Using Statistical Methods for Water Quality Management -- Contents -- List of Figures -- List of Tables -- Preface -- Part I Issues -- 1 Introduction -- 1.1 Conventions -- 1.2 The essentials -- 1.3 Meeting management's information needs -- 1.4 Water quality observations as random variables -- 1.5 Samples and populations -- 1.6 Special characteristics of water quality data -- 1.7 Data analysis protocols -- 2 Basic concepts of probability and statistics -- 2.1 Probability rules -- 2.2 Representing data -- 2.2.1 Types of data -- 2.2.2 Frequency, bar graph, and histogram -- 2.2.3 Describing the distribution of probability -- 2.2.4 Discrete versus continuous data -- 2.2.5 Summary statistics -- 2.3 Exploratory and graphical methods -- 2.4 Important distributions -- 2.5 Continuous distributions -- 2.5.1 Normal distribution -- 2.5.2 Lognormal distribution -- 2.5.3 Gamma distribution -- 2.5.4 Beta distribution -- 2.6 Discrete distributions -- 2.6.1 Binomial distribution -- 2.6.2 Poisson distribution -- 2.6.3 Negative binomial distribution -- 2.6.4 Hypergeometric distribution -- 2.6.5 Multinomial distribution -- 2.7 Sampling distributions -- 2.7.1 Student's t-distribution -- 2.7.2 Chi-square distribution -- 2.7.3 F-distribution -- 2.8 Statistical tables -- 2.9 Correlation and measures thereof -- 2.10 Statistical models and model parameters -- 2.11 Serial correlation, seasonality, trend, and scale -- 2.11.1 Effect of serial correlation -- 2.12 Regression -- 2.12.1 Applications to water quality -- 2.12.2 Nonparametric regression -- 2.13 Estimating model parameters -- 2.13.1 Point versus interval estimation -- 2.13.2 Interval estimates -- 2.13.3 Bias -- 2.13.4 Percentiles -- Problems -- Appendix: Conditional probabilities-The Monty Hall dilemma -- 3 Intervals -- 3.1 Confidence intervals -- 3.1.1 For means -- 3.1.2 For prediction -- 3.1.3 For percentiles.

3.2 Tolerance intervals -- 3.3 Credible intervals -- 3.4 Intervals literature -- Problems -- 4 Hypothesis testing -- 4.1 The classical approach -- 4.2 The main steps in the classical approach -- 4.2.1 What is meant by a decision ? -- 4.3 Commentary on the five steps -- 4.3.1 Step 1: State the hypothesis to be tested -- 4.3.2 Step 2: State the acceptable error risk -- 4.3.3 Step 3: Sampling -- 4.3.4 Step 4: Apply the decision rule -- 4.3.5 Step 5: Rejecting, and accepting (maybe!) -- 4.4 An "evidence-based" approach (for dichotomous data) -- 4.5 Power analysis -- 4.6 Multiple comparisons -- 4.7 Nonparametric tests -- 4.8 Randomization tests -- 4.9 Bayesian methods -- 4.10 Likelihood approach -- 4.11 Hypothesis testing literature -- 4.11.1 The point-null hypothesis testing debate -- 4.11.2 The p-value culture -- Problems -- 5 Detection -- 5.1 One-sided hypothesis tests -- 5.1.1 Defining detection probability -- 5.1.2 What burden of proof? -- 5.1.3 Decision rules and detection probabilities -- 5.1.4 Detection regions -- 5.1.5 Non-normality and unequal variances -- 5.2 Point-null hypothesis tests -- 5.2.1 Defining detection probability -- 5.2.2 Power depends on sample size -- 5.2.3 Decision rules and detection probabilities -- 5.2.4 Detection regions -- 5.3 Equivalence tests -- 5.3.1 Defining detection probability -- 5.3.2 Decision rules and detection probabilities -- 5.3.3 Equivalence tests versus point-null tests -- 5.3.4 Detection regions -- 5.3.5 Equivalence versus the Power Approach -- 5.3.6 A potted history of equivalence testing -- 5.4 p-value behavior under different hypotheses -- 5.5 What has been detected? -- 5.6 Tests and confidence intervals -- 5.6.1 A Bayesian alternative -- 5.7 Association or causation? -- Problems -- 6 Mathematics and calculation methods -- 6.1 Introduction.

