Medical Statistics and Computer Experiments

The following sections are included:

• Preface

• Preface to the First Edition

• Preface to the Second Edition

• Contents

• About the Editors

• Introduction

The following sections are included:

• Variables and Data
  • Types of variables
    • Continuous variable
    • Discrete variable
    • Ordinal variable
  • Structure and feature of data
    • Basic observed unit
    • Recording item

• Frequency Table and Histogram
  • Frequency table
    • Discrete-type frequency table
    • Continuous type frequency table
  • Frequency plot and histogram
    • Frequency plot for discrete variable — bar chart
    • Frequency plot for continuous variable — histogram
    • Frequency plot for ordinal variable — bar chart

• Measurement for Average Level of a Sample
  • Arithmetic mean
    • Raw data based approach
    • Frequency table based approach

• Geometric mean
  • Median
    • Raw data based approach
    • Frequency table based approach

• Measurement for Variation of a Sample
  • Range R
  • Q₃ − Q₁
  • Variance and standard deviation
  • Coefficient of variation

• Relative Measures and Standardization Approaches
  • Ratio, frequency and intensity
    • Ratio
    • Relative frequency
    • Intensity
  • Crude death rate and standardization
    • Direct standardization approach
    • Indirect standardization approach
    • Nature of crude mortality rate and standardized mortality rate

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Explanation of Probability and Related Concepts
  • Probability
  • Odds
  • Bayes' formula

• Distributional Characters of Random Variables
  • Probability function of discrete random variable
  • Probability density function of continuous random variable
  • Distribution function of random variable
  • Population mean and population variance

• Binomial Distribution
  • Probability function
  • Plot of probability function
  • Population mean and population variance
  • Discussion on cure rate

• Poisson Distribution
  • Probability function
  • Plot of probability function
  • Population mean and population variance
  • Additive property

• Normal Distribution
  • Probability density function
  • Area under the normal probability density curve
  • Determination of reference range
  • Normal approximation of binomial distribution and Poisson distribution
    • Correction for continuity
    • Normal approximation

• Basic Operation on Probability
  • Basic operations
    • Additive formula
    • Multiplicative formula
  • Proof of Bayes' formula
  • Calculation for probability function of a binomial distribution

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Distribution of Sample Mean
  • Distribution of sample mean from a population of normal distribution
  • Distribution of sample mean from a population with non-normal distribution

• t Distribution
  • Standard normal deviate and standard t deviate
  • The probability density and critical values of t distribution

• The Confidence Interval for Population Mean of a Normal Distribution

• Confidence Interval for the Difference between Two Population Means

• Confidence Intervals for Probability and the Difference between Two Probabilities
  • Confidence intervals for population probability
  • Confidence intervals for difference of two population probabilities

• The Sample Size for Estimation of Confidence Interval
  • Sample size for confidence interval of the mean of normal population
  • Sample size for confidence interval of the probability of binomial population

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Specific Logic and Main Steps of Hypothesis Testing
  • Set up the statistical hypotheses
  • Select statistics and calculate its current value
  • Determine the P-value
  • Decision and conclusion

• The t Test for One Group of Data under Completely Randomized Design

• The t Test for Data under Randomized Paired Design

• The Tests for Comparing Two Means Based on Two Groups of Data under Completely Randomized Design
  • Equal variances
  • Unequal variances

• The F-Test for Equal Variances of Two Groups of Data under Completely Randomized Design

• Z-Tests for Parameters of Binomial Distribution and Poisson Distribution (Large Sample)
  • Z-test for population probability of binomial distribution (large n)
    • One sample
    • Two samples
  • Z-tests for population mean of Poisson distribution (large λ)
    • Single observation
    • Two observations
    • Two “groups” of observations

• Comments on Real Reports

• Computer Experiments
  • Computer implement of t tests
  • Experiments on two types of error

• Thinking, Practice and Experiments

The following sections are included:

• Two Types of Error and Power
  • The probabilities of two types of error
  • Power

• The Four Elements Affecting Power
  • Larger difference leading to larger power
  • Smaller variation or larger sample size leading to larger power
  • Larger a leading to larger power

• The Quantitative Relation between Power and the Four Elements
  • The test for single sample mean
  • The test for two sample means
  • The test for two frequencies (large sample)

