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. 2026 May 17;67(10):1649–1663. doi: 10.1111/jcpp.70166

Examining bidirectional associations between overprotective parenting and adolescent problem behaviors: a multiverse analysis

Nele Flamant 1,✉, Bart Soenens 1, Henning Tiemeier 2, Joachim Waterschoot 1, Peter Prinzie 1
PMCID: PMC13535752  PMID: 42144234

Abstract

Background

Research on overprotective parenting in adolescence has predominantly used cross‐sectional designs, limiting attention to possible bidirectional associations between such parenting and adolescents' problem behaviors. The few longitudinal studies available have yielded mixed findings regarding the direction of effects between overprotective parenting and adolescent outcomes, which may stem from substantive and design‐related choices (e.g., type of informant). The present study systematically investigated the potentially bidirectional interplay between overprotective parenting and adolescents' functioning over time using a multiverse analysis.

Methods

We used two‐wave repeated‐measures data with a two‐year interval from the Flemish Study on Parenting, Personality, and Development (FSPPD), including 588 adolescents, both parents, and their teachers. Adolescents and their parents reported on parental overprotection. Moreover, adolescents, their parents, and teachers rated adolescents' externalizing and internalizing problem behaviors. Latent change models were estimated within a multiverse framework.

Results

The results provide stronger evidence that adolescents with more overprotective parents show, on average, a greater increase in problem behavior than adolescents with less overprotective parents, compared to evidence for the opposite direction of effects.

Conclusions

Overall, these findings highlight the potential of a multiverse analysis and mainly support a directional effect of overprotective parenting on more problematic adolescent functioning within the two‐year time interval studied.

Keywords: Overprotective parenting, problem behavior, adolescence, multiverse analysis, bidirectional associations

Introduction

Overprotective parenting has been a prominent topic in both research and the public discourse on contemporary parenting. Popular media frequently highlight the growing prevalence of overprotective parenting, especially in high‐income Western societies, and its potential role in hindering adolescents' psychosocial adjustment (Glass & Tabatsky, 2014; Haidt, 2024; Lukianoff & Haidt, 2018). Consistent with these concerns about the potentially detrimental role of overprotective parenting in youth development, numerous empirical studies have shown significant positive associations between overprotective parenting and various indicators of adolescents' maladjustment (Zhang & Ji, 2023). However, this research has relied almost exclusively on cross‐sectional designs (de Roo, Veenstra, & Kretschmer, 2022; Zhang & Ji, 2023), thereby not allowing conclusions to be drawn about the direction of effects in the associations obtained. Moreover, the few longitudinal studies available yielded mixed findings regarding the direction of associations between overprotective parenting and adolescent outcomes (e.g., Allmann, Klein, & Kopala‐Sibley, 2022; Arslan, Lucassen, Keijsers, & Stevens, 2023). These inconsistencies may stem from a range of substantive or methodological choices, also known as researchers' degrees of freedom (e.g., the type of informant to measure overprotection or outcomes).

To clarify these inconsistencies, a more systematic examination of the potentially bidirectional relations between overprotective parenting and adolescent maladjustment is needed. Instead of making several a priori decisions about the outcomes and study design choices, the current study adopts a multiverse approach (Steegen, Tuerlinckx, Gelman, & Vanpaemel, 2016). This approach enumerates a pre‐specified set of defensible methodological and design decisions and evaluates the resulting set of specifications when studying associations between variables. The distribution of effects across these specifications is then summarized, together with robustness metrics, to characterize sensitivity rather than rely on a single estimate. Unlike traditional longitudinal studies that depend on a set of a priori decisions, a multiverse analysis allows researchers to examine which study‐specific choices influence the strength and direction of these associations. In this study, we applied a multiverse approach to study associations between overprotective parenting and adolescent problem behaviors, utilizing a unique and rich two‐wave repeated‐measures dataset. This dataset includes measures of both internalizing problems (e.g., anxiety and depressive symptoms) and externalizing problems (e.g., aggression and delinquent behaviors), assessed using the ASEBA scales (Achenbach, 1991), as well as reports on both parenting and adolescent problem behavior from multiple informants (i.e., parents, adolescents, and teachers) (Prinzie et al., 2003).

Associations between overprotective parenting and adolescent problem behaviors

Overprotective parenting is characteristic of parents who are excessively involved in their adolescents' lives, providing a level of protection that is developmentally inappropriate (Thomasgard, Metz, Edelbrock, & Shonkoff, 1995). Specific overprotective practices include warning adolescents continuously about potential dangers, invading adolescents' privacy, infantilizing the adolescent, and premature problem‐solving (Chevrier, Soenens, Zimmermann, Skhirtladze, & Van Petegem, 2023; Kins & Soenens, 2013). According to Self‐Determination Theory (Ryan & Deci, 2017), overprotective parenting is detrimental because it would undermine adolescents' basic psychological needs for autonomy, relatedness, and competence (Schiffrin et al., 2014, 2019; Van Petegem, Antonietti, Eira Nunes, Kins, & Soenens, 2020). Additionally, overprotective parenting is often said to be harmful because it would hinder adolescents' development of resilience, emotion regulation, and self‐reliance in coping with stress and decision making (Mathijs, Mouton, Zimmermann, & Van Petegem, 2023; Ungar, 2009).

Recent reviews and meta‐analyses have shown that overprotective parenting and related parenting constructs (such as helicopter parenting, which also involves excessive parental involvement and premature problem solving) are indeed related positively to both internalizing and externalizing problem behavior (de Roo et al., 2022; Ryan et al., 2026; Zhang & Ji, 2023). However, this work consistently highlights the predominance of cross‐sectional studies as a limitation of this literature, and the authors urgently call for more longitudinal research. Indeed, building on long‐standing theories that parent–child dynamics are reciprocal in nature (Bell, 1968; Sameroff, 1975), it is equally plausible that child maladjustment precedes and contributes to parental overprotection. For instance, Hudson and Rapee (2004) proposed that parents of anxious children are more likely to engage in overinvolved parenting as a means of reducing and preventing child distress, an idea that was supported in an observational study involving a speech preparation task (Hudson, Doyle, & Gar, 2009). In addition, abundant support has been found for bidirectional relations between adolescent problem behavior and other forms of intrusive parenting (e.g., psychological control), indicating that parents tend to respond to adolescents' problem behavior by increasing parental overcontrol (Pinquart, 2017a, 2017b).

