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• About this Book
• Cover Page
• Halftitle Page
• Title Page
• Copyright Page
• Dedication
• Contents in brief
• Contents
• Preface
• About the authors
• Acknowledgments
• SaplingPlus for Statistics: Course Resources
• Online Resources for Students
• Chapter 1 Statistics and samples
• 1.1 What is statistics?
• 1.2 Sampling Populations
• Populations and samples
• Properties of good samples
• Random sampling
• How to take a random sample
• The sample of convenience
• Volunteer bias
• Data in the real world
• 1.3 Types of Data and Variables
• Categorical and numerical variables
• Explanatory and response variables
• 1.4 Frequency distributions and probability distributions
• 1.5 Types of studies
• 1.6 Summary
• Chapter 1 Problems
• Practice problems
• Assignment problems
• Interleaf 1 Correlation does not require causation
• Chapter 2 Displaying data
• 2.1 Guidelines for effective graphs
• How to draw a bad graph
• How to draw a good graph
• 2.2 Showing data for one variable
• Showing categorical data: frequency table and bar graph
• Making a good bar graph
• A bar graph is usually better than a pie chart
• Showing numerical data: frequency table and histogram
• Describing the shape of a histogram
• How to draw a good histogram
• Other graphs for numerical data
• 2.3 Showing association between two variables and differences between groups
• Showing association between categorical variables
• Showing association between numerical variables: scatter plot
• Showing association between a numerical and a categorical variable
• 2.4 Showing trends in time and space
• 2.5 How to make good tables
• Follow similar principles for display tables
• 2.6 How to make data files
• 2.7 Summary
• Chapter 2 Problems
• Practice problems
• Assignment problems
• Chapter 3 Describing data
• 3.1 Arithmetic mean and standard deviation
• The sample mean
• Variance and standard deviation
• Rounding means, standard deviations, and other quantities
• Coefficient of variation
• Calculating mean and standard deviation from a frequency table
• Effect of changing measurement scale
• 3.2 Median and interquartile range
• The median
• The interquartile range
• The box plot
• 3.3 How measures of location and spread compare
• Mean versus median
• Standard deviation versus interquartile range
• 3.4 Cumulative frequency distribution
• Percentiles and quantiles
• Displaying cumulative relative frequencies
• 3.5 Proportions
• Calculating a proportion
• The proportion is like a sample mean
• 3.6 Summary
• 3.7 Quick formula summary
• Table of formulas for descriptive statistics
• Chapter 3 Problems
• Practice problems
• Assignment problems
• Chapter 4 Estimating with uncertainty
• 4.1 The sampling distribution of an estimate
• Estimating mean gene length with a random sample
• The sampling distribution of Y¯
• 4.2 Measuring the uncertainty of an estimate
• Standard error
• The standard error of Y¯
• The standard error of Y¯ from data
• 4.3 Confidence intervals
• The 2SE rule of thumb
• 4.4 Error bars
• 4.5 Summary
• 4.6 Quick formula summary
• Standard error of the mean
• Chapter 4 Problems
• Practice problems
• Assignment problems
• Interleaf 2 Pseudoreplication
• Chapter 5 Probability
• 5.1 The probability of an event
• 5.2 Venn diagrams
• 5.3 Mutually exclusive events
• 5.4 Probability distributions
• Discrete probability distributions
• Continuous probability distributions
• 5.5 Either this or that: adding probabilities
• The addition rule
• The probabilities of all possible mutually exclusive outcomes add to one
• The general addition rule
• 5.6 Independence and the multiplication rule
• Multiplication rule
• “And” versus “or”
• Independence of more than two events
• 5.7 Probability trees
• 5.8 Dependent events
• 5.9 Conditional probability and Bayes’ theorem
• Conditional probability
• The general multiplication rule
• Sampling without replacement
• Bayes’ theorem
• 5.10 Summary
• Chapter 5 Problems
• Practice problems
• Assignment problems
• Chapter 6 Hypothesis testing
• 6.1 Making and using statistical hypotheses
• Null hypothesis
• Alternative hypothesis
• To reject or not to reject
• 6.2 Hypothesis testing: an example
• Stating the hypotheses
• The test statistic
• The null distribution
• Quantifying uncertainty: the P-value
• Draw the appropriate conclusion
• Reporting the results
• 6.3 Errors in hypothesis testing
• Type I and Type II errors
• 6.4 When the null hypothesis is not rejected
• The test
• Interpreting a nonsignificant result
• 6.5 One-sided tests
• 6.6 Hypothesis testing versus confidence intervals
• 6.7 Summary
• Chapter 6 Problems
• Practice problems
• Assignment problems
• Interleaf 3 Why statistical significance is not the same as biological importance
• Chapter 7 Analyzing proportions
• 7.1 The binomial distribution
• Formula for the binomial distribution
• Number of successes in a random sample
• Sampling distribution of the proportion
• 7.2 Testing a proportion: the binomial test
