Biostatistical Analysis

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Jerrold H. Zar Prentice Hall 2010
1 Data: Types and Presentation 1.1 Types of Biological Data . . . . . . . 1.2 Accuracy and Significant Figures . . 1.3 Frequency Distributions . . . . . . . 1.4 Cumulative Frequency Distributions 2 Populations and Samples 2.1 Populations.......... 2.2 Samples from Populations . . 2.3 Random Sampling . . . . . . . . 2.4 Parameters and Statistics. . . . . 2.5 Outliers........... 3 Measures of Central Tendency 3.1 The Arithmetic Mean ..... . 3.2 The Median . . . . . . . . . . . . . . 3.3 The Mode . . . . . . . . . . . . . . .

1 Data: Types and Presentation 1.1 Types of Biological Data . . . . . . . 1.2 Accuracy and Significant Figures . . 1.3 Frequency Distributions . . . . . . . 1.4 Cumulative Frequency Distributions 2 Populations and Samples 2.1 Populations.......... 2.2 Samples from Populations . . 2.3 Random Sampling . . . . . . . . 2.4 Parameters and Statistics. . . . . 2.5 Outliers........... 3 Measures of Central Tendency 3.1 The Arithmetic Mean ..... . 3.2 The Median . . . . . . . . . . . . . . 3.3 The Mode . . . . . . . . . . . . . . . 3.4 Other Measures of Central Tendency . 3.5 Coding Data . . . . . . . . . . . . . . . . . . . . . . . 4 Measures of Variability and Dispersion 4.1 The Range . . . . . . . . . . . . . 4.2 Dispersion Measured with Quantiles 4.3 The Mean Deviation . . . . . . . . . 4.4 The Variance . . . . . . . . . . . . . 4.5 The Standard Deviation . . . . . . . 4.6 The Coefficient of Variation .... . 4.7 Indices of Diversity. . ... . 4.8 Coding Data . . . . . . . . . . . . . . 5 Probabilities 5.1 Counting Possible Outcomes 5.2 Permutations......... 5.3 Combinations....... 5.4 Sets ................. . 5.5 Probability of an Event ...... . 5.6 Adding Probabilities . . . . . . . . 5.7 Multiplying Probabilities ..... 5.8 Conditional Probabilities. . . . . . 6 The Normal Distribution 6.1 Proportions of a Normal Distribution 6.2 The Distribution of Means ....... . xi xiii 1 2 5 6 14 16 16 16 17 18 19 21 21 24 27 28 30 33 33 35 37 37 41 42 42 46 49 49 51 55 57 59 60 63 63 66 68 72iv Contents 6.3 Introduction to Statistical Hypothesis Testing. . . . . . . . . . . .. 74 6.4 Confidence Limits ............................ 85 6.5 Symmetry and Kurtosis ......................... 87 6.6 Assessing Departures from Normality . . . . . . . . . . . . . . . .. 91 7 One-Sample Hypotheses 97 7.1 Two-Tailed Hypotheses Concerning the Mean ............ 97 7.2 One-Tailed Hypotheses Concerning the Mean ............ 103 7.3 Confidence Limits for the Population Mean ............. , 105 7.4 Reporting Variability Around the Mean . . . . . . . . . . . . . . .. 108 7.5 Reporting Variability Around the Median. . . . . . . . . . . . 112 7.6 Sample Size and Estimation of the Population Mean . . . . . . 114 7.7 Sample Size, Detectable Difference, and Power in Tests Concerning the Mean. . . . . . .. . . . . . 115 7.8 Sampling Finite Populations. . . . . . . . . . . 118 7.9 Hypotheses Concerning the Median .... . . 120 7.10 Confidence Limits for the Population Median . 120 7.11 Hypotheses Concerning the Variance ................. 121 7.12 Confidence Limits for the Population Variance . . . . . . . . . . .. 122 7.13 Power and Sample Size in Tests Concerning the Variance . . . 124 7.14 Hypotheses Concerning the Coefficient of Variation. . . . . . . .. 125 7.15 Confidence Limits for the Population Coefficient of Variation ... 126 7.16 Hypotheses Concerning Symmetry and Kurtosis . . . . . . . . . .. 126 8 Two-Sample Hypotheses 130 8.1 Testing for Difference Between Two Means. . . . . . . . . 130 8.2 Confidence Limits for Population Means. . . . . . . . . . . 142 8.3 Sample Size and Estimation of the Difference Between Two Population Means. . . . . . . . . . . . . . . . . . . . . 146 8.4 Sample Size, Detectable Difference, and Power in Tests for Difference between Two Means. . . . . . . . . . . . 147 8.5 Testing for Difference Between Two Variances. . . . . 151 8.6 Confidence Limits for Population Variances and the Population Variance Ratio. . . . . . . . . . . . . . . . . . . . . .. 157 8.7 Sample Size and Power in Tests for Difference Between Two Variances ............................. 158 8.8 Testing fOT Difference Between Two Coefficients of Variation ... 