The Statistics Tutor's Quick Guide to Commonly Used Statistical Tests

Total Page:16

File Type:pdf, Size:1020Kb

The Statistics Tutor's Quick Guide to Commonly Used Statistical Tests statstutor community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence Stcp-marshallowen-7 The Statistics Tutor’s www.statstutor.ac.uk © Ellen Marshall, University of Sheffield Reviewer: Jean Russell University of Sheffield Quick Guide to Commonly Used Statistical Tests Page 2 of 57 Contents CONTENTS .............................................................................................................................................................................................. 2 INTRODUCTION ........................................................................................................................................................................................ 3 TIPS FOR TUTORING ................................................................................................................................................................................... 4 SECTION 1 GENERAL INFORMATION ............................................................................................................................................................... 5 DATA TYPES ............................................................................................................................................................................................ 6 SUMMARY STATISTICS ................................................................................................................................................................................ 6 Summary of descriptive and graphical statistics ............................................................................................................................. 6 DECIDING ON APPROPRIATE STATISTICAL METHODS FOR RESEARCH .......................................................................................................................... 7 ORDINAL DATA ........................................................................................................................................................................................ 7 WHICH TEST SHOULD I USE? ....................................................................................................................................................................... 7 Common Single Comparison Tests ................................................................................................................................................... 7 Tests of association .......................................................................................................................................................................... 7 One scale dependent and several independent variables ............................................................................................................... 9 ASSUMPTION OF NORMALITY ..................................................................................................................................................................... 10 Statistical tests for normality ......................................................................................................................................................... 10 Non-parametric tests ..................................................................................................................................................................... 11 OTHER COMMON ASSUMPTIONS ................................................................................................................................................................. 12 For independent t-tests and ANOVA .............................................................................................................................................. 12 For repeated measures ANOVA ..................................................................................................................................................... 12 Independent observations ............................................................................................................................................................. 12 CONFIDENCE INTERVALS ........................................................................................................................................................................... 13 HYPOTHESIS TESTING ............................................................................................................................................................................... 14 MULTIPLE TESTING .................................................................................................................................................................................. 14 SAMPLE SIZE AND HYPOTHESIS TESTS ............................................................................................................................................................ 15 EFFECT SIZE ........................................................................................................................................................................................... 15 SECTION 2 THE MOST COMMON STATISTICAL TECHNIQUES USED ................................................................................................................. 16 INDEPENDENT T-TEST ............................................................................................................................................................................... 17 MANN-WHITNEY TEST ............................................................................................................................................................................ 18 PAIRED T-TEST ....................................................................................................................................................................................... 19 WILCOXON SIGNED RANK TEST ................................................................................................................................................................. 20 ONE-WAY ANOVA ............................................................................................................................................................................... 21 KRUSKAL-WALLIS TEST ............................................................................................................................................................................ 22 ONE-WAY ANOVA WITH REPEATED MEASURES (WITHIN SUBJECTS) .................................................................................................................... 23 FRIEDMAN TEST ..................................................................................................................................................................................... 24 TWO-WAY ANOVA ............................................................................................................................................................................... 25 CHI-SQUARED TEST ................................................................................................................................................................................ 