Statistical Tools for Analysis. Monitoring Objectives. Objectives Drive the Statistics Used

Transcription

1 Statistical Tools for Analysis Anne McFarland Texas Institute for Applied Environmental Research (TIAER) Tarleton State University Monitoring Objectives Focus Watershed Scale Evaluating BMP Effectiveness Quantifying Load Reductions Objectives Drive the Statistics Used Interested in totals, such as masses, volumes or long term chronic effects, then use means Interested in typical values or how often something occurs, then use medians 1

2 Measures of Central Tendency Mean or Average = (sum of values)/n (NOT resistant to outliers) Median = midpoint or mean of the two middle values (more resistant to outliers) Concentrations & Loads Concentrations represent a point in time Instantaneous effect Units generally mg/l Loads represent a mass over time Cumulative effects Units generally lbs/yr or kg/yr Measures of Central Tendency Geometric Mean = average of logarithmic values of a data set converted back to a real (base 10) number Geometric Mean = Antilog[(log(X i )+ log(x n ))/n of base 10 log Can also be calculated as: 2

3 Geometric Mean Why use it with bacteria data? Bacteria data are quite variable (can grow exponentially under the right conditions) Geometric mean is not overly influenced by a few very large values Example Dataset E. coli Sample (MPN/ Number 100 ml) Average 159 Geomean 43 Median 30 Concentrations to Loads Concentration (mg/l) Load (kg or lbs/day) Need Flow (cfs) or Volume (ft 3 /day) Upstream Downstream 3

4 Concentration to Load Concentration (mg/l) to Load (lbs/day) ** KNOW YOUR UNITS ** (concentration, mg/l) * (28.32 L/ft 3 ) * (volume, ft 3 /day) = mg/day (mg/day)/(1,000,000 mg/kg) = kg/day (kg/day)*( lbs/kg) = lbs/day Units and Conversion Factors Concentration to Load Bacteria Concentrations (cfu/100 ml) or (MPN/100 ml) Loads (cfu or MPN/day) ** See Examples in Handout ** What if loads are not important? Do you still need flow? 4

5 Flow vs Concentration Ln(Flow, cms) Example: Largely nonpoint source contributions North Bosque near Valley Mills Flow vs Concentration Ln(Flow, cms) Example: Largely point source contributions North Bosque below Stephenville Measures of Central Tendency Flow Weighted Average Where subscript i = individual observation c = concentration w = weighting factor or flow associated with the concentration 5

6 Flow Weighted Average Example: Concentration (mg/l) Flow (cfs) Flow wtd Avg = [(0.45*10) + (2.30*0.01) + (0.75*15)] ( ) = 0.63 mg/l Average (not flow wtd) = 1.17 mg/l Event Mean Concentration EMC = average pollutant concentration over the duration of a storm event EMC = total pollutant loading per event total runoff volume per event n = i 1 Where V = runoff volume over time i C = concentration at time i n = number of samples Flow in Statistical Analyses Flow used as: Weighting factor for concentrations in comparing Means (t Tests) Example: T test comparing concentrations at two stations Weighting factor (flow), gives more emphasis to concentrations at high than at low flows 6

7 Flow in Statistical Analyses Flow used as: Covariate in Analysis of Covariance (ANCOVA) for evaluating BMPs ANCOVA combines analysis of variance and regression analysis in comparing treatments Covariate (flow) an observed continuous variable that influences measured results that are not related to the treatment Flow in Statistical Analyses Flow used as: Flow adjustment variable for trend analysis Trend analysis evaluates changes over time Want to control for hydrologic changes in evaluating concentration changes over time Exploratory Data Analysis Become familiar with your data Graph it Develop summaries Plot relationships (if appropriate) 7

8 Exploratory Data Analysis Data Plots Hog Creek Assessing Water Quality Management Plan Implementation in the Middle and South Bosque River and Hog Creek Watershed, TIAER Report PR1104 Exploratory Data Analysis Explore the Distribution of the Data Types of plots Histogram with Normal Curve Normal Probability Plot Box and Whisker Plots 8

9 Test Parametric Assumptions Normality Shaprio Wilks Statistic Equal Variances Hartley s F test Independence Autocorrelation (correlation with time) Seasonality (correlograms) Parametric Statistics If data are not normal & parametric statistics are applied, what happens? Skewed data or with outliers Decreases the power of the test More likely to accept the Ho (for example no difference between means) when false Normal Data Distributions Log 9

10 Exploratory Data Analysis Are data transformations needed? Water Quality and Flow Data Generally follow a log distribution Log or Ln Transformation If transformed, parametric assumptions need to be tested on the transformed data. Nonparametric Tests Distribution Free (Not Assumption Free) Data do not need to follow a normal distribution, but if comparing more than one group, the shape should be the same Other Assumptions Equal Variances Independent, random samples Permutation Tests Newer test procedure in environmental statistics Allows analysis of means when data do not fit a normal distribution No distribution assumption required Can be used on moderate sized data sets avoiding Central Limit Theorem need for 50 to 70 observations Estimates means via resampling 10

