Comparison of Artificial Neural Networks with Logistic Regression in Prediction of Kidney Transplant Outcomes

Total Page:16

File Type:pdf, Size:1020Kb

Comparison of Artificial Neural Networks with Logistic Regression in Prediction of Kidney Transplant Outcomes View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Canberra Research Repository 2009 International Conference on Future Computer and Communication Comparison of Artificial Neural Networks with Logistic Regression in Prediction of Kidney Transplant Outcomes Fariba Shadabi Dharmendra Sharma Faculty of Information Sciences and Engineering Faculty of Information Sciences and Engineering University of Canberra, ACT, 2601, Australia University of Canberra, ACT, 2601, Australia [email protected] [email protected] good analytical alternative to logistic regression techniques [2], [4]. In this paper we discuss our experience of applying data Abstract— Predicting the outcome of a graft mining techniques for the purpose of predicting the outcome transplant with high level of accuracy is a challenging task. To answer the challenge, data of medical procedures or events. The case study described mining can play a significant role. The goal of this in this paper is from the kidney transplantation domain. In study is to compare the performances and features earlier works, we addressed some of the practical issues of an Artificially Intelligent (AI)-based data mining associated with the use of ANN in the prediction of kidney technique namely Artificial Neural Network with transplant outcomes [5], [6]. In this paper we compare the Logistic Regression as a standard statistical data performances and features of an ANN approach to logistic mining method to predict the outcome of kidney regression in predicting renal transplantation outcomes. transplants over a 2-year horizon. The methodology employed utilizes a dataset made II. RENAL TRANSPLANT CHALLENGES available to us from a kidney transplant database. The dataset embodies a number of important The first successful human organ transplantation was properties, which make it a good starting point for carried out on December 1954 by Dr. Joseph Murray, in the purpose of this research. Results reveal that in Brigham hospital, Boston. In this instance Richard Herrick most cases, the neural network technique from Northboro, Massachusetts was given a kidney from his outperforms logistic regression. This study healthy identical twin brother, Ronald. He survived another highlights that in some situations, different eight years before the original medical condition destroyed techniques can potentially be integrated to improve his new organ. the accuracy of predictions. With the advent of more sophisticated anti-rejection drugs and antibiotics, organ transplants from related or unrelated donors have become much more common. Over the last five Keywords-Logistice Regression, Neural Network decades, thousands of kidney and other vital organ grafts such as heart and liver transplants have been performed I. INTRODUCTION successfully by surgeons around the world. Data mining techniques can be employed to support Although anti-rejection drugs have helped to boost the clinical data analysis. The logistic regression model is a success rate of transplants, the biggest challenge that statistical data mining method, which has been used widely continues to face transplant patients and surgeons is the risk by researchers in medicine for many years [1]. Logistic of rejection of transplanted organs. Over the years, there has regression is popular mainly because it enables the been substantial research into methods to predict graft researcher to avoid the need for many precise assumptions outcomes and identify key parameters influencing the required by other regression methods. Recently AI-based success or failure of transplanted organs [7]. Successful data mining techniques such as Artificial Neural Networks kidney transplantation will not only extend the longevity (ANNs) have drawn the attention of computer scientists and and quality of life for the recipient but also reduce medical clinicians for intelligent information retrieval from clinical expenses and increase the access to donor kidneys by data sources [2], [3]. reducing the need for multiple kidney transplants in the one Both ANN and logistic regression techniques have the patient. ability to model non-linear relationships between dependent Until now most clinical prediction methods have largely and independent variables. However research shows that been focused upon the use of standard statistical models. with the growing power of ANN tools, ANN can often be a However, statistical techniques often do not provide 978-0-7695-3591-3/09 $25.00 © 2009 IEEE 543 DOI 10.1109/ICFCC.2009.139 adequate knowledge for solving highly complex clinical classifier is trained with each of the training sets. As a result prediction problems. ANN have the ability to provide good each classifier may produce a different prediction [16]. solutions in situations where large number of variables Bagging could offer a significant improvement in prediction contribute to an outcome but their individual influence is accuracy. It is especially useful for a classifier with poor weak and not well understood. Clinical data gathered from performance, or where a classifier has been presented with a patients who have undergone graft transplant surgery have small training sample set, or where small changes in the data this characteristic and are known to be complex [8], [9]. can result in large changes in the classifier predictions (low In this paper we compare a widely used ANN approach