Mixed Model Analysis of Variance
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APPLICATION of the TAGUCHI METHOD to SENSITIVITY ANALYSIS of a MIDDLE- EAR FINITE-ELEMENT MODEL Li Qi1, Chadia S
APPLICATION OF THE TAGUCHI METHOD TO SENSITIVITY ANALYSIS OF A MIDDLE- EAR FINITE-ELEMENT MODEL Li Qi1, Chadia S. Mikhael1 and W. Robert J. Funnell1, 2 1 Department of BioMedical Engineering 2 Department of Otolaryngology McGill University Montréal, QC, Canada H3A 2B4 ABSTRACT difference in the model output due to the change in the input variable is referred to as the sensitivity. The Sensitivity analysis of a model is the investigation relative importance of parameters is judged based on of how outputs vary with changes of input parameters, the magnitude of the calculated sensitivity. The OFAT in order to identify the relative importance of method does not, however, take into account the parameters and to help in optimization of the model. possibility of interactions among parameters. Such The one-factor-at-a-time (OFAT) method has been interactions mean that the model sensitivity to one widely used for sensitivity analysis of middle-ear parameter can change depending on the values of models. The results of OFAT, however, are unreliable other parameters. if there are significant interactions among parameters. Alternatively, the full-factorial method permits the This paper incorporates the Taguchi method into the analysis of parameter interactions, but generally sensitivity analysis of a middle-ear finite-element requires a very large number of simulations. This can model. Two outputs, tympanic-membrane volume be impractical when individual simulations are time- displacement and stapes footplate displacement, are consuming. A more practical approach is the Taguchi measured. Nine input parameters and four possible method, which is commonly used in industry. It interactions are investigated for two model outputs. -
Higher-Order Asymptotics
Higher-Order Asymptotics Todd Kuffner Washington University in St. Louis WHOA-PSI 2016 1 / 113 First- and Higher-Order Asymptotics Classical Asymptotics in Statistics: available sample size n ! 1 First-Order Asymptotic Theory: asymptotic statements that are correct to order O(n−1=2) Higher-Order Asymptotics: refinements to first-order results 1st order 2nd order 3rd order kth order error O(n−1=2) O(n−1) O(n−3=2) O(n−k=2) or or or or o(1) o(n−1=2) o(n−1) o(n−(k−1)=2) Why would anyone care? deeper understanding more accurate inference compare different approaches (which agree to first order) 2 / 113 Points of Emphasis Convergence pointwise or uniform? Error absolute or relative? Deviation region moderate or large? 3 / 113 Common Goals Refinements for better small-sample performance Example Edgeworth expansion (absolute error) Example Barndorff-Nielsen’s R∗ Accurate Approximation Example saddlepoint methods (relative error) Example Laplace approximation Comparative Asymptotics Example probability matching priors Example conditional vs. unconditional frequentist inference Example comparing analytic and bootstrap procedures Deeper Understanding Example sources of inaccuracy in first-order theory Example nuisance parameter effects 4 / 113 Is this relevant for high-dimensional statistical models? The Classical asymptotic regime is when the parameter dimension p is fixed and the available sample size n ! 1. What if p < n or p is close to n? 1. Find a meaningful non-asymptotic analysis of the statistical procedure which works for any n or p (concentration inequalities) 2. Allow both n ! 1 and p ! 1. 5 / 113 Some First-Order Theory Univariate (classical) CLT: Assume X1;X2;::: are i.i.d. -
Linear Mixed-Effects Modeling in SPSS: an Introduction to the MIXED Procedure
