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INTRODUCTION: Physical activity (PA) is recognized as one of the key lifestyle behaviors that reduces risk of developing dementia late in life. However, PA also leads to increased respiration, and in areas with high levels of air pollution, PA may increase exposure to pollutants linked with higher risk of developing dementia. Here, we investigate whether air pollution attenuates the association between PA and dementia risk. METHODS: This prospective cohort study included 35,562 adults 60 and older from the UK Biobank. Average acceleration magnitude (ACC(ave)) from wrist-worn accelerometers was used to assess PA levels. Air pollution levels (NO, NO(2), PM(10), PM(2.5), PM(2.5–10), and PM(2.5 absorbance)) were estimated with land use regression methods. Incident all-cause dementia was derived from inpatient hospital records and death registry data. RESULTS: In adjusted models, ACC(ave) was associated with reduced risk of developing dementia (HR [CI]=0.71 [0.60,0.83]), while air pollution variables were not associated with dementia risk. There were significant interactions between ACC(ave) and PM(2.5) (HR(interaction) [CI]=1.33 [1.13,1.57]) and PM(2.5 absorbance) (HR(interaction) [CI]=1.24 [1.07,1.45]) on incident dementia. At the lowest tertiles of pollution, ACC(ave) was associated with reduced risk of incident dementia (HR(PM 2.5) [CI]=0.66 [0.49,0.91], HR(PM 2.5 absorbance) [CI]=0.60 [0.44,0.81]). At the highest tertiles of these pollutants, there was no significant association of ACC(ave) with incident dementia (HR(PM 2.5) [CI]=0.88 [0.68,1.14], HR(PM 2.5 absorbance) [CI]=0.79 [0.60,1.04]). CONCLUSIONS: PA is associated with reduced risk of developing all-cause dementia. However, exposure to even moderate levels of air pollution attenuates the benefits of PA on risk of dementia.

It is well known that the ordinary least squares (OLS) regression estimator is not robust. Many robust regression estimators have been proposed and inferential methods based on these estimators have been derived. However, for two independent groups, let θj(X) be some conditional measure of location for the jth group, given X, based on some robust regression estimator. An issue that has not been addressed is computing a 1 − α confidence interval for θ1(X) − θ2(X) in a manner that allows both within group and between group hetereoscedasticity. The paper reports the finite sample properties of a simple method for accomplishing this goal. Simulations indicate that, in terms of controlling the probability of a Type I error, the method performs very well for a wide range of situations, even with a relatively small sample size. In principle, any robust regression estimator can be used. The simulations are focused primarily on the Theil–Sen estimator, but some results using Yohai’s MM-estimator, as well as the Koenker and Bassett quantile regression estimator, are noted. Data from the Well Elderly II study, dealing with measures of meaningful activity using the cortisol awakening response as a covariate, are used to illustrate that the choice between an extant method based on a nonparametric regression estimator, and the method suggested here, can make a practical difference.

Let ρj be Pearson's correlation between Y and Xj (j = 1, 2). A problem that has received considerable attention is testing H0: ρ1 = ρ2. A well-known concern, however, is that Pearson's correlation is not robust (e.g., Wilcox, 2005), and the usual estimate of ρj, rj has a finite sample breakdown point of only 1/n. The goal in this paper is to consider extensions to situations where Pearson's correlation is replaced by a particular robust measure of association. Included are results where there are p > 2 predictors and the goal to compare any two subsets of m < p predictors.

When comparing two independent groups, the shift function compares all of the quantiles in a manner that controls the probability of at least one Type I error, assuming random sampling only. Moreover, it provides a much more detailed sense of how groups compare, versus using a single measure of location, and the associated plot of the data can yield valuable insights. This note examines the small-sample properties of an ex- tension of the shift function where the goal is to compare the distributions of two specified linear sums of the random variables under study, with an emphasis on a two-by-two design. A very simple method controls the proba- bility of a Type I error. Moreover, very little power is lost versus comparing means when sampling is from normal distributions with equal variances.

