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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Presents readers with a user-friendly, non-technical introductionto statistics and the principles of plant and crop experimentation.Avoiding mathematical jargon, it explains how to plan and design anexperiment, analyse results, interpret computer output and presentfindings. Using specific crop and plant case studies, this guidepresents:
* The reasoning behind each statistical method is explained beforegiving relevant, practical examples
* Step-by-step calculations with examples linked to three computerpackages (MINITAB, GENSTAT and SAS)
* Exercises at the end of many chapters
* Advice on presenting results and report writing
Written by experienced lecturers, this text will be invaluable toundergraduate and postgraduate students studying plant sciences,including plant and crop physiology, biotechnology, plant pathologyand agronomy, plus ecology and environmental science students andthose wanting a refresher or reference book in statistics.
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Veröffentlichungsjahr: 2013
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Contents
Preface
Chapter 1 Basic Principles of Experimentation
1.1 INTRODUCTION
1.2 FIELD AND GLASSHOUSE EXPERIMENTS
1.3 CHOICE OF SITE
1.4 SOIL TESTING
1.5 SATELLITE MAPPING
1.6 SAMPLING
Chapter 2 Basic Statistical Calculations
2.1 INTRODUCTION
2.2 MEASUREMENTS AND TYPE OF VARIABLE
2.3 SAMPLES AND POPULATIONS
Chapter 3 Basic Data Summary
3.1 INTRODUCTION
3.2 FREQUENCY DISTRIBUTIONS (DISCRETE DATA)
3.3 FREQUENCY DISTRIBUTIONS (CONTINUOUS DATA)
3.4 DESCRIPTIVE STATISTICS
Chapter 4 The Normal Distribution, the t-Distribution and Confidence Intervals
4.1 INTRODUCTION TO THE NORMAL DISTRIBUTION
4.2 THE STANDARD NORMAL DISTRIBUTION
4.3 FURTHER USE OF THE NORMAL TABLES
4.4 USE OF THE PERCENTAGE POINTS TABLE (APPENDIX 2)
4.5 THE NORMAL DISTRIBUTION IN PRACTICE
4.6 INTRODUCTION TO CONFIDENCE INTERVALS
4.7 ESTIMATION OF THE POPULATION MEAN, µ
4.8 THE SAMPLING DISTRIBUTION OF THE MEAN
4.9 CONFIDENCE LIMITS FOR µ WHEN A IS KNOWN
4.10 CONFIDENCE LIMITS FOR µ WHEN σ IS UNKNOWN—USE OF THE t-DISTRIBUTION
4.11 DETERMINATION OF SAMPLE SIZE
4.12 ESTIMATION OF TOTAL CROP YIELD
Chapter 5 Introduction to Hypothesis Testing
5.1 THE STANDARD NORMAL DISTRIBUTION AND THE t-DISTRIBUTION
5.2 THE SINGLE SAMPLE t-TEST
5.3 THE P-VALUE
5.4 TYPE I AND TYPE II ERRORS
5.5 CHOICE OF LEVEL OF SIGNIFICANCE
5.6 THE USEFULNESS OF A TEST
5.7 ESTIMATION VERSUS HYPOTHESIS TESTING
5.8 THE PAIRED SAMPLES t-TEST
Chapter 6 Comparison of Two Independent Sample Means
6.1 INTRODUCTION
6.2 THE INDEPENDENT SAMPLES t-TEST
6.3 CONFIDENCE INTERVALS
6.4 THE THEORY BEHIND THE t-TEST
6.5 THE F-TEST
6.6 UNEQUAL SAMPLE VARIANCES
6.7 DETERMINATION OF SAMPLE SIZE FOR A GIVEN PRECISION
Chapter 7 Linear Regression and Correlation
7.1 BASIC PRINCIPLES OF SIMPLE LINEAR REGRESSION (SLR)
7.2 EXPERIMENTAL VERSUS OBSERVATIONAL STUDIES
7.3 THE CORRELATION COEFFICIENT
7.4 THE LEAST SQUARES REGRESSION LINE AND ITS ESTIMATION
7.5 CALCULATION OF RESIDUALS
7.6 THE GOODNESS OF FIT
7.7 CALCULATION OF THE CORRELATION COEFFICIENT
7.8 ASSUMPTIONS, HYPOTHESIS TESTS AND CONFIDENCE INTERVALS FOR SIMPLE LINEAR REGRESSION
7.9 TESTING THE SIGNIFICANCE OF A CORRELATION COEFFICIENT
Chapter 8 Curve Fitting
8.1 INTRODUCTION
8.2 POLYNOMIAL FITTING
8.3 QUADRATIC REGRESSION
8.4 OTHER TYPES OF CURVE
8.5 MULTIPLE LINEAR REGRESSION
Chapter 9 The Completely Randomised Design
9.1 INTRODUCTION
9.2 DESIGN CONSTRUCTION
9.3 PRELIMINARY ANALYSIS
9.4 THE ONE-WAY ANALYSIS OF VARIANCE MODEL
9.5 ANALYSIS OF VARIANCE
9.6 AFTER ANOVA
9.7 REPORTING RESULTS
9.8 THE COMPLETELY RANDOMISED DESIGN — UNEQUAL REPLICATION
9.9 DETERMINATION OF NUMBER OF REPLICATES PER TREATMENT
Chapter 10 The Randomised Block Design
10.1 INTRODUCTION
10.2 THE ANALYSIS IGNORING BLOCKS
10.3 THE ANALYSIS INCLUDING BLOCKS
10.4 USING THE COMPUTER
10.5 THE EFFECT OF BLOCKING
10.6 THE RANDOMISED BLOCKS MODEL
10.7 USING A HAND CALCULATOR TO FIND THE SUMS OF SQUARES
10.8 COMPARISON OF TREATMENT MEANS
10.9 REPORTING THE RESULTS
10.10 DECIDING HOW MANY BLOCKS TO USE
10.11 PLOT SAMPLING
Chapter 11 The Latin Square Design
11.1 INTRODUCTION
11.2 RANDOMISATION
11.3 INTERPRETATION OF COMPUTER OUTPUT
11.4 THE LATIN SQUARE MODEL
11.5 USING YOUR CALCULATOR
Chapter 12 Factorial Experiments
12.1 INTRODUCTION
12.2 ADVANTAGES OF FACTORIAL EXPERIMENTS