stability issue). The pseudo-code of a classifier ensemble known as Multi-layer Perceptron (MLP) networks with with bagging is shown in Table 1. logistic regression, with the challenge being to select the Table 1. The bagging approach right kidney from the available pool of organs for a particular patient, thereby maximizing the chances for the Given: Training set S (with n cases and their classes), learning algorithm L, number of successful transplantation. bootstrap samples T III. MATERIAL AND METHODS Process:1.Model Generation A. Dataset for i = 1 to T: First, confirm that you have the correct template for your Generate a new training set(a bootstrap paper size. This template has been tailored for output on the sample) with n cases, using random US-letter paper size. If you are using A4-sized paper, please drawing (with replacement) from S close this template and download the file for A4 paper Apply the learning algorithm L to the bootstrap format called “CPS_A4_format”. sample B. Artificial Neural Networks and logistic regression keep the resulting model M(i) for future use The inspiration of ANN came from the desire to simulate features of the brain, namely biological neural networks and 2.Prediction for a given test case learning systems, which show high power in pattern for i = 1 to T: recognition tasks and adaptability. ANN architectures are Predict class of case using M(i) generally divided into two categories namely feed-forward network (networks without any loops in their path) and Return class that appears most frequently feedback networks (networks with recursive loops). Different network architectures and training algorithms can affect networks capabilities and usually a lot of trial and IV. METHODOLOGY error experimentation is necessary to determine the optimal Before The following methodology was employed: network topologies and training parameters. Detailed 1. Pre-process the data set. This includes: extracting information about the foundations of ANNs can be found in the data from different tables, cleaning the data, [12], [13]. For the purpose of this study, multilayer, feed- transforming the nominal attributes into numeric forward networks were used to differentiate between attributes, and choosing the appropriate parameters successful and unsuccessful transplants. All neural network to be included in the dataset with the help of classifiers described in this paper were implemented using domain expert. the Delphi programming language. 2. Split the dataset for training and testing (with Binomial (or binary) logistic regression is an effective balanced distribution of success and failure cases). supervised learning method that can be used to estimate the 3. Perform classification using the MLP network and probability of a certain event occurring. MLP network coupled with bagging. C. Bagging 4. Perform classification using the logistic regression and logistic regression coupled with bagging. An ensemble consists of several individually trained 5. Assess and compare the predictive accuracy of the classifiers that are jointly used to solve a problem. The most classifiers. popular methods for generating training sets for classifiers are Bagging and Boosting. We conducted two main experiments: Bagging (bootstrapping aggregates) was originally proposed Experiment 1. Neural Networks by Breiman [15]. Bagging is a popular method for training A single MLP classifier: Here we constructed a component classifiers. This technique generates several single MLP classifier. This experiment is reported training sets using random drawing (with replacement) from in more details in our previous study [5]. the original training set. Consequently, in every new training set there are data points, which appear more than Neural network ensemble: In this experiment, we once while others, do not appear at all. Each individual employed the bagging strategy to generate different 544 training sets from the original training set, using rate also reached 76% with 89%-agreement among the random drawing with replacement technique. Here networks
Recommended publications
  • 6.5 Applications of Exponential and Logarithmic Functions 469
    6.5 Applications of Exponential and Logarithmic Functions 469 6.5 Applications of Exponential and Logarithmic Functions As we mentioned in Section 6.1, exponential and logarithmic functions are used to model a wide variety of behaviors in the real world. In the examples that follow, note that while the applications are drawn from many different disciplines, the mathematics remains essentially the same. Due to the applied nature of the problems we will examine in this section, the calculator is often used to express our answers as decimal approximations. 6.5.1 Applications of Exponential Functions Perhaps the most well-known application of exponential functions comes from the financial world. Suppose you have $100 to invest at your local bank and they are offering a whopping 5 % annual percentage interest rate. This means that after one year, the bank will pay you 5% of that $100, or $100(0:05) = $5 in interest, so you now have $105.1 This is in accordance with the formula for simple interest which you have undoubtedly run across at some point before. Equation 6.1. Simple Interest The amount of interest I accrued at an annual rate r on an investmenta P after t years is I = P rt The amount A in the account after t years is given by A = P + I = P + P rt = P (1 + rt) aCalled the principal Suppose, however, that six months into the year, you hear of a better deal at a rival bank.2 Naturally, you withdraw your money and try to invest it at the higher rate there.