Technical report Linear Mixed-Effects Modeling in SPSS: An Introduction to the MIXED Procedure Table of contents Introduction. 1 Data preparation for MIXED . 1 Fitting fixed-effects models . 4 Fitting simple mixed-effects models . 7 Fitting mixed-effects models . 13 Multilevel analysis . 16 Custom hypothesis tests . 18 Covariance structure selection. 19 Random coefficient models . 20 Estimated marginal means. 25 References . 28 About SPSS Inc. 28 SPSS is a registered trademark and the other SPSS products named are trademarks of SPSS Inc. All other names are trademarks of their respective owners. © 2005 SPSS Inc. All rights reserved. LMEMWP-0305 Introduction The linear mixed-effects models (MIXED) procedure in SPSS enables you to fit linear mixed-effects models to data sampled from normal distributions. Recent texts, such as those by McCulloch and Searle (2000) and Verbeke and Molenberghs (2000), comprehensively review mixed-effects models. The MIXED procedure fits models more general than those of the general linear model (GLM) procedure and it encompasses all models in the variance components (VARCOMP) procedure. This report illustrates the types of models that MIXED handles. We begin with an explanation of simple models that can be fitted using GLM and VARCOMP, to show how they are translated into MIXED. We then proceed to fit models that are unique to MIXED. The major capabilities that differentiate MIXED from GLM are that MIXED handles correlated data and unequal variances. Correlated data are very common in such situations as repeated measurements of survey respondents or experimental subjects. MIXED extends repeated measures models in GLM to allow an unequal number of repetitions. -
Multilevel and Mixed-Effects Modeling 391
386 Statistics with Stata , . d itl 770/ ""01' ncdctemp However the residuals pass tests for white noise 111uahtemp, compar e WI 1 /0 l' r : , , '12 19) (p = .65), and a plot of observed and predicted values shows a good visual fit (Figure . , Multilevel and Mixed-Effects predict uahhat2 solar, ENSO & C02" label variable uahhat2 "predicted from volcanoes, Modeling predict uahres2, resid label variable uahres2 "UAH residuals from ARMAX model" wntestq uahres2, lags(25) portmanteau test for white noise portmanteau (Q)statistic 21. 7197 Mixed-effects modeling is basically regression analysis allowing two kinds of effects: fixed =rob > chi2(25) 0.6519 effects, meaning intercepts and slopes meant to describe the population as a whole, just as in Figure 12.19 ordinary regression; and also random effects, meaning intercepts and slopes that can vary across subgroups of the sample. All of the regression-type methods shown so far in this book involve fixed effects only. Mixed-effects modeling opens a new range of possibilities for multilevel o models, growth curve analysis, and panel data or cross-sectional time series, "r~ 00 01 Albright and Marinova (2010) provide a practical comparison of mixed-modeling procedures uj"'! > found in Stata, SAS, SPSS and R with the hierarchical linear modeling (HLM) software 0>- m developed by Raudenbush and Bryck (2002; also Raudenbush et al. 2005). More detailed ~o c explanation of mixed modeling and its correspondences with HLM can be found in Rabe• ro (I! Hesketh and Skrondal (2012). Briefly, HLM approaches multilevel modeling in several steps, ::IN co I' specifying separate equations (for example) for levelland level 2 effects. -
How Differences Between Online and Offline Interaction Influence Social
Available online at www.sciencedirect.com ScienceDirect Two social lives: How differences between online and offline interaction influence social outcomes 1 2 Alicea Lieberman and Juliana Schroeder For hundreds of thousands of years, humans only Facebook users,75% ofwhom report checking the platform communicated in person, but in just the past fifty years they daily [2]. Among teenagers, 95% report using smartphones have started also communicating online. Today, people and 45%reportbeingonline‘constantly’[2].Thisshiftfrom communicate more online than offline. What does this shift offline to online socializing has meaningful and measurable mean for human social life? We identify four structural consequences for every aspect of human interaction, from differences between online (versus offline) interaction: (1) fewer how people form impressions of one another, to how they nonverbal cues, (2) greater anonymity, (3) more opportunity to treat each other, to the breadth and depth of their connec- form new social ties and bolster weak ties, and (4) wider tion. The current article proposes a new framework to dissemination of information. Each of these differences identify, understand, and study these consequences, underlies systematic psychological and behavioral highlighting promising avenues for future research. consequences. Online and offline lives often intersect; we thus further review how online engagement can (1) disrupt or (2) Structural differences between online and enhance offline interaction. This work provides a useful offline interaction -