12.3 MAIN EFFECTS AND INTERACTIONS
12.4 VARIETIES AS FACTORS
12.5 ANALYSIS OF A RANDOMISED BLOCKS FACTORIAL EXPERIMENT WITH TWO FACTORS
12.6 GENERAL ADVICE ON PRESENTATION
12.7 EXPERIMENTS WITH MORE THAN TWO FACTORS
12.8 CONFOUNDING
12.9 FRACTIONAL REPLICATION
Chapter 13 Comparison of Treatment Means
13.1 INTRODUCTION
13.2 TREATMENTS WITH NO STRUCTURE
13.3 TREATMENTS WITH STRUCTURE (FACTORIAL STRUCTURE)
13.4 TREATMENTS WITH STRUCTURE (LEVELS OF A QUANTITATIVE FACTOR)
13.5 TREATMENTS WITH STRUCTURE (CONTRASTS)
Chapter 14 Checking the Assumptions and Transformation of Data
14.1 THE ASSUMPTIONS
14.2 TRANSFORMATIONS
Chapter 15 Missing Values and Incomplete Blocks
15.1 INTRODUCTION
15.2 MISSING VALUES IN A COMPLETELY RANDOMISED DESIGN
15.3 MISSING VALUES IN A RANDOMISED BLOCK
15.4 OTHER TYPES OF EXPERIMENT
15.5 INCOMPLETE BLOCK DESIGNS
Chapter 16 Split Plot Designs
16.1 INTRODUCTION
16.2 USES OF THIS DESIGN
16.3 THE SKELETON ANALYSIS OF VARIANCE TABLES
16.4 AN EXAMPLE WITH INTERPRETATION OF COMPUTER OUTPUT
16.5 THE GROWTH CABINET PROBLEM
16.6 OTHER TYPES OF SPLIT PLOT EXPERIMENT
16.7 REPEATED MEASURES
Chapter 17 Comparison of Regression Lines and Analysis of Covariance
17.1 INTRODUCTION
17.2 COMPARISON OF TWO REGRESSION LINES
17.3 ANALYSIS OF COVARIANCE
17.4 ANALYSIS OF COVARIANCE RANDOMISED DESIGN
17.5 COMPARING SEVERAL REGRESSION LINES
17.6 CONCLUSION
Chapter 18 Analysis of Counts
18.1 INTRODUCTION
18.2 THE BINOMIAL DISTRIBUTION
18.3 CONFIDENCE INTERVALS FOR A PROPORTION
18.4 HYPOTHESIS TEST OF A PROPORTION
18.5 COMPARING TWO PROPORTIONS
18.6 THE CHI-SQUARE GOODNESS OF FIT TEST
18.7 r × c CONTINGENCY TABLES
18.8 2 × c CONTINGENCY TABLES: COMPARISON OF SEVERAL PROPORTIONS
18.9 2 × 2 CONTINGENCY TABLES: COMPARISON OF TWO PROPORTIONS
18.10 ASSOCIATION OF PLANT SPECIES
18.11 HETEROGENEITY CHI-SQUARE
Chapter 19 Some Non-parametric Methods
19.1 INTRODUCTION
19.2 THE SIGN TEST
19.3 THE WILCOXON SINGLE-SAMPLE TEST
19.4 THE WILCOXON MATCHED PAIRS TEST
19.5 THE MANN-WHITNEY U TEST
19.6 THE KRUSKAL-WALLIS TEST
19.7 FRIEDMAN’S TEST
Appendix 1 The normal distribution function
Appendix 2 Percentage points of the normal distribution
Appendix 3 Percentage points of the t-distribution
Appendix 4a 5 per cent points of the F-distribution
Appendix 4b 2.5 per cent points of the F-distribution
Appendix 4c 1 per cent points of the F-distribution
Appendix 4d 0.1 per cent points of the F-distribution
Appendix 5 Percentage points of the sample correlation coefficient (r) when the population correlation coefficient is 0 and n is the number of X, Y pairs
Appendix 6 5 per cent points of the Studentised range, for use in Tukey and SNK tests
Appendix 7 Percentage points of the chi-square distribution
Appendix 9 Critical values of T in the Wilcoxon signed rank or matched pairs test
Appendix 10 Critical values of U in the Mann–Whitney test
References
Further reading
Index
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Library of Congress Cataloging-in-Publication Data
Clewer, Alan G.Practical statistics and experimental design for plant and crop science / Alan G. Clewer and David H. Scarisbrick.p. cm.Includes bibliographical references and index.ISBN 0-471-89908-9 (hbk : alk. paper) – ISBN 0-471-89909-7 (pbk. : alk. paper)1. Botany–Research–Statistical methods. 2. Botany, Economic–Research–Statistical methods. 3. Biometry. I. Scarisbrick, D. H. II. Title.QK51. C58 2001580′.7′27–dc21
00-047313
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0 471 89908 9 (PPC)
ISBN 0 471 89909 7 (Pbk)
Preface
The references at the end of this book confirm that there are many textbooks on statistics for students who are interested in applied biology. Most of these cover the same subject material, although they vary quite widely in styles of presentation. Thus, it is important to answer the question—why has another book been written? It would be pointless to review the same topics on the design and analysis of experiments unless some original features can be detected by the reader in the 19 chapters that make up this book. One claim to originality is that this text is closely linked to the computer outputs from three commonly used statistical packages (Genstat 5—release 4.1, Minitab—release 12.1 and SAS—release 6.12). However, this in itself may not be sufficient to justify the vast amount of time required to produce another text, and provide a satisfactory answer to the above question.