To date, only a limited number of studies have examined associations between parental overprotection and adolescent problem behavior using longitudinal designs (e.g., Janssens, Oldehinkel, & Rosmalen, 2009; Jian, Qin, Gao, & Xu, 2025; Johnco et al., 2021). Overall, the findings from these longitudinal studies are inconclusive. For example, Janssens et al. (2009) found in a two‐wave study that perceived parental overprotection related to increases in adolescent‐reported somatic complaints over a two‐and‐a‐half‐year period. Similarly, Jian et al. (2025), using a three‐wave longitudinal design, found that child‐reported maternal overprotection predicted parent‐reported child anxiety. However, results varied depending on the informant: parent‐reported child anxiety predicted subsequent increases in parent‐reported overprotection, both for mothers and fathers, whereas reciprocal associations were observed between child‐reported anxiety and child‐reported maternal overprotection. In contrast, other longitudinal studies have failed to find evidence for prospective associations. For instance, Johnco et al. (2021) followed pre‐ and early adolescents annually over 3 years and examined bidirectional associations between parental overprotection and internalizing problems (anxiety and depression), but found no evidence for longitudinal effects in either direction, neither for parent‐reported nor child‐reported data. Taken together, these studies illustrate that findings from longitudinal studies are inconclusive. Most importantly, and inconsistent with the widespread assumption that overprotective parenting is harmful to adolescents' development, there is no systematic evidence for a negative impact of overprotective parenting on changes in adolescent development. Because the available longitudinal studies differ in many aspects, including the type of problem behavior examined and the informants used to measure parenting and adolescent outcomes, it is currently unclear whether the inconsistent findings are primarily due to data‐analytical choices, the specific variables examined, or other methodological decisions. Therefore, a multiverse analysis is indicated, as it provides a more systematic estimation of the potentially bidirectional associations between overprotective parenting and adolescents' problem behavior.

The merits of a multiverse analysis

When researchers collect data, they face numerous decisions about how to process the data. These decisions, often termed researchers' degrees of freedom (Simmons, Nelson, & Simonsohn, 2011), involve various substantive and methodological choices, including data cleaning methods, variable selection, alternative operationalizations of variables, and statistical approaches. Traditionally, research adheres to a single analytical pathway. However, when no consensus exists on which options yield the most accurate results, researchers may (intentionally or unintentionally) select analyses that produce the most favorable outcomes, potentially introducing bias and non‐replicable results. By contrast, a multiverse analysis systematically explores a vast number of analytical pathways, providing a more comprehensive understanding of the data and evaluating the consistency of research findings across different choices (Steegen et al., 2016). Some scholars argue that a multiverse analysis should be preferred over preregistration, as it allows researchers to assess the robustness of their results across various substantive, methodological, and analytical decisions (Oberauer & Lewandowsky, 2019).

The combination of inconclusive findings on the relationship between parental overprotection and adolescents' problem behavior, along with the numerous uncertainties in data processing in this field, underscores the value of a multiverse analysis. Using a multiverse approach, the present study focuses on five substantive and methodological decisions that may have contributed to the mixed findings regarding the relation between overprotective parenting and adolescents' problem behavior.

What type of problem behavior is included?

Studies have differed in terms of the type of problem behavior included. Whereas some studies have focused on broadband scales for internalizing (e.g., Arslan et al., 2023; Sentse, Veenstra, Lindenberg, Verhulst, & Ormel, 2009) or externalizing problems (e.g., Sentse et al., 2009), other studies focused on more specific subscales, such as anxiety (Jian et al., 2025), somatic complaints (e.g., Janssens et al., 2009), or delinquent behavior (e.g., Nelson, Padilla‐Walker, & McLean, 2020). A meta‐analysis (including mainly cross‐sectional studies) demonstrated that overprotective parenting related more strongly to internalizing problems than to externalizing problems (de Roo et al., 2022), demonstrating that the type of problem behavior may affect the results. Also within the domain of internalizing problems, the effects of overprotective parenting may differ depending on the level of generality at which these problems are examined. A meta‐analysis focusing specifically on associations between parental overprotection and internalizing problems indeed showed that the overall effect size was greater in studies relying on a broadband scale of internalizing problems compared to studies relying on specific measures of anxiety and depression (Zhang & Ji, 2023).

Who is reporting on parental overprotection?

Studies vary in the informants they include to assess overprotective parenting, with studies relying either on adolescent reports (e.g., Janssens et al., 2009), or parental reports (e.g., Allmann et al., 2022). Parental self‐reports of parenting tend to be more favorable than adolescent reports and, according to observational studies, less accurate compared to adolescent reports (e.g., Herbers, Garcia, & Obradović, 2017). Importantly, the type of informant may affect findings about the relationship between overprotective parenting and adolescent functioning. Zhang and Ji (2023) reported in their meta‐analysis that the link between overprotective parenting and adolescent depression was stronger when adolescents reported on the parenting behavior than when the parent was the reporter. This finding is consistent with the assumption that adolescents' perception of parenting, or their attributed psychological meaning to the parenting behavior, ultimately determines how adolescents feel and act (Soenens, Vansteenkiste, & Van Petegem, 2015). If, on the other hand, only adolescent (but not parental) reports of overprotective parenting relate to adolescent problem behavior, one may argue that the associations obtained are impacted by reporter bias. When longitudinal associations are observed across multiple informants, this provides evidence that the effects are not solely driven by a single reporter's subjective perception.

Who is reporting on adolescents' problem behaviors?

Similar to parenting, different informants can be used to report on adolescents' problem behavior, including adolescents themselves, their parents, or one of their teachers. Cross‐informant correlations have small to moderate effect sizes (Chana, Tampke, & Fite, 2021; De Los Reyes et al., 2015), with small correlations especially between parents and teachers, and between children and teachers (De Los Reyes et al., 2015). Cross‐informant correlations are typically higher for externalizing problems due to their greater visibility compared to internalizing problems (Chana et al., 2021). Meta‐analyses of cross‐sectional research demonstrated that the informant of the outcome also affects the association between overprotective parenting and problem behavior. de Roo et al. (2022) found that the link between overprotective parenting and internalizing problems was twice as strong when the outcome was self‐reported compared to when it was parent‐reported. However, it remains to be examined whether the type of informant also influences the nature and strength of longitudinal associations between overprotective parenting and adolescent problem behavior.

