67 Trend Analysis
Description: Time series trend analysis
Found in: State of the Ecosystem - Gulf of Maine & Georges Bank (2018+), State of the Ecosystem - Mid-Atlantic (2018+)
Indicator category: Extensive analysis, not yet published
Contributor(s): Sean Hardison, Charles Perretti, Geret DePiper
Data steward: NA
Point of contact: Kimberly Bastille, kimberly.bastille@noaa.gov
Public availability statement: NA
67.1 Methods
Summarizing trends for ecosystem indicators is desirable, but the power of statistical tests to detect a trend is hampered by low sample size and autocorrelated observations (see Nicholson and Jennings 2004; Wagner et al. 2013; Storch 1999). Prior to 2018, time series indicators in State of the Ecosystem reports were presented with trend lines based on a Mann-Kendall test for monotonic trends to test significance (p < 0.05) of both long term (full time series) and recent (2007–2016) trends, although not all time series were considered for trend analysis due to limited series lengths. There was also concern that a Mann-Kendall test would not account for any autocorrelation present in State of the Ecosystem (SOE) indicators.
In a simulation study (Hardison et al. 2019), we explored the effect of time series length and autocorrelation strength on statistical power of three trend detection methods: a generalized least squares model selection approach, the Mann-Kendall test, and Mann-Kendall test with trend-free pre-whitening. Methods were applied to simulated time series of varying trend and autocorrelation strengths. Overall, when sample size was low (N = 10) there were high rates of false trend detection, and similarly, low rates of true trend detection. Both of these forms of error were further amplified by autocorrelation in the trend residuals. Based on these findings, we selected a minimum series length of N = 30 for indicator time series before assessing trend.
We also chose to use a GLS model selection (GLS-MS) approach to evaluate indicator trends in the 2018 (and future) State of the Ecosystem reports, as this approach performed best overall in the simulation study. GLS-MS also allowed for both linear and quadratic model fits and quantification of uncertainty in trend estimates. The model selection procedure for the GLS approach fits four models to each time series and selects the best fitting model using AICc. The models are, 1) linear trend with uncorrelated residuals, 2) linear trend with correlated residuals, 3) quadratic trend with uncorrelated residuals, and 4) quadratic trend with correlated residuals. I.e., the models are of the form
\[ Y_t = \alpha_0 + \alpha_1X_t + \alpha_2X_t^2 + \epsilon_t\] \[\epsilon_t = \rho\epsilon_{t-1} + \omega_t\]
\[w_t \sim N(0, \sigma^2)\]
Where \(Y_t\) is the observation in time \(t\), \(X_t\) is the time index, \(\epsilon_t\) is the residual in time \(t\), and \(\omega_t\) is a normally distributed random variable. Setting \(\alpha_2 = 0\) yields the linear trend model, and \(\rho = 0\) yields the uncorrelated residuals model.
The best fit model was tested against the null hypothesis of no trend through a likelihood ratio test (p < 0.05). All models were fit using the R package nlme (Pinheiro et al. 2017) and AICc was calculated using the R package AICcmodavg (Mazerolle 2017). In SOE time series figures, significant positive trends were colored orange, and negative trends purple.
67.1.3 Data analysis
Code used for trend analysis can be found here.
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