Statistics for Environmental Science and Management - Chapter 1 ppt

1 The Role of Statistics in Environmental Science 2.2 Simple Random Sampling 2.3 Estimation of Population Means 2.4 Estimation of Population Totals 2.5 Estimation of Proportions 2.6 Samp

Statistics for Environmental Science and Management

Bryan F.J Manly

Statistical Consultant Western Ecosystem Technology Inc

Wyoming, USA

CHAPMAN & HALL/CRC

Boca Raton London New York Washington, D.C

Library of Congress Cataloging-in-Publication Data

Manly, Bryan F.J., 1944-

Statistics for environmental science and management / by Bryan F.J Manly

p cm

Includes bibliographical references and index

ISBN l-58488-029-5 (alk paper)

1 Environmental sciences - Statistical methods 2 Environmental

management - Statistical methods I Title

GE45.S73 M36 2000

CIP

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A great deal of intelligence can be invested in ignorance when the need for illusion is deep.

Saul Bellow

1 The Role of Statistics in Environmental Science

2.2 Simple Random Sampling

2.3 Estimation of Population Means

2.4 Estimation of Population Totals

2.5 Estimation of Proportions

2.6 Sampling and Non-Sampling Errors

2.7 Stratified Random Sampling

2.13 Choosing Sample Sizes

2.14 Unequal Probability Sampling

2.15 The Data Quality Objectives Process

2.16 Chapter Summary

3 Models for Data

3.1 Statistical Models

3.2 Discrete Statistical Distributions

3.3 Continuous Statistical Distributions

3.4 The Linear Regression Model

3.5 Factorial Analysis of Variance

3.6 Generalized Linear Models

3.7 Chapter Summary

4 Drawing Conclusions from Data

4.1 Introduction

4.2 Observational and Experimental Studies

4.3 True Experiments and Quasi-Experiments

4.4 Design-Based and Model-Based Inference

4.5 Tests of Significance and Confidence Intervals

5.2 Purposely Chosen Monitoring Sites

5.3 Two Special Monitoring Designs

5.4 Designs Based on Optimization

5.5 Monitoring Designs Typically Used

5.6 Detection of Changes by Analysis of Variance

5.7 Detection of Changes Using Control Charts

5.8 Detection of Changes Using CUSUM Charts

5.9 Chi-Squared Tests for a Change in a Distribution

5.10 Chapter Summary

6 Impact Assessment

6.1 Introduction

6.2 The Simple Difference Analysis with BACI Designs

6.3 Matched Pairs with a BACI Design

7.2 Problems with Tests of Significance

7.3 The Concept of Bioequivalence

7.4 Two-Sided Tests of Bioequivalence

8.4 Tests for Randomness

8.5 Detection of Change Points and Trends

8.6 More Complicated Time Series Models

8.8 Forecasting

8.9 Chapter Summary

9 Spatial Data Analysis

9.1 Introduction

9.2 Types of Spatial Data

9.3 Spatial Patterns in Quadrat Counts

9.4 Correlation Between Quadrat Counts

9.5 Randomness of Point Patterns

9.6 Correlation Between Point Patterns

9.7 Mantel Tests for Autocorrelation

10.4 Comparing the Means of Two or More Samples

10.5 Regression with Censored Data

10.6 Chapter Summary

11 Monte Carlo Risk Assessment

11.1 Introduction

11.2 Principles for Monte Carlo Risk Assessment

11.3 Risk Analysis Using a Spreadsheet Add-On

A2 Distributions for Sample Data

A3 Distributions of Sample Statistics

A4 Tests of Significance

A5 Confidence Intervals

A6 Covariance and Correlation

B1 The Standard Normal Distribution

B2 Critical Values for the t-Distribution

B3 Critical Values for the Chi-Squared Distribution

B4 Critical Values for the F-Distribution

B5 Critical Values for the Durbin-Watson Statistic

References

This book is intended to introduce environmental scientists andmanagers to the statistical methods that will be useful for them in theirwork A secondary aim was to produce a text suitable for a course instatistics for graduate students in the environmental science area Iwrote the book because it seemed to me that these groups shouldreally learn about statistical methods in a special way It is true thattheir needs are similar in many respects to those working in otherareas However, there are some special topics that are relevant toenvironmental science to the extent that they should be covered in anintroductory text, although they would probably not be mentioned at all

in such a text for a more general audience I refer to environmentalmonitoring, impact assessment, assessing site reclamation, censoreddata, and Monte Carlo risk assessment, which all have their ownchapters here