• Approximations for the binomial test
• 7.3 Estimating proportions
• Estimating the standard error of a proportion
• Confidence intervals for proportions—the Agresti–Coull method
• Confidence intervals for proportions—the Wald method
• 7.4 Deriving the binomial distribution
• 7.5 Summary
• 7.6 Quick formula summary
• Binomial distribution
• Proportion
• Agresti–Coull 95% confidence interval for a proportion
• Binomial test
• Chapter 7 Problems
• Practice problems
• Assignment problems
• Interleaf 4 Biology and the history of statistics
• Chapter 8 Fitting probability models to frequency data
• 8.1 χ2 Goodness-of-fit test: the proportional model
• Null and alternative hypotheses
• Observed and expected frequencies
• The χ2 test statistic
• The sampling distribution of χ2 under the null hypothesis
• Calculating the P-value
• Critical values for the χ2 distribution
• 8.2 Assumptions of the χ2 goodness-of-fit test
• 8.3 Goodness-of-fit tests when there are only two categories
• 8.4 Random in space or time: the Poisson distribution
• Formula for the Poisson distribution
• Testing randomness with the Poisson distribution
• Comparing the variance to the mean
• 8.5 Summary
• 8.6 Quick formula summary
• χ2 Goodness-of-fit test
• Test statistic: χ2
• Poisson distribution
• Chapter 8 Problems
• Practice problems
• Assignment problems
• Interleaf 5 Making a plan
• Chapter 9 Contingency analysis: Associations between categorical variables
• 9.1 Associating two categorical variables
• 9.2 Estimating association in 2×2 tables: relative risk
• Relative risk
• Reduction in risk
• 9.3 Estimating association in 2×2 tables: the odds ratio
• Odds
• Odds ratio
• Standard error and confidence interval for odds ratio
• Odds ratio vs. relative risk
• 9.4 The χ2 contingency test
• Hypotheses
• Expected frequencies assuming independence
• The χ2 statistic
• Degrees of freedom
• P-value and conclusion
• A shortcut for calculating the expected frequencies
• The χ2 contingency test is a special case of the χ2 goodness-of-fit test
• Assumptions of the χ2 contingency test
• Correction for continuity
• 9.5 Fisher’s exact test
• 9.6 Summary
• 9.7 Quick formula summary
• Confidence interval for relative risk
• Confidence interval for odds ratio
• The χ2 contingency test
• Fisher’s exact test
• Chapter 9 Problems
• Practice problems
• Assignment problems
• Review Problems 1
• Chapter 10 The normal distribution
• 10.1 Bell-shaped curves and the normal distribution
• 10.2 The formula for the normal distribution
• 10.3 Properties of the normal distribution
• 10.4 The standard normal distribution and statistical tables
• Using the standard normal table
• Using the standard normal to describe any normal distribution
• 10.5 The normal distribution of sample means
• Calculating probabilities of sample means
• 10.6 Central limit theorem
• 10.7 Normal approximation to the binomial distribution
• 10.8 Summary
• 10.9 Quick formula summary
• Z-standardization
• Normal approximation to the binomial distribution
• Chapter 10 Problems
• Practice problems
• Assignment problems
• Interleaf 6 Controls in medical studies
• Chapter 11 Inference for a normal population
• 11.1 The t-distribution for sample means
• Student’s t-distribution
• Finding critical values of the t-distribution
• 11.2 The confidence interval for the mean of a normal distribution
• The 95% confidence interval for the mean
• The 99% confidence interval for the mean
• 11.3 The one-sample t-test
• The effects of larger sample size: body temperature revisited
• 11.4 Assumptions of the one-sample t-test
• 11.5 Estimating the standard deviation and variance of a normal population
• Confidence limits for the variance
• Confidence limits for the standard deviation
• Assumptions
• 11.6 Summary
• 11.7 Quick formula summary
• Confidence interval for a mean
• One-sample t-test
• Confidence interval for variance
• Chapter 11 Problems
• Practice problems
• Assignment problems
• Chapter 12 Comparing two means
• 12.1 Paired sample versus two independent samples
• 12.2 Paired comparison of means
• Estimating mean difference from paired data
• Paired t-test
• Assumptions
• 12.3 Two-sample comparison of means
• Confidence interval for the difference between two means
• Two-sample t-test
• Assumptions
• Welch’s t-test
• 12.4 Using the correct sampling units
• 12.5 The fallacy of indirect comparison
• 12.6 Interpreting overlap of confidence intervals
• 12.7 Comparing variances
• The F-test of equal variances
• Levene’s test for homogeneity of variances
• 12.8 Summary
• 12.9 Quick formula summary
• Confidence interval for the mean difference (paired data)
• Paired t-test
• Standard error of difference between two means
• Confidence interval for the difference between two means (two samples)
• Two-sample t-test
• Welch’s confidence interval for the difference between two means
• Welch’s approximate t-test
• F-test
• Levene’s test
• Chapter 12 Problems
• Practice problems
• Assignment problems
• Interleaf 7 Which test should I use?