    [Show full text]
  • Chapter 10. Logistic Models
    www.ck12.org Chapter 10. Logistic Models CHAPTER 10 Logistic Models Chapter Outline 10.1 LOGISTIC GROWTH MODEL 10.2 CONDITIONS FAVORING LOGISTIC GROWTH 10.3 REFERENCES 207 10.1. Logistic Growth Model www.ck12.org 10.1 Logistic Growth Model Here you will explore the graph and equation of the logistic function. Learning Objectives • Recognize logistic functions. • Identify the carrying capacity and inflection point of a logistic function. Logistic Models Exponential growth increases without bound. This is reasonable for some situations; however, for populations there is usually some type of upper bound. This can be caused by limitations on food, space or other scarce resources. The effect of this limiting upper bound is a curve that grows exponentially at first and then slows down and hardly grows at all. This is characteristic of a logistic growth model. The logistic equation is of the form: C C f (t) = 1+ab−t = 1+ae−kt The above equations represent the logistic function, and it contains three important pieces: C, a, and b. C determines that maximum value of the function, also known as the carrying capacity. C is represented by the dashed line in the graph below. 208 www.ck12.org Chapter 10. Logistic Models The constant of a in the logistic function is used much like a in the exponential function: it helps determine the value of the function at t=0. Specifically: C f (0) = 1+a The constant of b also follows a similar concept to the exponential function: it helps dictate the rate of change at the beginning and the end of the function.
    [Show full text]
  • Neural Network Architectures and Activation Functions: a Gaussian Process Approach
    Fakultät für Informatik Neural Network Architectures and Activation Functions: A Gaussian Process Approach Sebastian Urban Vollständiger Abdruck der von der Fakultät für Informatik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. rer. nat. Stephan Günnemann Prüfende der Dissertation: 1. Prof. Dr. Patrick van der Smagt 2. Prof. Dr. rer. nat. Daniel Cremers 3. Prof. Dr. Bernd Bischl, Ludwig-Maximilians-Universität München Die Dissertation wurde am 22.11.2017 bei der Technischen Universtät München eingereicht und durch die Fakultät für Informatik am 14.05.2018 angenommen. Neural Network Architectures and Activation Functions: A Gaussian Process Approach Sebastian Urban Technical University Munich 2017 ii Abstract The success of applying neural networks crucially depends on the network architecture being appropriate for the task. Determining the right architecture is a computationally intensive process, requiring many trials with different candidate architectures. We show that the neural activation function, if allowed to individually change for each neuron, can implicitly control many aspects of the network architecture, such as effective number of layers, effective number of neurons in a layer, skip connections and whether a neuron is additive or multiplicative. Motivated by this observation we propose stochastic, non-parametric activation functions that are fully learnable and individual to each neuron. Complexity and the risk of overfitting are controlled by placing a Gaussian process prior over these functions. The result is the Gaussian process neuron, a probabilistic unit that can be used as the basic building block for probabilistic graphical models that resemble the structure of neural networks.
    [Show full text]
  • Neural Networks
    School of Computer Science 10-601B Introduction to Machine Learning Neural Networks Readings: Matt Gormley Bishop Ch. 5 Lecture 15 Murphy Ch. 16.5, Ch. 28 October 19, 2016 Mitchell Ch. 4 1 Reminders 2 Outline • Logistic Regression (Recap) • Neural Networks • Backpropagation 3 RECALL: LOGISTIC REGRESSION 4 Using gradient ascent for linear Recall… classifiers Key idea behind today’s lecture: 1. Define a linear classifier (logistic regression) 2. Define an objective function (likelihood) 3. Optimize it with gradient descent to learn parameters 4. Predict the class with highest probability under the model 5 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 6 Using gradient ascent for linear Recall… classifiers This decision function isn’t Use a differentiable function differentiable: instead: 1 T pθ(y =1t)= h(t)=sign(θ t) | 1+2tT( θT t) − sign(x) 1 logistic(u) ≡ −u 1+ e 7 Recall… Logistic Regression Data: Inputs are continuous vectors of length K. Outputs are discrete. (i) (i) N K = t ,y where t R and y 0, 1 D { }i=1 ∈ ∈ { } Model: Logistic function applied to dot product of parameters with input vector. 1 pθ(y =1t)= | 1+2tT( θT t) − Learning: finds the parameters that minimize some objective function. θ∗ = argmin J(θ) θ Prediction: Output is the most probable class. yˆ = `;Kt pθ(y t) y 0,1 | ∈{ } 8 NEURAL NETWORKS 9 Learning highly non-linear functions f: X à Y l f might be non-linear
