History of Biostatistics
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Confidence Intervals for the Population Mean Alternatives to the Student-T Confidence Interval
Journal of Modern Applied Statistical Methods Volume 18 Issue 1 Article 15 4-6-2020 Robust Confidence Intervals for the Population Mean Alternatives to the Student-t Confidence Interval Moustafa Omar Ahmed Abu-Shawiesh The Hashemite University, Zarqa, Jordan, [email protected] Aamir Saghir Mirpur University of Science and Technology, Mirpur, Pakistan, [email protected] Follow this and additional works at: https://digitalcommons.wayne.edu/jmasm Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons Recommended Citation Abu-Shawiesh, M. O. A., & Saghir, A. (2019). Robust confidence intervals for the population mean alternatives to the Student-t confidence interval. Journal of Modern Applied Statistical Methods, 18(1), eP2721. doi: 10.22237/jmasm/1556669160 This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Robust Confidence Intervals for the Population Mean Alternatives to the Student-t Confidence Interval Cover Page Footnote The authors are grateful to the Editor and anonymous three reviewers for their excellent and constructive comments/suggestions that greatly improved the presentation and quality of the article. This article was partially completed while the first author was on sabbatical leave (2014–2015) in Nizwa University, Sultanate of Oman. He is grateful to the Hashemite University for awarding him the sabbatical leave which gave him excellent research facilities. This regular article is available in Journal of Modern Applied Statistical Methods: https://digitalcommons.wayne.edu/ jmasm/vol18/iss1/15 Journal of Modern Applied Statistical Methods May 2019, Vol. -
STAT 22000 Lecture Slides Overview of Confidence Intervals
STAT 22000 Lecture Slides Overview of Confidence Intervals Yibi Huang Department of Statistics University of Chicago Outline This set of slides covers section 4.2 in the text • Overview of Confidence Intervals 1 Confidence intervals • A plausible range of values for the population parameter is called a confidence interval. • Using only a sample statistic to estimate a parameter is like fishing in a murky lake with a spear, and using a confidence interval is like fishing with a net. We can throw a spear where we saw a fish but we will probably miss. If we toss a net in that area, we have a good chance of catching the fish. • If we report a point estimate, we probably won’t hit the exact population parameter. If we report a range of plausible values we have a good shot at capturing the parameter. 2 Photos by Mark Fischer (http://www.flickr.com/photos/fischerfotos/7439791462) and Chris Penny (http://www.flickr.com/photos/clearlydived/7029109617) on Flickr. Recall that CLT says, for large n, X ∼ N(µ, pσ ): For a normal n curve, 95% of its area is within 1.96 SDs from the center. That means, for 95% of the time, X will be within 1:96 pσ from µ. n 95% σ σ µ − 1.96 µ µ + 1.96 n n Alternatively, we can also say, for 95% of the time, µ will be within 1:96 pσ from X: n Hence, we call the interval ! σ σ σ X ± 1:96 p = X − 1:96 p ; X + 1:96 p n n n a 95% confidence interval for µ. -
Epidemic Models
Chapter 9 Epidemic Models 9.1 Introduction to Epidemic Models Communicable diseases such as measles, influenza, and tuberculosis are a fact of life. We will be concerned with both epidemics, which are sudden outbreaks of a disease, and endemic situations, in which a disease is always present. The AIDS epidemic, the recent SARS epidemic, recurring influenza pandemics, and outbursts of diseases such as the Ebola virus are events of concern and interest to many peo- ple. The prevalence and effects of many diseases in less-developed countries are probably not as well known but may be of even more importance. Every year mil- lions, of people die of measles, respiratory infections, diarrhea, and other diseases that are easily treated and not considered dangerous in the Western world. Diseases such as malaria, typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable. We give a brief introduction to the modeling of epidemics; more thorough de- scriptions may be found in such references as [Anderson & May (1991), Diekmann & Heesterbeek (2000)]. This chapter will describe models for epidemics, and the next chapter will deal with models for endemic situations, but we begin with some general ideas about disease transmission. The idea of invisible living creatures as agents of disease goes back at least to the writings of Aristotle (384 BC–322 BC). It developed as a theory in the sixteenth century. The existence of microorganisms was demonstrated by van Leeuwenhoek (1632–1723) with the aid of the first microscopes. -
