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Statistical Modeling for Biomedical Researchers
Comment from the Stata technical groupThis new text is ideal for a one-semester graduate course in biostatistics and epidemiology for students in the health sciences. It assumes only a basic knowledge of statistics, such as that obtained from the standard one-semester introductory statistics course. Stata is used extensively throughout the text, making it possible to introduce computationally complex methods with little or no higher-level mathematics. As a result, the book focuses on concepts and model assumptions, rather than on the underlying mathematics. One appealing feature of the text is that when Stata output is displayed, the most important elements of the output are highlighted and explained in the notes that follow the output. This helps the reader make real sense of the output by providing the appropriate focus for the problem at hand. The text is also replete with examples featuring real medical data. Table of contentsPreface
1 Introduction
1.1 Algebraic Notation
1.2 Descriptive Statistics
1.2.1 Dot Plot
1.3 The Stata Statistical Software Package1.2.2 Sample Mean 1.2.3 Residual 1.2.4 Sample Variance 1.2.5 Sample Standard Deviation 1.2.6 Percentile and Median 1.2.7 Box Plot 1.2.8 Histogram 1.2.9 Scatter Plot
1.3.1 Downloading Data from My Web Site
1.4 Inferential Statistics1.3.2 Creating Dot Plots with Stata 1.3.3 Stata Command Syntax 1.3.4 Obtaining Interactive Help from Stata 1.3.5 Stata Log Files 1.3.6 Displaying Other Descriptive Statistics with Stata
1.4.1 Probability Density Function
1.5 Additional Reading1.4.2 Mean, Variance and Standard Deviation 1.4.3 Normal Distribution 1.4.4 Expected Value 1.4.5 Standard Error 1.4.6 Null Hypothesis, Alternative Hypothesis and P Value 1.4.7 95% Confidence Interval 1.4.8 Statistical Power 1.4.9 The z and Student's t Distributions 1.4.10 Paired t Test 1.4.11 Performing Paired t Tests with Stata 1.4.12 Independent t Test Using a Pooled Standard Error Estimate 1.4.13 Independent t Test using Separate Standard Error Estimates 1.4.14 Independent t Tests using Stata 1.4.15 The Chi-Squared Distribution 1.6 Exercises 2 Simple Linear Regression
2.1 Sample Covariance
2.2 Sample Correlation Coefficient 2.3 Population Covariance and Correlation Coefficient 2.4 Conditional Expectation 2.5 Simple Linear Regression Model 2.6 Fitting the Linear Regression Model 2.7 Historical Trivia: Origin of the Term Regression 2.8 Determining the Accuracy of Linear Regression Estimates 2.9 Ethylene Glycol Poisoning Example 2.10 95% Confidence Interval for y[x] = a + Bx Evaluated at x 2.11 95% Prediction Interval for the Response of a New Patient 2.12 Simple Linear Regression with Stata 2.13 Lowess Regression 2.14 Plotting a Lowess Regression Curve in Stata 2.15 Residual Analyses 2.16 Studentized Residual Analysis Using Stata 2.17 Transforming the x and y Variables
2.17.1 Stabilizing the Variance
2.18. Analyzing Transformed Data with Stata2.17.2 Correcting for Non-linearity 2.17.3 Example: Research Funding and Morbidity for 29 Diseases 2.19. Testing the Equality of Regression Slopes
2.19.1 Example: The Framingham Heart Study
2.20 Comparing Slope Estimates with Stata2.21 Additional Reading 2.22 Exercises 3 Multiple Linear Regression
3.1 The Model
3.2 Confounding Variables 3.3 Estimating the Parameters for a Multiple Linear Regression Model 3.4 R2 Statistic for Multiple Regression Models 3.5 Expected Response in the Multiple Regression Model 3.6 The Accuracy of Multiple Regression Parameter Estimates 3.7 Leverage 3.8 95% Confidence Interval for yhati 3.9 95% Prediction Intervals 3.10 Example: The Framingham Heart Study
3.10.1 Preliminary Univariate Analyses
3.11 Scatterplot Matrix Graphs
3.11.1 Producing Scatterplot Matrix Graphs with Stata
3.12 Modeling Interaction in Multiple Linear Regression
3.12.1 The Framingham Example
3.13 Multiple Regression Modeling of the Framingham Data3.14 Intuitive Understanding of a Multiple Regression Model
3.14.1 The Framingham Example
3.15 Calculating 95% Confidence and Prediction Intervals3.16 Multiple Linear Regression with Stata 3.17 Automatic Methods of Model Selection
3.17.1 Forward Selection using Stata
3.18 Collinearity3.17.2 Backward Selection 3.17.3 Forward Stepwise Selection 3.17.4 Backward Stepwise Selection 3.17.5 Pros and Cons of Automated Model Selection 3.19 Residual Analyses 3.20 Influence
