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      Cryptic multiple hypotheses testing in linear models: overestimated effect sizes and the winner's curse

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          Abstract

          Fitting generalised linear models (GLMs) with more than one predictor has become the standard method of analysis in evolutionary and behavioural research. Often, GLMs are used for exploratory data analysis, where one starts with a complex full model including interaction terms and then simplifies by removing non-significant terms. While this approach can be useful, it is problematic if significant effects are interpreted as if they arose from a single a priori hypothesis test. This is because model selection involves cryptic multiple hypothesis testing, a fact that has only rarely been acknowledged or quantified. We show that the probability of finding at least one ‘significant’ effect is high, even if all null hypotheses are true (e.g. 40% when starting with four predictors and their two-way interactions). This probability is close to theoretical expectations when the sample size ( N) is large relative to the number of predictors including interactions ( k). In contrast, type I error rates strongly exceed even those expectations when model simplification is applied to models that are over-fitted before simplification (low N/ k ratio). The increase in false-positive results arises primarily from an overestimation of effect sizes among significant predictors, leading to upward-biased effect sizes that often cannot be reproduced in follow-up studies (‘the winner's curse’). Despite having their own problems, full model tests and P value adjustments can be used as a guide to how frequently type I errors arise by sampling variation alone. We favour the presentation of full models, since they best reflect the range of predictors investigated and ensure a balanced representation also of non-significant results.

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          Null Hypothesis Testing: Problems, Prevalence, and an Alternative

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            Is Open Access

            Conclusions beyond support: overconfident estimates in mixed models

            Mixed-effect models are frequently used to control for the nonindependence of data points, for example, when repeated measures from the same individuals are available. The aim of these models is often to estimate fixed effects and to test their significance. This is usually done by including random intercepts, that is, intercepts that are allowed to vary between individuals. The widespread belief is that this controls for all types of pseudoreplication within individuals. Here we show that this is not the case, if the aim is to estimate effects that vary within individuals and individuals differ in their response to these effects. In these cases, random intercept models give overconfident estimates leading to conclusions that are not supported by the data. By allowing individuals to differ in the slopes of their responses, it is possible to account for the nonindependence of data points that pseudoreplicate slope information. Such random slope models give appropriate standard errors and are easily implemented in standard statistical software. Because random slope models are not always used where they are essential, we suspect that many published findings have too narrow confidence intervals and a substantially inflated type I error rate. Besides reducing type I errors, random slope models have the potential to reduce residual variance by accounting for between-individual variation in slopes, which makes it easier to detect treatment effects that are applied between individuals, hence reducing type II errors as well.
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              Individual recognition: it is good to be different.

              Individual recognition (IR) behavior has been widely studied, uncovering spectacular recognition abilities across a range of taxa and modalities. Most studies of IR focus on the recognizer (receiver). These studies typically explore whether a species is capable of IR, the cues that are used for recognition and the specializations that receivers use to facilitate recognition. However, relatively little research has explored the other half of the communication equation: the individual being recognized (signaler). Provided there is a benefit to being accurately identified, signalers are expected to actively broadcast their identity with distinctive cues. Considering the prevalence of IR, there are probably widespread benefits associated with distinctiveness. As a result, selection for traits that reveal individual identity might represent an important and underappreciated selective force contributing to the evolution and maintenance of genetic polymorphisms.
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                Author and article information

                Contributors
                +46-18-4712827 , forstmeier@orn.mpg.de
                holger.schielzeth@ebc.uu.se
                Journal
                Behav Ecol Sociobiol
                Behavioral Ecology and Sociobiology
                Springer-Verlag (Berlin/Heidelberg )
                0340-5443
                1432-0762
                19 August 2010
                19 August 2010
                January 2011
                : 65
                : 1
                : 47-55
                Affiliations
                [1 ]Max Planck Institute for Ornithology, Eberhard-Gwinner-Str., 82319 Seewiesen, Germany
                [2 ]Department of Evolutionary Biology, Evolutionary Biology Centre, Uppsala University, Norbyvägen 18D, SE-752 36 Uppsala, Sweden
                Author notes

                Communicated by L. Garamszegi

                Article
                1038
                10.1007/s00265-010-1038-5
                3015194
                21297852
                f941ed47-d0d3-469f-8c57-709c5227e156
                © The Author(s) 2010
                History
                : 12 May 2010
                : 23 July 2010
                : 29 July 2010
                Categories
                Original Paper
                Custom metadata
                © Springer-Verlag 2010

                Ecology
                model selection,multiple regression,bonferroni correction,generalised linear models,effect size estimation,multiple testing,publication bias,parameter estimation

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