Modern Statistics for the Life Sciences

by Grafen, Alan; Hails, Rosie

• ISBN13: 9780199252312
• ISBN10: 0199252319
• Format: Paperback
• Copyright: 2002-05-09
• Publisher: Oxford University Press

Summary

Model formulae represent a powerful methodology for describing, discussing, understanding, and performing the component of statistical tests known as linear statistics. It was developed for professional statisticians in the 1960s and has become increasingly available as the use of computers has grown and software has advanced. Modern Statistics for Life Scientists puts this methodology firmly within the grasp of undergraduates for the first time. The authors assume a basic knowledge of statistics--up to and including one and two sample t-tests and their non-parametric equivalents. They provide the conceptual framework needed to understand what the method does--but without mathematical proofs--and introduce the ideas in a simple and steady progression with worked examples and exercises at every stage. This innovative text offers students a single conceptual framework for a wide range of tests-including t-tests, oneway and multiway analysis of variance, linear and polynomial regressions, and analysis of covariance-that are usually introduced separately. More importantly, it gives students a language in which they can frame questions and communicate with the computers that perform the analyses. A companion website, www.oup.com/grafenhails, provides a wealth of worked exercises in the three statistical languages; Minitab, SAS, and SPSS. Appropriate for use in statistics courses at undergraduate and graduate levels, Modern Statistics for the Life Sciences is also a helpful resource for students in non-mathematics-based disciplines using statistics, such as geography, psychology, epidemiology, and ecology.

Author Biography

Degrees in Experimental Psychology, Economics and Zoology have exposed Professor Alan Grafen to various different statistical traditions, and also to his main research interest in how adaptive complexity arises through natural selection. He has been interested in statistics since he was an undergraduate, learned mathematical theory of statistics as a graduate student, and encountered modern statistics in the package GLIM as a research student. The impetus to produce a systematic introduction for undergraduates to model formulae and the General Linear Model came from his appointment in 1989 to a lectureship in Quantitative Biology at Oxford University. Degrees in Zoology, Pest Management and Population Dynamics led Dr Rosie Hails toward the more quantitative areas of ecology. Most of her research career has developed the theme of the potential impacts of biological invasions, with reference to both natural invasions and genetically modified organisms. In the early 1990s, she was involved in the first experiments monitoring the behaviour and population dynamics of transgenic plants in natural habitats across the UK with Professor Mick Crawley. More recently, at the NERC Centre for Ecology and Hydrology in Oxford, her research themes have included the dynamics of wildlife diseases as well as plants. In moving to Oxford, Dr Hails became involved in teaching Professor Alan Grafen's undergraduate course, principally through a position at St Anne's College.

