Regression discontinuity design for the study of health effects of exposures acting early in life

Regression discontinuity design (RDD) is a quasi-experimental approach to study the causal effect of an exposure on later outcomes by exploiting the discontinuity in the exposure probability at an assignment variable cut-off. With the intent of facilitating the use of RDD in the Developmental Origins of Health and Disease (DOHaD) research, we describe the main aspects of the study design and review the studies, assignment variables and exposures that have been investigated to identify short- and long-term health effects of early life exposures. We also provide a brief overview of some of the methodological considerations for the RDD identification using an example of a DOHaD study. An increasing number of studies investigating the effects of early life environmental stressors on health outcomes use RDD, mostly in the context of education, social and welfare policies, healthcare organization and insurance, and clinical management. Age and calendar time are the mostly used assignment variables to study the effects of various early life policies and programs, shock events and guidelines. Maternal and newborn characteristics, such as age, birth weight and gestational age are frequently used assignment variables to study the effects of the type of neonatal care, health insurance, and newborn benefits, while socioeconomic measures have been used to study the effects of social and welfare programs. RDD has advantages, including intuitive interpretation, and transparent and simple graphical representation. It provides valid causal estimates if the assumptions, relatively weak compared to other non-experimental study designs, are met. Its use to study health effects of exposures acting early in life has been limited to studies based on registries and administrative databases, while birth cohort data has not been exploited so far using this design. Local causal effect around the cut-off, difficulty in reaching high statistical power compared to other study designs, and the rarity of settings outside of policy and program evaluations hamper the widespread use of RDD in the DOHaD research. Still, the assignment variables’ cut-offs for exposures applied in previous studies can be used, if appropriate, in other settings and with additional outcomes to address different research questions.


Introduction
Since the first Barker's studies in the mid-1980s, focused on the influence of foetal undernutrition and growth during early periods of development on different short-and long-term health outcomes, 1 The assumption of no unmeasured confounding is probably the main obstacle to causal inference within the context of non-experimental studies, and several analytical and design approaches have been developed and extensively used in the past decades to control for confounding and obtain a potentially unbiased estimate of the treatment/exposure effect. These include, but are not limited to, multivariable regression analysis, exposure and outcome negative control, within-sibling design, and instrumental variable analysis. Natural experiments, such as epidemics and famines, have been historically used to establish causal relationships and to study the causes of health outcomes. 5,6 A series of studies of the Dutch "Hunger Winter" of 1944 on the health effects originating from the prenatal period, [7][8][9] were among the first applications of natural experiments in the context of birth cohort research. These studies exploited random or as-if random treatment or intervention assignments of the population that may stem from various natural or pseudo-natural sources. Methods typical applied in these settings involve instrumental variable analysis, difference-in-differences, interrupted time series, and regression discontinuity. 10 However, the relative rarity of "shock events" or strong exogenous instruments for exposures, the need of more complicated research designs, the involvement of small and more gradual effects, the small proportion of population affected by the exposure and the need of prompt data collection prevent the widespread use of natural experiments in life course epidemiology. Studies exploiting the geographical or temporal variations in medical prescriptions or population-level policies are more Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022 common. These studies use the "quasi-naturally" occurring variation in exposure to identify the effect of an event/exposure on an outcome of interest.
Here, we focus on regression discontinuity design (RDD), a quasi-experimental approach, applied in circumstances where an exogenous source of variation arises from a continuously measured assignment variable with a clearly defined cut-off point above or below which the population is at least partially assigned to a treatment or exposure. The assignment variable creates a discontinuity in the probability of the exposure at the threshold, where the direction and magnitude of the jump is a direct measure of the causal effect of the exposure on the outcome for subjects near the cutpoint. This approach has been widely applied in the context of natural experiments and by the end of the 20 th century the RDD has become popular in the econometrics, educational and social research exploiting the threshold rules often used by educational institutions, public and private insurance schemes, governmental welfare programs and social policies. 11,12 Although in many of these studies the main outcomes of interest were health outcomes, the examined program and policy interventions led to their publications mostly in the economics journals and their findings caused generally less attention in the health and epidemiology literature. [12][13][14] A growing number of birth cohorts and multiple birth cohort consortia are being established to study the effects of early life exposures on later health outcomes. The RDD is little used in this context. We thus describe the RDD, highlighting its underlying assumptions, advantages, limitations, and approaches for testing the assumptions and the validity of the design, and focus on the studies, assignment variables and interventions/exposures that have been investigated in the context of perinatal and pediatric epidemiology.