The book is not intended to be a complete introduction to statistics.Rather, it is assumed that readers have already taken a course orread a book on basic methods, covering the ideas of random variation,statistical distributions, tests of significance, and confidence intervals.For those who have done this some time ago, Appendix A is meant toprovide a quick refresher course

A number of people have contributed directly or indirectly to thisbook I must first mention Lyman McDonald of West Inc., Cheyenne,Wyoming, who first stimulated my interest in environmental statistics,

as distinct from ecological statistics Much of the contents of the bookare influenced by the discussions that we have had on mattersstatistical Jennifer Brown from the University of Canterbury in NewZealand has influenced the contents because we have shared theteaching of several short courses on statistics for environmentalscientists and managers Likewise, sharing a course on statistics forMSc students of environmental science with Caryn Thompson andDavid Fletcher has also had an effect on the book Other people aretoo numerous to name, so I would just like to thank generally thosewho have contributed data sets, helped me check references andequations, etc

Most of this book was written in the Department of Mathematicsand Statistics at the University of Otago As usual, the university wasgenerous with the resources that are needed for the major effort ofwriting a book, including periods of sabbatical leave that enabled me

to write large parts of the text without interruptions, and an excellentlibrary

However, the manuscript would definitely have taken longer tofinish if I had not been invited to spend part of the year 2000 as aVisiting Researcher at the Max Planck Institute for Limnology at Plön

in Germany This enabled me to write the final chapters and put thewhole book together I am very grateful to Winfried Lampert, theDirector of the Institute, for his kind invitation to come to Plön, and forallowing me to use the excellent facilities at the Institute while I wasthere

The Saul Bellow quotation above may need some explanation Itresults from attending meetings where an environmental matter isargued at length, with everyone being ignorant about the true facts ofthe case Furthermore, one suspects that some people there wouldprefer not to know the true facts because this would be likely to endthe arguments

Bryan F.J Manly

May 2000

CHAPTER 1 The Role of Statistics in Environmental Science

1.1 Introduction

In this chapter the role of statistics in environmental science isconsidered by examining some specific examples First, however, animportant point needs to be made The importance of statistics isobvious because much of what is learned about the environment isbased on numerical data Therefore the appropriate handling of data

is crucial Indeed, the use of incorrect statistical methods may makeindividuals and organizations vulnerable to being sued for largeamounts of money Certainly in the United States it appears thatincreasing attention to the use of statistical methods is driven by thefear of litigation

One thing that it is important to realize in this context is that there

is usually not a single correct way to gather and analyse data At bestthere may be several alternative approaches that are all about equallygood At worst the alternatives may involve different assumptions,and lead to different conclusions This will become apparent fromsome of the examples in this and the following chapters

1.2 Some Examples

The following examples demonstrate the non-trivial statisticalproblems that can arise in practice, and show very clearly theimportance of the proper use of statistical theory Some of theseexamples are revisited again in later chapters

For environmental scientists and resource managers there arethree broad types of situation that are often of interest:

(a) baseline studies intended to document the present state of theenvironment in order to establish future changes resulting, forexample, from unforeseen events such as oil spills;

(b) targeted studies designed to assess the impact of planned eventssuch as the construction of a dam, or accidents such as oil spills;and

(c) regular monitoring intended to detect trends and changes inimportant variables, possibly to ensure that compliance conditionsare being met for an industry that is permitted to discharge smallamounts of pollutants into the environment.

The examples include all of these types of situations

Example 1.1 The Exxon Valdez Oil Spill

Oil spills resulting from the transport of crude and refined oils occurfrom time to time, particularly in coastal regions Some very largespills (over 100,000 tonnes) have attracted considerable interest

around the world Notable examples are the Torrey Canyon spill in the English Channel in 1967, the Amoco Cadiz off the coast of Brittany, France in 1978, and the grounding of the Braer off the

Shetland Islands in 1993 These spills all bring similar challenges fordamage control for the physical environment and wildlife There isintense concern from the public, resulting in political pressures onresource managers There is the need to assess both short-term andlong-term environmental impacts Often there are lengthy legal cases

to establish liability and compensation terms

One of the most spectacular oil spills was that of the Exxon Valdez,

which grounded on Bligh Reef in Prince William Sound, Alaska, on 24March 1989, spilling more than 41 million litres of Alaska north slopecrude oil This was the largest spill up to that time in United States

coastal waters, although far from the size of the Amoco Cadiz spill.

