Week 10: Causality with Measured Confounding
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Hypothesis Testing and Likelihood Ratio Tests
Hypottthesiiis tttestttiiing and llliiikellliiihood ratttiiio tttesttts Y We will adopt the following model for observed data. The distribution of Y = (Y1, ..., Yn) is parameter considered known except for some paramett er ç, which may be a vector ç = (ç1, ..., çk); ç“Ç, the paramettter space. The parameter space will usually be an open set. If Y is a continuous random variable, its probabiiillliiittty densiiittty functttiiion (pdf) will de denoted f(yy;ç) . If Y is y probability mass function y Y y discrete then f(yy;ç) represents the probabii ll ii tt y mass functt ii on (pmf); f(yy;ç) = Pç(YY=yy). A stttatttiiistttiiicalll hypottthesiiis is a statement about the value of ç. We are interested in testing the null hypothesis H0: ç“Ç0 versus the alternative hypothesis H1: ç“Ç1. Where Ç0 and Ç1 ¶ Ç. hypothesis test Naturally Ç0 § Ç1 = ∅, but we need not have Ç0 ∞ Ç1 = Ç. A hypott hesii s tt estt is a procedure critical region for deciding between H0 and H1 based on the sample data. It is equivalent to a crii tt ii call regii on: a critical region is a set C ¶ Rn y such that if y = (y1, ..., yn) “ C, H0 is rejected. Typically C is expressed in terms of the value of some tttesttt stttatttiiistttiiic, a function of the sample data. For µ example, we might have C = {(y , ..., y ): y – 0 ≥ 3.324}. The number 3.324 here is called a 1 n s/ n µ criiitttiiicalll valllue of the test statistic Y – 0 . S/ n If y“C but ç“Ç 0, we have committed a Type I error. -
Data Collection: Randomized Experiments
9/2/15 STAT 250 Dr. Kari Lock Morgan Knee Surgery for Arthritis Researchers conducted a study on the effectiveness of a knee surgery to cure arthritis. Collecting Data: It was randomly determined whether people got Randomized Experiments the knee surgery. Everyone who underwent the surgery reported feeling less pain. SECTION 1.3 Is this evidence that the surgery causes a • Control/comparison group decrease in pain? • Clinical trials • Placebo Effect (a) Yes • Blinding • Crossover studies / Matched pairs (b) No Statistics: Unlocking the Power of Data Lock5 Statistics: Unlocking the Power of Data Lock5 Control Group Clinical Trials Clinical trials are randomized experiments When determining whether a treatment is dealing with medicine or medical interventions, effective, it is important to have a comparison conducted on human subjects group, known as the control group Clinical trials require additional aspects, beyond just randomization to treatment groups: All randomized experiments need a control or ¡ Placebo comparison group (could be two different ¡ Double-blind treatments) Statistics: Unlocking the Power of Data Lock5 Statistics: Unlocking the Power of Data Lock5 Placebo Effect Study on Placebos Often, people will experience the effect they think they should be experiencing, even if they aren’t actually Blue pills are better than yellow pills receiving the treatment. This is known as the placebo effect. Red pills are better than blue pills Example: Eurotrip 2 pills are better than 1 pill One study estimated that 75% of the -
The Practice of Causal Inference in Cancer Epidemiology
Vol. 5. 303-31 1, April 1996 Cancer Epidemiology, Biomarkers & Prevention 303 Review The Practice of Causal Inference in Cancer Epidemiology Douglas L. Weedt and Lester S. Gorelic causes lung cancer (3) and to guide causal inference for occu- Preventive Oncology Branch ID. L. W.l and Comprehensive Minority pational and environmental diseases (4). From 1965 to 1995, Biomedical Program IL. S. 0.1. National Cancer Institute, Bethesda, Maryland many associations have been examined in terms of the central 20892 questions of causal inference. Causal inference is often practiced in review articles and editorials. There, epidemiologists (and others) summarize evi- Abstract dence and consider the issues of causality and public health Causal inference is an important link between the recommendations for specific exposure-cancer associations. practice of cancer epidemiology and effective cancer The purpose of this paper is to take a first step toward system- prevention. Although many papers and epidemiology atically reviewing the practice of causal inference in cancer textbooks have vigorously debated theoretical issues in epidemiology. Techniques used to assess causation and to make causal inference, almost no attention has been paid to the public health recommendations are summarized for two asso- issue of how causal inference is practiced. In this paper, ciations: alcohol and breast cancer, and vasectomy and prostate we review two series of review papers published between cancer. The alcohol and breast cancer association is timely, 1985 and 1994 to find answers to the following questions: controversial, and in the public eye (5). It involves a common which studies and prior review papers were cited, which exposure and a common cancer and has a large body of em- causal criteria were used, and what causal conclusions pirical evidence; over 50 studies and over a dozen reviews have and public health recommendations ensued. -
Statistics and Causal Inference (With Discussion)
