'Statistical Irreproducibility' Does Not Improve with Larger Sample Size
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Exploring Bimodality in Introductory Computer Science Performance Distributions
EURASIA Journal of Mathematics, Science and Technology Education, 2018, 14(10), em1591 ISSN:1305-8223 (online) OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/93190 Exploring Bimodality in Introductory Computer Science Performance Distributions Ram B Basnet 1, Lori K Payne 1, Tenzin Doleck 2*, David John Lemay 2, Paul Bazelais 2 1 Colorado Mesa University, Grand Junction, Colorado, USA 2 McGill University, Montreal, Quebec, CANADA Received 7 April 2018 ▪ Revised 19 June 2018 ▪ Accepted 21 June 2018 ABSTRACT This study examines student performance distributions evidence bimodality, or whether there are two distinct populations in three introductory computer science courses grades at a four-year southwestern university in the United States for the period 2014-2017. Results suggest that computer science course grades are not bimodal. These findings counter the double hump assertion and suggest that proper course sequencing can address the needs of students with varying levels of prior knowledge and obviate the double-hump phenomenon. Studying performance helps to improve delivery of introductory computer science courses by ensuring that courses are aligned with student needs and address preconceptions and prior knowledge and experience. Keywords: computer science performance, coding, programming, double hump, grade distribution, bimodal distribution, unimodal distribution BACKGROUND In recent years, there has been a resurgence of interest in the practice of coding (Kafai & Burke, 2013), with many pushing for making it a core competency for students (Lye & Koh, 2014). There are inherent challenges in learning to code evidenced by high failure and dropout rates in programming courses (Ma, Ferguson, Roper, & Wood, 2011; (Qian & Lehman, 2017; Robins, 2010). -
If I Were Doing a Quick ANOVA Analysis
ANOVA Note: If I were doing a quick ANOVA analysis (without the diagnostic checks, etc.), I’d do the following: 1) load the packages (#1); 2) do the prep work (#2); and 3) run the ggstatsplot::ggbetweenstats analysis (in the #6 section). 1. Packages needed. Here are the recommended packages to load prior to working. library(ggplot2) # for graphing library(ggstatsplot) # for graphing and statistical analyses (one-stop shop) library(GGally) # This package offers the ggpairs() function. library(moments) # This package allows for skewness and kurtosis functions library(Rmisc) # Package for calculating stats and bar graphs of means library(ggpubr) # normality related commands 2. Prep Work Declare factor variables as such. class(mydata$catvar) # this tells you how R currently sees the variable (e.g., double, factor) mydata$catvar <- factor(mydata$catvar) #Will declare the specified variable as a factor variable 3. Checking data for violations of assumptions: a) relatively equal group sizes; b) equal variances; and c) normal distribution. a. Group Sizes Group counts (to check group frequencies): table(mydata$catvar) b. Checking Equal Variances Group means and standard deviations (Note: the aggregate and by commands give you the same results): aggregate(mydata$intvar, by = list(mydata$catvar), FUN = mean, na.rm = TRUE) aggregate(mydata$intvar, by = list(mydata$catvar), FUN = sd, na.rm = TRUE) by(mydata$intvar, mydata$catvar, mean, na.rm = TRUE) by(mydata$intvar, mydata$catvar, sd, na.rm = TRUE) A simple bar graph of group means and CIs (using Rmisc package). This command is repeated further below in the graphing section. The ggplot command will vary depending on the number of categories in the grouping variable. -
Healthy Volunteers (Retrospective Study) Urolithiasis Healthy Volunteers Group P-Value 5 (N = 110) (N = 157)
Supplementary Table 1. The result of Gal3C-S-OPN between urolithiasis and healthy volunteers (retrospective study) Urolithiasis Healthy volunteers Group p-Value 5 (n = 110) (n = 157) median (IQR 1) median (IQR 1) Gal3C-S-OPN 2 515 1118 (810–1543) <0.001 (MFI 3) (292–786) uFL-OPN 4 14 56392 <0.001 (ng/mL/mg protein) (10–151) (30270-115516) Gal3C-S-OPN 2 52 0.007 /uFL-OPN 4 <0.001 (5.2–113.0) (0.003–0.020) (MFI 3/uFL-OPN 4) 1 IQR, Interquartile range; 2 Gal3C-S-OPN, Gal3C-S lectin reactive osteopontin; 3 MFI, mean fluorescence intensity; 4 uFL-OPN, Urinary full-length-osteopontin; 5 p-Value, Mann–Whitney U-test. Supplementary Table 2. Sex-related difference between Gal3C-S-OPN and Gal3C-S-OPN normalized to uFL- OPN (retrospective study) Group Urolithiasis Healthy volunteers p-Value 5 Male a Female b Male c Female d a vs. b c vs. d (n = 61) (n = 49) (n = 57) (n = 100) median (IQR 1) median (IQR 1) Gal3C-S-OPN 2 1216 972 518 516 0.15 0.28 (MFI 3) (888-1581) (604-1529) (301-854) (278-781) Gal3C-S-OPN 2 67 42 0.012 0.006 /uFL-OPN 4 0.62 0.56 (9-120) (4-103) (0.003-0.042) (0.002-0.014) (MFI 3/uFL-OPN 4) 1 IQR, Interquartile range; 2 Gal3C-S-OPN, Gal3C-S lectin reactive osteopontin; 3MFI, mean fluorescence intensity; 4 uFL-OPN, Urinary full-length-osteopontin; 5 p-Value, Mann–Whitney U-test. -