26 ODDS AND RELATIVE RISK ........................................................................................................................................................................ 27 Odds ............................................................................................................................................................................................... 27 Odds Ratio ...................................................................................................................................................................................... 27 Relative Risk (RR) ........................................................................................................................................................................... 27 CORRELATION ........................................................................................................................................................................................ 28 PEARSON’S CORRELATION COEFFICIENT ......................................................................................................................................................... 28 RANKED CORRELATION COEFFICIENTS ............................................................................................................................................................ 30 Spearman’s Rank Correlation Coefficient ..................................................................................................................................... 30 Kendall’s Tau Rank Correlation Coefficient .................................................................................................................................... 30 PARTIAL CORRELATION ............................................................................................................................................................................. 30 REGRESSION .......................................................................................................................................................................................... 31 LINEAR REGRESSION ................................................................................................................................................................................ 31 LOGISTIC
Recommended publications
  • 14. Non-Parametric Tests You Know the Story. You've Tried for Hours To
    14. Non-Parametric Tests You know the story. You’ve tried for hours to normalise your data. But you can’t. And even if you can, your variances turn out to be unequal anyway. But do not dismay. There are statistical tests that you can run with non-normal, heteroscedastic data. All of the tests that we have seen so far (e.g. t-tests, ANOVA, etc.) rely on comparisons with some probability distribution such as a t-distribution or f-distribution. These are known as “parametric statistics” as they make inferences about population parameters. However, there are other statistical tests that do not have to meet the same assumptions, known as “non-parametric tests.” Many of these tests are based on ranks and/or are comparisons of sample medians rather than means (as means of non-normal data are not meaningful). Non-parametric tests are often not as powerful as parametric tests, so you may find that your p-values are slightly larger. However, if your data is highly non-normal, non-parametric test may detect an effect that is missed in the equivalent parametric test due to falsely large variances caused by the skew of the data. There is, of course, no problem in analyzing normally distributed data with non-parametric statistics also. Non-parametric tests are equally valid (i.e. equally defendable) and most have been around as long as their parametric counterparts. So, if your data does not meet the assumption of normality or homoscedasticity and if it is impossible (or inappropriate) to transform it, you can use non-parametric tests.
    [Show full text]
  • Summarize — Summary Statistics
    Title stata.com summarize — Summary statistics Description Quick start Menu Syntax Options Remarks and examples Stored results Methods and formulas References Also see Description summarize calculates and displays a variety of univariate summary statistics. If no varlist is specified, summary statistics are calculated for all the variables in the dataset. Quick start Basic summary statistics for continuous variable v1 summarize v1 Same as above, and include v2 and v3 summarize v1-v3 Same as above, and provide additional detail about the distribution summarize v1-v3, detail Summary statistics reported separately for each level of catvar by catvar: summarize v1 With frequency weight wvar summarize v1 [fweight=wvar] Menu Statistics > Summaries, tables, and tests > Summary and descriptive statistics > Summary statistics 1 2 summarize — Summary statistics Syntax summarize varlist if in weight , options options Description Main detail display additional statistics meanonly suppress the display; calculate only the mean; programmer’s option format use variable’s display format separator(#) draw separator line after every # variables; default is separator(5) display options control spacing, line width, and base and empty cells varlist may contain factor variables; see [U] 11.4.3 Factor variables. varlist may contain time-series operators; see [U] 11.4.4 Time-series varlists. by, collect, rolling, and statsby are allowed; see [U] 11.1.10 Prefix commands. aweights, fweights, and iweights are allowed. However, iweights may not be used with the detail option; see [U] 11.1.6 weight. Options Main £ £detail produces additional statistics, including skewness, kurtosis, the four smallest and four largest values, and various percentiles. meanonly, which is allowed only when detail is not specified, suppresses the display of results and calculation of the variance.
    [Show full text]
  • U3 Introduction to Summary Statistics
    Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Statistics • The collection, evaluation, and interpretation of data Introduction to Summary Statistics • Statistical analysis of measurements can help verify the quality of a design or process Summary Statistics Mean Central Tendency Central Tendency • The mean is the sum of the values of a set • “Center” of a distribution of data divided by the number of values in – Mean, median, mode that data set. Variation • Spread of values around the center – Range, standard deviation, interquartile range x μ = i Distribution N • Summary of the frequency of values – Frequency tables, histograms, normal distribution Project Lead The Way, Inc. Copyright 2010 1 Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Mean Central Tendency Mean Central Tendency x • Data Set μ = i 3 7 12 17 21 21 23 27 32 36 44 N • Sum of the values = 243 • Number of values = 11 μ = mean value x 243 x = individual data value Mean = μ = i = = 22.09 i N 11 xi = summation of all data values N = # of data values in the data set A Note about Rounding in Statistics Mean – Rounding • General Rule: Don’t round until the final • Data Set answer 3 7 12 17 21 21 23 27 32 36 44 – If you are writing intermediate results you may • Sum of the values = 243 round values, but keep unrounded number in memory • Number of values = 11 • Mean – round to one more decimal place xi 243 Mean = μ = = = 22.09 than the original data N 11 • Standard Deviation: Round to one more decimal place than the original data • Reported: Mean = 22.1 Project Lead The Way, Inc.
