Regression To The Mean Effect Research Articles

Regression to the mean effect for bi-variate dispersed count data

Regression to the mean (RTM) occurs when extreme observations upon remeasurement are found closer to the population mean. Interventions are usually applied to subjects based on a cut-off point to check it effectiveness. The average difference of the pre-post observations called the total effect is the sum of RTM and intervention effects and needs to be accounted for RTM to avoid incorrect conclusions and unbiasedly estimate the intervention effect. The existing methods assume equidispersion and strict positive correlation for bivariate count data. This study considers a bivariate Poisson distribution which takes both the dispersion and negative correlation into consideration. Expressions for the total effect are derived and partitioned into RTM and intervention effects. The maximum likelihood estimators of the RTM and intervention effects are found numerically, and their asymptotic properties are verified via simulations. Data on the English premier league are used to estimate the RTM and intervention effects.

Quantifying and Estimating Regression to the Mean Effect for Bivariate Beta-Binomial Distribution

ABSTRACT Subjects with initially extreme observations upon remeasurement are found closer to the population mean. This tendency of observations toward the mean is called regression to the mean (RTM) and can make natural variation in repeated data look like real change. Studies, where subjects are selected on a baseline criterion, should be guarded against the RTM effect to avoid erroneous conclusions. In an intervention study, the difference between pre-post variables is the combined effect of intervention/treatment and RTM. Thus, accounting for RTM is essential to accurately estimate the intervention effect. Many real-life examples are better modeled by a bivariate binomial model with varying probability of success. In this article, a bivariate beta-binomial distribution is used that allows the probability of success to vary from subject to subject. Expressions for the total, RTM, and treatment effect are derived, and their behavior is demonstrated graphically. Maximum likelihood estimators of RTM are derived, and their statistical properties are studied via Monte Carlo simulation. The proposed techniques are employed to estimate the RTM effect by utilizing data related to the Countway WM-class circulation.

Regression to the mean: Estimation and adjustment under the bivariate normal distribution

Regression to the mean (RTM) is a statistical phenomenon that happens when subjects having relatively high or low observations upon remeasurement are found closer to the population mean. RTM can erroneously influence the conclusion in a pre-post study design. Expressions are available for quantifying RTM when the distribution of pre-post variables is bivariate normal; however, these methods assume the pre and post observations are identically distributed and strictly positively correlated. This study generalizes previous results to include non-stationary, normally distributed random variables that are potentially negatively correlated, and also provides the decomposition of the conditional mean difference into RTM and unbiased intervention effects. In addition, the maximum likelihood estimators are derived and the unbiasedness, consistency and normality of these estimators are established. A simulation study is conducted to asses the accuracy of estimating the RTM and intervention effects using existing and the proposed methods. Data on the blood lead level in both intervention and placebo groups are used for decomposing the total change in blood lead level on pre-post occasions into RTM and intervention effects.

Avoiding pitfalls: Bayes factors can be a reliable tool for post hoc data selection in implicit learning.

Research on implicit processes has revealed problems with awareness categorizations based on nonsignificant results. Moreover, post hoc categorizations result in regression to the mean (RTM), by which aware participants are wrongly categorized as unaware. Using Bayes factors to obtain sensitive evidence for participants' lack of knowledge may deal with nonsignificance being nonevidential, but also may prevent regression-to-the-mean effects. Here, we examine the reliability of a novel Bayesian awareness categorization procedure. Participants completed a reward learning task followed by a flanker task measuring attention towards conditioned stimuli. They were categorized as B_Aware and B_Unaware of stimulus-outcome contingencies, and those with insensitive Bayes factors were deemed B_Insensitive. We found that performance for B_Unaware participants was below chance level using unbiased tests. This was further confirmed using a resampling procedure with multiple iterations, contrary to the prediction of RTM effects. Conversely, when categorizing participants using t tests, t_Unaware participants showed RTM effects. We also propose a group boundary optimization procedure to determine the threshold at which regression to the mean is observed. Using Bayes factors instead of t tests as a post hoc categorization tool allows evaluating evidence of unawareness, which in turn helps avoid RTM. The reliability of the Bayesian awareness categorization procedure strengthens previous evidence for implicit reward conditioning. The toolbox used for the categorization procedure is detailed and made available. Post hoc group selection can provide evidence for implicit processes; the relevance of RTM needs to be considered for each study and cannot simply be assumed to be a problem.