Statistics for Dummies Cheat Sheet from Statistics for Dummies, 2Nd Edition by Deborah Rumsey
Statistics For Dummies Cheat Sheet From Statistics For Dummies, 2nd Edition by Deborah Rumsey Copyright © 2013 & Trademark by John Wiley & Sons, Inc. All rights reserved. Whether you’re studying for an exam or just want to make sense of data around you every day, knowing how and when to use data analysis techniques and formulas of statistics will help. Being able to make the connections between those statistical techniques and formulas is perhaps even more important. It builds confidence when attacking statistical problems and solidifies your strategies for completing statistical projects. Understanding Formulas for Common Statistics After data has been collected, the first step in analyzing it is to crunch out some descriptive statistics to get a feeling for the data. For example: Where is the center of the data located? How spread out is the data? How correlated are the data from two variables? The most common descriptive statistics are in the following table, along with their formulas and a short description of what each one measures. Statistically Figuring Sample Size When designing a study, the sample size is an important consideration because the larger the sample size, the more data you have, and the more precise your results will be (assuming high- quality data). If you know the level of precision you want (that is, your desired margin of error), you can calculate the sample size needed to achieve it. To find the sample size needed to estimate a population mean (µ), use the following formula: In this formula, MOE represents the desired margin of error (which you set ahead of time), and σ represents the population standard deviation. -
Medicine in 18Th and 19Th Century Britain, 1700-1900
Medicine in 18th and 19th century Britain, 1700‐1900 The breakthroughs th 1798: Edward Jenner – The development of How had society changed to make medical What was behind the 19 C breakthroughs? Changing ideas of causes breakthroughs possible? vaccinations Jenner trained by leading surgeon who taught The first major breakthrough came with Louis Pasteur’s germ theory which he published in 1861. His later students to observe carefully and carry out own Proved vaccination prevented people catching smallpox, experiments proved that bacteria (also known as microbes or germs) cause diseases. However, this did not put an end The changes described in the Renaissance were experiments instead of relying on knowledge in one of the great killer diseases. Based on observation and to all earlier ideas. Belief that bad air was to blame continued, which is not surprising given the conditions in many the result of rapid changes in society, but they did books – Jenner followed these methods. scientific experiment. However, did not understand what industrial towns. In addition, Pasteur’s theory was a very general one until scientists begun to identify the individual also build on changes and ideas from earlier caused smallpox all how vaccination worked. At first dad bacteria which cause particular diseases. So, while this was one of the two most important breakthroughs in ideas centuries. The flushing toilet important late 19th C invention wants opposition to making vaccination compulsory by law about what causes disease and illness it did not revolutionise medicine immediately. Scientists and doctors where the 1500s Renaissance – flushing system sent waste instantly down into – overtime saved many people’s lives and wiped‐out first to be convinced of this theory, but it took time for most people to understand 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. -
Medicine in Eighteenth and Nineteenth Century Britain, C.1700-C.1900
Year 10 History Knowledge Organiser – Key topic 3: Medicine in eighteenth and nineteenth century Britain, c.1700-c.1900 2. Key dates 1. Key terms 1 1796 Edward Jenner tested his vaccine on James Phipps. He infected him with cowpox, and this 1 Microbes A living organism which is too small to see prevented him catching smallpox. without a microscope, this includes 2 1842 Edwin Chadwick published his ‘Report on the Sanitary Conditions of the Labouring Classes’. bacteria. 1847 James Simpson discovered that chloroform could be used as an anaesthetic. 2 The A movement of European intellectuals that 3 Enlightenment emphasised reason rather than tradition. 4 1848 First Public Health Act – set up Boards of Health but was not compulsory. 3 Decaying Matter Material, such as vegetables or animals, that has died and is rotting. 