Regression Discontinuity Design -basic concepts and frameworks
RDD, introduced in 1960s, 15 is a quasi-experimental design that shares similarities with randomized controlled trials, but lacks the completely random assignment to the intervention (intervention, treatment, or exposure, hereafter referred as exposure in general). This type of study design typically implies that a researcher or whoever imposes a certain policy, program, or clinical decision control the assignment to the exposure using an a priori, often administratively, decided criterion (e.g., an eligibility rule, or a clinical decision-making guideline). The exposure assignment in RDD studies is thus based on the cut-off value of a continuously measured variable, Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022 the assignment variable (also referred to in the literature as the "forcing", "rating" or the "running" variable), that creates a discontinuity in the probability of the exposure at the cut-off point. The assignment variable can be any continuous variable measured before the exposure, provided that individuals cannot manipulate the value of this variable to systematically place themselves above or below the cut-off. The stronger is the individuals' inability to control their own value of the assignment variable the more valid is the design to identify the causal effects. As the cut-off value in the assignment variable is typically determined by an administrative decision, clinical guideline, or some "shock event", it is unrelated to the baseline pre-exposure characteristics of the individuals near the cut-off. This implies that individuals just below the known cut-off are on average similar in all observed and unobserved baseline characteristics to those just above the cut-off except for the exposure of interest, i.e., they are exchangeable. The dissimilarity (i.e. lack of exchangeability) between these two groups typically increases far from the cut-off rendering the validity of the design plausible for relatively narrow windows around the assignment variable cut-off. If the assumption of exchangeability holds, any difference in the outcome on the two sides of the cut-off will be caused by the exposure. The magnitude of the discontinuity in the outcome at the cut-off represents thus the average treatment effect around the cut-off point, where exposed and unexposed individuals are most similar.
The RDD design, formalized by a causal diagram -a Direct Acyclic Graph (DAG), 16,17 is represented in Figure 1. The panel A of Figure 1 shows that the assignment variable (X) determines the eligibility status (D) for an exposure / treatment (E) and is a direct cause of an outcome of interest (Y). U denotes measured and unmeasured confounding factors. In a standard causal inference approach the assignment variable represents a confounding variable in the path between E and Y, which can be controlled for, but which conditioning is not enough to block the backdoor confounding path through other measured and unmeasured variables (EUY). In the close window around the cut-off (c), the assignment variable still determines the eligibility status, but it does not directly affect the outcome and is not affected by other observed and unobserved variables. This means that the eligibility becomes independent of other factors and that it can be used as an instrument for the exposure at the cut-off.
The main condition and assumptions of RDD can be therefore summarized as follows: Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022 i) A continuous pre-exposure variable with a clearly defined cut-off value for the exposure assignment ("Assignment rule condition"); ii) No sign of manipulation of the "eligibility criteria" to be below or above the assignment variable cut-off; iii) Individuals close to the cut-off are exchangeable. This assumption implies also that the outcome probability function is continuous at the cut-off in the absence of the exposure.
The power of RDD lays in the fact that it can provide valid causal estimates under relatively weak assumptions compared to other observational study designs, 11,18,19 and that most of these assumptions can be tested empirically.
( Figure 1 here) There are two main conceptual and inference frameworks in RDD that involve slightly different estimation procedures and interpretation of the causal effects: the continuity-based approach, and the local randomization approach. In this paper we focus only on the conceptual commonalities of the two frameworks, while the analytical estimation methods of the two approaches are discussed in detail elsewhere. 20,21 Briefly, the main assumption behind both frameworks is that individuals just below the cut-off are comparable to the individuals just above the cut-off in all relevant characteristics, except for their exposure status. The general logic of measuring a difference in regression lines at the cut-off applies to both frameworks but the main differences are the estimation methods and statistical inference. The continuity-based framework, the most widely used method in practice, is based on the continuity of average potential outcomes near the cut-off, and it typically uses polynomial methods to approximate the regression functions on the two sides of the cut-off (least-squares methods to a polynomial of the observed outcome on the assignment variable). 19,20,[22][23][24] For the estimation it is possible to use either all the observations available around the cut-off (parametric methods) or only observations in a selected window around the cutoff (local or non-parametric methods). According to the local randomization framework, instead, RDD is seen as a randomized experiment near the cut-off, which imposes somewhat stronger underlying assumptions than the continuity-based approach (instead of the continuity of the unknown regression functions at the cut-off, one assumes random assignment in a narrow window around the cut-off with the assignment variable being unrelated to the average potential outcomes). 21,[25][26][27] The estimation is then based on standard methods for random experiments, such as finite-sample Fisher's methods, or on the Neyman's approach with large-sample approximations. 21