    [Show full text]
  • Descriptive Statistics
    Descriptive Statistics Fall 2001 Professor Paul Glasserman B6014: Managerial Statistics 403 Uris Hall Histograms 1. A histogram is a graphical display of data showing the frequency of occurrence of particular values or ranges of values. In a histogram, the horizontal axis is divided into bins, representing possible data values or ranges. The vertical axis represents the number (or proportion) of observations falling in each bin. A bar is drawn in each bin to indicate the number (or proportion) of observations corresponding to that bin. You have probably seen histograms used, e.g., to illustrate the distribution of scores on an exam. 2. All histograms are bar graphs, but not all bar graphs are histograms. For example, we might display average starting salaries by functional area in a bar graph, but such a figure would not be a histogram. Why not? Because the Y-axis values do not represent relative frequencies or proportions, and the X-axis values do not represent value ranges (in particular, the order of the bins is irrelevant). Measures of Central Tendency 1. Let X1,...,Xn be data points, such as the result of n measurements or observations. What one number best characterizes or summarizes these values? The answer depends on the context. Some familiar summary statistics are these: • The mean is given by the arithemetic average X =(X1 + ···+ Xn)/n.(No- tation: We will often write n Xi for X1 + ···+ Xn. i=1 n The symbol i=1 Xi is read “the sum from i equals 1 upto n of Xi.”) 1 • The median is larger than one half of the observations and smaller than the other half.
    [Show full text]
  • Measures of Dispersion for Multidimensional Data
    European Journal of Operational Research 251 (2016) 930–937 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Computational Intelligence and Information Management Measures of dispersion for multidimensional data Adam Kołacz a, Przemysław Grzegorzewski a,b,∗ a Faculty of Mathematics and Computer Science, Warsaw University of Technology, Koszykowa 75, Warsaw 00–662, Poland b Systems Research Institute, Polish Academy of Sciences, Newelska 6, Warsaw 01–447, Poland article info abstract Article history: We propose an axiomatic definition of a dispersion measure that could be applied for any finite sample of Received 22 February 2015 k-dimensional real observations. Next we introduce a taxonomy of the dispersion measures based on the Accepted 4 January 2016 possible behavior of these measures with respect to new upcoming observations. This way we get two Available online 11 January 2016 classes of unstable and absorptive dispersion measures. We examine their properties and illustrate them Keywords: by examples. We also consider a relationship between multidimensional dispersion measures and mul- Descriptive statistics tidistances. Moreover, we examine new interesting properties of some well-known dispersion measures Dispersion for one-dimensional data like the interquartile range and a sample variance. Interquartile range © 2016 Elsevier B.V. All rights reserved. Multidistance Spread 1. Introduction are intended for use. It is also worth mentioning that several terms are used in the literature as regards dispersion measures like mea- Various summary statistics are always applied wherever deci- sures of variability, scatter, spread or scale. Some authors reserve sions are based on sample data. The main goal of those characteris- the notion of the dispersion measure only to those cases when tics is to deliver a synthetic information on basic features of a data variability is considered relative to a given fixed point (like a sam- set under study.
    [Show full text]
  • Krishnan Correlation CUSB.Pdf
    SELF INSTRUCTIONAL STUDY MATERIAL PROGRAMME: M.A. DEVELOPMENT STUDIES COURSE: STATISTICAL METHODS FOR DEVELOPMENT RESEARCH (MADVS2005C04) UNIT III-PART A CORRELATION PROF.(Dr). KRISHNAN CHALIL 25, APRIL 2020 1 U N I T – 3 :C O R R E L A T I O N After reading this material, the learners are expected to : . Understand the meaning of correlation and regression . Learn the different types of correlation . Understand various measures of correlation., and, . Explain the regression line and equation 3.1. Introduction By now we have a clear idea about the behavior of single variables using different measures of Central tendency and dispersion. Here the data concerned with one variable is called ‗univariate data‘ and this type of analysis is called ‗univariate analysis‘. But, in nature, some variables are related. For example, there exists some relationships between height of father and height of son, price of a commodity and amount demanded, the yield of a plant and manure added, cost of living and wages etc. This is a a case of ‗bivariate data‘ and such analysis is called as ‗bivariate data analysis. Correlation is one type of bivariate statistics. Correlation is the relationship between two variables in which the changes in the values of one variable are followed by changes in the values of the other variable. 3.2. Some Definitions 1. ―When the relationship is of a quantitative nature, the approximate statistical tool for discovering and measuring the relationship and expressing it in a brief formula is known as correlation‖—(Craxton and Cowden) 2. ‗correlation is an analysis of the co-variation between two or more variables‖—(A.M Tuttle) 3.