ARIADNE (Axion Resonant Interaction Detection Experiment): an NMR-Based Axion Search, Snowmass LOI
ARIADNE (Axion Resonant InterAction Detection Experiment): an NMR-based Axion Search, Snowmass LOI Andrew A. Geraci,∗ Aharon Kapitulnik, William Snow, Joshua C. Long, Chen-Yu Liu, Yannis Semertzidis, Yun Shin, and Asimina Arvanitaki (Dated: August 31, 2020) The Axion Resonant InterAction Detection Experiment (ARIADNE) is a collaborative effort to search for the QCD axion using techniques based on nuclear magnetic resonance. In the experiment, axions or axion-like particles would mediate short-range spin-dependent interactions between a laser-polarized 3He gas and a rotating (unpolarized) tungsten source mass, acting as a tiny, fictitious magnetic field. The experiment has the potential to probe deep within the theoretically interesting regime for the QCD axion in the mass range of 0.1- 10 meV, independently of cosmological assumptions. In this SNOWMASS LOI, we briefly describe this technique which covers a wide range of axion masses as well as discuss future prospects for improvements. Taken together with other existing and planned axion experi- ments, ARIADNE has the potential to completely explore the allowed parameter space for the QCD axion. A well-motivated example of a light mass pseudoscalar boson that can mediate new interactions or a “fifth-force” is the axion. The axion is a hypothetical particle that arises as a consequence of the Peccei-Quinn (PQ) mechanism to solve the strong CP problem of quantum chromodynamics (QCD)[1]. Axions have also been well motivated as a dark matter candidatedue to their \invisible" nature[2]. Axions can couple to fundamental fermions through a scalar vertex and a pseudoscalar vertex with very weak coupling strength. -
Statistical Analysis 8: Two-Way Analysis of Variance (ANOVA)
Statistical Analysis 8: Two-way analysis of variance (ANOVA) Research question type: Explaining a continuous variable with 2 categorical variables What kind of variables? Continuous (scale/interval/ratio) and 2 independent categorical variables (factors) Common Applications: Comparing means of a single variable at different levels of two conditions (factors) in scientific experiments. Example: The effective life (in hours) of batteries is compared by material type (1, 2 or 3) and operating temperature: Low (-10˚C), Medium (20˚C) or High (45˚C). Twelve batteries are randomly selected from each material type and are then randomly allocated to each temperature level. The resulting life of all 36 batteries is shown below: Table 1: Life (in hours) of batteries by material type and temperature Temperature (˚C) Low (-10˚C) Medium (20˚C) High (45˚C) 1 130, 155, 74, 180 34, 40, 80, 75 20, 70, 82, 58 2 150, 188, 159, 126 136, 122, 106, 115 25, 70, 58, 45 type Material 3 138, 110, 168, 160 174, 120, 150, 139 96, 104, 82, 60 Source: Montgomery (2001) Research question: Is there difference in mean life of the batteries for differing material type and operating temperature levels? In analysis of variance we compare the variability between the groups (how far apart are the means?) to the variability within the groups (how much natural variation is there in our measurements?). This is why it is called analysis of variance, abbreviated to ANOVA. This example has two factors (material type and temperature), each with 3 levels. Hypotheses: The 'null hypothesis' might be: H0: There is no difference in mean battery life for different combinations of material type and temperature level And an 'alternative hypothesis' might be: H1: There is a difference in mean battery life for different combinations of material type and temperature level If the alternative hypothesis is accepted, further analysis is performed to explore where the individual differences are. -
The Method of Maximum Likelihood for Simple Linear Regression