Sharp and fuzzy regression discontinuity design
In the RDD, the exposure is determined by the assignment rule either completely (deterministically) or partially (probabilistically). When the assignment rule perfectly determines the exposure (from 0 to 1 at the cut-off), the regression discontinuity takes a sharp design. This means that the exposure assignment and the actual exposure status coincide, i.e., all individuals above the cut-off are assigned to an exposure and are exposed, while all those below the cut-off are assigned to the unexposed group, with no cross-over effects. If the assignment rule affects the probability of exposure creating a discontinuous change at the threshold, without an extreme 0 to 1 jump, regression discontinuity takes a fuzzy design. In this setting, there are exposed and unexposed individuals both above and below the cut-off, but the probability of being exposed jumps discontinuously at the cut-off. 11,18 The sharp and fuzzy RDD are shown with hypothetical examples in Figure 2. Sharp RDD is generally appropriate in scenarios of imposed policy measures and RDD in time (time as the assignment variable). Panel A shows a simulated sharp RDD where, for example, a family poverty index can be used as an assignment variable, which value of at least 30 is used as an eligibility criteria for governmental conditional cash transfer at childbirth. Given that all families above this cut-off receive the benefit, while those below the cut-off do not receive it, there is a sharp 0-to-1 jump at the cut-off. As we expect that families with a poverty index just below 30 are very similar to those just above the cut-off, except from the conditional cash transfer, we could estimate the effect of conditional cash transfer on later child outcomes around this cut-off.
Panel B in Figure 2 depicts a hypothetical example (simulated data) of a fuzzy RDD motivated by the study of Daysal et al. 28 The authors investigated the effect of the obstetrician supervision of deliveries on the short-term infant health outcomes, using a national rule of 37 gestational weeks (259 days) at delivery for obstetrician's instead of midwife's delivery supervision. In this scenario, Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022 the probability of obstetrician supervision does not "jump" from 1 to 0, meaning that not all the deliveries before 37 completed gestational weeks are supervised by an obstetrician and that some deliveries after 37 gestational weeks are however under the care of an obstetrician for reasons other than prematurity, such as complications during delivery and slow delivery progression. Thus, the simulated discontinuity in the probability of the exposure (obstetrician delivery supervision) represents a fuzzy type of the RDD.
( Figure 2 here) The sharp and the fuzzy RDDs are similar in a way that the identification procedure, i.e., the main condition of the continuity at the cut-off, is identical, and the general logic of evaluating the difference in regression lines at the cut-off applies to both. However, the estimation procedure differs between the two designs due to crossovers at the cut-off or non-compliance present in the fuzzy design. In addition to the abovementioned general RDD assumptions, the estimation of the causal effects in the fuzzy RDD requires some additional assumptions, as discussed below in section 4.
Several modifications to the two general RDD settings have been proposed in literature. In particular, kink RDD is the extension of the sharp or fuzzy design where rather than the jump in the exposure, the cut-off determines the change in the first derivative. 29 The main assumptions of the kink RDD are the same of fuzzy and sharp RDD, i.e., the individuals on the either side of the cut-off are similar in the baseline characteristics, but instead of estimating a jump in the intercept, kink RDD estimates a change in slope at the cut-off. RDD with multiple cut-offs in the assignment variable, and RDD with composite assignment rules dependent on multiple assignment variables have been also addressed in literature. [30][31][32] A special case of RDD is a RDD in time where the assignment variable is a calendar time and a known date of intervention, policy change or some other type of shock event is used as a cut-off. 33 This type of RDD is very similar to an interrupted time series or a simple pre-post comparison, with the difference that RDD relies on a bandwidth estimator, and is, thus, useful if an effect in a narrow window around the cut-off is meaningful.

Testing the validity and assumptions of regression discontinuity design
Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022

9
The main assumptions of the RDD and their potential to be tested empirically are summarized in Table 1.

Assignment rule condition
The main condition of RDD is the existence of a continuous pre-exposure variable with a clearly defined cut-off value for the exposure assignment. This implies that the assignment variable cannot be influenced by the exposure and that it temporally precedes exposure. The assignment rule is an administratively imposed or recommended value of the assignment variable that determines the eligibility for a policy or program implementation, the introduction of an invention or treatment, or a guideline-based clinical decision. This condition can be empirically tested by plotting the relationship between the exposure and the assignment variables. A discontinuous change in the probability of the exposure at the assignment variable cut-off ( Figure 2) suggests the possibility of using RDD but is not enough to determine its internal validity. In the case of sharp RDD this discontinuity is defined as the jump in the probability of the exposure from 0 to 1 and is relatively easy to visualize. In fuzzy RDD the observed discontinuities in the exposure at the cut-off are often relatively small, and the smaller the jump in the probability of the exposure is, the weaker the validity of the design is.
In addition to the exposure discontinuity at the cut-off, a graphical presentation of the relationship between the exposure and the assignment variables allows the examination of discontinuities in the exposure at locations other than the cut-off. Any such additional discontinuities threat the validity of RDD as they may indicate other exposures or interventions that could confound the estimate of the causal effect of the exposure of interest. Every RDD analysis must have a clear description of the assignment rule, and the underlying assignment process and alternative hypotheses must be excluded by providing evidence (often only theoretical) that the same cut-off value of the assignment variable is not used to assign the individuals to other exposures that could affect the outcome.
In identifying possible RDD settings it is very important not to mistake the discontinuity with the non-linearity. It is possible, that two local linear regressions fitted on the two sides of the cut-off produce spurious jump in the predicted probability of the exposure when the underlying relationship is non-linear. 34 This is true also for the final step of RDD when looking at the discontinuity in the outcome and will be discussed more in detail below. Another issue to have in mind when plotting the exposure probability as a function of the assignment variable is the width of bins in the assignment variable within which the mean exposure probability is calculated. Too narrow bins allow visualizing the underlying data pattern but can at the same time hide or underestimate the magnitude of the jump. On the other hand, wide bins eliminate noise in the data but hide the underlying data distribution. The impact of the bin width on the graphical presentation of the relationship between the exposure and the assignment variables is shown in Figure 3 using the same data of Figure 2b. To avoid subjective selection of the bin width and potential manipulations in graphical presentations, formal tests and data-driven bin width selection are standard procedures in RDD analyses. 11,35 ( Figure 3 here)

