Inference in Practice
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Statistical Inferences Hypothesis Tests
STATISTICAL INFERENCES HYPOTHESIS TESTS Maªgorzata Murat BASIC IDEAS Suppose we have collected a representative sample that gives some information concerning a mean or other statistical quantity. There are two main questions for which statistical inference may provide answers. 1. Are the sample quantity and a corresponding population quantity close enough together so that it is reasonable to say that the sample might have come from the population? Or are they far enough apart so that they likely represent dierent populations? 2. For what interval of values can we have a specic level of condence that the interval contains the true value of the parameter of interest? STATISTICAL INFERENCES FOR THE MEAN We can divide statistical inferences for the mean into two main categories. In one category, we already know the variance or standard deviation of the population, usually from previous measurements, and the normal distribution can be used for calculations. In the other category, we nd an estimate of the variance or standard deviation of the population from the sample itself. The normal distribution is assumed to apply to the underlying population, but another distribution related to it will usually be required for calculations. TEST OF HYPOTHESIS We are testing the hypothesis that a sample is similar enough to a particular population so that it might have come from that population. Hence we make the null hypothesis that the sample came from a population having the stated value of the population characteristic (the mean or the variation). Then we do calculations to see how reasonable such a hypothesis is. We have to keep in mind the alternative if the null hypothesis is not true, as the alternative will aect the calculations. -
The Focused Information Criterion in Logistic Regression to Predict Repair of Dental Restorations
THE FOCUSED INFORMATION CRITERION IN LOGISTIC REGRESSION TO PREDICT REPAIR OF DENTAL RESTORATIONS Cecilia CANDOLO1 ABSTRACT: Statistical data analysis typically has several stages: exploration of the data set; deciding on a class or classes of models to be considered; selecting the best of them according to some criterion and making inferences based on the selected model. The cycle is usually iterative and will involve subject-matter considerations as well as statistical insights. The conclusion reached after such a process depends on the model(s) selected, but the consequent uncertainty is not usually incorporated into the inference. This may lead to underestimation of the uncertainty about quantities of interest and overoptimistic and biased inferences. This framework has been the aim of research under the terminology of model uncertainty and model averanging in both, frequentist and Bayesian approaches. The former usually uses the Akaike´s information criterion (AIC), the Bayesian information criterion (BIC) and the bootstrap method. The last weigths the models using the posterior model probabilities. This work consider model selection uncertainty in logistic regression under frequentist and Bayesian approaches, incorporating the use of the focused information criterion (FIC) (CLAESKENS and HJORT, 2003) to predict repair of dental restorations. The FIC takes the view that a best model should depend on the parameter under focus, such as the mean, or the variance, or the particular covariate values. In this study, the repair or not of dental restorations in a period of eighteen months depends on several covariates measured in teenagers. The data were kindly provided by Juliana Feltrin de Souza, a doctorate student at the Faculty of Dentistry of Araraquara - UNESP. -
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Monte Carlos study on Power Rates of Some Heteroscedasticity detection Methods in Linear Regression Model with multicollinearity problem O.O. Alabi, Kayode Ayinde, O. E. Babalola, and H.A. Bello Department of Statistics, Federal University of Technology, P.M.B. 704, Akure, Ondo State, Nigeria Corresponding Author: O. O. Alabi, [email protected] Abstract: This paper examined the power rate exhibit by some heteroscedasticity detection methods in a linear regression model with multicollinearity problem. Violation of unequal error variance assumption in any linear regression model leads to the problem of heteroscedasticity, while violation of the assumption of non linear dependency between the exogenous variables leads to multicollinearity problem. Whenever these two problems exist one would faced with estimation and hypothesis problem. in order to overcome these hurdles, one needs to determine the best method of heteroscedasticity detection in other to avoid taking a wrong decision under hypothesis testing. This then leads us to the way and manner to determine the best heteroscedasticity detection method in a linear regression model with multicollinearity problem via power rate. In practices, variance of error terms are unequal and unknown in nature, but there is need to determine the presence or absence of this problem that do exist in unknown error term as a preliminary diagnosis on the set of data we are to analyze or perform hypothesis testing on. Although, there are several forms of heteroscedasticity and several detection methods of heteroscedasticity, but for any researcher to arrive at a reasonable and correct decision, best and consistent performed methods of heteroscedasticity detection under any forms or structured of heteroscedasticity must be determined. -
The Effects of Simplifying Assumptions in Power Analysis
