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Contents Preface ix List of figures xi List of tables xiii 1 Ashort history of statistics in the law 1 1.1 History 1 1.2 Some recent uses of statistics in forensic science 3 1.3 What is probability? 4 2 Data types, location and dispersion 7 2.1 Types of data 7 2.2 Populations and samples 9 2.3 Distributions 9 2.4 Location 11 2.5 Dispersion 13 2.6 Hierarchies of variation 14 3 Probability 17 3.1 Aleatory probability 17 One throw of a six-sided die 17 A single throw with more than one outcome of interest 18 Two six-sided dice 19 3.2 Binomial probability 21 3.3 Poisson probability 24 3.4 Empirical probability 25 Modelled empirical probabilities 25 Truly empirical probabilities 27 4 The normal distribution 29 4.1 The normal distribution 29 4.2 Standard deviation and standard error of the mean 30 4.3 Percentage points of the normal distribution 32 4.4 The t-distribution and the standard error of the mean 34 4.5 t-testing between two independent samples 36 4.6 Testing between paired observations 40 4.7 Confidence, significance and p-values 42 5 Measures of nominal and ordinal association 45 5.1 Association between discrete variables 45 5.2 ¿2 test for a 2 x 2 table 46 5.3 Yules Q 48 5.4 ¿2 tests for greater than 2 x 2 tables 49 5.5 "2 and Cramers V2 50 5.6 The limitations of ¿2 testing 51 5.7 Interpretation and conclusions 52 6 Correlation 55 6.1 Significance tests for correlation coefficients 59 6.2 Correlation coefficients for non-linear data 60 6.3 The coefficient of determination 63 6.4 Partial correlation 63 6.5 Partial correlation controlling for two or more covariates 69 7 Regression and calibration 75 7.1 Linear models 75 7.2 Calculation of a linear regression model 78 7.3 Testing 'goodness of fit' 80 7.4 Testing coefficients a and b 81 7.5 Residuals 83 7.6 Calibration 85 A linear calibration model 86 Calculation of a confidence interval for a point 89 7.7 Points to remember 91 8 Evidence evaluation 95 8.1 Verbal statements of evidential value 95 8.2 Evidence types 96 8.3 The value of evidence 97 8.4 Significance testing and evidence evaluation 102 9 Conditional probability and Bayes' theorem 105 9.1 Conditional probability 105 9.2 Bayes' theorem 108 9.3 The value of evidence 112 10 Relevance and the formulation of propositions 117 10.1 Relevance 117 10.2 Hierarchy of propositions 118 10.3 Likelihood ratios and relevance 120 10.4 The logic of relevance 122 10.5 The formulation of propositions 123 10.6 What kind of propositions can we not evaluate? 124 11 Evaluation of evidence in practice 129 11.1 Which database to use 129 Type and geographic factors 129 DNA and database selection 131 11.2 Verbal equivalence of the likelihood ratio 133 11.3 Some common criticisms of statistical approaches 136 12 Evidence evaluation examples 139 12.1 Blood group frequencies 139 12.2 Trouser fibres 141 12.3 Shoe types 144 12.4 Airweapon projectiles 148 12.5 Height description from eyewitness 150 13 Errors in interpretation 153 13.1 Statistically based errors of interpretation 153 Transposed conditional 153 Defender's fallacy 155 Another match error 155 Numerical conversion error 156 13.2 Methodological errors of interpretation 157 Different level error 157 Defendant's database fallacy 158 Independence assumption 158 14 DNA I 161 14.1 Loci and alleles 161 14.2 Simple case genotypic frequencies 162 14.3 Hardy-Weinberg equilibrium 164 14.4 Simple case allelic frequencies 166 14.5 Accounting for sub-populations 168 15 DNA II 171 15.1 Paternity - mother and father unrelated 171 15.2 Database searches and value of evidence 174 15.3 Discussion 176 16 Sampling and sample size estimation 179 16.1 Estimation of a mean 179 16.2 Sample sizes for t-tests 181 Two sample t-test 181 One sample t-test 183 16.3 How many drugs to sample 184 16.4 Concluding comments 188 17 Epilogue 191 17.1 Graphical models and Bayesian Networks 192 Graphical models 192 Bayesian networks 194 17.2 Kernel density estimation 195 17.3 Multivariate continuous matching 196 Appendices A Worked solutions to questions 199 B Percentage points of the standard normal distribution 225 C Percentage points of t-distributions 227 D Percentage points of ¿2-distributions 229 E Percentage points of beta-beta distributions 231 F Percentage points of F-distributions 233 G Calculating partial correlations using Excel software 235 References 239 Index 245
Library of Congress Subject Headings for this publication:
Forensic sciences -- Statistical methods.
Forensic statistics.
Evidence (Law) -- Statistical methods.