C.6 Study Question 6: Is there seasonality in the concentrations?
Most statistical tests assume statistical independence of the sample data. Temporally dependent groundwater data violate the assumption of independence. High levels of variability unrelated to the long-term temporal trend also make it difficult to identify statistically significant long-term trends and to estimate attenuation rates and remediation time frames (McHugh et al. 2011).
When temporal variability exists because of the distribution of the timing of the sample collection, the distribution exhibits time dependence or autocorrelationCorrelation of values of a single variable data set over successive time intervals (Unified Guidance). The degree of statistical correlation either (1) between observations when considered as a series collected over time from a fixed sampling point (temporal autocorrelation) or (2) within a collection of sampling points when considered as a function of distance between distinct locations (spatial autocorrelation). (for example, a cyclical pattern of data affected by the seasons). To verify statistical independence, demonstrate a low correlationAn estimate of the degree to which two sets of variables vary together, with no distinction between dependent and independent variables (USEPA 2013b). between the concentration and the time of the sampling event. Evaluate the cyclical change (seasonal variation) to adequately understand the varianceThe square of the standard deviation (EPA 1989); a measure of how far numbers are separated in a data set. A small variance indicates that numbers in the dataset are clustered close to the mean. of the population. A cyclical pattern can biasSystematic deviation between a measured (observed) or computed value and its true value. Bias is affected by faulty instrument calibration and other measurement errors, systematic errors during data collection, and sampling errors such as incomplete spatial randomization during the design of sampling programs (Unified Guidance). the variance of the distribution or create a slope of the concentration that is not monotonic. You can evaluate the temporal evaluation and adjust the distribution to evaluate the trend accordingly.
This question is usually relevant in the remediation, monitoring, and closure stages of the project life cycle.
Selecting and Characterizing the Data Set
Refer to Section 3.4: Common Statistical Assumptions for further discussion of how the following requirements may affect statistical analysis results.
- Check for outliersValues unusually discrepant from the rest of a series of observations (Unified Guidance). using box plots, probability plots, Dixon's test, or Rosner's test.
- Check for autocorrelation between seasonal sampling events.
- The ability to detect trends can be impacted by aggregating data across wells.
- In general, you can obtain better detection of trends using longer records of data, but in many cases, attenuation rates will differ based on remedial methods.
- See also Section 4.1: Considerations for Statistical Analysis.
If the test does not assume a distribution, then no testing of the distribution is necessary. However, if a substantial number of nondetectsLaboratory analytical result known only to be below the method detection limit (MDL), or reporting limit (RL); see "censored data" (Unified Guidance). are present, then the test cannot indicate autocorrelation. The samples should cover multiple years with an observable seasonal pattern each year. Each season should include at least three measurements.
When the objective is to determine if there is a temporal change or pattern to the data, simple graphical procedures can reveal significant trends. However, if cyclical effects complicate the pattern of the data then consider other statistical methods listed below to answer this study question.
Statistical Methods and Tools
Determine if there is a significant cyclical pattern in the data that creates autocorrelation between the samples; statistical independence of the data is a key assumption for many statistical tests. When the objective is to determine if there is a temporal change or pattern to single series data, use the sample autocorrelation function or Rank von Neumann ratio test to identify correlated samples from specific seasons. When the objective is to determine if there is a temporal change or pattern to a group of wells, use time series plots or the Kruskal-Wallis test to identify correlated samples related to specific seasons.
Sample Autocorrelation Function
- Data must follow normal distributionSymmetric distribution of data (bell-shaped curve), the most common distribution assumption in statistical analysis (Unified Guidance). for this test.
- A minimum of 8 to 10 measurements are recommended, although a greater number of measurements may be necessary to obtain the desired confidence levelDegree of confidence associated with a statistical estimate or test, denoted as (1 – alpha) (Unified Guidance)..
- This test is very sensitive to extreme values (outliers).
Over several measurements, calculate an autocorrelation coefficient. If any coefficient exceeds the critical value, then assume samples are dependent.
Plot the autocorrelation coefficient over the number of lags overlaid (plus or minus the critical value) to identify the significance of the dependent nature of the samples. If the shape of the function is sinusoidal, then the data exhibit seasonal fluctuation. Adjust the values for seasonality. If seasonality occurs, change the frequency of sampling or adjust the data set.
- This test requires no distribution assumptions.
- This test cannot handle a substantial number of tied values or nondetect values.
- A minimum of 10 to 12 observations from a single well is recommended.
Since this test does not assume a distribution, no testing of the distribution is necessary. However, a substantial number of nondetects will cause the test to lose its validity for autocorrelation.
Seasonality is one of many reasons for temporal correlation. By evaluating von Neumann ratio and comparing it to a lower critical point, you can identify evidence for temporal correlation at a selected level of significance. Withne's test sufficient evidence of autocorrelation, you must adjust the data to evaluate the trend of the observations. If this occurs, the frequency of sampling should change or adjust the data set.
- Standardize the concentration on y-axis versus time on x-axis.
- Assign values to nondetects.
- Plot parallel lines for several wells.
The standardized concentration is assigned by subtracting the meanThe arithmetic average of a sample set that estimates the middle of a statistical distribution (Unified Guidance). concentration of each well from the concentration and dividing by the standard deviation. You can visually identify temporal patterns by plotting the standardized concentrations of several wells over time. Seasonal fluctuations will show up in the time series as parallel traces. Since this is a qualitative method, adjust for seasonal variations in evaluating the trend.
- As a nonparametricStatistical test that does not depend on knowledge of the distribution of the sampled population (Unified Guidance). test, normality of the data is not required.
- At least three groups of data (see Section 5.8.2) must be present.
If applied to a group of wells, little to no spatial variation should exist.
This test identifies temporal differences among sample periods or nonstationarity (change in means over time). Evaluate the temporal effects of individual sampling events or cyclical events (seasons) by aggregating concentrations across monitoring wells for each sampling date, as described above. The Kruskal-Wallis test confirms whether medianThe 50th percentile of an ordered set of samples (Unified Guidance). measurement levels differ by season, thus indicating the presence of seasonality.
If acceptable under the regulatory program, change the sample frequency as a simple remedy for temporal correlation. If the test identifies a significant temporal effect, adjust the data set to account for significant seasonal correlations (seasonality). Otherwise, use specific statistical tests that have a seasonal test method. For example, you can perform a seasonal Mann-Kendall test on each group (season), then combine the S statistics of the groups to calculate the overall S statistic. You can identify a significant trend over time at one location when the absolute value of S is greater than the critical point. This result indicates that the mean is not stationaryA distribution whose population characteristics do not change over time or space (Unified Guidance). at this sampling pointA specific spatial location from which groundwater is being sampled., despite the seasonal fluctuations.
Interpretation of Results and Associated Uncertainty
The temporal trend analyses explain whether and how contaminant concentrations are changing over time. As discussed above, a nonparametric test such as the seasonal Mann-Kendall test does not require assumptions regarding the data distribution. However, significant seasonal fluctuations in the data will cause false negativeIn hypothesis testing, if the alternative hypothesis (Hᴀ) is true but is rejected in favor of the null hypothesis (H₀) which is not true, then a false negative (Type II, β) error has occurred (Unified Guidance). errors in the statistical test methods. Temporally dependent and autocorrelated data generally contain both a truly random and nonrandom component. Only strong correlations are likely to affect the results of further statistical testing. See also Section 4.5.1 and Section 4.6.2.
Related Study Questions
Study Question 5: Is there a trend in contaminant concentrations?
Key Words: Temporal Trends, Remediation, Monitoring, Closure, Cyclic or Periodic Change
Publication Date: December 2013