    [Show full text]
  • Lecture 5: Logistic Regression 1 MLE Derivation
    CSCI 5525 Machine Learning Fall 2019 Lecture 5: Logistic Regression Feb 10 2020 Lecturer: Steven Wu Scribe: Steven Wu Last lecture, we give several convex surrogate loss functions to replace the zero-one loss func- tion, which is NP-hard to optimize. Now let us look into one of the examples, logistic loss: given d parameter w and example (xi; yi) 2 R × {±1g, the logistic loss of w on example (xi; yi) is defined as | ln (1 + exp(−yiw xi)) This loss function is used in logistic regression. We will introduce the statistical model behind logistic regression, and show that the ERM problem for logistic regression is the same as the relevant maximum likelihood estimation (MLE) problem. 1 MLE Derivation For this derivation it is more convenient to have Y = f0; 1g. Note that for any label yi 2 f0; 1g, we also have the “signed” version of the label 2yi − 1 2 {−1; 1g. Recall that in general su- pervised learning setting, the learner receive examples (x1; y1);:::; (xn; yn) drawn iid from some distribution P over labeled examples. We will make the following parametric assumption on P : | yi j xi ∼ Bern(σ(w xi)) where Bern denotes the Bernoulli distribution, and σ is the logistic function defined as follows 1 exp(z) σ(z) = = 1 + exp(−z) 1 + exp(z) See Figure 1 for a visualization of the logistic function. In general, the logistic function is a useful function to convert real values into probabilities (in the range of (0; 1)). If w|x increases, then σ(w|x) also increases, and so does the probability of Y = 1.
    [Show full text]
  • An Introduction to Logistic Regression: from Basic Concepts to Interpretation with Particular Attention to Nursing Domain
    J Korean Acad Nurs Vol.43 No.2, 154 -164 J Korean Acad Nurs Vol.43 No.2 April 2013 http://dx.doi.org/10.4040/jkan.2013.43.2.154 An Introduction to Logistic Regression: From Basic Concepts to Interpretation with Particular Attention to Nursing Domain Park, Hyeoun-Ae College of Nursing and System Biomedical Informatics National Core Research Center, Seoul National University, Seoul, Korea Purpose: The purpose of this article is twofold: 1) introducing logistic regression (LR), a multivariable method for modeling the relationship between multiple independent variables and a categorical dependent variable, and 2) examining use and reporting of LR in the nursing literature. Methods: Text books on LR and research articles employing LR as main statistical analysis were reviewed. Twenty-three articles published between 2010 and 2011 in the Journal of Korean Academy of Nursing were analyzed for proper use and reporting of LR models. Results: Logistic regression from basic concepts such as odds, odds ratio, logit transformation and logistic curve, assumption, fitting, reporting and interpreting to cautions were presented. Substantial short- comings were found in both use of LR and reporting of results. For many studies, sample size was not sufficiently large to call into question the accuracy of the regression model. Additionally, only one study reported validation analysis. Conclusion: Nurs- ing researchers need to pay greater attention to guidelines concerning the use and reporting of LR models. Key words: Logit function, Maximum likelihood estimation, Odds, Odds ratio, Wald test INTRODUCTION The model serves two purposes: (1) it can predict the value of the depen- dent variable for new values of the independent variables, and (2) it can Multivariable methods of statistical analysis commonly appear in help describe the relative contribution of each independent variable to general health science literature (Bagley, White, & Golomb, 2001).