Applied Statistics Lecture Notes Kosuke Imai Department of Politics Princeton University February 2, 2008 Making statistical inferences means to learn about what you do not observe, which is called parameters, from what you do observe, which is called data. We learn the basic principles of statistical inference from a perspective of causal inference, which is a popular goal of political science research. Namely, we study statistics by learning how to make causal inferences with statistical methods. 1 Statistical Framework of Causal Inference What do we exactly mean when we say “An event A causes another event B”? Whether explicitly or implicitly, this question is asked and answered all the time in political science research. The most commonly used statistical framework of causality is based on the notion of counterfactuals (see Holland, 1986). That is, we ask the question “What would have happened if an event A were absent (or existent)?” The following example illustrates the fact that some causal questions are more difficult to answer than others. Example 1 (Counterfactual and Causality) Interpret each of the following statements as a causal statement. 1. A politician voted for the education bill because she is a democrat. 2. A politician voted for the education bill because she is liberal. 3. A politician voted for the education bill because she is a woman. In this framework, therefore, the fundamental problem of causal inference is that the coun- terfactual outcomes cannot be observed, and yet any causal inference requires both factual and counterfactual outcomes. This idea is formalized below using the potential outcomes notation. -
Chapter 4: Fisher's Exact Test in Completely Randomized Experiments
1 Chapter 4: Fisher’s Exact Test in Completely Randomized Experiments Fisher (1925, 1926) was concerned with testing hypotheses regarding the effect of treat- ments. Specifically, he focused on testing sharp null hypotheses, that is, null hypotheses under which all potential outcomes are known exactly. Under such null hypotheses all un- known quantities in Table 4 in Chapter 1 are known–there are no missing data anymore. As we shall see, this implies that we can figure out the distribution of any statistic generated by the randomization. Fisher’s great insight concerns the value of the physical randomization of the treatments for inference. Fisher’s classic example is that of the tea-drinking lady: “A lady declares that by tasting a cup of tea made with milk she can discriminate whether the milk or the tea infusion was first added to the cup. ... Our experi- ment consists in mixing eight cups of tea, four in one way and four in the other, and presenting them to the subject in random order. ... Her task is to divide the cups into two sets of 4, agreeing, if possible, with the treatments received. ... The element in the experimental procedure which contains the essential safeguard is that the two modifications of the test beverage are to be prepared “in random order.” This is in fact the only point in the experimental procedure in which the laws of chance, which are to be in exclusive control of our frequency distribution, have been explicitly introduced. ... it may be said that the simple precaution of randomisation will suffice to guarantee the validity of the test of significance, by which the result of the experiment is to be judged.” The approach is clear: an experiment is designed to evaluate the lady’s claim to be able to discriminate wether the milk or tea was first poured into the cup. -
Bayesian Causal Inference
Bayesian Causal Inference Maximilian Kurthen Master’s Thesis Max Planck Institute for Astrophysics Within the Elite Master Program Theoretical and Mathematical Physics Ludwig Maximilian University of Munich Technical University of Munich Supervisor: PD Dr. Torsten Enßlin Munich, September 12, 2018 Abstract In this thesis we address the problem of two-variable causal inference. This task refers to inferring an existing causal relation between two random variables (i.e. X → Y or Y → X ) from purely observational data. We begin by outlining a few basic definitions in the context of causal discovery, following the widely used do-Calculus [Pea00]. We continue by briefly reviewing a number of state-of-the-art methods, including very recent ones such as CGNN [Gou+17] and KCDC [MST18]. The main contribution is the introduction of a novel inference model where we assume a Bayesian hierarchical model, pursuing the strategy of Bayesian model selection. In our model the distribution of the cause variable is given by a Poisson lognormal distribution, which allows to explicitly regard discretization effects. We assume Fourier diagonal covariance operators, where the values on the diagonal are given by power spectra. In the most shallow model these power spectra and the noise variance are fixed hyperparameters. In a deeper inference model we replace the noise variance as a given prior by expanding the inference over the noise variance itself, assuming only a smooth spatial structure of the noise variance. Finally, we make a similar expansion for the power spectra, replacing fixed power spectra as hyperparameters by an inference over those, where again smoothness enforcing priors are assumed. -
Articles Causal Inference in Civil Rights Litigation
VOLUME 122 DECEMBER 2008 NUMBER 2 © 2008 by The Harvard Law Review Association ARTICLES CAUSAL INFERENCE IN CIVIL RIGHTS LITIGATION D. James Greiner TABLE OF CONTENTS I. REGRESSION’S DIFFICULTIES AND THE ABSENCE OF A CAUSAL INFERENCE FRAMEWORK ...................................................540 A. What Is Regression? .........................................................................................................540 B. Problems with Regression: Bias, Ill-Posed Questions, and Specification Issues .....543 1. Bias of the Analyst......................................................................................................544 2. Ill-Posed Questions......................................................................................................544 3. Nature of the Model ...................................................................................................545 4. One Regression, or Two?............................................................................................555 C. The Fundamental Problem: Absence of a Causal Framework ....................................556 II. POTENTIAL OUTCOMES: DEFINING CAUSAL EFFECTS AND BEYOND .................557 A. Primitive Concepts: Treatment, Units, and the Fundamental Problem.....................558 B. Additional Units and the Non-Interference Assumption.............................................560 C. Donating Values: Filling in the Missing Counterfactuals ...........................................562 D. Randomization: Balance in Background Variables ......................................................563 -