    [Show full text]
  • Summary Statistics, Distributions of Sums and Means
    Summary statistics, distributions of sums and means Joe Felsenstein Department of Genome Sciences and Department of Biology Summary statistics, distributions of sums and means – p.1/18 Quantiles In both empirical distributions and in the underlying distribution, it may help us to know the points where a given fraction of the distribution lies below (or above) that point. In particular: The 2.5% point The 5% point The 25% point (the first quartile) The 50% point (the median) The 75% point (the third quartile) The 95% point (or upper 5% point) The 97.5% point (or upper 2.5% point) Note that if a distribution has a small fraction of very big values far out in one tail (such as the distributions of wealth of individuals or families), the may not be a good “typical” value; the median will do much better. (For a symmetric distribution the median is the mean). Summary statistics, distributions of sums and means – p.2/18 The mean The mean is the average of points. If the distribution is the theoretical one, it is called the expectation, it’s the theoretical mean we would be expected to get if we drew infinitely many points from that distribution. For a sample of points x1, x2,..., x100 the mean is simply their average ¯x = (x1 + x2 + x3 + ... + x100) / 100 For a distribution with possible values 0, 1, 2, 3,... where value k has occurred a fraction fk of the time, the mean weights each of these by the fraction of times it has occurred (then in effect divides by the sum of these fractions, which however is actually 1): ¯x = 0 f0 + 1 f1 + 2 f2 + ..
    [Show full text]
  • Numerical Summary Values for Quantitative Data 35
    3.1 Numerical summary values for quantitative data 35 Chapter 3 Descriptive Statistics II: Numerical Summary Values 3.1 Numerical summary values for quantitative data For many purposes a few well–chosen numerical summary values (statistics) will suffice as a description of the distribution of a quantitative variable. A statistic is a numerical characteristic of a sample. More formally, a statistic is a numerical quantity computed from the values of a variable, or variables, corresponding to the units in a sample. Thus a statistic serves to quantify some interesting aspect of the distribution of a variable in a sample. Summary statistics are particularly useful for comparing and contrasting the distribution of a variable for two different samples. If we plan to use a small number of summary statistics to characterize a distribution or to compare two distributions, then we first need to decide which aspects of the distribution are of primary interest. If the distributions of interest are essentially mound shaped with a single peak (unimodal), then there are three aspects of the distribution which are often of primary interest. The first aspect of the distribution is its location on the number line. Generally, when speaking of the location of a distribution we are referring to the location of the “center” of the distribution. The location of the center of a symmetric, mound shaped distribution is clearly the point of symmetry. There is some ambiguity in specifying the location of the center of an asymmetric, mound shaped distribution and we shall see that there are at least two standard ways to quantify location in this context.
