5. Dummy-Variable Regression and Analysis of Variance
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A Simple Censored Median Regression Estimator
Statistica Sinica 16(2006), 1043-1058 A SIMPLE CENSORED MEDIAN REGRESSION ESTIMATOR Lingzhi Zhou The Hong Kong University of Science and Technology Abstract: Ying, Jung and Wei (1995) proposed an estimation procedure for the censored median regression model that regresses the median of the survival time, or its transform, on the covariates. The procedure requires solving complicated nonlinear equations and thus can be very difficult to implement in practice, es- pecially when there are multiple covariates. Moreover, the asymptotic covariance matrix of the estimator involves the density of the errors that cannot be esti- mated reliably. In this paper, we propose a new estimator for the censored median regression model. Our estimation procedure involves solving some convex min- imization problems and can be easily implemented through linear programming (Koenker and D'Orey (1987)). In addition, a resampling method is presented for estimating the covariance matrix of the new estimator. Numerical studies indi- cate the superiority of the finite sample performance of our estimator over that in Ying, Jung and Wei (1995). Key words and phrases: Censoring, convexity, LAD, resampling. 1. Introduction The accelerated failure time (AFT) model, which relates the logarithm of the survival time to covariates, is an attractive alternative to the popular Cox (1972) proportional hazards model due to its ease of interpretation. The model assumes that the failure time T , or some monotonic transformation of it, is linearly related to the covariate vector Z 0 Ti = β0Zi + "i; i = 1; : : : ; n: (1.1) Under censoring, we only observe Yi = min(Ti; Ci), where Ci are censoring times, and Ti and Ci are independent conditional on Zi. -
Cluster Analysis, a Powerful Tool for Data Analysis in Education
International Statistical Institute, 56th Session, 2007: Rita Vasconcelos, Mßrcia Baptista Cluster Analysis, a powerful tool for data analysis in Education Vasconcelos, Rita Universidade da Madeira, Department of Mathematics and Engeneering Caminho da Penteada 9000-390 Funchal, Portugal E-mail: [email protected] Baptista, Márcia Direcção Regional de Saúde Pública Rua das Pretas 9000 Funchal, Portugal E-mail: [email protected] 1. Introduction A database was created after an inquiry to 14-15 - year old students, which was developed with the purpose of identifying the factors that could socially and pedagogically frame the results in Mathematics. The data was collected in eight schools in Funchal (Madeira Island), and we performed a Cluster Analysis as a first multivariate statistical approach to this database. We also developed a logistic regression analysis, as the study was carried out as a contribution to explain the success/failure in Mathematics. As a final step, the responses of both statistical analysis were studied. 2. Cluster Analysis approach The questions that arise when we try to frame socially and pedagogically the results in Mathematics of 14-15 - year old students, are concerned with the types of decisive factors in those results. It is somehow underlying our objectives to classify the students according to the factors understood by us as being decisive in students’ results. This is exactly the aim of Cluster Analysis. The hierarchical solution that can be observed in the dendogram presented in the next page, suggests that we should consider the 3 following clusters, since the distances increase substantially after it: Variables in Cluster1: mother qualifications; father qualifications; student’s results in Mathematics as classified by the school teacher; student’s results in the exam of Mathematics; time spent studying. -
The Detection of Heteroscedasticity in Regression Models for Psychological Data