Table of Contents

Why use this book	xi
How to use this book	xii
How to teach this text	xiv
An introduction to analysis of variance	1	(21)
Model formulae and geometrical pictures	1	(1)
General Linear Models	1	(1)
The basic principles of ANOVA	2	(8)
What happens when we calculate a variance?	3	(1)
Partitioning the variability	4	(4)
Partitioning the degrees of freedom	8	(1)
F-ratios	9	(1)
An example of ANOVA	10	(6)
Presenting the results	14	(2)
The geometrical approach for an ANOVA	16	(3)
Summary	19	(1)
Exercises	20	(2)
Melons	20	(1)
Dioecious trees	21	(1)
Regression	22	(25)
What kind of data are suitable for regression?	22	(1)
How is the best fit line chosen?	23	(3)
The geometrical view of regression	26	(2)
Regression-an example	28	(5)
Confidence and prediction intervals	33	(2)
Confidence intervals	33	(1)
Prediction intervals	33	(2)
Conclusions from a regression analysis	35	(5)
A strong relationship with little scatter	35	(1)
A weak relationship with lots of noise	36	(2)
Small datasets and pet theories	38	(1)
Significant relationships-but that is not the whole story	39	(1)
Unusual observations	40	(2)
Large residuals	40	(1)
Influential points	41	(1)
The role of X and Y-does it matter which is which?	42	(3)
Summary	45	(1)
Exercises	45	(2)
Does weight mean fat?	45	(1)
Dioecious trees	46	(1)
Models, parameters and GLMs	47	(9)
Populations and parameters	47	(1)
Expressing all models as linear equations	48	(4)
Turning the tables and creating datasets	52	(3)
Influence of sample size on the accuracy of parameter estimates	54	(1)
Summary	55	(1)
Exercises	55	(1)
How variability in the population will influence our analysis	55	(1)
Using more than one explanatory variable	56	(20)
Why use more than one explanatory variable?	56	(3)
Leaping to the wrong conclusion	56	(1)
Missing a significant relationship	57	(2)
Elimination by considering residuals	59	(2)
Two types of sum of squares	61	(4)
Eliminating a third variable makes the second less informative	62	(2)
Eliminating a third variable makes the second more informative	64	(1)
Urban Foxes-an example of statistical elimination	65	(3)
Statistical elimination by geometrical analogy	68	(4)
Partitioning and more partitioning	68	(3)
Picturing sequential and adjusted sums of squares	71	(1)
Summary	72	(1)
Exercises	73	(3)
The cost of reproduction	73	(2)
Investigating obesity	75	(1)
Designing experiments-keeping it simple	76	(20)
Three fundamental principles of experimental design	76	(9)
Replication	76	(2)
Randomisation	78	(2)
Blocking	80	(5)
The geometrical analogy for blocking	85	(3)
Partitioning two categorical variables	85	(1)
Calculating the fitted model for two categorical variables	86	(2)
The concept of orthogonality	88	(4)
The perfect design	88	(3)
Three pictures of orthogonality	91	(1)
Summary	92	(1)
Exercises	93	(3)
Growing carnations	93	(2)
The dorsal crest of the male smooth newt	95	(1)
Combining continuous and categorical variables	96	(14)
Reprise of models fitted so far	96	(1)
Combining continuous and categorical variables	97	(5)
Looking for a treatment for leprosy	97	(2)
Sex differences in the weight-fat relationship	99	(3)
Orthogonality in the context of continuous and categorical variables	102	(2)
Treating variables as continuous or categorical	104	(2)
The general nature of General Linear Models	106	(1)
Summary	107	(1)
Exercises	108	(2)
Conservation and its influence on biomass	108	(1)
Determinants of the Grade Point Average	109	(1)
Interactions--getting more complex	110	(17)
The factorial principle	110	(2)
Analysis of factorial experiments	112	(3)
What do we mean by an interaction?	115	(2)
Presenting the results	117	(10)
Factorial experiments with insignificant interactions	117	(3)
Factorial experiments with significant interactions	120	(3)
Error bars	123	(4)
Extending the concept of interactions to continuous variables	127	(5)
Mixing continuous and categorical variables	127	(2)
Adjusted Means (or least square means in models with continuous variables)	129	(1)
Confidence intervals for interactions	130	(1)
Interactions between continuous variables	131	(1)
Uses of interactions	132	(2)
Is the story simple or complicated?	133	(1)
Is the best model additive?	133	(1)
Summary	134	(1)
Exercises	134	(2)
Antidotes	134	(1)
Weight, fat and sex	135	(1)
Checking the models I: independence	136	(17)
Heterogeneous data	137	(5)
Same conclusion within and between subsets	140	(1)
Creating relationships where there are none	140	(1)
Concluding the opposite	141	(1)
Repeated measures	142	(5)
Single summary approach	142	(3)
The multivariate approach	145	(2)
Nested data	147	(1)
Detecting non-independence	148	(3)
Germination of tomato seeds	149	(2)
Summary	151	(1)
Exercises	151	(2)
How non-independence can inflate sample size enormously	151	(1)
Combining data from different experiments	152	(1)
Checking the models II: the other three asumptions	153	(33)
Homogeneity of variance	153	(2)
Normality of error	155	(2)
Linearity/additivity	157	(1)
Model criticism and solutions	157	(16)
Histogram of residuals	158	(2)
Normal probability plots	160	(3)