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Public Access Theses and Dissertations from Education and Human Sciences, College of the College of Education and Human Sciences (CEHS) 4-2011 The Effects of Simplifying Assumptions in Power Analysis Kevin A. Kupzyk University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/cehsdiss Part of the Educational Psychology Commons Kupzyk, Kevin A., "The Effects of Simplifying Assumptions in Power Analysis" (2011). Public Access Theses and Dissertations from the College of Education and Human Sciences. 106. https://digitalcommons.unl.edu/cehsdiss/106 This Article is brought to you for free and open access by the Education and Human Sciences, College of (CEHS) at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Public Access Theses and Dissertations from the College of Education and Human Sciences by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Kupzyk - i THE EFFECTS OF SIMPLIFYING ASSUMPTIONS IN POWER ANALYSIS by Kevin A. Kupzyk A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Psychological Studies in Education Under the Supervision of Professor James A. Bovaird Lincoln, Nebraska April, 2011 Kupzyk - i THE EFFECTS OF SIMPLIFYING ASSUMPTIONS IN POWER ANALYSIS Kevin A. Kupzyk, Ph.D. University of Nebraska, 2011 Adviser: James A. Bovaird In experimental research, planning studies that have sufficient probability of detecting important effects is critical. Carrying out an experiment with an inadequate sample size may result in the inability to observe the effect of interest, wasting the resources spent on an experiment. -
Introduction to Hypothesis Testing
Introduction to Hypothesis Testing OPRE 6301 Motivation . The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief, or hypothesis, about a parameter. Examples: Is there statistical evidence, from a random sample of potential customers, to support the hypothesis that more than 10% of the potential customers will pur- chase a new product? Is a new drug effective in curing a certain disease? A sample of patients is randomly selected. Half of them are given the drug while the other half are given a placebo. The conditions of the patients are then mea- sured and compared. These questions/hypotheses are similar in spirit to the discrimination example studied earlier. Below, we pro- vide a basic introduction to hypothesis testing. 1 Criminal Trials . The basic concepts in hypothesis testing are actually quite analogous to those in a criminal trial. Consider a person on trial for a “criminal” offense in the United States. Under the US system a jury (or sometimes just the judge) must decide if the person is innocent or guilty while in fact the person may be innocent or guilty. These combinations are summarized in the table below. Person is: Innocent Guilty Jury Says: Innocent No Error Error Guilty Error No Error Notice that there are two types of errors. Are both of these errors equally important? Or, is it as bad to decide that a guilty person is innocent and let them go free as it is to decide an innocent person is guilty and punish them for the crime? Or, is a jury supposed to be totally objective, not assuming that the person is either innocent or guilty and make their decision based on the weight of the evidence one way or another? 2 In a criminal trial, there actually is a favored assump- tion, an initial bias if you will. -
Power of a Statistical Test
Power of a Statistical Test By Smita Skrivanek, Principal Statistician, MoreSteam.com LLC What is the power of a test? The power of a statistical test gives the likelihood of rejecting the null hypothesis when the null hypothesis is false. Just as the significance level (alpha) of a test gives the probability that the null hypothesis will be rejected when it is actually true (a wrong decision), power quantifies the chance that the null hypothesis will be rejected when it is actually false (a correct decision). Thus, power is the ability of a test to correctly reject the null hypothesis. Why is it important? Although you can conduct a hypothesis test without it, calculating the power of a test beforehand will help you ensure that the sample size is large enough for the purpose of the test. Otherwise, the test may be inconclusive, leading to wasted resources. On rare occasions the power may be calculated after the test is performed, but this is not recommended except to determine an adequate sample size for a follow-up study (if a test failed to detect an effect, it was obviously underpowered – nothing new can be learned by calculating the power at this stage). How is it calculated? As an example, consider testing whether the average time per week spent watching TV is 4 hours versus the alternative that it is greater than 4 hours. We will calculate the power of the test for a specific value under the alternative hypothesis, say, 7 hours: The Null Hypothesis is H0: μ = 4 hours The Alternative Hypothesis is H1: μ = 6 hours Where μ = the average time per week spent watching TV. -
Confidence Intervals and Hypothesis Tests
Chapter 2 Confidence intervals and hypothesis tests This chapter focuses on how to draw conclusions about populations from sample data. We'll start by looking at binary data (e.g., polling), and learn how to estimate the true ratio of 1s and 0s with confidence intervals, and then test whether that ratio is significantly different from some baseline value using hypothesis testing. Then, we'll extend what we've learned to continuous measurements. 2.1 Binomial data Suppose we're conducting a yes/no survey of a few randomly sampled people 1, and we want to use the results of our survey to determine the answers for the overall population. 2.1.1 The estimator The obvious first choice is just the fraction of people who said yes. Formally, suppose we have samples x1,..., xn that can each be 0 or 1, and the probability that each xi is 1 is p (in frequentist style, we'll assume p is fixed but unknown: this is what we're interested in finding). We'll assume our samples are indendepent and identically distributed (i.i.d.), meaning that each one has no dependence on any of the others, and they all have the same probability p of being 1. Then our estimate for p, which we'll callp ^, or \p-hat" would be n 1 X p^ = x : n i i=1 Notice thatp ^ is a random quantity, since it depends on the random quantities xi. In statistical lingo,p ^ is known as an estimator for p. Also notice that except for the factor of 1=n in front, p^ is almost a binomial random variable (in particular, (np^) ∼ B(n; p)). -
What Is Statistic?
