Book•
Generalized Linear Models
01 Jan 1983-
TL;DR: In this paper, a generalization of the analysis of variance is given for these models using log- likelihoods, illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables), and gamma (variance components).
Abstract: The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log- likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components).
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TL;DR: In this article, a Bayesian approach to hypothesis testing, model selection, and accounting for model uncertainty is presented, which is straightforward through the use of the simple and accurate BIC approximation, and it can be done using the output from standard software.
Abstract: It is argued that P-values and the tests based upon them give unsatisfactory results, especially in large samples. It is shown that, in regression, when there are many candidate independent variables, standard variable selection procedures can give very misleading results. Also, by selecting a single model, they ignore model uncertainty and so underestimate the uncertainty about quantities of interest. The Bayesian approach to hypothesis testing, model selection, and accounting for model uncertainty is presented. Implementing this is straightforward through the use of the simple and accurate BIC approximation, and it can be done using the output from standard software. Specific results are presented for most of the types of model commonly used in sociology. It is shown that this approach overcomes the difficulties with P-values and standard model selection procedures based on them. It also allows easy comparison of nonnested models, and permits the quantification of the evidence for a null hypothesis of interest, such as a convergence theory or a hypothesis about societal norms.
6,100 citations
22 Apr 2014
TL;DR: The generalized additive model (GA) as discussed by the authors is a generalization of the generalized linear model, which replaces the linear model with a sum of smooth functions in an iterative procedure called local scoring algorithm.
Abstract: Likelihood-based regression models such as the normal linear regression model and the linear logistic model, assume a linear (or some other parametric) form for the covariates $X_1, X_2, \cdots, X_p$. We introduce the class of generalized additive models which replaces the linear form $\sum \beta_jX_j$ by a sum of smooth functions $\sum s_j(X_j)$. The $s_j(\cdot)$'s are unspecified functions that are estimated using a scatterplot smoother, in an iterative procedure we call the local scoring algorithm. The technique is applicable to any likelihood-based regression model: the class of generalized linear models contains many of these. In this class the linear predictor $\eta = \Sigma \beta_jX_j$ is replaced by the additive predictor $\Sigma s_j(X_j)$; hence, the name generalized additive models. We illustrate the technique with binary response and survival data. In both cases, the method proves to be useful in uncovering nonlinear covariate effects. It has the advantage of being completely automatic, i.e., no "detective work" is needed on the part of the statistician. As a theoretical underpinning, the technique is viewed as an empirical method of maximizing the expected log likelihood, or equivalently, of minimizing the Kullback-Leibler distance to the true model.
5,700 citations
TL;DR: Introduction Continuous Outcomes Binary Outcomes Testing and Fit Ordinal Outcomes Nominal outcomes Limited Outcomes Count Outcomes Conclusions
Abstract: Introduction Continuous Outcomes Binary Outcomes Testing and Fit Ordinal Outcomes Nominal Outcomes Limited Outcomes Count Outcomes Conclusions
5,248 citations
TL;DR: This study provides a working guide to boosted regression trees (BRT), an ensemble method for fitting statistical models that differs fundamentally from conventional techniques that aim to fit a single parsimonious model.
Abstract: Summary 1 Ecologists use statistical models for both explanation and prediction, and need techniques that are flexible enough to express typical features of their data, such as nonlinearities and interactions 2 This study provides a working guide to boosted regression trees (BRT), an ensemble method for fitting statistical models that differs fundamentally from conventional techniques that aim to fit a single parsimonious model Boosted regression trees combine the strengths of two algorithms: regression trees (models that relate a response to their predictors by recursive binary splits) and boosting (an adaptive method for combining many simple models to give improved predictive performance) The final BRT model can be understood as an additive regression model in which individual terms are simple trees, fitted in a forward, stagewise fashion 3 Boosted regression trees incorporate important advantages of tree-based methods, handling different types of predictor variables and accommodating missing data They have no need for prior data transformation or elimination of outliers, can fit complex nonlinear relationships, and automatically handle interaction effects between predictors Fitting multiple trees in BRT overcomes the biggest drawback of single tree models: their relatively poor predictive performance Although BRT models are complex, they can be summarized in ways that give powerful ecological insight, and their predictive performance is superior to most traditional modelling methods 4 The unique features of BRT raise a number of practical issues in model fitting We demonstrate the practicalities and advantages of using BRT through a distributional analysis of the short-finned eel ( Anguilla australis Richardson), a native freshwater fish of New Zealand We use a data set of over 13 000 sites to illustrate effects of several settings, and then fit and interpret a model using a subset of the data We provide code and a tutorial to enable the wider use of BRT by ecologists
4,787 citations
TL;DR: New normal linear modeling strategies are presented for analyzing read counts from RNA-seq experiments, and the voom method estimates the mean-variance relationship of the log-counts, generates a precision weight for each observation and enters these into the limma empirical Bayes analysis pipeline.