The paper considers the problem of comparing measures of lo- cation associated with two dependent groups when values are missing at random, with an emphasis on robust measures of location. It is known that simply imputing missing values can be unsatisfactory when testing hypothe- ses about means, so the goal here is to compare several alternative strategies that use all of the available data. Included are results on comparing means and a 20% trimmed mean. Yet another method is based on the usual median but differs from the other methods in a manner that is made obvious. (It is somewhat related to the formulation of the Wilcoxon-Mann-Whitney test for independent groups.) The strategies are compared in terms of Type I error probabilities and power.

Motivated by a situation encountered in the Well Elderly 2 study, the paper considers the problem of robust multiple comparisons based on K independent tests associated with 2K independent groups. A simple strategy is to use an extension of Dunnett’s T3 procedure, which is designed to control the probability of one or more Type I errors. However, this method and related techniques fail to take into account the overall pattern of p-values when making decisions about which hypotheses should be rejected. The paper suggests a multiple comparison procedure that does take the overall pattern into account and then describes general situations where this alternative approach makes a practical difference in terms of both power and the probability of one or more Type I errors. For reasons summarized in the paper, the focus is on 20% trimmed means, but in principle the method considered here is relevant to any situation where the Type I error probability of the individual tests can be controlled reasonably well.

Sedentary behavior (SB) is associated with cardiometabolic disease and mortality, but its association with dementia is currently unclear. This study investigates whether SB is associated with incident dementia regardless of engagement in physical activity (PA). A total of 146,651 participants from the UK Biobank who were 60 years or older and did not have a diagnosis of dementia (mean [SD] age: 64.59 [2.84] years) were included. Self-reported leisure-time SBs were divided into two domains: time spent watching television (TV) or time spent using a computer. A total of 3,507 individuals were diagnosed with all-cause dementia over a mean follow-up of 11.87 (±1.17) years. In models adjusted for a wide range of covariates, including time spent in PA, time spent watching TV was associated with increased risk of incident dementia (HR [95% CI] = 1.24 [1.15 to 1.32]) and time spent using a computer was associated with decreased risk of incident dementia (HR [95% CI] = 0.85 [0.81 to 0.90]). In joint associations with PA, TV time and computer time remained significantly associated with dementia risk at all PA levels. Reducing time spent in cognitively passive SB (i.e., TV time) and increasing time spent in cognitively active SB (i.e., computer time) may be effective behavioral modification targets for reducing risk of dementia regardless of engagement in PA.

Let p1,…, pJ denote the probability of a success for J independent random variables having a binomial distribution and let p(1) ≤ … ≤ p(J) denote these probabilities written in ascending order. The goal is to make a decision about which group has the largest probability of a success, p(J). Let p̂1,…, p̂J denote estimates of p1,…,pJ, respectively. The strategy is to test J − 1 hypotheses comparing the group with the largest estimate to each of the J − 1 remaining groups. For each of these J − 1 hypotheses that are rejected, decide that the group corresponding to the largest estimate has the larger probability of success. This approach has a power advantage over simply performing all pairwise comparisons. However, the more obvious methods for controlling the probability of one more Type I errors perform poorly for the situation at hand. A method for dealing with this is described and illustrated.

A basic method for comparing two independent groups is in terms of the probability that a randomly sampled observation from the first group is less than a randomly sampled observation from the second group. The Wilcoxon–Mann–Whitney test is based on an estimate of this probability, but it uses an incorrect estimate of the standard error when the distributions differ. Numerous methods have been derived that are aimed at dealing with this issue. The goal here is to suggest a method for estimating this probability, given the value of a covariate. A well-known quantile regression estimator provides a way of dealing with this issue. The paper reports simulation results on how well this method performs.

When dealing with regression, a well known concern is that a few bad leverage points can result in a poor fit to the bulk of the data. This is the case even when using various robust estimators, which is known as contamination bias. Currently, a relatively e ective method for detecting bad leverage points is based in part on the least median of squares regression estimator. This note suggests a modification of this method that is better able to detect bad leverage points. The modification also provides a substantially better technique for dealing with contamination bias.