The publicity surrounding it was enormous and the costs for cleanup,damage assessment and compensation have been considerable atnearly $US12,000 per barrel lost, compared with the more typical

$US5,000 per barrel, for which the typical sale price is only about

$US15 (Wells et al., 1995, p 5) Figure 1.1 shows the path of the oilthrough Prince William Sound and the western Gulf of Alaska

There were many targeted studies of the Exxon Valdez spill related

to the persistence and fate of the oil and the impact on fisheries andwildlife Here only three of these studies, concerned with the shorelineimpact of the oil, are considered The investigators used differentstudy designs and all met with complications that were not foreseen

in advance of sampling The three studies are Exxon's Shoreline

Ecology Program (Page et al., 1995; Gilfillan et al., 1995), the Oil Spill Trustees' Coastal Habitat Injury Assessment (Highsmith et al., 1993; McDonald et al., 1995), and the Biological Monitoring Survey (Houghton et al., 1993) The summary here owes much to a paper

presented by Harner et al (1995) at an International Environmetrics

Conference in Kuala Lumpur, Malaysia

Figure 1.1 The path of the oil spill from the Exxon Valdez that occurred on

24 March (day 1) until 18 May 1989 (day 56), through Prince William Soundand the western Gulf of Alaska

The Exxon Shoreline Ecology Program

The Exxon Shoreline Ecology Program started in 1989 with thepurposeful selection of a number of heavily oiled sites along theshoreline that were to be measured over time in order to determinerecovery rates Because these sites are not representative of theshoreline potentially affected by oil they were not intended to assessthe overall damage

In 1990, using a stratified random sampling design of a type that

is discussed in Chapter 2, the study was enlarged to include manymore sites Basically, the entire area of interest was divided into anumber of short segments of shoreline Each segment was thenallocated to one of 16 strata based on the substrate type (exposedbedrock, sheltered bedrock, boulder/cobble, and pebble/gravel) andthe degree of oiling (none, light, moderate, and heavy) For example,the first stratum was exposed bedrock with no oiling Finally, four sites

were chosen from each of the 16 strata for sampling to determine theabundances of more than a thousand species of animals and plants.

A number of physical variables were also measured at each site.The analysis of the data collected from the Exxon ShorelineEcology Program was based on the use of what are calledgeneralized linear models for species counts These models aredescribed in Chapter 3, and here it suffices to say that the effects ofoiling were estimated on the assumption that the model used for eachspecies was correct, with an allowance being made for differences inphysical variables between sites

A problem with the sampling design was that the initial allocation

of shoreline segments to the 16 strata was based on the information

in a geographical information system (GIS) However, this resulted insome sites being misclassified, particularly in terms of oiling levels.Furthermore, sites were not sampled if they were near an active eaglenest or human activity The net result was that the samplingprobabilities used in the study design were not quite what they weresupposed to be The investigators considered that the effect of thiswas minor However, the authors of the National Oceanic andAtmospheric Administrations guidance document for assessing thedamage from oil spills argue that this could be used in an attempt to

discredit the entire study (Bergman et al., 1995, Section F) It is

therefore an example of how a minor deviation from the requirements

of a standard study design may lead to potentially very seriousconsequences

The Oil Spill Trustees' Coastal Habitat Injury Assessment

The Exxon Valdez Oil Spill Trustee Council was set up to oversee the

allocation of funds from Exxon for the restoration of Prince WilliamSound and Alaskan waters Like the Exxon Shoreline EcologyProgram, the 1989 Coastal Habitat Injury Assessment study that wasset up by the Council was based on a stratified random samplingdesign of a type that will be discussed in Chapter 3 There were 15strata used, with these defined by five habitat types, each with threelevels of oiling Sample units were shoreline segments with varyinglengths, and these were selected using a GIS system, withprobabilities proportional to their lengths

Unfortunately, so many sites were misclassified by the GIS systemthat the 1989 study design had to be abandoned in 1990 Instead,each of the moderately and heavily oiled sites that were sampled in

1989 was matched up with a comparable unoiled control site based

on physical characteristics, to give a paired comparison design The

investigators then considered whether the paired sites weresignificantly different with regard to species abundance.