    [Show full text]
  • ED441031.Pdf
    DOCUMENT RESUME ED 441 031 TM 030 830 AUTHOR Fahoome, Gail; Sawilowsky, Shlomo S. TITLE Review of Twenty Nonparametric Statistics and Their Large Sample Approximations. PUB DATE 2000-04-00 NOTE 42p.; Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 24-28, 2000). PUB TYPE Information Analyses (070)-- Numerical/Quantitative Data (110) Speeches/Meeting Papers (150) EDRS PRICE MF01/PCO2 Plus Postage. DESCRIPTORS Monte Carlo Methods; *Nonparametric Statistics; *Sample Size; *Statistical Distributions ABSTRACT Nonparametric procedures are often more powerful than classical tests for real world data, which are rarely normally distributed. However, there are difficulties in using these tests. Computational formulas are scattered tnrous-hout the literature, and there is a lack of avalialalicy of tables of critical values. This paper brings together the computational formulas for 20 commonly used nonparametric tests that have large-sample approximations for the critical value. Because there is no generally agreed upon lower limit for the sample size, Monte Carlo methods have been used to determine the smallest sample size that can be used with the large-sample approximations. The statistics reviewed include single-population tests, comparisons of two populations, comparisons of several populations, and tests of association. (Contains 4 tables and 59 references.)(Author/SLD) Reproductions supplied by EDRS are the best that can be made from the original document. Review of Twenty Nonparametric Statistics and Their Large Sample Approximations Gail Fahoome Shlomo S. Sawilowsky U.S. DEPARTMENT OF EDUCATION Mice of Educational Research and Improvement PERMISSION TO REPRODUCE AND UCATIONAL RESOURCES INFORMATION DISSEMINATE THIS MATERIAL HAS f BEEN GRANTED BY CENTER (ERIC) This document has been reproduced as received from the person or organization originating it.
    [Show full text]
  • Detecting Trends Using Spearman's Rank Correlation Coefficient
    Environmental Forensics 2001) 2, 359±362 doi:10.1006/enfo.2001.0061, available online at http://www.idealibrary.com on Detecting Trends Using Spearman's Rank Correlation Coecient Thomas D. Gauthier Sciences International Inc, 6150 State Road 70, East Bradenton, FL 34203, U.S.A. Received 23 August 2001, Revised manuscript accepted 1 October 2001) Spearman's rank correlation coecient is a useful tool for exploratory data analysis in environmental forensic investigations. In this application it is used to detect monotonic trends in chemical concentration with time or space. # 2001 AEHS Keywords: Spearman rank correlation coecient; trend analysis; soil; groundwater. Environmental forensic investigations can involve a The idea behind the rank correlation coecient is variety of statistical analysis techniques. A relatively simple. Each variable is ranked separately from lowest simple technique that can be used for exploratory data to highest e.g. 1, 2, 3, etc.) and the dierence between analysis is the Spearman rank correlation coecient. In ranks for each data pair is recorded. If the data are this technical note, I describe howthe Spearman rank correlated, then the sum of the square of the dierence correlation coecient can be used as a statistical tool between ranks will be small. The magnitude of the sum to detect monotonic trends in chemical concentrations is related to the signi®cance of the correlation. with time or space. The Spearman rank correlation coecient is calcu- Trend analysis can be useful for detecting chemical lated according to the following equation: ``footprints'' at a site or evaluating the eectiveness of Pn an installed or natural attenuation remedy.
    [Show full text]
  • 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. ..........................................................................
    [Show full text]
  • Friedman Test
    community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence stcp-marquier-FriedmanR The following resources are associated: The R dataset ‘Video.csv’ and R script ‘Friedman.R’ Friedman test in R (Non-parametric equivalent to repeated measures ANOVA) Dependent variable: Continuous (scale) but not normally distributed or ordinal Independent variable: Categorical (Time/ Condition) Common Applications: Used when several measurements of the same dependent variable are taken at different time points or under different conditions for each subject and the assumptions of repeated measures ANOVA have not been met. It can also be used to compare ranked outcomes. Data: The dataset ‘Video’ contains some results from a study comparing videos made to aid understanding of a particular medical condition. Participants watched three videos (A, B, C) and one product demonstration (D) and were asked several Likert style questions about each. These were summed to give an overall score for each e.g. TotalAGen below is the total score of the ordinal questions for video A. Friedman ranks each participants responses The Friedman test ranks each person’s score from lowest to highest (as if participants had been asked to rank the methods from least favourite to favourite) and bases the test on the sum of ranks for each column. For example, person 1 gave C the lowest Total score of 13 and A the highest so SPSS would rank these as 1 and 4 respectively. As the raw data is ranked to carry out the test, the Friedman test can also be used for data which is already ranked e.g.
    [Show full text]