159 8.9 Confidence Limits for the Difference Between Two Coefficients of Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 162 8.10 Nonparametric Statistical Methods . . . . . . . . . . . . . . 162 8.11 Two-Sample Rank Testing. . . . . . . . . . . . . . . . . . . 163 8.12 Testing for Difference Between Two Medians. . . . . . . 172 8.13 Two-Sample Testing of Nominal-Scale Data. . . . . . . . 174 8.14 Testing for Difference Between Two Diversity Indices. . 174 8.15 Coding Data . . . . . . . . . . . . . . . . . . . . . . . . . 176 9 Paired-Sample Hypotheses 9.1 Testing Mean Difference Between Paired Samples ... 9.2 Confidence Limits for the Population Mean Difference 9.3 Power, Detectable Difference and Sample Size in Paired-Sample Testing of Means ............. . 179 179 182 182Contents v 9.4 Testing for Difference Between Variances from Two Correlated Populations. . . . . . . . . . . . . . . . . . 9.5 Paired-Sample Testing by Ranks . . . . . . . . . . . . 9.6 Confidence Limits for the Population Median Difference 10 Multisample Hypotheses and the Analysis of Variance 10.1 Single-Factor Analysis of Variance ............. . 10.2 Confidence Limits for Population Means. . . . . . . . . . . 10.3 Sample Size. Detectable Difference, and Power in Analysis of Variance .......................... . 10.4 Nonparametric Analysis of Variance .......... . 10.5 Testing for Difference Among Several Medians. . . . . . . . . 10.6 Homogeneity of Variances ..................... . 10.7 Homogeneity of Coefficients of Variation ............ . 10.8 Coding Data . . . . . . . . . . . . . . . . . . . . . . . 10.9 Multisample Testing for Nominal-Scale Data .......... . 11 Multiple Comparisons 11.1 Testing All Pairs of Means . . . . . . . . . . . . . . . . . . . . . 11.2 Confidence Intervals for Multiple Comparisons ........ . 11.3 Testing a Control Mean Against Each Other Mean . . . . . . . 11.4 Multiple Contrasts . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Nonparametric Multiple Comparisons ................ . 11.6 Nonparametric Multiple Contrasts . . . . . . . . . . . . . 11.7 Multiple Comparisons Among Medians .. 11.8 Multiple Comparisons Among Variances 12 Two-Factor Analysis of Variance 12.1 Two-Factor Analysis of Variance with Equal Replication ..... 12.2 Two-Factor Analysis of Variance with Unequal Replication .. 12.3 Two-Factor Analysis of Variance Without Replication. 12.4 Two-Factor Analysis of Variance with Randomized Blocks or Repeated Measures . . . . . . . . . . . . . . . 12.5 Multiple Comparisons and Confidence Intervals in Two-Factor Analysis of Variance ................ . 12.6 Sample Size, Detectable Difference, and Power in Two-Factor Analysis of Variance .......... . 12.7 Nonparametric Randomized-Block or Repeated-Measures Analysis of Variance ...................... . 12.8 Dichotomous Nominal-Scale Data in Randomized Blocks or from Repeated Measures . . . . . . . . . . . . . . . . . . . 12.9 Multiple Comparisons with Dichotomous Randomized-Block or Repeated-Measures Data. . . . . . . . .. . ..... . 12.10 Introduction to Analysis of Covariance ............. . 13 Data Transformations 13.1 The Logarithmic Transformation ................ . 13.2 The Square-Root Transformation ................ . 13.3 The Arcsine Transformation ............ . 13.4 Other Transformations . . . . . . . . . . . . . . . . . . . . . . . 182 183 188 189 190 206 207 214 219 220 221 224 224 226 227 232 234 237 239 243 244 244 249 250 265 267 270 274 275 277 281 283 284 286 287 291 291 295vi Contents 14 Multiway Factorial Analysis of Variance 296 14.1 Three-Factor Analysis of Variance . . . . .. .... . . . . . . . 296 14.2 The Latin-Square Experimental Design . .. .. . . . . . . . . . 299 14.3 Higher-Order Factorial Analysis of Variance . . . . . . . . . . . 303 14.4 Multiway Analysis of Variance with Blocks or Repeated Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 14.5 Factorial Analysis of Variance with Unequal Replication . . . . .. 304 14.6 Multiple Comparisons and Confidence Intervals in Multiway Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . 305 14.7 Power. Detectable Difference, and Sample Size in Multiway Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . 305 15 Nested (Hierarchical) Analysis of Variance 307 15.1 Nesting Within One Factor ............ ........ 307 15.2 Nesting in Factorial Experimental Designs. . .. ........ 313 15.3 Multiple Comparisons and Confidence Intervals ........ 