    [Show full text]
  • Fm 3-34 Engineer Operations Headquarters, Department
    FM 3-34 April 2009 ENGINEER OPERATIONS DISTRIBUTION RESTRICTION: Approved for public release; distribution is unlimited. HEADQUARTERS, DEPARTMENT OF THE ARMY This publication is available at Army Knowledge Online (AKO) <www.us.army.mil> and the General Dennis J. Reimer Training and Doctrine Digital Library at <www.train.army.mil>. *FM 3-34 Field Manual No. Headquarters 3-34 Department of the Army Washington, DC, 2 April 2009 Engineer Operations Contents Page PREFACE .............................................................................................................. v INTRODUCTION .................................................................................................. vii Chapter 1 THE OPERATIONAL ENVIRONMENT ............................................................. 1-1 Understanding the Operational Environment ..................................................... 1-1 The Military Variable ........................................................................................... 1-4 Spectrum of Requirements ................................................................................. 1-5 Support Spanning the Levels of War.................................................................. 1-7 Engineer Soldiers ............................................................................................... 1-9 Chapter 2 ENGINEERING IN UNIFIED ACTION ............................................................... 2-1 SECTION I–THE ENGINEER REGIMENT ........................................................ 2-1 The
    [Show full text]
  • Notes: Logistic Functions I. Use, Function, and General Graph If
    Notes: Logistic Functions I. Use, Function, and General Graph If growth begins slowly, then increases rapidly and eventually levels off, the data often can be model by an “S-curve”, or a logistic function. The logistic function describes certain kinds of growth. These functions, like exponential functions, grow quickly at first, but because of restrictions that place limits on the size of the underlying population, eventually grow more slowly and then level off. For real numbers a, b, and c, the function is a logistic function. If a >0, a logistic function increases when b >0 and decreases when b < 0. Logistic Growth Function Equation: x-Intercept: none y-intercept: Horizontal Asymptote: The number, , is called the limiting value or the upper limit of the function because the graph of a logistic growth function will have a horizontal asymptote at y = c. As is clear from the graph above, the characteristic S-shape in the graph of a logistic function shows that initial exponential growth is followed by a period in which growth slows and then levels off, approaching (but never attaining) a maximum upper limit. Logistic functions are good models of biological population growth in species which have grown so large that they are near to saturating their ecosystems, or of the spread of information within societies. They are also common in marketing, where they chart the sales of new products over time; in a different context, they can also describe demand curves: the decline of demand for a product as a function of increasing price can be modeled by a logistic function, as in the figure below.
    [Show full text]
  • Sigmoid Functions in Reliability Based Management 2007 15 2 67 2.1 the Nature of Performance Growth
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Periodica Polytechnica (Budapest University of Technology and Economics) Ŕ periodica polytechnica Sigmoid functions in reliability based Social and Management Sciences management 15/2 (2007) 67–72 doi: 10.3311/pp.so.2007-2.04 Tamás Jónás web: http://www.pp.bme.hu/so c Periodica Polytechnica 2007 RESEARCH ARTICLE Received 2008-09-14 Abstract 1 Introduction This paper introduces the theoretical background of a few A sigmoid function is a mathematical function that produces possible applications of the so-called sigmoid-type functions in a curve having an "S" shape, and is defined by the manufacturing and service management. 1 σλ,x (x) (1) An extended concept of reliability, derived from fuzzy theory, 0 λ(x x0) = 1 e− − is discussed here to illustrate how reliability based management + formula. The sigmoid function is also called sigmoidal curve decisions can be made consistent, when handling of weakly de- [1] or logistic function. The interpretation of sigmoid-type func- fined concepts is needed. tions – I use here – is that any function that can be transformed I demonstrate how performance growth can be modelled us- into σλ,x (x) through substitutions and linear transformations ing an aggregate approach with support of sigmoid-type func- 0 is considered as a sigmoid-type one. There are several well- tions. known applications of sigmoid-type functions. A few examples Suitably chosen parameters of sigmoid-type functions allow are: threshold function in neural networks [2], approximation these to be used as failure probability distribution and survival of Gaussian probability distribution, logistic regression [3], or functions.