6.2 Formulae and calculation methods for important distributions -- 6.2.1 Normal distribution -- 6.2.2 Lognormal distribution -- 6.2.3 Gamma distribution -- 6.2.4 Beta distribution -- 6.2.5 Binomial distribution -- 6.2.6 Poisson distribution -- 6.2.7 Negative binomial distribution -- 6.2.8 Hypergeometric distribution -- 6.2.9 Multinomial distribution -- 6.2.10 Student's t-distribution -- 6.2.11 Chi-square distribution -- 6.2.12 F-distribution -- Problems -- 6.3 Calculating confidence and tolerance limits on normal percentiles -- 6.3.1 Two-sided equi-tailed confidence limits on normal percentiles -- 6.3.2 One-sided confidence (tolerance) limits on normal percentiles -- 6.3.3 Two-sided tolerance limits on percentiles -- 6.3.4 Calculating noncentral t-probabilities -- Problems -- 6.4 Development of the detection formulae -- 6.4.1 One-sided test, unknown variance -- 6.4.2 One-sided test, known variance -- 6.4.3 Point-null hypothesis test, unknown variance -- 6.4.4 Point-null hypothesis test, known variance -- 6.4.5 Equivalence hypothesis, known variance -- 6.4.6 Inequivalence hypothesis, known variance -- 6.4.7 (In equivalence hypotheses, unknown variance -- 6.4.8 Detection regions for equivalence tests and the Power Approach -- 6.4.9 Calculating the incomplete noncentral t-probabilities -- Problems -- Part II Problems and Solutions -- 7 Formulating water quality standards -- 7.1 Setting the scene -- 7.1.1 Explicit and narrative statements -- 7.2 Key questions -- 7.3 Enforceability -- 7.4 Requirements for enforceability -- Problems -- 8 Percentile standards (and the Reverend Bayes) -- 8.1 Two forms of percentile standards -- 8.2 Calculating nonparametric sample percentiles -- 8.2.1 Comparing the alternatives -- 8.2.2 Median versus geometric mean -- 8.3 Percentiles versus maxima -- 8.4 Percentile compliance rules.

8.4.1 Developing rules using the Classical approach -- 8.4.2 Developing rules using the Bayesian approach -- 8.4.3 Comparing the Classical and Bayesian results -- 8.4.4 Conclusion -- Problems -- 9 Microbial water quality and human health -- 9.1 The key elements -- 9.1.1 Hazard identification -- 9.1.2 Exposure assessment -- 9.1.3 Dose-response -- 9.1.4 Risk profiling -- 9.2 Risk communication -- 9.3 Some useful mathematical details -- 9.3.1 Added-zeroes distributions for dichotomous data -- 9.3.2 Hockey stick distribution -- 9.3.3 Loglogistic distribution -- 9.3.4 Dose-response curves -- 9.3.5 Conditional dose-response relationships -- 9.3.6 What if the distribution of pathogens is not Poisson? -- 9.3.7 Comparing conditional and average-dose risks -- Problems -- 10 MPNs and microbiology -- 10.1 MPNs in standard tables -- 10.2 MPN "confidence intervals" -- 10.3 Sampling an MPN distribution -- 10.4 Calculating exact MPNs -- 10.4.1 Considering each set separately -- 10.4.2 Combining the sets -- Problems -- 11 Trends, impacts, concordance, detection limits -- 11.1 Detecting and analyzing trends -- 11.1.1 Trend slope estimator -- 11.1.2 Seasonal Kendall trend test -- 11.1.3 Two anomalies -- 11.1.4 Multi-site trends -- a Bayesian approach -- 11.2 Impacts and nominal data -- 11.3 Concordance assessment -- 11.3.1 Cohen's kappa for dichotomous variables -- 11.3.2 Continuous variables-Lin's concordance correlation coefficient -- 11.4 Detection limits -- 11.4.1 Type A censoring -- 11.4.2 Type B censoring -- 11.4.3 Handling "less than" data -- Problems -- Appendix -- 12 Answers to exercises -- References -- Author Index -- Topic Index -- Appendix A Statistical Tables -- A.1 Normal distribution -- A.1.1 Cumulative unit normal distribution -- A.1.2 Percentiles of the unit normal distribution -- A.2 t-distribution -- A.2.1 Cumulative Student's t-distribution.

A.2.2 Percentiles of Student's t-distribution -- A.3 Binomial distribution -- A.3.1 Cumulative binomial probabilities -- A.4 Normal confidence and tolerance intervals -- A.4.1 k factor for two-sided 95% confidence intervals on a normal percentile -- A.4.2 k factor for one-sided confidence (or tolerance) intervals on a normal percentile -- A.4.3 k factor for two-sided tolerance intervals on a normal percentile -- List of Figures -- Accurate observations are both precise and unbiased -- Bar graph of MPN data -- Cumulative distribution function of MPN data -- Symmetry and skewness -- Lowess fit through data for the Ngakaroa Stream -- Boxplot for somatic coliphage in recreational waters -- Probability density function (pdf) for the unit normal distribution -- Cumulative distribution function (CDF) for the unit normal distribution -- Normal and lognormal probability density functions -- A variety of shapes for the pdf of the two-parameter gamma distribution -- A variety of shapes for the pdf of the two-parameter beta distribution -- Probability mass functions for three common discrete distributions -- Student t-distributions and the unit normal distribution -- Probability density functions for the x2- and F-distributions -- Five possibilities for reporting areas under the t-distribution -- Possible linear correlations -- Field Raynes effluent suspended solids data and serial correlation structure -- Geometric mean confidence limits as a function of sample size -- "Error bars'' -- Prediction intervals and confidence intervals for linear regression -- Confidence limits for the 95%ile as a function of sample size -- Highest density and equi-tails regions -- Critical region for a one-sided test using the unit normal distribution -- Critical region for a two-sided test using the unit normal distribution -- Binomial distribution for n = 20 and t = 0.5.

Negative binomial distribution for k = 5 and t = 0.5.