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9780521553292

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

by
  • ISBN13:

    9780521553292

  • ISBN10:

    0521553296

  • Format: Hardcover
  • Copyright: 1997-01-28
  • Publisher: 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.

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

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