Which parent is the focus of investigation?

Studies vary in their treatment of parental gender, with some focusing only on maternal overprotection (Van Petegem et al., 2020), others focusing on maternal and paternal overprotection separately (e.g., Jian et al., 2025), and some assessing parental overprotective parenting more generally (e.g., Johnco et al., 2021). The choice between maternal, paternal, or a combined assessment of parental overprotection may significantly impact findings. Studies have shown not only that mothers tend to be more overprotective than fathers (Scharf, Rousseau, & Bsoul, 2017), but also that associations between overprotective parenting and children's problem behavior are more pronounced for maternal compared to paternal overprotection (de Roo et al., 2022; Zhang & Ji, 2023). The scarcity of longitudinal studies examining parental gender makes it currently not possible to draw solid conclusions about the role of parental gender in associations between overprotective parenting and adolescent maladjustment.

Is measurement error controlled for?

To date, most studies on overprotective parenting have relied on manifest variables in their analyses. Yet, estimates of associations between latent factors tend to be more precise and robust, as they account for measurement error and associated noise (Cole & Preacher, 2014). Therefore, in the current multiverse analysis, we will consider the use of manifest versus latent variables as a fifth factor potentially affecting the nature of associations between overprotective parenting and adolescent outcomes.

The current study

The current study aims to systematically estimate the bidirectional associations between overprotective parenting and adolescents' problem behavior, using a multiverse analysis approach within a rich two‐wave repeated‐measures dataset. The primary aim of a multiverse analysis is twofold: to assess the robustness of results across a range of model specifications and to identify analytic choices that influence conclusions, rather than to interpret any specific set of modeling choices. Within this multiverse analysis, Latent Change Models (LCM; Hertzog & Nesselroade, 2003; McArdle & Nesselroade, 1994) are estimated. Latent change models reliably capture between‐person differences in latent factors representing both initial levels (i.e., intercepts) and intra‐individual changes over time (i.e., slopes). Applied to the current study, the LCM approach will result in latent intercepts of parental overprotection and adolescents' problem behavior at T1, as well as latent slopes representing intra‐individual changes in these constructs between two waves of assessment, separated by a 2‐year interval. This interval covered the transition from early to middle adolescence, a transition during which overprotective parenting may be particularly relevant because adolescents further assert independence (Soenens, Vansteenkiste, & Beyers, 2019). Importantly, the results thus reflect between‐person differences in intra‐individual change across early to middle adolescence (approximately ages 14–16) and should not be generalized to longer or shorter time intervals or to other age groups.

LCMs include three distinct types of bivariate associations. First, the level‐with‐level association reflects whether adolescents whose parents are more overprotective at T1 also show higher levels of problem behavior at T1 (i.e., between‐person differences in initial level). Next, the change‐with‐change association captures whether adolescents whose parents increase (or decrease) in parental overprotection over time also show higher increases (or decreases) in problem behavior over time (i.e., between‐person differences in intra‐individual change). Finally, the level‐to‐change regressions reflect whether individual differences in one variable at T1 are associated with subsequent change over time in the other variable (i.e., association between between‐person differences in initial level with between‐person differences in intra‐individual change). Specifically, in the current LCM two‐level‐to‐change regressions are estimated: (1) the regression from parental overprotection at T1 to change in problem behavior over time (from T1 to T2), and (2) the regression from problem behavior at T1 to change in parental overprotection over time. Because our primary interest lies in the direction of effects of the associations between overprotective parenting and adolescents' problem behavior, the level of change in regression is of primary importance. Of secondary interest are the level‐with‐level and change‐with‐change associations, which suggest that parental overprotection and adolescents' problem behavior tend to go hand in hand, but do not provide information about the reciprocal nature of these associations.

To assess the robustness and stability of the associations in the LCM, the multiverse analysis allows for the examination of how various substantive and methodological decisions affect the nature and strength of the associations. We focus on five potential sources of influence, that is, (a) the type of problem behavior (i.e., internalizing versus externalizing) and the breadth of the assessment of problem behavior (i.e., broadband versus narrowband scales), (b) the type of informant used to measure parental overprotection (i.e., adolescents, mothers, and fathers, and every possible combination of these reports), (c) the type of informant used to measure adolescent problem behavior (i.e., adolescents, mothers, fathers, and teachers, and every possible combination of these reports), (d) the gender of the targeted parent (i.e., maternal or paternal overprotection, or a combined measure aggregated across parental gender), and (e) the statistical treatment of measurement error (i.e., models using latent factors or manifest variables).

Method

Procedure and participants

The Flemish Study on Parenting, Personality and Development (FSPPD; Prinzie et al., 2003) is an ongoing multi‐informant longitudinal study that was initiated in 1999. The study was approved by the Ethics Committee of KU Leuven (OT 98/12 ZKA 2922). In total 684 families participated. All participants were asked to sign informed consent prior to participation. For the present study, we will rely on Wave 5 (referred to as Time 1 in this paper; 2007; M age = 13.8) and Wave 6 (referred to as Time 2 in this paper; 2009; M age = 15.8) of the study, as overprotection was measured in these two waves. To be included in the analytical sample, at least one informant from the family (i.e., adolescent, mother, father, or teacher) had to participate at either T1 or T2. The final sample consists of 588 Belgian adolescents, their parents, and teachers. Appendix S1 provides more detailed information on the data collection procedure, the missingness in the data, and demographic information.

Measures

All measures were assessed at both time points. Perceptions of maternal and paternal overprotection were assessed by adolescents and both parents. Adolescent' problem behavior was rated by adolescents, both parents and the teacher. A more detailed overview of the number of items and Cronbach's alpha for each scale can be found in Table S1.

Overprotective parenting

Overprotection was measured using the overprotection subscale from the Egna Minnen Beträffande Uppfostran (EMBU; Perris, Jacobsson, Lindström, von Knorring, & Perris, 1980). Adolescents completed the EMBU about their mother and father, and mothers and fathers completed the EMBU about themselves and their partner. All versions consist of 12 items, which were scored on a 4‐point Likert scale (1 = no, never; 4 = yes, most of the time). Example items are: “Do you think that your mother/father is too worried about things happening to you?”, and “If your child has a little secret, do you want your child to tell you?”. Across the different informants and parental gender, Cronbach's alphas were on average .71 and .73 at T1 and T2, respectively (range = .67 to .77).