A simulation-based comparison of two methods for determining the treatment effect in children diagnosed with hyperkinetic disorder

In order to show the effect of treatment, the change between two repeated psychometric measurements at the individual level should be estimated. The simplest method is to calculate the absolute difference between two measurements. However, measurements obtained in a clinical setting are often influenced by other changes not related to the treatment. One of the typical sources of error is regression to the mean (RTM). Iachina and Bilenberg (2012) propose a new method to adjust to the RTM effect. In this work, we will evaluate the performance of absolute difference method and adjusted method in a simulation study.

Best (but oft-forgotten) practices: identifying and accounting for regression to the mean in nutrition and obesity research

Regression to the mean (RTM) is a statistical phenomenon where initial measurements of a variable in a nonrandom sample at the extreme ends of a distribution tend to be closer to the mean upon a second measurement. Unfortunately, failing to account for the effects of RTM can lead to incorrect conclusions on the observed mean difference between the 2 repeated measurements in a nonrandom sample that is preferentially selected for deviating from the population mean of the measured variable in a particular direction. Study designs that are susceptible to misattributing RTM as intervention effects have been prevalent in nutrition and obesity research. This field often conducts secondary analyses of existing intervention data or evaluates intervention effects in those most at risk (i.e., those with observations at the extreme ends of a distribution). To provide best practices to avoid unsubstantiated conclusions as a result of ignoring RTM in nutrition and obesity research. We outlined best practices for identifying whether RTM is likely to be leading to biased inferences, using a flowchart that is available as a web-based app at https://dustyturner.shinyapps.io/DecisionTreeMeanRegression/. We also provided multiple methods to quantify the degree of RTM. Investigators can adjust analyses to include the RTM effect, thereby plausibly removing its biasing influence on estimating the true intervention effect. The identification of RTM and implementation of proper statistical practices will help advance the field by improving scientific rigor and the accuracy of conclusions. This trial was registered at clinicaltrials.gov as NCT00427193.

Regression to the mean for the bivariate binomial distribution.

Regression to the mean (RTM) occurs when subjects having relatively high or low measurements are remeasured and found closer to the population mean. This phenomenon can potentially lead to an inaccurate conclusion in a pre-post study design. Expressions are available for quantifying RTM when the distribution of pre and post observations are bivariate normal and bivariate Poisson. However, situations exist where the response variables are the number of successes in a fixed number of trials and follow the bivariate binomial distribution. In this article, expressions for quantifying RTM effects are derived when the underlying distribution is the bivariate binomial. Unlike the normal and Poisson distributions, the correlation between pre and post observations can be either negative or positive under the bivariate binomial distribution and the severity of RTM is greater in the former case. The percentage relative difference is used to highlight the differences in quantifying RTM under the bivariate binomial distribution and normal and Poisson approximations to the bivariate binomial distribution. Expressions for estimating RTM using the method of maximum likelihood along with its asymptotic distribution are derived. A simulation study is conducted to empirically assess the statistical properties of the RTM estimator and its asymptotic distribution. Data examples using the number of obese individuals and the number of nonconforming cardboard cans are discussed.

Quantifying the regression to the mean effect in Poisson processes.

Regression to the mean (RTM) can occur whenever an extreme observation is selected from a population and a later observation is closer to the population mean. A consequence of this phenomenon is that natural variability can be mistaken as real change. Simple expressions are available to quantify RTM when the underlying distribution is bivariate normal. However, there are many real-world situations, which are better approximated as a Poisson process. Examples include the number of hard disk failures during a year, the number of cargo ships damaged by waves, daily homicide counts in California, and the number of deaths per quarter attributable to acquired immune deficiency syndrome in Australia. In this paper, we derive expressions for quantifying RTM effects for the bivariate Poisson distribution for both the homogeneous and inhomogeneous cases. Statistical properties of our derivations have been evaluated through a simulation study. The asymptotic distributions of RTM estimators have been derived. The RTM effect for the number of people killed in road accidents in different regions of New South Wales (Australia) is estimated using maximum likelihood.