• Chapter 13 Handling violations of assumptions
• 13.1 Detecting deviations from normality
• Graphical methods
• Formal test of normality
• 13.2 When to ignore violations of assumptions
• Violations of normality
• Unequal standard deviations
• 13.3 Data transformations
• Log transformation
• Other transformations
• Confidence intervals with transformations
• Avoid multiple testing with transformations
• 13.4 Nonparametric alternatives to one-sample and paired t-tests
• Sign test
• The Wilcoxon signed-rank test
• 13.5 Comparing two groups: the Mann–Whitney U-test
• Tied ranks
• Large samples and the normal approximation
• 13.6 Assumptions of nonparametric tests
• 13.7 Type I and Type II error rates of nonparametric methods
• 13.8 Permutation tests
• Assumptions of permutation tests
• 13.9 Summary
• 13.10 Quick formula summary
• Transformations
• Back-transformations
• Sign test
• Mann–Whitney U-test
• Chapter 13 Problems
• Practice problems
• Assignment problems
• Review Problems 2
• Chapter 14 Designing experiments
• 14.1 Lessons from clinical trials
• Design components
• 14.2 How to reduce bias
• Simultaneous control group
• Randomization
• Blinding
• 14.3 How to reduce the influence of sampling error
• Replication
• Balance
• Blocking
• Extreme treatments
• 14.4 Experiments with more than one factor
• 14.5 What if you can’t do experiments?
• Match and adjust
• 14.6 Choosing a sample size
• Plan for precision
• Plan for power
• Plan for data loss
• 14.7 Summary
• 14.8 Quick formula summary
• Planning for precision
• Planning for power
• Chapter 14 Problems
• Practice problems
• Assignment problems
• Interleaf 8 Data dredging
• Chapter 15 Comparing means of more than two groups
• 15.1 The analysis of variance
• Hypotheses
• ANOVA in a nutshell
• ANOVA tables
• Partitioning the sum of squares
• Calculating the mean squares
• The variance ratio, F
• Variation explained: R^2
• ANOVA with two groups
• 15.2 Assumptions and alternatives
• The robustness of ANOVA
• Data transformations
• Nonparametric alternatives to ANOVA
• 15.3 Planned comparisons
• Planned comparison between two means
• 15.4 Unplanned comparisons
• Testing all pairs of means using the Tukey–Kramer method
• Assumptions
• 15.5 Fixed and random effects
• 15.6 ANOVA with randomly chosen groups
• ANOVA calculations
• Variance components
• Repeatability
• Assumptions
• 15.7 Summary
• 15.8 Quick formula summary
• Analysis of variance (ANOVA)
• Kruskal–Wallis test
• Planned confidence interval for the difference between two of k means
• Planned test of the difference between two of k means
• Tukey–Kramer test of all pairs of means
• Repeatability and variance components
• Chapter 15 Problems
• Practice problems
• Assignment problems
• Interleaf 9 Experimental and statistical mistakes
• Chapter 16 Correlation between numerical variables
• 16.1 Estimating a linear correlation coefficient
• The correlation coefficient
• Standard error
• Approximate confidence interval
• 16.2 Testing the null hypothesis of zero correlation
• 16.3 Assumptions
• 16.4 The correlation coefficient depends on the range
• 16.5 Spearman’s rank correlation
• Procedure for large n
• Assumptions of Spearman’s correlation
• 16.6 The effects of measurement error on correlation
• 16.7 Summary
• 16.8 Quick formula summary
• Shortcuts
• Covariance
• Correlation coefficient
• Confidence interval (approximate) for a population correlation
• The t-test of zero linear correlation
• Spearman’s rank correlation
• Spearman’s rank correlation test
• Correlation corrected for measurement error
• Chapter 16 Problems
• Practice problems
• Assignment problems
• Interleaf 10 Publication bias
• Chapter 17 Regression
• 17.1 Linear regression
• The method of least squares
• Formula for the line
• Calculating the slope and intercept
• Populations and samples
• Predicted values
• Residuals
• Standard error of slope
• Confidence interval for the slope
• 17.2 Confidence in predictions
• Confidence intervals for predictions
• Extrapolation