Problem behavior

Adolescents' internalizing and externalizing problems were assessed with the Dutch translations of the ASEBA (Achenbach, 1991; Verhulst, Van der Ende, & Koot, 1996). Adolescents completed the Youth Self‐Report (YSR), parents the Child Behavior Checklist (CBCL), and teachers the Teacher Report Form (TRF), all rated on a 3‐point scale (0 = not true, 2 = often true). Internalizing problems comprised anxious/depressed (e.g., “cries a lot”), withdrawn/depressed (e.g., “prefers to be alone”), and somatic complaints (e.g., “suffers from dizziness”) subscales; externalizing problems comprised aggressive (e.g., “fights a lot”) and rule‐breaking behavior (e.g., “lies or cheats”). Reliability was high across informants, with average Cronbach's alphas of .86 for overall internalizing and .88 for overall externalizing problems at both T1 and T2 (ranges = .82–.90).

Plan of analysis

Measurement invariance

Prior to testing our research questions, we examined measurement invariance across time and across informants for all constructs. Direct comparisons involving teacher reports could not be conducted, as teacher‐reported items, although substantially overlapping in content, were not exact one‐to‐one matches with adolescent or parent‐reported items.

Latent change modeling

To conduct the LCMs within a multiverse framework, several steps were carried out. First, univariate LCMs were estimated for each study variable, which provide insight into whether variables increase or decrease over time and whether there is variation in the level and change parameters. When latent factors were used (instead of manifest variables), the univariate LCM also included a measurement model. For broadband problem behavior, subscales were used as indicators to create latent scores. For overprotection and the narrowband scales of problem behavior, item parcels served as indicators (mixing items with stronger and weaker loadings; Little, Cunningham, Shahar, & Widaman, 2002). In case of multiple informants (e.g., parents and teacher), a higher‐order latent factor was created using those informants' latent scores. In a second step, multivariate LCMs were estimated, modeling overprotection and problem behavior together. The parameters representing the levels and changes were then saved for the next step. Herein, structural associations were estimated between the level and change parameters of overprotection and problem behavior, while controlling for child's gender, child's age, family structure (intact versus not intact), and parental education level. This process was repeated for all the different model combinations in the multiverse framework (see below). Latent change models (LCMs) were estimated in R using the lavaan package (Rosseel, 2012). Models were estimated using maximum likelihood with robust standard errors (MLR), and missing data were handled using full information maximum likelihood (FIML). The R code used to estimate the latent change model within the multiverse framework is available on OSF (https://osf.io/ga3vw/overview?view_only=e5689863978b40fd972a76c6e52dddde). Model fit was assessed using CFI, RMSEA, and SRMR, with CFI ≥ .90 and RMSEA/SRMR ≤ .08 indicating acceptable fit (Hu & Bentler, 1999; Kline, 2005).

Multiverse framework

To build the data multiverse, all combinations of the five key substantive and methodological decisions were considered. Table 1 presents the possible choices for the five factors (i.e., operationalization of problem behavior, informant of overprotection, informant of problem behavior, targeted parent, and accounting for measurement error). As can be seen in Table 1, there are 7 (operationalization of problem behavior) × 7 (informant of overprotection) × 15 (informant of problem behavior) × 3 (targeted parent) × 2 (measurement error) = 4,410 possible combinations in total. Because the central aim of a multiverse approach is to interpret results in an aggregated manner, assessing the robustness and stability of conclusions across a wide range of reasonable analytical decisions, there is no need to correct for multiple testing. Rather, multiverse analysis has been proposed specifically as a response to the implicit multiple‐comparison problem inherent in testing research questions (Steegen et al., 2016). In contrast, statistical power is not evaluated at the level of the multiverse as a whole but instead pertains to each individual model estimated within it. Prior simulation research suggests that sample sizes exceeding ~400 are typically sufficient to achieve adequate power for complex SEMs (Wolf et al., 2013); therefore, given our sample size of 588, the estimated models were likely adequately powered.

Table 1.

The multiverse framework

  1. Operationalization of problem behavior (O)
    1. O1: Internalizing problems (broadband – internalizing problems)
    2. O2: Anxious/depressed (narrowband – internalizing problems)
    3. O3: Withdrawn/depressed (narrowband – internalizing problems)
    4. O4: Somatic complaints (narrowband – internalizing problems)
    5. O5: Externalizing problems (broadband – externalizing problems)
    6. O6: Aggressive behavior (narrowband – externalizing problems)
    7. O7: Rule‐breaking behavior (narrowband – externalizing problems)
  2. Informant overprotection (IO)
    1. IO1: Adolescent‐reported
    2. IO2: Mother reported
    3. IO3: Father‐reported
    4. IO4: Multi‐informant: adolescent & mother
    5. IO5: Multi‐informant: adolescent & father
    6. IO6: Multi‐informant: mother & father (i.e., parent‐reported)
    7. IO7: Multi‐informant: adolescent, mother & father
  3. Informant problem behavior (IP)
    1. IP1: Adolescent‐reported
    2. IP2: Mother reported
    3. IP3: Father‐reported
    4. IP4: Teacher‐reported
    5. IP5: Multi‐informant: adolescent & mother
    6. IP6: Multi‐informant: adolescent & father
    7. IP7: Multi‐informant: adolescent & teacher
    8. IP8: Multi‐informant: mother & teacher
    9. IP9: Multi‐informant: father & teacher
    10. IP10: Multi‐informant: mother & father (i.e., parent‐reported)
    11. IP11: Multi‐informant: adolescent, mother & father
    12. IP12: Multi‐informant: adolescent, mother & teacher
    13. IP13: Multi‐informant: adolescent, father & teacher
    14. IP14: Multi‐informant: mother, father & teacher (i.e., adult‐reported)
    15. IP15: Multi‐informant: adolescent, mother, father & teacher
  4. Targeted parent (T)
    1. T1: Maternal overprotection
    2. T2: Paternal overprotection
    3. T3: Parental overprotection (latent combination of maternal and paternal) overprotection
  5. Controlling for Measurement error (M)
    1. M1: Latent factors
    2. M2: Manifest variables
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The results and specifications (e.g., which informant was used) for each model were saved. The overall results were then evaluated in terms of effect sizes, confidence intervals and p‐values. To evaluate the effects, we examined the distribution of effect estimates across the multiverse by examining the mean, median, and interquartile range, thereby capturing not only the average effect but also its variability across model specifications. In addition, we assessed the proportion of effect estimates exceeding a pre‐specified smallest effect size of interest (SESOI; β ≥ .10), corresponding to a small effect in psychological research (Cohen, 1988). To further evaluate robustness, we also examined the proportion of estimates with a reverse sign, defined as the proportion of specifications yielding effects in the opposite direction of the a priori expected (positive) associations.

