Experiments in Ecology : Their Logical Design and Interpretation Using Analysis of Variance

Acknowledgements xvii Introduction 1(6) A framework for investigating biological patterns and processes 7(17) Introduction 7(1) Observations 8(2) Models, theories, explanations 10(2) Models of physiological stress 10(1) Models based on competition 10(1) Grazing models 10(1) Models to do with hazards 11(1) Models of failure of recruitment 11(1) Numerous competing models 12(1) Hypotheses, predictions 13(2) Null hypotheses 15(1) Experiments and their interpretation 16(1) What to do next? 17(2) Measurements, gathering data and a logical structure 19(2) A consideration: why are you measuring things? 21(1) Conclusion: a plea for more thought 22(2) Populations, frequency distributions and samples 24(26) Introduction 24(1) Variability in measurements 24(1) Observations and measurements as frequency distributions 25(2) Defining the population to be observed 27(3) The need for samples 30(1) The location parameter 30(3) Sample estimate of the location parameter 33(1) The dispersion parameter 34(2) Sample estimate of the dispersion parameter 36(1) Degrees of freedom 37(1) Representative sampling and accuracy of samples 38(6) Other useful parameters 44(6) Skewness 44(3) Kurtosis 47(3) Statistical tests of null hypotheses 50(15) Why a statistical test? 50(1) An example using coins 51(4) The components of a statistical test 55(2) Null hypothesis 55(1) Test statistic 56(1) Region of rejection and critical value 56(1) Type I error or rejection of a true null hypothesis 57(1) Statistical test of a theoretical biological example 58(4) Transformation of a normal distribution to the standard normal distribution 59(3) One- and two-tailed null hypotheses 62(3) Statistical tests on samples 65(35) Repeated sampling 65(5) The standard error from the normal distribution of sample means 70(1) Confidence intervals for a sampled mean 70(3) Precision of a sample estimate of the mean 73(1) A contrived example of use of the confidence interval of sampled means 74(2) Student's t-distribution 76(1) Increasing precision of sampling 77(4) The chosen probability used to construct the confidence interval 78(1) The sample size (n) 78(2) The variance of the population (σ2) 80(1) Description of sampling 81(1) Student's t-test for a mensurative hypothesis 82(2) Goodness-of-fit, mensurative experiments and logic 84(3) Type I and Type II errors in relation to a null hypothesis 87(4) Determining the power of a simple statistical test 91(6) Probability of Type I error 92(1) Size of experiment (n) 93(2) Variance of the population 95(2) `Effect size' 97(1) Power and alternative hypotheses 97(3) Simple experiments comparing the means of two populations 100(40) Paired comparisons 100(4) Confounding and lack of controls 104(2) Unpaired experiments 106(1) Standard error of the difference between two means 107(7) Independence of samples 108(1) Homogeneity of variances 109(5) Allocation of sample units to treatments 114(4) Interpretation of a simple ecological experiment 118(6) Power of an experimental comparison of two populations 124(4) Alternative procedures 128(4) Binomial (sign) test for paired data 128(2) Other alternative procedures 130(2) Are experimental comparisons of only two populations useful? 132(8) The wrong population is being sampled 132(5) Modifications to the t-test to compare more than two populations 137(2) Conclusion 139(1) Analysis of variance 140(58) Introduction 140(1) Data collected to test a single-factor null hypothesis 141(2) Partitioning of the data: the analysis of variation 143(2) A linear model 145(4) What do the sums of squares measure? 149(3) Degrees of freedom 152(1) Mean squares and test statistic 153(1) Solution to some problems raised earlier 154(1) So what happens with real data? 155(1) Unbalanced data 156(1) Machine formulae 157(1) Interpretation of the result 157(1) Assumptions of analysis of variance 158(1) Independence of data 159(20) Positive correlation within samples 160(6) Negative correlation within samples 166(2) Negative correlation among samples 168(4) Positive correlation among samples 172(7) Dealing with non-independence 179(2) Heterogeneity of variances 181(3) Tests for heterogeneity of variances 183(1) Quality control 184(3) Transformations of data 187(7) Square-root transformation of counts (or Poisson data) 188(1) Log transformation for rates, ratios, concentrations and other data 189(3) Arc-sin transformation of percentages and proportions 192(1) No transformation is possible 192(2) Normality of data 194(1) The summation assumption 195(3) More analysis of variance 198(45) Fixed or random factors 198(6) Interpretation of fixed or random factors 204(5) Power of an analysis of a fixed factor 209(7) Non-central F-ratio and power 209(2) Influences of α, n, σ2e and Ai values 211(3) Construction of an alternative hypothesis 214(2) Power of an analysis of a random factor 216(7) Central F-ratios and power 216(2) Influences of α, n, σ2e, σ2A