08:48 Saturday 19th September, 2015 See updates and corrections at http://www.stat.cmu.edu/~cshalizi/mreg/ Lecture 6: The Method of Maximum Likelihood for Simple Linear Regression 36-401, Fall 2015, Section B 17 September 2015 1 Recapitulation We introduced the method of maximum likelihood for simple linear regression in the notes for two lectures ago. Let's review. We start with the statistical model, which is the Gaussian-noise simple linear regression model, defined as follows: 1. The distribution of X is arbitrary (and perhaps X is even non-random). 2. If X = x, then Y = β0 + β1x + , for some constants (\coefficients", \parameters") β0 and β1, and some random noise variable . 3. ∼ N(0; σ2), and is independent of X. 4. is independent across observations. A consequence of these assumptions is that the response variable Y is indepen- dent across observations, conditional on the predictor X, i.e., Y1 and Y2 are independent given X1 and X2 (Exercise 1). As you'll recall, this is a special case of the simple linear regression model: the first two assumptions are the same, but we are now assuming much more about the noise variable : it's not just mean zero with constant variance, but it has a particular distribution (Gaussian), and everything we said was uncorrelated before we now strengthen to independence1. Because of these stronger assumptions, the model tells us the conditional pdf 2 of Y for each x, p(yjX = x; β0; β1; σ ). (This notation separates the random variables from the parameters.) Given any data set (x1; y1); (x2; y2);::: (xn; yn), we can now write down the probability density, under the model, of seeing that data: n n (y −(β +β x ))2 Y 2 Y 1 − i 0 1 i p(yijxi; β0; β1; σ ) = p e 2σ2 2 i=1 i=1 2πσ 1See the notes for lecture 1 for a reminder, with an explicit example, of how uncorrelated random variables can nonetheless be strongly statistically dependent. -
Use of the Kurtosis Statistic in the Frequency Domain As an Aid In
lEEE JOURNALlEEE OF OCEANICENGINEERING, VOL. OE-9, NO. 2, APRIL 1984 85 Use of the Kurtosis Statistic in the FrequencyDomain as an Aid in Detecting Random Signals Absmact-Power spectral density estimation is often employed as a couldbe utilized in signal processing. The objective ofthis method for signal ,detection. For signals which occur randomly, a paper is to compare the PSD technique for signal processing frequency domain kurtosis estimate supplements the power spectral witha new methodwhich computes the frequency domain density estimate and, in some cases, can be.employed to detect their presence. This has been verified from experiments vith real data of kurtosis (FDK) [2] forthe real and imaginary parts of the randomly occurring signals. In order to better understand the detec- complex frequency components. Kurtosis is defined as a ratio tion of randomlyoccurring signals, sinusoidal and narrow-band of a fourth-order central moment to the square of a second- Gaussian signals are considered, which when modeled to represent a order central moment. fading or multipath environment, are received as nowGaussian in Using theNeyman-Pearson theory in thetime domain, terms of a frequency domain kurtosis estimate. Several fading and multipath propagation probability density distributions of practical Ferguson [3] , has shown that kurtosis is a locally optimum interestare considered, including Rayleigh and log-normal. The detectionstatistic under certain conditions. The reader is model is generalized to handle transient and frequency modulated referred to Ferguson'swork for the details; however, it can signals by taking into account the probability of the signal being in a be simply said thatit is concernedwith detecting outliers specific frequency range over the total data interval. -
Auditor: an R Package for Model-Agnostic Visual Validation and Diagnostics
auditor: an R Package for Model-Agnostic Visual Validation and Diagnostics APREPRINT Alicja Gosiewska Przemysław Biecek Faculty of Mathematics and Information Science Faculty of Mathematics and Information Science Warsaw University of Technology Warsaw University of Technology Poland Faculty of Mathematics, Informatics and Mechanics [email protected] University of Warsaw Poland [email protected] May 27, 2020 ABSTRACT Machine learning models have spread to almost every area of life. They are successfully applied in biology, medicine, finance, physics, and other fields. With modern software it is easy to train even a complex model that fits the training data and results in high accuracy on test set. The problem arises when models fail confronted with the real-world data. This paper describes methodology and tools for model-agnostic audit. Introduced tech- niques facilitate assessing and comparing the goodness of fit and performance of models. In addition, they may be used for analysis of