Psychological Test and Assessment Modeling, Volume 58, 2016 (4), 567-592 The detection of heteroscedasticity in regression models for psychological data Andreas G. Klein1, Carla Gerhard2, Rebecca D. Büchner2, Stefan Diestel3 & Karin Schermelleh-Engel2 Abstract One assumption of multiple regression analysis is homoscedasticity of errors. Heteroscedasticity, as often found in psychological or behavioral data, may result from misspecification due to overlooked nonlinear predictor terms or to unobserved predictors not included in the model. Although methods exist to test for heteroscedasticity, they require a parametric model for specifying the structure of heteroscedasticity. The aim of this article is to propose a simple measure of heteroscedasticity, which does not need a parametric model and is able to detect omitted nonlinear terms. This measure utilizes the dispersion of the squared regression residuals. Simulation studies show that the measure performs satisfactorily with regard to Type I error rates and power when sample size and effect size are large enough. It outperforms the Breusch-Pagan test when a nonlinear term is omitted in the analysis model. We also demonstrate the performance of the measure using a data set from industrial psychology. Keywords: Heteroscedasticity, Monte Carlo study, regression, interaction effect, quadratic effect 1Correspondence concerning this article should be addressed to: Prof. Dr. Andreas G. Klein, Department of Psychology, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 6, 60629 Frankfurt; email: [email protected] 2Goethe University Frankfurt 3International School of Management Dortmund Leipniz-Research Centre for Working Environment and Human Factors 568 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel Introduction One of the standard assumptions underlying a linear model is that the errors are inde- pendently identically distributed (i.i.d.). -
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. -
Testing Hypotheses for Differences Between Linear Regression Lines
United States Department of Testing Hypotheses for Differences Agriculture Between Linear Regression Lines Forest Service Stanley J. Zarnoch Southern Research Station e-Research Note SRS–17 May 2009 Abstract The general linear model (GLM) is often used to test for differences between means, as for example when the Five hypotheses are identified for testing differences between simple linear analysis of variance is used in data analysis for designed regression lines. The distinctions between these hypotheses are based on a priori assumptions and illustrated with full and reduced models. The studies. However, the GLM has a much wider application contrast approach is presented as an easy and complete method for testing and can also be used to compare regression lines. The for overall differences between the regressions and for making pairwise GLM allows for the use of both qualitative and quantitative comparisons. Use of the Bonferroni adjustment to ensure a desired predictor variables. Quantitative variables are continuous experimentwise type I error rate is emphasized. SAS software is used to illustrate application of these concepts to an artificial simulated dataset. The values such as diameter at breast height, tree height, SAS code is provided for each of the five hypotheses and for the contrasts age, or temperature. However, many predictor variables for the general test and all possible specific tests. in the natural resources field are qualitative. Examples of qualitative variables include tree class (dominant, Keywords: Contrasts, dummy variables, F-test, intercept, SAS, slope, test of conditional error. codominant); species (loblolly, slash, longleaf); and cover type (clearcut, shelterwood, seed tree). Fraedrich and others (1994) compared linear regressions of slash pine cone specific gravity and moisture content Introduction on collection date between families of slash pine grown in a seed orchard. -
Analysis of Variance and Analysis of Variance and Design of Experiments of Experiments-I
Analysis of Variance and Design of Experimentseriments--II MODULE ––IVIV LECTURE - 19 EXPERIMENTAL DESIGNS AND THEIR ANALYSIS Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 Design of experiment means how to design an experiment in the sense that how the observations or measurements should be obtained to answer a qqyuery inavalid, efficient and economical way. The desigggning of experiment and the analysis of obtained data are inseparable. If the experiment is designed properly keeping in mind the question, then the data generated is valid and proper analysis of data provides the valid statistical inferences. If the experiment is not well designed, the validity of the statistical inferences is questionable and may be invalid. It is important to understand first the basic terminologies used in the experimental design. Experimental unit For conducting an experiment, the experimental material is divided into smaller parts and each part is referred to as experimental unit. The experimental unit is randomly assigned to a treatment. The phrase “randomly assigned” is very important in this definition. Experiment A way of getting an answer to a question which the experimenter wants to know. Treatment Different objects or procedures which are to be compared in an experiment are called treatments. Sampling unit The object that is measured in an experiment is called the sampling unit. This may be different from the experimental unit. 3 Factor A factor is a variable defining a categorization. A factor can be fixed or random in nature. • A factor is termed as fixed factor if all the levels of interest are included in the experiment. -
