Fundamentals of Mathematics in Medical Research: Theory and Cases is a comprehensive guide to the mathematical principles and methods used in medical research. This book is structured to facilitate learning and application and gives a solid foundation to readers.
The book is divided into multiple parts that explain basic concepts in a progressive way.
- Part I covers real-valued functions of one or more variables with geometric representations to establish a core understanding of advanced mathematics
- Part II covers inferential methods of probability and statistics from basic probability to parametric and nonparametric tests
- Part III covers correlation theory and advanced analysis of real-valued functions
- Part IV covers multivariable analysis for solving complex problems with an explanation of Markov Chain models
- An Appendix provides solutions for all exercises and along with Fortran 90 programs, Python scripts and Linux scripts to explore the mathematical concepts explained in the book.
Key Features:
- Introduction and Conclusion in Every Chapter
- Exercises and Solutions
- Program codes and scripts
- Comprehensive Coverage of Mathematics for Academics and Research
- Real-World Cases
Readership
Researchers, students, and professionals in medicine and allied fields.Table of Contents
- CONTENTS
- FOREWORD I
- FOREWORD II
- PREFACE
- ACKNOWLEDGEMENTS
- PART I SINGLE AND MULTIVARIABLE FUNCTIONS
CHAPTER 1 FUNCTIONS USING A SINGLE VARIABLE
- 1.1. INTRODUCTION
- 1.2. FUNCTIONS
- 1.3. PROPERTIES OF REAL-VALUED FUNCTIONS
- 1.4. TRANSFORMATIONS OF REAL-VALUED FUNCTIONS
- 1.4.1. Translation
- 1.4.2. Scaling
- 1.5. REAL-VALUED FUNCTIONS
- 1.5.1. Polynomial Real-Valued Functions
- 1.5.2. Piecewise Real-Valued Functions
- 1.5.3. Absolute Value Real-Valued Functions
- 1.6. CONCLUSION
- 1.7. REMARKS
- 1.8. EXERCISES
CHAPTER 2 FUNCTIONS OF SEVERAL VARIABLES
- 2.1. INTRODUCTION
- 2.2. FUNCTIONS
- 2.3. PROPERTIES OF THE RVF
- 2.4. RVF
- 2.4.1. Polynomial RVF
- 2.4.2. Absolute Value RVF
- 2.5. HIGH-DIMENSIONAL RVF
- 2.6. VECTOR-VALUED FUNCTIONS
- 2.7. CONCLUSION
- 2.8. REMARKS
- 2.9. EXERCISES
- PART II INFERENCE BASED ON DATA
CHAPTER 3 PROBABILITY
- 3.1. INTRODUCTION
- 3.2. PROBABILITY
- 3.3. CONDITIONAL PROBABILITY
- 3.4. TOTAL PROBABILITY THEOREM
- 3.5. BAYES’ THEOREM
- 3.6. PROBABILISTIC FUNCTIONS
- 3.6.1. Normal Distribution Function
- 3.6.2. Applications
- 3.6.3. Binomial Distribution Function
- 3.6.4. Applications
- 3.6.5. Poisson Distribution Function
- 3.6.6. Applications
- 3.7. CASES
- 3.8. CONCLUSION
- 3.9. EXERCISES
CHAPTER 4 ESTIMATION AND DECISION THEORY
- 4.1. INTRODUCTION
- 4.2. STATISTICAL HYPOTHESIS AND SIGNIFICANCE
- 4.3. ERRORS TYPE I AND TYPE II
- 4.4. QUALITY CONTROL TEST
- 4.4.1. Sensitivity and Specificity
- 4.4.2. Precision and Accuracy
- 4.4.3. Receiver Operating Characteristic Curves
- 4.5. CONCLUSION
- 4.6. REMARKS
- 4.7. EXERCISES
CHAPTER 5 NON-PARAMETRIC STATISTIC METHODS
- 5.1. INTRODUCTION
- 5.2. SAMPLES
- 5.2.1. Related Samples
- 5.2.2. Independent Samples
- 5.2.3. Sample Size Determination
- 5.3. HYPOTHESIS
- 5.3.1. Scientific Hypothesis
- 5.3.2. Statistical Hypothesis
- 5.4. ONE SAMPLE METHODS
- 5.5. TWO SAMPLE METHODS
- 5.6. METHODS
- 5.6.1. Two Sample Chi-squared Test
- 5.6.2. One Sample Runs Test
- 5.6.3. One Sample Binomial Test
- 5.6.4. Two Sample Kolmogorov Test
- 5.6.5. Two Sample U Test
- 5.7. CONCLUSION
- 5.8. REMARKS
- 5.9. EXERCISES
- PART III ANALYSIS BY REGRESSION
CHAPTER 6 CORRELATION THEORY
- 6.1. INTRODUCTION
- 6.2. LINEAR CORRELATION ON R2
- 6.3. NON-LINEAR CORRELATION ON R2
- 6.4. MULTIPLE CORRELATION ON R3
- 6.5. LINEAR AND NON-LINEAR CORRELATION
- 6.5.1. Linear Correlation Coefficient
- 6.5.2. Non-Linear Correlation Coefficient
- 6.6. CASES
- 6.7. CONCLUSION
- 6.8. REMARKS
- 6.9. EXERCISES
CHAPTER 7 CURVE FITTING
- 7.1. INTRODUCTION
- 7.2. THE METHOD OF LEAST SQUARES
- 7.3. LINEAR LEAST SQUARES METHOD
- 7.3.1. Linear Relationship between Two Variables
- 7.3.2. Linear Relationship between Three Variables
- 7.3.3. Linear Relationship between Multiple Variables
- 7.4. NONLINEAR LEAST SQUARES METHOD
- 7.4.1. Nonlinear Relationship between Two Variables
- 7.4.2. Nonlinear Relationship between Multiple Variables
- 7.5. CONCLUSION
- 7.6. REMARKS
- 7.7. EXERCISES
CHAPTER 8 MULTIVARIATE ANALYSIS OF VARIANCE
- 8.1. INTRODUCTION
- 8.2. FISHER STATISTICAL TEST
- 8.3. LATIN SQUARES
- 8.4. COMPARISON OF VARIANCE
- 8.4.1. Fisher Test in One-Factor
- 8.5. MULTIVARIATE ANALYSIS OF VARIANCE
- 8.5.1. Fisher Test in Two-Factors
- 8.5.2. Fisher Test in Three-Factors
- 8.5.3. Fisher Test in Four-Factors
- 8.6. CASES
- 8.7. CONCLUSION
- 8.8. REMARKS
- 8.9. EXERCISES
- PART IV USING A STOCHASTIC MODEL
CHAPTER 9 DISCRETE-TIME MARKOV CHAINS
- 9.1. INTRODUCTION
- 9.2. PROCESS OF DISCRETE-TIME MARKOV CHAINS
- 9.3. MARKOV CHAIN MODEL
- 9.4. CALCULATING THE STATIONARY DISTRIBUTION
- 9.5. STEADY STATE VECTOR
- 9.6. EIGENVALUES AND EIGENVECTORS
- 9.7. CASES
- 9.8. CONCLUSION
- 9.9. REMARKS
- 9.10. EXERCISES
- SOLUTIONS
- APPENDIX
- REFERENCES
- SUBJECT INDEX
