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

by A. J. Underwood

• ISBN13: 9780521556965
• ISBN10: 0521556961
• Format: Paperback
• Copyright: 1997-01-28
• Publisher: Cambridge University Press

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.

Table of Contents

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