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

by A. J. Underwood

ISBN13: 9780521553292

ISBN10: 0521553296

Format: Hardcover

Pub. Date: 1997-01-28

Publisher(s): Cambridge University Press

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Summary

Ecological theories and hypotheses are usually complex because of natural variability in space and time, which often makes the design of experiments difficult. This book describes how to design ecological experiments from a statistical basis using analysis of variance, so reliable conclusions can be drawn. The logical procedures that lead to a need for experiments are described, followed by an introduction to simple statistical tests. This leads to a detailed account of analysis of variance, looking at procedures, assumptions, and problems. One-factor analysis is extended to nested (hierarchical) designs and factorial analysis. Finally, some regression methods for examining relationships between variables are covered. Examples of ecological experiments are used throughout to illustrate the procedures and examine problems. This book will be invaluable to practicing ecologists as well as advanced students involved in experimental design.

Acknowledgements

xvii

Introduction

(6)

A framework for investigating biological patterns and processes

(17)

Introduction

(1)

Observations

(2)

Models, theories, explanations

(2)

Models of physiological stress

(1)

Models based on competition

(1)

Grazing models

(1)

Models to do with hazards

(1)

Models of failure of recruitment

(1)

Numerous competing models

(1)

Hypotheses, predictions

(2)

Null hypotheses

(1)

Experiments and their interpretation

(1)

What to do next?

(2)

Measurements, gathering data and a logical structure

(2)

A consideration: why are you measuring things?

(1)

Conclusion: a plea for more thought

(2)

Populations, frequency distributions and samples

(26)

Introduction

(1)

Variability in measurements

(1)

Observations and measurements as frequency distributions

(2)

Defining the population to be observed

(3)

The need for samples

(1)

The location parameter

(3)

Sample estimate of the location parameter

(1)

The dispersion parameter

(2)

Sample estimate of the dispersion parameter

(1)

Degrees of freedom

(1)

Representative sampling and accuracy of samples

(6)

Other useful parameters

(6)

Skewness

(3)

Kurtosis

(3)

Statistical tests of null hypotheses

(15)

Why a statistical test?

(1)

An example using coins

(4)

The components of a statistical test

(2)

Null hypothesis

(1)

Test statistic

(1)

Region of rejection and critical value

(1)

Type I error or rejection of a true null hypothesis

(1)

Statistical test of a theoretical biological example

(4)

Transformation of a normal distribution to the standard normal distribution

(3)

One- and two-tailed null hypotheses

(3)

Statistical tests on samples

(35)

Repeated sampling

(5)

The standard error from the normal distribution of sample means

(1)

Confidence intervals for a sampled mean

(3)

Precision of a sample estimate of the mean

(1)

A contrived example of use of the confidence interval of sampled means

(2)

Student's t-distribution

(1)

Increasing precision of sampling

(4)

The chosen probability used to construct the confidence interval

(1)

The sample size (n)

(2)

The variance of the population (σ2)

(1)

Description of sampling

(1)

Student's t-test for a mensurative hypothesis

(2)

Goodness-of-fit, mensurative experiments and logic

(3)

Type I and Type II errors in relation to a null hypothesis

(4)

Determining the power of a simple statistical test

(6)

Probability of Type I error

(1)

Size of experiment (n)

(2)

Variance of the population

(2)

`Effect size'

(1)

Power and alternative hypotheses

(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

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

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