and a 218(2) Construction of an alternative hypothesis 220(3) Alternative analysis of ranked data 223(1) Multiple comparisons to identify the alternative hypothesis 224(19) Introduction 224(1) Problems of excessive Type I error 225(1) A priori versus a posteriori comparisons 226(1) A priori procedures 227(7) A posteriori comparisons 234(9) Nested analyses of variance 243(53) Introduction and need 243(2) Hurlbert's `pseudoreplication' 245(1) Partitioning of the data 245(5) The linear model 250(4) Degrees of freedom and mean squares 254(5) Tests and interpretation: what do the nested bits mean? 259(9) F-ratio of appropriate mean squares 259(1) Solution to confounding 260(1) Multiple comparisons 261(1) Variability among replicated units 261(7) Pooling of nested components 268(5) Rationale and procedure 268(1) Pooling, Type II and Type I errors 269(4) Balanced sampling 273(2) Nested analyses and spatial pattern 275(4) Nested analysis and temporal pattern 279(4) Cost-benefit optimization 283(6) Calculation of power 289(2) Residual variance and an `error' term 291(5) Factorial experiments 296(62) Introduction 296(4) Partitioning of variation when there are two experimental factors 300(5) Appropriate null hypotheses for a two-factor experiment 305(1) A linear model and estimation of components by mean squares 306(6) Why do a factorial experiment? 312(6) Information about interactions 313(3) Efficiency and cost-effectiveness of factorial designs 316(2) Meaning and interpretation of interactions 318(5) Interactions of fixed and random factors 323(8) Multiple comparisons for two factors 331(4) When there is a significant interaction 331(1) When there is no significant interaction 331(2) Control of experiment-wise probability of Type I error 333(2) Three or more factors 335(1) Interpretation of interactions among three factors 335(5) Power and detection of interactions 340(2) Spatial replication of ecological experiments 342(2) What to do with a mixed model 344(2) Problems with power in a mixed analysis 346(1) Magnitudes of effects of treatments 347(8) Magnitudes of effects of fixed treatments 348(1) Some problems with such measures 348(3) Magnitudes of components of variance of random treatments 351(4) Problems with estimates of effects 355(3) Summation and interactions 355(1) Comparisons among experiments or areas 356(1) Conclusions on magnitudes of effects 357(1) Construction of any analysis from general principles 358(27) General procedures 358(3) Constructing the linear model 361(1) Calculating the degrees of freedom 362(2) Mean square estimates and F-ratios 364(6) Designs seen before 370(5) Designs with two factors 370(1) Designs with three factors 370(5) Construction of sums of squares using orthogonal designs 375(1) Post hoc pooling 375(2) Quasi F-ratios 377(1) Multiple comparisons 378(2) Missing data and other practicalities 380(5) Loss of individual replicates 382(1) Missing sets of replicates 383(2) Some common and some particular experimental designs 385(34) Unreplicated randomized blocks design 385(4) Tukey's test for non-additivity 389(2) Split-plot designs 391(10) Latin squares 401(2) Unreplicated repeated measures 403(5) Asymmetrical controls: one factor 408(1) Asymmetrical controls: fixed factorial designs 409(5) Problems with experiments on ecological competition 414(1) Asymmetrical analyses of random factors in environmental studies 415(4) Analyses involving relationships among variables 419(59) Introduction to linear regression 419(3) Tests of null hypotheses about regressions 422(2) Assumptions underlying regression 424(7) Independence of data at each X 425(2) Homogeneity of variances at each X 427(1) X values are not fixed 428(1) Normality of errors in Y 429(2) Analysis of variance and regression 431(1) How good is the regression? 431(3) Multiple regressions 434(5) Polynomial regressions 439(5) Other, non-linear regressions 444(1) Introduction to analysis of covariance 444(3) The underlying models for covariance 447(10) Regression in each treatment 448(1) A common regression in each treatment 449(5) The total regression, all data combined 454(3) The procedures: making adjustments 457(5) Interpretation of the analysis 462(2) The assumptions needed for an analysis of covariance 464(7) Assumptions in regressions 464(1) Assumptions in analysis of variance 465(1) Assumptions specific to an analysis of covariance 466(5) Alternatives when regressions differ 471(3) A two-factor scenario 471(2) The Johnson-Neyman technique 473(1) Comparisons of regressions 474(1) Extensions of analysis of covariance to other designs 474(4) More than one covariate 475(1) Non-linear relationships 476(1) More than one experimental factor 476(2) Conclusions: where to from here? 478(8) Be logical, be eco-logical 478(2) Alternative models and hypotheses 480(1) Pilot experiments: all experiments are preliminary 481(1) Repeated experimentation 481(3) Criticisms and the growth of knowledge 484(2) References 486(10) Author index 496(3) Subject index 499