When dealing with the association between some random variable and two covariates, extensive experience with smoothers indicates that often a linear model poorly reflects the nature of the association. A simple approach via quantile grids that reflects the nature of the association is given. The two main goals are to illustrate this approach can make a practical difference, and to describe R functions for applying it. Included are comments on dealing with more than two covariates.

A fundamental way of characterizing how two independent compares compare is in terms of the probability that a randomly sampled observation from the first group is less than a randomly sampled observation from the second group. The paper suggests a bivariate analog and investigates methods for computing confidence intervals. An interaction for a two-by-two design is investigated as well.

Being able to remove or weigh down the influence of outlier data is desirable for any statistical models. While Magnetic and ElectroEncephaloGraphic (MEEG) data are often averaged across trials per condition, it is becoming common practice to use information from all trials to build statistical linear models. Individual trials can, however, have considerable weight and thus bias inferential results (effect sizes as well as thresholded t/F/p maps). Here, rather than looking for univariate outliers, defined independently at each measurement point, we apply the principal component projection (PCP) method at each channel, deriving a single weight per trial at each channel independently. Using both synthetic data and open EEG data, we show (1) that PCP is efficient at detecting a large variety of outlying trials; (2) how PCP-based weights can be implemented in the context of the general linear model (GLM) with accurate control of type 1 family-wise error rate; and (3) that our PCP-based Weighted Least Square (WLS) approach increases the statistical power of group analyses as well as a much slower Iterative Reweighted Least Squares (IRLS), although the weighting scheme is markedly different. Together, our results show that WLS based on PCP weights derived from whole trial profiles is an efficient method to weigh down the influence of outlier EEG data in linear models.Data availabilityall data used are publicly available (CC0), all code (simulations and data analyzes) is also available online in the LIMO MEEG GitHub repository (MIT license).

Perceiving speech in noise (SIN) is important for health and well-being and decreases with age. Musicians show improved speech-in-noise abilities and reduced age-related auditory decline, yet it is unclear whether short term music engagement has similar effects. In this randomized control trial we used a pre-post design to investigate whether a 12-week music intervention in adults aged 50-65 without prior music training and with subjective hearing loss improves well-being, speech-in-noise abilities, and auditory encoding and voluntary attention as indexed by auditory evoked potentials (AEPs) in a syllable-in-noise task, and later AEPs in an oddball task. Age and gender-matched adults were randomized to a choir or control group. Choir participants sang in a 2-hr ensemble with 1-hr home vocal training weekly; controls listened to a 3-hr playlist weekly, attended concerts, and socialized online with fellow participants. From pre- to post-intervention, no differences between groups were observed on quantitative measures of well-being or behavioral speech-in-noise abilities. In the choir group, but not the control group, changes in the N1 component were observed for the syllable-in-noise task, with increased N1 amplitude in the passive condition and decreased N1 latency in the active condition. During the oddball task, larger N1 amplitudes to the frequent standard stimuli were also observed in the choir but not control group from pre to post intervention. Findings have implications for the potential role of music training to improve sound encoding in individuals who are in the vulnerable age range and at risk of auditory decline.

In a regression context, consider p independent variables and a single dependent variable. The paper addresses two goals. The first is to determine the extent it is reasonable to make a decision about whether the largest estimate of the Winsorized correlations corresponds to the independent variable that has the largest population Winsorized correlation. The second is to determine the extent it is reasonable to decide that the order of the estimates of the Winsorized correlations correctly reflects the true ordering. Both goals are addressed by testing relevant hypotheses. Results in Wilcox (in press) suggest using a multiple comparisons procedure designed specifically for the situations just described, but execution time can be quite high. A modified method for dealing with this issue is proposed.

For a binary random variable Y, let p(x) = P(Y = 1 | X = x) for some covariate X. The goal of computing a confidence interval for p(x) is considered. In the logistic regression model, even a slight departure difficult to detect via a goodness-of-fit test can yield inaccurate results. The accuracy of a confidence interval can deteriorate as the sample size increases. The goal is to suggest an alternative approach based on a smoother, which provides a more flexible approximation of p(x).

