Statistics and Measurement Concepts with OpenStat

Transcription

1 Statistics and Measurement Concepts with OpenStat

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3 William Miller Statistics and Measurement Concepts with OpenStat

4 William Miller Urbandale, Iowa USA ISBN ISBN (ebook) DOI / Springer New York Heidelberg Dordrecht London Library of Congress Control Number: # 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 (

5 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 edu/%7egcramsey/gallery.html and Joachim Verhagen 7ejcdverha/scijokes Urbandale, Iowa, USA William G. Miller v

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7 Contents 1 Basic Statistics... 1 Introduction... 1 Symbols Used in Statistics Probability Concepts... 4 Additive Rules of Probability The Law of Large Numbers... 6 Multiplication Rule of Probability Permutations and Combinations Conditional Probability Bayesian Statistics Maximum Liklihood (Adapted from S. Purcell, 10 A Simple Example of MLE Analytic MLE Numerical MLE Other Practical Considerations Log-Likelihood Model Identification Local Minima Probabilty as an Area Sampling Descriptive Statistics The Mean Variance and Standard Deviation Estimating Population Parameters: Mean and Standard Deviation The Standard Error of the Mean Testing Hypotheses for Differences Between or Among Means The Nature of Scientific Investigation Decision Risks Hypotheses Related to a Single Mean vii

8 viii Contents Determining Type II Error and Power of the Test Sample Size Requirements for the Test of One Mean Confidence Intervals for a Sample Mean Frequency Distributions The Normal Distribution Model The Median Skew Kurtosis The Binomial Distribution The Poisson Distribution The Chi-Squared Distribution The F Ratio Distribution The Student t Test The Product Moment Correlation Testing Hypotheses for Relationships Among Variables: Correlation Scattergrams Transformation to z Scores Simple Linear Regression The Least-Squares Fit Criterion The Variance of Predicted Scores The Variance of Errors of Prediction Testing Hypotheses Concerning the Pearson Product Moment Correlation Hypotheses About Correlations in One Population Test That the Correlation Equals a Specific Value Testing Equality of Correlations in Two Populations Differences Between Correlations in Dependent Samples Partial and Semi-Partial Correlations Partial Correlation Semi-Partial Correlation Autocorrelation Series Introduction Multiple Regression The Linear Regression Equation Least Squares Calculus Finding a Change in Y Given a Change in X for Y ¼ f(x) Relative Change in Y for a Change in X The Concept of a Derivative Some Rules for Differentiating Polynomials Geometric Interpretation of a Derivative Finding the Value of X for Which f(x) Is Least

9 Contents ix A Generalization of the Last Example Partial Derivatives Least Squares Regression for Two or More Independent Variables Matrix Form for Normal Equations Using Raw Scores Matrix Form for Normal Equations Using Deviation Scores Matrix Form for Normal Equations Using Standardized Scores Hypothesis Testing in Multiple Regression Testing the Significance of the Multiple Regression Coefficient The Standard Error of Estimate Testing the Regression Coefficients Testing the Difference Between Regression Coefficients Analysis of Variance Theory of Analysis of Variance The Completely Randomized Design Introduction A Graphic Representation Null Hypothesis of the Design Summary of Data Analysis Model and Assumptions Fixed and Random Effects Analysis of Variance: The Two-Way, Fixed-Effects Design Stating the Hypotheses Interpreting Interactions Random Effects Models One Between, One Repeated Design Introduction The Research Design Theoretical Model Assumptions Summary Table Population Parameters Estimated Two Factor Repeated Measures Analysis Nested Factors Analysis of Variance Design The Research Design The Variance Model The ANOVA Summary Table A, B and C Factors with B Nested in A Latin and Greco-Latin Square Designs Some Theory The Latin Square Plan 1 by B. J. Winer Plan

10 x Contents Plan 3 Latin Squares Design Analysis of Greco-Latin Squares Plan 5 Latin Square Design Plan 6 Latin Squares Design Plan 7 for Latin Squares Plan 9 Latin Squares Analysis of Variance Using Multiple Regression Methods A Comparison of ANOVA and Regression Effect Coding Orthogonal Coding Dummy Coding Two Factor ANOVA by Multiple Regression Analysis of Covariance by Multiple Regression Analysis Sums of Squares by Regression The General Linear Model Canonical Correlation Introduction Eigenvalues and Eigenvectors The Canonical Analysis Discriminant Function/MANOVA Theory Cluster Analyses Theory Hierarchical Cluster Analysis Path Analysis Theory Factor Analysis The Linear Model Non-Parametric Statistics Contingency Chi-Square Spearman Rank Correlation Mann Whitney U Test Fisher s Exact Test Kendall s Coefficient of Concordance Kruskal-Wallis One-Way ANOVA Wilcoxon Matched-Pairs Signed Ranks Test Cochran Q Test Sign Test Friedman Two Way ANOVA Probability of a Binomial Event Runs Test Kendall s Tau and Partial Tau The Kaplan-Meier Survival Test Kolmogorov-Smirnov Test

11 Contents xi 7 Statistical Process Control Introduction XBAR Chart Range Chart S Control Chart CUSUM Chart pchart Defect (Non-Conformity) c Chart Defects Per Unit u Chart Linear Programming Introduction Calculation Implementation in Simplex Measurement Test Theory Scales of Measurement Nominal Scales Ordinal Scales of Measurement Interval Scales of Measurement Ratio Scales of Measurement Reliability, Validity and Precision of Measurement Reliability The Kuder: Richardson Formula 20 Reliability Validity Concurrent Validity Predictive Validity Discriminate Validity Construct Validity Content Validity Effects of Test Length Composite Test Reliability Reliability by ANOVA Sources of Error: An Example A Hypothetical Situation Item and Test Analysis Procedures Classical Item Analysis Methods Item Discrimination Item Difficulty The Item Analysis Program Item Response Theory The One Parameter Logistic Model Estimating Parameters in the Rasch Model: Prox. Method

12 xii Contents Item Banking and Individualized Testing Measuring Attitudes, Values, Beliefs Methods for Measuring Attitudes Affective Measurement Theory Thurstone Paired Comparison Scaling Successive Interval Scaling Procedures Guttman Scalogram Analysis Likert Scaling Semantic Differential Scales Behavior Checklists Codifying Personal Interactions Bibliography Index

13 List of Figures Fig. 1.1 Maximum liklihood estimation Fig. 1.2 Maximum liklihood estimation Fig. 1.3 Local minima Fig. 2.1 Distribution of sample means Fig. 2.2 Sample size estimation Fig. 2.3 Power curves Fig. 2.4 Null and alternate hypotheses for sample means Fig. 2.5 Sample plot of test scores Fig. 2.6 Sample proportions of test scores Fig. 2.7 Sample sumulative probabilities of test scores Fig. 2.8 Sample probability plot Fig. 2.9 A poisson distribution Fig Chi-squared distribution with 4 of freedom Fig t distribution with 2 of freedom Fig t distribution with 100 of freedom Fig. 3.1 A negative correlation plot Fig. 3.2 Scattergram of two variables Fig. 3.3 Scattergram of a negative relationship Fig. 3.4 Scattergram of two variables with low relationship Fig. 3.5 A simulated negative correlation plot Fig. 3.6 X versus Y plot of five values Fig. 3.7 Plot for a correlation of Fig. 3.8 Single sample tests dialog form Fig. 3.9 Form for comparison of correlations Fig The autocorrelation dialog Fig The moving average dialog Fig Plot of smoothed points using moving averages Fig Plot of residuals obtained using moving averages Fig Polynomial regression smoothing form Fig Plot of polynomial smoothed points xiii

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