Lack of manipulation in the assignment variable
After examining the presence of discontinuity in the exposure at the cut-off, it is important to test two fundamental assumptions determining the validity of the design. The first assumption of no manipulation in the assignment variable means that the cut-off value is exogenous -unrelated to the individuals' value of the assignment variable, and at the same time individuals' assignment variable values are not determined by the imposed cut-off in the assignment variable. In practice, when there is a benefit in receiving an exposure/treatment, the manipulation in assignment variable occurs when the treatment assignment rule is public knowledge and individuals just barely qualifying for a desired exposure/treatment manage to cross the cut-off, with few individuals remaining just below the threshold (imagine a cut-off for final high-school grade for entering university and individuals with a bit more effort manage to cross it in the last moment). In a setting like this, individuals in a narrow window below and above the cut-off are no longer similar and the internal validity of the RDD is compromised. On the other hand, even if individuals are unable to precisely manipulate the assignment variable, the variation in treatment/exposure near the cutoff may still not be as good as random. 11 This can happen with administrative procedures that nonrandomly affect the position of individuals in the assignment variable near the cut-off rendering the continuity assumption less plausible. 36,37 It is also important to note that the assumption of no precise manipulation in the assignment variable is not an issue in designs where assignment variables are completely exogenous with no possibility of manipulation in either direction, such as RDD in time (e.g., age, calendar year). Thus, one of the most important steps in performing a valid RDD is a complete knowledge of the data generation process underlying the assignment rule and the understanding of the assignment variable's susceptibility to manipulation.
There is a formal test of the control over the assignment variable that complements the knowledge of the exact assignment mechanism. A strategically manipulated position in the assignment variable creates a discontinuity in the distribution of the assignment variable at the cut-off. For example, if a desirable treatment is assigned if individuals are above the cut-off, and individuals can precisely control over the assignment variable, we expect the density of the assignment variable to be close to zero just below the cut-off and positive just above the cut-off. This can be tested both visually using a graphical representation and formally using the McCrary density test. 38 The basic idea behind the McCrary test is that the marginal density of the assignment variable without manipulation should be continuous around the cut-off. This can be visualized graphically with the density plot of the assignment variable, where any discontinuity/jump in the density near the cut-off is suggestive of some degree of manipulation ( Figure 4, the graphs are generated using simulated data and the STATA rddensity command 39 ). The McCrary test is implemented as a Wald test of the null hypothesis that the discontinuity is zero. 38 As pointed out in the original McCrary paper, this test will be informative only when the manipulation of the assignment variable is monotonic, i.e., manipulation shifts individuals in one direction only. Several other modifications and refinements of the assignment variable manipulation test have been proposed in literature and are widely implemented in the existing statistical applications. [39][40][41] ( Figure 4 here)