There is an extensive literature dealing with inferences about the probability of success. A minor goal in this note is to point out when certain recommended methods can be unsatisfactory when the sample size is small. The main goal is to report results on the two-sample case. Extant results suggest using one of four methods. The results indicate when computing a 0.95 confidence interval, two of these methods can be more satisfactory when dealing with small sample sizes.

To summarise skewed (asymmetric) distributions, such as reaction times, typically the mean or the median are used as measures of central tendency. Using the mean might seem surprising, given that it provides a poor measure of central tendency for skewed distributions, whereas the median provides a better indication of the location of the bulk of the observations. However, the sample median is biased: with small sample sizes, it tends to overestimate the population median. This is not the case for the mean. Based on this observation, Miller (1988) concluded that "sample medians must not be used to compare reaction times across experimental conditions when there are unequal numbers of trials in the conditions". Here we replicate and extend Miller (1988), and demonstrate that his conclusion was ill-advised for several reasons. First, the median's bias can be corrected using a percentile bootstrap bias correction. Second, a careful examination of the sampling distributions reveals that the sample median is median unbiased, whereas the mean is median biased when dealing with skewed distributions. That is, on average the sample mean estimates the population mean, but typically this is not the case. In addition, simulations of false and true positives in various situations show that no method dominates. Crucially, neither the mean nor the median are sufficient or even necessary to compare skewed distributions. Different questions require different methods and it would be unwise to use the mean or the median in all situations. Better tools are available to get a deeper understanding of how distributions differ: we illustrate the hierarchical shift function, a powerful alternative that relies on quantile estimation. All the code and data to reproduce the figures and analyses in the article are available online.

When comparing two independent groups, a possible appeal of the quantile shift measure of effect size is that its magnitude takes into account situations where one or both distributions are skewed. Extant results indicate that a percentile bootstrap method performs reasonably well given the goal of making inferences about this measure of effect size. The goal here is to suggest a method for making inferences about this measure of effect size when there is a covariate. The method is illustrated with data dealing with the wellbeing of older adults.

This paper reviews and offers tutorials on robust statistical methods relevant to clinical and experimental psychopathology researchers. We review the assumptions of one of the most commonly applied models in this journal (the general linear model, GLM) and the effects of violating them. We then present evidence that psychological data are more likely than not to violate these assumptions. Next, we overview some methods for correcting for violations of model assumptions. The final part of the paper presents 8 tutorials of robust statistical methods using R that cover a range of variants of the GLM (t-tests, ANOVA, multiple regression, multilevel models, latent growth models). We conclude with recommendations that set the expectations for what methods researchers submitting to the journal should apply and what they should report.

The percentile bootstrap is the Swiss Army Knife of statistics: it is a non-parametric method based on data-driven simulations. It can be applied to many statistical problems, as a substitute to standard parametric approaches, or in situations where parametric methods do not exist. In this tutorial, we cover R code to implement the percentile bootstrap in a few situations: one-sample estimation and the comparison of two independent groups for measures of central tendency (means and trimmed means) and spread. For each example, we explain how to derive a bootstrap distribution, and how to get a confidence interval and a p value from that distribution. We also demonstrate how to run a simulation to assess the behaviour of the bootstrap. In some situations, the bootstrap performs poorly, such as when making inferences about the mean. But for other purposes, it is the only known method that works well over a broad range of situations, such as when comparing medians and there are tied (duplicated) values. More broadly, combining the percentile bootstrap with robust estimators, i.e. estimators that are not overly sensitive to outliers, the bootstrap can help users gain a deeper understanding of their data, relative to conventional methods.

This paper introduces the R package WRS2 that implements various robust statistical methods. It elaborates on the basics of robust statistics by introducing robust location, dispersion, and correlation measures. The location and dispersion measures are then used in robust variants of independent and dependent samples t tests and ANOVA, including between-within subject designs and quantile ANOVA. Further, robust ANCOVA as well as robust mediation models are introduced. The paper targets applied researchers; it is therefore kept rather non-technical and written in a tutorial style. Special emphasis is placed on applications in the social and behavioral sciences and illustrations of how to perform corresponding robust analyses in R. The R code for reproducing the results in the paper is given in the Supplementary Materials.

