Statistical Data Analysis Explained
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Preface xiii
Acknowledgements xv
About the authors xvii
1 Introduction 1
1.1 The Kola Ecogeochemistry Project 5
1.1.1 Short description of the Kola Project survey area 6
1.1.2 Sampling and characteristics of the different sample materials 9
1.1.3 Sample preparation and chemical analysis 11
2 Preparing the Data for Use in R and DAS+R 13
2.1 Required data format for import into R and DAS+R 14
2.2 The detection limit problem 17
2.3 Missing values 20
2.4 Some "typical" problems encountered when editing a laboratory data report file to a DAS+R file 21
2.4.1 Sample identification 22
2.4.2 Reporting units 22
2.4.3 Variable names 23
2.4.4 Results below the detection limit 23
2.4.5 Handling of missing values 24
2.4.6 File structure 24
2.4.7 Quality control samples 25
2.4.8 Geographical coordinates, further editing and some unpleasant limitations of spreadsheet programs 25
2.5 Appending and linking data files 25
2.6 Requirements for a geochemical database 27
2.7 Summary 28
3 Graphics to Display the Data Distribution 29
3.1 The one-dimensional scatterplot 29
3.2 The histogram 31
3.3 The density trace 34
3.4 Plots of the distribution function 35
3.4.1 Plot of the cumulative distribution function (CDF-plot) 35
3.4.2 Plot of the empirical cumulative distribution function (ECDF-plot) 36
3.4.3 The quantile-quantile plot (QQ-plot) 36
3.4.4 The cumulative probability plot (CP-plot) 39
3.4.5 The probability-probability plot (PP-plot) 40
3.4.6 Discussion of the distribution function plots 41
3.5 Boxplots 41
3.5.1 The Tukey boxplot 42
3.5.2 The log-boxplot 44
3.5.3 The percentile-based boxplot and the box-and-whisker plot 46
3.5.4 The notched boxplot 47
3.6 Combination of histogram, density trace, one-dimensional scatterplot, boxplot, and ECDF-plot 48
3.7 Combination of histogram, boxplot or box-and-whisker plot, ECDF-plot, and CP-plot 49
3.8 Summary 50
4 Statistical Distribution Measures 51
4.1 Central value 51
4.1.1 The arithmetic mean 51
4.1.2 The geometric mean 52
4.1.3 The mode 52
4.1.4 The median 52
4.1.5 Trimmed mean and other robust measures of the central value 53
4.1.6 Influence of the shape of the data distribution 53
4.2 Measures of spread 56
4.2.1 The range 56
4.2.2 The interquartile range (IQR) 56
4.2.3 The standard deviation 57
4.2.4 The median absolute deviation (MAD) 57
4.2.5 Variance 58
4.2.6 The coefficient of variation (CV) 58
4.2.7 The robust coefficient of variation (CVR) 59
4.3 Quartiles, quantiles and percentiles 59
4.4 Skewness 59
4.5 Kurtosis 59
4.6 Summary table of statistical distribution measures 60
4.7 Summary 60
5 Mapping Spatial Data 63
5.1 Map coordinate systems (map projection) 64
5.2 Map scale 65
5.3 Choice of the base map for geochemical mapping 66
5.4 Mapping geochemical data with proportional dots 68
5.5 Mapping geochemical data using classes 69
5.5.1 Choice of symbols for geochemical mapping 70
5.5.2 Percentile classes 71
5.5.3 Boxplot classes 71
5.5.4 Use of ECDF- and CP-plot to select classes for mapping 74
5.6 Surface maps constructed with smoothing techniques 74
5.7 Surface maps constructed with kriging 76
5.7.1 Construction of the (semi)variogram 76
5.7.2 Quality criteria for semivariograms 79
5.7.3 Mapping based on the semivariogram (kriging) 79
5.7.4 Possible problems with semivariogram estimation and kriging 80
5.8 Colour maps 82
5.9 Some common mistakes in geochemical mapping 84
5.9.1 Map scale 84
5.9.2 Base map 84
5.9.3 Symbol set 84
5.9.4 Scaling of symbol size 84
5.9.5 Class selection 86
5.10 Summary 88
6 Further Graphics for Exploratory Data Analysis 91
6.1 Scatterplots (xy-plots) 91
6.1.1 Scatterplots with user-defined lines or fields 92
6.2 Linear regression lines 93
6.3 Time trends 95
6.4 Spatial trends 97
6.5 Spatial distance plot 99