314 15.4 Power. Detectable Difference. and Sample Size in Nested Analysis of Variance ................................ 315 16 Multivariate Analysis of Variance 316 16.1 The Multivariate Normal Distribution . . . . . . . . . . . . . . . . . 316 16.2 Multivariate Analysis of Variance Hypothesis Testing . . . . . . .. 319 16.3 Further Analysis ........... . . . . . . .. 326 16.4 Other Experimental Designs ...................... 326 17 Simple Linear Regression 328 17.1 Regression versus Correlation. . . . . . . . . . . . . . . . . . . 32B 17.2 The Simple Linear Regression Equation . . 330 17.3 Testing the Significance of a Regression . . 337 17.4 Interpretations of Regression Functions . . 341 17.5 Confidence Intervals in Regression. . . . . 342 17.6 Inverse Prediction .............. . . . . . . . 347 17.7 Regression with Replication and Testing for Linearity. . . . . .. 349 17.8 Power and Sample Size in Regression ................ 355 17.9 Regression Through the Origin . . . . . . . . . . . . . . . . . . .. 355 17.10 Data Transformations in Regression . . . . . . . . . . . . . . . .. 357 17.11 The Effect of Coding Data. . . . . . . .. .............. 361 18 Comparing Simple Linear Regression Equations IB.l Comparing Two Slopes .......... . 18.2 Comparing Two Elevations . . . . . . . . IB.3 Comparing Points on Two Regression Lines. 18.4 Comparing More Than Two Slopes ... . 18.5 Comparing More Than Two Elevations .... . IB.6 Multiple Comparisons Among Slopes ..... . IB.7 Multiple Comparisons Among Elevations ... . 18.8 MUltiple Comparisons of Points Among Regression Lines. 18.9 An Overall Test for Coincidental Regressions ........... . 19 Simple Linear Correlation 19.1 The Correlation Coefficient .......... . 19.2 Hypotheses About the Correlation Coefficient 363 363 367 371 372 375 375 376 377 378 379 379 383Contents vii 19.3 Confidence Intervals for the Population Correlation Coefficient .. 386 19.4 Power and Sample Size in Correlation . . . . . . . . . . . . . . . .. 386 19.5 Comparing Two Correlation Coefficients ............... 390 19.6 Power and Sample Size in Comparing Two Correlation Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 19.7 Comparing More Than Two Correlation Coefficients ........ 393 19.8 Multiple Comparisons Among Correlation Coefficients ....... 396 19.9 Rank Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 398 19.10 Weighted Rank Correlation ....................... 402 19.11 Correlation with Nominal-Scale Data ................. 405 19.12 Intraclass Correlation ...... . . . . . . . . . . . . .. 411 19.13 Concordance Correlation .... . . . . . . . . . . . . . . . 414 19.14 The Effect of Coding . . . . . . . . . . . . . . . . . . . .. 417 20 Multiple Regression and Correlation 419 20.1 Intermediate Computational Steps . . . . . . . . . . . . . . . . . .. 420 20.2 The Multiple-Regression Equation . . . . . . . . . . . . . . . . . . . 423 20.3 Analysis of Variance of Multiple Regression or Correlation . . . . . 426 20.4 Hypotheses Concerning Partial Regression Coefficients . . . . . . . 430 20.5 Standardized Partial Regression Coefficients .... . . . . . . . . . 433 20.6 Selecting Independent Variables ........ ; .---.--:. . . . . . . . 433 20.7 Partial Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 20.8 Predicting Y Values .................... ...... 440 20.9 Testing Difference Between Two Partial Regression Coefficients . . . . . . . . . . . . . . . . . . . . . . 442 20.10 "Dummy" Variables . . . . . . . . . . . . . . . . . .. 443 20.11 Interaction of Independent Variables. . . . . . . . . . . .. 444 20.12 Comparing Multiple Regression Equations .............. 444 20.13 Multiple Regression Through The Origin ............... 446 20.14 Nonlinear Regression .......................... 447 20.15 Descriptive Versus Predictive Models .. . . . . . . . . . . 448 20.16 Concordance: Rank Correlation Among Several Variables . . . . . 449 21 Polynomial Regression 21.1 Polynomial Curve Fitting 21.2 Quadratic Regression .. 22 Testing for Goodness of Fit 22.1 Chi-Square Goodness of Fit for Two Categories 22.2 Chi-Square Correction for Continuity .. . . . .. . ....... . 22.3 Chi-Square Goodness of Fit for More Than Two Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.4 Subdividing Chi-Square Goodness of Fit ............... . 22.5 Chi-Square Goodness of Fit with Small Frequencies ........ . 