4. DATA ANALYSIS Data analysis begins in the monitoring program group of observations selected from the target design phase. Those responsible for monitoring population. In the case of water quality should identify the goals and objectives for monitoring, descriptive statistics of samples are monitoring and the methods to be used for used almost invariably to formulate conclusions or analyzing the collected data. Monitoring statistical inferences regarding populations objectives should be specific statements of (MacDonald et al., 1991; Mendenhall, 1971; measurable results to be achieved within a stated Remington and Schork, 1970; Sokal and Rohlf, time period (Ponce, 1980b). Chapter 2 provides 1981). A point estimate is a single number an overview of commonly encountered monitoring representing the descriptive statistic that is objectives. Once goals and objectives have been computed from the sample or group of clearly established, data analysis approaches can observations (Freund, 1973). For example, the be explored. mean total suspended solids concentration during baseflow is 35 mg/L. Point estimates such as the Typical data analysis procedures usually begin mean (as in this example), median, mode, or with screening and graphical methods, followed geometric mean from a sample describe the central by evaluating statistical assumptions, computing tendency or location of the sample. The standard summary statistics, and comparing groups of data. deviation and interquartile range could likewise be The analyst should take care in addressing the used as point estimates of spread or variability. issues identified in the information expectations report (Section 2.2). By selecting and applying The use of point estimates is warranted in some suitable methods, the data analyst responsible for cases, but in nonpoint source analyses point evaluating the data can prevent the “data estimates of central tendency should be coupled rich)information poor syndrome” (Ward 1996; with an interval estimate because of the large Ward et al., 1986). spatial and temporal variability of nonpoint source pollution (Freund, 1973). For example, the sample This chapter provides detailed information on the mean and standard deviation could be used to statistical analysis of environmental monitoring report that the mean total suspended solids data. The first section of the chapter is intended concentration during baseflow is 35 ± 10 mg/L for both the manager and data analyst. Its goal is using a 95 percent confidence interval. Stated in to acquaint the reader with key concepts and issues other words, there is a 95 percent chance that the related to data analysis. This section also provides actual mean baseflow concentration is between 25 recommendations for selecting statistical and 45 mg/L. There is a 5 percent chance that the procedures for routine analyses and can be used to mean baseflow concentration is outside this range. guide the reader in selecting additional portions of The confidence interval is a function of the the chapter for more in-depth reading. variability of the data, the number of observations, and the probability (e.g., 95 percent) selected by 4.1 INTRODUCTION the data analyst. This sort of estimation can be useful in developing baseline information, 4.1.1 Estimation and Hypothesis Testing developing or verifying models, or determining the load of a single nonpoint source runoff event. Instead of presenting every observation collected, the data analyst usually summarizes major Evaluating the effectiveness of controls and characteristics with a few descriptive statistics. changing environmental conditions is one of the Descriptive statistics include any characteristic key monitoring program objectives described in designed to summarize an important feature of a Chapter 2. In addition to summarizing key data set or sample (Freund, 1973). The reader statistics that describe the central tendency and should note that a sample in this context refers to a spread of water quality variables and biological 4-1 Data Analysis Chapter 4 metrics, statistical analysis usually involves error is equal to the significance level (α) of the hypothesis testing. Two common types of test and is selected by the data analyst. In most hypothesis testing done in environmental cases, managers or analysts define 1-α to be in the monitoring are step changes and monotonic trends. range of 0.90 to 0.99 (e.g., a confidence level of 90 Step changes are typically evaluated when to 99 percent), although there have been comparing at least two different sample environmental applications where 1-α has been set populations such as an impacted site and a to 0.80. Selecting a 95 percent confidence level reference site or when comparing one sample implies that the analyst will incorrectly reject the population to an action level. Step changes can Ho (i.e., a false positive) 5 percent of the time. also be evaluated when comparing samples collected during different time periods. Type II error depends on the significance level, Monotonic trends (e.g., consistently increasing