• 17.3 Testing hypotheses about a slope
• The t-test of regression slope
• The ANOVA approach
• 17.4 Regression toward the mean
• 17.5 Assumptions of regression
• Outliers
• Detecting nonlinearity
• Detecting non-normality and unequal variance
• 17.6 Transformations
• 17.7 The effects of measurement error on regression
• 17.8 Regression with nonlinear relationships
• A curve with an asymptote
• Quadratic curves
• Formula-free curve fitting
• 17.9 Logistic regression: fitting a binary response variable
• 17.10 Summary
• 17.11 Quick formula summary
• Shortcuts
• Regression slope
• Regression intercept
• Confidence interval for the regression slope
• Confidence interval for the predicted mean Y at a given X (confidence bands)
• Confidence interval for the predicted individual Y at a given X (prediction intervals)
• The t-test of a regression slope
• The ANOVA method for testing zero slope
• Chapter 17 Problems
• Practice problems
• Assignment problems
• Interleaf 11 Meta-analysis
• Review Problems 3
• Chapter 18 Analyzing multiple factors
• 18.1 ANOVA and linear regression are linear models
• Modeling with linear regression
• Generalizing linear regression
• Linear models
• 18.2 Analyzing experiments with blocking
• Analyzing data from a randomized block design
• Model formula
• Fitting the model to data
• 18.3 Analyzing factorial designs
• Model formula
• Testing the factors
• The importance of distinguishing fixed and random factors
• 18.4 Adjusting for the effects of a covariate
• Testing interaction
• Fitting a model without an interaction term
• 18.5 Assumptions of linear models
• 18.6 Summary
• Chapter 18 Problems
• Practice problems
• Assignment problems
• Interleaf 12 Using species as data points
• Chapter 19 Computer-intensive methods
• 19.1 Hypothesis testing using simulation
• 19.2 Bootstrap standard errors and confidence intervals
• Bootstrap standard error
• Confidence intervals by bootstrapping
• Bootstrapping with multiple groups
• Assumptions and limitations of the bootstrap
• 19.3 Summary
• Chapter 19 Problems
• Practice problems
• Assignment problems
• Chapter 20 Likelihood
• 20.1 What is likelihood?
• 20.2 Two uses of likelihood in biology
• Phylogeny estimation
• Gene mapping
• 20.3 Maximum likelihood estimation
• Probability model
• The likelihood formula
• The maximum likelihood estimate
• Likelihood-based confidence intervals
• 20.4 Versatility of maximum likelihood estimation
• Probability model
• The likelihood formula
• The maximum likelihood estimate
• Bias
• 20.5 Log-likelihood ratio test
• Likelihood ratio test statistic
• Testing a population proportion
• 20.6 Summary
• 20.7 Quick formula summary
• Likelihood
• Likelihood-based confidence interval for a single parameter
• Log-likelihood ratio test for a single parameter
• Chapter 20 Problems
• Practice problems
• Assignment problems
• Chapter 21 Survival analysis
• 21.1 Survival curves
• Calculation summary
• Confidence intervals
• Median survival time
• Assumptions
• 21.2 Compare survival curves
• Hazard ratio
• Hazard ratio calculation
• Logrank test
• Assumptions
• 21.3 Summary
• 21.4 Quick formula summary
• Hazard ratio
• 95% Confidence interval for the hazard ratio
• Logrank test
• Chapter 21 Problems
• Practice problems
• Assignment problems
• Notes
• Statistical tables
• Using statistical tables
• Statistical Table A: The χ2 distribution
• Statistical Table B: The standard normal (Z) distribution
• Statistical Table C: Student’s t-distribution
• Statistical Table D: The F-distribution
• Statistical Table E: Mann–Whitney U-distribution
• Statistical Table F: Tukey–Kramer q-distribution
• Statistical Table G: Critical values for the Spearman’s rank correlation
• Literature cited
• Answers to practice problems
• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Review 1
• Chapter 10
• Chapter 11
• Chapter 12
• Chapter 13
• Review 2
• Chapter 14
• Chapter 15
• Chapter 16
• Chapter 17
• Review 3
• Chapter 18
• Chapter 19
• Chapter 20
• Chapter 21
• Index
• glossary
• Back Cover