcomes for partici-pants and nonparticipants at the cut-point only identifies the mean impact of the program locally at the cut-point. In other words, the estimated impact, if valid, only applies to the observations at or close to the cut-point. In the widely hypothesized situation of heterogeneous effects of the program, this local effect might be very different from the effect for observations that are far away from the cut-point.

Overview

Example

The intuition behind the RDD is well illustrated using the evaluation of merit-based scholarships. The main problem with estimating the causal effect of such an intervention is the homogeneity of performance to the assignment of treatment (e.g. scholarship award). Since high-performing students are more likely to be awarded the merit scholarship and continue performing well at the same time, comparing the outcomes of awardees and non-recipients would lead to an upward bi…

Methodology

The two most common approaches to estimation using an RDD are non-parametric and parametric (normally polynomial regression).
The most common non-parametric method used in the RDD context is a local linear regression. This is of the form:
where is the treatment cutoff and is a binary variable equal to one if . Letting be the bandwidth of …

Required assumptions

Regression discontinuity design requires that all potentially relevant variables besides the treatment variable and outcome variable be continuous at the point where the treatment and outcome discontinuities occur. One sufficient, though not necessary, condition is if the treatment assignment is "as good as random" at the threshold for treatment. If this holds, then it guarantees that those who j…

Advantages

• When properly implemented and analysed, the RDD yields an unbiased estimate of the local treatment effect. The RDD can be almost as good as a randomised experiment in measuring a treatment effect.
• RDD, as a quasi-experiment, does not require ex-ante randomisation and circumvents ethical issues of random assignment.

Disadvantages

• The estimated effects are only unbiased if the functional form of the relationship between the treatment and outcome is correctly modelled. The most popular caveats are non-linear relationships that are mistaken as a discontinuity.
• Contamination by other treatments. Suppose another treatment occurs at the same cutoff value of the same assignment variable. In that case, the measured discontinuity in the outcome variable may be partially attributed to this other treat…

Extensions

The identification of causal effects hinges on the crucial assumption that there is indeed a sharp cut-off, around which there is a discontinuity in the probability of assignment from 0 to 1. In reality, however, cutoffs are often not strictly implemented (e.g. exercised discretion for students who just fell short of passing the threshold) and the estimates will hence be biased.
In contrast to the sharp regression discontinuity design, a fuzzy regression discontinuity design …