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. -
This Is Dr. Chumney. the Focus of This Lecture Is Hypothesis Testing –Both What It Is, How Hypothesis Tests Are Used, and How to Conduct Hypothesis Tests
TRANSCRIPT: This is Dr. Chumney. The focus of this lecture is hypothesis testing –both what it is, how hypothesis tests are used, and how to conduct hypothesis tests. 1 TRANSCRIPT: In this lecture, we will talk about both theoretical and applied concepts related to hypothesis testing. 2 TRANSCRIPT: Let’s being the lecture with a summary of the logic process that underlies hypothesis testing. 3 TRANSCRIPT: It is often impossible or otherwise not feasible to collect data on every individual within a population. Therefore, researchers rely on samples to help answer questions about populations. Hypothesis testing is a statistical procedure that allows researchers to use sample data to draw inferences about the population of interest. Hypothesis testing is one of the most commonly used inferential procedures. Hypothesis testing will combine many of the concepts we have already covered, including z‐scores, probability, and the distribution of sample means. To conduct a hypothesis test, we first state a hypothesis about a population, predict the characteristics of a sample of that population (that is, we predict that a sample will be representative of the population), obtain a sample, then collect data from that sample and analyze the data to see if it is consistent with our hypotheses. 4 TRANSCRIPT: The process of hypothesis testing begins by stating a hypothesis about the unknown population. Actually we state two opposing hypotheses. The first hypothesis we state –the most important one –is the null hypothesis. The null hypothesis states that the treatment has no effect. In general the null hypothesis states that there is no change, no difference, no effect, and otherwise no relationship between the independent and dependent variables. -
The Scientific Method: Hypothesis Testing and Experimental Design
Appendix I The Scientific Method The study of science is different from other disciplines in many ways. Perhaps the most important aspect of “hard” science is its adherence to the principle of the scientific method: the posing of questions and the use of rigorous methods to answer those questions. I. Our Friend, the Null Hypothesis As a science major, you are probably no stranger to curiosity. It is the beginning of all scientific discovery. As you walk through the campus arboretum, you might wonder, “Why are trees green?” As you observe your peers in social groups at the cafeteria, you might ask yourself, “What subtle kinds of body language are those people using to communicate?” As you read an article about a new drug which promises to be an effective treatment for male pattern baldness, you think, “But how do they know it will work?” Asking such questions is the first step towards hypothesis formation. A scientific investigator does not begin the study of a biological phenomenon in a vacuum. If an investigator observes something interesting, s/he first asks a question about it, and then uses inductive reasoning (from the specific to the general) to generate an hypothesis based upon a logical set of expectations. To test the hypothesis, the investigator systematically collects data, either with field observations or a series of carefully designed experiments. By analyzing the data, the investigator uses deductive reasoning (from the general to the specific) to state a second hypothesis (it may be the same as or different from the original) about the observations. -
Hypothesis Testing – Examples and Case Studies
Hypothesis Testing – Chapter 23 Examples and Case Studies Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a test statistic. 3. Use the test statistic to determine the p-value. 4. The result is statistically significant if the p-value is less than or equal to the level of significance. Often media only presents results of step 4. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 2 23.2 Testing Hypotheses About Proportions and Means If the null and alternative hypotheses are expressed in terms of a population proportion, mean, or difference between two means and if the sample sizes are large … … the test statistic is simply the corresponding standardized score computed assuming the null hypothesis is true; and the p-value is found from a table of percentiles for standardized scores. Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 3 Example 2: Weight Loss for Diet vs Exercise Did dieters lose more fat than the exercisers? Diet Only: sample mean = 5.9 kg sample standard deviation = 4.1 kg sample size = n = 42 standard error = SEM1 = 4.1/ √42 = 0.633 Exercise Only: sample mean = 4.1 kg sample standard deviation = 3.7 kg sample size = n = 47 standard error = SEM2 = 3.7/ √47 = 0.540 measure of variability = [(0.633)2 + (0.540)2] = 0.83 Copyright ©2005 Brooks/Cole, a division of Thomson Learning, Inc. 4 Example 2: Weight Loss for Diet vs Exercise Step 1. -
The Theory of the Design of Experiments
The Theory of the Design of Experiments D.R. COX Honorary Fellow Nuffield College Oxford, UK AND N. REID Professor of Statistics University of Toronto, Canada CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C. C195X/disclaimer Page 1 Friday, April 28, 2000 10:59 AM Library of Congress Cataloging-in-Publication Data Cox, D. R. (David Roxbee) The theory of the design of experiments / D. R. Cox, N. Reid. p. cm. — (Monographs on statistics and applied probability ; 86) Includes bibliographical references and index. ISBN 1-58488-195-X (alk. paper) 1. Experimental design. I. Reid, N. II.Title. III. Series. QA279 .C73 2000 001.4 '34 —dc21 00-029529 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W.