## 5.2 Confidence Limits One of the strengths of statistics is that they quantify uncertainty about data. Confidence limits (sometimes called "confidence intervals") clearly illustrate that uncertainty, thus, regulators often require them. For example, confidence limits may be used to compare groundwater monitoring data to a fixed threshold, such as a compliance criterionGeneral term used in this document to identify a groundwater concentration that is relevant to a project; used instead of designations such as Groundwater Protection Standard, clean-up standard, or clean-up level., or for placing an upper limit on backgroundNatural or baseline groundwater quality at a site that can be characterized by upgradient, historical, or sometimes cross-gradient water quality (Unified Guidance).. Confidence limits are the maximum and minimum values bracketing the statistic of interest (usually the arithmetic or geometric meanA summary statistic calculated by multiplying the data values and taking the Nth root, where N is the sample size (science-dictionary.org 2013).) based on the distribution of the data (usually the normal or lognormalA dataset that is not normally distributed (symmetric bell-shaped curve) but that can be transformed using a natural logarithm so that the data set can be evaluated using a normal-theory test (Unified Guidance). distribution) at a certain confidence levelDegree of confidence associated with a statistical estimate or test, denoted as (1 – alpha) (Unified Guidance). (usually 95%). In other words, confidence limits are the maximum or minimum values above or below which you are confident (at a selected confidence level) that the statistic will occur. Confidence limits can be parametricA statistical test that depends upon or assumes observations from a particular probability distribution or distributions (Unified Guidance). or nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance).. For the calculation of parametric confidence limits, the underlying statistical distribution must be known in order to select the appropriate confidence limit. Certain more robust methods (e.g., calculation of robust confidence limits) may permit the calculation of confidence limits without removal of outliersValues unusually discrepant from the rest of a series of observations (Unified Guidance). within background data (USEPA 1999).

Parametric and Nonparametric Confidence Limits

Confidence limits can be parametric or nonparametric. For the calculation of parametric confidence limits, the underlying statistical distribution must be known in order to select the appropriate confidence limit.

### 5.2.1 Determining Which Confidence Limits Are Needed

When using confidence limits, you must determine if one-sided or two-sided confidence limits are needed. This determination ensures that confidence limits are not over- or underestimated. If you are comparing data to a criterion and only need to know whether concentrations fall above or below a criterion, then only one of the confidence levels is of interest. The two-sided approach is appropriate when assessing the uncertainty of hydraulic parameters, such as the hydraulic conductivity estimates of a well.

Confidence intervals are often applied in the following scenarios:

• Compliance or assessment monitoring where it is assumed that concentrations do not exceed a criterion and you must determine if concentrations have exceeded the criterion. In this case, a calculated lower confidence level (LCL) exceeding the standard, indicates confidence that the measured concentrations are above the criterion.
• Corrective action sites where it is assumed that concentrations exceed a criterion and confirmation must be provided that the site media have been remediated to concentrations below the criterion. In this case, a calculated upper confidence level (UCL) below the criterion, indicates that the criterion has been met.
• To determine the strength of evidence for an upward or downward trend in data, two sided confidence limits may be calculated for the estimated slope of the trend line. The calculation of two-sided confidence limits that do not include the value zero, are indicative of evidence of a trend at the selected confidence level (such as 95%).

Before calculating confidence limits, the data should be examined to evaluate what distribution fits the data, whether the underlying assumptions for constructing confidence limits are valid, and whether the selected confidence level is appropriate for the planned application (that is, the question you are trying to answer). Confidence limits may be constructed in several ways, depending on the distribution of the data and the question of interest, when assessing environmental data. Some common applications of confidence limits are listed below:

### 5.2.2 Confidence Interval Around a Normal Mean

If the data are normally distributed, if the data pass normality tests (such as probability plots or the Shapiro-Wilk test), or are reasonably symmetric, choose the confidence interval around a normal mean. This method estimates the upper and lower confidence limits (UCL and LCL) around the arithmetic mean of a data set based on an underlying normal distributionSymmetric distribution of data (bell-shaped curve), the most common distribution assumption in statistical analysis (Unified Guidance). model. Construct a one-sided test instead of a two-sided test if that is most appropriate. These confidence intervals are most appropriate when comparing concentration means to criteria.

### 5.2.3 Confidence Interval Around Lognormal Geometric Mean

Typical environmental data are not normally distributed but instead are heavily right-skewed. One way to handle these data is to transform them logarithmically. The transformed lognormal data may fit a normal distribution. The log-transformed data are no longer in the arithmetic domain, but the logarithmic domain.

Sometimes, it may seem easiest to simply log-transform the data, calculate the arithmetic mean of the log-transformed data, construct a confidence interval around this value, and then back-transform the confidence levels back to obtain the correct confidence interval. Unfortunately, this approach results in a confidence interval around the geometric mean, not the arithmetic mean, which usually results in an underestimate of the true mean. Be aware that a confidence interval calculated in this way may not meet regulations applicable to the site.

### 5.2.4 Confidence Intervals Around Lognormal Arithmetic Mean

Confidence intervals about the arithmetic mean, the statistic commonly required by regulations, are useful for skewed, lognormal data. This method is appropriate when you need to compare your data to an arithmetic mean and the data fit a normal distribution when log-transformed. Be aware that the available procedures for constructing this type of confidence interval can produce unacceptable results. Land’s procedure is commonly used, but if the lognormal data have a high coefficient of variation, consider a bootstrapA computerized method for assigning measures of accuracy to sample estimates. This technique allows estimation of the sample distribution of almost any statistic using only very simple methods. Bootstrap methods are generally superior to ANOVA for small data sets or where sample distributions are nonnormal (USEPA 2010). confidence interval around the arithmetic mean.

### 5.2.5 Confidence Interval Around Upper Percentile

Sometimes you must construct confidence intervals around a percentile. For example, if the criterion is a concentration that represents the 90th percentile, then a confidence interval around the upper 90th percentile should be calculated. If the standard is a fixed criterion, such as a “not to exceed” maximum, then it is appropriate to use a confidence interval around a high percentile, such as the upper 95th or 99th percentiles. Be cautious when selecting a percentile as it may be extremely difficult to demonstrate corrective action success if too high a percentile is selected.

### 5.2.6 Nonparametric Confidence Interval Around a Median or Percentile

If your data do not fit a normal, lognormal, or other distribution, or if there are too many nondetects, use of a nonparametric confidence interval is appropriate. Nonparametric methods do not assume a particular distribution. Unfortunately, this generally results in wider confidence intervals and the need for larger data sets for making confident decisions. This method is appropriate when comparing concentrations to a percentile, such as the medianThe 50th percentile of an ordered set of samples (Unified Guidance). (50th percentile) or 90th percentile. If you need to compare concentrations to a maximum criterion, a large percentile, such as the 95th or 99th percentile may be applied.

### 5.2.7 Confidence Interval Band Around Linear Regression Lines

If a linear trend is present in your data, you can describe the uncertainty in these data by constructing a confidence band around the trend line over the rangeThe difference between the largest value and smallest value in a dataset (NIST/SEMATECH 2012). of the data set. The confidence band is constructed of the individual confidence intervals around the mean as a function of time, not an upper percentile. This method is most appropriate for cases where the fixed criterion represents a mean concentration and not an explicit upper percentile or “not to exceed” value.

Publication Date: December 2013

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