5 1854 John Snow mapped the spread of disease around the Broad Street pump to prove that cholera was caused by dirty water. 4 Bacteriology The study of bacteria. 6 1858 The Great Stink near the Houses of Parliament prompted action on sewage. 5 Tuberculosis A disease which affects the lungs causing serious difficulties in breathing. 7 1861 Louis Pasteur published the Germ Theory 6 Cholera A waterborne disease which killed many 1865 Joseph Lister used carbolic acid for the first time. He wrapped up a leg after an operation in th 8 people in the 19 century by causing the acid-soaked bandages and the wound healed cleanly. body to become dehydrated. 9 1875 Second Public Health Act, made government intervention in public health compulsory. -
Confidence Intervals
Confidence Intervals PoCoG Biostatistical Clinic Series Joseph Coll, PhD | Biostatistician Introduction › Introduction to Confidence Intervals › Calculating a 95% Confidence Interval for the Mean of a Sample › Correspondence Between Hypothesis Tests and Confidence Intervals 2 Introduction to Confidence Intervals › Each time we take a sample from a population and calculate a statistic (e.g. a mean or proportion), the value of the statistic will be a little different. › If someone asks for your best guess of the population parameter, you would use the value of the sample statistic. › But how well does the statistic estimate the population parameter? › It is possible to calculate a range of values based on the sample statistic that encompasses the true population value with a specified level of probability (confidence). 3 Introduction to Confidence Intervals › DEFINITION: A 95% confidence for the mean of a sample is a range of values which we can be 95% confident includes the mean of the population from which the sample was drawn. › If we took 100 random samples from a population and calculated a mean and 95% confidence interval for each, then approximately 95 of the 100 confidence intervals would include the population mean. 4 Introduction to Confidence Intervals › A single 95% confidence interval is the set of possible values that the population mean could have that are not statistically different from the observed sample mean. › A 95% confidence interval is a set of possible true values of the parameter of interest (e.g. mean, correlation coefficient, odds ratio, difference of means, proportion) that are consistent with the data. 5 Calculating a 95% Confidence Interval for the Mean of a Sample › ‾x‾ ± k * (standard deviation / √n) › Where ‾x‾ is the mean of the sample › k is the constant dependent on the hypothesized distribution of the sample mean, the sample size and the amount of confidence desired. -
P Values and Confidence Intervals Friends Or Foe
P values and Confidence Intervals Friends or Foe Dr.L.Jeyaseelan Dept. of Biostatistics Christian Medical College Vellore – 632 002, India JPGM WriteCon March 30-31, 2007, KEM Hospital, Mumbai Confidence Interval • The mean or proportion observed in a sample is the best estimate of the true value in the population. • We can combine these features of estimates from a sample from a sample with the known properties of the normal distribution to get an idea of the uncertainty associated with a single sample estimate of the population value. Confidence Interval • The interval between Mean - 2SE and Mean + 2 SE will be 95%. • That is, we expect that 95% CI will not include the true population value 5% of the time. • No particular reason for choosing 95% CI other than convention. Confidence Interval Interpretation: The 95% CI for the sample mean is interpreted as a range of values which contains the true population mean with probability 0.95%. Ex: Mean serum albumin of the population (source) was 35 g/l. Calculate 95% CI for the mean serum albumin using each of the 100 random samples of size 25. Scatter Plot of 95% CI: Confidence intervals for mean serum albumin constructed from 100 random samples of size 25. The vertical lines show the range within which 95% of sample means are expected to fall. Confidence Interval • CI expresses a summary of the data in the original units or measurement. • It reflects the variability in the data, sample size and the actual effect size • Particularly helpful for non-significant findings. Relative Risk 1.5 95% CI 0.6 – 3.8 Confidence Interval for Low Prevalence • Asymptotic Formula: p ± 1.96 SE provides symmetric CI. -
Student's T Distribution
