Comparisons of two quantile regression smoothers

The paper compares the small-sample properties of two non-parametric quantile regression estimators. 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, which apparently has not been studied via simulations. The motivation for this paper stems from the Well Elderly 2 study, a portion of which was aimed at understanding the association between the cortisol awakening response and two measures of stress. COBS indicated what appeared be an usual form of curvature. The modified running interval smoother gave a strikingly different estimate, which raised the issue of how it compares to COBS in terms of mean squared error and bias as well as its ability to avoid a spurious indication of curvature. R functions for applying the methods were used in conjunction with default settings for the various optional arguments. The 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.Comment: 18 pp, 5 figure

ANCOVA: A global test based on a robust measure of location or quantiles when there is curvature

For two independent groups, let $M_j(x)$ be some conditional measure of location for the $j$th group associated with some random variable $Y$, given that some covariate $X=x$. When $M_j(x)$ 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 $H_0$: $M_1(x)=M_2(x)$ 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 $H_0$ for each $x \in \{x_1, \ldots, x_p\}$. This paper suggests a method for testing the global hypothesis $H_0$: $M_1(x)=M_2(x)$ for $\forall x \in \{x_1, \ldots, x_p\}$ 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. 2 Data from the Well Elderly 2 study illustrate that testing the global hypothesis can make a practical difference.Comment: 23 pp 2 Figure

A Note on Inferences About the Probability of Success

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

Regression: Determining Which of p Independent Variables Has the Largest or Smallest Correlation With the Dependent Variable, Plus Results on Ordering the Correlations Winsorized

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 a) 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

Inferences About the Probability of Success, Given the Value of a Covariate, Using a Nonparametric Smoother

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)

Bivariate Analogs of the Wilcoxon–Mann–Whitney test and the Patel–Hoel Method for Interactions

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

Regression When There Are Two Covariates: Some Practical Reasons for Considering Quantile Grids

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

Robust ANCOVA, Curvature, and the Curse of Dimensionality

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

Identifying Which of \u3cem\u3eJ\u3c/em\u3e Independent Binomial Distributions Has the Largest Probability of Success

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