The answer is more closely linked to our concern about the misuse of statistics by many students and their lack of understanding of the basic principles that underlie the statistical techniques they refer to in their dissertations. Because it is now all too easy to carry out inappropriate analyses by computer, the advent of statistical packages has diluted many students’ understanding and interest in the basic principles which are the foundation stone of good design. They often look for an analytical method that seems to ‘fit’ their data. This frequently results in problems of interpretation; the design of experiments and the related data analysis should not be treated as separate components of the experimental process. Design is more important than analysis, because without a good design, the analysis is meaningless. Matching experimental data with a program given in a textbook on computing and statistics after the experiment has been completed is rarely successful and should be avoided.
As external examiners we frequently find computer summaries of data analyses in the appendices of dissertations, although when faced with simple questions the majority of students seem to have very little understanding of the terms and figures given in their printouts. For example, the ANOVA summary usually includes a mean square (MS) column. It is now quite rare for students to give the correct translation of the error (residual) mean square and even more rare for them to know that this is also an estimate of variance. There is always muddle in relation to the constants quoted for simple regression equations, and discussion of the usefulness of slope and intercept is usually missing mainly because the role of the constants and equations is rarely understood.
Thus, the book has mainly been written to demonstrate both the usefulness of statistics (like previous texts), and also to provide a clear explanation of terms, figures and symbols given in computer printouts. We decided not to link these discussions to one statistical package in order to illustrate the diversity of layouts that are used to provide a summary of the same results. For most statistical procedures the three packages have much output in common, but some are better than others for certain options. For the examples for which we give computer outputs, we use the package that we consider illustrates the main points we are trying to make. The book also encourages students to review the underlying principles of many statistical tests before using them in their research work; this point should be noted by supervisors! When interviewing students in relation to their data-handling methods it is sometimes the supervision that can be faulted, especially when students confess that they simply browsed a recommended text in order to find a statistical example which had similarities to their own design and experimental results.
Students reading this book should initially work through the text and exercise examples using a hand calculator. This technique assists understanding and interpretation. When interpretive skills have been achieved, results can then be confirmed by studying the computer outputs provided. Future data can then be immediately analysed using computer packages.
The classical textbooks describe how to do calculations by hand using the correction factor method. This method is no longer needed due to the widespread use of computers and hand calculators. However, we do explain it in the chapter on one-way analysis of variance. In subsequent chapters we show how the various sums of squares can be calculated using the standard deviation function provided by most calculators.
Biological statistics (biometry) is relevant to all areas encompassed by the general subject title Applied Plant Biology. It is an essential tool which is used to uncover or discover scientific information contained within raw (new) data. Like applied plant sciences, biometry is a broad-based subject. It is concerned with all aspects of experimentation (design, sampling methods, data analysis, interpretation and discussion) which are relevant to the research objectives. Experiments should always be designed in such a way that questions posed by the objectives have a good chance of being answered. This implies that the method of analysis and statistical tests are determined before the experiment is carried out.
This book introduces students to the important role of biometry in applied plant science research. Because many readers will have little prior knowledge of biological statistics, the first five chapters are used to summarise the basic principles which underpin simple statistical concepts. Although a more advanced treatise is provided in later chapters, the mathematical theory underpinning the techniques described is mainly ignored. Instead, the text provides some description and justification of the most important calculations which are widely used in statistics testing, and a detailed review of computer output which is now commonly presented by students in their dissertations. For each output an interpretation is given, and for many, how most entries can be found from a hand calculator. A novel feature of the book is the inclusion of examples showing how the sums of squares for the various terms in the most popular analysis of variance models are partitioned without using algebra.
Many biology students feel uneasy with statistical calculations and equations, and are rarely concerned with acquiring an understanding of the mathematical theory of statistics. They are mainly interested in learning how to apply a range of statistical tools to design, analyse and then interpret their results. Although this approach to biological statistics is commonplace, it is the authors’ view that effort should still be made to understand some of the background calculations associated with many tests used to compare treatment means. This is an important part of comprehending statistical methods; comprehension is still important even in an era when computers usually undertake the time-consuming labour of arithmetic calculations.
When statistical analyses were carried out using hand-operated calculators, the amount of data collected and analysed was mainly controlled by the sluggishness of desk machinery. Because there were no mountains of computer output to file and review, raw data and results were usually pondered and discussed in great detail. Packaged computer programs have revolutionised data-handling techniques. They have removed most of the drudgery from the analysis of large experiments which compare a range of treatments, and eliminated arithmetic errors. In addition, because most programs provide facilities for reviewing raw data, it is now much easier to check that assumptions which underpin many statistical tests are really true before proceeding with an analysis. An initial study using graphs, scatter diagrams and tables is helpful in deciding whether a particular statistical method is really appropriate for the new data being examined.
It could be argued that statistical tables are no longer required to carry out statistical tests when the P-value is given in the computer output. Nevertheless, to understand the P-value, a familiarity with the underlying distribution of the test statistic is required, so we include statistical tables in Appendices 1 to 10 inclusive. They are also required for those readers without a computer or relevant software.
The authors hope that this book will assist students and researchers in crop and plant sciences to explain in simple language the objectives of simple statistical tests, and achieve an understanding of the principles of experimentation. This book can be read at several levels and so will be useful for a wide range of readers. It will be useful for teachers. It will be useful for users of statistical packages who want to interpret the output. It will be useful for researchers who want to know how to design a simple experiment and analyse and present the results. It will also be useful for those who want to know a little of the background theory needed to justify the procedures and how the calculations are performed.