Plotting the residuals against the fitted values	163	(3)
Transformations affect homogeneity and normality simultaneously	166	(1)
Plotting the residuals against each continuous explanatory variable	167	(1)
Solutions for nonlinearity	168	(4)
Hints for looking at residual plots	172	(1)
Predicting the volume of merchantable wood: an example of model criticism	173	(5)
Selecting a transformation	178	(2)
Summary	180	(1)
Exercises	181	(5)
Stabilising the variance	181	(1)
Stabilising the variance in a blocked experiment	181	(2)
Lizard skulls	183	(1)
Checking the `perfect' model	184	(2)
Model selection t: principles of model choice and designed experiments	186	(23)
The problem of model choice	186	(3)
Three principles of model choice	189	(6)
Economy of variables	189	(2)
Multiplicity of p-values	191	(1)
Considerations of marginality	192	(1)
Model choice in the polynomial problem	193	(2)
Four different types of model choice problem	195	(1)
Orthogonal and near orthogonal designed experiments	196	(5)
Model choice with orthogonal experiments	196	(2)
Model choice with loss of orthogonality	198	(3)
Looking for trends across levels of a categorical variable	201	(4)
Summary	205	(1)
Exercises	206	(3)
Testing polynomials requires sequential sums of squares	206	(1)
Partitioning a sum of squares into polynomial components	207	(2)
Model selection II; datasets with several explanatory variables	209	(23)
Economy of variables in the context of multiple regression	210	(7)
R-squared and adjusted R-squared	210	(3)
Prediction Intervals	213	(4)
Multiplicity of p-values in the context of multiple regression	217	(3)
The enormity of the problem	217	(1)
Possible solutions	217	(3)
Automated mode) selection procedures	220	(5)
How stepwise regression works	220	(1)
The stepwise regression solution to the whale watching problem	221	(4)
Whale Watching: using the GLM approach	225	(3)
Summary	228	(1)
Exercises	229	(3)
Finding the best treatment for cat fleas	229	(2)
Multiplicity of p-values	231	(1)
Random effects	232	(23)
What are random effects?	232	(2)
Distinguishing between fixed and random factors	232	(2)
Why does it matter?	234	(1)
Four new concepts to deal with random effects	234	(4)
Components of variance	234	(1)
Expected mean square	235	(1)
Nesting	236	(1)
Appropriate Denominators	237	(1)
A one-way ANOVA with a random factor	238	(3)
A two-level nested ANOVA	241	(3)
Nesting	241	(3)
Mixing random and fixed effects	244	(3)
Using mock analyses to plan an experiment	247	(5)
Summary	252	(1)
Exercises	253	(2)
Examining microbial communities on leaf surfaces	253	(1)
How a nested analysis can solve problems of non-independence	254	(1)
Categorical data	255	(26)
Categorical data: the basics	255	(3)
Contingency table analysis	255	(2)
When are data truly categorical?	257	(1)
The Poisson distribution	258	(7)
Two properties of a Poisson process	258	(1)
The mathematical description of a Poisson distribution	259	(2)
The dispersion test	261	(4)
The chi-squared test in contingency tables	265	(4)
Derivation of the chi-squared formula	265	(2)
Inspecting the residuals	267	(2)
General linear models and categorical data	269	(9)
Using contingency tables to illustrate orthogonality	269	(2)
Analysing by contingency table and GLMs	271	(5)
Omitting important variables	276	(1)
Analysing uniformity	277	(1)
Summary	278	(1)
Exercises	279	(2)
Soya beans revisited	279	(1)
Fig trees in Costa Rica	280	(1)
What lies beyond?	281	(4)
Generalised Linear Models	281	(2)
Multiple y variables, repeated measures and within-subject factors	283	(1)
Conclusions	284	(1)
Answers to exercises	285	(32)
Chapter 1	285	(2)
Chapter 2	287	(1)
Chapter 3	288	(1)
Chapter 4	289	(3)
Chapter 5	292	(2)
Chapter 6	294	(1)
Chapter 7	295	(3)
Chapter 8	298	(1)
Chapter 9	299	(9)
Chapter 10	308	(2)
Chapter 11	310	(3)
Chapter 12	313	(1)
Chapter 13	314	(3)
Revision section: The basics	317	(15)
Populations and samples	317	(1)
Three types of variability: of the sample, the population and the estimate	318	(4)
Variability of the sample	318	(1)
Variability of the population	319	(1)
Variability of the estimate	319	(3)
Confidence intervals: a way of precisely representing uncertainty	322	(2)
The null hypothesis-taking the conservative approach	324	(3)
Comparing two means	327	(4)
Two sample t-test	327	(1)
Alternative tests	328	(1)
One and two tailed tests	329	(2)
Conclusion	331	(1)
Appendix 1: The meaning of p-values and confidence intervals	332	(3)
What is a p-value?	332	(2)
What is a confidence interval?	334	(1)
Appendix 2: Analytical results about variances of sample means	335	(4)
Introducing the basic notation	335	(1)
Using the notation to define the variance of a sample	335	(1)
Using the notation to define the mean of a sample	336	(1)
Defining the variance of the sample mean	336	(1)
To illustrate why the sample variance must be calculated with n - 1 in its denominator (rather than n) to be an unbiased estimate of the population variance	337	(2)
Appendix 3: Probability distributions	339	(4)
Some gentle theory	339	(2)
Confirming simulations	341	(2)
Bibliography	343	(2)
Index	345