or sample size, and variability, and which alternative decreasing concentrations) are typically evaluated hypothesis is true. The power of a test (1-β) is when the analyst is investigating long-term defined as the probability of correctly rejecting Ho gradual changes over time. when Ho is false. In general, for a fixed sample size, α and β vary inversely. For a fixed value of The null hypothesis (Ho) is the root of hypothesis α, β can be reduced by increasing the sample size testing. Traditionally, null hypotheses are (Remington and Schork, 1970). Figure 4-1 statements of no change, no effect, or no illustrates this relationship. Suppose this interest is difference. For example, the flow-averaged mean in testing whether there is a significant difference total suspended solids concentration after BMP between the means from two independent random implementation is equal to the flow-averaged samples. As the difference in the two sample mean total suspended solids concentration before means increases (as indicated on the x-axis), the BMP implementation. The alternative hypothesis probability of rejecting Ho, the power, increases. If (Ha) is counter to the null hypothesis, traditionally the real difference between the two sample means being statements of change, effect, or difference. is zero, the probability of rejecting Ho is equal to Upon rejecting Ho, Ha would be accepted. the significance level, α. Figure 4-1A shows the Regardless of the statistical test selected for general relationship between α and β if α is analyzing the data, the analyst must select the changed. Figure 4-1B shows the relationship significance level of the test. That is, the analyst between α and β if the sample size is increased. must determine what error level is acceptable. There are two types of errors in hypothesis testing: Type I: The null hypothesis (Ho) is rejected when Ho is Table 4-1. Errors in hypothesis testing. really true. State of affairs in the population Type II: The null hypothesis Decision Ho is True Ho is False (Ho) is accepted when Ho is really false. Accept Ho 1-α β (Confidence level) (Type II error) Table 4-1 depicts these errors, with the magnitude of Type I Reject Ho α 1-β errors represented by α and (Significance level) (Power) the magnitude of Type II (Type I error) errors represented by β. The probability of making a Type I 4-2 Chapter 4 Data Analysis Figure 4-1. Comparison of α and β. 4-3 Data Analysis Chapter 4 4.1.2 Characteristics of Environmental 4.1.3 Recommendations for Selecting Data Statistical Methods The selected statistical method must match the The statistical methods discussed in this manual type of environmental data collected and the include parametric and nonparametric procedures. decisions to be made. Although summarizing the Parametric procedures assume that the data being mean annual dissolved oxygen concentration analyzed have a specific distribution (usually along an impaired stream might provide an normal), and they are appropriate when the indication of habitat quality, evaluating the underlying distribution is known (or is assumed minimum dissolved oxygen during summer with confidence). For data with an unknown months over the same time period might have a distribution, nonparametric methods should be greater impact on subsequent management used since these methods do not require that the decisions since that is when critical conditions data have a defined distribution. often occur. Environmental managers and data analysts must collectively determine which Nonparametric methods can directly handle special statistical methods will result in the most useful data commonly found in the nonpoint source area, information for decision makers. such as censored data or outliers. Censored data are those observations without an exact numerical The selection of appropriate statistical methods value, such as a value of less than 10 μg/L (<10 must be based on the attributes of the data μg/L) or not-detected (ND). Censored data often (Harcum, 1990). Two main types of attributes appear in laboratory reports when the important to environmental monitoring are data concentration being analyzed is lower than the record limitations and statistical characteristics. detection limit or higher than the allowable range Common data record limitations include missing for a particular type of laboratory equipment or values, changing sampling frequencies over time, procedure (Dakins et al., 1996; Gilliom and Helsel, different numbers of samples during different 1986). Censored data can cause problems in sampling periods, measurement uncertainty, parametric methods because these methods often censored data (e.g., “less-thans”), small sample require that all data have numerical values. In this sizes, and outliers. Data limitations are, for the case, nonparametric methods can be used because most part, human-induced attributes that often they often deal with the ranking of the data, not the result in less reliable observations and less data themselves. For example, for data “below the information for a given data set.