ACKNOWLEDGEMENTS
We wish to thank BASF for permission to reproduce some diagrams and tables. We also thank MINITAB, SAS and Genstat for permission to include computer output. SAS is a registered trade mark of the SAS Institute Inc.
Chapter 1
Basic Principles of Experimentation
1.1 INTRODUCTION
The principles of experimentation can be studied by students who enjoy pure mathematics, and by those who wish to use mathematical principles as tools only for the design and analysis of their experiments. When carrying out research work at an experimental station or university, the applied biologist may be able to discuss the layout and analysis of experimental work with a professional statistician. However, when working in isolated rural areas (especially in developing countries), the experimenter must demonstrate a basic understanding of statistics and also have the confidence to solve design and data-handling problems. Even when professional support is available, it is still essential that the researcher is aware of the wide range of statistical procedures used by applied biologists. More importantly, he or she must have acquired sufficient statistical skills to interpret and discuss experimental results which are analysed using computer packages such as Genstat, Minitab and SAS.
Experimental objectives must be clearly and concisely stated at the outset of an investigational programme. Before starting a field or greenhouse experiment, it is wise to purchase a diary. The first page should be used to give a clear exposition of research objectives. The diary should contain a summary of previous cropping, a description of the treatments, detailed site plans and daily observations. When an experiment is written up, these observations may help to explain results that may at first seem to be anomalous.
1.2 FIELD AND GLASSHOUSE EXPERIMENTS
There are two main systems of experimentation used to explore the effect(s) of experimental treatments on plant development and yield (Figure 1.1). In both, treatments may be applied before, at, as well as after sowing. For example, the objectives of an experiment may be to compare different seed dressings, fertiliser placement at drilling, or the post-emergence application of a plant growth regulator at defined growth stages. The first descriptive system shown in Figure 1.1 is widely used by agronomists at arable research centres and farm demonstration sites. However, if the chosen treatments result in a significant increase in yield, the agronomist may be unable to fully explain the field results. It is impossible to provide an in-depth discussion of morphological and physiological factors affected by the treatments unless some additional measurements such as light interception, leaf area index, or crop growth rate are taken during the growing season (System 2).
Figure 1.1. Systems of experimentation
System 1 is also rather risky because the results of a season’s technical work (site mapping, cultivations, sowing, pot and plot maintenance), are solely dependent on data collected at final harvest. It is frustrating when treatment effects which were clearly visible during mid-season are obscured by lodging and seed loss due to thunderstorm damage during the ripening period. Similar end-of-season losses can also occur in glasshouse trials using System 1. In a hydroponic study on the response of wheat to varying concentrations of sulphur, the ears were destroyed by field mice which invaded the glasshouse cubicle just two days before the planned final harvest date. A trial which needed a great deal of maintenance time in relation to monitoring and topping up the different nutrient solutions only provided advice on rodent control! If measurements such as number of tillers per plant and sulphur concentrations in leaves and stems had been taken during vegetative development (System 2), some useful discussions of the effects of sulphur depletion on wheat development may have resulted.
The second system provides useful background information which may be used to interpret and discuss the final results. For example, when regular descriptive samples are taken the effects of treatments on plant morphology and the reproductive components of yield can be determined. If more sophisticated facilities are available, it may also be helpful to carry out physiological measurements such as chlorophyll concentration, light interception and photosynthesis. Subsequently, yield data can be analysed and interpreted in much more detail; additional confidence may then result in the reliability of a new product or variety. If final yield data are lost from System 2, the researcher may still have sufficient information in his or her diary to create a scientific report on vegetative development, flowering, and the early stages of reproductive development. Data collected during the experimental growing season may also be used to look for possible relationships between measurements such as soil temperature, rainfall, emergence and plant development using simple correlation techniques (Chapter 7).
As a disadvantage, the introduction of crop sampling increases experimental costs. Growth analysis is especially tedious and time consuming, and it is essential to ponder original experimental objectives before investing time in measuring components such as leaf area and numbers of seeds in pods or spikelets. The applied plant scientist may enjoy philately as a hobby, but must avoid the temptation to file vast quantities of descriptive data unless the scientific reasons are clearly defined. Computer packages will analyse data and produce means and standard errors. However, they are unable to guide the researcher on interpretative skills and sampling procedures or indicate whether the time invested in measuring a particular parameter is really worthwhile.
1.3 CHOICE OF SITE
The choice of field and the siting of a trial are probably the most important decisions made by the researcher. Incorrect choice of site may mean that the experimental results are difficult to interpret or are even meaningless. There are several common-sense starting points. It is essential to choose a uniform site as trials are designed to detect differences between small plot areas. Thus any variations in soil texture or pH within a site may partly or completely mask treatment effects. Local knowledge is clearly important, and it may be wise to consult historical records of drainage systems and former field boundaries. Previous site management must be researched because many legume and oilseed crops should be sown only once in a 5-year rotation — the principles of crop rotation must be adhered to when the experimental objectives are being defined.
An accessible site makes for ease of sampling, although it is worth remembering that trials situated near parking areas and footpaths may be subjected to damage by trespassers. It is best to avoid compacted headlands and wooded areas as downdraughts from trees can cause severe lodging. In order to minimise edge effects it is usually advisable to surround the experimental site with the crop species being studied. Frequently, plots will be positioned so that they can be sprayed with protective agrochemicals using the tramlines of a surrounding commercial crop.