Development of Crash Modification Factors for Installation of High-Tension Cable Median Barriers

Cross-median crashes are among the most hazardous collision types on freeways. The primary countermeasure to reduce such crashes is the installation of a median barrier, with a cable barrier being a widely used alternative. As of 2014, approximately 317 mi of cable median barriers had been installed on freeways throughout Michigan. To assess the safety impacts of this installation program, a comprehensive before-and-after crash analysis was conducted; this analysis included a manual crash report review to identify all median-related crashes. Safety performance functions were developed for roadway segments before and after cable barrier installation, as well as for similar control segments where no barrier had been installed. The empirical Bayes (EB) method was used to develop crash modification factors as a function of median width. To explore the effects of regression to the mean (RTM), the EB results were compared with those from a naïve before–after analysis. The results showed that cable barriers significantly reduced the frequency of fatal and severe injury crashes, although substantial increases in less severe crashes were also observed. The cable barrier was found to have greater impacts on crashes across nearly all severity levels when used on narrow medians (i.e., 26 to 50 ft) compared with wider medians (50 to 94 ft). In addition, reductions in fatal and serious injuries were less pronounced when compared with prior research (especially for wider medians), which was largely attributed to RTM effects. Ultimately, this study provides important guidance to deal with the expected safety impacts (and related costs) associated with cable median barrier installation.

The need to control for regression to the mean in social psychology studies.

It is common in repeated measurements for extreme values at the first measurement to approach the mean at the subsequent measurement, a phenomenon called regression to the mean (RTM). If RTM is not fully controlled, it will lead to erroneous conclusions. The wide use of repeated measurements in social psychology creates a risk that an RTM effect will influence results. However, insufficient attention is paid to RTM in most social psychological research. Notable cases include studies on the phenomena of social conformity and unrealistic optimism (Klucharev et al., 2009, 2011; Sharot et al., 2011, 2012b; Campbell-Meiklejohn et al., 2012; Kim et al., 2012; Garrett and Sharot, 2014). In Study 1, 13 university students rated and re-rated the facial attractiveness of a series of female faces as a test of the social conformity effect (Klucharev et al., 2009). In Study 2, 15 university students estimated and re-estimated their risk of experiencing a series of adverse life events as a test of the unrealistic optimism effect (Sharot et al., 2011). Although these studies used methodologies similar to those used in earlier research, the social conformity and unrealistic optimism effects were no longer evident after controlling for RTM. Based on these findings we suggest several ways to control for the RTM effect in social psychology studies, such as adding the initial rating as a covariate in regression analysis, selecting a subset of stimuli for which the participant' initial ratings were matched across experimental conditions, and using a control group.

Do leukocyte telomere length dynamics depend on baseline telomere length? An analysis that corrects for ‘regression to the mean’

Leukocyte telomere length (LTL) shortens with age. Longitudinal studies have reported accelerated LTL attrition when baseline LTL is longer. However, the dependency of LTL attrition on baseline LTL might stem from a statistical artifact known as regression to the mean (RTM). To our knowledge no published study of LTL dynamics (LTL and its attrition rate) has corrected for this phenomenon. We illustrate the RTM effect using replicate LTL measurements, and show, using simulated data, how the RTM effect increases with a rise in stochastic measurement variation (representing LTL measurement error), resulting in spurious increasingly elevated dependencies of attrition on baseline values. In addition, we re-analyzed longitudinal LTL data collected from four study populations to test the hypothesis that LTL attrition depends on baseline LTL. We observed that the rate of LTL attrition was proportional to baseline LTL, but correction for the RTM effect reduced the slope of the relationship by 57% when measurement error was low (coefficient of variation ~2%). A modest but statistically significant effect remained however, indicating that high baseline LTL is associated with higher LTL attrition even when correcting for the RTM effect. Baseline LTL explained 1.3% of the variation in LTL attrition, but this effect, which differed significantly between the study samples, appeared to be primarily attributable to the association in men (3.7%).

Defining multiple criteria for meaningful outcome in routine outcome measurement using the Health of the Nation Outcome Scales