Exchangeability around the assignment variable cut-off
The exchangeability assumption, also called the continuity assumption, implies that individuals just above and below the cut-off are similar with respect to the distribution of observed and unobserved factors, except from the treatment/exposure, and thus they have the same expected outcome if subject to the same exposure level. 19 This assumption was formalized using the Rubin's potential outcomes framework of causal inference, 42,43 that for each individual imagines a pair of potential outcomes, one for what would occur if the individual were exposed to the treatment/exposure, and one if not exposed. If it were possible to observe both potential outcomes simultaneously, we would be able to estimate the causal effect of the treatment/exposure as the difference between the two potential outcomes at an individual level. As this is possible only in theory, average treatment effects are estimated over populations. The exchangeability/continuity assumption in the RDD setting is, thus, formalized as the continuity of the potential outcomes at the area of the cut-off, which allows us to use the average observed outcomes of individuals just below the cut-off as a counterfactual for those that are just above the cut-off.
As the continuity assumption involves continuous conditional expectation functions of the potential outcomes through the cutoff, and the potential outcomes are unobservable, it is not directly testable. However, there is a way to test its implications. If the continuity assumption holds and individuals immediately below the threshold are a valid counterfactual for those immediately above the threshold, then the distribution of observed baseline characteristics is also expected to have similar distributions in these two groups. In a valid RDD, these distributions tend to be the same as we approach the cut-off (narrower bandwidths around the cut-off). As a result, we can indirectly test the continuity assumption by looking at the distribution of observed baseline covariates that by definition are not influenced by treatment/exposure, and should not be changing at the cut-off. Any discontinuity of the observables at the cut-off indicates a violation in the underlying assumption and calls into question the RDD validity.
In practice, it is advisable to perform both a graphical inspection and a formal testing. The graphical representation is a series of simple plots of the relationship between the baseline covariates not affected by treatment/exposure and the assignment variable. For example, in a hypothetical study of the effect of the type of neonatal care on offspring outcomes where 2500 grams is a birth weight cut-off for intensive neonatal care, we should examine the distribution of the observed maternal baseline characteristics (e.g., age, education level, pre-pregnancy body mass index (BMI), household income) around the 2500 grams of birth weight ( Figure 5, data from the NINFEA birth cohort 44 , which is approved by the local Ethical Committee -approval n. 45, and subsequent amendments). Discontinuity in any of the baseline observables around the cut-off, in our hypothetical example maternal pre-pregnancy BMI, threats the validity of the design.
( Figure 5 here) As there are several approaches to estimation in RDD (see below for details and further readings), to test the balance of covariates at the cut-off one should follow the same visualization, test statistic, and inference procedures used in the main RD analysis for the outcome of interest. For example, nonparametric local polynomial regression-discontinuity estimation techniques could be applied if the estimation is done by the continuity approach, 20,[22][23][24]45 or methods from the classical analysis of experiments and potential outcomes could be used in the case of local randomization approach. 21,[25][26][27] Independently of the RRD conceptual framework, if there are many covariates to test, some discontinuities may be observed by random chance only. In this setting, the multiple testing problem may become an issue with an over-rejection of the null hypotheses. To control the family-wise error rate or false-discovery rate one may apply the Bonferroni or Benjamini-Hochberg multiple testing corrections, respectively. 46,47 As we are generally not interested in a single covariate test, but in an overall joint null hypothesis of the baseline covariates balance, alternative approaches have been proposed to combine the multiple tests into a single test statistic, 48,49 but these are seldom used in the current RDD literature.
Covariate adjustment has been proposed and broadly used with the aim of improving the validity of the RRD design in situations with imbalances in the distribution of the baseline covariates at the cut-off. It has been, however, shown that the validity of the RDD cannot be fixed by the covariate adjustment and that additional strong assumptions are needed to interpret covariateadjusted RDD estimates as causal. 50,51 In general, the adjustment for covariates in RDD can be useful to improve efficiency and statistical power, with the treatment point estimate being stable to the adjustment. 50 Additional assumptions for fuzzy RDD Settings with a sharp RDD are unfortunately not so common in practice. Imperfect exposure/treatment take-up and factors other than the assignment rule that affect the probability of assignment are frequent issues that call for the use of a fuzzy RDD. In a fuzzy RDD, the change in the probability of being treated/exposed at the cut-off is always less than one, and this RDD setting has a close analogy to the noncompliance in randomized controlled trials, where one is interested in both the effect of being assigned to treatment and the effect of receiving treatment on the outcome of interest. The former involves the estimation of the average "intention-to-treat" effect and has the properties of a sharp RDD (perfect compliance to assignment), while the latter requires additional assumptions (imperfect compliance to treatment/exposure take-up). The fuzzy RDD estimates the local average treatment effect (LATE) 52 , which is the average treatment effect for the compliers. The estimation is similar to the Wald's treatment effect in an instrumental variable (IV) design (it can be recovered using a two-stage least-squares), 19 and the additional underlying assumptions are the assumptions of the IV design: exclusion restrictions, monotonicity, and relevance. 52 The exclusion restriction condition requires that any effect of the proposed instrument, in the case of RDD a binary variable indicating the assignment, on the outcome is exclusively through its potential effect on exposure. This assumption is untestable and subject-matter knowledge on the assignment mechanism must be applied to rule out the possibility of any direct effect of the assignment rule on the outcome of interest. This assumption, however, often holds in the RDD settings.
The monotonicity or "no defiers" assumption implies a monotonic relationship between the binary variable indicating the assignment, and exposure. In other words, the assignment rule must not increase the exposure in some people and decrease in others, but in every individual exposure assignment must either leave the exposure status unchanged or change it in the same direction of the assignment. Although this assumption is also untestable, its plausibility should be investigated by common sense and observed data patterns. 53 The relevance assumption requires the assignment binary variable (instrument) to be associated with the exposure, and it is empirically verifiable. In the IV setting with the two-stage least squares estimator, it comprises the first stage estimates, i.e., the predicted value of exposure based on the instrument. The same applies to the fuzzy RDD where it is equally important to verify that a discontinuity exists in the relationship between the binary assignment variable and the exposure variable. As with an IV, the stronger the first stage, that is, the larger the discontinuity at the cutoff is, the more efficient and less prone to bias the estimates from the second stage are. The assumption can be tested graphically ( Figure 2) or formally using, for example, F-statistics. As a rule of thumb, the instrument is declared weak if the F-statistic is less than 10. 54