When undertaking an agronomic research programme, it is valuable to have growth room and glasshouse facilities in addition to field sites. This enables detailed physiological work to be continued during the winter months. In theory, research glasshouses should provide a reasonably uniform environment in which light intensity, temperature and daylength can be controlled by computer technology. In practice, the glasshouse can be a more variable environment to work in than the field. There are frequently wide and sudden fluctuations in temperature between overcast and sunny periods even in the winter months, and variation in light profiles especially when neighbouring research cubicles are using a different daylength regime. In order to minimise these problems, it is important to re-randomise the position of the growing containers on the greenhouse benches from time to time, and surround the experimental unit with guard pots to reduce edge effects. It is essential to monitor pests and diseases — mildew, botrytis, aphids and white fly are common problems in the glasshouse, and while biological systems for insect control can be used, it is wise to have insurance supplies of agrochemicals in store, especially for the control of fungal diseases.
1.4 SOIL TESTING
Large experimental sites are more likely to include variations in soil texture, thus for most arable crop investigations the total experimental area rarely exceeds 1 ha. Larger plots are usually necessary in grassland or grazing studies in order to accommodate fencing, gangways, weighing pens and sufficient replication of the livestock assigned to individual grazing treatments.
Regardless of size it is essential to provide soil analyses before the treatments are applied. The purpose is to assess the adequacy, surplus or deficiency of available nutrients for crop growth. A standard soil analysis package measures soil pH and estimates available concentrations of phosphorus, potassium and magnesium. Some minor elements such as boron and copper can also be measured using soil samples, while others, for example manganese, are usually assessed from plant samples. For nutrient and pH assessment, soil is usually removed from the top 10–20 cm. Although it may be valuable to study nutrient levels in deeper profiles, it is often backbreaking work if traditional soil augers are used. Mechanical soil sampling equipment is available, but this is more expensive than hand-operated augers and rarely available at Experimental Stations in developing countries.
The number of soil samples must be sufficiently large to be representative of the experimental site. Within each site samples should be taken across a W pattern; 6–10 separate samples being taken along each arm of the W. Samples are usually bulked into one bag. The bulked sample comprises around 1 kg of soil which is taken to represent an entire site containing approximately 2000 t soil/ha to a ploughing depth of 20 cm. Clearly, using a small bulked sample is not a precise measure of nutrient or acidity levels and the problem of accuracy is increased by the use of very small soil subsamples for nutrient analysis. Although the researcher may have invested a great deal of time and physical effort in order to obtain a representative soil sample, the amount of soil used by the analytical chemist is often tiny. For example, when determining ammonium nitrate and nitrite nitrogen levels from fresh soil, a subsample of 20 g soil is used. For potassium, magnesium, sodium and manganese only 5 g of dried soil is required when using ammonium acetate extraction techniques. As a result it is important to assess the number of subsamples in relation to the variation between replicates analyses. If the level of variation between three or four replicates is high additional subsamples must be analysed, although for some analyses such as sulphate-sulphur this decision will be expensive. Unfortunately, all soil laboratory procedures are time consuming and costly, but this must not deter the researcher from clearly defining the pH, soil texture and nutrient status of the experimental site prior to drilling.
1.5 SATELLITE MAPPING
The careful control of inputs or precision farming is not new, and many farmers have selectively applied some agrochemicals for many years using their local knowledge of individual fields. For example, patch spraying with graminicides or the application of lime to sections of fields or headlands can often be cost effective.
New technology has been developed which may supersede conventional soil-sampling procedures for assessing the causes of yield variations that occur within most large arable fields. It is based on satellite-controlled navigation systems or GPS (Global Positioning Satellites). GP yield monitoring systems can be fitted to combine harvesters, with up to 500 grain weight checks/ha during harvest. The computer system can create a colour-coded yield map of each field. This shows yield variations across large areas, which would have been difficult to detect from auger samples. It has resulted in the development of management systems for the precise application of nutrients, and as a result manufacturers have now launched computer-controlled fertiliser spreaders.
The success of satellite precision farming systems will ultimately be controlled by their cost effectiveness, and the reliability of microchips linked to machinery which is designed to work under field conditions. For experimentation, it must be remembered that the most common reason for yield variations within a field is soil type. Boundaries between different soil textures were formerly defined by hedgerows, thus it may be easier to consult historical farm maps before using satellite technology. At present this exciting technology is of little value for choosing the position of an experimental site within a field. Background variation is still best defined by conventional soil analysis and historical research, and ultimately controlled by the correct choice of experimental design.
1.6 SAMPLING
A crop sample is a small portion of a population taken for detailed study. It may be a length of row, quadrat area, a number of plants or pots taken at random. Hopefully, it will be representative (large) enough to inform the researcher what he or she needs to know about the whole population of a particular treatment.
Some crop-sampling procedures provide excellent exercises for improving physical fitness! Hand lifting, bagging and labelling samples of potatoes and sugar beet requires stamina on hot summer days. Locating and then removing quadrat areas of oilseed rape from dense, tangled and lodged crops is a challenge of patience and technical skill for the research agronomist. It is important to prepare bags and labels before going into the field, and to check that each plot has been allocated a sample bag (using the field plan) before starting work. Luggage labels made from cardboard are ideal — if felt tip pens are used check that the ink is waterproof. If cold storage facilities are not available, it is best to process samples block by block — assuming a block design has been used (Chapter 10). This minimises problems associated with wilting and loss of dry matter.
As statistical packages are now readily available, it is tempting to carry out data analyses before examining the form of distribution that is associated with the sampled data. This examination is highly recommended. Crop experimentation encompasses both discrete and continuous distributions. Variables such as number of branches, number of pods per plant and number of seeds per pod have discrete distributions, while those such as plant height and dry weight have continuous distributions (Chapters 2 and 3).
