Confidence Interval
What Is Confidence Interval?
A confidence interval, in statistics, refers to the probability that a population parameter will fall between two set values for a certain proportion of times. Confidence intervals measure the degree of uncertainty or certainty in a sampling method. A confidence interval can take any number of probabilities, with the most common being a 95% or 99% confidence level.
In statistics, a confidence interval is a type of estimate computed from the statistics of the observed data. This proposes a range of plausible values for an unknown parameter. The interval has an associated confidence level that the true parameter is in the proposed range.
Example with diagram:
Formula
Interval estimation can be contrasted with point estimation. A point estimate is a single value given as the estimate of a population parameter that is of interest, for example, the mean of some quantity. An interval estimate specifies instead a range within which the parameter is estimated to lie. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates.
Confidence intervals were introduced to statistics by Jerzy Neyman in a paper published in 1937.
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.
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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%).
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 distribution 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.
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.
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 bootstrap confidence interval around the arithmetic mean.
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.
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 median (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.
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 range 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.