    [Show full text]
  • Learning Activation Functions in Deep Neural Networks
    UNIVERSITÉ DE MONTRÉAL LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS FARNOUSH FARHADI DÉPARTEMENT DE MATHÉMATIQUES ET DE GÉNIE INDUSTRIEL ÉCOLE POLYTECHNIQUE DE MONTRÉAL MÉMOIRE PRÉSENTÉ EN VUE DE L’OBTENTION DU DIPLÔME DE MAÎTRISE ÈS SCIENCES APPLIQUÉES (GÉNIE INDUSTRIEL) DÉCEMBRE 2017 c Farnoush Farhadi, 2017. UNIVERSITÉ DE MONTRÉAL ÉCOLE POLYTECHNIQUE DE MONTRÉAL Ce mémoire intitulé : LEARNING ACTIVATION FUNCTIONS IN DEEP NEURAL NETWORKS présenté par : FARHADI Farnoush en vue de l’obtention du diplôme de : Maîtrise ès sciences appliquées a été dûment accepté par le jury d’examen constitué de : M. ADJENGUE Luc-Désiré, Ph. D., président M. LODI Andrea, Ph. D., membre et directeur de recherche M. PARTOVI NIA Vahid, Doctorat, membre et codirecteur de recherche M. CHARLIN Laurent, Ph. D., membre iii DEDICATION This thesis is dedicated to my beloved parents, Ahmadreza and Sholeh, who are my first teachers and always love me unconditionally. This work is also dedicated to my love, Arash, who has been a great source of motivation and encouragement during the challenges of graduate studies and life. iv ACKNOWLEDGEMENTS I would like to express my gratitude to Prof. Andrea Lodi for his permission to be my supervisor at Ecole Polytechnique de Montreal and more importantly for his enthusiastic encouragements and invaluable continuous support during my research and education. I am very appreciated to him for introducing me to a MITACS internship in which I have developed myself both academically and professionally. I would express my deepest thanks to Dr. Vahid Partovi Nia, my co-supervisor at Ecole Poly- technique de Montreal, for his supportive and careful guidance and taking part in important decisions which were extremely precious for my research both theoretically and practically.
    [Show full text]
  • Applied Machine Learning
    Applied Machine Learning Annalisa Marsico OWL RNA Bionformatics group Max Planck Institute for Molecular Genetics Free University of Berlin 13 Mai, SoSe 2015 Neural Networks Motivation: the non-linear hypothesis x x x Goal: learn the non-linear problem x x x x x One solution: add a lot of non-linear features x as input to the logistic function x x x x x x x x x = ( + + + + x + + + ⋯ . ) x x g = sigmoid function If more than two dimensions, including polynomial feature not a good idea It might lead to over fitting. Neural Networks • Solution : SVM with projection in a high-dimensional space. Kernel ‘trick’. • New solution: Neural Networks to learn non-linear complex hypothesis Neural Networks (NNs) – Biological motivation • Originally motivated by the desire of having machines which can mimic the brain • Widely used in the 80s and early 90s, popularity diminished in the late 90s • Recent resurgence: more computational power to run large-scale NNs The ‘brain’ algorithm • Fascinating hypothesis (Roe et al. 1992): the brain functions through a single learning algorithm Auditory cortex X Neurons Schematic representation of a single neuron in the brain - Neuron is a computational unit ‘Inputs’ (dendrites) Synapse Node of Cell body Ranvier Schwann cell Myelin sheath Cell nucleus Communication between neurons Modeling neurons Logistic unit Activation function is often a sigmoid 1 ≡ ℎ = ≡ () 1 + There exist other activation function: step function, linear function Logistic unit representation Bias unit, =1 () Sigmoid
    [Show full text]
  • Logistic Regression
    CHAPTER 5 Logistic regression Logistic regression is the standard way to model binary outcomes (that is, data yi that take on the values 0 or 1). Section 5.1 introduces logistic regression in a simple example with one predictor, then for most of the rest of the chapter we work through an extended example with multiple predictors and interactions. 5.1 Logistic regression with a single predictor Example: modeling political preference given income Conservative parties generally receive more support among voters with higher in- comes. We illustrate classical logistic regression with a simple analysis of this pat- tern from the National Election Study in 1992. For each respondent i in this poll, we label yi =1ifheorshepreferredGeorgeBush(theRepublicancandidatefor president) or 0 if he or she preferred Bill Clinton (the Democratic candidate), for now excluding respondents who preferred Ross Perot or other candidates, or had no opinion. We predict preferences given the respondent’s income level, which is characterized on a five-point scale.1 The data are shown as (jittered) dots in Figure 5.1, along with the fitted logistic regression line, a curve that is constrained to lie between 0 and 1. We interpret the line as the probability that y =1givenx—in mathematical notation, Pr(y =1x). We fit and display the logistic regression using the following R function calls:| fit.1 <- glm (vote ~ income, family=binomial(link="logit")) Rcode display (fit.1) to yield coef.est coef.se Routput (Intercept) -1.40 0.19 income 0.33 0.06 n=1179,k=2 residual deviance = 1556.9, null deviance = 1591.2 (difference = 34.3) 1 The fitted model is Pr(yi =1)=logit− ( 1.40 + 0.33 income).
    [Show full text]