Recombining Partitions Via Unimodality Tests
Working Paper 13-07 Departamento de Estadística Statistics and Econometrics Series (06) Universidad Carlos III de Madrid March 2013 Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624-98-48 RECOMBINING PARTITIONS VIA UNIMODALITY TESTS Adolfo Álvarez (1) , Daniel Peña(2) Abstract In this article we propose a recombination procedure for previously split data. It is based on the study of modes in the density of the data, since departing from unimodality can be a sign of the presence of clusters. We develop an algorithm that integrates a splitting process inherited from the SAR algorithm (Peña et al., 2004) with unimodality tests such as the dip test proposed by Hartigan and Hartigan (1985), and finally, we use a network configuration to visualize the results. We show that this can be a useful tool to detect heterogeneity in the data, but limited to univariate data because of the nature of the dip test. In a second stage we discuss the use of multivariate mode detection tests to avoid dimensionality reduction techniques such as projecting multivariate data into one dimension. The results of the application of multivariate unimodality tests show that is possible to detect the cluster structure of the data, although more research can be oriented to estimate the proper fine-tuning of some parameters of the test for a given dataset or distribution. Keywords: Cluster analysis, unimodality, dip test. (1) Álvarez, Adolfo, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] . (2) Peña, Daniel, Departamento de Estadística, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe (Madrid), Spain, e-mail: [email protected] . -
Violin Plots: a Box Plot-Density Trace Synergism Author(S): Jerry L
Violin Plots: A Box Plot-Density Trace Synergism Author(s): Jerry L. Hintze and Ray D. Nelson Source: The American Statistician, Vol. 52, No. 2 (May, 1998), pp. 181-184 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2685478 Accessed: 02/09/2010 11:01 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=astata. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to The American Statistician. http://www.jstor.org StatisticalComputing and Graphics ViolinPlots: A Box Plot-DensityTrace Synergism JerryL. -
Statistical Analysis in JASP
Copyright © 2018 by Mark A Goss-Sampson. All rights reserved. This book or any portion thereof may not be reproduced or used in any manner whatsoever without the express written permission of the author except for the purposes of research, education or private study. CONTENTS PREFACE .................................................................................................................................................. 1 USING THE JASP INTERFACE .................................................................................................................... 2 DESCRIPTIVE STATISTICS ......................................................................................................................... 8 EXPLORING DATA INTEGRITY ................................................................................................................ 15 ONE SAMPLE T-TEST ............................................................................................................................. 22 BINOMIAL TEST ..................................................................................................................................... 25 MULTINOMIAL TEST .............................................................................................................................. 28 CHI-SQUARE ‘GOODNESS-OF-FIT’ TEST............................................................................................. 30 MULTINOMIAL AND Χ2 ‘GOODNESS-OF-FIT’ TEST. .......................................................................... -
Beanplot: a Boxplot Alternative for Visual Comparison of Distributions
