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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.

Purpose To provide a reference for study design comparing intraocular lens (IOL) power calculation formulas, to show that the standard deviation (SD) of the prediction error (PE) is the single most accurate measure of outcomes, and to provide the most recent statistical methods to determine P values for type 1 errors. Setting Baylor College of Medicine, Houston, Texas, and University of Southern California, Los Angeles, California, USA. Design Retrospective consecutive case series. Methods Two datasets comprised of 5200 and 13 301 single eyes were used. The SDs of the PEs for 11 IOL power calculation formulas were calculated for each dataset. The probability density functions of signed and absolute PE were determined. Results None of the probability distributions for any formula in either dataset was normal (Gaussian). All the original signed PE distributions were not normal, but symmetric and leptokurtotic (heavy tailed) and had higher peaks than a normal distribution. The absolute distributions were asymmetric and skewed to the right. The heteroscedastic method was much better at controlling the probability of a type I error than older methods. Conclusions (1) The criteria for patient and data inclusion were outlined; (2) the appropriate sample size was recommended; (3) the requirement that the formulas be optimized to bring the mean error to zero was reinforced; (4) why the SD is the single best parameter to characterize the performance of an IOL power calculation formula was demonstrated; and (5) and using the heteroscedastic statistical method was the preferred method of analysis was shown.

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.

This chapter covers one-way, two-way, and three-way designs where one or more factors involve dependent groups. R functions for dealing with global tests as well as multiple comparisons are included. Multivariate rank-based methods are covered as well.

Chapter 6 is generally concerned with issues related to multivariate data. The chapter begins with basic techniques for measuring the overall dispersion of a data cloud and how to measure the extent a particular point is nested within the cloud. These methods have practical value when testing hypotheses. Next, robust multivariate measures of location and scatter are introduced, followed by multivariate outlier detection techniques. Included are recent results related to situations where the number of variables is large. Confidence regions for robust multivariate measures of location, which take into account the overall structure of the data when dealing with outliers, are described. Inferential methods in the one-sample case are described, including an extension of Hotelling's method to trimmed means. Multivariate extensions of the Wilcoxon–Mann–Whitney test are covered, one of which is based on a projection of the data that helps characterize the extent two groups differ. The notion of data depth can be used to compare groups, which is based in part on a multivariate notion of ranks. This is followed by methods for comparing dependent groups. The chapter concludes with recent advances relevant to robust principal components, multivariate discriminate analysis, as well as some recent results related to functional data. R functions are described and illustrated.

Chapter 9 describes robust measures of association. Several types are available and their relative merits are discussed. Inferential methods based on these measures of association are described as well. Although Pearson's correlation is not robust, a method for testing the hypothesis that Pearson's correlation is equal to zero is included that allows heteroscedasticity. Methods for comparing measures of association, in the independent case, are described. In terms of detecting an association, there are two methods that test the hypothesis that the regression surface is a horizontal plane. That is, these methods are sensitive to curvature, which might be missed when using Pearson's correlation or even the robust measures of association covered in this chapter. R functions are described and illustrated that provide access to all of the measures of association.

This chapter describes how to estimate the measures of location and scatter that were introduced in Chapter 2 . Some related estimators are described and their relative merits are summarized. Included is a summary of some methods for estimating distributions, including kernel density estimators, which have practical value for reasons illustrated in subsequent chapters. Strategies for detecting outliers are introduced. Practical concerns about detecting outliers via the mean and variance are described and illustrated. Theoretically sound methods for estimating the standard errors of robust estimators are described, which are not intuitive based on standard training in an applied statistics course. R functions for estimating robust measures of location, and their standard errors, are described and illustrated.

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.