Sampling schemes should only be agreed after a careful assessment has been made of plant establishment. They must avoid bias, but at the same time take into account variability in establishment possibly caused by poor drill calibration. The number of plants in individual rows of adjacent plots of winter wheat is shown in Table 1.1.
There are large differences between rows within each plot with rows 5 and 3 having the lowest plant populations. The position of these two rows varied according to the direction of drilling, the problem having mainly arisen due to shallower coulter depth. In this experiment the background variation in establishment was high, and as a result a random sampling system based on a length of one row would have been inappropriate. Instead, a quadrat area encompassing eight rows was randomly removed on each sampling occasion following establishment. It is always important to determine the reliability of the sampling system by examining the level of variation in plant establishment between replicate samples taken from the same treatment.
The morphology of individual crop plants varies widely in an unevenly established crop. Between-plant variation (plot background variation) is especially problematic to sampling procedures in poorly established precision-sown or transplanted crops such as maize and tobacco. Sampling difficulties encountered in many field experiments are clearly illustrated by data obtained from two experimental plots of winter oilseed rape. Quadrat samples (0.33 m2) were randomly taken when the lowermost terminal raceme buds were yellowing. The individual dry weights of all plants were recorded and the background variation is summarised in Table 1.2.
Table 1.1. The number of winter wheat plants in six plots and individual rows (replicate 1) of 0.5 m prior to the application of husbandry treatments
Table 1.2. Ranges of plant dry weights (g)
For both nitrogen fertiliser treatments the numbers of plants varied widely, and each quadrat contained a number of very small and large specimens. Possible treatment differences may not have been detected if only small plants had been subsampled from each quadrat for growth analysis. It is essential to use a subsampling system which eliminates bias because in practice there is always a tendency to avoid large plants as their subsequent analysis in the crop laboratory is time consuming.
The wide range of plant-to-plant variation shown in Table 1.2 cannot be quantified accurately if only small numbers of plants are studied. Yet subsample sizes in many research papers on rapeseed agronomy and physiology have only consisted of 3–20 plants per treatment. A detailed statistical study of data given in Table 1.2 indicated that for the sample mean plant weight to be within 1 g of the true weight (with 95% confidence), a random sample of approximately 600 plants would be required! As this is time consuming it again raises the question of the value of overcollecting descriptive data unless the accuracy of a sampling system is known. In System 2 it may be better to minimise the time invested in crop description (traditional growth analysis), in order to study in more detail environmental and physiological parameters affecting canopy development and crop yield. The latter approach (crop modelling), has been made more accessible with the availability of field recording equipment which is directly linked to computers.
Although statisticians will always recommend random sampling, this may not be practical in tall, high-density plots of crops such as oilseed rape without causing damage to surrounding areas. In this situation it may be wiser to adopt a ‘step-ladder’ sampling system in which a quadrat enclosing inner plot rows is first removed from a uniform area at either the top or bottom of each experimental plot. After leaving a discard distance a second quadrat can be removed on a later date using the first sampled area as a working base. This process can then be repeated. An unsampled, undisturbed area of plot must be left for commercial yield assessment. However, the importance of selecting the sampling units in such a way that they shall be as representative as possible of the entire population cannot be overemphasised.
Chapter 2
Basic Statistical Calculations
2.1 INTRODUCTION
When all statistical analyses were carried out using hand-operated calculators, the amount of experimental data collected and analysed was partly controlled by the sluggishness of early desk machinery. Because each analysis may have taken many minutes or even hours to complete, raw data were pondered in detail before calculations were attempted.
Package computer programs have revolutionised attitudes to data-collection and data-handling systems in applied biological research. Sadly, they also seem to have diluted many students’ understanding of basic statistics. It is now far too easy to rely uncritically on computer output and to carry out sophisticated analyses which may be inappropriate and lead to misleading conclusions.
Computer technology has greatly improved presentational but not interpretative skills. For example, during oral examinations many students are unable to explain basic statistical terms such as standard error and variance, even when exquisitely presented tables and figures created by computer technology include summaries of statistical tests. Their attitude now seems to be ‘don’t think, use the computer’, and if the output looks good then include it in the dissertation to impress the examiner!
The main objective of this chapter is to provide a definition and clear understanding of some basic statistical terms which are commonly used when analysing data collected from field and glasshouse experiments. For simplicity, only a small number of observations is included in the analyses so that the reader can check the results using a hand calculator.
2.2 MEASUREMENTS AND TYPE OF VARIABLE
The unit on which measurements are made may be a whole plot, a small area of a plot, a single plant, a stem, a leaf, etc. Suppose the experimental unit is an individual plant. For each, measurements may be made on several variables, such as height, weight, leaf area or number of internodes. Variables may be discrete, continuous or categorical.
A continuous variable is one that can take any value in a certain range. For instance, plant height is a continuous variable. If one plant has a height of 20 cm and another a height of 21 cm, it is possible to find a third plant with a height of between 20 and 21 cm. For continuous variables, measurements are approximate because they have to be rounded off to a whole number or to a fixed number of decimal places.
A discrete variable is one which can only take certain values. An example is the number of seeds in a pod. This number must be an integer such as 0, 1, 2, 3, etc. We cannot have a pod with 2.1 seeds.
A categorical variable is formed when data are classified into categories. For example, each plant measured could be classified according to variety. In this case variety is a category, sometimes called a classification variable. If the varieties are given names, there is no natural order. If there were three varieties, we could assign the numbers 1, 2 and 3 to them, but it would be meaningless to do any calculations on these numbers. However, it may be meaningful to count the number of plants of each variety. These data may be summarised in a table or a bar chart.