• Estimation of Sample Size for the Tests in Common Use
  • The test for single sample mean
  • The test for two sample means
  • The test for two frequencies (large sample)

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Chi-Square Distribution and Pearson's Goodness-of-Fit Test
  • Chi-square distribution
  • The χ² test for goodness of fit (large sample)

• The χ² Test for Comparison between Two Independent Sample Frequencies
  • Setting up the testing hypotheses
  • Calculating the current value of statistic
    • The expect frequencies
    • Calculating the value of statistic
  • Calculating degrees of freedom
  • Determine the P-value and make conclusion

• The χ² Tests for Binary Variable under a Paired Design
  • The χ² test for independence between two binary variables
  • The χ² test for comparison between two sample frequencies

• The χ² Test for R × C Contingency Table
  • The χ² test for whole R × C table
  • The χ² tests for split R × C tables
  • Measurement of association for R × C table

• The χ² Test for Confirming a Supposed Distribution
  • Test hypothesis
  • Estimating the parameter π
  • Calculating the probabilities and expected frequencies
  • Calculating χ² value and degrees of freedom
  • Decision and conclusion

• Hypothesis Testing for Two Standardized Rates
  • Test for direct standardized rates
  • Test for SMR from indirect standardization

• Fisher's Exact Test for 2 × 2 Table
  • Setting up the testing hypotheses
  • The conditional probabilities
  • P-value and conclusion

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Concept of Linear Correlation
  • Independent random sample of a bi-variable distribution
  • Scatter diagram

• Correlation Coefficient
  • Population correlation coefficient
  • Sample correlation coefficient
  • Calculation of sample correlation coefficient

• Inference on Correlation Coefficient
  • Hypothesis test
  • Interval estimation

• Rank Correlation
  • Spearman's rank correlation coefficient
  • When there are more ties in rank

• Caution in Analysis of Linear Correlation
  • The importance of scatter diagram
  • No correlation for non-random sample
  • Be careful about pooling data
  • Interpretation of correlation coefficient

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Statistical Description of Linear Regression
  • Linear regression equation
  • Regression coefficient and its calculation

• Statistical Inference on Regression
  • Hypothesis test for regression equation
    • Analysis of variance for regression equation
    • t-test for regression coefficient
  • Determination coefficient
  • Confidence interval for β

• The Application of Linear Regression and the Pre-Requisites
  • Two interval estimations
    • Confidence interval for μ_y|x
    • Prediction interval for Y
  • Whether two data sets can be pooled for regression?
    • Are they parallel?
    • Can be pooled?
  • Linear regression through origin
  • On the basic assumptions

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Through Linear Regression
  • Linear regression after transformation of X
  • Not recommend any transformation of Y

• Nonlinear Regression
  • Least squared estimate for nonlinear regression
  • Goodness of fit for nonlinear regression

• Polynomial Regression

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Essential Conception of Experimental Design
  • Three elements of an experimental design
    • Treatment
    • Subject
    • Effect
  • Experimental error and its characteristics
    • Random error and its characteristics
    • Non-random error and its characteristics
  • The statistical principles of experimental design
    • Control
    • Randomization
    • Replication

• Statistical Principle in Clinical Trials
  • The choice of the outcome variables
    • Primary variable and secondary variable
    • Compound variable
    • Global assessment variable
  • The choice of the control groups
    • Placebo control
    • Active control
    • Dose-response control
  • The important skill for avoiding bias — blinding method
  • Data management
    • Database
    • Further check of CRF
    • Double input
    • Visual checking
    • Computer checking
  • Duty and task of statistician
    • Duty of statistician
    • Prepare statistical analysis plan
    • Statistical analysis of the data
    • Write statistical analysis report

• Randomization Techniques
  • The generation of random number
    • Random number table
    • Random number generator
  • Random sampling, random grouping and random ordering
    • Random sampling
    • Random grouping
    • Random ordering
    • Stratified randomization

• Comments on Some Medical Examples

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• One-Way Analysis of Variance — The Completely Random Design
  • The completely random design
  • The underlying concepts of ANOVA
  • Two assumptions on analysis of variance
    • Bartlett's test
    • Test for normality and transformations
  • Hypothesis test of means (one-way analysis of variance)
  • Multiple comparisons
    • LSD-t test (least significant difference t test)
    • SNK-q test (Student–Newman–Keuls q test)