Other sensitivity analyses and validity checks in RDD
The main RDD analysis consists of visually depicting any discontinuity in the outcome of interest by plotting the distribution of the outcome at the two sides of the assignment variable cut-off ( Figure 6, simulated data). Any jump in the distribution of the outcome at the assignment variable cut-off can be estimated either using local polynomials, 20,[22][23][24] or by other techniques of the local randomization approach. 21,27 To strengthen the validity of RDD several additional checks are strongly advised and are summarized below.

Sensitivity to bandwidth choice
The most frequently used RDD estimation methods are non-parametric or local methods that consider only observations in a selected window around the cut-off. Optimal bandwidth size can be selected either ad hoc using common sense and previous knowledge or by data-driven algorithms. 55 In practice, the bandwidth size depends on data availability around the cut-off.
Ideally, one would like to use a very narrow window around the cut-off, but this comes at the cost of less precise estimates and lower external validity in most of the RDD settings. 56 Sensitivity analysis with alternative specifications of bandwidth size to check the robustness of the estimated effects is a standard in RDD.

Sensitivity to observations near the cut-off
Even if there is no evidence of manipulation in the assignment variable, the observations very near the cut-off are likely to be the most influential when fitting local polynomials. The sensitivity test is also called "donut hole" approach and consists of repeating the analysis on different subsamples where observations are removed in a symmetric distance around the cutoff, starting with those closest to the cut-off and then increasing the distance around cut-off in the attempt to understand the sensitivity of the results to those observations. 57,58

Specification of the response function
Model misspecification is an issue in any analysis, and RDD effects are unbiased only if the functional form of the relationship between the assignment variable and the outcome variable is correctly modeled. One of the frequent issues in RDD is a nonlinear relationship between the assignment variable and the outcome that can be misinterpreted as a discontinuity. Although linear regressions are normally employed in RDD, when the true functional form is unknown it is recommended to include alternative specifications, for example higher order polynomials, in the regression models, 59 and to check the robustness of the effect estimates to multiple specifications. 11 The goodness-of-fit tests can be performed assess model misspecifications. However, it has been shown that higher-order polynomials can lead to overfitting and bias, 60 and it is generally recommended to fit local linear regressions with linear and quadratic forms only, or alternatively to use a local linear nonparametric regression. 19

RDD applications in perinatal and pediatric epidemiology
Three review articles so far evaluated the application of RDD in healthcare research. [12][13][14] The most recent and the only systematic review, 12  the cut-off rules (program eligibility, legislation cut-offs, date of sudden events, and clinical decision-making rules). 12 From the studies provided in the systematic review 12 we identified studies conducted in the context of perinatal, childhood and adolescent epidemiology with the aim to understand the potential of promoting the use of RDD in the existing birth cohort consortia. We also updated the search until August 13, 2022, using PubMed, with a more specific search strategy detailed in Supplementary   Table 1.
Of 325 studies from the previous systematic review, 12 we considered 108 studies potentially relevant for perinatal and childhood epidemiology (Supplementary Table 2). The additional Pubmed search identified 92 studies, of which 60 were considered relevant for the current review (Supplementary Table 3 Interestingly, about 40% of studies published in the past three and a half years focused on clinical settings and shock events, the thematic fields which were quite neglected in the past RDD studies on perinatal and pediatric epidemiology. This is also due to the recent COVID-19 pandemic, which was the focus of several recent RDD studies. [61][62][63][64][65] Despite an increasing number of RDD studies in the field of perinatal and childhood epidemiology, most of the studies were setting-specific evaluating the effect of specific programs, policies, and sudden events, and are, thus, difficult to implement or replicate using data of birth cohorts.
However, there are some previous applications that used data that are typically collected in birth cohorts and may serve as motivating examples for future studies. Table 2 summarizes some of the assignment variables used for identification of discontinuities in perinatal epidemiology that could be replicated or extended in future birth cohort research. (