Simple Linear Regression: Straight Line Regression Between an Outcome Variable (Y ) and a Single Explanatory Or Predictor Variable (X)
1 Introduction to Regression \Regression" is a generic term for statistical methods that attempt to fit a model to data, in order to quantify the relationship between the dependent (outcome) variable and the predictor (independent) variable(s). Assuming it fits the data reasonable well, the estimated model may then be used either to merely describe the relationship between the two groups of variables (explanatory), or to predict new values (prediction). There are many types of regression models, here are a few most common to epidemiology: Simple Linear Regression: Straight line regression between an outcome variable (Y ) and a single explanatory or predictor variable (X). E(Y ) = α + β × X Multiple Linear Regression: Same as Simple Linear Regression, but now with possibly multiple explanatory or predictor variables. E(Y ) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: A special case is polynomial regression. 2 3 E(Y ) = α + β1 × X + β2 × X + β3 × X + ::: Generalized Linear Model: Same as Multiple Linear Regression, but with a possibly transformed Y variable. This introduces considerable flexibil- ity, as non-linear and non-normal situations can be easily handled. G(E(Y )) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: In general, the transformation function G(Y ) can take any form, but a few forms are especially common: • Taking G(Y ) = logit(Y ) describes a logistic regression model: E(Y ) log( ) = α + β × X + β × X + β × X + ::: 1 − E(Y ) 1 1 2 2 3 3 2 • Taking G(Y ) = log(Y ) is also very common, leading to Poisson regression for count data, and other so called \log-linear" models. -
THE ONE-SAMPLE Z TEST
10 THE ONE-SAMPLE z TEST Only the Lonely Difficulty Scale ☺ ☺ ☺ (not too hard—this is the first chapter of this kind, but youdistribute know more than enough to master it) or WHAT YOU WILL LEARN IN THIS CHAPTERpost, • Deciding when the z test for one sample is appropriate to use • Computing the observed z value • Interpreting the z value • Understandingcopy, what the z value means • Understanding what effect size is and how to interpret it not INTRODUCTION TO THE Do ONE-SAMPLE z TEST Lack of sleep can cause all kinds of problems, from grouchiness to fatigue and, in rare cases, even death. So, you can imagine health care professionals’ interest in seeing that their patients get enough sleep. This is especially the case for patients 186 Copyright ©2020 by SAGE Publications, Inc. This work may not be reproduced or distributed in any form or by any means without express written permission of the publisher. Chapter 10 ■ The One-Sample z Test 187 who are ill and have a real need for the healing and rejuvenating qualities that sleep brings. Dr. Joseph Cappelleri and his colleagues looked at the sleep difficul- ties of patients with a particular illness, fibromyalgia, to evaluate the usefulness of the Medical Outcomes Study (MOS) Sleep Scale as a measure of sleep problems. Although other analyses were completed, including one that compared a treat- ment group and a control group with one another, the important analysis (for our discussion) was the comparison of participants’ MOS scores with national MOS norms. Such a comparison between a sample’s mean score (the MOS score for par- ticipants in this study) and a population’s mean score (the norms) necessitates the use of a one-sample z test. -
Chapter 2 Simple Linear Regression Analysis the Simple