3.20.1 DFBETA Influence Statistic
3.21 Residual and Influence Analyses Using Stata3.20.2 Cook's Distance 3.20.3 The Framingham Example 3.22 Additional Reading 3.23 Exercises 4 Simple Logistic Regression
4.1 Example: APACHE Score and Mortality in Patients with Sepsis
4.2 Sigmoidal Family of Logistic Regression Curves 4.3 The Log Odds of Death Given a Logistic Probability Function 4.4 The Binomial Distribution 4.5 Simple Logistic Regression Model 4.6 Generalized Linear Model 4.7 Contrast Between Logistic and Linear Regression 4.8 Maximum Likelihood Estimation
4.8.1 Variance of Maximum Likelihood Parameter Estimates
4.9 Statistical Tests and Confidence Intervals
4.9.1 Likelihood Ratio Tests
4.10 Sepsis Example4.9.2 Quadratic Approximations to the Log Likelihood Ratio Function 4.9.3 Score Tests 4.9.4 Wald Tests and Confidence Intervals 4.9.5 Which Test Should You Use? 4.11 Logistic Regression with Stata 4.12 Odds Ratios and the Logistic Regression Model 4.13 95% Confidence Interval for the Odds Ratio Associated with a Unit Increase in x
4.13.1 Calculating this Odds Ratio with Stata
4.14 Logistic Regression with Grouped Response Data4.15 95% Confidence Interval for the P[x] 4.16 95% Confidence Intervals for Proportions 4.17 Example: The Ibuprofen in Sepsis Trial 4.18 Logistic Regression with Grouped Data Using Stata 4.19 Simple 2x2 Case-Control Studies
4.19.1 Example: The Ille-et-Vilaine Study of Esophageal Cancer and Alcohol
4.20 Logistic Regression Models for 2x2 Contingency Tables4.19.2 Review of Classical Case-Control Theory 4.19.3 95% Confidence Interval for the Odds Ratio: Woolf's Method 4.19.4 Test of the Null Hypothesis that the Odds Ratio Equals One 4.19.5 Test of the Null Hypothesis that Two Proportions are Equal
4.20.1 Nuisance Parameters
4.21 Creating a Stata Data File4.20.2 95% Confidence Interval for the Odds Ratio: Logistic Regression 4.22 Analyzing Case-Control Data with Stata 4.23 Regressing Disease Against Exposure 4.24 Additional Reading 4.25 Exercises 5 Multiple Logistic Regression
5.1 Mantel–Haenszel Estimate of an Age-Adjusted Odds Ratio
5.2 Mantel–Haenszel x2 Statistic for Multiple 2x2 Tables 5.3 95% Confidence Interval for the Age-Adjusted Odds Ratio 5.4 Breslow and Day's Test for Homogeneity 5.5 Calculating the Mantel–Haenszel Odds Ratio using Stata 5.6 Multiple Logistic Regression Model 5.7 95% Confidence Interval for an Adjusted Odds Ratio 5.8 Logistic Regression for Multiple 2x2 Contingency Tables 5.9 Analyzing Multiple 2x2 Tables with Stata 5.10 Handling Categorical Variables in Stata 5.11 Effect of Dose of Alcohol on Esophageal Cancer Risk
5.11.1 Analyzing Model with Stata
5.12 Effect of Dose of Tobacco on Esophageal Cancer Risk5.13 Deriving Odds Ratios from Multiple Parameters 5.14 The Standard Error of a Weighted Sum of Regression Coefficients 5.15 Confidence Intervals for Weighted Sums of Coefficients 5.16 Hypothesis Tests for Weighted Sums of Coefficients 5.17 The Estimated Variance-Covariance Matrix 5.18 Multiplicative Models of Two Risk Factors 5.19 Multiplicative Model of Smoking, Alcohol, and Esophageal Cancer 5.20 Fitting a Multiplicative Model with Stata 5.21 Model of Two Risk Factors with Interaction 5.22 Model of Alcohol, Tobacco, and Esophageal Cancer with Interaction Terms 5.23 Fitting a Model with Interaction using Stata 5.24 Model Fitting: Nested Models and Model Deviance 5.25 Effect Modifiers and Confounding Variables 5.26 Goodness-of-Fit Tests
5.26.1 The Pearson x2 Goodness-of-Fit Statistic
5.27 Hosmer–Lemeshow Goodness-of-Fit Test
5.27.1 An Example: The Ille-et-Vilaine Cancer Data Set
5.28 Residual and Influence Analysis
5.28.1 Standardized Pearson Residual
5.29 Using Stata for Goodness-of-Fit Tests and residual Analyses5.28.2 DFBETA Influence Statistic 5.28.3 Residual Plots of the Ille-et-Vilaine Data on Esophageal Cancer 5.30 Frequency Matched Case–Control Studies 5.31 Conditional Logistic Regression 5.32 Analyzing Data with Missing Values
5.32.1 Cardiac Output in the Ibuprofen in Sepsis Study
5.33 Additional Reading5.32.2 Modeling Missing Values with Stata 5.34 Exercises 6 Introduction to Survival Analysis
6.1 Survival and Cumulative Mortality Functions