• Two-Way Analysis of Variance — The Randomized Complete-Block Design
  • The randomized complete-block design
  • Two-way analysis of variance

• Three-Way Analysis of Variance: The Latin-Square Design
  • Design of Latin squares
    • Randomly permute the rows
    • Randomly permute the columns
  • Analysis of variance

• Factorial Experiments and Analysis of Variance
  • Introduction
  • Experimental design
  • Analysis of variance

• Split-Plot Designs and Analysis of Variance
  • Introduction
  • Experimental design
    • Completely randomized split-plot design
    • Randomized complete-block split-plot design
  • Analysis of variance
    • Completely randomized split-plot design
    • Randomized complete-block split-plot design

• Cross-Over Design and Analysis of Variance
  • Introduction
  • Experimental design
  • Analysis of variance

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Wilcoxon's Signed Rank Test
  • The test for small sample
  • The test for large sample (normal approximation)
  • Construction of the table for critical values

• Wilcoxon's Rank Sum Test for Comparing the Locations of Two Distributions
  • The test for small sample
  • The test for large sample (normal approximation)
  • The construction of the table for critical value

• Hypothesis Testing for the Locations of More Than 2 Populations
  • Kruskal–Wallis test for the data from a completely random design
  • Friedman test for the data from a randomized block design
  • Multiple comparison of mean ranks in k groups

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Basic Procedure of Multiple Regression
  • The model
  • Basic procedure with an example

• Multiple Correlation
  • Simple correlation coefficient
  • Multiple correlation coefficient
  • Partial correlation coefficient
  • Test of correlation

• Selection of Independent Variables
  • Criteria for comparing models
    • Multiple correlation squared R²
    • Adjusted R² or MSE
    • Mallows' C_p
  • Algorithms for variable selection
    • All possible subsets methods
    • Backward elimination method
    • Forward selection method
    • Stepwise selection method

• Further Topics in Multiple Regression
  • Model diagnostics and remedies
  • Multi-collinearity
  • Outliers and influential observations
  • Interaction effects
  • Dummy variables for groups of data

• Path Analysis

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Multivariate Statistical Description
  • Mean vector
  • Covariance matrix
  • Correlation matrix
  • Multivariate normal distribution

• Comparison between Two Mean Vectors — Hotelling's T² Test
  • Test for single mean vector μ = μ₀
  • Test for two mean vectors μ₁ = μ₂

• Comparisons among Several Multivariate Means — Multivariate Analysis of Variance
  • Decomposition of variation (matrix)
  • Wilks' Λ*

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Basic Ideas of Discriminant Analysis

• Fisher's Discriminant Analysis
  • Discrimination of two groups
  • Discrimination of multiple groups

• Bayesian Discriminant Analysis
  • Bayesian criterion
  • Classification function
  • A prior probability
  • A posterior probability

• Stepwise Discriminant Analysis

• Retrospective and Prospective Validation

• Considerations in Applications
  • The quality of training data is the key for a good discriminant analysis
  • The more groups, the less accurate of the classification rule
  • Fisher's and Bayesian discriminations are the same for two class classification
  • Prospective validation

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Logistic Regression Model
  • Basic concept of logistic regression model
  • Parameter estimation of logistic regression
  • The hypothesis testing for logistic model
    • The test for goodness of fit
    • The likelihood ratio test
    • The score test
    • Wald test
  • More examples

• Conditional Logistic Regression

• Multinomial Logistic Regression Model
  • Ordinal multinomial logistic model
  • Multinomial logistic model

• Application of Logistic Regression
  • Screening of disease-related risk factors
  • Adjustment of confounding factors
  • Prediction and discrimination
  • The attentions should be paid

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Meaning of Clustering
  • Comparison of cluster analysis and discriminant analysis
  • Cluster analysis
  • Cluster statistics

• Hierarchical Cluster

• Fast Cluster

• Variable Cluster

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Basic Concepts of Principal Component Analysis

• Searching for the Summary Variable
  • Definition of the principal components
  • Properties of principal components