Abstract: New normal linear modeling strategies are presented for analyzing read counts from RNA-seq experiments. The voom method estimates the mean-variance relationship of the log-counts, generates a precision weight for each observation and enters these into the limma empirical Bayes analysis pipeline. This opens access for RNA-seq analysts to a large body of methodology developed for microarrays. Simulation studies show that voom performs as well or better than count-based RNA-seq methods even when the data are generated according to the assumptions of the earlier methods. Two case studies illustrate the use of linear modeling and gene set testing methods.
4,475 citations
References
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01 Jan 1972
TL;DR: The drum mallets disclosed in this article are adjustable, by the percussion player, as to balance, overall weight, head characteristics and tone production of the mallet, whereby the adjustment can be readily obtained.
Abstract: The drum mallets disclosed are adjustable, by the percussion player, as to weight and/or balance and/or head characteristics, so as to vary the "feel" of the mallet, and thus also the tonal effect obtainable when playing upon kettle-drums, snare-drums, and other percussion instruments; and, typically, the mallet has frictionally slidable, removable and replaceable, external balancing mass means, positionable to serve as the striking head of the mallet, whereby the adjustment as to balance, overall weight, head characteristics and tone production may be readily obtained. In some forms, the said mass means regularly serves as a removable and replaceable striking head; while in other forms, the mass means comprises one or more thin elongated tubes having a frictionally-gripping fit on an elongated mallet body, so as to be manually slidable thereon but tight enough to avoid dislodgment under normal playing action; and such a tubular member may be slidable to the head-end of the mallet to serve as a striking head or it may be slidable to a position to serve as a hand grip; and one or more such tubular members may be placed in various positions along the length of the mallet. The mallet body may also have a tapered element at the head-end to assure retention of mass members especially of enlarged-head types; and the disclosure further includes such heads embodying a relatively hard inner portion and a relatively soft outer covering.
10,148 citations
01 May 1972
TL;DR: In this paper, the authors used iterative weighted linear regression to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation.
Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Blackwell Publishing and Royal Statistical Society are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series A (General). SUMMARY The technique of iterative weighted linear regression can be used to obtain maximum likelihood estimates of the parameters with observations distributed according to some exponential family and systematic effects that can be made linear by a suitable transformation. A generalization of the analysis of variance is given for these models using log-likelihoods. These generalized linear models are illustrated by examples relating to four distributions; the Normal, Binomial (probit analysis, etc.), Poisson (contingency tables) and gamma (variance components). The implications of the approach in designing statistics courses are discussed.
8,793 citations
Book•
01 Jan 1959
TL;DR: The general decision problem, the Probability Background, Uniformly Most Powerful Tests, Unbiasedness, Theory and First Applications, and UNbiasedness: Applications to Normal Distributions, Invariance, Linear Hypotheses as discussed by the authors.
Abstract: The General Decision Problem.- The Probability Background.- Uniformly Most Powerful Tests.- Unbiasedness: Theory and First Applications.- Unbiasedness: Applications to Normal Distributions.- Invariance.- Linear Hypotheses.- The Minimax Principle.- Multiple Testing and Simultaneous Inference.- Conditional Inference.- Basic Large Sample Theory.- Quadratic Mean Differentiable Families.- Large Sample Optimality.- Testing Goodness of Fit.- General Large Sample Methods.
6,480 citations