What is Statistic? OPRE 6301 In today’s world. ...we are constantly being bombarded with statistics and statistical information. For example: Customer Surveys Medical News Demographics Political Polls Economic Predictions Marketing Information Sales Forecasts Stock Market Projections Consumer Price Index Sports Statistics How can we make sense out of all this data? How do we differentiate valid from flawed claims? 1 What is Statistics?! “Statistics is a way to get information from data.” Statistics Data Information Data: Facts, especially Information: Knowledge numerical facts, collected communicated concerning together for reference or some particular fact. information. Statistics is a tool for creating an understanding from a set of numbers. Humorous Definitions: The Science of drawing a precise line between an unwar- ranted assumption and a forgone conclusion. The Science of stating precisely what you don’t know. 2 An Example: Stats Anxiety. A business school student is anxious about their statistics course, since they’ve heard the course is difficult. The professor provides last term’s final exam marks to the student. What can be discerned from this list of numbers? Statistics Data Information List of last term’s marks. New information about the statistics class. 95 89 70 E.g. Class average, 65 Proportion of class receiving A’s 78 Most frequent mark, 57 Marks distribution, etc. : 3 Key Statistical Concepts. Population — a population is the group of all items of interest to a statistics practitioner. — frequently very large; sometimes infinite. E.g. All 5 million Florida voters (per Example 12.5). Sample — A sample is a set of data drawn from the population. -
Structured Statistical Models of Inductive Reasoning
CORRECTED FEBRUARY 25, 2009; SEE LAST PAGE Psychological Review © 2009 American Psychological Association 2009, Vol. 116, No. 1, 20–58 0033-295X/09/$12.00 DOI: 10.1037/a0014282 Structured Statistical Models of Inductive Reasoning Charles Kemp Joshua B. Tenenbaum Carnegie Mellon University Massachusetts Institute of Technology Everyday inductive inferences are often guided by rich background knowledge. Formal models of induction should aim to incorporate this knowledge and should explain how different kinds of knowledge lead to the distinctive patterns of reasoning found in different inductive contexts. This article presents a Bayesian framework that attempts to meet both goals and describe 4 applications of the framework: a taxonomic model, a spatial model, a threshold model, and a causal model. Each model makes probabi- listic inferences about the extensions of novel properties, but the priors for the 4 models are defined over different kinds of structures that capture different relationships between the categories in a domain. The framework therefore shows how statistical inference can operate over structured background knowledge, and the authors argue that this interaction between structure and statistics is critical for explaining the power and flexibility of human reasoning. Keywords: inductive reasoning, property induction, knowledge representation, Bayesian inference Humans are adept at making inferences that take them beyond This article describes a formal approach to inductive inference the limits of their direct experience. -
Statistical Inference: Paradigms and Controversies in Historic Perspective
Jostein Lillestøl, NHH 2014 Statistical inference: Paradigms and controversies in historic perspective 1. Five paradigms We will cover the following five lines of thought: 1. Early Bayesian inference and its revival Inverse probability – Non-informative priors – “Objective” Bayes (1763), Laplace (1774), Jeffreys (1931), Bernardo (1975) 2. Fisherian inference Evidence oriented – Likelihood – Fisher information - Necessity Fisher (1921 and later) 3. Neyman- Pearson inference Action oriented – Frequentist/Sample space – Objective Neyman (1933, 1937), Pearson (1933), Wald (1939), Lehmann (1950 and later) 4. Neo - Bayesian inference Coherent decisions - Subjective/personal De Finetti (1937), Savage (1951), Lindley (1953) 5. Likelihood inference Evidence based – likelihood profiles – likelihood ratios Barnard (1949), Birnbaum (1962), Edwards (1972) Classical inference as it has been practiced since the 1950’s is really none of these in its pure form. It is more like a pragmatic mix of 2 and 3, in particular with respect to testing of significance, pretending to be both action