Finally, we conducted a meta‐analytic regression to summarize how estimated effects varied across the five included model specifications (i.e., operationalization of problem behavior, informant overprotection, informant problem behavior, targeted parent, and whether measurement error was controlled for). Because the different model estimates were derived from a common dataset, this approach violates the meta‐analytic assumption that effect sizes are statistically independent. Accordingly, the meta‐analysis was used as a descriptive tool and as a summary analysis, rather than as a stand‐alone inferential test (cf. Modecki et al., 2020). To mitigate the impact of dependence between estimates, we applied cluster‐robust variance estimation. Analyses were conducted using the metafor package in R (Viechtbauer, 2010).

Results

Descriptive statistics and correlations between the study variables for the different informants are presented in Table S2.

Measurement invariance

First, measurement invariance across time was examined. The majority of models converged as specified, and results supported (partial) scalar invariance across all constructs. Second, measurement invariance across informants was tested. The majority of models converged as specified, and results supported metric, but not scalar, invariance across constructs, indicating that latent means cannot be directly compared across informants.

Preliminary LCM analyses

For overprotective parenting, the average intercept at T1 was equal to 1.98. On average, there was a small decline in parental overprotection across the period of 2 years (M = −.05), with significant changes observed in 76.2% of the models. For internalizing problems and externalizing problems, the average intercepts were, respectively, 0.25 and 0.23. Over 2 years, internalizing problems decreased on average by −0.01 (with significant changes in 0.02% of models), whereas externalizing problems increased by 0.004 (with significant changes in 51.1% of univariate models). In 82.9% of the univariate LCMs, we found significant variance in the level and change parameters, indicating that families differed in (a) their baseline levels of overprotection and adolescents' problem behavior and (b) how parental overprotection and adolescents' problem behavior changed over time. The models using manifest variables were fully saturated (CFI = 1.00, RSMEA = 0.00, and SRMR = 0.00). The models using latent factors showed acceptable fits for overprotective parenting, with, on average, CFI = .95, RMSEA = .05, and SRMR = .05. The average fit indices for problem behavior were CFI = .96, RMSEA = .06, and SRMR = .06. Detailed univariate LCM results are presented in Table S3 (manifest models) and Table S4 (latent models).

Next, multivariate Latent Change Models were conducted to estimate and save the latent level and change parameters for the sake of further analyses. Again, the multivariate models using manifest variables were fully saturated, with a perfect fit. For the multivariate LCMs using latent factors, the fit indices were acceptable (on average: CFI = .87, RMSEA = .05, and SRMR = .08).

Multiverse analysis

Finally, the structural associations between the level and change parameters of overprotection and problem behavior were estimated, controlling for child's gender, child's age, family structure and parental education level. This third step was repeated for all possible combinations (i.e., n = 4,410) in the multiverse framework. Because we relied on saved latent level and change scores from the multivariate LCMs, all 4,410 models conducted in the multiverse analysis were saturated (CFI = 1.00, RSMEA = 0.00, SRMR = 0.00) and thus included in the results.

Overall effects

Figure 1 presents the average estimate, the 95% CI and the proportion of the models that were significant for each of the three different types of examined associations. First, the level‐with‐level association was significant in 83.1% of the models (n = 3,665). On average, the standardized estimate (β) was equal to .22 (median = .21, IQR = .17), of which 83.3% exceeded the pre‐specified SESOI of .10 (n = 3,676). Of all estimates, 1.6% had a sign reversed to the expected positive direction (n = 72), and all significant associations were positive. This indicates that adolescents whose parents are more overprotective at T1 also display more problem behavior at T1. Second, the change‐with‐change association was significant in 63.8% of the models (n = 2,813), with a mean β of .16 (median = .14, IQR = .18), of which 63.7% exceeded the pre‐specified SESOI of .10 (n = 2,810). Of all estimates, 14.1% had a sign reversed to the expected positive direction (n = 623). Of the significant effects, the association was positive in 95.6% of the time (n = 2,689). This indicates that adolescents whose parents display an increase (or decrease) in overprotective parenting from T1 to T2 also report a simultaneous increase (or decrease) in problem behavior. Finally, the level‐to‐change regressions give insight into the direction of the association between parental overprotection and adolescents' problem behavior. The effect of the level of overprotective parenting on change in problem behavior was significant in 55.4% of the models (n = 2,445). The standardized effect was on average .11 (median = .10, IQR = .10), of which 50.6% exceeded the pre‐specified SESOI of .10 (n = 2,234). Of all estimates, 8.1% had a sign reversed to the expected positive direction (n = 359). Of the significant associations, 99.2% were positive (n = 2,426). This indicates that more than half of the tested models suggest that adolescents whose parents are more overprotective at T1 show, on average, a stronger increase in problem behavior 2 years later, compared to adolescents whose parents are less overprotective. Testing the opposite direction of association, the regression of level of problem behavior on change in overprotective parenting was significant in 20.5% of the models (n = 905). The average estimate was equal to −.01 (median = .02, IQR = .09), of which only 6.9% exceeded the pre‐specified SESOI of .10 (n = 305). Of all estimates, 40.1% had a sign reversed to the expected positive direction (n = 1,767). Of the significant associations, 41.5% of the cases were positive (n = 376). This indicates that most of the models suggest that adolescents' levels of problem behavior at T1 are not associated with increases or decreases in overprotective parenting from T1 to T2.

Figure 1.

