R. Pitt April 5, 2007 Module 5: Statistical Analyses INTRODUCTION .......................................................................................................................... 2 GENERAL STEPS IN THE ANALYSIS OF DATA .................................................................. 2 EXPERIMENTAL DESIGN .................................................................................................................... 3 Sample size................................................................................................................................... 4 Determination of Outliers ............................................................................................................ 5 SELECTION OF STATISTICAL PROCEDURES ........................................................................................ 5 Statistical Power .......................................................................................................................... 5 Comparison Tests......................................................................................................................... 5 Data Associations and Model Building........................................................................................ 7 EXPLORATORY DATA ANALYSES ...................................................................................................... 8 Basic Data Plots........................................................................................................................... 8 Probability Plots .......................................................................................................................... 8 Digidot Plot................................................................................................................................ 13 Scatterplots................................................................................................................................. 14 Grouped Box and Whisker Plots................................................................................................ 17 COMPARING MULTIPLE SETS OF DATA WITH GROUP COMPARISON TESTS...................................... 19 Simple Comparison Tests with Two Groups .............................................................................. 20 Comparisons of Many Groups ................................................................................................... 23 DATA ASSOCIATIONS ...................................................................................................................... 23 Correlation Matrices.................................................................................................................. 24 Hierarchical Cluster Analyses ...................................................................................................26 Principal Component Analyses (PCA) and Factor Analyses..................................................... 29 ANALYSIS OF TRENDS IN RECEIVING WATER INVESTIGATIONS....................................................... 33 Preliminary Evaluations before Trend Analyses are Used........................................................ 34 Statistical Methods Available for Detecting Trends................................................................... 35 Example of Long-Term Trend Analyses for Lake Rönningesjön, Sweden ................................. 35 EXAMPLE STORMWATER DATA ANALYSIS.................................................................... 50 SAMPLING EFFORT AND BASIC DATA PRESENTATIONS ................................................................... 50 SUMMARY OF DATA ........................................................................................................................ 55 Data Summaries......................................................................................................................... 55 EXPLORATORY DATA ANALYSIS OF RAINFALL AND RUNOFF CHARACTERISTICS FOR URBAN AREAS57 EVALUATION OF DATA GROUPINGS AND ASSOCIATIONS................................................................. 64 Exploratory Data Analyses ........................................................................................................ 64 Simple Correlation Analyses...................................................................................................... 68 Complex Correlation Analyses................................................................................................... 72 Model Building........................................................................................................................... 78 “Outliers” and Extreme Observations....................................................................................... 99 STATISTICAL EVALUATION OF A WATER TREATMENT CONTROL DEVICE; THE UPFLOW FILTER ....................................................................................................................................... 107 CONTROLLED EXPERIMENTS ......................................................................................................... 107 ACTUAL STORM EVENT MONITORING........................................................................................... 108 OTHER EXPLORATORY DATA METHODS USED TO EVALUATE STORMWATER CONTROLS.............. 114 EVALUATION OF BACTERIA DECAY COEFFICIENTS FOR FATE ANALYSES ..... 120 FATE MECHANISMS FOR MICROORGANISMS.................................................................................. 120 DECAY RATE CURVES OF LAKE MICROORGANISMS ...................................................................... 125 REFERENCES ........................................................................................................................... 127 APPENDIX A: FACTORIAL ANALYSES EXAMPLES...................................................... 130 EXAMPLES OF AN EXPERIMENTAL DESIGN USING FACTORIAL ANALYSES: SEDIMENT SCOUR....... 130 Introduction.............................................................................................................................. 130 Experimental Design................................................................................................................ 132 Results ...................................................................................................................................... 132 EXAMPLE USING FACTORIAL ANALYSES TO EVALUATE EXISTING DATA: LAKE TUSCALOOSA WATER QUALITY 139 Introduction.............................................................................................................................. 139 Experimental Design for Lake Tuscaloosa .............................................................................. 141 Experimental Design and Factorial Analysis for the North River site .................................... 141 Summary................................................................................................................................... 143 FACTORIAL ANALYSIS USED IN MODELING THE FATES OF POLYCYCLIC AROMATIC HYDROCARBONS (PAHS) AFFECTING TREATABILITY OF STORMWATER................................................................................ 145 Abstract .................................................................................................................................... 145 Introduction.............................................................................................................................. 145 Methodology............................................................................................................................. 146 Results ...................................................................................................................................... 147 Conclusions.............................................................................................................................. 151 Acknowledgements ................................................................................................................... 152 APPENDIX B: EXAMPLES FOR SPECIFIC STATISTICAL TESTS ............................... 153 PROBABILITY PLOT PREPARATION USING EXCEL .......................................................................... 153 COMPARISONS OF TWO SETS OF DATA USING EXCEL .................................................................... 162 Paired Tests: ............................................................................................................................ 162 Independent Tests: ................................................................................................................... 163 EXAMPLE OF ANOVA USING EXCEL............................................................................................. 164 EXAMPLE REGRESSION ANALYSIS USING EXCEL........................................................................... 165 OTHER STATISTICAL TESTS AVAILABLE IN EXCEL........................................................................ 169 WILCOXON RANK-SUM TEST ........................................................................................................ 170 Introduction Statistical analyses are a critical component of research. The analyses that are to be conducted for a specific research activity must be carefully thought out in advance of any data collection and be an integral component of the experimental design activities. This module reviews a number of statistical tests that have been useful for a variety of water quality projects conducted by the author. The field of statistical