The bootstrap is a versatile technique that relies on data-driven simulations to make statistical inferences. When combined with robust estimators, the bootstrap can afford much more powerful and flexible inferences than is possible with standard approaches such as t-tests on means. In this tutorial, we use detailed illustrations of bootstrap simulations to give readers an intuition of what the bootstrap does and how it can be applied to solve many practical problems, such as building confidence intervals for many aspects of the data. In particular, we illustrate how to build confidence intervals for measures of location, including measures of central tendency, in the one-sample case, for two independent and two dependent groups. We also demonstrate how to compare correlation coefficients using the bootstrap and to perform simulations to determine if the bootstrap is fit for purpose for a particular application. Our approach is to suggest and motivate what could be done in a situation, with an understanding that various options are valid, though they may help answer different questions about a dataset. The tutorial also addresses two widespread misconceptions about the bootstrap: that it makes no assumptions about the data, and that it leads to robust inferences on its own. The tutorial focuses on detailed graphical descriptions, with data and code available online to reproduce the figures and analyses in the article (OSF: https://osf.io/8b4t5/; GitHub: https://github.com/GRousselet/bootstrap).

When dealing with a logistic regression model, there is a simple method for estimating the strength of the association between the jth covariate and the dependent variable when all covariates are entered into the model. There is the issue of determining whether the jth independent variable has a stronger or weaker association than the kth independent variable. This note describes a method for dealing with this issue that was found to perform reasonably well in simulations.

There is a substantial collection of robust analysis of covariance (ANCOVA) methods that effectively deals with non-normality, unequal population slope parameters, outliers, and heteroscedasticity. Some are based on the usual linear model and others are based on smoothers (nonparametric regression estimators). However, extant results are limited to one or two covariates. A minor goal here is to extend a recently-proposed method, based on the usual linear model, to situations where there are up to six covariates. The usual linear model might provide a poor approximation of the true regression surface. The main goal is to suggest a method, based on a robust smoother, for dealing with curvature when there are three or four covariates. The results include perspectives on the curse of dimensionality. Perspectives on the use of a linear model versus a smoother are given.

The paper describes a nonparametric analog of Cohen's d, Q. It is established that a confidence interval for Q can be computed via a method for computing a confidence interval for the median of D = X1 − X2, which in turn is related to making inferences about P(X1 < X2).

For J ⩾ 2 independent groups, the article deals with testing the global hypothesis that all J groups have a common population median or identical quantiles, with an emphasis on the quartiles. Classic rank-based methods are sometimes suggested for comparing medians, but it is well known that under general conditions they do not adequately address this goal. Extant methods based on the usual sample median are unsatisfactory when there are tied values except for the special case J = 2. A variation of the percentile bootstrap used in conjunction with the Harrell–Davis quantile estimator performs well in simulations. The method is illustrated with data from the Well Elderly 2 study.

For robust measures of location associated with J dependent groups, various methods have been proposed that are aimed at testing the global hypothesis of a common measure of location applied to the marginal distributions. A criticism of these methods is that they do not deal with outliers in a manner that takes into account the overall structure of the data. Location estimators have been derived that deal with outliers in this manner, but evidently there are no simulation results regarding how well they perform when the goal is to test the some global hypothesis. The paper compares four bootstrap methods in terms of their ability to control the Type I error probability when the sample size is small, two of which were found to perform poorly. The choice of location estimator was found to be important as well. Indeed, for several of the estimators considered here, control over the Type I error probability was very poor. Only one estimator performed well when using the first of two general approaches that might be used. It is based on a variation of the (affine equivariant) Donoho-Gasko trimmed mean. For the second general approach, only a skipped estimator performed reasonably well. (It removes outliers via a projection method and averages the remaining data.) Only one bootstrap method was found to perform well when using the first approach. A different bootstrap method is recommended when using the second approach.