Figure 1

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Overall results of the multiverse analysis. The first number represents the overall standardized estimate, the 95% CI is presented in square brackets. The proportion of significant association is presented in brackets

The role of the researcher's degrees of freedom

The associations between overprotective parenting and problem behavior, according to the different substantive and methodological choices, are presented in Figures 2, 3, 4, 5. Specifically, Figure 2 shows the results for the level‐with‐level association, while Figure 3 presents the results for the change‐with‐change association. Figure 4 displays the findings for the effect of overprotective parenting on problem behavior, and Figure 5 depicts the findings for the effect of problem behavior on overprotective parenting. Importantly, the variability observed across specifications should not be interpreted as a limitation of the approach, but rather as information about the sensitivity of the findings to specific modeling decisions. In this context, the average effect reported in the multiverse does not represent a “best” estimate of the true effect size. Instead, it provides a descriptive summary statistic of the set of effect estimates obtained across the analytical specifications included in the multiverse. Accordingly, inference should focus primarily on consistency, direction, and variability of effects across specifications rather than on the mean alone. As shown in Figures 2, 3, 4, 5, differences resulting from substantive and methodological choices were minimal, with the strength of the associations remaining relatively consistent across the different approaches.

Figure 2.

Figure 2

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Forest plot for the association between level of overprotection and level of problem behavior according to the different substantive and methodological choices. This figure summarizes results from the multiverse analysis aggregated by substantive and methodological choices. For each operationalization, the figure shows (A) the proportion of models using that operationalization in which the level‐with‐level association was statistically significant, (B) the mean estimate (β) with its 95% confidence interval, (C) the median estimate with the IQR across those models, (D) the proportion of models that exceeded .10, which was pre‐specified as the effect size of interest, and (E) the proportion of estimates with a sign reversed to the expected (positive) direction. For example, when internalizing problems were operationalized using the broadband scale, the level‐with‐level association was statistically significant in 90% of cases., with a mean estimate of β = .26 (95% CI [.17, .35]), a median of .26 (IQR .13–.30). 83% of the models had an estimate that exceeded .10, and only 2% of the models had a negative sign

Figure 3.

Figure 3

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Forest plot for the association between change in overprotection and change in problem behavior according to the different substantive and methodological choices. This figure summarizes results from the multiverse analysis aggregated by substantive and methodological choices. For each operationalization, the figure shows (A) the proportion of models using that operationalization in which the change‐with‐change association was statistically significant, (B) the mean estimate (β) with its 95% confidence interval, (C) the median estimate with the IQR across those models, (D) the proportion of models that exceeded .10, which was pre‐specified as the effect size of interest, and (E) the proportion of estimates with a sign reversed to the expected (positive) direction

Figure 4.

Figure 4

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Forest plot for the effect of level of overprotection on change in problem behavior according to the different substantive and methodological choices. This figure summarizes results from the multiverse analysis aggregated by substantive and methodological choices. For each operationalization, the figure shows (A) the proportion of models using that operationalization in which the level‐on‐change regression from overprotection to problem behavior was statistically significant, (B) the mean estimate (β) with its 95% confidence interval, (C) the median estimate with the IQR across those models, (D) the proportion of models that exceeded .10, which was pre‐specified as the effect size of interest, and (E) the proportion of estimates with a sign reversed to the expected (positive) direction

Figure 5.

Figure 5

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Forest plot for the effect of level of problem behavior on change in overprotection according to the different substantive and methodological choices. This figure summarizes results from the multiverse analysis aggregated by substantive and methodological choices. For each operationalization, the figure shows (A) the proportion of models using that operationalization in which the level‐on‐change regression from problem behavior to overprotection was statistically significant, (B) the mean estimate (β) with its 95% confidence interval, (C) the median estimate with the IQR across those models, (D) the proportion of models that exceeded .10, which was pre‐specified as the effect size of interest, and (E) the proportion of estimates with a sign reversed to the expected (positive) direction

Yet, visual inspection of Figures 2, 3, 4, 5 showed that there were a couple of notable patterns in the strength of associations between overprotective parenting and problem behaviors. First, associations were more pronounced for externalizing problems than for internalizing problems, a tendency most clearly visible at the level of change‐with‐change associations (Figure 3). Second, across all associations, the largest effect estimates were observed when multi‐informant scores were used for overprotective parenting that included the adolescent's assessment (Figures 2, 3, 4, 5). In addition, the regression of level of overprotection on change in problem behavior was substantially lower for father‐reported overprotection (Figure 4). Further, teacher‐reported problem behavior yielded the lowest estimates across all examined associations (Figures 2, 3, 4, 5). Finally, across all associations, the largest estimates were observed for models using latent factors compared to models using manifest variables (Figures 2, 3, 4, 5).

Meta‐analytical regression

Finally, to summarize the findings, we conducted a meta‐analysis for each examined association with the estimated effects as the dependent variable, and the five substantive and methodological choices as predictors. Results are presented in Table S5. Overall, results confirmed findings that controlling for measurement error was the biggest source of variation in model estimates. In contrast, the operationalization of the targeted parent (i.e., maternal, paternal, or parental overprotection) had the smallest influence on the effect size estimates. Regarding the level of overprotection on change in problem behavior regression, findings confirmed that relying on only father‐report (for overprotection and/or problem behavior) or teacher report (for problem behavior) often yielded smaller effect sizes compared to multi‐informant reports, adolescent reports, or mother‐reports.

To complement the meta‐analytic summary, we visualized the multiverse of model results using a specification curve with a decision dashboard (see Figure 6 for the effect of level overprotection on change in problem behavior, and Figures S1–S3 for the level‐with‐level association, change‐with‐change association, and level problem behavior on change in overprotection regression, respectively). All model estimates were ordered from smallest to largest and displayed along the x‐axis, allowing inspection of the full distribution of effects. To facilitate interpretation of systematic patterns, estimates were grouped into bins, and for each bin, we identified the modeling choice that was most common within that bin relative to how often it appeared across the full set of model specifications. This base‐rate adjusted dashboard provides a descriptive overview of how specific combinations of substantive and methodological decisions were associated with smaller versus larger effect sizes across the multiverse of model specifications.

Figure 6.

