Estimating Explanatory Power in a Simple Regression Model Via Smoothers

Consider the regression model Y = γ(X) + ε , where γ(X) is some conditional measure of location associated with Y , given X. Let Υ̂ be some estimate of Y, given X, and let τ2 (Y) be some measure of variation. Explanatory power is η2 = τ2 (Υ̂) /τ2(Y) . When γ(X) = β0 + β1X and τ2(Y) is the variance of Y , η2 = ρ2 , where ρ is Pearson\u27s correlation. The small-sample properties of some methods for estimating a robust analog of explanatory power via smoothers is investigated. The robust version of a smoother proposed by Cleveland is found to be best in most cases

Inferences About the Components of a Generalized Additive Model

A method for making inferences about the components of a generalized additive model is described. It is found that a variation of the method, based on means, performs well in simulations. Unlike many other inferential methods, switching from a mean to a 20% trimmed mean was found to offer little or no advantage in terms of both power and controlling the probability of a Type I error

Comparing Two Independent Groups Via a Quantile Generalization of the Wilcoxon-Mann-Whitney Test

The Wilcoxon-Mann-Whitney test, as well as modern improvements, are based in part on an estimate of p = P(D \u3c 0), where D = X−Y and X and Y are independent random variables; a common goal is to test H0: p = 0.5. This corresponds to testing H0: ξ0.5, where ξ0.5 is the 0.5 quantile of the distribution of D. If the distributions associated with X and Y do not differ, then D has a symmetric distribution about zero. In particular, ξq + ξ1-q = 0 for any q ≤ 0.5, where ξq is the qth quantile. Methods aimed at testing H0: p = 0.5 are generalized by suggesting a method for testing H0: ξq + ξ1-q = 0, q \u3c 0.

On a Test of Independence via Quantiles that is Sensitive to Curvature

Let (Yi ,Xi ) , i =1,..., n , be a random sample from some p+1 variate distribution where Xi is a vector having length p. Many methods for testing the hypothesis that Y is independent of X are relatively insensitive to a broad class of departures from independence. Power improvements focus on the median of Y or some other quantile and test the hypothesis that the regression surface is a horizontal plane versus some unknown form. A wild bootstrap method (Stute et al. 1998) can be used based on quantiles, but with small or moderate sample sizes, control over the probability of a Type I error can be unsatisfactory when sampling from asymmetric distributions. He and Zhu (2003) is readily adapted to testing the hypothesis that the conditional & gamma; quantile of Y does not depend on X where critical values are determined via simulations. A modification is suggested that avoids the need for simulations to obtain critical values, and perform wells in terms of Type I errors even when sampling from asymmetric distributions

An Omnibus Test When Using a Regression Estimator With Multiple Predictors

In quantile regression, the goal is to estimate theγ quantile of Y given values for p predictors. Methods for making inferences about the individual slope parameters have been proposed, some of which have been found to perform very well in simulations. But for an omnibus test that all slope parameters are zero, it appears that little is known about how best to proceed. For the special case γ =.5, a drop-in-dispersion test has been recommended, but it requires a large sample size to control the probability of a Type I error and it assumes that the usual error term is homoscedastic. The article suggests an alternative method that performs well in simulations, it allows heteroscedasticity, and it can be used when γ ≠ .5

Comparing the Strength of Association of Two Predictors via Smoothers or Robust Regression Estimators

Consider three random variables, Y , X1 and X2, having some unknown trivariate distribution and let n2j (j = 1, 2) be some measure of the strength of association between Y and Xj. When n2j is taken to be Pearson’s correlation numerous methods for testing Ho : n21 = n22 have been proposed. However, Pearson’s correlation is not robust and the methods for testing H0 are not level robust in general. This article examines methods for testing H0 based on a robust fit. The first approach assumes a linear model and the second approach uses a nonparametric regression estimator that provides a flexible way of dealing with curvature. The focus is on the Theil-Sen estimator and Cleveland’s LOESS smoother. It is found that a basic percentile bootstrap method avoids Type I errors that exceed the nominal level. However, situations are identified where this approach results in Type I error probabilities well below the nominal level. Adjustments are suggested for dealing with this problem

ANCOVA: A Robust Omnibus Test Based On Selected Design Points

Many robust analogs of the classic analysis of covariance method have been proposed. One approach, when comparing two independent groups, uses selected design points and then compares the groups at each design point using some robust method for comparing measures of location. So, if K design points are of interest, K tests are performed. There are rather obvious ways of performing, instead, an omnibus test that for all K points, no differences between the groups exist. One of the main results here is that several variations of these methods can perform very poorly in simulations. An alternative approach, based in part on the usual sample median, is suggested and found to perform reasonably well in simulations. It is noted that when using other robust measures of location, the method can be unsatisfactory

Inferences about the Population Mean: Empirical Likelihood versus Bootstrap-t

The problem of making inferences about the population mean, μ, is considered. Known theoretical results suggest that a Bartlett corrected empirical likelihood method is preferable to two basic bootstrap techniques: a symmetric two-sided bootstrap-t and an equal-tailed bootstrap-t. However, simulations in this study indicate that, when the sample size is small, these two bootstrap methods are generally better in terms of Type I errors and probability coverage. As the sample size increases, situations are found where the Bartlett corrected empirical likelihood method performs better than the equal-tailed bootstrap-t, but the symmetric bootstrap-t gives the best results. None of the four methods considered are always satisfactory in terms of probability coverage or Type I errors, particularly when dealing with skewed distributions where the expected proportion of points flagged as outliers is somewhat high. If this proportion is 0.14, for example, all four methods can be unsatisfactory even with n=300, but if sampling from a symmetric distribution or a skewed distribution with relatively light tails the results suggest using a symmetric two-sided bootstrap-t method

On Flexible Tests of Independence and homoscedasticity

Consider the nonparametric regression model Y = m(X) + τ(X)ε , where X and ε are independent random variables, ε has a mean of zero and variance σ2, τ is some unknown function used to model heteroscedasticity, and m(X) is an unknown function reflecting some conditional measure of location associated with Y, given X. Detecting dependence, by testing the hypothesis that m(X) does not vary with X, has the potential of being more sensitive to a wider range of associations compared to using Pearson\u27s correlation. This note has two goals. The first is to point out situations where a certain variation of an extant test of this hypothesis fails to control the probability of a Type I error, but another variation avoids this problem. The successful variation provides a new test of H0:τ(X) ≡ 1, the hypothesis that the error term is homoscedastic, which has the potential of higher power versus a method recently studied by Wilcox (2006). The second goal is to report some simulation results on how this method performs