The article provides perspectives on p values, null hypothesis testing, and alternative techniques in light of modern robust statistical methods. Null hypothesis testing and p values can provide useful information provided they are interpreted in a sound manner, which includes taking into account insights and advances that have occurred during the past 50 years. There are, of course, limitations to what null hypothesis testing and p values reveal about data. But modern advances make it clear that there are serious limitations and concerns associated with conventional confidence intervals, standard Bayesian methods, and commonly used measures of effect size. Many of these concerns can be addressed using modern robust methods.

During initiation of a piqué turn, a dancer generates impulse to achieve the desired lateral translation and whole-body rotation. The goal of this study was to determine how individuals regulate impulse generation when initiating piqué turns with increased rotational demands. Skilled dancers (n=10) performed single (∼360°) and double (∼720°) piqué turns from a stationary position. Linear and angular impulse generated by the push and turn legs were quantified using ground reaction forces and compared across turn conditions as a group and within a dancer using probability-based statistical methods. The results indicate that as the rotation demands of the piqué turn increased, the net angular impulse generated increased whereas net lateral impulse decreased. Early during turn initiation, the free moment contributed to angular impulse generation. Later during turn initiation, horizontal reaction forces were controlled to generate angular impulse. As rotational demands increased, the moment applied increased primarily from redirection of the horizontal reaction force (RFh) at the push leg and a combination of RFh magnitude and moment arm increases at the turn leg. RFh at each leg were coordinated to limit unwanted net linear impulse. Knowledge of observed subject-specific mechanisms is important to inform the design of turning performance training tools.

The small-sample properties of two non-parametric quantile regression estimators are compared. The first is based on constrained B-spline smoothing (COBS) and the other is based on a variation and slight extension of a running interval smoother. R functions for applying the methods were used in conjunction with default settings for the various optional arguments. Results indicate that the modified running interval smoother has practical value. Manipulation of the optional arguments might impact the relative merits of the two methods, but the extent to which this is the case remains unknown.

For two independent groups, let Mj(x) be some conditional measure of location for the jth group associated with some random variable Y , given that some covariate X = x. When Mj(x) is a robust measure of location, or even some conditional quantile of Y , given X, methods have been proposed and studied that are aimed at testing H0: M1(x) = M2(x) that deal with curvature in a flexible manner. In addition, methods have been studied where the goal is to control the probability of one or more Type I errors when testing H0 for each x ∈ {x1, . . . , xp}. This paper suggests a method for testing the global hypothesis H0: M1(x) = M2(x) for ∀x ∈ {x1, . . . , xp} when using a robust or quantile location estimator. An obvious advantage of testing p hypotheses, rather than the global hypothesis, is that it can provide information about where regression lines differ and by how much. But the paper summarizes three general reasons to suspect that testing the global hypothesis can have more power. Data from the Well Elderly 2 study illustrate that testing the global hypothesis can make a practical difference.

To summarise skewed (asymmetric) distributions, such as reaction times, typically the mean or the median are used as measures of central tendency. Using the mean might seem surprising, given that it provides a poor measure of central tendency for skewed distributions, whereas the median provides a better indication of the location of the bulk of the observations. However, the sample median is biased: with small sample sizes, it tends to overestimate the population median. This is not the case for the mean. Based on this observation, Miller (1988) concluded that “sample medians must not be used to compare reaction times across experimental conditions when there are unequal numbers of trials in the conditions.” Here we replicate and extend Miller (1988), and demonstrate that his conclusion was ill-advised for several reasons. First, the median’s bias can be corrected using a percentile bootstrap bias correction. Second, a careful examination of the sampling distributions reveals that the sample median is median unbiased, whereas the mean is median biased when dealing with skewed distributions. That is, on average the sample mean estimates the population mean, but typically this is not the case. In addition, simulations of false and true positives in various situations show that no method dominates. Crucially, neither the mean nor the median are sufficient or even necessary to compare skewed distributions. Different questions require different methods and it would be unwise to use the mean or the median in all situations. Better tools are available to get a deeper understanding of how distributions differ: we illustrate a powerful alternative that relies on quantile estimation. All the code and data to reproduce the figures and analyses in the article are available online.