Using the reliable and clinically significant change approach, we aimed to identify meaningful outcome indicators for the Health of the Nation Outcome Scales (HoNOS) and to combine them in a single model. We applied these indicators to the 1-year outcome of two large samples of people attending community mental health services in Italy (cohort 1) and the Netherlands (cohort 2). Data were drawn from two studies on routine outcome assessment. The criteria for meaningful outcome were defined on both study cohorts and both language versions of the scale. The model combined (a) two criteria for adequate change (at least 4 or 8 points change), (b) two cut-offs for clinically significant change (a total score of 10 was the threshold between mild and moderate illness, 13 between moderate and severe illness), and (c) a method for classifying stable subjects in three degrees of severity (stable in mild, moderate or severe illness). Results were compared with those given by the effect size (ES) and analysis of variance and covariance (ANOVA and ANCOVA). For the proposed approach the outcome of cohort 1 was better than cohort 2, with 65-67% of its subjects showing a positive outcome compared to only 45-46%. The other reference methods (ES and ANOVA), however, showed a greater improvement for cohort 2. ANCOVA indicated that the differences were due to regression to the mean (RTM) which showed opposite effects across the two cohorts. The proposed approach proved valuable and generalizable for interpreting outcome on HoNOS, scarcely influenced by the RTM effect. Its introduction could benefit outcome evaluation and management.

Single group, pre- and post-test research designs: Some methodological concerns

This article provides two illustrations of some of the factors that can influence findings from pre- and post-test research designs in evaluation studies, including regression to the mean (RTM), maturation, history and test effects. The first illustration involves a re-analysis of data from a study by Marsden (2004), in which pre-test scores are plotted against gain scores to demonstrate RTM effects. The second illustration is a methodological review of single group, pre- and post-test research designs (pre-experiments) that evaluate causal relationships between intervention and outcome. Re-analysis of Marsden’s prior data shows that learners with higher baseline scores consistently made smaller gains than those with lower baseline scores, demonstrating that RTM is clearly observable in single group, pre-post test designs. Our review found that 13% of the sample of 490 articles were evaluation studies. Of these evaluation studies, about half used an experimental design. However, a quarter used a single group, pre-post test design, and researchers using these designs did not mention possible RTM effects in their explanations, although other explanatory factors were mentioned. We conclude by describing how using experimental or quasi-experimental designs would have enabled researchers to explain their findings more accurately, and to draw more useful implications for pedagogy.

Empirical Bayes method in the study of traffic safety via heterogeneous negative multinomial model

In the study of traffic safety, expected crash frequencies across sites are generally estimated via the negative binomial model, assuming time invariant safety. Since the time invariant safety assumption may be invalid, Hauer (1997) proposed a modified empirical Bayes (EB) method. Despite the modification, no attempts have been made to examine the generalisable form of the marginal distribution resulting from the modified EB framework. Because the hyper-parameters needed to apply the modified EB method are not readily available, an assessment is lacking on how accurately the modified EB method estimates safety in the presence of the time variant safety and regression-to-the-mean (RTM) effects. This study derives the closed form marginal distribution, and reveals that the marginal distribution in the modified EB method is equivalent to the negative multinomial (NM) distribution, which is essentially the same as the likelihood function used in the random effects Poisson model. As a result, this study shows that the gamma posterior distribution from the multivariate Poisson–gamma mixture can be estimated using the NM model or the random effects Poisson model. This study also shows that the estimation errors from the modified EB method are systematically smaller than those from the comparison group method by simultaneously accounting for the RTM and time variant safety effects. Hence, the modified EB method via the NM model is a generalisable method for estimating safety in the presence of the time variant safety and the RTM effects.

The sensitivity of estimates of regression to the mean

Estimations of the effectiveness of remedial treatments in road safety analysis are frequently bedevilled by the problem of regression to the mean (RTM). The number of accidents x observed at a site in the “before” period is a “noisy” quantity: x is Poisson distributed about an (unknown) true mean m for that site, so that x = m + e. Sites selected for treatment tend to have a positive random error component e, which will on average be zero in the “after” period, even if no treatment is applied. Methods for estimating RTM usually require some assumption about the underlying (prior) between-site distribution of the true means f 0( m): for example, in the empirical Bayes method, a gamma distribution is assumed. The paper considers the impact of different assumptions for this distribution and, indeed, whether any distributional form needs to be assumed. Using Markov Chain Monte Carlo methods, a variety of distributional forms are assumed for f 0( m) and applied to each of a number of real data sets, including that from a major study on the effectiveness of speed cameras. It is shown that, in some cases, the size of the estimated RTM effect can be quite sensitive to the choice of distribution.