11 Statistical Designs Major Statistical Designs for Watershed Studies: Single Watersheds Before After & Upstream Downstream Trend or Step Trend Analysis Paired Watersheds Before After Control Impact (BACI) Single Watershed: Before After Temporal Before and After Practice Implementation Source Diagram: NRCS, NWQH Monitoring Station Single Watershed: Before After Statistical Approach Two Sample t test comparing means Before and After Assumes samples random, independent, normally distributed and with equal variances Wilcoxon Rank Sum Test (nonparametric) Source Diagram: NRCS, NWQH Monitoring Station 11

12 Single Watershed: Before After Statistical Approach Assumes similar climate and hydrology between the Before and After periods Monitoring covariates, such as flow, can aid in adjusting for differences Source Diagram: NRCS, NWQH Monitoring Station Single Watershed: Above Below Spatial Above and Below Practice Location (upstream downstream) Practice Source Diagram: NRCS, NWQH Single Watershed: Above Below Statistical Approach Paired t test of above and below observations Observations are independent Difference in paired observations normally distributed Sign Test and Signed Rank Test (nonparametric) Observations still need to be independent, but normal distribution not required 12

13 Single Watershed: Above Below One of the Disadvantages Differences may be do to inherent differences within the watershed and not the practice Practice Single Watershed: Above Below Before monitoring helps account for inherent differences within the watershed Before Monitoring After Monitoring No Practice Practice Source Diagram: NRCS, NWQH Paired Watersheds Paired watershed or BACI approach (Before After Control Impact) Two (or more) Watersheds Involves a Calibration & Treatment period At least one watershed serves as a control 13

14 Paired Watersheds Control Treatment Calibration Period Station A Control Station B Treatment Treatment Period Station A Station B Source Diagram: NRCS, NWQH Evaluating BMPs Paired Watershed or BACI (Before After Control Impact) Average storm flow or storm volume used as covariate nonpoint source contributions (volume will vary with each runoff event) Paired Watershed Approach Source: J.D. Hewlett Principles of Forest Hydrology. The University of Georgia Press. 14

15 Paired Watersheds Statistical Approach ANOCOVA analysis of covariance Requires develop of a significant regression relationship between control and treatment watersheds during both calibration and treatment periods Paired Watersheds Controlling Phosphorus in Runoff from Long term Dairy Waste Application Fields. AWRA, 2004 McFarland and Hauck Paired Watersheds Controlling Phosphorus in Runoff from Long term Dairy Waste Application Fields. AWRA, 2004 McFarland and Hauck 15

16 Detecting Water Quality Changes Before & After BMP Implementation Garry L. Grabow, Jean Spooner, Laura A. Lombardo, and Daniel E. Line (1999) NCSU Water Quality Group Biological and Agricultural Engineering Department NWQEP NOTES Newsletter #95 July 1999 J. C. Clausen and J. Spooner Paired Watershed Study Design. Office of Water, U.S. Environmental Protection Agency, Washington, DC. EPA 841 F p Paired Watersheds Disadvantages Assumes there is a quantifiable relationship in water quality between the two watersheds Parametric assumptions may be violated Variances in water quality data unlikely to be equal between time periods Data may not be normally distributed Data may not be independent (serially correlated) Trend Analysis Subgroup Index (year) 16

17 Trend Analysis Lots of available methods, but most have limited application because Datasets have Censored data (< values) Have data gaps Are not normally distributed Are not independent (serial correlation and/or seasonality) Trend Stations Statistical Approach Seasonal Kendall test (preferred method) Nonparametric test, so data do not need to be normally distributed Can accept censored data (< values) Can accept some data gaps Can deal with seasonality Trend Analysis Additional Considerations: Changes in analytical methods over time Changes in reporting limits over time Limit of Quantitation (LOQs, < or leftcensored values) Influences from flow or hydrologic variability 17

18 Trend Analysis Dataset Preparation Evaluate & adjust for variability in reporting limits Check for seasonal influences and use seasonal tests, if appropriate Evaluate & adjust for variability in flow The Seasonal Kendall (SK) test for trend was developed by the U.S. Geological Survey and has become the most frequently used test for trend in the environmental sciences. Report PDF (173 KB) Downloads Directory Scientific Investigations Report Computer Program for the Kendall Family of Trend Tests By Dennis R. Helsel, David K. Mueller, and James R. Slack Statistical Data Analysis Look at your data Remember your objectives Test assumptions Consider the need for data transformations (Log or Ln) Flow adjust, if needed Consider seasonality or correlation with time 18

19 Major References USDA NRCS Chapter 4, Statistical Designs. In: National Handbook of Water Quality Monitoring, Title 450 Technology TS/stelprdb pdf Helsel & Hirsch Statistical Method in Water Resources. USGS, Techniques for Water Resource Investigations Practical Stats 19