and evidence oriented, which is hard to fulfill in a consistent manner. To keep our minds on track we do not single out this as a separate paradigm, but will discuss this at the end. A main concern through the history of statistical inference has been to establish a sound scientific framework for the analysis of sampled data. Concepts were initially often vague and disputed, but even after their clarification, various schools of thought have at times been in strong opposition to each other. When we try to describe the approaches here, we will use the notions of today. All five paradigms of statistical inference are based on modeling the observed data x given some parameter or “state of the world” , which essentially corresponds to stating the conditional distribution f(x|(or making some assumptions about it). -
Understanding Statistical Hypothesis Testing: the Logic of Statistical Inference
Review Understanding Statistical Hypothesis Testing: The Logic of Statistical Inference Frank Emmert-Streib 1,2,* and Matthias Dehmer 3,4,5 1 Predictive Society and Data Analytics Lab, Faculty of Information Technology and Communication Sciences, Tampere University, 33100 Tampere, Finland 2 Institute of Biosciences and Medical Technology, Tampere University, 33520 Tampere, Finland 3 Institute for Intelligent Production, Faculty for Management, University of Applied Sciences Upper Austria, Steyr Campus, 4040 Steyr, Austria 4 Department of Mechatronics and Biomedical Computer Science, University for Health Sciences, Medical Informatics and Technology (UMIT), 6060 Hall, Tyrol, Austria 5 College of Computer and Control Engineering, Nankai University, Tianjin 300000, China * Correspondence: [email protected]; Tel.: +358-50-301-5353 Received: 27 July 2019; Accepted: 9 August 2019; Published: 12 August 2019 Abstract: Statistical hypothesis testing is among the most misunderstood quantitative analysis methods from data science. Despite its seeming simplicity, it has complex interdependencies between its procedural components. In this paper, we discuss the underlying logic behind statistical hypothesis testing, the formal meaning of its components and their connections. Our presentation is applicable to all statistical hypothesis tests as generic backbone and, hence, useful across all application domains in data science and artificial intelligence. Keywords: hypothesis testing; machine learning; statistics; data science; statistical inference 1. Introduction We are living in an era that is characterized by the availability of big data. In order to emphasize the importance of this, data have been called the ‘oil of the 21st Century’ [1]. However, for dealing with the challenges posed by such data, advanced analysis methods are needed. -
Introduction to Statistical Methods Lecture 1: Basic Concepts & Descriptive Statistics
Introduction to Statistical Methods Lecture 1: Basic Concepts & Descriptive Statistics Theophanis Tsandilas [email protected] !1 Course information Web site: https://www.lri.fr/~fanis/courses/Stats2019 My email: [email protected] Slack workspace: stats-2019.slack.com !2 Course calendar !3 Lecture overview Part I 1. Basic concepts: data, populations & samples 2. Why learning statistics? 3. Course overview & teaching approach Part II 4. Types of data 5. Basic descriptive statistics Part III 6. Introduction to R !4 Part I. Basic concepts !5 Data vs. numbers Numbers are abstract tokens or symbols used for counting or measuring1 Data can be numbers but represent « real world » entities 1Several definitions in these slides are borrowed from Baguley’s book on Serious Stats Example Take a look at this set of numbers: 8 10 8 12 14 13 12 13 Context is important in understanding what they represent ! The age in years of 8 children ! The grades (out of 20) of 8 students ! The number of errors made by 8 participants in an experiment from a series of 100 experimental trials ! A memorization score of the same individual over repeated tests !7 Sampling The context depends on the process that generated the data ! collecting a subset (sample) of observations from a larger set (population) of observations of interest The idea of taking a sample from a population is central to understanding statistics and the heart of most statistical procedures. !8 Statistics A sample, being a subset of the whole population, won’t necessarily resemble it. Thus, the information the sample provides about the population is uncertain.