• Computation and Interpretation of Principal Components
  • Computational procedures
  • The number of principal components
  • Correlation between principal components and variables

• Principal Component Analysis in Regression

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Factor Model

• Derivation of Factors
  • Principal factor method
  • Iterative principal factor method

• Factor Pattern Plot and Factor Rotation
  • Factor pattern plot
  • Factor rotation

• Factor Score and Application of Factor Patterns
  • Factor score and principal components
  • Common factors and latent variables
  • Factor analysis and structure efficiency

• Confirmative Factor Analysis

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Canonical Correlation
  • Definition of canonical correlation variables
  • Calculations of canonical correlation coefficients and linear combination coefficients
  • Canonical correlation structures
  • Rotation of canonical variables and adjusted canonical correlation coefficient
  • Redundancy of canonical variables
  • Regression on canonical variables

• Correspondence Analysis
  • Computation procedure
  • The use of correspondence analysis
  • Application — Analysis of contraceptive methods used by various groups of people
  • Relationship between correspondence analysis and canonical correlation analysis

• Canonical Discriminant Analysis

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• The Basic Concept of Survival Analysis
  • The record of follow up data (data construction)
  • Survival time (complete data, censored data)

• The Product-Limit Method for One Group of Survival Data
  • Survival function and survival curve
  • The basic requirements by K-M method

• The Log-Rank Test and Breslow Test for Comparing Two Survival Data Sets
  • The log-rank test
  • Breslow test

• The Cox Regression
  • A brief introduction
    • The Cox model
    • Estimation and test for regression coefficients
  • Application of Cox regression
    • Data structure
    • Preliminary screening of the covariates
    • Cox regression model with multiple covariate
    • Risk ratio
    • Standard coefficient
  • The estimation of survival function

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Log-Linear Models for Contingency Tables
  • Log-linear models for 3-dimensional contingency tables
  • Combining contingency tables
  • Model selection for higher dimensional contingency tables

• Poisson Regression

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Examples of Repeated Measurements

• Imperfect Analysis and Its Origins

• Analysis of Summary Measures Approach

• General Linear Models for Repeated Measurements
  • Test for sphericity of variance-covariance matrix
  • Univariate analysis of variance
  • Analysis of variance for data with orthogonal transformation
  • Multivariate analysis of variance

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Introduction

• Design and Analysis of Qualitative Responses

• Design and Analysis of Quantitative Responses

• Group Sequential Schemes
  • Continuous responses
  • Discrete responses
  • Examples

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Design of the Study

• Sampling Methods and Estimation of Population Parameters
  • Simple random sampling
    • Sampling method
    • Calculation of sample means and proportions
    • Calculation of standard errors
    • Population parameters and confidence intervals
  • Systematic sampling (mechanical sampling)
    • Sampling method
    • Calculation of standard error
    • Estimation of confidence intervals of population parameters
  • Stratified sampling
    • Sampling method
    • Calculation of sample mean and sample proportion
    • Calculation of standard errors
    • Estimation of population parameters and confidence intervals
  • Cluster sampling
    • Sampling methods
    • Calculation of sample mean and proportion
    • Calculation of standard errors

• Estimation of Sample Size
  • Sample size in cluster sampling
    • Sample size needed for estimating population proportion
    • Sample size needed for estimating population mean
  • Sample size in simple random sampling
    • Sample size needed for infinite population
    • Sample size needed for finite population
  • Sample size in stratified sampling
    • Sample size needed to estimate population probability
    • Sample size needed to estimate population mean

• The Current Life Table
  • Construction and formulas of abridged current life table
    • Age-specific death rate {column 4)
    • Age-specific probability of death q_x {column 5)
    • Number of survivors, l_x {column 6) and number of deaths, d_x (column 7)
    • Number of person-years lived in the interval (x, x + n)L_x (column 8)
    • Total number of person-years lived beyond age x, T_x (column 9)
    • Life expectancy at age x, e_x (column 10)
  • Analysis of life table
    • Probability of death q_x
    • Number of survivals l_x
    • Number of deaths d_x
    • Life expectancy e_x

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Study Design
  • Study population
  • Control population