6.6 Spiderplots (normalised multi-element diagrams) 101
6.7 Scatterplot matrix 102
6.8 Ternary plots 103
6.9 Summary 106
7 Defining Background and Threshold, Identification of Data Outliers and Element Sources 107
7.1 Statistical methods to identify extreme values and data outliers 108
7.1.1 Classical statistics 108
7.1.2 The boxplot 109
7.1.3 Robust statistics 110
7.1.4 Percentiles 111
7.1.5 Can the range of background be calculated? 112
7.2 Detecting outliers and extreme values in the ECDF- or CP-plot 112
7.3 Including the spatial distribution in the definition of background 114
7.3.1 Using geochemical maps to identify a reasonable threshold 114
7.3.2 The concentration-area plot 115
7.3.3 Spatial trend analysis 118
7.3.4 Multiple background populations in one data set 119
7.4 Methods to distinguish geogenic from anthropogenic element sources 120
7.4.1 The TOP/BOT-ratio 120
7.4.2 Enrichment factors (EFs) 121
7.4.3 Mineralogical versus chemical methods 128
7.5 Summary 128
8 Comparing Data in Tables and Graphics 129
8.1 Comparing data in tables 129
8.2 Graphical comparison of the data distributions of several data sets 133
8.3 Comparing the spatial data structure 136
8.4 Subset creation - a mighty tool in graphical data analysis 138
8.5 Data subsets in scatterplots 141
8.6 Data subsets in time and spatial trend diagrams 142
8.7 Data subsets in ternary plots 144
8.8 Data subsets in the scatterplot matrix 146
8.9 Data subsets in maps 147
8.10 Summary 148
9 Comparing Data Using Statistical Tests 149
9.1 Tests for distribution (Kolmogorov-Smirnov and Shapiro-Wilk tests) 150
9.1.1 The Kola data set and the normal or lognormal distribution 151
9.2 The one-sample t-test (test for the central value) 154
9.3 Wilcoxon signed-rank test 156
9.4 Comparing two central values of the distributions of independent data groups 157
9.4.1 The two-sample t-test 157
9.4.2 The Wilcoxon rank sum test 158
9.5 Comparing two central values of matched pairs of data 158
9.5.1 The paired t-test 158
9.5.2 The Wilcoxon test 160
9.6 Comparing the variance of two data sets 160
9.6.1 The F-test 160
9.6.2 The Ansari-Bradley test 160
9.7 Comparing several central values 161
9.7.1 One-way analysis of variance (ANOVA) 161
9.7.2 Kruskal-Wallis test 161
9.8 Comparing the variance of several data groups 161
9.8.1 Bartlett test 161
9.8.2 Levene test 162
9.8.3 Fligner test 162
9.9 Comparing several central values of dependent groups 163
9.9.1 ANOVA with blocking (two-way) 163
9.9.2 Friedman test 163
9.10 Summary 164
10 Improving Data Behaviour for Statistical Analysis: Ranking and Transformations 167
10.1 Ranking/sorting 168
10.2 Non-linear transformations 169
10.2.1 Square root transformation 169
10.2.2 Power transformation 169
10.2.3 Log(arithmic)-transformation 169
10.2.4 Box-Cox transformation 171
10.2.5 Logit transformation 171
10.3 Linear transformations 172
10.3.1 Addition/subtraction 172
10.3.2 Multiplication/division 173
10.3.3 Range transformation 174
10.4 Preparing a data set for multivariate data analysis 174
10.4.1 Centring 174
10.4.2 Scaling 174
10.5 Transformations for closed number systems 176
10.5.1 Additive logratio transformation 177
10.5.2 Centred logratio transformation 178
10.5.3 Isometric logratio transformation 178
10.6 Summary 179
11 Correlation 181
11.1 Pearson correlation 182
11.2 Spearman rank correlation 183
11.3 Kendall-tau correlation 184
11.4 Robust correlation coefficients 184
11.5 When is a correlation coefficient significant? 185
11.6 Working with many variables 185
11.7 Correlation analysis and inhomogeneous data 187
11.8 Correlation results following additive logratio or centred logratio transformations 189
11.9 Summary 191
12 Multivariate Graphics 193
12.1 Profiles 193
12.2 Stars 194
12.3 Segments 196
12.4 Boxes 197
12.5 Castles and trees 198
12.6 Parallel coordinates plot 198