Student’s t distribution - supplement to chap- ter 3 For large samples, X¯ − µ Z = n √ (1) n σ/ n has approximately a standard normal distribution. The parameter σ is often unknown and so we must replace σ by s, where s is the square root of the sample variance. This new statistic is usually denoted with a t. X¯ − µ t = n√ (2) s/ n If the sample is large, the distribution of t is also approximately the standard normal. What if the sample is not large? In the absence of any further information there is not much to be said other than to remark that the variance of t will be somewhat larger than 1. If we assume that the random variable X has a normal distribution, then we can say much more. (Often people say the population is normal, meaning the distribution of the rv X is normal.) So now suppose that X is a normal random variable with mean µ and variance σ2. The two parameters µ and σ are unknown. Since a sum of independent normal random variables is normal, the distribution of Zn is exactly normal, no matter what the sample size is. Furthermore, Zn has mean zero and variance 1, so the distribution of Zn is exactly that of the standard normal. The distribution of t is not normal, but it is completely determined. It depends on the sample size and a priori it looks like it will depend on the parameters µ and σ. With a little thought you should be able to convince yourself that it does not depend on µ. -
Multiple Random Variables
Multiple random variables Multiple random variables We essentially always consider multiple random variables at once. The key concepts: Joint, conditional and marginal distri- −→ butions, and independence of RVs. Let X and Y be discrete random variables. Joint distribution: −→ pXY(x,y) = Pr(X = x and Y = y) Marginal distributions: −→ pX(x) = Pr(X = x) = y pXY(x,y) p (y) = Pr(Y = y) = p (x,y) Y !x XY Conditional distributions! : −→ pX Y=y(x) = Pr(X =x Y = y) = pXY(x,y) / pY(y) | | Example Sample a couple who are both carriers of some disease gene. X = number of children they have Y = number of affected children they have x pXY(x,y) 012345pY(y) 0 0.160 0.248 0.124 0.063 0.025 0.014 0.634 1 0 0.082 0.082 0.063 0.034 0.024 0.285 y 2 0 0 0.014 0.021 0.017 0.016 0.068 3 0 0 0 0.003 0.004 0.005 0.012 4 0 0 0 0 0.000 0.001 0.001 5 0 0 0 0 0 0.000 0.000 pX(x) 0.160 0.330 0.220 0.150 0.080 0.060 Pr( Y = y X = 2 ) | x pXY(x,y) 012 345pY(y) 0 0.160 0.248 0.124 0.063 0.025 0.014 0.634 1 0 0.082 0.082 0.063 0.034 0.024 0.285 y 2 000.014 0.021 0.017 0.016 0.068 3 000 0.003 0.004 0.005 0.012 4 000 0 0.000 0.001 0.001 5 000 0 0 0.000 0.000 pX(x) 0.160 0.330 0.220 0.150 0.080 0.060 y 012345 Pr( Y=y X=2 ) 0.564 0.373 0.064 0.000 0.000 0.000 | Pr( X = x Y = 1 ) | x pXY(x,y) 012345pY(y) 0 0.160 0.248 0.124 0.063 0.025 0.014 0.634 1 0 0.082 0.082 0.063 0.034 0.024 0.285 y 2 0 0 0.014 0.021 0.017 0.016 0.068 3 0 0 0 0.003 0.004 0.005 0.012 4 0 0 0 0 0.000 0.001 0.001 5 0 0 0 0 0 0.000 0.000 pX(x) 0.160 0.330 0.220 0.150 0.080 0.060 x 012345 Pr( X=x Y=1 ) 0.000 0.288 0.288 0.221 0.119 0.084 | Independence Random variables X and Y are independent if p (x,y) = p (x) p (y) −→ XY X Y for every pair x,y. -
Statistical Tests, P-Values, Confidence Intervals, and Power
Statistical Tests, P-values, Confidence Intervals, and Power: A Guide to Misinterpretations Sander GREENLAND, Stephen J. SENN, Kenneth J. ROTHMAN, John B. CARLIN, Charles POOLE, Steven N. GOODMAN, and Douglas G. ALTMAN Misinterpretation and abuse of statistical tests, confidence in- Introduction tervals, and statistical power have been decried for decades, yet remain rampant. A key problem is that there are no interpreta- Misinterpretation and abuse of statistical tests has been de- tions of these concepts that are at once simple, intuitive, cor- cried for decades, yet remains so rampant that some scientific rect, and foolproof. Instead, correct use and interpretation of journals discourage use of “statistical significance” (classify- these statistics requires an attention to detail which seems to tax ing results as “significant” or not based on a P-value) (Lang et the patience of working scientists. This high cognitive demand al. 1998). One journal now bans all statistical tests and mathe- has led to an epidemic of shortcut definitions and interpreta- matically related procedures such as confidence intervals (Trafi- tions that are simply wrong, sometimes disastrously so—and mow and Marks 2015), which has led to considerable discussion yet these misinterpretations dominate much of the scientific lit- and debate about the merits of such bans (e.g., Ashworth 2015; erature. Flanagan 2015). In light of this problem, we provide definitions and a discus- Despite such bans, we expect that the statistical methods at sion of basic statistics that are more general and critical than issue will be with us for many years to come. We thus think it typically found in traditional introductory expositions.