• Measures of Disease Occurrence
  • Cumulative incidence probability
    • Approximation of the incidence probability
    • The method of cohort life table for cumulative incidence probability
  • Person-time incidence rate
  • Age-specific person-time rate
  • Two-dimensional person-year rate by age-period cross classification

• Analysis of Data from Prospective Studies
  • Relative risk
    • Mantel–Haenszel χ² test
    • Method based on binomial distribution and normal approximation
  • Attributable risk
  • Population attributable risk
  • Analysis of dose-response relationship
  • Stratified analysis
    • Mantel-Haenzel stratified χ² test
    • Adjusted relative risk, RR_α
    • Confidence interval for adjusted relative risk

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Designs of Case-Control Studies
  • Types of designs
    • Design for group comparison
    • Design for matched comparison
  • Determination of sample size
    • Sample size for group comparison
    • Sample size required for matched comparison
  • Selection of cases and controls
  • Bias and avoidance

• Analysis of Data from Design for Group Comparison
  • Analysis of data for a single 2 × 2 table
    • Calculation of odds ratio
    • Hypothesis testing for odds ratio
    • Confidence interval for odds ratio
    • Population attributable risk
  • Analysis for stratified 2 × 2 Tables
    • M-H adjustment
    • Logarithmic transformation adjustment
  • Analysis of dose-response for several exposure categories

• Analysis of Matched Data
  • Analysis of data with 1:1 matching
  • Analysis of data with l:m matching
  • Analysis of data with l:m_i matching

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Basic Concepts
  • Genetic terminology
  • Hardy-Weinberg equilibrium
  • Linkage and linkage equilibrium

• Segregation Analysis
  • Simple segregation analysis
    • Ascertainment
    • Segregation analysis of dominant loci
    • Segregation analysis of recessive loci
  • Regressive logistic model
    • Regressive logistic model
    • Genetic analysis
    • Estimation and hypothesis testing of parameters

• Linkage Analysis
  • Introduction
  • The LODS method
  • The affected sib pair (ASP) method

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Statistical Evaluation of Measuring Devices
  • Validity
  • Reliability
  • Responsibility to change

• Statistics for Describing Reliability
  • Reliability for simple repeated measurements
  • Reliability of complex repeated measurements

• Statistics Describing Criteria Validity
  • Criteria validity of quantitative measurements
  • Criteria validity of qualitative measurements
  • Criteria validity of ordered measurements

• Reliability, Validity and Responsiveness to Change of Scales
  • Indices evaluating reliability of scales
    • Test-retest reliability
    • Split-half reliability
    • Chronbach's α coefficient
  • Indices evaluating validity of scales
    • Face validity
    • Content validity
    • Criteria validity
    • Construct validity
    • Discriminant validity
  • Indices evaluating responsibility to change

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Design and Data Layout

• Measures Frequently Used in Diagnostic Test
  • Sensitivity
  • Specificity
  • Youden's index
  • Positive likelihood ratio
  • Negative likelihood ratio
  • Positive predictive value
  • Negative predictive value

• Analysis of ROC Curve
  • Calculating ROC operating point
  • Plottting ROC curve
  • Calculating the area under ROC curves
  • Comparing the areas under ROC curve

• Decision Making on Diagnostic and Screening Test

• Computer Experiments

• Think, Practice and Experiments

The following sections are included:

• Basic Notions
  • The definition of meta-analysis
  • Effect magnitude
  • Homogeneity test, fix effect model and random effect model
  • The design and planning of a meta-analysis
    • Definition of the problem
    • Retrieval strategy
    • Assessing research literatures
    • Data analysis
    • Interpretation of the result and prospect of further research

• Statistical Methods Commonly Used in Meta-Analysis
  • Collecting and arranging the data
  • Data presentation in meta-analysis
  • Mergence of effect magnitudes
    • Weighted combination
    • Homogeneity test
    • The 95% confidence interval of mean effect magnitude
  • Mergence of odds ratio
    • Weighted combination
    • Homogeneity test
    • The 95% confidence interval of OR

• Computer Experiments

• Thinking, Practice and Experiments

The following sections are included:

• Appendix A Introduction of Statistical Analysis System (SAS)

• Appendix B Statistical Table

• Appendix C Data Sets of Some Real Medical Examples

• Appendix D Answers to Exercises

• Index