Red light for red-light cameras?: A meta-analysis of the effects of red-light cameras on crashes

A meta-analysis has been conducted on the effects of red-light cameras (RLCs) on intersection crashes. The size and direction of results reported from studies included in the meta-analysis are strongly affected by study methodology. The studies that have controlled for most confounding factors yield the least favourable results. Based on these studies, installation of RLCs leads to an overall increase in the number of crashes by about 15%. Rear-end collisions increase by about 40% and right angle collisions, which are the target crashes for RLC, are reduced by about 10%. All effects are, however, non-significant. Meta-regression analysis shows that results are more favourable when there is a lack of control for regression to the mean (RTM). An interaction is found between control for RTM and control for those spillover effects that result from the tendency of RLCs to affect crash levels in nearby intersections without RLC. In studies controlling for RTM, additional control for spillover effects reduces the favourability of results still further. Studies controlling for both RTM and spillover effects tend also to control for more additional factors than other studies. It is likely that the results are affected by additional moderator variables, which could not be investigated in this meta-analysis. RLCs may reduce crashes under some conditions, but on the whole RLCs do not seem to be a successful safety measure.

Estimating regression to the mean and true effects of an intervention in a four‐wave panel study

First, to analyse whether a taxation-related decrease in spirit prices had a similar effect on spirit consumption for low-, medium- and high-level drinkers. Secondly, as the relationship between baseline values and post-intervention changes is confounded with regression to the mean (RTM) effects, to apply different approaches for estimating the RTM effect and true change. Consumption of spirits and total alcohol consumption were analysed in a four-wave panel study (one pre-intervention and three post-intervention measurements) of 889 alcohol consumers sampled from the general population of Switzerland. Two correlational methods, one method quantitatively estimating the RTM effect and one growth curve approach based on hierarchical linear models (HLM), were used to estimate RTM effects among low-, medium- and high-level drinkers. Adjusted for RTM effects, high-level drinkers increased consumption more than lighter drinkers in the short term, but this was not a persisting effect. Changes in taxation affected mainly light and moderate drinkers in the long term. All methods concurred that RTM effects were present to a considerable degree, and methods quantifying the RTM effect or adjusting for it yielded similar estimates. Intervention studies have to consider RTM effects both in the study design and in the evaluation methods. Observed changes can be adjusted for RTM effects and true change can be estimated. The recommended method, particularly if the aim is to estimate change not only for the sample as a whole, but for groups of drinkers with different baseline consumption levels, is growth curve modelling. If reliability of measurement instruments cannot be increased, the incorporation of more than one pre-intervention measurement point may be a valuable adjustment of the study design.

Regression to the mean in substance use disorder treatment research

Regression to the mean (RTM) refers to the tendency for a group of cases that differ from the population mean to move (regress) towards the mean, on average, when re-assessed, if scores at the two points are less than perfectly correlated. This paper considers factors that affect the magnitude of RTM and how RTM may impact findings from primary studies and reviews of substance use disorder (SUD) treatment. The paper is guided largely by A Primer on Regression Artifacts by Campbell and Kenny. It reviews potential RTM effects in three areas of SUD treatment research. One is the extent to which within-group improvement in comparative treatment trials, including 'placebo effects', is a function of RTM. The second is the vulnerability of treatment evaluations employing non-equivalent control group designs to RTM and biased estimates of treatment effects when matching, or statistical equating is used to adjust for pre-existing group differences. The final issue is the impact of RTM in syntheses of research findings on SUD treatments. In particular, the tendency for later studies of a particular intervention to have smaller treatment effect sizes relative to earlier studies is considered as an RTM phenomenon. RTM is a pervasive, but often unrecognized phenomenon that can bias findings in SUD treatment studies and in systematic reviews of that research. SUD treatment researchers should be aware of RTM, take any available steps to reduce it, and try to diagnose whether it is still affecting research findings.

HOW TO SHOW THAT UNICORN MILK IS A CHRONOBIOTIC: THE REGRESSION-TO-THEMEAN STATISTICAL ARTIFACT

Few chronobiologists may be aware of the regression-to-the-mean (RTM) statistical artifact, even though it may have far-reaching influences on chronobiological data. With the aid of simulated measurements of the circadian rhythm phase of body temperature and a completely bogus stimulus (unicorn milk), we explain what RTM is and provide examples relevant to chronobiology. We show how RTM may lead to erroneous conclusions regarding individual differences in phase responses to rhythm disturbances and how it may appear as though unicorn milk has phase-shifting effects and can successfully treat some circadian rhythm disorders. Guidelines are provided to ensure RTM effects are minimized in chronobiological investigations. (Chronobiology International, 18(6), 1041–1053, 2001)