22.6 Heterogeneity Chi-Square Testing for Goodness of Fit ....... . 22.7 The Log-Likelihood Ratio for Goodness of Fit ........... . 22.8 Kolmogorov-Smirnov Goodness of Fit . . . . . . . . . . . . . . . . . 23 Contingency Tables 23.1 Chi-Square Analysis of Contingency Tables . 23.2 Visualizing Contingency-Table Data ..... 458 458 463 466 467 469 470 472 473 474 478 481 490 492 494viii Contents 23.3 23.4 23.5 23.6 23.7 23.8 2 x 2 Contingency Tables ............. . Contingency Tables with Small Frequencies ... . Heterogeneity Testing of 2 x 2 Tables ...... . Subdividing Contingency Tables ......... . The Log-Likelihood Ratio for Contingency Tables Multidimensional Contingency Tables ...... . 24 Dichotomous Variables 24.1 Binomial Probabilities . . . . . . . . . . . . . . 24.2 The Hypergeometric Distribution. . . . . . . . 24.3 Sampling a Binomial Population ....... . 24.4 Goodness of Fit for the Binomial Distribution. 24.5 The Binomial Test and One-Sample Test of a Proportion ..... 24.6 The Sign Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.7 Power, Detectable Difference, and Sample Size for the Binomial and Sign Tests . . . . . . . . . . . . . . . . . . . . . . 24.8 Confidence Limits for a Population Proportion . . . . . . . 24.9 Confidence Interval for a Population Median ....... . 24.10 Testing for Difference Between Two Proportio~y.......... . 24.11 Confidence Limits for the Difference BetweeI1Proportions .... . 24.12 Power, Detectable Difference, and Sample Size in Testing Difference Between Two Proportions . 24.13 Comparing More Than Two Proportions ... . 24.14 Multiple Comparisons for Proportions .... . 24.15 Trends Among Proportions . . . . . . . . . . . 24.16 The Fisher Exact Test ............. . 24.17 Paired-Sample Testing of Nominal-Scale Data 24.18 Logistic Regression . . . . . . . . . . . . . . . . 25 Testing for Randomness 25.1 Poisson Probabilities . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Confidence Limits for the Poisson Parameter . . . . . . . . . 25.3 Goodness of Fit for the Poisson Distribution . . . . . . . . . . 25.4 The Poisson Distribution for the Binomial Test. . ........ . 25.5 Comparing Two Poisson Counts ................... . 25.6 Serial Randomness of Nominal-Scale Categories . . . . . . . . 25.7 Serial Randomness of Measurements: Parametric Testing ... 25.8 Serial Randomness of Measurements: Nonparametric Testing 26 Circular Distributions: Descriptive Statistics 26.1 Data on a Circular Scale ........... . 26.2 Graphical Presentation of Circular Data . . 26.3 Trigonometric Functions . . . . . . . . . . . 26.4 The Mean Angle ...... . . . . . . . . . 26.5 Angular Dispersion . . . . . . . . . . . . . . 26.6 The Median and Modal Angles . . . . . . . 26.7 Confidence Limits for the Population Mean and Median Angles .. . . . . . . ...... . 26.8 Axial Data . . . . . . . . . . . . . . . . . . 26.9 The Mean of Mean Angles. . ...... . 497 503 504 506 508 510 518 519 524 526 529 532 537 539 543 548 549 551 552 555 557 559 561 569 577 585 585 587 589 592 595 597 599 601 60S 605 607 610 612 615 617 618 619 621Contents ix 27 Circular Distributions: Hypothesis Testing 624 27.1 Testing Significance of the Mean Angle ........... 624 27.2 Testing Significance of the Median Angle . . . . . . . . . . 629 27.3 Testing Symmetry Around the Median Angle . . . . . . . . 631 27.4 Two-Sample and Multisample Testing of Mean Angles ....... 632 27.5 Nonparametric Two-Sample and Multisample Testing of Angles 637 27.6 Two-Sample and Multisample Testing of Median Angles ... 642 27.7 Two-Sample and Multisample Testing of Angular Distances. 642 27.8 Two-Sample and MuItisample Testing of Angular Dispersion. 644 27.9 Parametric Analysis of the Mean of Mean Angles ....... 645 27.10 Nonparametric Analysis of the Mean of Mean Angles . . . . . . .. 646 27.11 Parametric Two-Sample Analysis of the Mean of Mean Angles. .. 647 27.12 Nonparamctric Two-Sample Analysis of the Mean of Mean Angles 649 27.13 Parametric Paired-Sample Testing with Angles . . . . . 652 27.14 Nonparametric Paired-Sample Testing with Angles. . . . . . . 654 27.15 Parametric Angular Correlation and Regression . 654 27.16 Nonparametric Angular Correlation . . . . . . . . . . . . . . . 660 27.17 Goodness-of-Fit Testing for Circular Distributions . . . . . . . 662 27.18 Serial Randomness of Nominal-Scale Categories on a etfCle. 665

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