7.Advantages and limitations of the RDD in perinatal and pediatric epidemiology
The RDD is a study design that gained an increasing popularity in health research due to many advantages over other non-experimental study designs. Its estimates and validity checks can be easily presented using simple graphical representations that improve transparency and integrity of the results. The interpretation of the results is intuitive and straightforward, and its validity and the underlying assumptions are relatively weak compared to other study designs and analytical approaches, and many of them can be tested empirically. The RDD circumvents ethical issues of random assignment, and, if the underlying assumptions are met and credible, it can be almost as good as a randomized experiment in measuring treatment effect. To date, most of the RDD studies have primarily used linear regression models for continuous outcomes, but its application is also generalizable to binary, time-to-event, and count outcomes. 66 Most of the previous perinatal and pediatric epidemiology RDD studies were conducted on the data recovered from registries and administrative databases, which often lack important details and individual-level data. The existing birth cohorts collect a plenty of a very detailed data on parental characteristics, pregnancy outcomes, newborn, infant and later childhood long-term outcomes that have been rarely exploited using RDD. There are several reasons for this, including some of the main limitations of the RDD.
While the RDD has strong internal validity, its external validity is often considered the main caveat. The RD estimate of the treatment effect is limited to the subpopulation of individuals at the discontinuity cut-off and is uninformative about the effect anywhere else. In the sharp RDD the treatment effect is interpreted as the average treatment effect at the cut-off, and it can be generalized and approximated to the average treatment effect only with certain additional assumptions. 67 In the fuzzy RDD the local average treatment effect is estimated at the cut-off, and it is even less generalizable because it is inferred only to the compliers, i.e., the subpopulation of individuals who comply with the assignment rule at the cut-off. In addition to the nature of the estimated effect, the external validity of RDD is further threatened by often setting-specific research questions that cannot be extrapolated and replicated in different populations (e.g., country-specific policies). The utility of RDD also depends on the practical and clinical relevance of the cut-off being studied.
The estimation in RDD implies that we need adequate power for estimating regression line on both sides of the cut-off, i.e., a lot of observations near the cut-off. It has been shown that RDD needs up to three times as many participants as a randomized experiment to have the equivalent power. 68,69 The power will be additionally reduced by the selection of a relatively narrow bandwidth around the cut-off, which is needed to maintain the local randomization, and by small

Conclusions
The regression discontinuity design is a powerful approach for causal inference in perinatal and pediatric epidemiology that has several advantages over other non-experimental study designs, including strong internal validity and a relatively weak and testable assumptions. Its widespread

Simulated data
Regression discontinuity design applied in perinatal epidemiology and birth cohort research | August, 2022 Figure 5. Graphical representation of RD for predetermined covariates for a hypothetical example of birth weight as an assignment variable for an intensive neonatal care. Data from the NINFEA birth cohort. Figure 6. Discontinuity in the outcome (weight at 18 months of age) at the assignment variable cut-off (birth weight). Simulated data.