6.2 Right Censored Data 6.3 Kaplan–Meier Survival Curves 6.4 An Example: Genetic Risk of Recurrent Intracerebral Hemorrhage 6.5 95% Confidence Intervals for Survival Functions 6.6 Cumulative Mortality Function 6.7 Censoring and Bias 6.8 Logrank Test 6.9 Using Stata to Derive Survival Functions and the Logrank Test 6.10 Logrank Test for Multiple Patient Groups 6.11 Hazard Functions 6.12 Proportional Hazards 6.13 Relative Risks and Hazard Ratios 6.14 Proportional Hazards Regression Analysis 6.15 Hazard Regression Analysis of the Intracerebral Hemorrhage Data 6.16 Proportional Hazards Regression Analysis with Stata 6.17 Tied Failure Times 6.18 Additional Reading 6.19 Exercises 7 Hazard Regression Analysis
7.1 Proportional Hazards Model
7.2 Relative Risks and Hazard Ratios 7.3 95% Confidence Intervals and Hypothesis Tests 7.4 Nested Models and Model Deviance 7.5 An Example: The Framingham Heart Study
7.5.1 Univariate Analyses
7.6 Cox–Snell Generalized Residuals and Proportional Hazards Models7.5.2 Multiplicative Model of DBP and Gender on Risk of CHD 7.5.3 Using Interaction Terms to Model the Effects of Gender and DBP on CHD 7.5.4 Adjusting for Confounding Variables 7.5.5 Interpretation 7.5.6 Alternate Models 7.7 Proportional Hazards Regression Analysis using Stata 7.8 Stratified Proportional Hazards Models 7.9 Survival Analysis with Ragged Study Entry
7.9.1 Kaplan–Meier Survival Curve and the Logrank Test with Ragged Entry
7.10 Hazard Regression Models with Time Dependent Covariates7.9.2 Age, Sex, and CHD in the Framingham Heart Study 7.9.3 Proportional Hazards Regression Analysis with Ragged Entry 7.9.4 Survival Analysis with Ragged Entry using Stata
7.10.1 Cox–Snell Residuals for Models with Time-Dependent Covariates
7.11 Modeling Time-Dependent Covariates with Stata7.10.2 Testing the Proportional Hazards Assumption 7.10.3 Alternative Models 7.12 Additional Reading 7.13 Exercises 8 Introduction to Poisson Regression: Inferences on Morbidity and Mortality Rates
8.1 Elementary Statistics Involving Rates
8.2 Calculating Relative Risks from Incidence Data Using Stata 8.3 The Binomial and Poisson Distributions 8.4 Simple Poisson Regression for 2x2 Tables 8.5 Poisson Regression and the Generalized Linear Model 8.6 Contrast Between Poisson, Logistic, and Linear Regression 8.7 Simple Poisson Regression with Stata 8.8 Poisson Regression and Survival Analysis
8.8.1 Recoding Survival Data on Patients as Patient-Year Data
8.9 Converting the Framingham Survival Data Set to Person-Time Data8.8.2 Converting Survival Records to Person-Years of Follow-Up using Stata 8.10 Simple Poisson Regression with Multiple Data Records 8.11 Poisson Regression with a Classification Variable 8.12 Applying Simple Poisson Regression to the Framingham Data 8.13 Additional Reading 8.14 Exercises 9 Multiple Poisson Regression
9.1 Multiple Poisson Regression Model
9.2 An Example: The Framingham Heart Study
9.2.1 A Multiplicative Model of Gender, Age and Coronary Heart Disease
9.3 Using Stata to Perform Poisson Regression9.2.2 A Model of Age, Gender and CHD with Interaction Terms 9.2.3 Adding Confounding Variables to the Model 9.4 Residual Analyses for Poisson Regression Models
9.4.1 Deviance Residuals
9.5 Residual Analysis of Poisson Regression Models Using Stata9.6 Additional Reading 9.7 Exercises 10 Fixed Effects Analysis of Variance
10.1 One-Way Analysis of Variance
10.2 Multiple Comparisons 10.3 Reformulating Analysis of Variance as a Linear Regression Model 10.4 Non-parametric Methods 10.5 Kruskal-Wallis Test 10.6 Example: A Polymorphism in the Estrogen Receptor Gene 10.7 One-Way Analyses of Variance Using Stata 10.8 Two-Way Analysis of Variance, Analysis of Covariance, and Other Models 10.9 Additional Reading 10.10 Exercises 11 Repeated-Measures Analysis of Variance
11.1 Example: Effect of Race and Dose of Isoproterenol on Blood Flow
11.2 Exploratory Analysis of Repeated Measures Data Using Stata 11.3 Response Feature Analysis 11.4 Example: The Isoproterenol Data Set 11.5 Response Feature Analysis Using Stata 11.6 The Area-Under-the-Curve Response Feature 11.7 Generalized Estimating Equations 11.8 Common Correlation Structures 11.9 GEE Analysis and the Huber-White Sandwich Estimator 11.10 Example: Analyzing the Isoproterenol Data with GEE 11.11 Using Stata to Analyze the Isoproterenol Data Set Using GEE 11.12 GEE Analyses with Logistic or Poisson Models 11.13 Additional Reading 11.14 Exercises Appendix A Summary of Stata Commands Used in this Text
References
Index
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