Statistics and Measurement Concepts with OpenStat

Statistics and Measurement Concepts with OpenStat

William Miller Statistics and Measurement Concepts with OpenStat

William Miller Urbandale, Iowa USA ISBN 978-1-4614-5742-8 ISBN 978-1-4614-5743-5 (ebook) DOI 10.1007/978-1-4614-5743-5 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012952927 # Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface To the hundreds of graduate students and users of my statistics programs. Your encouragement, suggestions and patience have kept me motivated to maintain my interest in statistics and measurement. To my wife who has endured my hours of time on the computer and wonders why I would want to create free material. To my dogs Tuffy, Annie, Lacy and Heidi who sit under my desk for hours at a time while I pursue this hobby. To Gary C. Ramseyer s First Internet Gallery of Statistics Jokes http://www.ilstu. edu/%7egcramsey/gallery.html and Joachim Verhagen http://www.xs4all.nl/% 7ejcdverha/scijokes Urbandale, Iowa, USA William G. Miller v

Contents 1 Basic Statistics... 1 Introduction... 1 Symbols Used in Statistics................................ 1 Probability Concepts... 4 Additive Rules of Probability.............................. 5 The Law of Large Numbers... 6 Multiplication Rule of Probability........................... 7 Permutations and Combinations..... 7 Conditional Probability................................... 8 Bayesian Statistics...... 10 Maximum Liklihood (Adapted from S. Purcell, http://statgen.iop.kcl.ac.uk/bgim/mle/sslike_1.html)... 10 A Simple Example of MLE.... 11 Analytic MLE... 13 Numerical MLE...................................... 13 Other Practical Considerations............................ 14 Log-Likelihood...... 15 Model Identification.... 15 Local Minima........................................ 16 Probabilty as an Area.................................... 17 Sampling... 17 2 Descriptive Statistics... 19 The Mean... 19 Variance and Standard Deviation..... 22 Estimating Population Parameters: Mean and Standard Deviation.... 23 The Standard Error of the Mean... 25 Testing Hypotheses for Differences Between or Among Means... 27 The Nature of Scientific Investigation... 27 Decision Risks... 28 Hypotheses Related to a Single Mean........................ 30 vii

viii Contents Determining Type II Error and Power of the Test................ 33 Sample Size Requirements for the Test of One Mean...... 36 Confidence Intervals for a Sample Mean... 39 Frequency Distributions... 40 The Normal Distribution Model... 42 The Median... 43 Skew...... 44 Kurtosis.............................................. 44 The Binomial Distribution... 45 The Poisson Distribution.................................. 46 The Chi-Squared Distribution... 47 The F Ratio Distribution... 48 The Student t Test..................................... 49 3 The Product Moment Correlation... 53 Testing Hypotheses for Relationships Among Variables: Correlation.................................... 54 Scattergrams...... 54 Transformation to z Scores................................ 56 Simple Linear Regression... 61 The Least-Squares Fit Criterion............................. 62 The Variance of Predicted Scores... 65 The Variance of Errors of Prediction.......... 66 Testing Hypotheses Concerning the Pearson Product Moment Correlation....... 67 Hypotheses About Correlations in One Population... 67 Test That the Correlation Equals a Specific Value... 68 Testing Equality of Correlations in Two Populations...... 70 Differences Between Correlations in Dependent Samples... 72 Partial and Semi-Partial Correlations......................... 73 Partial Correlation... 74 Semi-Partial Correlation... 75 Autocorrelation...... 76 Series... 84 Introduction......................................... 84 4 Multiple Regression... 87 The Linear Regression Equation............ 87 Least Squares Calculus... 90 Finding a Change in Y Given a Change in X for Y ¼ f(x)... 92 Relative Change in Y for a Change in X... 93 The Concept of a Derivative..... 94 Some Rules for Differentiating Polynomials.................. 95 Geometric Interpretation of a Derivative...................... 97 Finding the Value of X for Which f(x) Is Least............... 98

Contents ix A Generalization of the Last Example... 99 Partial Derivatives... 100 Least Squares Regression for Two or More Independent Variables... 101 Matrix Form for Normal Equations Using Raw Scores.... 103 Matrix Form for Normal Equations Using Deviation Scores........ 104 Matrix Form for Normal Equations Using Standardized Scores... 105 Hypothesis Testing in Multiple Regression... 106 Testing the Significance of the Multiple Regression Coefficient.... 106 The Standard Error of Estimate... 107 Testing the Regression Coefficients.... 107 Testing the Difference Between Regression Coefficients........... 109 5 Analysis of Variance... 111 Theory of Analysis of Variance............................. 111 The Completely Randomized Design... 112 Introduction... 112 A Graphic Representation............................... 113 Null Hypothesis of the Design... 113 Summary of Data Analysis.............................. 113 Model and Assumptions................................ 114 Fixed and Random Effects................................ 115 Analysis of Variance: The Two-Way, Fixed-Effects Design.... 115 Stating the Hypotheses... 118 Interpreting Interactions.... 119 Random Effects Models.... 120 One Between, One Repeated Design......................... 121 Introduction......................................... 121 The Research Design... 121 Theoretical Model... 123 Assumptions................... 123 Summary Table..... 124 Population Parameters Estimated.... 124 Two Factor Repeated Measures Analysis...................... 125 Nested Factors Analysis of Variance Design... 125 The Research Design... 125 The Variance Model.... 126 The ANOVA Summary Table... 126 A, B and C Factors with B Nested in A...... 126 Latin and Greco-Latin Square Designs........................ 127 Some Theory... 127 The Latin Square.... 128 Plan 1 by B. J. Winer................................. 130 Plan 2... 130