Beanplot: A Boxplot Alternative for Visual Comparison of Distributions Peter Kampstra VU University Amsterdam Abstract This introduction to the R package beanplot is a (slightly) modified version of Kamp- stra(2008), published in the Journal of Statistical Software. Boxplots and variants thereof are frequently used to compare univariate data. Boxplots have the disadvantage that they are not easy to explain to non-mathematicians, and that some information is not visible. A beanplot is an alternative to the boxplot for visual comparison of univariate data between groups. In a beanplot, the individual observations are shown as small lines in a one-dimensional scatter plot. Next to that, the estimated density of the distributions is visible and the average is shown. It is easy to compare different groups of data in a beanplot and to see if a group contains enough observations to make the group interesting from a statistical point of view. Anomalies in the data, such as bimodal distributions and duplicate measurements, are easily spotted in a beanplot. For groups with two subgroups (e.g., male and female), there is a special asymmetric beanplot. For easy usage, an implementation was made in R. Keywords: exploratory data analysis, descriptive statistics, box plot, boxplot, violin plot, density plot, comparing univariate data, visualization, beanplot, R, graphical methods, visu- alization. 1. Introduction There are many known plots that are used to show distributions of univariate data. There are histograms, stem-and-leaf-plots, boxplots, density traces, and many more. Most of these plots are not handy when comparing multiple batches of univariate data. For example, com- paring multiple histograms or stem-and-leaf plots is difficult because of the space they take. -
UNIMODALITY and the DIP STATISTIC; HANDOUT I 1. Unimodality Definition. a Probability Density F Defined on the Real Line R Is C
UNIMODALITY AND THE DIP STATISTIC; HANDOUT I 1. Unimodality Definition. A probability density f defined on the real line R is called unimodal if the following hold: (i) f is nondecreasing on the half-line ( ,b) for some real b; (ii) f is nonincreasing on the half-line (−∞a, + ) for some real a; (iii) For the largest possible b in (i), which∞ exists, and the smallest possible a in (ii), which also exists, we have a b. ≤ Here a largest possible b exists since if f is nondecreasing on ( ,bk) −∞ for all k and bk b then f is also nondecreasing on ( ,b). Also b < + since f is↑ a probability density. Likewise a smallest−∞ possible a exists∞ and is finite. In the definition of unimodal density, if a = b it is called the mode of f. It is possible that f is undefined or + there, or that f(x) + as x a and/or as x a. ∞ ↑ ∞ If a<b↑ then f is constant↓ on the interval (a,b) and we can take it to be constant on [a,b], which will be called the interval of modes of f. Any x [a,b] will be called a mode of f. When a = b the interval of modes∈ reduces to the singleton a . { } Examples. Each normal distribution N(µ,σ2) has a unimodal density with a unique mode at µ. Each uniform U[a,b] distribution has a unimodal density with an interval of modes equal to [a,b]. Consider a−1 −x the gamma density γa(x) = x e /Γ(a) for x > 0 and 0 for x 0, where the shape parameter a > 0. -
Field Guide to Continuous Probability Distributions
Field Guide to Continuous Probability Distributions Gavin E. Crooks v 1.0.0 2019 G. E. Crooks – Field Guide to Probability Distributions v 1.0.0 Copyright © 2010-2019 Gavin E. Crooks ISBN: 978-1-7339381-0-5 http://threeplusone.com/fieldguide Berkeley Institute for Theoretical Sciences (BITS) typeset on 2019-04-10 with XeTeX version 0.99999 fonts: Trump Mediaeval (text), Euler (math) 271828182845904 2 G. E. Crooks – Field Guide to Probability Distributions Preface: The search for GUD A common problem is that of describing the probability distribution of a single, continuous variable. A few distributions, such as the normal and exponential, were discovered in the 1800’s or earlier. But about a century ago the great statistician, Karl Pearson, realized that the known probabil- ity distributions were not sufficient to handle all of the phenomena then under investigation, and set out to create new distributions with useful properties. During the 20th century this process continued with abandon and a vast menagerie of distinct mathematical forms were discovered and invented, investigated, analyzed, rediscovered and renamed, all for the purpose of de- scribing the probability of some interesting variable. There are hundreds of named distributions and synonyms in current usage. The apparent diver- sity is unending and disorienting. Fortunately, the situation is less confused than it might at first appear. Most common, continuous, univariate, unimodal distributions can be orga- nized into a small number of distinct families, which are all special cases of a single Grand Unified Distribution. This compendium details these hun- dred or so simple distributions, their properties and their interrelations. -