Chapter 2 Simple Linear Regression Analysis The simple linear regression model We consider the modelling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as a simple linear regression model. When there are more than one independent variables in the model, then the linear model is termed as the multiple linear regression model. The linear model Consider a simple linear regression model yX01 where y is termed as the dependent or study variable and X is termed as the independent or explanatory variable. The terms 0 and 1 are the parameters of the model. The parameter 0 is termed as an intercept term, and the parameter 1 is termed as the slope parameter. These parameters are usually called as regression coefficients. The unobservable error component accounts for the failure of data to lie on the straight line and represents the difference between the true and observed realization of y . There can be several reasons for such difference, e.g., the effect of all deleted variables in the model, variables may be qualitative, inherent randomness in the observations etc. We assume that is observed as independent and identically distributed random variable with mean zero and constant variance 2 . Later, we will additionally assume that is normally distributed. The independent variables are viewed as controlled by the experimenter, so it is considered as non-stochastic whereas y is viewed as a random variable with Ey()01 X and Var() y 2 . Sometimes X can also be a random variable. -
Pubh 7405: REGRESSION ANALYSIS
PubH 7405: REGRESSION ANALYSIS Review #2: Simple Correlation & Regression COURSE INFORMATION • Course Information are at address: www.biostat.umn.edu/~chap/pubh7405 • On each class web page, there is a brief version of the lecture for the day – the part with “formulas”; you can review, preview, or both – and as often as you like. • Follow Reading & Homework assignments at the end of the page when & if applicable. OFFICE HOURS • Instructor’s scheduled office hours: 1:15 to 2:15 Monday & Wednesday, in A441 Mayo Building • Other times are available by appointment • When really needed, can just drop in and interrupt me; could call before coming – making sure that I’m in. • I’m at a research facility on Fridays. Variables • A variable represents a characteristic or a class of measurement. It takes on different values on different subjects/persons. Examples include weight, height, race, sex, SBP, etc. The observed values, also called “observations,” form items of a data set. • Depending on the scale of measurement, we have different types of data. There are “observed variables” (Height, Weight, etc… each takes on different values on different subjects/person) and there are “calculated variables” (Sample Mean, Sample Proportion, etc… each is a “statistic” and each takes on different values on different samples). The Standard Deviation of a calculated variable is called the Standard Error of that variable/statistic. A “variable” – sample mean, sample standard deviation, etc… included – is like a “function”; when you apply it to a target element in its domain, the result is a “number”. For example, “height” is a variable and “the height of Mrs. -
Tests of Hypotheses Using Statistics
Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables . -
ANALYSIS of VARIANCE and MISSING OBSERVATIONS in COMPLETELY RANDOMIZED, RANDOMIZED BLOCKS and LATIN SQUARE DESIGNS a Thesis
ANALYSIS OF VARIANCE AND MISSING OBSERVATIONS IN COMPLETELY RANDOMIZED, RANDOMIZED BLOCKS AND LATIN SQUARE DESIGNS A Thesis Presented to The Department of Mathematics Kansas State Teachers College, Emporia, Kansas In Partial Fulfillment of the Requirements for the Degree Master of Science by Kiritkumar K. Talati May 1972 c; , ACKNOWLEDGEMENTS Sincere thanks and gratitude is expressed to Dr. John Burger for his assistance, patience, and his prompt attention in all directions in preparing this paper. A special note of appreciation goes to Dr. Marion Emerson, Dr. Thomas Davis, Dr. George Poole, Dr. Darrell Wood, who rendered assistance in the research of this paper. TABLE OF CONTENTS CHAPTER I. INTRODUCTION • • • • • • • • • • • • • • • • 1 A. Preliminary Consideration. • • • • • • • 1 B. Purpose and Assumptions of the Analysis of Variance • • • • • • • • •• 1 C. Analysis of Covariance • • • • • • • • • 2 D. Definitions. • • • • • • • • • • • • • • ) E. Organization of Paper. • • • • • • • • • 4 II. COMPLETELY RANDOMIZED DESIGN • • • • • • • • 5 A. Description............... 5 B. Randomization. • • • • • • • • • • • • • 5 C. Problem and Computations •••••••• 6 D. Conclusion and Further Applications. •• 10 III. RANDOMIZED BLOCK DESIGN. • • • • • • • • • • 12 A. Description............... 12 B. Randomization. • • • • • • • • • • • • • 1) C. Problem and Statistical Analysis • • • • 1) D. Efficiency of Randomized Block Design as Compared to Completely Randomized Design. • • • • • • • • • •• 20 E. Missing Observations • • • • • • • • •• 21 F.