12.7 Summary 200
13 Multivariate Outlier Detection 201
13.1 Univariate versus multivariate outlier detection 201
13.2 Robust versus non-robust outlier detection 204
13.3 The chi-square plot 205
13.4 Automated multivariate outlier detection and visualisation 205
13.5 Other graphical approaches for identifying outliers and groups 208
13.6 Summary 210
14 Principal Component Analysis (PCA) and Factor Analysis (FA) 211
14.1 Conditioning the data for PCA and FA 212
14.1.1 Different data ranges and variability, skewness 212
14.1.2 Normal distribution 213
14.1.3 Data outliers 213
14.1.4 Closed data 214
14.1.5 Censored data 215
14.1.6 Inhomogeneous data sets 215
14.1.7 Spatial dependence 215
14.1.8 Dimensionality 216
14.2 Principal component analysis (PCA) 216
14.2.1 The scree plot 217
14.2.2 The biplot 219
14.2.3 Mapping the principal components 220
14.2.4 Robust versus classical PCA 221
14.3 Factor analysis 222
14.3.1 Choice of factor analysis method 224
14.3.2 Choice of rotation method 224
14.3.3 Number of factors extracted 224
14.3.4 Selection of elements for factor analysis 225
14.3.5 Graphical representation of the results of factor analysis 225
14.3.6 Robust versus classical factor analysis 229
14.4 Summary 231
15 Cluster Analysis 233
15.1 Possible data problems in the context of cluster analysis 234
15.1.1 Mixing major, minor and trace elements 234
15.1.2 Data outliers 234
15.1.3 Censored data 235
15.1.4 Data transformation and standardisation 235
15.1.5 Closed data 235
15.2 Distance measures 236
15.3 Clustering samples 236
15.3.1 Hierarchical methods 236
15.3.2 Partitioning methods 239
15.3.3 Model-based methods 240
15.3.4 Fuzzy methods 242
15.4 Clustering variables 242
15.5 Evaluation of cluster validity 244
15.6 Selection of variables for cluster analysis 246
15.7 Summary 247
16 Regression Analysis (RA) 249
16.1 Data requirements for regression analysis 251
16.1.1 Homogeneity of variance and normality 251
16.1.2 Data outliers, extreme values 253
16.1.3 Other considerations 253
16.2 Multiple regression 254
16.3 Classical least squares (LS) regression 255
16.3.1 Fitting a regression model 255
16.3.2 Inferences from the regression model 256
16.3.3 Regression diagnostics 259
16.3.4 Regression with opened data 259
16.4 Robust regression 260
16.4.1 Fitting a robust regression model 261
16.4.2 Robust regression diagnostics 262
16.5 Model selection in regression analysis 264
16.6 Other regression methods 266
16.7 Summary 268
17 Discriminant Analysis (DA) and Other Knowledge-Based Classification Methods 269
17.1 Methods for discriminant analysis 269
17.2 Data requirements for discriminant analysis 270
17.3 Visualisation of the discriminant function 271
17.4 Prediction with discriminant analysis 272
17.5 Exploring for similar data structures 275
17.6 Other knowledge-based classification methods 276
17.6.1 Allocation 276
17.6.2 Weighted sums 278
17.7 Summary 280
18 Quality Control (QC) 281
18.1 Randomised samples 282
18.2 Trueness 282
18.3 Accuracy 284
18.4 Precision 286
18.4.1 Analytical duplicates 287
18.4.2 Field duplicates 289
18.5 Analysis of variance (ANOVA) 290
18.6 Using maps to assess data quality 293
18.7 Variables analysed by two different analytical techniques 294
18.8 Working with censored data - a practical example 296
18.9 Summary 299
19 Introduction to R and Structure of the DAS+R Graphical User Interface 301
19.1 R 301
19.1.1 Installing R 301
19.1.2 Getting started 302
19.1.3 Loading data 302
19.1.4 Generating and saving plots in R 303
19.1.5 Scatterplots 305
19.2 R-scripts 307
19.3 A brief overview of relevant R commands 311
19.4 DAS+R 315
19.4.1 Loading data into DAS+R 316
19.4.2 Plotting diagrams 316
19.4.3 Tables 317
19.4.4 Working with "worksheets" 317
19.4.5 Groups and subsets 317
19.4.6 Mapping 318
19.5 Summary 318
References 321
Index 337