On Multivariate Unimodal Distributions
ON MULTIVARIATE UNIMODAL DISTRIBUTIONS By Tao Dai B. Sc. (Mathematics), Wuhan University, 1982 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF STATISTICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1989 © Tao Dai, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £ tasW> tj-v*> The University of British Columbia Vancouver, Canada Date ./ v^y^J. DE-6 (2/88) ABSTRACT In this thesis, Kanter's representation of multivariate unimodal distributions is shown equivalent to the usual mixture of uniform distributions on symmetric, compact and convex sets. Kanter's idea is utilized in several contexts by viewing multivariate distributions as mixtures of uniform distributions on sets of various shapes. This provides a unifying viewpoint of what is in the literature and .gives some important new classes of multivariate unimodal distributions. The closure properties of these new classes under convolution, marginality and weak convergence, etc. and their relationships with other notions of multivariate unimodality are discussed. Some interesting examples and their 2- or 3-dimensional pictures are presented. -
Chapter 6: Graphics
Chapter 6 Graphics Modern computers with high resolution displays and graphic printers have rev- olutionized the visual display of information in fields ranging from computer- aided design, through flow dynamics, to the spatiotemporal attributes of infec- tious diseases. The impact on statistics is just being felt. Whole books have been written on statistical graphics and their contents are quite heterogeneous– simple how-to-do it texts (e.g., ?; ?), reference works (e.g., Murrell, 2006) and generic treatment of the principles of graphic presentation (e.g., ?). There is even a web site devoted to learning statistics through visualization http: //www.seeingstatistics.com/.Hence,itisnotpossibletobecomprehensive in this chapter. Instead, I focus on the types of graphics used most often in neu- roscience (e.g., plots of means) and avoid those seldom used in the field (e.g., pie charts, geographical maps). The here are four major purposes for statistical graphics. First, they are used to examine and screen data to check for abnormalities and to assess the distributions of the variables. Second, graphics are very useful aid to exploratory data analysis (??). Exploratory data analysis, however, is used for mining large data sets mostly for the purpose of hypothesis generation and modification, so that use of graphics will not be discussed here. Third, graphics can be used to assess both the assumptions and the validity of a statistical model applied to data. Finally, graphics are used to present data to others. The third of these purposes will be discussed in the appropriate sections on the statistics. This chapter deals with the first and last purposes–examining data and presenting results. -
Large Time Unimodality for Classical and Free Brownian Motions with Initial Distributions
ALEA, Lat. Am. J. Probab. Math. Stat. 15, 353–374 (2018) DOI: 10.30757/ALEA.v15-15 Large time unimodality for classical and free Brownian motions with initial distributions Takahiro Hasebe and Yuki Ueda Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan E-mail address: [email protected] URL: https://www.math.sci.hokudai.ac.jp/~thasebe/ Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan E-mail address: [email protected] Abstract. We prove that classical and free Brownian motions with initial distri- butions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar re- sults are shown for a symmetric stable process with index 1 and a positive stable process with index 1/2. We also prove that free Brownian motion with initial sym- metric unimodal distribution is unimodal, and discuss strong unimodality for free convolution. 1. Introduction A Borel measure µ on R is unimodal if there exist a R and a function f : R [0, ) which is non-decreasing on ( ,a) and non-increasing∈ on (a, ), such that→ ∞ −∞ ∞ µ(dx)= µ( a )δ + f(x) dx. (1.1) { } a The most outstanding result on unimodality is Yamazato’s theorem (Yamazato, 1978) saying that all classical selfdecomposable distributions are unimodal. After this result, in Hasebe and Thorbjørnsen (2016), Hasebe and Thorbjørnsen proved the free analog of Yamazato’s result: all freely selfdecomposable distributions are unimodal.