2.3 SAMPLES AND POPULATIONS
One of the main objectives of statistical analysis is to find out as much as possible about a population. Most populations are far too large to be measured. For example, suppose the population under study is a field of wheat. You may want to know the average yield per plant. As resources are not available to measure every wheat plant, a random sample can be taken. The average yield of these plants is an estimate of the mean yield of all plants in the field. The estimate calculated could be ‘a long way’ from the true value. A statement is needed of how close the sample mean is likely to be to the population mean. For example, it would be helpful to state with 95% confidence that the mean yield per plant lies between 25 and 30 g. The calculation of a confidence interval requires some background theory, and in the following discussion a population of field plants is assumed.
x is the symbol used for yield, and Σ is the summation sign. It means add up all the xs (the yields).
As it would be impractical to assess the yield of 2.5 million individual plants, an estimate of μ can be found by taking a sample from the population. To be unbiased and representative of the population, the plants to be chosen for inclusion in the sample must be selected at random. In this way all individuals in the population have an equal chance of being included in the sample.
is the symbol for sample mean and its formula is
2.3.1 Median
If there are n numbers, the median is the (n + l)/2 ranked number. If n is odd, this is the middle number after sorting them in order of magnitude, and if n is even it is the average of the middle two.
The data of the last example after sorting are: 11.6, 12.5, 14.8, 15.2 and 17.4. The median is therefore 14.8. If 13.1 is added, the median is the average of 13.1 and 14.8, namely 13.95.
The median is preferred to the mean when the distribution is very skew (nonsymmetrical). For instance, if 17.4 is replaced by 37.4 in the original data set, the median is still 14.8, but the mean is 18.3.
The distribution is positively skewed when there is a small proportion of unusually high values which normally results in the mean being larger than the median. The distribution is negatively skewed when there is a small proportion of unusually low values which normally results in the mean being smaller than the median.
2.3.2 Population Variance
The population variance is denoted by σ2 and it is the average of the squared deviations from the population mean. It is a measure of the variation in the values and the formula is
It cannot be calculated because it is impossible to measure all the x values. It is estimated by calculating the sample variance.
2.3.3 Sample Variance
An unbiased estimator of the population variance is the sample variance, denoted by s2. Its formula is
where is the sample mean and n is the number of sample observations.
To calculate s2, we find the sum of the squares of the deviations from the sample mean and divide by the degrees of freedom (n – 1). Division by n would give a biased estimate of σ2. The sample variance is used in hypothesis testing and in calculating confidence intervals (Chapters 4 and 5).
2.3.4 Degrees of Freedom
Only n – 1 of the deviations are free to vary. Once the sample is taken, the sample mean is fixed. If n – 1 of the deviations from the sample mean are calculated, the nth deviation is fixed, as all n deviations must add to zero. As a result, there are n – 1 degrees of freedom (df) associated with this estimator of population variance. If the population mean was known and substituted for in the formula for s2, there would be n df because in this situation where n – 1 of the deviations from μ are known, the nth deviation from μ cannot be predicted. The sum of the deviations of the n sample values from the sample mean is zero, but the sum of the deviations of the n sample values from the population mean is not zero.
Table 2.1. Corrected sums of squares for Example 2.2
2.3.5 Corrected Sum of Squares
A measure of variation which is frequently used in later chapters is the corrected sum of squares. This is the sum of the squares of the deviations from the sample mean: it is denoted by Sxx and its formula is
For Example 2.2,
2.3.6 The Computational Formula for Sxx
When calculating the deviations from the sample mean by hand, rounding-off errors will occur if is not recorded to a sufficient number of decimal places. If a large number of decimal places are used, the calculations become tedious. This problem can be avoided by using the following alternative formula:
Using this version, the corrected sum of squares is the uncorrected sum of squares minus (the square of the sum divided by n). (Σx)2/n is called the correction factor and denoted by CF.
Table 2.2. Sums of squares for Example 2.2
x
x
2
14.8
219.04
15.2
231.04
17.4
302.76
11.6
134.56
12.5
156.25
Total
71.5
1043.65
2.3.7 Standard Deviation
The standard deviation is the square root of the variance and is measured in the original units. If the x values are measured in cm, the variance is in cm2. As this is a difficult term to work with, the problem is removed by taking the square root. Thus, the standard deviation is in cm.
For Example 2.2,
For most distributions which are fairly symmetrical, about 95% of the population lies within two standard deviations of the mean.
2.3.8 The Coefficient of Variation (CV)
The CV is the standard deviation expressed as a percentage of the mean. It is independent of the units of measurement.
For the population
For a sample
For Example 2.2,
The concept of coefficient of variation can be better understood by considering the following two data sets:
They both have the same variation. You should verify that their sample standard deviations are both 1.645. However, the variation within set I is very large in relation to its mean of 4.38. This is expressed by a coefficient of variation of 37.56%. Set II is not very variable in relation to its mean of 104.38. The CV is only 1.58%.
The calculations described so far are very important. Practise them until confident that you understand the concepts of mean, variance and standard deviation. In research, you may be dealing with large amounts of data and will use a computer to do the calculations. Practise using a hand calculator with small data sets, to help you understand and interpret computer output.
2.3.9 The Hand calculator
There are many makes and models to choose from. You should obtain a calculator which gives means and standard deviations. Look for a model with SD mode or STATS mode. The buttons labelled σn–1 or s will give the sample standard deviation.
2.3.10 The Weighted Mean
The values of n1, n2 and n3 are the weights.
2.3.11 The Harmonic Mean
In Chapters 9 and 10 you will learn how to compare several treatment means using a Least Significant Difference test. The calculated LSD value assumes that each mean is based on the same number of values called the number of replications per treatment. A solution used by SAS when the number of replications is unequal is to use the harmonic mean of the number of replications and not the ordinary or arithmetic mean.