Figure 6

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Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the effect of level of overprotection on change in problem behavior. The top panel displays the specification curve, with each point representing the standardized regression coefficient (β) from one model specification examining the association between parental overprotection at T1 and change in adolescents' problem behavior from T1 to T2. Model specifications are sorted from low to high effect sizes. Dashed horizontal lines indicate a null association (β = 0). Points are colored by statistical significance (p < .05 vs. p ≥ .05). The lower panel presents a decision dashboard summarizing the substantive and methodological choices underlying each specification, including the operationalization of problem behavior, informant source for overprotection and problem behavior, targeted parent, and treatment of measurement error. Colors indicate the most frequently selected option within each specification bin (base‐rate adjusted). Together, the figure illustrates the robustness and variability of the association across a wide range of plausible analytic decisions

Discussion

Concerns about the impact of overprotective parenting on youth development are widely discussed in popular media (Glass & Tabatsky, 2014; Haidt, 2024; Lukianoff & Haidt, 2018). Although meta‐analyses have indeed identified a link between overprotective parenting and adolescents' problem behavior (de Roo et al., 2022; Zhang & Ji, 2023), much of the empirical evidence is based on cross‐sectional studies, limiting inferences on the directionality of these associations. The few longitudinal studies available yielded inconsistent findings, leaving it unclear whether overprotective parenting precedes problem behavior, co‐occurs with it, or follows it. To address this ambiguity, the present study employs a multiverse analysis, systematically testing the robustness of associations between overprotection and problem behavior across multiple substantive and methodological decisions.

The concurrent relationships between overprotective parenting and adolescents' problem behavior

This multiverse analysis of overprotective parenting and problem behavior examined two concurrent (i.e., level‐with‐level and change‐with‐change) and two directional associations (i.e., level‐to‐change in both directions). Regarding the concurrent associations, a small to moderate positive cross‐sectional association between overprotective parenting at T1 and adolescent problem behavior at T1 was found. This finding aligns with a large body of literature showing positive relations between overprotective parenting and both internalizing and externalizing problems (de Roo et al., 2022; Zhang & Ji, 2023). Beyond these cross‐sectional associations, our findings also indicate associations at the level of intra‐individual change. Specifically, adolescents whose parents show increases in overprotective parenting also, on average, show increases in problem behavior. Of particular interest, however, are the relations between baseline levels of overprotective parenting and changes in problem behavior (and vice versa), as these relationships are indicative of the temporal ordering of these variables.

The level‐to‐change regression of overprotective parenting on adolescents' problem behavior

More than half of the tested models (55%) supported a positive association between overprotective parenting at T1 and subsequent increases in adolescents' problem behavior, indicating that adolescents whose parents are more overprotective parents tended, on average, to show greater increases in problem behavior from early to middle adolescence. Although one might interpret the percentage of 55 from a “glass half‐empty” perspective, we argue this is relatively high, especially given that the multiverse analysis employed a notably conservative statistical test of the effect. Indeed, many of the models included in the multiverse analysis would likely not have been selected in studies examining a single analytical approach. Results further suggest that these ‘less obvious models’ tended to be associated with smaller effect sizes and were less likely to yield statistically significant effects (see Figures 4 and 6). This was, for instance, the case when teachers reported on internalizing problems, consistent with research suggesting that teachers' assessments of adolescents' internalizing problems are less accurate (De Los Reyes et al., 2015; see Figure S4 or Figure 6) or when, for instance, fathers reported on maternal overprotection (see Figure 6). In contrast, the largest effect sizes were observed in models that included multi‐informant reports (including adolescent reports), externalizing problems, and controlled for measurement error (see Figure 6).

Although small in terms of effect size, the effect of overprotective parenting on adolescents' problem behavior was rather consistent, with effect sizes showing minimal variation across the different modeling choices in the multiverse analysis. With one exception, all models produced overall effect sizes ranging between .07 and .13. This suggests that inconsistencies in earlier studies may reflect the small effect of parental overprotection on adolescents' problem behavior, which underpowered studies may fail to detect (i.e., type II error). A few notable findings are worth discussing. First, higher levels of overprotective parenting are associated with subsequent increases in various forms of problem behavior, including depressive and anxiety symptoms, somatic complaints, and aggression (see Figure S5). To better understand this observed multifinality, future research could explore potential moderating factors such as temperament traits (e.g., Ryan & Ollendick, 2018), personality (e.g., Deutz et al., 2021) and coping strategies (e.g., Flamant et al., 2022). A second notable finding is that effects of overprotective parenting were relatively consistent across parental gender, with both maternal and paternal overprotection being positively related to increases in problem behaviors (see Figure 4, subsection Targeted parent). Third, although results for teacher‐reported outcomes should be interpreted cautiously, as more than 60% of the teacher‐reported data at T2 were missing, models with teacher‐reported internalizing problems showed smaller effect sizes than models based on parent or adolescent reports. For externalizing problems, effect sizes based on teacher reports were comparable to those based on parent or adolescent reports (see Figure S4).

The level‐to‐change regression of adolescents' problem behavior on parental overprotection

Finally, this study examined whether between‐person differences in adolescents' problem behavior at T1 were associated with between‐person differences in subsequent changes in parental overprotection. Overall, average effects across the various substantive and methodological choices ranged from −0.06 to 0.04, indicating little evidence for this direction (see Figure 5). Several factors not accounted for in this multiverse analysis may explain why some previous studies did observe that adolescents' problem behavior precedes overprotection (e.g., Gao, Hou, Hao, & Yu, 2023), while others, including the present study, did not (e.g., Johnco et al., 2021; Nelson et al., 2020). First, the two‐year interval in the current repeated‐measures study may have been too long to detect associations between adolescents' problem behavior and subsequent changes in parental overprotection. Prior studies suggest that associations in the parent‐to‐adolescent and adolescent‐to‐parent directions may operate at different timeframes. For instance, a recent study examining parent‐adolescent conflict and adolescents' ill‐being showed that the time gap between parent–child conflict predicting ill‐being was substantially longer compared to the time gap between ill‐being predicting conflict (Bülow et al., 2025).

Second, although previous meta‐analyses on parental overprotection found no evidence for a moderating effect of age in cross‐sectional associations between overprotective parenting and problem behavior (de Roo et al., 2022; Zhang & Ji, 2023), age could still play a critical role at the level of associations across time. One possible explanation is that parents may perceive early or middle adolescents' problem behavior (compared to other age groups) as more normative and transient, limiting subsequent worrying and overprotective behavior. Although this hypothesis is speculative at the moment, it may help explain why studies have found that child anxiety predicted increases in parental overprotection (Jian et al., 2025), whereas early adolescent anxiety did not (Johnco et al., 2021), and may also account for the absence of a systematic association between adolescents' problem behavior and changes in parental overprotection in the present study, which focused on early to middle adolescence.