Supplementary Material
Supplementary  3 Clinical Obstetrician supervision of preterm birth Gestational age Seven-and 28-day mortality, Apgar score Belenkiy (2010) 4 Healthcare/Insurance Health insurance Age Obstetric treatment intensity Garrouste (2011) 5 Healthcare/Insurance Reimbursement eligibility Down syndrome risk score Amniocentesis and fetal health Almond (2011) 6 Healthcare/Insurance Length of hospital stay Clock time Maternal and newborn health De La Mata (2012) 7 Healthcare/Insurance Medicaid Family income Healthcare utilization, health status, obesity, school sickness absence Koch (2013) 8 Healthcare/Insurance Public health insurance for children Family income Healthcare utilization and expenditure Camacho (2013) 9 Healthcare/Insurance Subsidized Regime health insurance for the poor Poverty index Neonatal health (birthweight, Apgar score), prenatal care Palmer (2015) 10 Healthcare/Insurance Public health insurance for preschool children Age Healthcare utilization, expenditure, substitution (crowdout) Koch (2015) 11 Healthcare/Insurance Public health insurance Family income Parents' self-reported health and preventive care usage Han (2016) 12 Healthcare/Insurance Children's Medical Subsidy Program Age Healthcare utilization and expenditure Bhowmick (2016) 13 Healthcare/Insurance Community health worker programme Population Pregnancy and child health outcomes Laughery (2016) 14 Healthcare/Insurance Health Professional Shortage Area designation Number of GPs per 10,000 population Hospitalizations, mortality, prenatal care, neonatal health Lee (2017) 15 Healthcare/Insurance Medicaid plan (fee for service vs managed care) Birthweight Hospital readmission, length of stay, mortality Bernal (2017) 16 Healthcare/Insurance Seguro Integral de Salud (social health insurance) Household Targeting Index (welfare  index) Vaccines, birth control  36 Social and welfare programs Group-based credit programmes for the poor Acres of land owned by household Contraceptive use and fertility   37 Social and welfare programs BabyFirst home visit programme Family Stress Checklist score Family social support, parental mental health, parenting outcomes Ludwig (2007) 38 Social and welfare programs Head Start County-level poverty index Child mortality Urquieta (2009) 39 Social and welfare programs Oportunidades poverty alleviation programme Poverty index Skilled attendance at delivery Alam (2011) 40 Social and welfare programs Female School Stipend Program (conditional cash transfer) District literacy rate Sexual and fertility decisions (early marriage and childbearing Rosero (2011) 41 Social and welfare programs Early childhood programmes (home visits and childcare centres) for poor families) Programme proposal quality score Multiple child health and development measures; maternal stress and depression de Brauw (2011) 42 Social and welfare programs Comunidades Solidarias Rurales Municipal poverty score Prenatal and postnatal care, skilled attendance, birth at health facility Janssens (2011) 43 Social and welfare programs Women's empowerment and health education programme  45 Social and welfare programs Unemployment Subsidy and retraining Welfare index Children's weight, height, BMI, Apgar score Bor (2013) 46 Social and welfare programs Extension of eligibility for Child Support Grant Date of birth Time to first pregnancy from age 14 (teenage pregnancy) Tibone (2013) 47 Social and welfare programs US foreign aid policy change Month and year of conception Abortion rates You (2013) 48 Social and welfare programs Formal microcredit Predicted probability of borrowing microcredit Child malnutrition (BMI, anemia, zinc deficiency) González (2013) 49 Social and welfare programs Universal child benefit Calendar time Incidence of conceptions and abortions Carneiro (2014) 50 Social and welfare programs Head Start Family income Health measures from CNLSY longitudinal survey Sun (2014) 51 Social and welfare programs Increased women's bargaining power following divorce reform Month and year of conception Sex ratio of second children following firstborn girls; birth spacing; child caloric intake; husband's alcohol and cigarette consumption Filmer (2014) 52 Social and welfare programs Scholarships for poor children Dropout risk score Teenage pregnancy Cogneau (2015) 53 Social and welfare programs National boundaries Distance from border Children's height-for age, access to safe water Carranza Barona (2015) 54 Social and welfare programs Bono de Desarrollo Humano (conditional cash transfer) Selben welfare index Exclusive breastfeeding in the first six months of life El-Kogali (2015) 55 Social and welfare programs Community development programme District poverty level Child growth and nutrition Beuchert (2016) 56 Social and welfare programs Maternity leave policy change Calendar time Hospital visits Deepti Thomas (2016) 57 Social and welfare programs National Rural Employment Guarantee Act State development index Child and maternal mortality, vaccinations Sachdeva (2016) 58 Social and welfare programs National rural road construction programme Population Prenatal care and contraception, healthcare supply Cygan-Rehm (2016) 59 Social and welfare programs Parental benefit based on net earnings Calendar time Fertility and birth spacing You (2016) 60 Social and welfare programs Access to microcredit Propensity to borrow from rural microcredit schemes  63 Social and welfare programs Reform of maternity leave legislation Month of birth Long-term sickness Tang (2017) 64 Social and welfare programs Head Start Calendar time Child cognitive development and behaviors Guldi, (2018) 65 Social and welfare programs Supplemental Security Income benefit Birthweight Infant mortality, child motor skill development, parenting behaviors, inequalities Rahman (2018) 66 Social and welfare programs Safe motherhood scheme (conditional cash transfer)  109 Clinical Obstetrician supervision of preterm birth Gestational age Mortality, fetal distress, emergency Csection Hutcheon (2020) 110 Clinical Antenatal corticosteroid administration Gestational age Early child development score Brilli (2020) 111 Clinical Neonatal care for high birthweight newborns Birthweight Neonatal intensive care, use of antibiotics, infant mortality Bommer (2020) 112 Clinical Routine probiotics supplementation Gestational age Anthropometric development, lateonset sepsis among moderately preterm newborns Song (2020) 113 Clinical  137 Social and welfare programs Free access to swimming pools Age Children's participation in swimming Velasco (2020) 138 Social and welfare programs Conditional cash transfer Socioeconomic status index Contraceptive behavior among women of childbearing age Chuard (2020) 139 Social and welfare programs Policy reform on the duration of paid parental leave: the share of mothers who work up to the 32nd week of pregnancy Calendar time Birthweight, gestational length, Apgar scores Brauw (2020) 140 Social and welfare programs Conditional cash transfer Distance to cluster based on municipality level poverty rate and the severe stunting rate among first graders Maternal health service utilization Batyra (2021) 141 Social and welfare programs Changes in minimum-age-at-marriage laws Age at law implementation Teenage marriage Alfaro-Hudak (2022) 142 Social and welfare programs The Supplemental Nutrition Assistance Program Poverty-income ratio Cardiometabolic risk factors in children and adolescents González (2022) 143 Social and welfare programs Universal child benefit (cash transfer) Date of birth Birth weight, still birth, early neonatal death in the first 24 hours, normality of birth, C-section, weeks of gestation. Gao (2022) 144 Social and welfare programs Reproductive health policy Year of birth Income, years of schooling Xu (2022) 145 Social and welfare programs China's New Rural Pension programme Age of elderly Physical and mental health of rural children Hong (2019) 146 Education Universal pre-kindergarten program Date of birth Healthcare utilization and recorded diagnoses (physical health conditions) Courtin (2019) 147 Education Education reform that raised the minimum school leaving age