Chapter 3
Basic Data Summary
3.1 INTRODUCTION
In Chapter 2 we discussed how to summarise a few sample numbers in the form of basic statistics such as mean and standard deviation. In practice, we often have a large amount of data to summarise. Before submitting data to sophisticated statistical analyses it is advisable to obtain a numerical and graphical summary to check for ‘outliers’ and to display the distribution.
3.2 FREQUENCY DISTRIBUTIONS (DISCRETE DATA)
In a set of discrete data, many of the values are repeated. The distribution can be summarised in a frequency table and illustrated in a line diagram or bar chart. The calculation of the mean is made easier by multiplying each distinct x-value by its frequency (the number of times it appears) and the formula for the sample mean becomes:
Table 3.1. Frequency table for Example 3.1
Table 3.2. Calculations for Example 3.1
Figure 3.1. Line diagram for data of Example 3.1. Note: a line diagram is often presented as a bar chart for greater visual impact
3.2.1 The Mode
The mode is the most frequent number. In this example it is 4.
3.2.2 The Median
3.2.3 Use of Spreadsheets
You can easily carry out the calculations involved in finding the mean and standard deviation of a frequency distribution by entering the data for x and f in the first two columns of a spreadsheet. You can then instruct the computer to calculate the entries for xf and x2f in the next two columns. The sums of columns 2, 3 and 4 can be found and entered in the formulae used to find and s2.
3.3 FREQUENCY DISTRIBUTIONS (CONTINUOUS DATA)
Consider variables such as height and weight which are measured on a continuous scale. No two plants will have exactly the same weight if measurements are made with sufficient accuracy. Important features can be seen if the data are grouped into classes. If weights are recorded to the nearest gram, this is a form of grouping. A plant whose weight is recorded as 80 g has a real weight between 79.5 and 80.5 g, provided the scales are not biased.
Notice that several plots have yields of 64 g. This is because a form of grouping has already been applied by recording the weights to the nearest gram. These plots have yields of between 63.5 and 64.5 g.
The grouping into classes is arbitrary. We could have used smaller or larger class intervals, or made the lower boundary of the first class different from 39.5. Also, we could have had classes of unequal width.
3.3.1 The Histogram
The tally column in Table 3.3 gives an idea of the distribution of weights. A clearer representation of the distribution is obtained by drawing a histogram and Figure 3.2 shows a histogram for the data of Example 3.2.
Table 3.3. Grouped frequency table for data of Example 3.2
Figure 3.2. Histogram of data of Example 3.2 produced by Minitab
Once you have entered your data into a computer, you can easily obtain a histogram displayed on the screen. Outliers (unusual data values which may be due to recording errors) are easily identified and should be investigated. A character histogram (Figure 3.3) is often better at showing skewness and outliers.
Figure 3.3. Character histogram of data for Example 3.2
Do not confuse a bar chart with a histogram. The bars of a bar chart represent distinct categories, such as different varieties. The bars do not have to be contiguous. The bars of a histogram represent the values of a continuous variable divided into class intervals and are contiguous (no gaps between them).
3.3.2 Quartiles and Ranges
The lower quartile (Q1) is such that 25% of the observations are less than Q1.
The upper quartile (Q3) is such that 75% of the observations are less than Q3 and 25% are above
The range
This is the largest value minus the smallest value and it is very sensitive to outliers.
3.3.3 Other Graphical Methods
A boxplot is a graphical device for displaying the quartiles and the range of a distribution. A boxplot for the data of Example 3.2 produced by Minitab is shown in Figure 3.4.
Other visual methods of distribution summary are the dot plot and the stem and leaf plot. Figures 3.5 and 3.6 show Minitab versions of these for Example 3.2.
The stem-and-leaf plot is similar to the character histogram but gives more information. The first column shows cumulative frequency, the second column is the stem and the third column is the leaf which represents the final digit. For this example it shows that the two smallest yields are 42 and 44 g. The next highest yield is 49 g and the highest yield is 108 g. Twenty-three yields are less than or equal to 64 g and 35 yields are less than or equal to 68 g. Thirty-nine yields are greater than or equal to 75 g and 28 yields are greater than or equal to 80 g. Six yields are in the median class from 70 to 73. This diagram can also show whether any bias has taken place during measurement when recording the final digit. In this example an eight appears as the final digit 13 times whereas a zero only appears six times. Should we regard this as suspicious?
Figure 3.4. Boxplot for data of Example 3.2
Figure 3.5. Dotplot for data of Example 3.2 showing 80 yields (g)
These graphical methods are useful for exploratory data analysis. They enable you to compare several distributions and check for symmetry or lack of it (skewness). They are particularly useful in highlighting any unusual values (outliers) which may or may not be due to recording errors.
3.4 DESCRIPTIVE STATISTICS
While graphical methods give visual summaries of a distribution, descriptive statistics give numerical summaries of central tendency and dispersion and can be used to compare distributions and carry out further analyses.
Figure 3.6. Stem-and-leaf plot for data of Example 3.2 showing 80 yields (g)
Chapter 4
The Normal Distribution, the t-Distribution and Confidence Intervals
4.1 INTRODUCTION TO THE NORMAL DISTRIBUTION
Suppose you are interested in the distribution of plant heights in a field; call the total collection of heights the population of interest. An idea of plant height distribution can be obtained by taking a large random sample of plants and drawing a histogram and the shape of the histogram will depend on the widths of the classes chosen and their class boundaries. If you were able to measure all plants in the field, the heights could be grouped into a very large number of classes. Each class width would be very small. As a result, the mid-points of the tops of the rectangles could be joined to form a smooth curve.
For many variables such as plant height and yield the smoothed-out histogram is approximately symmetrical about the mean, and about 95% of the population observations lie within 2 standard deviations of the mean. The common occurrence of such distributions has led to the importance of a theoretical curve used to describe them. It is called the normal distribution (Figure 4.1).