the similarity of residuals and for identification of outliers and influential observations. The examination is carried out by diagnostic scores and visual verification. Presented methods were implemented in the auditor package for R. Due to flexible and con- sistent grammar, it is simple to validate models of any classes. K eywords machine learning, R, diagnostics, visualization, modeling 1 Introduction Predictive modeling is a process that uses mathematical and computational methods to forecast outcomes. arXiv:1809.07763v4 [stat.CO] 26 May 2020 Lots of algorithms in this area have been developed and are still being develop. Therefore, there are countless possible models to choose from and a lot of ways to train a new new complex model. -
Mixed Model Methodology, Part I: Linear Mixed Models
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/271372856 Mixed Model Methodology, Part I: Linear Mixed Models Technical Report · January 2015 DOI: 10.13140/2.1.3072.0320 CITATIONS READS 0 444 1 author: jean-louis Foulley Université de Montpellier 313 PUBLICATIONS 4,486 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Football Predictions View project All content following this page was uploaded by jean-louis Foulley on 27 January 2015. The user has requested enhancement of the downloaded file. Jean-Louis Foulley Mixed Model Methodology 1 Jean-Louis Foulley Institut de Mathématiques et de Modélisation de Montpellier (I3M) Université de Montpellier 2 Sciences et Techniques Place Eugène Bataillon 34095 Montpellier Cedex 05 2 Part I Linear Mixed Model Models Citation Foulley, J.L. (2015). Mixed Model Methodology. Part I: Linear Mixed Models. Technical Report, e-print: DOI: 10.13140/2.1.3072.0320 3 1 Basics on linear models 1.1 Introduction Linear models form one of the most widely used tools of statistics both from a theoretical and practical points of view. They encompass regression and analysis of variance and covariance, and presenting them within a unique theoretical framework helps to clarify the basic concepts and assumptions underlying all these techniques and the ones that will be used later on for mixed models. There is a large amount of highly documented literature especially textbooks in this area from both theoretical (Searle, 1971, 1987; Rao, 1973; Rao et al., 2007) and practical points of view (Littell et al., 2002). -
5. Dummy-Variable Regression and Analysis of Variance
Sociology 740 John Fox Lecture Notes 5. Dummy-Variable Regression and Analysis of Variance Copyright © 2014 by John Fox Dummy-Variable Regression and Analysis of Variance 1 1. Introduction I One of the limitations of multiple-regression analysis is that it accommo- dates only quantitative explanatory variables. I Dummy-variable regressors can be used to incorporate qualitative explanatory variables into a linear model, substantially expanding the range of application of regression analysis. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 2 2. Goals: I To show how dummy regessors can be used to represent the categories of a qualitative explanatory variable in a regression model. I To introduce the concept of interaction between explanatory variables, and to show how interactions can be incorporated into a regression model by forming interaction regressors. I To introduce the principle of marginality, which serves as a guide to constructing and testing terms in complex linear models. I To show how incremental I -testsareemployedtotesttermsindummy regression models. I To show how analysis-of-variance models can be handled using dummy variables. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 3 3. A Dichotomous Explanatory Variable I The simplest case: one dichotomous and one quantitative explanatory variable. I Assumptions: Relationships are additive — the partial effect of each explanatory • variable is the same regardless of the specific value at which the other explanatory variable is held constant. The other assumptions of the regression model hold. • I The motivation for including a qualitative explanatory variable is the same as for including an additional quantitative explanatory variable: to account more fully for the response variable, by making the errors • smaller; and to avoid a biased assessment of the impact of an explanatory variable, • as a consequence of omitting another explanatory variables that is relatedtoit.