Limitations

A first limitation of the present study concerns the use of a two‐wave repeated‐measures design. Although this design permits the examination of reciprocal associations between overprotective parenting and adolescents' problem behavior within the specific timeframe studied, it does not allow for firm conclusions regarding the directionality of these associations more generally, let alone regarding causal processes at the within‐person level. Future research would benefit from incorporating additional measurement waves and at various timescales (including monthly, weekly, daily, and momentary intervals) to more adequately capture within‐person processes and to permit stronger inferences about directionality.

Second, the data used in the current study were collected in 2007 (T1) and 2009 (T2). Societal trends, technological advancements, and other external factors that have occurred since then potentially limit the generalizability of the findings to present‐day contexts. For instance, popular media often claims that parents have become increasingly overprotective over the past few decades (Glass & Tabatsky, 2014), which might lead to a more normalized and benign appraisal of overprotective parenting, thereby reducing its negative effects on youth. In addition, since the data collection, other measurements of parental overprotection have been developed (e.g., Kins & Soenens, 2013). Whereas these newer scales provide a purer measure of parental overprotection, the EMBU also entails items that assess general controlling parenting. Future research would do well to replicate our findings with present‐day data.

A third limitation of the present study concerns measurement invariance. Although (partial) scalar invariance across time was supported for all constructs, some higher‐order and lower‐order models did not converge, suggesting that longitudinal measurement properties of certain constructs should be interpreted with caution. Although we could not demonstrate scalar invariance across all informants, the measures did demonstrate metric invariance, which is sufficient to answer our research questions about structural associations between constructs. Indeed, although constructs were measured on a comparable scale across informants, systematic differences in item intercepts precluded direct comparisons of latent means (which was not the goal of the current study). Comparisons involving teacher reports were further limited because teacher‐reported items, while substantively overlapping with adolescent‐ or parent‐reported items, were not exact one‐to‐one matches. Future research should employ multi‐informant measures with equivalent item content and intercepts to enable robust cross‐informant latent mean comparisons.

Further, to assess overprotective parenting and adolescents' problem behavior, the current study relied solely on questionnaires. Although questionnaire‐based research offers insightful information, it is inherently limited by individual interpretations and social desirability effects. Therefore, future studies incorporating observational assessments could provide a more objective and robust complement to the current questionnaire‐based study.

Finally, the relatively high educational background of our sample may affect the generalizability of our findings. It remains uncertain whether our results extend to lower socioeconomic contexts. Future research should explore whether the associations observed in this study hold across diverse socioeconomic backgrounds.

Conclusion

Using a unique and rich two‐wave repeated‐measures dataset, the current study applied a multiverse approach to examine associations between parental overprotection and adolescents' problem behavior. The study provided evidence for positive associations between parental overprotection and adolescents' problem behavior at the level of baseline interindividual differences and at the level of interindividual differences in intra‐individual change over time. Of particular interest, the findings indicated a small but robust association between parental overprotection at T1 and subsequent increases in adolescents' problem behavior, with little to no evidence for a systematic association in the opposite direction.

Ethical considerations

The study was approved by the Ethics Committee of the Catholic University of Leuven (12‐01‐98; Ref. OT 98/12 ZKA 2922). All participants and parents signed an active informed consent prior to participation.

Key points.

What's known?

  • Overprotective parenting is often considered to be one of the major contributors to youth mental health problems.

  • Yet, empirical research is predominantly cross‐sectional, with the limited number of longitudinal studies yielding inconsistent results.

What's new?

  • This study adopted a multiverse approach to examine bidirectional relationships between overprotective parenting and adolescent maladjustment more systematically.

  • The findings suggest that overprotective parenting is associated with increases in adolescents' problem behavior, but not the other way around.

What's relevant?

  • Effect sizes are small, suggesting that overprotective parenting is not a key contributor to increases in youth mental health problems.

  • Future research should gain more insight into the underlying mechanisms explaining the link between overprotective parenting and youth functioning.

Supporting information

Appendix S1. Information on the data collection procedure, the missingness in the data, and demographic background of the sample.

Figure S1. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the level‐with‐level association between overprotection and problem behavior.

Figure S2. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the change‐with‐change association between overprotection and problem behavior.

Figure S3. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the effect of level of problem behavior on change in overprotection.

Figure S4. Forest plot for the effect of level of overprotection on change in problem behavior according to the operationalization of problem behavior and the informant of problem behavior.

Figure S5. Forest plot for the effect of level of overprotection on change in problem behavior according to the operationalization of problem behavior.

Table S1. Number of items and reliability for the ASEBA scales.

Table S2. Means, standard deviations and Pearson correlations between the study variables.

Table S3. Parameter estimates of the univariate LCMs using manifest variables.

Table S4. Parameter estimates and fit indices of the univariate LCMs using latent factors.

Table S5. Results of a meta‐regression with robust variance estimation and post‐hoc comparisons.

JCPP-67-1649-s001.docx (1.4MB, docx)

Acknowledgments

This study was supported by an FWO Odysseus grant (G0DCB23N). The authors have declared that they have no competing or potential conflicts of interest.

Conflict of interest statement: No conflicts declared.

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Appendix S1. Information on the data collection procedure, the missingness in the data, and demographic background of the sample.

Figure S1. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the level‐with‐level association between overprotection and problem behavior.

Figure S2. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the change‐with‐change association between overprotection and problem behavior.

Figure S3. Specification Curve with binned decision dashboard displaying how the different combinations of substantive and methodological choices relate to effect size for the effect of level of problem behavior on change in overprotection.

Figure S4. Forest plot for the effect of level of overprotection on change in problem behavior according to the operationalization of problem behavior and the informant of problem behavior.

Figure S5. Forest plot for the effect of level of overprotection on change in problem behavior according to the operationalization of problem behavior.

Table S1. Number of items and reliability for the ASEBA scales.

Table S2. Means, standard deviations and Pearson correlations between the study variables.

Table S3. Parameter estimates of the univariate LCMs using manifest variables.

Table S4. Parameter estimates and fit indices of the univariate LCMs using latent factors.

Table S5. Results of a meta‐regression with robust variance estimation and post‐hoc comparisons.

JCPP-67-1649-s001.docx (1.4MB, docx)

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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