Showing papers by "Robert Tibshirani published in 1987"

Generalized Additive Models: Some Applications

Abstract: Generalized additive models have the form η(x) = α + σ fj (x j ), where η might be the regression function in a multiple regression or the logistic transformation of the posterior probability Pr(y = 1 | x) in a logistic regression. In fact, these models generalize the whole family of generalized linear models η(x) = β′x, where η(x) = g(μ(x)) is some transformation of the regression function. We use the local scoring algorithm to estimate the functions fj (xj ) nonparametrically, using a scatterplot smoother as a building block. We demonstrate the models in two different analyses: a nonparametric analysis of covariance and a logistic regression. The procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in a standard linear analysis. A variety of inferential tools have been developed to aid the analyst in assessing the relevance and significance of the estimated functions: these include confidence curves, degrees of freedom estimates, and approximat...

Local Likelihood Estimation

Abstract: A scatterplot smoother is applied to data of the form {(x 1, y 1), (x 2, y 2, …, (xn, yn )} and uses local fitting to estimate the dependence of Y on X. A simple example is the running lines smoother, which fits a least squares line to the y values falling in a window around each x value. The value of the estimated function at x is given by the value of the least squares line at x. A smoother generalizes the least squares line, which assumes that the dependence of Y on X is linear. In this article, we extend the idea of local fitting to likelihood-based regression models. One such application is to the class of generalized linear models (Nelder and Wedderburn 1972). We enlarge this class by replacing the covariate form β0 + xβ1 with an unspecified smooth function s(x). This function is estimated from the data by a technique we call local likelihood estimation. The method consists of maximum likelihood estimation for β0 and β1, applied in a window around each x value. Multiple covariates are incor...

Bootstrap confidence intervals and bootstrap approximations

Abstract: The BCa bootstrap procedure (Efron 1987) for constructing parametric and nonparametric confidence intervals is considered. Like the bootstrap, this procedure can be applied to complicated problems in a wide range of situations. For models indexed by a scalar parameter θ with efficient estimator , the BCa procedure relies on the existence of a transformation g(·) such that () is approximately normally distributed with standard deviation 1 + ag(θ), although explicit knowledge of g(·) is not required. In this article, we show how to construct this transformation by generalizing the one given by Efron (1987, sec. 10) for translation families. This construction consists of the composition of a variance-stabilizing transformation and a skewness-reducing transformation. It produces a new interval, the BCa interval, that is asymptotically equivalent to the BCa interval and can be computed without bootstrap sampling. We also derive from this construction an accurate approximation to the bootstrap distribu...

Non-Parametric Logistic and Proportional Odds Regression

Abstract: SUMMARY We describe the additive non-parametric logistic regression model of the form logit[P(x)] ==a+ -fj(xj), where P(x) = P(y = 1 1 x) for a 0-1 variable y, x is a vector of p covariates, and the f; are general real-valued functions. Each of the f; can be chosen to be either linear, general non-linear (estimated by a scatterplot smoother) or step functions for discrete covariates. The functions are estimated simultaneously using the "local scoring algorithm". The model can be used as an exploratory tool for uncovering the form of covariate effects or it can be used in a more formal manner in model building. We also describe the additive proportional odds model logit[yk(x)] = Ik)-fj(X1) for ordinal response data. Here Yk is the probability of the response being at most k: yk(X) = P( Y ? k I x). Both these models are motivated and described in detail, and several examples are given.

Short-term economic change and the utilization of mental health facilities in a metropolitan area.

Abstract: The relationship between economic change and mental disorder has been examined by several investigators over the past century. The purpose of this paper is to explore this basic relationship and determine its direction in a large Metropolitan area. A time series analysis was undertaken to investigate whether or not unemployment and indicators of mental health are related over time. Trends in the following four indicators of mental health were examined: 1. Number of admissions to all psychiatric facilities serving Metropolitan Toronto. 2. Number of admissions to one provincial psychiatric hospital (Queen Street Mental Health Centre) in Toronto, which serves a chronic population. 3. Number of discharges from Queen Street Mental Health Centre. 4. Number of admissions plus those assessed and not admitted to Queen Street Mental Health Centre. In order to assess the possible delayed effects of unemployment, correlation analyses were computed for several lag times. "Lag time" is defined here as the time delay between unemployment and its potential effects on mental health indicators. Lag times used were zero, three, six and twelve months. The best equation found was for a six month lag, indicating an inverse relationship; as unemployment increases, admissions and discharges decrease. Results suggest that there are no simple relationships between the dependent and independent variables. Observed trends may be due to much wider exogenous factors such as hospital capacity and changing admission criteria.

Generalized Additive Models, Cubic Splines and Penalized Likelihood.

Abstract: : Generalized additive models extended the class of generalized linear models by allowing an arbitrary smooth function for any or all of the covariates The functions are established by the local scoring procedure, using a smoother as a building block in an iterative algorithm This paper utilizes a cubic spline smoother in the algorithm and show how the resultant procedure can be view as a method for automatically smoothing a suitably defined partial residual, and more formally, a method for maximizing a penalized likelihood The authors also examine convergence of the inner (backfitting) loop in this case and illustrate these ideas with some binary response data Keywords: Spline; Non-parametric regression