Dr. Almonroeder is uniquely qualified to author a text on applied research as it relates to the rehabilitation professions. As a practicing clinician, researcher, and faculty member, he has the rare ability to describe the complexities of research and statistics in a relevant and understandable way to bridge the gap between research and clinical practice. Nathan Short, PhD, OTD, OTR/L, CHT, Huntington University, USA Advanced Statistics for Physical and Occupational Therapy Advanced Statistics for Physical and Occupational Therapy explains the basis for statistical analyses that are commonly used to answer clinical research questions related to physical and occupational therapy. This textbook provides a resource to help students and faculty in physical and occupational therapy graduate programs understand the basis for common statistical analyses and be able to apply these techniques in their own research. This textbook provides readers with the basis for common statistical analyses, including t-tests, analysis of variance, regression, and nonparametric tests. Each chapter includes step-by-step tutorials with corresponding example data sets explaining how to conduct these statistical analyses using Statistical Package for the Social Sciences (SPSS) software and the Excel Analysis ToolPak, as well as how to identify and interpret relevant output and report results. Advanced Statistics for Physical and Occupational Therapy is key reading for students in physical therapy, occupational therapy, sport performance, and sport rehabilitation gradu­ ate programs as well as students in athletic training courses, applied statistics in sport, and research methods in sport modules. This new text will also be of interest to practicing clinicians who hope to better understand the research they are reading and/or are interested in starting to conduct their own clinical research. Thomas Gus Almonroeder, PT, DPT, PhD, is an assistant professor in the Department of Health Professions at the University of Wisconsin – La Crosse, USA. Advanced Statistics for Physical and Occupational Therapy Thomas Gus Almonroeder Cover image: © kali9 / Getty images First published 2022 by Routledge 605 Third Avenue, New York, NY 10158 and by Routledge 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2022 Thomas Gus Almonroeder The right of Thomas Gus Almonroeder to be identified as author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Names: Almonroeder, Thomas Gus, author. Title: Advanced statistics for physical and occupational therapy / Thomas Gus Almonroeder. Description: First edition. | New York, NY : Routledge, 2022. | Includes bibliographical references and index. Identifiers: LCCN 2021061866 (print) | LCCN 2021061867 (ebook) | ISBN 9781032017129 (hardback) | ISBN 9781032017112 (paperback) | ISBN 9781003179757 (ebook) Subjects: LCSH: Physical therapy—Statistical aspects. | Occupational therapy. | Statistics. Classification: LCC RM695 .A36 2022 (print) | LCC RM695 (ebook) | DDC 615.8/2—dc23/eng/20211220 LC record available at https://lccn.loc.gov/2021061866 LC ebook record available at https://lccn.loc.gov/2021061867 ISBN: 978-1-032-01712-9 (hbk) ISBN: 978-1-032-01711-2 (pbk) ISBN: 978-1-003-17975-7 (ebk) DOI: 10.4324/9781003179757 Typeset in Times New Roman by Apex CoVantage, LLC Access the Support Material: www.routledge.com/9781032017112 Contents List of Figures ix List of Tables xii List of Boxes xiii Preface xiv Glossary xvi 1 Introduction to fundamental concepts 1 2 Descriptive statistics 11 3 The normal distribution 31 4 Probability and statistical inference 40 5 t-tests 58 6 Introduction to analysis of variance 77 7 Factorial analysis of variance 98 8 Correlation 112 9 Simple linear regression 131 10 Multiple regression 149 11 Logistic regression 168 12 Non-parametric tests 179 13 Effect sizes and confidence intervals 188 viii Contents 14 Sample size estimation 195 15 Analysis of covariance 201 16 Multivariate analysis of variance 204 Data Sets 207 Appendixes 216 References 234 Index 239 Figures 1.1 Diagram depicting sampling error. 4 1.2 Table highlighting the attributes of nominal, ordinal, interval, and ratio variables. 8 2.1 Histogram of 6 Minute Walk Test distances for a hypothetical sample of 50 subjects. 14 2.2 Stem-and-leaf plot of resting heart rate values for a hypothetical sample of 40 subjects. 16 2.3 Subplot A includes an example of a unimodal distribution (one peak), while subplot B includes an example of a bimodal distribution (two peaks). 17 2.4 Subplot A includes an example of a positively skewed distribution (‘tails off’ to the right), while subplot B includes an example of a negatively skewed distribution (‘tails off’ to the left). 18 2.5 Examples of a leptokurtic distribution and a platykurtic distribution. 19 2.6 Example of a frequency distribution with the normal distribution curve superimposed (left side of the figure). 20 2.7 Examples of how the locations of the mean, median, and mode differ for positively (subplot A) and negatively (subplot B) skewed distributions. 21 2.8 Examples of two distributions of scores with the same means but different degrees of variability. 23 2.9 Examples of two different distributions of scores. 24 2.10 Resting heart rates for 15 individuals. 25 2.11 Box plot based on the resting heart rate data from Figure 2.10. 25 3.1 Graph of the normal distribution curve. 32 3.2 Standard normal curves with different areas designated. 35 3.3 Standard normal curve highlighting the 68–95–99.7 rule. 36 3.4 Normal distribution curves for birth weights of babies born in the United States (μ = 7.5 pounds, σ = 1.3 pounds). 37 3.5 Relevant areas of the distributions for Learning Activity 3.1. 39 4.1 Hypothetical distribution of body mass index (BMI) values for adults in the United States population. 42 4.2 Two superimposed distributions of sample means. 44 4.3 Hypothetical confidence intervals based on 20 random samples of size n. 47 4.4 90%, 95%, and 99% confidence intervals (CIs) based on the hypothetical hemoglobin data. 48 4.5 Table highlighting each of the four potential scenarios whenever a hypothesis test is conducted. 50