There are two aspects of the analysis of the data from this studythat are unusual First, the results of comparing site pairs (oiled andunoiled) were summarised as p-values (probabilities of observingdifferences as large as those seen on the hypothesis that oiling had

no effect) These p-values were then combined using a meta-analysiswhich is a method for combining data that is described in Chapter 4.This method for assessing the evidence was used because each sitepair was thought to be an independent study of the effects of oiling.The second unusual aspect of the analysis was the weighting ofresults that was used for one of the two methods of meta-analysis thatwas employed By weighting the results for each site pair by thereciprocal of the probability of the pair being included in the study, itwas possible to make inferences with respect to the entire set ofpossible pairs in the study region This was not a particularly simpleprocedure to carry out because inclusion probabilities had to beestimated by simulation It did, however, overcome the problemsintroduced by the initial misclassification of sites

The Biological Monitoring Survey

The Biological Monitoring Survey was instigated by the NationalOceanic and Atmospheric Administration to study differences inimpact between oiling alone and oiling combined with high pressurehot water washing at sheltered rocky sites Thus there were threecategories of sites used Category 1 sites were unoiled Category 2sites were oiled but not washed Category 3 sites were oiled andwashed Sites were subjectively selected, with unoiled ones beingchosen to match those in the other two categories Oiling levels werealso classified as being light or moderate/heavy depending on theirstate when they were laid out in 1989 Species counts andpercentage cover were measured at sampled sites

Randomization tests were used to assess the significance of thedifferences between the sites in different categories because of theextreme nature of the distributions found for the recorded data Thesetypes of test are discussed in Chapter 4 Here it is just noted that thehypothesis tested is that an observation was equally likely to haveoccurred for a site in any one of the three categories These tests cancertainly provide valid evidence of differences between the categories.However, the subjective methods used to select sites allow theargument to be made that any significant differences were due to theselection procedure rather than the oiling or the hot water treatment

Another potential problem with the analysis of the study is that itmay have involved pseudoreplication (treating correlated data asindependent data), which is also defined and discussed in Chapter 4.This is because sampling stations along a transect on a beach weretreated as if they provided completely independent data, although infact some of these stations were in close proximity In reality,observations taken close together in space can be expected to bemore similar than observations taken far apart Ignoring this fact mayhave led to a general tendency to conclude that sites in the differentcategories differed when this was not really the case.

General Comments on the Three Studies

The three studies on the Exxon Valdez oil spill took different

approaches and lead to answers to different questions The ExxonShoreline Ecology Program was intended to assess the impact ofoiling over the entire spill zone by using a stratified random samplingdesign A minor problem is that the standard requirements of thesampling design were not quite followed because of sitemisclassification and some restrictions on sites that could be sampled.The Oil Trustees' Coastal Habitat Study was badly upset by sitemisclassification in 1989, and was therefore converted to a pairedcomparison design in 1990 to compare moderately or heavily oiledsites with subjectively chosen unoiled sites This allowed evidence forthe effect of oiling to be assessed, but only at the expense of acomplicated analysis involving the use of simulation to estimate theprobability of a site being used in the study, and a special method tocombine the results for different pairs of sites The BiologicalMonitoring Survey focussed on assessing the effects of hot waterwashing, and the design gives no way for making inferences to theentire area affected by the oil spill

All three studies are open to criticism in terms of the extent towhich they can be used to draw conclusions about the overall impact

of the oil spill in the entire area of interest For the Exxon CoastalEcology Program and the Trustees' Coastal Habitat InjuryAssessment, this was the result of using stratified random samplingdesigns for which the randomization was upset to some extent As a

case study the Exxon Valdez oil spill should, therefore, be a warning

to those involved in oil spill impact assessment in the future aboutproblems that are likely to occur with this type of design Anotheraspect of these two studies that should give pause for thought is thatthe analyses that had to be conducted were rather complicated and