Secondhand smoke exposure is a major public health risk that is especially harmful to the developing brain, but it is unclear if early exposure affects brain structure during middle age and older adulthood. Here we analyzed brain MRI data from the UK Biobank in a population-based sample of individuals (ages 44-80) who were exposed (n=2,510) or unexposed (n=6,079) to smoking around birth. We used robust statistical models, including quantile regressions, to test the effect of perinatal smoke exposure (PSE) on cortical surface area (SA), thickness, and subcortical volume. We hypothesized that PSE would be associated with cortical disruption in primary sensory areas compared to unexposed (PSE-) adults. After adjusting for multiple comparisons, SA was significantly lower in the pericalcarine (PCAL), inferior parietal (IPL), and regions of the temporal and frontal cortex of PSE+adults; these abnormalities were associated with increased risk for several diseases, including circulatory and endocrine conditions. Sensitivity analyses conducted in a hold-out group of healthy participants (exposed, n=109, unexposed, n=315) replicated the effect of PSE on SA in the PCAL and IPL. Collectively our results show a negative, long term effect of PSE on sensory cortices that may increase risk for disease later in life.

Secondhand smoke exposure is a major public health risk that is especially harmful to the developing brain, but it is unclear if early exposure affects brain structure during middle age and older adulthood. Here we analyzed brain MRI data from the UK Biobank in a population-based sample of individuals (ages 44–80) who were exposed (n = 2510) or unexposed (n = 6079) to smoking around birth. We used robust statistical models, including quantile regressions, to test the effect of perinatal smoke exposure (PSE) on cortical surface area (SA), thickness, and subcortical volumes. We hypothesized that PSE would be associated with cortical disruption in primary sensory areas compared to unexposed (PSE−) adults. After adjusting for multiple comparisons, SA was significantly lower in the pericalcarine (PCAL), inferior parietal (IPL), and regions of the temporal and frontal cortex of PSE+ adults; these abnormalities were associated with increased risk for several diseases, including circulatory and endocrine conditions. Sensitivity analyses conducted in a hold-out group of healthy participants (exposed, n = 109, unexposed, n = 315) replicated the effect of PSE on SA in the PCAL and IPL. Collectively our results show a negative, long term effect of PSE on sensory cortices that may increase risk for disease later in life.

The article provides perspectives on p values, null hypothesis testing, and alternative techniques in light of modern robust statistical methods. Null hypothesis testing and p values can provide useful information provided they are interpreted in a sound manner, which includes taking into account insights and advances that have occurred during the past 50 years. There are, of course, limitations to what null hypothesis testing and p values reveal about data. But modern advances make it clear that there are serious limitations and concerns associated with conventional confidence intervals, standard Bayesian methods, and commonly used measures of effect size. Many of these concerns can be addressed using modern robust methods.

For two independent groups, let $$M_j(\mathbf {X})$$Mj(X) be some conditional measure of location for the jth group associated with some random variable Y given $$\mathbf {X}=(X_1, X_2)$$X=(X1,X2). Let $$\Omega =\{\mathbf {X}_1, \ldots , \mathbf {X}_K\}$$Ω={X1,ź,XK} be a set of K points to be determined. An extant technique can be used to test $$H_0$$H0: $$M_1(\mathbf {X})=M_2(\mathbf {X})$$M1(X)=M2(X) for each $$\mathbf {X} \in \Omega $$XźΩ without making any parametric assumption about $$M_j(\mathbf {X})$$Mj(X). But there are two general reasons to suspect that the method can have relatively low power. The paper reports simulation results on an alternative approach that is designed to test the global hypothesis $$H_0$$H0: $$M_1(\mathbf {X})=M_2(\mathbf {X})$$M1(X)=M2(X) for all $$\mathbf {X} \in \Omega $$XźΩ. The main result is that the new method offers a distinct power advantage. Using data from the Well Elderly 2 study, it is illustrated that the alternative method can make a practical difference in terms of detecting a difference between two groups.