x Contents Plan 3 Latin Squares Design... 131 Analysis of Greco-Latin Squares......................... 132 Plan 5 Latin Square Design......... 133 Plan 6 Latin Squares Design... 134 Plan 7 for Latin Squares............................... 134 Plan 9 Latin Squares........ 135 Analysis of Variance Using Multiple Regression Methods... 136 A Comparison of ANOVA and Regression... 136 Effect Coding... 137 Orthogonal Coding..... 138 Dummy Coding...... 140 Two Factor ANOVA by Multiple Regression...... 141 Analysis of Covariance by Multiple Regression Analysis.......... 144 Sums of Squares by Regression... 146 The General Linear Model... 146 Canonical Correlation.................................... 147 Introduction......................................... 147 Eigenvalues and Eigenvectors... 148 The Canonical Analysis.................................. 151 Discriminant Function/MANOVA.... 152 Theory... 152 Cluster Analyses... 153 Theory... 153 Hierarchical Cluster Analysis..... 153 Path Analysis... 154 Theory... 154 Factor Analysis..... 155 The Linear Model..................................... 155 6 Non-Parametric Statistics... 159 Contingency Chi-Square... 160 Spearman Rank Correlation... 160 Mann Whitney U Test................................... 160 Fisher s Exact Test...... 161 Kendall s Coefficient of Concordance........................ 161 Kruskal-Wallis One-Way ANOVA........... 161 Wilcoxon Matched-Pairs Signed Ranks Test...... 162 Cochran Q Test... 163 Sign Test... 163 Friedman Two Way ANOVA... 164 Probability of a Binomial Event..... 164 Runs Test...... 165 Kendall s Tau and Partial Tau... 166 The Kaplan-Meier Survival Test... 166 Kolmogorov-Smirnov Test... 166

Contents xi 7 Statistical Process Control... 167 Introduction... 167 XBAR Chart... 167 Range Chart........................................... 168 S Control Chart... 168 CUSUM Chart... 168 pchart... 169 Defect (Non-Conformity) c Chart..... 170 Defects Per Unit u Chart.................................. 170 8 Linear Programming... 171 Introduction... 171 Calculation...... 172 Implementation in Simplex... 173 9 Measurement... 175 Test Theory... 175 Scales of Measurement... 176 Nominal Scales.... 176 Ordinal Scales of Measurement... 176 Interval Scales of Measurement............................. 177 Ratio Scales of Measurement... 178 Reliability, Validity and Precision of Measurement... 179 Reliability.... 179 The Kuder: Richardson Formula 20 Reliability................ 181 Validity... 184 Concurrent Validity... 185 Predictive Validity... 185 Discriminate Validity.... 185 Construct Validity... 186 Content Validity.... 187 Effects of Test Length... 188 Composite Test Reliability...... 189 Reliability by ANOVA... 190 Sources of Error: An Example... 190 A Hypothetical Situation... 191 Item and Test Analysis Procedures................... 199 Classical Item Analysis Methods... 200 Item Discrimination................................... 200 Item Difficulty...... 201 The Item Analysis Program................................ 202 Item Response Theory................................... 203 The One Parameter Logistic Model.......................... 205 Estimating Parameters in the Rasch Model: Prox. Method........ 206

xii Contents Item Banking and Individualized Testing..... 209 Measuring Attitudes, Values, Beliefs......................... 210 Methods for Measuring Attitudes... 211 Affective Measurement Theory... 213 Thurstone Paired Comparison Scaling....... 214 Successive Interval Scaling Procedures..................... 217 Guttman Scalogram Analysis...... 221 Likert Scaling........................................ 225 Semantic Differential Scales... 226 Behavior Checklists................................... 228 Codifying Personal Interactions........................... 229 Bibliography... 231 Index... 235

List of Figures Fig. 1.1 Maximum liklihood estimation... 12 Fig. 1.2 Maximum liklihood estimation... 16 Fig. 1.3 Local minima... 17 Fig. 2.1 Distribution of sample means... 32 Fig. 2.2 Sample size estimation... 34 Fig. 2.3 Power curves... 36 Fig. 2.4 Null and alternate hypotheses for sample means... 37 Fig. 2.5 Sample plot of test scores... 41 Fig. 2.6 Sample proportions of test scores... 41 Fig. 2.7 Sample sumulative probabilities of test scores... 42 Fig. 2.8 Sample probability plot... 46 Fig. 2.9 A poisson distribution... 47 Fig. 2.10 Chi-squared distribution with 4 of freedom... 48 Fig. 2.11 t distribution with 2 of freedom... 50 Fig. 2.12 t distribution with 100 of freedom... 50 Fig. 3.1 A negative correlation plot... 54 Fig. 3.2 Scattergram of two variables... 55 Fig. 3.3 Scattergram of a negative relationship... 55 Fig. 3.4 Scattergram of two variables with low relationship.... 56 Fig. 3.5 A simulated negative correlation plot... 61 Fig. 3.6 X versus Y plot of five values... 62 Fig. 3.7 Plot for a correlation of 1.0... 63 Fig. 3.8 Single sample tests dialog form... 70 Fig. 3.9 Form for comparison of correlations... 74 Fig. 3.10 The autocorrelation dialog... 78 Fig. 3.11 The moving average dialog... 79 Fig. 3.12 Plot of smoothed points using moving averages... 80 Fig. 3.13 Plot of residuals obtained using moving averages... 80 Fig. 3.14 Polynomial regression smoothing form... 81 Fig. 3.15 Plot of polynomial smoothed points... 81 xiii

xiv List of Figures Fig. 3.16 Plot of residuals from polynomial smoothing... 82 Fig. 3.17 Auto and partial autocorrelation plot... 85 Fig. 4.1 A simple function map... 91 Fig. 4.2 A function map in three dimensions... 92 Fig. 4.3 The minimum of a function derivative... 99